03 — KERNEL: the fast path, its cost law, its determinism, and the route ladder that prices T1#

NEPTUNE · LINUS (systems gate: dispatch count, memory layout, wall-clock against the FLOP model). Written 2026-09-05 at HEAD 99777ab under CHARTER.md; every READ below is pinned @ 99777ab unless another SHA is printed. Evidence classes as CHARTER §3.

Preface — what this book decides, and the verse that decides it first. This book decides one thing: whether the sentence "faster to train" in its T1 form (GPU-seconds-to-floor, CHARTER §1) can ever be earned by the shape's one triangular solve, and by which route. It does not decide accuracy (books 01, 04), sample efficiency (book 02), or any bed (book 04). The verse that decides first is 03.1, the route ladder: five links — fused Triton kernel, chunked torch solve, CSR two-stage path, dense solve_triangular with $n$ declared per $s$, and the withdrawal sentence — each with its own kill and its own price, ordered so that the cheapest refutation is read first. Every later verse is one rung of that ladder made precise: the MAC and byte models (03.2–03.3), the wall-clock crossover predicted with the counter as the point estimate (03.4), the memory ceilings the certified device imposes and the table the fused route restores (03.5), the measured increment read as the per-op floor it is (03.6), the cost-law experiment specified so that every price becomes RUN (03.7), T1 operationalised (03.8), the determinism regime and what the kernel must guarantee (03.9–03.10), the kernel design as a specification with no code written (03.11–03.14), the Sherman–Morrison route for candidate moves (03.15), the CSR path (03.16) with the Mapper schedule that would feed it (03.19, written beside 03.16 for that reason), the blockwise published control (03.17), and the re-certification protocol for a stack that, as of this session, has no CUDA torch installed at all (03.18). No number here is a result: no kernel exists in the tree, no CEQ arm has trained, and the only cost numbers the record owns are per-op floors taken on torch 2.5.1+cu121 at ab5b485. The census rows this book closes are B2 (kernel half), B13, B14, B15 and B25 (per-step half); the closing tables map each to its verse and list every kill cheapest first.

Notation used throughout. $s$ positions, $d$ head dimension, $h$ heads, $n$ sequences per step (the record's batch axis), $C$ chunk length, $K=s/C$ chunks, $B_M$ query-tile rows. $P$ the causal row-stochastic operator of regime S (docs/CEQ_SHAPE.md Definition 2), $M=I-\gamma P$, $z=M^{-1}V$, $O=(1-\gamma)Pz$. MACs are multiply-accumulates; FLOP $=2\,$MAC. $u$ is unit roundoff: $2^{-24}$ (float32), $2^{-8}$ (bfloat16), $2^{-53}$ (float64); $\gamma_k := ku/(1-ku)$ is the standard-model constant (DERIVED, not the horizon dial — the subscript disambiguates). Geometry of record: $s=64$, $d=16$, $h=1$, $n=2048$, float32.

Part A — the operator, its arithmetic, its bytes, its ceilings#

03.1 — The route ladder: five links, each with a kill and a price#

Statement. The read $O=(1-\gamma)P\,(I-\gamma P)^{-1}V$ is computed by exactly one of five routes, tried in the order below; a route is shipped only after its own kill has been run and not fired, and T1 is licensed only on a shipped route whose cost law is RUN (03.7).

link	route	what it computes	depth (serial rounds)	MACs per head, increment over the $s^2d$ softmax head	memory class
L1	fused Triton kernel (03.11–03.14)	online causal softmax tiles feeding a chunked block-triangular solve; $P$ never materialised	$s/C$ chunk steps	$+s^2d/2$ (exact, and independent of $C$: 03.11)	$O(sd)$ retained per sequence
L2	chunked torch solve (03.11 in torch ops)	per chunk-row $k$: the $C\times(k{+}1)C$ logit block, exact softmax, `solve_triangular` on the $C\times C$ diagonal block, one `matmul`	$s/C$ launches $\times\,4$ kernels	$+s^2d/2$ (exact, and independent of $C$: 03.11)	$O(sC)$ transient, $O(s^2)$ retained unless checkpointed per chunk
L3	CSR two-stage path (03.16)	a certified causal tile schedule; Neumann $K$ hops on the sparse $P$ with the union certificate	$K$	$+K\cdot\mathrm{nnz}(P)\,d$	$O(\mathrm{nnz})$
L4	dense `solve_triangular` with $n$ declared per $s$ (03.5)	`torch.linalg.solve_triangular` on the materialised $[n,h,s,s]$ operator	$s$ inside cuBLAS `trsm`	$+s^2d/2$	$O(s^2)$ retained, $C_{\rm OP}\in[3.823,5.823]$ tensors
L5	withdrawal	—	—	—	—

L5 is the sentence: one solve is not cheaper than the stack it replaces on this device; the T1 sentence is withdrawn and A1/A2 stand alone.

Hypotheses. Regime S: $P\ge0$ row-stochastic, causal, absorbing rows as identity rows, $\gamma\in[0,1)$ so $M$ is lower-triangular with diagonal $1-\gamma P_{ii}\in[1-\gamma,1)$ (READ docs/CEQ_SHAPE.md:414-435). Geometry at which the table is non-vacuous: $s\in\{64,256,1024,4096\}$, $d\in\{16,32,64,128\}$ (the only head dimensions a tl.dot tile admits on this stack, READ ceq/mz_kernel.py:41), $n$ per $s$ from 03.5, $h=1$ at the record's geometry. L1 and L2 assume a chunk length $C$ dividing $s$; L3 assumes a schedule with a certificate (L-CERT); L4 assumes the operator fits. Head-dimension admissibility for L1 is narrower than the tl.dot list: the fused route additionally requires a $(B_M,C)$ pair inside 03.3's SRAM budget under the solve stage's $B_M=C$ (03.12), which admits $C=128$ at $d=16$, $C=64$ at $d\in\{32,64\}$ and $C=32$ at $d=128$ — the pairs are printed in 03.3's admissible table and no other $(B_M,C,d)$ triple is an L1 hypothesis. $d=128$ is admissible for L1 only at $C=32$, at an arithmetic intensity of $16$ FLOP/B, i.e. memory-bound by a factor $4.66$ against the ridge.

Evidence. The MAC column: DERIVED in 03.2 and 03.11. The increment of L4 at $s=64$: RUN[NEPTUNE] $2.514-1.473=1.041$ ms per step at $n=2048$, float32, forward and backward, READ docs/sources/judge/sec_shape.md:243, READ docs/CEQ_SHAPE.md:590 — a per-op floor with no producer in the tree (P-1; the producer is N-02, READ docs/PLAN.md:536-541). L1, L2, L3 above $s=64$: NOT MEASURED — needs the kernel (L1, 03.11–03.14), the chunked torch route (L2, 03.11), the schedule and its certificate (L3, 03.16); the record marks all three NOT MEASURED — needs a chunked kernel (READ docs/CEQ_SHAPE.md:585-587). No .py in ceq/, scale/ or scripts/ calls solve_triangular: RUN this session, grep -rn solve_triangular --include=*.py . returns the two copies of tests/foreman/test_operator_is_a_triangular_solve.py (:62, :105) and nothing else — even L4 exists in the tree only as a test helper. The chunked pattern is owned: CITED [V] yang-2024-deltanet Eq. 10 (per-chunk unit-lower-triangular inverse by forward substitution, READ docs/sources/sweep/sweep_linrec.md:24-33); the reason no state-space dual shortens L1 or L2 below $O(s^2 d)$ is DERIVED here in three lines and rests on no citation (the CITED [V] hu-2025-ssdtheory of READ docs/sources/sweep/sweep_linrec.md:26-30 is inadmissible for a load-bearing statement under L-EQ and is demoted to a concurrence, carrying nothing):

DERIVED (rank of the causal row-softmax, and why no fixed-size state crosses a chunk). (1) Write $S=QK^\top/\sqrt d$, $E=\exp(S)$ entrywise, $L$ the causal $0/1$ mask, $D=\mathrm{diag}((E\odot L)\mathbb 1)$, so $P=D^{-1}(E\odot L)$. $E\odot L$ is lower-triangular with diagonal $e^{S_{ii}}>0$, hence nonsingular; $D^{-1}$ is a positive diagonal, which preserves rank. $\operatorname{rank}(P)=s$ exactly, for every $Q,K$, every $d$ and every $s$ — including $d=1$. A dual that carries a state of dimension $r<s$ therefore cannot reproduce $P$ exactly at any $s$. (2) The chunk-crossing question is the off-diagonal block $P_{kl}$, $l<k$, which is unmasked: $P_{kl}=D_k^{-1}\exp(S_{kl})$ with $\operatorname{rank}(S_{kl})\le d$. For $\operatorname{rank}(A)=r$ the Hadamard power obeys $\operatorname{rank}(A^{\circ t})\le\binom{r+t-1}{t}$ (each entry of $A^{\circ t}$ is a degree-$t$ monomial in the $r$ latent features), so truncating $\exp(A)=\sum_{t\ge0}A^{\circ t}/t!$ at order $T$ gives a matrix of rank at most $\sum_{t=0}^{T}\binom{r+t-1}{t}$. At $d=16$: $T=0$ gives $1$, $T=1$ gives $17$, $T=2$ gives $153$. A running state of width $r<C$ therefore forces $T\le1$ at every $C\ge32$. (3) The $T=1$ truncation error in the max norm is at least $\big(e^{a}-1-a\big)$ at $a=\max_{ij}|S_{kl,ij}|$. Numeric instance at this book's geometry ($d=16$, $C=128$, logits of order $a=3$ after the $1/\sqrt d$ scaling): the rank-$17$ truncation misses by $e^{3}-4=16.1$ against entries bounded by $e^{3}=20.1$ — an $80\,\%$ miss, not an approximation. The inter-chunk term therefore stays a dense $C^2d$ GEMM per $(k,l)$ pair and $s^2d/2$ in total, on both L1 and L2.

The blockwise evaluation of the same operator is occupied: CITED [V] zhao-2026-structuredsparse (READ docs/sources/sweep/sweep_resolvent.md:218-226); its exponent $O(n^{4/3}d)$ is [V]-graded and is NOT MEASURED — needs the equation-level read owed in 03.17, so it appears in no ladder row and prices nothing; 03.17 makes the route the fellow on the class alone.

Mechanism. P-4 (a route that exists only in prose is scaffolding that does not exist — the grep above is the check), M-8 (each link priced at its own rate, never by a cross-link ratio), P-8 (every unmeasured price carries its direction in the headline), V-9 (a replacement whose kill equals the verse's is not a replacement — each link below has a different kill).

Kill (a ratio inside one interleaved block, not an absolute millisecond count). An absolute threshold on the re-measurement is unreachable and is struck here: $2.2\times1.041=2.290$ ms was the N-02 threshold (READ docs/PLAN.md:539), and the un-synchronised-to-synchronised change on this arm has a known sign and no measured magnitude — the un-synchronised clock under-reports the device by an amount no number in this repository bounds for solve_triangular (READ V20_R15_JOURNAL.md:53 @ 99777ab) — so an absolute millisecond threshold read off the un-synchronised floor is decidable against nothing and fires or spares by an accident of the queue (V-11).

The $2.0\times$–$6.6\times$ figure is not this arm's and bounds nothing here. READ scale/m3_flops.py:101-121 reads it off the per-row gate settle arm — a Python loop over t_max steps whose Gram copy is $268$ MB at $(s,n)=(64,8192)$ — as that arm's measured wall-clock ratios ($3.48$ at $s=16$; $11.19$ and $7.43$ at $s=64$) against its own flat FLOP ratio $1.700$, i.e. "the model is optimistic by 2.0x at the pilot geometry and by 4.4x to 6.6x at the shipped one" (READ scale/m3_flops.py:110). It is the optimism of a FLOP model for that arm, not a timer bias measured on solve_triangular, which that file never calls. Carrying it onto the solve increment is M-8 — a constant carried across arms, the mechanism this verse invokes against others — and it is struck from every threshold in this book; it survives only where the per-row gate arm is named beside it. The two hazards that do attach to the solve increment are (h1) the absent torch.cuda.synchronize() in the producing harness, sign known and magnitude unmeasured (READ V20_R15_JOURNAL.md:53 @ 99777ab), and (h2) the $1.65\times$ session-to-session drift on identical code and geometry, measured ($2.0775$ against $3.4372$ s/step, READ scale/m3_flops.py:117-121). The frozen kill is the gap-invariant increment ratio $$\rho_\Delta:=\frac{t_{\rm solve+Pz}-t_{PV}}{t_{PV}}\quad\text{both arms synchronised, in the \emph{same} interleaved block of 03.7,}$$ which is invariant to first order under a dispatch overhead common to the two arms — which is how (h1) is neutralised, both arms being timed in the same block under the same regime; the residual is second order and is bounded by no number in this repository, so this kill's consequence is always "the record's cost number is void", never "the shape is slow" (M-3). What survives is (h2): the reference ratio was taken in a different session from the re-measurement, and $1.65\times$ is the only measured session-to-session factor on identical code, so the threshold is (h2) times the reference and nothing else enters it. Reference, from the same un-synchronised session and quoted as the un-synchronised floor it is: $\rho_\Delta=1.041/1.473=0.707$ at $n=2048$, $2.312/2.347=0.985$ at $n=4096$, $4.542/5.374=0.845$ at $n=8192$ — a spread of $1.39\times$ across the record's own three points, which is why the threshold is registered per $n$ and not pooled. Frozen: $\rho_\Delta>1.65\times0.707=1.167$ at $(s,n)=(64,2048)$ (DERIVED: (h2)'s measured drift times the reference) — then the record's only cost number is void and every link's price is re-derived before anything else runs. The earlier $2.2\times0.707=1.555$ carried N-02's $2.2$, a factor with no derivation on this arm, and is struck with it. $1.041$ ms is quoted nowhere in this book as a synchronised expectation; it is an un-synchronised per-op floor (03.6). Instrument: N-02 run inside 03.7's interleaved block with the $PV$ control in the same block; price $\approx1$ GPU-min DERIVED (READ docs/PLAN.md:540), one evening; reachable on the certified device once 03.18 step 1 lands. N-02's own PASS band of $\pm12\,\%$ of $1.041$ ms (READ docs/PLAN.md:540) is refused as written and does not appear in this canon: $\pm12\,\%$ is the clock's non-stationarity band for two readings taken under the same synchronisation regime (READ docs/CEQ_SHAPE.md:2680), and it cannot bracket a reading taken across the un-synchronised-to-synchronised change whose magnitude is unmeasured; the $\pm12\,\%$ band applies to $\rho_\Delta$ between two synchronised sessions and to nothing else. A CORRECTIONS row against docs/PLAN.md:539-540 is owed and named in "Attacks answered". Planted negative, on which the kill must fire (V-15). The same interleaved block with the exact solve replaced by the $K=16$ Neumann iteration on the same operator — a known-slow route whose increment the record has already measured at $25.409-1.473=23.936$ ms against the same $PV$ control at $n=2048$, i.e. $\rho_\Delta=16.25$, $13.9\times$ the frozen $1.167$ (DERIVED from the RUN pair of 03.6) — must fire the kill; an instrument that reads $\rho_\Delta\le1.167$ on it is refused. The synchronised-versus-un-synchronised comparison is not this verse's plant: it condemns the timer rather than the number, and it lives as 03.7's kill (ii), where a positive control belongs.

If killed. The ladder's order survives any single kill; what dies is the link. The replacement chain, each link with its own kill, derived here:

L1 killed (parity or price, 03.12's kill) → L2, the same recurrence $M_{kk}z_k=V_k+\gamma\sum_{l<k}P_{kl}z_l$ (03.11; the sign is plus — see 03.11's corrected display and its $\gamma=0.6$ row-sum check) in torch ops with per-chunk torch.utils.checkpoint; Hypotheses: $C\in\{32,64,128\}$ divides $s$; Evidence: NOT MEASURED — needs the chunked torch route (03.11); Kill: at $s=4096$, $n$ per 03.5's dense column, wall-clock above the serial L4 solve at every $C$ (N-11's kill, READ docs/PLAN.md:781-787), price "minutes once built" [ASSUMED], cheaper than L1's kill because no kernel is written.
L2 killed → L3, Neumann on a certified sparse $P$ with $\delta_{\rm union}^{O}=\varepsilon/(1-\gamma)+\gamma^{K+1}$ on the shipped $O$ read and $\delta_{\rm union}^{z}=\varepsilon/(1-\gamma)+\gamma^{K+1}/(1-\gamma)$ on a bare $z$ read, each times $\|V\|_\infty$ (03.16, where the $(1-\gamma)$ cancellation is derived); Hypotheses: a schedule exists with $\delta_{\rm union}^{O}\|V\|_\infty<\mathrm{sd}(\text{label})$; Evidence: the do-nothing schedule and the fill-in guard exist in the tree (READ ceq/mz_kernel.py:13-21); Kill: one exceedance of the vector bound on 1,024 drawn cells (K-I, READ docs/PLAN.md:806), price $\approx2.6$ s at $s=64$ DERIVED — cheaper than L2's kill.
L3 killed → L4, dense solve_triangular with $n$ per $s$ from 03.5; Hypotheses: $n\le$ the residency column; Evidence: RUN[NEPTUNE] $1.041$ ms at $s=64$; Kill: the S-66 exponent interval in $s$ excludes $2$ toward $3$ (K-9, READ docs/PLAN.md:723), price $\ge1{,}248$ GPU-s of timed work DERIVED in 03.7 (the $206$–$537$ GPU-s of READ V20_R15_THEORY_TABLE.md:107 @ 99777ab prices four harness edits, not this sweep) — cheaper in evenings than L3's schedule construction and more decisive, since it kills every long-context price at once.
L4 killed → L5, the Terminal.

Terminal. When L1–L4 are dead: "The shape's read is one exact triangular solve of depth $s$ costing $+s^2d/2$ MACs over the softmax head it contains, priced only at $s=64$, $n\le8192$ on the certified device as a per-op floor taken on an un-synchronised clock whose bias has a known sign and no measured magnitude; no long-context price exists; the T1 sentence is withdrawn and A1/A2 stand alone." Every MAC statement in 03.2 survives as a MAC statement (M-3: MACs and seconds are stated separately).

03.2 — The per-step MAC model: one solve against $L$ stacked attention layers#

Statement. Per head and per sequence, forward only, causal half counted: $$\mathrm{MAC}_{\rm softmax}(s,d)=\tfrac{s^2d}{2}\ (QK^\top)+\tfrac{s^2d}{2}\ (PV)=s^2d,\qquad \mathrm{MAC}_{\rm shape}(s,d)=s^2d+\tfrac{s^2d}{2}\ (\text{solve})=\tfrac{3}{2}s^2d,$$ the read $O=(1-\gamma)Pz$ replacing $PV$ at equal cost. Against the depth skyline of $L=\lfloor\log_2 t^\star\rfloor+2$ layers of the same head (CHARTER §1), per-layer projections included at $4d_{\rm m}^2 s$ MACs each with $d_{\rm m}=hd$: $$\frac{\mathrm{MAC}_{\rm shape}}{\mathrm{MAC}_{\rm sky}}=\frac{\tfrac32 s^2 d+4d_{\rm m}^2 s}{L\,(s^2d+4d_{\rm m}^2 s)}\;\xrightarrow{\;s\gg d_{\rm m}\;}\;\frac{3}{2L}.$$ Instances (DERIVED, $d=d_{\rm m}=16$, $h=1$): at $t^\star=8$, $L=5$; the attention-only ratio is $0.300$; with projections at $s=64$: $(1.5\cdot65536+65536)/(5\cdot(65536+65536))=0.250$; at $s=4096$: $(1.5\cdot2.684\times10^8+4.19\times10^6)/(5\cdot2.726\times10^8)=0.298$.

The matched-parameter skyline, at a head dimension the stack admits. Ruling 3 shrinks the stack's width so that $L\cdot4d_{\rm m}'^2=4d_{\rm m}^2$, i.e. $d_{\rm m}'=d_{\rm m}/\sqrt L$. At the record's $(d_{\rm m},L)=(16,5)$ this returns $d_{\rm m}'=16/\sqrt5=7.155$, which is not an admissible head dimension — 03.1's Hypotheses admit $d\in\{16,32,64,128\}$ only (READ ceq/mz_kernel.py:41, _POW2_HEAD_DIMS = (16, 32, 64, 128)), so a skyline at $d_{\rm m}'=7.155$ can be neither built nor timed on the certified device and any A2 or T1 denominator constructed from it is a formula with no arm (V-10). The matched skyline is therefore registered at the depth-and-width pair that lands on an admissible $d$: $$(d_{\rm m},L,d_{\rm m}')=(32,\,4,\,16),\qquad L\cdot4d_{\rm m}'^2=4\cdot4\cdot256=4096=4\cdot32^2=4d_{\rm m}^2\ \text{exactly matched},$$ with $L=4=\lfloor\log_2 4\rfloor+2$ the skyline depth at $t^\star=4$ (READ MATHEMATICS.md:284-296, hop_4→4). At that pair the skyline's attention MACs are $L\,s^2d_{\rm m}'=\sqrt L\,s^2d_{\rm m}=2\,s^2 d_{\rm m}$. The matched ratio is printed at both limits, each labelled, and the projection term is dropped nowhere without the word asymptote beside it: with projections at $s=64$, $d_{\rm m}=32$ — a registered geometry — $(1.5\cdot4096\cdot32+4\cdot32^2\cdot64)/(2\cdot4096\cdot32+4\cdot32^2\cdot64)=458752/524288=\mathbf{0.875}$; with projections at $s=4096$, $8.2208\times10^8/1.09052\times10^9=\mathbf{0.754}$; and $\tfrac{3}{2\sqrt L}=\mathbf{0.750}$ only as the $s\gg d_{\rm m}$ attention-only asymptote, which is the ratio at neither registered geometry. The same discipline on the forbidden $(d_{\rm m},L)=(16,5)$ pair reads $(1.5\cdot65536+65536)/(\sqrt5\cdot65536+65536)=163840/212074=0.773$ with projections at $s=64$ against $3/(2\sqrt5)=0.671$ attention-only — a gap of $0.102$ between the two, the size of the error a bare asymptote makes when it is quoted at $s=64$; both are printed here as the arithmetic of a pair A-14 forbids, and neither is a threshold anywhere in this canon. The shape's per-step arithmetic is below the skyline's under both matchings, and above the depth-1 head's by exactly $1.5\times$ — the unmatched instances above are printed as the unmatched controls they are, and the matched row is the only one an A2 or T1 denominator may use.

Hypotheses. Dense causal softmax; one head; the solve by substitution (no explicit inverse, which would cost $s^3/6$); $L$ from $t^\star$ by MATHEMATICS.md §2 (READ MATHEMATICS.md:284-296: hop_2→3, hop_4→4, hop_8→5, hop_16→6); the matched-width variant assumes head dimension scales with $d_{\rm m}$ at fixed $h$ and lands on an admissible $d\in\{16,32,64,128\}$ — which fixes the matched pair at $(d_{\rm m},L,d_{\rm m}')=(32,4,16)$ and forbids the $(16,5)$ pair as a matched denominator anywhere in this book or in book 04's A2/T1 rows. Non-vacuous for every $s\ge d_{\rm m}$; at $s<d_{\rm m}$ the projections dominate and the ratio tends to $1/L$.

Evidence. DERIVED (the four lines above). The record's own statement of the increment, "$+s^2d/2$ MACs, depth $s$, $3s^2d/2$ total, $+50\,\%$": READ docs/CEQ_SHAPE.md:2591, READ docs/sources/judge/proposition_ledger.md:24. The measured floor at $s=64$: solve$+Pz$ $2.514$ ms against $PV$ $1.473$ ms, ratio $1.707$ against the MAC ratio $2.0$ for that pair and $1.5$ for the whole head — the operator at this geometry is not arithmetic-bound (03.3). The skyline's step price the record carries: $7.6$ s per 150-step cell [ASSUMED linear in depth] (READ docs/CEQ_SHAPE.md:2298) against $1.681$ s for the shape (READ docs/CEQ_SHAPE.md:2293), ratio $0.221$ — consistent in sign with the unmatched $0.250$ and not a measurement. $7.6=5\times1.524$ s is five copies of the unmatched depth-1 cell at $d_{\rm m}=16$ (READ docs/CEQ_SHAPE.md:2298 for the $7.6$ s row; READ docs/CEQ_SHAPE.md:2292 for the $1.524$ s softmax control cell it is five copies of); it is not a matched-parameter skyline price and is never used as an A2 or T1 denominator (Ruling 3). The matched skyline at $(d_{\rm m},L,d_{\rm m}')=(32,4,16)$ has no step price: NOT MEASURED — needs 03.7's sweep with a fourth arm at $L=4$, $d_{\rm m}'=16$.

Mechanism. M-3 (MACs and depth stated separately; a MAC ratio is never a seconds ratio), P-8 (the ratio's direction is "at most" for the skyline, since a fused softmax stack's constant factors are the best in the field), R-SKY and D-1 (the skyline row is printed beside the shape's on every cost line, never omitted).

Kill (one pair, named once). N-09's reading (READ docs/PLAN.md:732-736), read on the solve$+Pz$-to-$PV$ pair and on no other: $t_{\rm solve+Pz}/t_{PV}$ above $3$ at any $s\le1024$ under 03.7's protocol. That pair's MAC prediction is $2.0$ ($\tfrac32s^2d$ against $\tfrac12s^2d$) and its floor at $s=64$ reads $1.707$, so the frozen $3$ is $3/1.707=1.757\times$ the floor reading and $1.5\times$ the MAC prediction. The earlier parenthesis "(MAC prediction $1.0$ for that pair)" named the solve-alone-to-$PV$ pair, whose reading is $1.041/1.473=0.707$ and against which the same $3$ would be $4.24\times$ — two referents for one frozen number, and the second is struck: solve-alone against $PV$ is never this kill's pair, and $0.707$ appears in this book only as 03.1's $\rho_\Delta$ reference. Threshold frozen at $3$ on the solve$+Pz$-to-$PV$ ratio. Instrument: S-66/N-09; price inside 03.7's DERIVED $\ge1{,}248$ GPU-s; decidable at $N=8$ repeats per point. Planted negative: a for-loop row-by-row substitution in Python at $s=256$ must read a ratio above $3$ (the latency chain made explicit) — the instrument must condemn a known-slow route.

If killed. The MAC model stands (it is arithmetic) and the seconds model becomes 03.4's two-term law $t_{\rm solve}=\tfrac{s^2 d}{2R_{\rm eff}}+\tfrac{s}{C}\tau_{\rm chunk}$ with $C=s$ for L4 — the depth term is the replacement statement; Hypotheses: $\tau_{\rm chunk}$ constant across $s$; Evidence: NOT MEASURED — needs N-09's ratio at four $s$ (03.7); Kill: the fitted $\tau_{\rm chunk}$ interval contains $0$ (then latency is not the cause and the excess is dispatch, P-8) — price $0$ GPU-s from S-66's journal, cheaper than the verse's kill.

Terminal. "Per head the shape costs $\tfrac32 s^2 d$ MACs against $s^2 d$ for one softmax head, $L s^2 d$ for the unmatched depth-$L$ skyline and $\sqrt L\,s^2 d_{\rm m}=2\,s^2d_{\rm m}$ for the matched skyline at $(d_{\rm m},L,d_{\rm m}')=(32,4,16)$; this is arithmetic, not a price, and licenses no T1 sentence."

03.3 — Bytes and arithmetic intensity: the materialised route is memory-bound, the fused route need not be#

Statement. Per sequence and head, forward, with $P$ materialised in float32 (routes L2 without recompute, L4): $$\text{bytes}_{\rm mat}\ \ge\ 3\cdot 4s^2\ (\text{write }P,\ \text{read }P\text{ twice}),\qquad I_{\rm mat}=\frac{2\cdot\tfrac32 s^2 d}{12\,s^2}=\frac{d}{4}\ \text{FLOP/B},$$ which reads $4$ FLOP/B at $d=16$ and $32$ FLOP/B at $d=128$. With $P$ recomputed on-chip from $B_M$-row query tiles streaming $K$, $V$, $z$ (route L1): $$\text{bytes}_{\rm fused}=\frac{s}{B_M}\cdot\frac{s}{2}\cdot 3\cdot4d=\frac{6\,s^2 d}{B_M},\qquad I_{\rm fused}=\frac{3s^2d}{6s^2d/B_M}=\frac{B_M}{2}\ \text{FLOP/B},$$ independent of $d$ and $s$: $64$ FLOP/B at $B_M=128$. The device ridge point is $R_{\rm peak}/\mathrm{BW}$; with the two [ASSUMED] constants below it reads $74.5$ FLOP/B. $I_{\rm fused}$ is read at the $B_M$ the solve stage admits, not at a free $B_M$: 03.12 fixes $B_M=C$ in the solve stage, so the SRAM inequality below decides $B_M$, and $B_M=256$ — the smallest tile that would reach the ridge, $B_M\ge2\times74.5=149$ — is inadmissible at every $d$ ($B_M=C=256$ at $d=16$ needs $327{,}680$ B $=328$ KB against a $100$ KB budget). The admissible readings are:

$d$	largest admissible $(B_M,C)$ with $B_M=C$	SRAM $4(4Cd+C^2)$	$I_{\rm fused}=B_M/2$	against the $74.5$ FLOP/B ridge
16	$(128,128)$	$98{,}304$ B $=98.3$ KB	$64$ FLOP/B	$85.9\,\%$ — within $14.1\,\%$, compute-bound side
32	$(64,64)$	$49{,}152$ B $=49.2$ KB	$32$ FLOP/B	$42.9\,\%$ — memory-bound
64	$(64,64)$	$81{,}920$ B $=81.9$ KB	$32$ FLOP/B	$42.9\,\%$ — memory-bound
128	$(32,32)$	$69{,}632$ B $=69.6$ KB	$16$ FLOP/B	$21.5\,\%$ — memory-bound

So the materialised route is memory-bound at every admissible $d\le128$, and the fused route comes within $14.1\,\%$ of the ridge at $d=16$ only, where the solve stage admits $B_M=C=128$; at $d\ge32$ the fused route is memory-bound too, by $2.3\times$ to $4.7\times$. The "within $14\,\%$" sentence is a $d=16$ sentence and carries its $(B_M,C)=(128,128)$ in every quotation.

Hypotheses. $P$ stored dense (the causal half not exploited by storage — true of [n,h,s,s] tensors); one forward read of $P$ for $Pz$ and one for the solve; float32 throughout; SRAM holds a $B_M\times d$ query tile plus a $C\times d$ key/value/$z$ tile plus the $B_M\times C$ score tile in float32: $$\mathrm{SRAM}(B_M,C,d)=4\big(2B_Md+2Cd+B_MC\big)\ \text{bytes}\ \le\ 100{,}000\ \text{B}\quad\text{[ASSUMED, sm\_8.9 product documentation]},$$ evaluated at one chunk length throughout — the earlier reading "$4(4096+2048+16384)=90$ KB at $B_M=128$, $C=64$, $d=16$" mixed two $C$ ($2Cd$ at $C=64$, $B_MC$ at $C=128$) and is struck: at $B_M=128$, $C=64$, $d=16$ the correct value is $4(4096+2048+8192)=57{,}344$ B $=57.3$ KB, and at $B_M=128$, $C=128$, $d=16$ it is $4(4096+4096+16384)=98{,}304$ B $=98.3$ KB, inside the budget. The general solution at $d=16$, $B_M=128$ is $16{,}384+640C\le100{,}000$, i.e. $C\le130$ — so $C=128$ is admissible and 03.11's earlier prediction that "$C=64$ is the largest $C$ whose $B_M\times C$ float32 score tile with $B_M=128$ fits" is false and is repaired in 03.11. Under the solve stage's $B_M=C$ (03.12) the inequality is $4(4Cd+C^2)\le100{,}000$ and the admissible pairs are those of the Statement's table. At $d=128$ the earlier figure "$4(32768+16384+16384)=262$ KB" mixed $C$ again (its correct value at $B_M=128$, $C=64$ is $4(32768+16384+8192)=229{,}376$ B $=229$ KB) and the pair it declared admissible, $B_M=64$, $C=32$, exceeds this verse's own budget: $4(16384+8192+2048)=106{,}496$ B $=106.5$ KB against $100$ KB — a Hypothesis satisfied by nothing (V-10). The largest admissible pair at $d=128$ is $(B_M,C)=(32,32)$ at $69{,}632$ B $=69.6$ KB with $B_M=C$, or $(64,16)$ at $86{,}016$ B $=86.0$ KB where the solve stage's $B_M=C$ constraint is relaxed for a softmax-only read; both are memory-bound.

Evidence. DERIVED (the six lines). Device constants: SM clock $3105$ MHz and memory clock $8001$ MHz RUN this session (nvidia-smi --query-gpu=clocks.max.sm,clocks.max.memory); CUDA-core count $3072$ and bus width $128$ bit [ASSUMED, product specification, not exposed by nvidia-smi]; hence $R_{\rm peak}=3072\cdot2\cdot3.105\times10^9=19.08$ TFLOP/s and $\mathrm{BW}=16\cdot16\times10^9=256$ GB/s, ridge $74.5$ FLOP/B, all DERIVED from one RUN and two ASSUMED — NOT MEASURED — needs a GEMM-peak and a copy-bandwidth microbenchmark (03.18's certificate extension). The floor's achieved rate: at $n=2048$, $s=64$, $d=16$ the $PV$ pair's $6.71\times10^7$ MACs forward, taken as $3\times$ for forward and backward, are $4.03\times10^8$ FLOP in $1.473$ ms $=0.273$ TFLOP/s, $1.4\,\%$ of the assumed peak; the solve increment reads $0.387$ TFLOP/s (DERIVED from the RUN floor) — the operator at $s=64$ is neither compute- nor bandwidth-bound but launch- and latency-bound — the same regime the per-row gate settle arm's FLOP-model optimism reads from the other side (READ scale/m3_flops.py:101-121), whose $2.0\times$–$6.6\times$ is that arm's number and is never a bound on this one (M-8, 03.1). The record's own reason the softmax arm is $O(s)$ in activations and the materialised arm $O(s^2)$: READ ceq/sizing.py:1-24 (measured ratio $1.44\times$ at $s=128$ to $8.06\times$ at $s=2048$).

Mechanism. P-8 (the ridge-point constants are assumed and say so in the headline), V-22 (no constant crosses to another device), M-3 (the $s=64$ rates are not scaled to $s=4096$ — they are quoted as the regime they show).

Kill. Measured $I$-gap: under 03.7, the fused kernel's forward at $s=4096$, $d=16$, $B_M=128$, $n=1024$ reads more than $2\times$ the bytes-predicted time $\text{bytes}_{\rm fused}\cdot n/\mathrm{BW}=6\cdot4096^2\cdot16\cdot1024/128=1.28849\times10^{10}$ B, $/2.56\times10^{11}$ B/s $=\mathbf{50.33}$ ms and more than $2\times$ the FLOP-predicted time $3s^2d\,n/R_{\rm peak}=8.246\times10^{11}/1.908\times10^{13}=43.2$ ms (frozen: $2\times$ the larger, $\mathbf{100.7}$ ms). The earlier $51.5$ ms is struck: it corresponds to $\mathrm{BW}=250.2$ GB/s and not to this verse's own $\mathrm{BW}=16\cdot16\times10^9=256$ GB/s, and it carried the frozen kill to $103$ ms and the plant to $412$ ms. Instrument: the kernel's own timing row (03.7 protocol); price seconds once the kernel exists; NOT MEASURED until 03.12. Planted negative, on which the kill must fire (V-15): the same kernel with the $K$, $V$, $z$ tile reload forced per query row instead of per query tile — a deliberate $B_M=1$ traffic pattern with the arithmetic unchanged — must read above $100.7$ ms, since its bytes prediction is $128\times$ the $B_M=128$ figure, $6.44$ s, i.e. $64\times$ the frozen threshold (DERIVED from $I_{\rm fused}=B_M/2$); a kernel that reads inside $100.7$ ms on that plant is not being measured and the instrument is refused. Separate tracking control, which is not a plant and on which the kill must not fire: the same kernel at $B_M=16$ must read within $2\times$ of its own bytes prediction at $B_M=16$, $8\times50.33=402.6$ ms — the model tracking a memory-bound configuration. The earlier text offered this control alone and called it the planted negative; a control on which the kill cannot fire is a positive control, and the verse now carries both, labelled.

If killed. The two-term model of 03.4 with a launch-and-latency term replaces the roofline: $t=\max(\text{FLOP}/R_{\rm eff},\text{bytes}/\mathrm{BW})+n_{\rm launch}\tau_{\rm launch}+(s/C)\tau_{\rm chunk}$, fitted from the S-66 journal with $R_{\rm eff}$, $\tau_{\rm launch}$, $\tau_{\rm chunk}$ free; Hypotheses: additive terms; Evidence: NOT MEASURED — needs S-66; Kill: R-squared below $0.99$ across the four $s$ (the certificate's own bar, READ scripts/k_cert.py:94, MIN_R2 = 0.99) — then every price is a measured point and no law exists (M-5.2's kill, READ docs/PLAN.md:745-749); price $0$ GPU-s from the journal.

Terminal. "Bytes and FLOP per sequence are stated per route as arithmetic; no wall-clock is derived from them on this device; every seconds figure is a measured point."

03.4 — The wall-clock crossover in $s$: where the depth-$s$ chain becomes visible, and the prediction with its counter#

Statement. Seconds per step for the solve on route $r$ obey, to first order, $$t_r(s,n)=\underbrace{\frac{n\,h\,s^2 d}{2R_r}}_{\text{arithmetic}}+\underbrace{\frac{s}{C_r}\,\tau_r\cdot\Big\lceil\frac{nh}{\mathcal P}\Big\rceil}_{\text{the serial chain, hidden only while }nh\ge\mathcal P},$$ with $C_{\rm L4}=b$ (the cuBLAS trsm block, [U]), $C_{\rm L1}=C$, $\mathcal P$ the number of chains the device runs concurrently [ASSUMED] $=$ SMs $\times$ resident programs. The crossover $s^\dagger$ at which the chain term equals the arithmetic term is $s^\dagger=2R_r\tau_r\lceil nh/\mathcal P\rceil/(C_r\,d\,n h)$. Prediction (the half that flatters, L-SIGN): on L4 at the dense residency $n$ of 03.5 the exponent in $s$ fitted across $\{64,256,1024,4096\}$ has an interval containing $2$ and excluding $3$ (Bet L, READ docs/PLAN.md:923). Counter, and under D-CALIB the point estimate: the interval excludes $2$ toward $3$, because at $s=4096$ the dense route runs at $n=8$ where $nh=8\ll\mathcal P$ and the chain is exposed on every step; the chunked fused route L1 at $n=1024$ hides it, so the same sweep on L1 reads an exponent interval containing $2$. The book's own prediction is the counter for L4 and the prediction for L1, and the two are one statement about occupancy, not two bets.

Hypotheses. The two terms add (no overlap of the chain with the arithmetic beyond $\mathcal P$ concurrent chains); $\tau_r$ independent of $s$; $R_r$ independent of $s$ above the launch floor; $n$ per $s$ as 03.5; $h=1$. Non-vacuous only when the sweep runs at two $n$ per $s$ (03.7 requires $n\in\{8,32,128\}$ at every $s\le1024$ so that $b$ and $c$ in $t=a n^b s^c$ separate).

Evidence. DERIVED. The record's occupancy fact in the same direction: the per-row lane's measured wall-clock ratio ran $3.48$ at $s=16$ and $7.43$–$11.19$ at $s=64$ against a flat FLOP ratio $1.700$ (READ scale/m3_flops.py:101-121), i.e. the gap widens with $s$ when a Python-level chain is exposed. The record's C17 confound: run order the strongest correlate of seconds at $\rho=+0.7029$, $p=0.0024$ (READ V20_R15_JOURNAL.md:53 @ 99777ab) — the reason no exponent from the record is admissible. The L4 chain length inside cuBLAS: NOT MEASURED — needstrsm_dispatchread from the profiler (N-01's field,READ docs/PLAN.md:529-534). The device's concurrency width, stated so that no kill of this verse infers it. 03.3's Evidence carries the CUDA-core count $3072$ [ASSUMED, product specification, not exposed by nvidia-smi]; sm_8.9 places $128$ CUDA cores per SM, so the SM count is $3072/128=\mathbf{24}$ [ASSUMED, product specification] (DERIVED from one assumed count and one assumed per-SM width — nvidia-smi --query-gpu= exposes neither, RUN this session). With at least one resident program per SM, $\mathcal P=\text{SMs}\times\text{resident programs}\ \ge\ 24$ under this verse's own definition. The number of resident programs, and hence $\mathcal P$ itself: NOT MEASURED — needs the occupancy sweep at fixed $s$ over $n\in\{1,2,4,\dots,2048\}$ (03.7 row 4); the lower bound $24$ is the only concurrency figure this book asserts, and it is asserted to forbid an inference, never to license one.

Mechanism. D-3 (a dial that does not vary: the record never varied $s$, so its exponent is unidentified at $n=1$ of the independent variable, READ V20_R15_THEORY_TABLE.md:100-103 @ 99777ab), M-3 (a pilot at $s=64$ scaled by the FLOP term understates the bill $2\times$–$3\times$), D-7 / L-SIGN (the counter stands beside the prediction with equal specificity), D-CALIB (the counter is the point estimate: 9 checked, 9 adverse).

Kill. For the counter (which this book carries as the estimate): the L4 exponent interval at $n=8$ across the four $s$ contains $2$ — then the chain is hidden at $nh=8$ and the fused route's occupancy argument buys nothing on the depth axis (it still buys the memory table of 03.5). The inference "$\mathcal P\le8$ on this device" that this kill once drew from that reading is struck, because it contradicts this book's own device constants: 03.4 defines $\mathcal P=$ SMs $\times$ resident programs, and $24$ SMs (Evidence below) with at least one resident program each gives $\mathcal P\ge24$, so $\mathcal P\le8$ and the $3072$-core reading cannot both hold (V-22, M-3). What the exponent reading licenses is a statement about the sweep, not about $\mathcal P$: the chain is not visible in seconds at $nh=8$, which is consistent with $\mathcal P\ge24$ and needs no concurrency count at all. $\mathcal P$ is measured by the occupancy row of 03.7 (row 4: the shape at $s=64$, $n\in\{1,2,4,\dots,2048\}$, $12$ cells) — the $n$ at which seconds stop being flat in $n$ — and is never inferred from an exponent. For the prediction on L1: the L1 exponent interval at $n=1024$ excludes $2$ toward $3$ — then chunking did not hide the chain and $C$ is too small or $\tau_{\rm L1}$ too large; 03.11's rule — the largest $C$ inside 03.3's SRAM budget, $C=128$ at $d=16$ — is re-fitted against the measured $\tau_{\rm L1}$, which is the only quantity left that can oppose a large $C$ once the MAC total is known $C$-independent. Thresholds frozen: a two-sided $95\,\%$ bootstrap interval over $1{,}000$ resamples of the $N=8$ per-point medians. Instrument: S-66 as specified in 03.7; price $\ge1{,}248$ GPU-s DERIVED (03.7's cell count) for L4, L1 NOT MEASURED. Planted negatives, one per direction, because the verse carries two kills and each needs a plant that fires its own. (i) For the prediction's kill (an interval excluding $2$ toward $3$): the Python for-loop row-by-row substitution at $s\in\{64,256\}$ must read an exponent interval excluding $2$ toward $3$ — the instrument must see a chain when one is planted. (ii) For the counter's kill (an interval containing $2$), which is the estimate this verse carries and which the clause (i) plant reads in the opposite direction and therefore cannot fire: a batched trsm at the occupancy row's top — $s=64$, $n=2048$, $h=1$, so $nh=2048\gg\mathcal P$ at any $\mathcal P$ the device admits ($\mathcal P\ge24$ by the SM count) — must read an exponent interval containing $2$, since at that $nh$ every chain is hidden by construction and the arithmetic term alone is left. An instrument that reads an interval excluding $2$ there cannot see a hidden chain as hidden, and the counter's kill is refused on it.

If killed (the counter falsified: the chain hidden at $n=8$): the crossover is not on the depth axis but on the memory axis, and the statement becomes 03.5's: L4 is the shipped route at every $s$ where its residency column admits the bed's $n$, and L1 is needed only above that column; Hypotheses: 03.5's byte model; Evidence: READ COSTS.md:101-115 for the residency rows that exist; Kill: N-20's reading — reserved above the $0.90$ budget $7{,}726{,}694{,}400$ B $=7.196$ GiB at the bed's $(s,n)$, 03.5's re-frozen threshold (READ docs/PLAN.md:763-767), price $\approx3$ GPU-min DERIVED, cheaper than S-66.

Terminal. "The exponent of seconds in $s$ is unmeasured on every route; the record's only $s$ is $64$; no crossover is claimed; every price above $s=64$ carries the word band and the direction at least."

03.5 — The $n$ and $s$ ceilings on the certified device, and the table the fused route restores#

Statement. With the record's byte model on the certified device, whose hard limit is the total VRAM $8{,}585{,}216{,}000$ B $=7.996$ GiB (READ results/k_cert_local.json, memory.cells[*].total_vram) and whose table budget is $0.90$ of it, $7{,}726{,}694{,}400$ B $=7.196$ GiB — the two are different numbers and this verse never uses one where it means the other, $$\text{bytes}(n,s,d)=1.256\,\big[C_{\rm OP}\,n s^2\cdot4+2\,n s d\cdot4+17.874\,n s d\cdot4\big],\qquad C_{\rm OP}\in[3.823,5.823],$$ the largest power-of-two $n$ per $s$ at $d=16$ is (DERIVED this session, reproducing READ docs/CEQ_SHAPE.md:2320-2326 at the upper constant):

$s$	L4, $C_{\rm OP}=5.823$	L4, $C_{\rm OP}=3.823$	L1 (no operator term; $z$ retained: $3nsd\cdot4$)	dense operator alone at $n=2048$
64	32768	32768	65536	$0.034$ GB
256	2048	4096	16384	$0.54$ GB
1024	128	256	4096	$8.6$ GB
4096	8	16	1024	$137.4$ GB

The dense control at $n=2048$, $s=4096$ is $2048\cdot4096^2\cdot4=1.374\times10^{11}$ B and runs nowhere on this device (READ docs/sources/design/refute_instrument_occvac.md:343). The $n=16384$ reproduction was dropped because the complex-valued workhorse arm reserved $10.578$ GiB under a training loop (Ruling 8, READ V17K_RULINGS.md:328-339 @ 99777ab; READ COSTS.md:107-115) and the $n=32768$ one at $13.969$ GiB; both are the operator term at $8$ bytes per element. The fused route lifts the $s=4096$ ceiling from $n=8$ to $n=1024$ ($128\times$) and the $s=1024$ ceiling from $128$ to $4096$ ($32\times$), at $6.55$ GiB predicted in each case.

Hypotheses. The two constants in the byte model have two different provenances and are cited apart, because attributing them to one fit is P-1 (a number quoted at a source that does not carry it) and M-8 (a constant carried across objects). (a) The residual constant $17.874$ is [FITTED] at R-squared $0.996373$ by the two-parameter least squares that separates the residual and operator terms on the sequence sweep, on ceq/lm.py::TinyLM at $B=4$, $d=256$, $L=4$, $H=4$ (READ COSTS.md:89, the row C_RESIDUAL (fp32) | 18 | 17.874 | 0.996373; $C_{\rm OP}=3.823$ is the same fit's other coefficient, READ COSTS.md:90). (b) The reserved-over-allocated ratio $1.256$ is not part of that fit and is not at those lines: it is READ COSTS.md:223, where it is described in the source's own words as "the worst reserved/allocated ratio measured over this box's resident shapes" — a different object (allocator reserve against allocation, not a byte-model coefficient), a different provenance (a maximum over measured shapes, not a least squares), and no R-squared. The earlier Hypotheses line of this verse attributed $1.256$ to the COSTS.md:84-99 fit and is struck. Carrying either constant to the shape at $d_{\rm m}=16$ is M-8 and is [ASSUMED] until N-18 re-solves them (READ docs/PLAN.md:751-755), and the two are [ASSUMED] for different reasons: (a) because it was fitted on another model at another width, (b) because a worst-case ratio over one box's resident shapes is not a law and its own source applies it to a T4 budget (READ COSTS.md:224-225), which V-22 forbids pooling with this device's. The L1 column assumes exactly one retained $s\times d$ float32 tensor beyond the softmax arm's ($z$) plus the two operator-side activations the record already counts. The L1 column is non-vacuous only once 03.14's backward exists (without it autograd retains $P$ and L1 collapses to the L4 column).

Evidence. DERIVED (the arithmetic re-run this session; the four L4 upper-constant entries equal the record's $32768/2048/128/8$). The residency rows that exist: READ COSTS.md:101-115 (softmax resident to $n=32768$ at $4.908$ GiB reserved; arm_smprime not resident at $16384$ under a training loop). The certificate's memory cells: READ results/k_cert_local.json (memory.cells, resident: false at arm_smprime, n = 32768; reserved $8{,}057{,}257{,}984$ B at $n=16384$ outside a training loop). $C_{\rm OP}$'s two ends: $3.823$ fitted at R-squared $0.996373$ (READ COSTS.md:87-91), $+2$ retained tensors [ASSUMED].

Mechanism. P-8 (Ruling 8's lesson: a point fitted through PCIe paging is not a point), V-22 (a T4's $13.50$ GiB budget is never pooled with this device's), M-8 (the constants are the signed arm's), P-1 (the table's producer is N-18/N-20; until they run the table is a formula).

Kill, frozen at the budget the table is built against. N-20's reading (READ docs/PLAN.md:763-767), re-registered: reserved bytes above $7{,}726{,}694{,}400$ B $=7.196$ GiB — the $0.90$ budget every row of the table above is solved at — at any $(s,n)$ the table admits, on the route the row names, under a training loop with synchronize(). Frozen. The device's total $8{,}585{,}216{,}000$ B $=7.996$ GiB stands separately as the hard limit whose exceedance is an allocator failure, not a reading: a row that reserves above it produces no cell at all.

Why the threshold moved. The earlier kill froze at the $7.996$ GiB total while the table refused rows at the $0.90$ budget, opening a band of $8{,}585{,}216{,}000-7{,}726{,}694{,}400=858{,}521{,}600$ B — $11.1\,\%$ of the budget — in which the table refuses a row and the kill cannot fire (V-11 on the kill, V-10 on the table's own gate; M-2, since the two numbers were frozen against different quantities). The band is not hypothetical: the record's own certificate holds a cell inside it — READ results/k_cert_local.json, memory.cells, arm_smprime, $s=64$, $n=16384$, reserved $8{,}057{,}257{,}984$ B $=7.504$ GiB with resident: true, fits: true, above the budget and below the total. That cell is this kill's reachability witness: it is a real draw on the registered bed at which the re-frozen kill fires and the old one could not.

Instrument: the certificate's residency cell with the shape as the arm (03.18); price $\approx3$ GPU-min DERIVED; decidable on any draw (the bytes do not depend on the data). Planted negatives, and the arithmetic refusal relabelled as the prediction it is. The clause this verse carried before — "the table must refuse $(4096,16)$ on L4 at the upper constant ($7.96\times10^9$ B predicted against the $7.727\times10^9$ B budget)" — is not a plant and is struck as one: the refusal is arithmetic the formula performs before any device is touched ($1.256[5.823\cdot16\cdot4096^2\cdot4+2\cdot16\cdot4096\cdot16\cdot4+17.874\cdot16\cdot4096\cdot16\cdot4]=7.958\times10^{9}$ B against $7.727\times10^{9}$ B, DERIVED and re-verified this session), so it is satisfied by construction and makes the kill — reserved bytes on the device above the threshold — fire on nothing (V-10, V-15). It survives as the prediction the plant tests. The two plants are both device readings: (P1) L4 at $(s,n)=(4096,16)$, the row the table refuses, run on the certified triple under a training loop with synchronize(), must read torch.cuda.max_memory_reserved above $7{,}726{,}694{,}400$ B or raise torch.cuda.OutOfMemoryError — a formula that refuses a row the device then runs resident inside the budget is a formula, not a ceiling, and the kill is refused as insensitive; (P2) the certificate's arm_smprime $n=16384$ cell above, re-run as a row this table admits, must fire the kill at $7.504$ GiB — a plant taken from the record rather than constructed. A row inside the table that the device then runs resident is the model reading pessimistic and is recorded as such, and a row outside it that runs resident by more than $10\,\%$ voids the constant (N-18's $[0.9,1.1]$ clause).

If killed (a row the table admits does not fit): the constants are re-solved per route by two-parameter least squares on a sequence sweep $128$–$2048$ (N-18), and the table is re-derived from measured constants; Hypotheses: R-squared at least $0.99$; Evidence: NOT MEASURED — needs N-18; Kill: R-squared below $0.99$ or measured-over-predicted outside $[0.9,1.1]$ — then no memory law exists for the shape and $n$ is found per $s$ by bisection on the device (price $\approx9$ GPU-min DERIVED, READ docs/PLAN.md:755), cheaper than the verse's kill in decisions because it never fits a law.

Terminal. "No memory law is claimed for the shape; $n$ is declared per $s$ by measurement on the certified device; the dense control at $n=2048$, $s=4096$ does not exist on any device this programme owns."

Part B — the cost law as a specification, and what T1 means (B14, B25)#

03.6 — The measured increment is a per-op floor and licenses no ratio#

Statement. The only cost numbers the record owns for the solve are the per-op increments $$\Delta t(n)=t_{\rm solve+Pz}-t_{PV}=1.041,\ 2.312,\ 4.542\ \text{ms per step at } n=2048,4096,8192$$ ($2.514-1.473$, $4.659-2.347$, $9.916-5.374$; RUN[NEPTUNE], float32, forward and backward, $s=64$, $d=16$), and the Neumann arm at $K=1$ and $K=16$: $3.001$ ms and $25.409$ ms at $n=2048$, against the solve's $2.514$ ms.

The Neumann arm is a two-term reading and is never a constant per hop. The two figures are irreconcilable under any constant-per-hop model and were printed side by side without one: $16\times3.001=48.016$ ms against the measured $25.409$ ms, a factor $1.890$. The RUN pair fixes both terms exactly and both are DERIVED from the same pair and from nothing else: $$t_{\rm Neumann}(K)=a+bK,\qquad b=\frac{25.409-3.001}{16-1}=\mathbf{1.4939}\ \text{ms per hop},\qquad a=3.001-b=\mathbf{1.5071}\ \text{ms fixed},$$ and the same marginal read against the $PV$ control instead of against the $K=1$ cell gives $(25.409-1.473)/16=\mathbf{1.496}$ ms per hop, agreeing with $b$ to $0.15\,\%$ — the fixed term is the $PV$-shaped work the arm performs once, $a=1.507$ ms against the control's $1.473$ ms. Which term each sentence uses is printed with the sentence, and no sentence uses both. (i) The licensed comparison of this verse — "at $s=64$ the exact solve is cheaper than one Neumann hop on this device" — is a comparison of totals at $K=1$: $2.514$ ms against $3.001$ ms, a margin of $0.487$ ms, and it uses neither $a$ nor $b$ alone. (ii) A cell at $K=16$ is priced from the $K=16$ total $25.409$ ms and from no other term, so a $150$-step cell reads $150\times0.025409=3.811$ s (DERIVED). (iii) Reading $25.409$ ms as a per-hop figure and multiplying — $16\times25.409\ \mathrm{ms}=0.407$ s — is a units error; it appears nowhere in this book, and both neighbouring books record it as failing to reproduce their $5.12$ s InfSA-16 cell by a factor $12.6$ (READ docs/canon/04_BEDS_AND_INSTRUMENTS.md:183, :781, READ docs/canon/06_PREDICTIONS.md:847, :849; MARS filed those readings at 04:164 and 06:349, which are 04.10's Statement and 06's C-20 row and carry no such number — the pins are re-resolved here and the old two are printed in "Attacks answered"). Under (ii) the $150$-step cell is $3.811$ s and not $5.12$ s either, so the $5.12$ s figure is not derivable from this pair under any of the three readings, which is the sentence both neighbouring books already carry and which this verse now supplies the arithmetic for. The extrapolation to a $K$ this pair does not bracket is NOT MEASURED — needs the Neumann arm swept over $K$ inside 03.7's interleaved block: two points fix a line and certify no linearity, and 03.16's $K\ge131$ regime is $8.2\times$ outside the measured range. Per sequence the increment is $0.508$, $0.565$, $0.554$ μs — linear in $n$ to within $11\,\%$ (DERIVED). Each is a floor on the end-to-end increment: the timer was un-synchronised and the arms were not interleaved, so the bias has a known sign (the un-synchronised clock under-reports the device) and no measured magnitude on this arm. The $2.0\times$–$6.6\times$ figure this verse once attached to these numbers is not theirs: it is the per-row gate settle arm's wall-clock ratios ($3.48$, $11.19$, $7.43$) against that arm's own flat FLOP ratio $1.700$ — a Python loop over t_max steps with a $268$ MB Gram copy, whose file never calls solve_triangular (READ scale/m3_flops.py:101-121, headline at :110). Carrying it here is M-8, a constant carried across arms; it is struck from this verse and from every threshold in this book, and the two hazards that do attach are named in 03.1 as (h1) the absent synchronize() and (h2) the $1.65\times$ session-to-session drift on identical code and geometry (READ scale/m3_flops.py:117-121). They license the sentence "at $s=64$ the exact solve is cheaper than one Neumann hop on this device" and no ratio against any other arm and no number at any other $s$.

Hypotheses. One box (RTX 4060 Laptop, torch 2.5.1+cu121, ab5b485); $s=64$; the same session for every pair (the $1.65\times$ session-to-session drift of READ scale/m3_flops.py:117-121 forbids cross-session pairs); the increment measured at the shipped $s$ and never scaled from a smaller $s$ (M-3).

Evidence. READ docs/sources/judge/sec_shape.md:243, READ docs/sources/judge/sec_apparatus.md:91, READ docs/CEQ_SHAPE.md:431, :590, READ docs/PLAN.md:539 — every one of them carries "producer owed to scripts/k_cert.py": there is no results/ row for these three numbers. The grep that establishes this is re-run without the extension filter, because the filtered form did not establish it. The earlier evidence line — "RUN this session: grep -rl 2.514 --include=*.txt --include=*.json results/ returns nothing" — is struck: results/ holds $172$ entries of which $57$ are *.jsonl (RUN this session: ls results/ | wc -l, ls results/*.jsonl | wc -l), and both --include filters exclude every one of them, so the command searched two thirds of the directory and its silence proved nothing (V-16: silence read as a pass). Re-run unfiltered and as a fixed string, RUN this session: grep -rlF 2.514 results/ returns one file, results/cameron_aggregators.jsonl, and it is not the producer — the hit is 2.5148398876190186, an entry of the causal.tau array of a record whose fields are {agent: "cameron", tag: "smoke", s: 128, d: 8, k: 4, draws: 12, ...} (RUN this session, first line of the file), a threshold statistic at $s=128$, $d=8$ carrying no timing field of any kind, no secs, no n=2048 and no ms. (The unescaped grep -rl 2.514 results/ MARS quotes returns $13$ files, since . matches any character; the fixed-string form is the one this verse runs and quotes.) The conclusion stands on the re-run command and not on the filtered one: no results/ record produces $2.514$, $4.659$ or $9.916$ ms, so under P-1 they are prose until N-02 runs. The per-row gate arm's FLOP-model optimism and the $1.65\times$ session-to-session drift, in that order: READ scale/m3_flops.py:110, READ scale/m3_flops.py:117-121 — the first belongs to that arm and to no other, the second is measured on identical code and is the only drift factor this book freezes a threshold from. The synchronisation hazard: READ V20_R15_JOURNAL.md:53 @ 99777ab; the record's own harness contains no torch.cuda.synchronize (READ docs/CEQ_SHAPE.md:1301-1303).

Mechanism. B14 closed here as a statement: P-8 (a floor headlined as a price), P-1 (no live producer), M-3 (pilot-to-geometry scaling), C17 (un-synchronised timer, run-order confound), V-23 (the softmax control's determinism and timing were never taken under the same protocol — the plural "the arms" covers one arm).

Kill. N-02's synchronised re-measurement reading $\rho_\Delta=(t_{\rm solve+Pz}-t_{PV})/t_{PV}>1.167$ at $(s,n)=(64,2048)$, both arms inside one interleaved block (frozen at 03.1's derivation: (h2)'s measured $1.65\times$ session-to-session drift times the un-synchronised reference ratio $1.041/1.473=0.707$) — the floor was not a floor of this device but of that session, and every derived price in docs/CEQ_SHAPE.md §5.7 is re-derived before any card runs. The absolute form of this kill — "above $2.2\times1.041=2.290$ ms" — is struck, and the earlier $1.555$ with it: the first because the un-synchronised-to-synchronised change on this arm has a known sign and an unmeasured magnitude, so no millisecond count read off the floor decides anything about the shape (V-11); the second because its multiplier $2.2$ was N-02's card constant with no derivation on this arm. The kill is a ratio because a ratio taken inside one block is invariant to first order under an overhead common to both arms; $1.041$ ms is an un-synchronised floor and is quoted as nothing else. Instrument: N-02 under 03.7's bracket with the $PV$ control in the same block; price $\approx1$ GPU-min DERIVED; decidable on torch.randn data at $N=8$ repeats (no bed, no seed rule needed). Planted negative, on which the kill must fire: the same block with the exact solve replaced by the $K=16$ Neumann iteration, whose increment this verse's own RUN pair puts at $25.409-1.473=23.936$ ms against the $PV$ control at $n=2048$, i.e. $\rho_\Delta=16.25$ against the frozen $1.167$ (DERIVED) — an instrument that does not fire there is refused (V-15). The un-synchronised-beside-synchronised reading is required of the microbenchmark as a timer sanity control (the former smaller on at least $6$ of $8$ repeats; equality on all $8$ refuses the instrument, V-16) and is labelled as the control it is, not as this kill's plant; as a kill it lives in 03.7 (ii).

If killed. The increment is re-based on the new measurement and the sentence "cheaper than one hop" is re-decided by the same run: if one hop reads below the solve, the Neumann route re-enters as a cost path with its certificate $\delta\,\|V\|_\infty$ printed and $1/(1-\hat\gamma)$ beside it (S-41, READ docs/PLAN.md:557-564); Hypotheses: $\gamma<1$, $P$ row-stochastic; Evidence: the certificate identity attained to $10^{-15}$ at $K\in\{1,2,4,8,16\}$ (RUN[M], READ docs/CEQ_SHAPE.md:2574); Kill: one exceedance of the vector bound on 1,024 draws, $\approx2.6$ s DERIVED — cheaper than N-02.

Terminal. "The solve's price at $s=64$ is an un-synchronised per-op floor of $1.041$ ms per step at $n=2048$ on one box and one build, with no producer; it is quoted with the words floor and at least, never as a synchronised expectation, and never as a ratio against another arm — the one ratio this book freezes, $\rho_\Delta$, is between the solve and its own $PV$ control inside a single synchronised interleaved block and is not derived from this floor."

03.7 — The cost-law experiment as a specification: synchronised, interleaved, order-recorded#

Statement. A price is RUN if and only if it comes from a cell of the following experiment; every other price in the canon is a floor or a band.

Timer. torch.cuda.synchronize() immediately before the start and immediately after the end of every timed block; two warm-up steps discarded; median of at least $12$ timed steps and at least $3.0$ s of timed work per cell; median over $3$ child processes (the certificate's own rule, READ COSTS.md:66-69); CUBLAS_WORKSPACE_CONFIG=:4096:8 set before the first CUDA call.
Arms and interleaving. Arms $\{$softmax control, shape on the softmax corner (route named), the matched skyline at $(d_{\rm m},L,d_{\rm m}')=(32,4,16)$ per 03.2 — never the unmatched depth-5 stack at $d_{\rm m}'=7.155$, which no tl.dot tile admits, one Neumann hop$\}$ run interleaved ABCD·ABCD·…, never blocked by arm; $N=8$ repeats per (arm, $s$, $n$) cell.
Order. The execution order of all cells is drawn once from a seeded permutation (seed journalled); the run index is a field of every cell; the Spearman $\rho$ of seconds against run index is computed per arm and pooled, with its $p$.
Geometry. $s\in\{64,256,1024,4096\}$; $n\in\{8,32,128\}$ at every $s\le1024$, and at $s=4096$ the route's residency column (03.5: $8$ on L4, up to $1024$ on L1); $d=16$, $h=1$; forward and backward and Adam.step; plus the occupancy row at $s=64$, $n\in\{1,2,4,\dots,2048\}$ for the shape.
Fit. $\log t=\log a+b\log n+c\log s$ per arm by OLS on per-repeat medians; the interval on $c$ from $1{,}000$ bootstrap resamples over the $N=8$ repeats per point, two-sided $95\,\%$; $c$ reported once at fixed $n=8$ across all four $s$ and once at $n=128$ across $s\le1024$.
Journal. results/cost/s_sweep.jsonl, fields: arm, route, C, s, n, d, run_index, seed, secs_sync, secs_unsync, peak_alloc, peak_reserved, cublas_workspace, torch, cuda, driver, git_head, producer_cmd.

Hypotheses. The certified device (03.18) with a CUDA build of torch; the --seq-len flag the record priced and never made (scripts/v15_r1.py:137 is a module constant, READ docs/CEQ_SHAPE.md:1301); clock non-stationarity $\pm12\,\%$ accepted as the quoting precision (READ docs/CEQ_SHAPE.md:2680). Non-vacuous when at least two $n$ per $s$ exist, which is why row 4 departs from the record's single-$n$-per-$s$ point set.

Evidence. The protocol's elements are the record's own repairs, each with a source: two synchronize() calls and a --seq-len flag priced at $206$–$537$ GPU-s band only (READ V20_R15_THEORY_TABLE.md:107, :184 @ 99777ab); randomised or blocked execution order, "a fifth edit no office ever priced" (READ V20_R15_THEORY_TABLE.md:184 @ 99777ab); the run-order correlation to beat, $\rho=+0.7029$, $p=0.0024$ (READ V20_R15_JOURNAL.md:53 @ 99777ab); the point set the record proposed, $\{(64,2048),(256,2048),(1024,128),(4096,8)\}$ (READ docs/PLAN.md:727), which confounds $n$ with $s$ and is repaired by row 4. Price, DERIVED from rows 1, 2 and 4 directly — the $206$–$537$ GPU-s band prices the harness edits and not this sweep. The band's own source states what it prices: "instrument edits: 4 — one argparse line, one substitution at :137, two cuda.synchronize()" (READ V20_R15_THEORY_TABLE.md:184 @ 99777ab; READ V20_R15_IT13_MERCURY.md:146 @ 99777ab). Quoting it as the price of a timed sweep is P-8 in the flattering direction and is struck here. The cell count, DERIVED from row 2's four arms (softmax control, shape, matched skyline, one Neumann hop — the earlier count line in this verse read three arms and is corrected) and row 4's geometry:

block	cells	arithmetic
$s\in\{64,256,1024\}\times n\in\{8,32,128\}$	$36$	$4$ arms $\times\,3\,n\,\times\,3\,s$
$s=4096$ at the route's residency $n$	$4$	$4$ arms $\times\,1\,n$
occupancy row, shape only, $s=64$, $n\in\{1,2,\dots,2048\}$	$12$	$1$ arm $\times\,12\,n$
total	$52$

Row 1 requires $\ge3.0$ s of timed work per cell and a median over $3$ child processes; row 2 requires $N=8$ repeats per cell. The lower bound on timed work alone is therefore $52\times8\times3.0=1{,}248$ GPU-s counting one process per repeat, and $52\times8\times3\times3.0=3{,}744$ GPU-s counting each repeat medianed over three child processes — $2.32\times$ to $6.97\times$ the band's top of $537$ GPU-s, before the two discarded warm-up steps per cell, process start-up, Triton compilation and the $s=4096$ cells' allocator pressure. The sweep's price is quoted as $\ge1{,}248$ GPU-s DERIVED (at least; $\ge3{,}744$ GPU-s under the three-process reading) in this book and nowhere as $206$–$537$ GPU-s. Both numbers stand in the same line wherever the sweep is priced.

Mechanism. C17 (the timer), R-14/K-9 (no price at $s>64$ before the sweep), M-3 (no scaling from $s=64$), D-3 (the dial varies), M-2 (thresholds frozen here, never refitted to the sweep's own data), V-15 (a planted order confound below).

Kill. Three frozen numbers, any one of which condemns the instrument rather than the shape: (i) $|\rho_{\rm order}|\ge0.25$ with $p\le0.05$ on the pooled $52$ cells $\times\,8$ repeats $=416$ timed readings ($36\times8=288$ at $s\le1024$, $4\times8=32$ at $s=4096$, $12\times8=96$ occupancy) — the confound survived and every price from that run is a band. The pool sizes and the critical value, stated rather than asserted: the per-arm pools are $9$ cells $\times\,8=72$ readings at $s\le1024$ plus $1\times8=8$ at $s=4096$, i.e. $80$ readings for each of the softmax control, the matched skyline and the Neumann arm, and $80+96=176$ for the shape, which carries the occupancy row alone; the smallest per-arm pool is therefore $80$ and the $2/\sqrt{n-1}$ rule reads $2/\sqrt{79}=0.225$ there, against $2/\sqrt{415}=0.098$ pooled. The threshold is frozen at $0.25$, at or above both ($0.25>0.225>0.098$), so it fires on no arm below its own critical value. The earlier line "the critical value $2/\sqrt{66}=0.246$ at $n=64$" is struck: $2/\sqrt{66}=0.2462$ is the rule at $n=67$, at $n=64$ it reads $2/\sqrt{63}=0.2520$, and the frozen $0.25$ sat between the two with no pool of either size in this experiment.

Why this threshold is not book 04's, and what the disagreement costs. READ docs/canon/04_BEDS_AND_INSTRUMENTS.md:170 — 04.10's Kill; MARS filed it at :151, which is 04.9 clause 6 and carries no $\rho$, and the pin is re-resolved here — freezes the same confound on the same interleaved design at Spearman $|\rho|>0.5$ with $p<0.05$ on the $N\ge16$ interleaved timings, against this verse's $|\rho|\ge0.25$ with $p\le0.05$. The two numbers are not reconciled into one, and the reason is printed rather than asserted: they are computed from two different $n$ on two different instruments. Book 04's is the arena's timing journal — $8$ seeds $\times\ 3$ interleaved arms $=24$ arm-cells (04.10's own arm-cell reading of $N$) — where $2/\sqrt{n-1}=2/\sqrt{23}=0.417$ and the $p=0.05$ boundary of the $t$-transform $t=\rho\sqrt{(n-2)/(1-\rho^2)}$ sits at $\rho=0.404$; book 04's $0.5$ is therefore above both, at its own $n$. This verse's is the $s$-sweep — smallest per-arm pool $80$ readings — where $2/\sqrt{79}=0.225$ and the $p=0.05$ boundary sits at $\rho=0.220$; the frozen $0.25$ is above both, at its own $n$. Each book's bar is the conservative one for its own instrument, and neither is transportable to the other's $n$ (V-22 on a threshold, M-2). The cost of carrying two bars is stated and not softened: a journal reading $\rho=+0.35$ is condemned by this book and passed by book 04 — MARS's number, and it holds. What separates them is power, not disagreement about the confound: at $n=80$ a $\rho$ of $0.35$ gives $t=3.300$, $p=0.0014$, and both of this verse's clauses fire; at $n=24$ the same $\rho$ gives $t=1.752$, $p=0.094$, and both of book 04's clauses spare it — so the two books deliver opposite verdicts on the same number because the same number carries $67\times$ different evidence on the two journals. What is not defensible and is recorded as owed: a single canon-wide $\rho$ bar cannot exist while the two instruments differ in $n$ by $3.3\times$, so a CORRECTIONS.md row is owed against 04_BEDS_AND_INSTRUMENTS.md:170 and this verse's kill (i) jointly, adopting the rule "$|\rho|$ above the larger of $2/\sqrt{n-1}$ and the $p=0.05$ boundary at that journal's own $n$, both printed in the cell" in place of two frozen constants. This repairer edits only book 03 and makes no git write; the row is named in "Attacks answered" and the two constants stand until the coordinator writes it; (ii) secs_unsync not smaller than secs_sync on at least $6$ of $8$ repeats of any cell — the timer is not measuring the device; (iii) R-squared of the fit below $0.99$ per arm — no law, points only. Instrument: this experiment; price $\ge1{,}248$ GPU-s DERIVED (Evidence above); decidable on torch.randn data at $N=8$. Planted negative (V-15): one extra block run in monotone order after a $60$ s compute burn on the device must read $|\rho_{\rm order}|\ge0.25$ — the instrument must detect the confound it was built against, on a plant.

If killed on (iii) (R-squared of the power-law fit below $0.99$ on any arm — the branch this verse lacked, and the reason it lacked one is that (iii) condemns the law, not the instrument, so the (i)/(ii) branch below does not answer it): the power law $t=a\,n^b s^c$ is abandoned per arm and replaced by the two-term additive model of 03.3's and 03.4's If-killed, fitted on the same journal, $$t=\max\!\Big(\frac{\text{FLOP}}{R_{\rm eff}},\frac{\text{bytes}}{\mathrm{BW}}\Big)+n_{\rm launch}\tau_{\rm launch}+\frac{s}{C}\,\tau_{\rm chunk}\Big\lceil\frac{nh}{\mathcal P}\Big\rceil,$$ with $R_{\rm eff}$, $\tau_{\rm launch}$, $\tau_{\rm chunk}$ free and every point quoted individually beside the fit, never summarised by an exponent. Hypotheses: the three terms add; $\tau_{\rm launch}$ and $\tau_{\rm chunk}$ independent of $s$ and $n$ over the swept range; the same $52$ cells, refitted, with no new run. Evidence: NOT MEASURED — needs 03.7's own journal,results/cost/s_sweep.jsonl, refitted; the record's own reason to expect a launch term rather than a power law is 03.3's $0.273$ TFLOP/s at $s=64$, $1.4\,\%$ of the assumed peak (DERIVED there). Kill: the additive fit's R-squared also below $0.99$ on any arm, or the fitted $\tau_{\rm chunk}$ interval containing $0$ and the fitted $\tau_{\rm launch}$ interval containing $0$ — then neither a power law nor an additive law describes this device's seconds, no cost law of any form exists in this canon, and every price is quoted as a single measured cell with its run_index, seed and git_head, which is this verse's Terminal in force. Price $0$ GPU-s (a refit of the journal already paid for), which is cheaper than the verse's own $\ge1{,}248$ GPU-s and strictly more decisive, since it ends the search for a law rather than re-running it.

If killed on (i) or (ii) (the instrument condemned): the replacement is a blocked design — each (arm, $s$, $n$) cell repeated in $8$ blocks, every block a fresh child process with its own warm-up, block index a covariate, and the price quoted as the within-block paired difference against the softmax control run in the same block; Hypotheses: block-to-block drift is common to both arms; Evidence: the certificate already takes medians over $3$ child processes (READ COSTS.md:66-69); Kill: the paired difference's sign disagrees across at least $3$ of $8$ blocks — then no price is quoted for that cell at all; price the same $\ge1{,}248$ GPU-s, $0$ new evenings. This kill is more decisive than the verse's (a sign flip is a coarser and more damning reading than a $\rho$).

Terminal. "No seconds figure above $s=64$ exists in the canon; the record's it.3 ratios stay withdrawn (C17); prices are floors and bands with the direction in the headline."

03.8 — T1 operationalised: GPU-seconds-to-floor, and the kernel's half of it#

Statement. For an arm $a$ on a registered bed, the distance-to-floor target is book 04's, adopted here verbatim and not restated in other words: $\tau$ is a level of the ladder $\{0.9,0.8,0.7,0.5\}$ in NRMSE-to-oracle units, registered before the run, and the level reported is the first that both arms reach inside the step budget (READ docs/canon/04_BEDS_AND_INSTRUMENTS.md:165, verse 04.10). The earlier reading of this verse — "book 04 sets $\varepsilon_N=\mathrm{MDE}_N$ at the realised paired sd" — attributed to book 04 a target book 04 does not carry and is struck: $\mathrm{MDE}_N$ is 04.9 clause 4's contrast bar for A1/A2, not 04.10's T1 target, and freezing a T1 threshold against one while measuring it against the other leaves the number with no referent (M-2, V-17, P-7). With $\tau$ the adopted target, $$T1(a)=\sum_{k=1}^{k^\star(a)}t_{\rm step}(a,k),\qquad k^\star(a)=\min\{k:\ \mathrm{NRMSE}_k(a)/\mathrm{NRMSE}_0(a)\le\tau\ \text{on held-out}\},$$ so that $T1(a)/T1(b)=\big(k^\star(a)/k^\star(b)\big)\cdot\big(\bar t_{\rm step}(a)/\bar t_{\rm step}(b)\big)$: the draws-to-floor ratio (T2, book 02) times the per-step ratio this book owns.

Denominator 1 — depth-1 softmax, read from the fitted law and not from three roundings of it. Three born books print three values for the depth-1 softmax whole-step price at $n=2048$ and the T1 threshold is frozen from it: $0.010160$ s here (as $1.524/150$), $0.010165$ s at READ docs/canon/06_PREDICTIONS.md:403, $0.010142$ s [MEASURED] at READ docs/canon/08_ARCHITECTURE.md:552 (MARS filed the last two at 06:349 and 08:462, which are 06's C-20 row and an 08 table rule; both pins are re-resolved here). The spread $0.010142$–$0.010165$ is $0.227\,\%$. This book quotes one value, with one class and one pin: the evaluation of the fitted law itself, $\exp(-12.1852)\cdot2048^{0.9963}=\mathbf{0.01016513}$ s per step, [FITTED] at R-squared $0.999998$ over five points (READ COSTS.md:73), re-evaluated RUN this session. $0.010165$ s is that law to six figures; $0.010160$ s is the $150$-step cell price $1.524$ s divided back out and is a rounding of the same law; $0.010142$ s is the single measured point of results/k_cert_local.json throughput.points, which is a datum the law was fitted through and not a competing law, and is quoted as such wherever it appears. The shape's step is that law plus 03.6's increment: $$\bar t_{\rm step}(\text{shape})=0.01016513+0.001041=0.01120613\ \text{s},\qquad \frac{\bar t_{\rm step}(\text{shape})}{\bar t_{\rm step}(\text{softmax})}=\mathbf{1.10241}\quad(\text{DERIVED}),$$ against $1.681/1.524=1.10302$ from the two [FITTED+RUN] cell prices (READ docs/CEQ_SHAPE.md:2292-2293), which differ in the third figure and are printed beside it as the cell reading they are. T1 against depth-1 softmax holds iff $k^\star_{\rm shape}/k^\star_{\rm softmax}<1/1.10241=\mathbf{0.9071}$ at that geometry, on that box, under the per-op floor, with $\tau$ from the ladder — the threshold is unchanged at three figures under either reading ($1/1.10302=0.9066$), and it is frozen at $0.907$ from the law. A CORRECTIONS.md row is owed against 06_PREDICTIONS.md:403-404 and 08_ARCHITECTURE.md:552 so that all three books quote the law's evaluation with the measured point labelled as a fitted-through datum; this repairer edits only book 03 and the row is named in "Attacks answered".

Denominator 2 — the matched skyline, re-derived. The earlier threshold $4.52=1.681/7.6$ used $7.6=5\times1.524$ s, five copies of the unmatched depth-1 cell at $d_{\rm m}=16$ (READ docs/CEQ_SHAPE.md:2298 for the $7.6$ s row, READ docs/CEQ_SHAPE.md:2292 for the $1.524$ s cell, [ASSUMED linear in depth]), which CHARTER §1 and Ruling 3 forbid as a T1 denominator: T1 is read at matched parameters. $4.52$ is struck as a threshold and printed below only as the unmatched control it is. The matched skyline is 03.2's $(d_{\rm m},L,d_{\rm m}')=(32,4,16)$, exactly matched at $4d_{\rm m}^2=L\cdot4d_{\rm m}'^2=4096$ parameters per layer-block, and its step price is NOT MEASURED — needs 03.7's sweep with the matched skyline arm. Until that sweep runs, the second threshold is frozen from the arithmetic ratio, DERIVED and labelled as arithmetic (M-3: a MAC ratio is not a seconds ratio, and this threshold is quoted as the MAC-based bound it is): $$\frac{\mathrm{MAC}_{\rm shape}}{\mathrm{MAC}_{\rm sky,matched}}=\frac{\tfrac32s^2d_{\rm m}+4d_{\rm m}^2s}{\sqrt L\,s^2d_{\rm m}+4d_{\rm m}^2s}=0.875\ (s=64),\qquad 0.754\ (s=4096),$$ so T1 against the matched skyline holds iff $k^\star_{\rm shape}/k^\star_{\rm sky}<1/0.875=1.143$ at $s=64$ (frozen), and $<1.327$ at $s=4096$. The threshold is re-frozen once, from the measured $\bar t_{\rm step}$ ratio the first time 03.7 runs the matched skyline arm, with a supersede marker (L-G2). Beside it, and never as a threshold: the unmatched control reads $1.681/7.6=0.221$, i.e. $4.52$, a factor $3.96$ more permissive than the matched arithmetic threshold — which is the size of the flattery Ruling 3 removes.

Hypotheses. Both arms reach the registered ladder level $\tau$ at all (an arm that never reaches it has $T1=\infty$ and the row reads NOT REACHED, never a ratio — 04.10's own VOID clause); matched parameters per Ruling 3 (counts in the header), which for denominator 2 means the $(32,4,16)$ pair and no other; the step price constant over training (the record's clock drifts $\pm12\,\%$, so the ratio is quoted to two figures); the same bed, seed rule and $N$ for both arms; $s=64$, $n=2048$ with $d_{\rm m}=16$ for denominator 1 and $d_{\rm m}=32$ for denominator 2 — at any other geometry $\bar t_{\rm step}$ comes only from 03.7. The two denominators are read at two different $d_{\rm m}$ and their thresholds are never pooled or compared to each other (V-22).

Evidence. Denominator 1: DERIVED from two [FITTED+RUN] cell prices ($1.681$ s at READ docs/CEQ_SHAPE.md:2293, $1.524$ s at READ docs/CEQ_SHAPE.md:2292). Denominator 2: DERIVED from the MAC identity of 03.2 at the matched pair; its seconds form is NOT MEASURED — needs 03.7's sweep with the matched skyline arm at $L=4$, $d_{\rm m}'=16$. The target $\tau$: READ docs/canon/04_BEDS_AND_INSTRUMENTS.md:165 (verse 04.10), adopted verbatim; MARS filed this citation as :147, which is 04.9 clause 3, and the governing line is :165. $k^\star$: NOT MEASURED — needs the arena on an admitted bed (book 04, S-62) and the training curve (book 02); no CEQ arm has trained (READ docs/CEQ_SHAPE.md:2942-2943). The definition matches CHARTER §1's T1 verbatim: wall-clock from initialisation to a fixed distance-to-floor at matched parameters, ratio against base self-attention and against the depth skyline.

Mechanism. B25 closed as a definition with its kill: P-7 (the phrase "faster to train" now has a referent), V-17 (the distance is in the bed's own floor units, L-FLOOR), M-8 (each arm at its own measured step price), D-1/R-SKY (the skyline's row is the second denominator and is printed in every T1 row).

Kill, with its bed, its draw, its seed rule and its price from the step budget the ladder level requires. $k^\star_{\rm shape}/k^\star_{\rm softmax}\ge0.907$, with $k^\star$ read at the ladder level $\tau$ of 04.10 (frozen from the per-step ratio $1.10241$; re-frozen once, when 03.7 replaces it by a RUN ratio, and the re-freezing is journalled with a supersede marker, L-G2).

The bed, named. BED-S as registered in READ docs/canon/04_BEDS_AND_INSTRUMENTS.md:27 (verse 04.2), at $s=64$, $d_{\rm m}=16$, $n=2048$, $t^\star=8$, $m=8$, $K=2$ — and BED-S does not admit at the design point: 04.2's Terminal reads "BED-S at the design point does not admit" (READ docs/canon/04_BEDS_AND_INSTRUMENTS.md:53). This kill is therefore journalled NOT EXECUTABLE (no admitted bed) today — V-16's third outcome, never a pass and never a firing — and becomes decidable on whichever object 04.2's own chain produces (link 1's dial, link 2's bed_1 cross-check, link 3's repricing). Naming the bed is what makes that readable; the earlier kill named a geometry, no bed, no draw and no seed rule, and was decidable against nothing (CHARTER §2 reachability rule). Draw and seed rule: the eight seeds $0$–$7$ of READ docs/canon/04_BEDS_AND_INSTRUMENTS.md:147 (04.9 clause 1), deduplicated, one thread lane read from the journal, both arms on byte-identical draws (train batch at seed, eval at seed+12345); $N=8$ arm-cells per arm.

Price, from the step budget and not from a $150$-step cell pair. The earlier price — "inside the $\approx34$ s bed-cell pair at $N=8$ (READ docs/CEQ_SHAPE.md:2295)" — is struck: :2295 prices "one bed-cell pair, 8 seeds each … two invocations", i.e. two arms $\times$ eight seeds $\times$ $150$ steps, whereas this kill needs $k^\star$ for both arms, the first step at which the held-out NRMSE ratio reaches $\tau$, which the $150$-step budget bounds only if $\tau$ is reached inside it. The price is therefore $$\text{GPU-s}\ =\ 8\big(k^\star_{\rm shape}\cdot0.01120613+k^\star_{\rm softmax}\cdot0.01016513\big)+4.0\ \text{s},$$ DERIVED from the two step prices above and the arena's $4.0$ s fixed cost per invocation (READ docs/CEQ_SHAPE.md:2294). At the record's registered cell length of $150$ steps (READ scale/e_ladder.py:140, steps=150) and both arms running the budget out, that is $8\times150\times0.02137126+4.0=\mathbf{29.6}$ s as a lower bound, not a price: $k^\star$ is NOT MEASURED — needs the training curve (book 02) and an admitted bed (04.2), and the figure scales linearly in $k^\star$, so a $\tau$ needing $1{,}500$ steps costs $260$ s and one needing $10^4$ costs $\approx29$ GPU-min. Book 04's own re-priced pair, $2\times8\times1.680+4.0=\approx30.9$ s (READ docs/canon/04_BEDS_AND_INSTRUMENTS.md:183), is the same $150$-step object at the shape's cell time for both arms and is likewise a lower bound here. The $\approx34$ s figure appears in this book only as the struck two-invocation reading of a $150$-step pair.

The third reading of this kill, printed as an outcome. The kill has three readings and not two: (a) $k^\star_{\rm shape}/k^\star_{\rm softmax}\ge0.907$ — fired, T1 against depth-1 withdrawn; (b) below $0.907$ — not fired; (c) NOT REACHED — at least one arm fails to reach any level of the ladder $\{0.9,0.8,0.7,0.5\}$ inside the registered step budget, so $T1=\infty$ for that arm, the ratio is undefined, and 04.10's VOID clause applies: the row reads NOT REACHED with the reached-level column empty and the step budget printed, and it is neither a firing nor a sparing. A journal that reports (a) or (b) on a cell where either arm did not reach $\tau$ is refused.

Instrument: the arena's per-step held-out NRMSE trace (book 04's identity manifest carries eval_trace, READ docs/canon/04_BEDS_AND_INSTRUMENTS.md:233); decidable at $N=8$ with the realised sd printed (M-3). Planted negative: the softmax control against itself at a different seed must read a ratio inside $[0.8,1.25]$ on at least $6$ of $8$ pairs — a T1 instrument that separates an arm from its own copy is refused.

If killed (the shape needs at least $0.907\times$ the softmax control's draws): T1 against depth-1 is withdrawn and the remaining T1 sentence is against the matched skyline $(32,4,16)$ with threshold $1.143$ at $s=64$; Hypotheses: the matched skyline reaches the same ladder level $\tau$ (Bet D's prediction; its counter is the point estimate, READ docs/PLAN.md:915), and the matched pair is buildable — it is, at $d_{\rm m}'=16$, unlike the $7.155$ the naive matching returns; Evidence: the MAC identity of 03.2 until 03.7 measures the arm; Kill: $k^\star_{\rm shape}/k^\star_{\rm sky}\ge1.143$ at $s=64$ (frozen) — a different number, on a different denominator arm, at a different $d_{\rm m}$, so it is not the verse's kill renamed; price $\approx64.8$ s for the skyline's eight seeds [ASSUMED] — more decisive because it is the last T1 denominator, and $3.96\times$ stricter than the struck unmatched $4.52$.

Terminal. "T1 is defined against 04.10's ladder target $\tau\in\{0.9,0.8,0.7,0.5\}$, its two kill numbers are frozen at $0.907$ (depth-1 softmax, seconds) and $1.143$ (the matched $(32,4,16)$ skyline, arithmetic), and no T1 number exists; 'faster to train' is not licensed by any per-step figure (CHARTER §1); a per-step ratio of $1.10241$ against depth-1 softmax — the fitted law $\exp(-12.1852)\cdot2048^{0.9963}=0.01016513$ s plus 03.6's increment, one value with one class and one pin, against $1.10302$ from the two cell prices — is the record's only measured direction, which is slower per step; the T1 kill has no admitted bed (BED-S does not admit at the design point) and is journalled NOT EXECUTABLE, its third reading is NOT REACHED, and its price is $8(k^\star_{\rm shape}\cdot0.01120613+k^\star_{\rm softmax}\cdot0.01016513)+4.0$ s, at least $29.6$ s and unbounded above in an unmeasured $k^\star$; and the $4.52$ once frozen against a five-copy unmatched stack is withdrawn as a threshold and survives only as an unmatched control."

Part C — determinism (B13)#

03.9 — The determinism regime: three outcomes, one observation, one hole#

Statement. Every route reports determinism as a three-valued reading per quantity — bitwise ($\max|\Delta|=0.0$ over $8$ repeats), drifting ($\max|\Delta|>0$), not executable (raises under use_deterministic_algorithms(True)) — for four quantities: softmax forward, solve forward, solve backward, softmax backward. No file under results/ carries a solve_triangular quantity, so no row of the four is a reading. The record's only determinism producer is one cell of results/k_cert_local.json written for a different arm — arm: "arm_smprime", device: "cuda", reduction_length: 64, $n=512$, $8$ repeats — whose quantities are forward, gradient, hop and cumsum, and which is printed below beside the rows it does not fill. The table the canon owns is therefore:

quantity	reading	producer, at `99777ab`
solve forward, strict mode	`NOT MEASURED — needs N-01`	none; the narrative sentence has no `results/` row (P-1)
solve backward, strict mode	`NOT MEASURED — needs N-01`	none; same absence
softmax forward (the control)	`NOT MEASURED — needs N-01`	none; V-23, never tested
softmax backward (the control)	`NOT MEASURED — needs N-01`	none; V-23, never tested
`arm_smprime` forward, flag off	bitwise, $\max	\Delta
`arm_smprime` forward, flag on	bitwise, $\max	\Delta
`arm_smprime` gradient, flag off	bitwise, $\max	\Delta
`arm_smprime` gradient, flag on	not executable — `RuntimeError: cumsum_cuda_kernel does not have a deterministic implementation`	`READ results/k_cert_local.json:1253-1255`
`arm_smprime` hop, flag off / flag on	bitwise, $\max	\Delta
`cumsum`, flag on	not executable — same `RuntimeError`	`READ results/k_cert_local.json:1262-1264`

What the record owns in prose only, and what N-01 exists to convert into the first two rows: a solve_triangular bitwise forward-and-backward reading over $8$ repeats under strict mode asserted in three narrative documents on one box and one build, with no torch-documented guarantee and no results/ row behind any of the three (P-1) — it is quoted in Evidence as narrative and is not a reading in the table above. cumsum raises in the backward of the scan and path-product arms, and that one is a row, on the arm_smprime cell. Ruling 1's regime is therefore the only one the canon can run: CUDA with warn_only=True; bitwise required for replay and every deciding forward cell; training's backward inherits the hole and its noise floor is measured once (two identical-seed chunks, $|\Delta|$ final loss) and printed beside every training number.

Hypotheses. torch 2.5.1+cu121, CUBLAS_WORKSPACE_CONFIG=:4096:8, one process, one stream, reduction length $64$ (READ COSTS.md:143-156); the installed stack is a different object (03.18). The reading is a repeatability observation on one box, never a contract (V-16 forbids reading silence as a pass; a documented guarantee is NOT FOUND in the torch documentation, [U]).

Evidence. The observation: READ docs/sources/judge/sec_apparatus.md:260, READ docs/CEQ_SHAPE.md:594-596, READ docs/sources/judge/sec_priorart.md:231-233 — all RUN[NEPTUNE], no producer (P-1; the producer is N-01, READ docs/PLAN.md:529-534). The hole: RuntimeError: cumsum_cuda_kernel does not have a deterministic implementation, raised by the backward of cumprod (READ COSTS.md:151-155; READ results/k_cert_local.json determinism.flag_on.gradient.executable = false). Ruling 1 verbatim: READ V17K_RULINGS.md:39-45 @ 99777ab. The control's omission: V-23 (READ docs/sources/judge/sec_apparatus.md:260, "the control softmax was never tested"). The training noise floor for the record's arms: $\delta_{\rm nrmse}=0.0$ bitwise on $6$ of $6$ identical-seed pairs with the flag off (READ results/v17k_r4_floor.jsonl; READ docs/sources/judge/sec_record.md:106); for the shape: NOT MEASURED — needs N-07 (03.10).

Mechanism. B13 closed here as a statement of what is and is not known: V-16 (the three-outcome table refuses a collapsed pass), V-23 (the control is in the table), P-1 (producer owed), M-10 / M-16 (the thread lane is read from the journal, never the machine; torch.set_num_threads(8) is a certificate field).

Kill. N-01's table reads $\max|\Delta|>0$ on the solve forward under either flag regime, or raises on the solve backward under the flag (frozen: any non-zero, any raise). Instrument: scripts/k_cert.py::determinism_at_64 extended by one quantity (READ scripts/k_cert.py:579, N-01), $8$ repeats, $n=512$; price $\approx1$ GPU-s [RUN class]; decidable on torch.randn operands. Planted negative: the certificate's own must-fire — a fabricated non-deterministic reduction must read not bitwise (READ tests/gate0/test_g06_kcert.py:116); and cumsum under the flag must raise (READ docs/CEQ_SHAPE.md:1872: determinism as three outcomes with cumsum required to raise, so a run reporting pass on an operator that raised is refused; docs/PLAN.md:552 was cited for this sentence and does not carry it).

If killed (the solve drifts or raises): the shape's forward deciding cells run under Ruling 1's warn_only=True and replay is re-defined as agreement within the measured forward floor — $\max|\Delta|$ over $8$ repeats printed as the tolerance of every deciding cell, with the cell's decision margin required to exceed $10\times$ that floor; Hypotheses: the drift is stationary across repeats; Evidence: the record's thread-count floor precedent, $2.345\times10^{-3}$ (READ docs/CEQ_SHAPE.md:2679); Kill: a deciding cell whose margin is below $10\times$ the floor is journalled NO READING, never a verdict — price $0$ GPU-s (read from the same $8$ repeats), cheaper than the verse's kill.

Terminal. "Determinism of the solve is an observation on torch 2.5.1+cu121 at ab5b485 with no producer and no guarantee; the canon runs Ruling 1's regime; a deciding cell prints its forward floor beside its margin or is not a deciding cell."

03.10 — What the kernel must guarantee: a fixed-order forward, and a priced deterministic backward#

Statement. The fused kernel (03.12) is admissible only if: (a) its forward contains no atomic operation and no data-dependent tile order — every reduction (row max, row sum, the $\sum_{l<k}P_{kl}z_l$ accumulation, the diagonal substitution) is a fixed-order loop over chunk index $l=0,\dots,k$ and row index $i=0,\dots,C-1$, so the forward is bitwise by construction across repeats on one device and one build; (b) its backward offers two modes with separately measured prices: deterministic — $dQ$ accumulated by a second program that loops over key tiles per query tile (no tl.atomic_add), $dK$, $dV$ accumulated inside the program that owns a key tile; and fast — $dQ$ by atomics — with the fast mode forbidden on any deciding cell and on the identical-seed floor run; (c) the identical-seed floor for the shape (N-07) is measured in both modes and printed.

Hypotheses. One device, one build, one stream; Triton's tl.dot and tl.sum are deterministic for a fixed tile shape and fixed loop order on a fixed build ([U]: not a documented guarantee — the same status as solve_triangular's, and measured the same way, $8$ repeats); tl.atomic_add on float32 is order-nondeterministic (this is the mechanism by which flash-style backwards drift and is the reason the fast mode is excluded from deciding cells).

Evidence. The design constraint is DERIVED from Ruling 1 (READ V17K_RULINGS.md:39-45 @ 99777ab) and from the record's own kernel discipline in the tree: ceq/mz_kernel.py fuses the online softmax with one accumulator and one normaliser across resolutions and guards the empty-row divide and the negative block index structurally (READ ceq/mz_kernel.py:1-25), forward only (READ THEORY.md:223-224 @ 99777ab: kernels#22 is forward-only; training through it requires a backward that does not exist). The deterministic-backward price: NOT MEASURED — needs the kernel (03.14); the flash-attention precedent that a deterministic $dQ$ pass costs a second sweep over key tiles is [U] and is not load-bearing — the kill below measures it.

Mechanism. Ruling 1, V-16 (three outcomes per mode), V-23 (both modes and the softmax control in one table), P-8 (the deterministic mode's price carries "at least" until measured).

Kill. The kernel's forward reads $\max|\Delta|>0$ over $8$ repeats at any $(s,n)\in\{(64,2048),(4096,1024)\}$ — the fixed-order claim is false on this build and the kernel is not admissible for deciding cells (frozen: any non-zero). Instrument: N-01's harness with the kernel as the route; price $\approx1$ GPU-s per geometry; decidable on torch.randn. Planted negative, on the forward, which is the mode the kill claims. The verse's earlier plant — the fast backward mode reading $\max|\Delta|>0$ on $dQ$ — exercises a path clause (b) forbids on every deciding cell, so it cannot make this kill fire: the kill reads the forward and the plant perturbs the backward (V-15). The forward plant is the same kernel compiled with the fixed-order claim deliberately broken and the arithmetic otherwise unchanged, in either of two forms, both of which must read $\max|\Delta|>0$ on at least $1$ of $8$ repeats at $(s,n)=(4096,1024)$: (F1) a data-dependent tile order — the off-diagonal loop over $l$ visited in an order derived from the row maxima $m_k$ rather than in the fixed order $l=0,\dots,k-1$, which changes the float32 summation order per draw; (F2) a tl.atomic_add inserted in the row-sum reduction that forms $\ell_k$, which is order-nondeterministic by the same mechanism that makes the fast backward drift. A kernel reading bitwise on both plants is not being measured for order sensitivity at all and the instrument is refused. Separate sensitivity control, kept and relabelled: the fast backward mode must read $\max|\Delta|>0$ on $dQ$ on at least $1$ of $8$ repeats at $s=4096$ — it certifies that the harness can see atomic drift on this build, it is not this kill's plant, and its own kill lives in clause (b)'s two-mode price table.

If killed (the forward drifts): the route for deciding cells drops to L2 (the torch chunked route, whose forward is solve_triangular plus matmul, both observed bitwise on the certified build) and the kernel is used for training only, under warn_only with its floor printed; Hypotheses: 03.9's bitwise observation was taken at reduction length $64$ (READ results/k_cert_local.json, determinism.reduction_length = 64, and determinism.flag_on.forward.bitwise = true, max_abs = 0.0 over $8$ repeats at $n=512$), and $C$ is the reduction length of the diagonal-block solve, so $C=32$ and $C=64$ are inside the observed regime and $C=128$ is outside it and is re-measured before any deciding cell runs at that $C$; Evidence: 03.9; Kill: L2's own forward drifts at $C=128$ — then $C\le64$ is the ceiling, $0$ GPU-s from the same table, cheaper than the verse's kill.

Terminal. "A kernel whose forward is not bitwise on the certified build decides nothing; training may use it under warn_only with the identical-seed floor printed beside every number; replay is on the torch route."

Part D — the kernel design as a specification (no code is written)#

03.11 — The chunked block-triangular recurrence#

Statement. Partition positions into $K=s/C$ chunks. With $P_{kl}\in\mathbb R^{C\times C}$ the $(k,l)$ block ($P_{kl}=0$ for $l>k$) and $M_{kk}=I-\gamma P_{kk}$, $$M_{kk}\,z_k=V_k+\gamma\sum_{l<k}P_{kl}\,z_l,\qquad k=0,\dots,K-1,$$ and $z$ so obtained equals $M^{-1}V$ exactly (block forward substitution; DERIVED: the block rows of $Mz=V$ read $M_{kk}z_k+\sum_{l<k}(-\gamma P_{kl})z_l=V_k$, and moving the off-diagonal term across the equals sign turns $-\gamma P_{kl}$ into $+\gamma P_{kl}$). The sign is plus. An earlier display of this verse read $V_k-\gamma\sum_{l<k}P_{kl}z_l$; it contradicted this verse's own DERIVED line and 03.13(iii), and it is struck. Numeric check (DERIVED against RUN[J], RUN[M], READ docs/CEQ_SHAPE.md:314): at $\gamma=0.6$, $V=\mathbb 1$, $M^{-1}$ has every row sum exactly $1/(1-\gamma)=2.5$, so $z=2.5\,\mathbb 1$; the plus recurrence returns, at $C=1$, $(1-\gamma P_{ii})\cdot2.5=1+\gamma\cdot2.5\,(1-P_{ii})$, an identity, while the minus recurrence returns $1/(1+\gamma)=0.625$ — wrong by a factor $2.5/0.625=4.0$. Costs per head and sequence (MACs): - off-diagonal accumulation: $\sum_{k}k\,C^2d=C^2d\,K(K-1)/2=s^2d/2-sCd/2$ exactly (not $\approx s^2d/2$), all GEMMs, depth $1$ per chunk step; - diagonal solve by in-tile substitution: $C^2d/2$ per chunk, $sCd/2$ total, depth $C$ per chunk step on-chip; - total, exactly: $(s^2d/2-sCd/2)+sCd/2=s^2d/2$, independent of $C$. The increment of exact block forward substitution over the softmax head it contains is the same $s^2d/2$ as dense substitution's, and the earlier total $s^2d/2+sCd/2$ over-counted by $sCd/2$ ($=2.10\times10^6$ MACs at $s=4096$, $C=64$, $d=16$) and contradicted 03.2's $\tfrac32s^2d$ for the whole shape. The over-count is struck from this verse, from 03.1's ladder table and from this verse's Terminal. - or the explicit block inverse $N_{kk}=M_{kk}^{-1}$ by nilpotent doubling — write $M_{kk}=D_k(I-\tilde N_k)$ with $D_k=\mathrm{diag}(1-\gamma P_{ii})$ and $\tilde N_k$ strictly lower, $\tilde N_k^{\,C}=0$, so $(I-\tilde N_k)^{-1}=\prod_{j=0}^{k^\dagger-1}(I+\tilde N_k^{2^j})$ exactly, with $k^\dagger=\lceil\log_2C\rceil$. The GEMM count includes the squarings that produce the powers being multiplied: each factor needs $\tilde N_k^{2^{j+1}}=(\tilde N_k^{2^j})^2$, so the schedule is $k^\dagger-1$ squarings plus $k^\dagger$ products, $2k^\dagger-1\approx2\log_2C$ tile GEMMs of $C^3$ MACs each, i.e. $\mathbf{2sC^2\log_2C}$ total across the $K=s/C$ chunks — twice the $sC^2\log_2C$ this verse printed before, which counted the products and not the squarings. Computed for all chunks in parallel, after which each chunk step is one $C\times C$ by $C\times d$ GEMM.

Depth: substitution variant $K\cdot C=s$ serial row-steps but every step on-chip; inverse variant $K+\log_2 C$ dependent GEMMs. Which variant: the inverse costs $2sC^2\log_2C$ against the substitution's $sCd/2$ — the inverse is cheaper in MACs only when $\mathbf{4C\log_2C<d}$, which no admissible $(C,d)$ satisfies and none comes near: at the most favourable admissible pair $(C,d)=(16,128)$ it reads $4\cdot16\cdot4=256<128$, failing by a factor $2$ rather than at the equality $128<128$ the halved count produced. The substitution variant is the specification at $d\le128$, and the doubling is the fallback when the on-chip serial loop is the measured bottleneck — a conclusion the corrected count strengthens, since the marginal case disappears.

Chunk length, re-derived. The sentence "total MACs $s^2d/2+sCd/2$ favour small $C$" is struck: the exact total is $s^2d/2$, independent of $C$, so arithmetic exerts no pull on $C$ at all. So does the total serial row-step count $K\cdot C=s$. The trade-off left is between the chunk-step count $s/C$ (fewer dependent steps favours large $C$) and SRAM (03.3's $4(4Cd+C^2)\le100{,}000$ B under the solve stage's $B_M=C$), with the tile-admissibility floor $C\ge16$. Large $C$ therefore wins up to the budget, and the earlier prediction "$C=64$ ... the largest $C$ whose $B_M\times C$ float32 score tile with $B_M=128$ fits 03.3's SRAM budget" is false — that reading mixed two chunk lengths inside one expression; at $B_M=128$, $d=16$ the budget solves to $16{,}384+640C\le100{,}000$, i.e. $C\le130$, and at $B_M=C$ to $4(64C+C^2)\le100{,}000$, i.e. $C\le129$. The specification fixes $C\in\{32,64,128\}$ for the sweep and predicts $C=128$ at $d=16$ ($98{,}304$ B $=98.3$ KB, $I_{\rm fused}=64$ FLOP/B), $C=64$ at $d\in\{32,64\}$ and $C=32$ at $d=128$ — in every case the largest admissible $C$, because nothing but SRAM opposes it.

Hypotheses. $\gamma<1$ so every $M_{kk}$ is invertible with diagonal in $[1-\gamma,1)$ (lower_triangular_isUnit, diag_one_sub_smul_pos [M], READ docs/CEQ_SHAPE.md:612); the block identity resolvent_fromBlocks [M] (READ docs/CEQ_SHAPE.md:613) is the theorem the recurrence instantiates; $C\mid s$; exact softmax rows available per chunk-row (03.12 supplies them in one pass). At $\gamma=0$ every $M_{kk}=I$ and $z=V$ bitwise (the P1 argument: $0\cdot x=0$, $v-0=v$, $v/1=v$ exact, READ docs/sources/design/design_theory.md:130-134).

Evidence. DERIVED (every line above). The record's own statement of the same costs: "$+s^2d/2+sCd/2+sC^2/3$, depth $s/C$" (READ docs/CEQ_SHAPE.md:585; N-11 READ docs/PLAN.md:781-787) — the $sC^2/3$ there is an explicit LU-style inverse; the doubling above replaces it by $2sC^2\log_2C$ with exact nilpotent termination, and the substitution variant removes it. The record's $+sCd/2$ term is the same double-count corrected in the Statement and is not carried here: $C^2dK(K-1)/2+KC^2d/2=s^2d/2$ exactly (DERIVED, two lines). Prior art owning the per-chunk triangular inverse: CITED [V] yang-2024-deltanet Eq. 10; the in-tile masked-row loop that computes it inside a Triton program: CITED [V] yang-2024-fla (the flash-linear-attention repository's chunk kernels, READ docs/sources/sweep/sweep_linrec.md Q14 row). Why no state carries across chunks: the DERIVED rank block of 03.1's Evidence (a causal row-softmax is exactly rank $s$ because $E\odot L$ is lower-triangular with positive diagonal; the off-diagonal block's rank-$17$ Hadamard truncation at $d=16$ misses by $80\,\%$ at logits of order $3$) — CITED [V] hu-2025-ssdtheory concurs and carries nothing (L-EQ). Wall-clock: NOT MEASURED — needs the chunked route (L2 in torch, L1 in Triton).

Mechanism. M-8 (the recurrence is priced as an increment over the head it contains), M-3 (MACs and depth separately), P-4 (no route exists until it is FOUND under results/ with route = chunked and its $C$), P-7 (the delta over DeltaNet is stated narrowly: the inter-chunk term is a dense product because the shape keeps the softmax normaliser bitwise at $\gamma=0$).

Kill. Two, in order of price: (i) parity — $\max|z_{\rm chunk}-z_{\rm serial}|$ in float32 at $s\in\{1024,4096\}$ above 03.13's single-signed-solve bound $\gamma_{s+1}(1+\gamma)\|V\|_\infty/(1-\gamma)^2$ (frozen: $6.110\times10^{-5}(1+\gamma)\|V\|_\infty/(1-\gamma)^2$ at $s=1024$, $2.443\times10^{-4}(1+\gamma)\|V\|_\infty/(1-\gamma)^2$ at $s=4096$, reading $1.161\times10^{-2}\|V\|_\infty$ and $4.641\times10^{-2}\|V\|_\infty$ at $\gamma=0.9$). The earlier threshold $2\gamma_{s+1}\|V\|_\infty/(1-\gamma)$ is struck: it is 03.13(iv)'s two-solve $V^\pm$ bound, and every route this book ships (L1, L2, L4) performs one signed solve, for which the correct constant is $9.50\times$ larger at $\gamma=0.9$ — the old threshold fired on a correct float32 route. Beyond the corrected bound the route is refused (L-CERT: an uncertified route ships no $\delta$); price seconds; planted negative: dropping block $(k,l)$ for one $l<k$ must read $\ge0.1\,\gamma\,\|V\|_\infty$ on random $V$ (an $O(1)$ miss). (ii) price — at $s=4096$, wall-clock above the serial L4 solve at every $C\in\{32,64,128\}$ at the same $n$ (N-11's kill), price "minutes once built" [ASSUMED].

If killed (parity fails at every $C$ — an implementation defect, not a theorem's): the doubling variant replaces the in-tile substitution (its GEMMs are the same tl.dot the off-diagonal term uses, removing the masked-row loop as a suspect), and it runs the same plus-sign recurrence, restated here so the replacement inherits nothing wrong: $$z_k=N_{kk}\Big(V_k+\gamma\sum_{l<k}P_{kl}z_l\Big),\qquad N_{kk}=M_{kk}^{-1}=\Big(\prod_{j=0}^{k^\dagger-1}\big(I+\tilde N_k^{2^j}\big)\Big)D_k^{-1},\qquad k^\dagger=\lceil\log_2C\rceil,$$ with $D_k=\mathrm{diag}(1-\gamma P_{ii})$ and $\tilde N_k$ strictly lower, $\tilde N_k^{\,C}=0$, at $2k^\dagger-1$ tile GEMMs per chunk ($k^\dagger-1$ squarings, $k^\dagger$ products) and $2sC^2\log_2C$ MACs in total. Hypotheses: $\tilde N_k^{\,C}=0$ exactly in floating point (true: strictly lower-triangular products stay strictly lower-triangular regardless of rounding, DERIVED); Evidence: NOT MEASURED — needs the variant; Kill: the same corrected single-solve parity bound $\gamma_{s+1}(1+\gamma)\|V\|_\infty/(1-\gamma)^2$, price seconds — equal price but more decisive, since a failure now isolates the tile loads rather than the loop. If price fails at every $C$: L4 is the shipped route and this verse closes NOT NEEDED on this device (N-11's own clause).

Terminal. "The block recurrence $M_{kk}z_k=V_k+\gamma\sum_{l<k}P_{kl}z_l$ is an exact identity costing exactly $+s^2d/2$ MACs — independent of $C$ — in $s/C$ chunk steps; the chunk length is decided by chunk-step count against SRAM alone; it is a specification, no implementation exists, and no seconds are attached to it."

03.12 — Fusion with the online causal softmax: tiles, statistics, and the one-pass schedule#

Statement. One Triton program per (sequence, head), persistent over chunk-rows $k=0,\dots,K-1$ (grid $=[\,n h\,]$; no per-chunk launch). For chunk-row $k$ with query tile $Q_k\in\mathbb R^{C\times d}$ — so $B_M=C$ in the solve stage, which is the constraint that decides the arithmetic intensity 03.3 reports for this route: 03.3's fused intensity is read at $B_M=C$, not at a free $B_M$, and its admissible pairs are the ones this stage admits ($C=128$ at $d=16$, giving $I_{\rm fused}=64$ FLOP/B and the $85.9\,\%$-of-ridge reading; $C=64$ at $d\in\{32,64\}$ and $C=32$ at $d=128$, both memory-bound). The softmax-only read, which runs no solve, may use $B_M=2C$ where SRAM admits it ($B_M=128$, $C=64$, $d=16$: $57{,}344$ B $=57.3$ KB) and is priced separately; no intensity of this route is ever quoted at a $B_M$ the solve stage does not admit. The five steps:

Off-diagonal pass over $l=0,\dots,k-1$: $S_{kl}=Q_kK_l^\top/\sqrt d$; online statistics $m_k\leftarrow\max(m_k,\mathrm{rowmax}\,S_{kl})$, rescale $\ell_k$ and the accumulator $A_k$ by $e^{m_{\rm old}-m_k}$; $\ell_k\mathrel{+}=\mathrm{rowsum}\,e^{S_{kl}-m_k}$; $A_k\mathrel{+}=e^{S_{kl}-m_k}\,z_l$ — the flash-attention loop with $z_l$ in place of $V_l$ (tl.dot, fp32 accumulate).
Diagonal tile, with the accumulator rescaled before anything else: compute $S_{kk}$ masked causally ($j\le i$); set $m_k^{\rm new}=\max(m_k,\mathrm{rowmax}\,S_{kk})$; rescale both $\ell_k\leftarrow\ell_k\,e^{m_k-m_k^{\rm new}}$ and $A_k\leftarrow A_k\,e^{m_k-m_k^{\rm new}}$ — this rescale of $A_k$ is a numbered step of the specification and is not optional. An earlier reading of this verse rescaled $A_k$ only inside step 1's off-diagonal loop and took "final $m_k,\ell_k$ after including it" at step 2, which leaves $A_k$ carrying $m_k^{\rm old}$ while $\ell_k$ carries $m_k^{\rm new}$; since $S_{ii}=q_i\!\cdot\!k_i/\sqrt d$ lies in every row's diagonal tile, the diagonal tile holds the row maximum in the common case, and steps 3 and 5 are then wrong by the factor $e^{m_k^{\rm new}-m_k^{\rm old}}$ at every $\gamma>0$. Then set $m_k\leftarrow m_k^{\rm new}$, $\ell_k\mathrel{+}=\mathrm{rowsum}\,e^{S_{kk}-m_k}$, and $\tilde P_{kk}=e^{S_{kk}-m_k}/\ell_k$ (exact row-normalised, since the causal row of chunk $k$ ends inside chunk $k$).
Right-hand side: $r_k=V_k+\gamma A_k/\ell_k$ — plus, per 03.11's corrected recurrence (the rescaled accumulator divided once by the final normaliser gives $\sum_{l<k}P_{kl}z_l$ exactly in exact arithmetic, and that term moves across the equals sign with a sign change). An earlier reading displayed $V_k-\gamma A_k/\ell_k$ and is struck.
Solve $(I-\gamma\tilde P_{kk})z_k=r_k$ by the in-tile substitution of 03.11 ($C$ masked row-steps over a $C\times d$ register tile); write $z_k$ to HBM (float32).
Read: $O_k=(1-\gamma)\big(\sum_{l<k}P_{kl}z_l+\tilde P_{kk}z_k\big)=(1-\gamma)\big(A_k/\ell_k+\tilde P_{kk}z_k\big)$ — no second pass, since $A_k$ is already the off-diagonal part of $Pz$; write $O_k$, $m_k$, $\ell_k$.

Memory layout: $Q,K,V,z,O$ as [n, h, s, d] with unit stride in $d$ (the sq_d = 1 convention of READ ceq/mz_kernel.py:44-52), tiles loaded coalesced along $d$; $z$ always float32 regardless of autocast (03.13); $m,\ell$ as [n, h, s] float32; $P$ never materialised. Bytes per sequence: 03.3's $6s^2d/B_M$ plus $5sd\cdot4$ for the five row tensors.

Hypotheses. $\gamma=0$ parity: at $\gamma=0$ step 3 returns $r_k=V_k$ bitwise (the $+\gamma A_k/\ell_k$ term is $+0\cdot x=+0.0$, and $v+0.0=v$ exactly; the diagonal is exactly $1$) and step 5 is the flash-attention read with the same accumulation order as the softmax-only program — so parity is bitwise against the kernel's own $\gamma=0$ path by construction, and against ceq/lm.py's naive softmax it is the fused-versus-naive gap the record already names ($1.110223\times10^{-16}$ on $19/64$ entries, READ docs/sources/judge/sec_record.md:80: a row-max subtraction, a mechanism and not a tolerance). Tile admissibility: $d\in\{16,32,64,128\}$, $C\in\{32,64,128\}$, and the pair $(B_M,C)=(C,C)$ inside 03.3's budget $4(4Cd+C^2)\le100{,}000$ B — which admits $(128,128)$ at $d=16$, $(64,64)$ at $d\in\{32,64\}$ and $(32,32)$ at $d=128$ and nothing larger. The kernel's parity target for B-J (torch.equal at $\gamma=0$, READ docs/CEQ_SHAPE.md:2567) is the lane's own softmaxAttn, so the bind is filed against the torch route and the kernel carries the fused gap as a printed number, never as bitwise.

Evidence. DERIVED (the schedule). The single-accumulator, single-normaliser fusion across tiles is already the record's own pattern in READ ceq/mz_kernel.py:1-12; the two structural guards (empty-row divide, negative block index) are READ ceq/mz_kernel.py:13-21 and are inherited verbatim (the causal diagonal tile never leaves $\ell_k=0$ because $j=i$ is always included — a DERIVED reason the empty-row guard is satisfied by construction here, stated so that V-10 does not read it as a test). The rescaling identity in step 3 is the online-softmax algebra ([U] — the flash-attention lineage is not in references.bib; the key is owed to book 05's bibliography repair and no sentence here rests on it: step 3 is four lines of DERIVED arithmetic). Timing: NOT MEASURED — needs the kernel.

Mechanism. V-24 (the parity bind ships its rejection region: $\gamma=0.5\Rightarrow2.3002850040264393$, READ docs/CEQ_SHAPE.md:2567), V-3 (parity by construction is declared as such), P-4 (the kernel is NAMED until a results/ row with route = fused exists), L-CERT (route, C, B_M are manifest fields, READ docs/sources/design/design_instrument.md:420).

Kill. In price order: (i) $\gamma=0$ parity against the kernel's own softmax-only program not torch.equal (frozen: any entry) — a construction error; price seconds; planted negative: $\gamma=0.5$ must read $\max|O(0.5)-O(0)|\ge1$ at $s=64$ (the record's $2.3003$ regime). Kill (i) is blind by construction to the step-2 rescale defect: at $\gamma=0$ the mis-scaled term is multiplied by zero and torch.equal reads True on a kernel whose $A_k$ carries the wrong row maximum (V-3, V-10). Kill (i-b) exists for that defect and for no other: (i-b) $\gamma=0.5$ parity against L4's $z$ at $(s,n)=(64,2048)$ outside 03.13's single-signed-solve bound $\gamma_{65}(1+\gamma)\|V\|_\infty/(1-\gamma)^2=3.874\times10^{-6}\cdot1.5\,\|V\|_\infty/0.25=2.325\times10^{-5}\|V\|_\infty$ (frozen: one exceedance) — the kernel is refused; price seconds, one launch, the cheapest kill in this verse. Planted negative: the same kernel with step 2's $A_k$ rescale removed must exceed that bound on at least $1{,}000$ of $1{,}024$ drawn cells at $\gamma=0.5$ (the omitted rescale is an $e^{m^{\rm new}-m^{\rm old}}$ factor, an $O(1)$ miss whenever the diagonal tile holds the row maximum — which it does on the majority of rows for logits of order $1$). (ii) parity against L4's $z$ outside 03.13's single-signed-solve bound at $(s,n)=(4096,64)$ — refused (L-CERT); the bound is $\gamma_{4097}(1+\gamma)\|V\|_\infty/(1-\gamma)^2$, reading $4.641\times10^{-2}\|V\|_\infty$ at $\gamma=0.9$. (iii) price: forward and backward wall-clock at $(4096,1024)$ above $2\times$ the roofline of 03.3 (frozen $100.7$ ms forward), or above the matched $(32,4,16)$ fused-softmax stack's time at the same $(s,n)$ — the kernel buys nothing on this device; price seconds once built; NOT MEASURED until then. The four kills are strictly cheaper than 03.11's price kill because each needs one launch.

If killed on (i) — $\gamma=0$ parity against the kernel's own softmax-only program not torch.equal. This is a construction error and not a defect of the schedule: at $\gamma=0$ step 3 returns $r_k=V_k$ by the exactness of $+0.0$ and step 5 is the flash read, so a mismatch means the two programs do not share an accumulation order — a tile shape, a loop bound or an accumulator dtype differing between the $\gamma$ and the $\gamma=0$ paths. The replacement is the single-path schedule: one program with $\gamma$ as a runtime scalar and no $\gamma=0$ specialisation anywhere, so the corner is the same instruction stream with a zero operand. Hypotheses: tl.dot and tl.sum fixed-shape determinism per 03.10; one accumulator, one normaliser, one dtype (float32) across both corners. Evidence: the record's own single-accumulator pattern, READ ceq/mz_kernel.py:1-12; NOT MEASURED — needs the kernel. Kill, its own and different: the single-path program's $\gamma=0$ output not torch.equal to the torch route's $\gamma=0$ output up to the fused-versus-naive gap the record already prints — $\max|\Delta|>1.110223\times10^{-16}$ on more than $19/64$ entries at $s=64$ (READ docs/sources/judge/sec_record.md:80), one exceedance — which is a bound on a different object (the torch route, not the kernel's own second program) and can fire where kill (i) cannot, since a single-path program has no second program to agree with. Price seconds, one launch, no compile of a second variant; more decisive than (i) because it removes the object whose disagreement (i) reads. If it fires, no fused program reproduces the corner bitwise on this build, 03.12's Terminal is in force, and L2 is the deciding route.

If killed on (i-b) — the $\gamma=0.5$ reading exceeds $2.325\times10^{-5}\|V\|_\infty$, i.e. the step-2 rescale is wrong or absent. The replacement is the two-pass schedule: a first pass over $l=0,\dots,k$ including the diagonal tile computing $m_k$ and $\ell_k$ only, and a second pass recomputing the exponentials against the final $m_k$ to form $A_k$ and $\tilde P_{kk}$ — no running rescale of $A_k$ exists to omit, at the cost of one extra pass over the causal half ($+s^2d/2$ MACs, $+6s^2d/B_M$ bytes, i.e. the fused route's arithmetic rises from $\tfrac32s^2d$ to $2s^2d$ and its intensity is unchanged at $B_M/2$). Hypotheses: the exponentials recompute bitwise from $Q,K,m_k$ in the same order (03.14's recompute clause); SRAM unchanged, since no accumulator is carried across the passes. Evidence: NOT MEASURED — needs the variant; the recompute-from-$m,\ell$ pattern is already required by 03.14's backward and is not new machinery. Kill, its own and different: the two-pass forward's price at $(4096,1024)$ above $\tfrac43\times$ the one-pass forward's — the arithmetic ratio $2s^2d/\tfrac32s^2d$ — which is a price threshold on a different quantity from (i-b)'s parity bound and fires where a rescale bug cannot; price seconds once built; if it fires, L2, where softmax supplies the normalisation and no online statistic exists to mis-scale.

If killed on (ii) — parity against L4's $z$ at $(4096,64)$ outside $4.641\times10^{-2}\|V\|_\infty$. The fused route is refused for deciding cells (L-CERT) and the route drops to L2, whose diagonal solve is solve_triangular on the certified build rather than an in-tile substitution, so the failing object is replaced rather than re-tuned. Hypotheses: $C\in\{32,64,128\}$ divides $s$; per-chunk checkpoint; the same plus recurrence of 03.11. Evidence: NOT MEASURED — needs L2; solve_triangular's float32 agreement with the dense inverse is the record's RUN[coord] $1.7763568394002505\times10^{-15}$ in float64 at $s=64$ (READ docs/CEQ_SHAPE.md:430). Kill, its own: L2's own chunk-versus-serial parity outside the same 03.13(iv) bound at $(4096,64)$ — one exceedance — which now convicts the recurrence rather than the kernel, since the tile loop is gone; price seconds, no Triton compile, cheaper than (ii); if it fires, 03.11's Terminal is in force and L4 is the shipped route.

If killed on (iii): L2 — the same five steps as torch ops per chunk-row ($C\times(k{+}1)C$ logits via matmul, softmax, solve_triangular on the $C\times C$ block, one matmul for $A_k$), per-chunk checkpoint so retained memory is $O(sd)+O(sC)$; Hypotheses: $K$ launches of $4$ kernels each are cheaper than the fused program's latency chain — the counter, under D-CALIB the estimate, is that they are not at $s=4096$ (at the predicted $C=128$, $K=32$ and $128$ launches at $\tau_{\rm launch}\approx10$ μs [ASSUMED] is $1.3$ ms per step, and at $C=32$, $K=128$ and $512$ launches is $5.1$ ms — both below the $43$ ms arithmetic floor at $n=1024$, so L2 is not launch-bound at any admissible $C$ and the estimate is that L2 ships); Evidence: NOT MEASURED — needs L2; Kill: L2 slower than L4 at every $C$ at the L4 residency $n$ (N-11), price minutes — cheaper than (iii) because no Triton compile is involved.

Terminal. "The fused schedule is an exact re-association of the softmax read and the block recurrence; it exists as a specification with its manifest fields and its three kills; no launch has occurred."

03.13 — Numerical stability of the solve in float32 and bfloat16#

Statement. Let $M=I-\gamma P$ with $P\ge0$ row-stochastic and causal. Then (i) $M$ is a lower-triangular M-matrix, row diagonally dominant: $|M_{ii}|-\sum_{j<i}|M_{ij}|=(1-\gamma P_{ii})-\gamma(1-P_{ii})=1-\gamma>0$; (ii) $M^{-1}\ge0$ with every row sum exactly $1/(1-\gamma)$; (iii) for $V\ge0$ forward substitution has no cancellation (every term $r_i-\sum_{j<i}(-\gamma P_{ij})z_j$ adds non-negative quantities), so the computed $\hat z$ satisfies $|\hat z-z|\le\gamma_{s+1}\,z$ componentwise; (iv) for the single signed solve every shipped route performs — L1, L2 and L4 all run one forward substitution on a signed $V$, not two — the standard triangular result $|\hat z-z|\le\gamma_{s+1}|M^{-1}||M||z|$ gives, using $M^{-1}\ge0$ with $\|M^{-1}\|_\infty=1/(1-\gamma)$, $\||M|\|_\infty=\max_i(1+\gamma-2\gamma P_{ii})\le1+\gamma$ and $\|z\|_\infty\le\|V\|_\infty/(1-\gamma)$, $$\boxed{\ \|\hat z-z\|_\infty\ \le\ \gamma_{s+1}\,\frac{(1+\gamma)\,\|V\|_\infty}{(1-\gamma)^2}\ },\qquad \gamma_{s+1}=\frac{(s{+}1)u}{1-(s{+}1)u},$$ which reads, at $\gamma=0.9$: float32 $7.36\times10^{-4}\|V\|_\infty$ ($s=64$), $2.91\times10^{-3}$ ($256$), $1.16\times10^{-2}$ ($1024$), $4.64\times10^{-2}$ ($4096$); float64 $1.37\times10^{-12}$ to $8.64\times10^{-11}$; bfloat16 $64.7\,\|V\|_\infty$ at $s=64$ and undefined ($(s{+}1)u\ge1$) at $s\ge256$. (iv′) Only if a route splits $V=V^+-V^-$, solves both halves and subtracts — which no shipped route does — does the tighter $\|\hat z-z\|_\infty\le2\gamma_{s+1}\|V\|_\infty/(1-\gamma)$ hold, reading $7.75\times10^{-5}$ ($s=64$) to $4.89\times10^{-3}$ ($s=4096$) at $\gamma=0.9$. The two-solve split is not a Hypothesis of L1, L2 or L4, and (iv′) is therefore never a threshold in this canon. The earlier text proved (iv′) and then froze it as the kill threshold for the single signed solve in 03.11, 03.13 and 03.14; the ratio of the two constants is $(1+\gamma)/(2(1-\gamma))=9.50$ at $\gamma=0.9$, so a correct float32 route at $s=4096$ reading $4.64\times10^{-2}\|V\|_\infty$ fired all three kills. All three thresholds are re-frozen on (iv). Rule: operands ($Q,K,V$, the exponentials) may be bfloat16 under autocast; the accumulation of $S$, $A_k$, $\ell_k$, the substitution and $z$ are float32, always. The committor head at $\gamma=1$ on the transient block $I-\hat Q$ has $V=R_k\mathbb 1\ge0$ and dominance $1-\|\hat Q\|_\infty=\min_i\hat P_{i,\mathcal A}>0$ (mass to a declared boundary), so (iii) applies directly with $1/(1-\gamma)$ replaced by $1/\min_i\hat P_{i,\mathcal A}$, and (iv) with $(1+\gamma)/(1-\gamma)^2$ replaced by $2/(\min_i\hat P_{i,\mathcal A})^2$.

Hypotheses. Standard floating-point model $\mathrm{fl}(a\circ b)=(a\circ b)(1+\delta)$, $|\delta|\le u$; the substitution's inner products accumulated in the same precision as $u$ names; $\gamma<1$ (or, at $\gamma=1$, a declared boundary reachable from every transient row); rounding of $P$'s entries themselves (the softmax) is not in the bound — it is the operator's own precision and enters as $\delta P$ with $\|\delta z\|_\infty\le\gamma\|\delta P\|_\infty\|V\|_\infty/(1-\gamma)^2$ (DERIVED from $M^{-1}$'s row sum applied twice). Non-vacuous at every $s$ in float32; vacuous in bfloat16 at every $s\ge256$, which is the content.

Evidence. DERIVED: (i) from row-stochasticity; (ii) RUN[J], RUN[M] row sum $2.5$ at $\gamma=0.6$ with and without boundary rows (READ docs/CEQ_SHAPE.md:314); (iii)–(iv′) by the standard backward-error analysis of substitution (Higham, Accuracy and Stability of Numerical Algorithms, 2nd ed., §8.2 — [U], no bib key; the derivation is written out here and does not rest on the citation: each computed $\hat z_i$ is the exact solution of a system with $|\Delta M|\le\gamma_{s+1}|M|$, so $|\hat z-z|\le\gamma_{s+1}|M^{-1}||M||z|$ componentwise; for an M-matrix with $V\ge0$ the perturbation cannot cancel, which is (iii); taking $\infty$-norms of the componentwise bound with the three factors above is (iv), four lines, no cancellation assumed). Numeric instance (L-EQ): the record's solve_triangular against the dense inverse reads $1.7763568394002505\times10^{-15}$ at $s=64$ in float64 (RUN[coord], READ docs/CEQ_SHAPE.md:430), inside the float64 single-signed-solve bound $\gamma_{65}(1+\gamma)\|V\|_\infty/(1-\gamma)^2=1.37\times10^{-12}\|V\|_\infty$ at $\gamma=0.9$ — a margin of $7.7\times10^{2}$, and the margin is quoted as the weaker check it is: the correct single-solve bound is $9.50\times$ looser at $\gamma=0.9$ than the two-solve (iv′) whose $10\times$ margin the earlier text quoted, so this instance certifies less than it appeared to. The bfloat16 row of the record's sizing table (operator term $3.341$ bytes per element under autocast, READ COSTS.md:87-99) is the reason the dtype rule costs $4.0$ bytes per element for $z$ and the scores, and N-19's [U] sentence "no bf16 path for solve_triangular" (READ docs/PLAN.md:757-761) is irrelevant to the kernel: the specification never solves in bfloat16 by rule, not by absence of a path.

Mechanism. P-10 (the two-solve theorem (iv′) was cited beyond its hypotheses as the bound of the single signed solve; the two are now separated and only (iv) is a threshold), V-17 (the parity threshold is in the bound's units, $(1+\gamma)\|V\|_\infty/(1-\gamma)^2$, never a bare $10^{-6}$ — N-11's "$10^{-6}$ in float32" is repaired here, since at $s=4096$ a correct float32 kernel can legitimately miss $10^{-6}$ by $4.6\times10^{4}$), V-10 (the bound is not satisfied by construction: a dropped block misses it by $O(1)$), L-CERT ($\delta_{\rm float}$ is a printed certificate line beside $\delta_{\rm route}$).

Supersession of book 08's two constants, marked as L-G2 requires. A neighbouring born book still carries both numbers this verse and 03.1 replace, and the two books cannot both hold. (1) The parity threshold. READ docs/canon/08_ARCHITECTURE.md:562 (08.20's If-killed, Replacement 1 — the chunked block-triangular solve) freezes "parity $10^{-6}$ against the serial solve" for the same chunked route, and READ docs/canon/08_ARCHITECTURE.md:138 (08.4's Replacement 2 — the dense inverse per chunk) freezes "parity against the serial solve worse than $10^{-6}$ in float32 at any $C\in\{32,64,128\}$". Both are superseded by 03.13(iv): a correct float32 single-signed forward substitution at $s=4096$, $\gamma=0.9$ reads up to $\gamma_{4097}(1+\gamma)\|V\|_\infty/(1-\gamma)^2=4.64\times10^{-2}\|V\|_\infty$, which misses $10^{-6}$ by $4.6\times10^{4}$, so a bare $10^{-6}$ condemns a correct kernel on arithmetic alone (V-17, a threshold imported out of its units). Supersede marker: supersedes: 08_ARCHITECTURE.md:138, :562 (parity 1e-6, float32) — superseded by 03_KERNEL.md 03.13(iv), $\gamma_{s+1}(1+\gamma)\|V\|_\infty/(1-\gamma)^2$; both 08 cells keep their text and this marker beside them (L-G2: a journal never moves and a superseded cell keeps its marker). (2) The plant's tolerance. READ docs/canon/08_ARCHITECTURE.md:560 (08.20's Kill) requires as its planted negative that "the dense per-op microbenchmark must reproduce $1.041$ ms within $\pm12\,\%$ at $n=2048$", while 03.1 and 03.6 freeze the kill at the ratio $\rho_\Delta>1.167$, which is $1.65\times$ the un-synchronised reference $0.707$ under the declared $2.0\times$–$6.6\times$ gap that 03.1 and 03.6 return to the per-row gate settle arm — so 08's plant demands a synchronised re-measurement land inside $\pm12\,\%$ of a number this book quotes only as an un-synchronised floor with a known-sign, unmeasured-magnitude bias. Supersede marker: supersedes: 08_ARCHITECTURE.md:560 (reproduce 1.041 ms ±12%) — superseded by 03_KERNEL.md 03.1/03.6, $\rho_\Delta$ inside one interleaved block, $\pm12\,\%$ applying to $\rho_\Delta$ between two synchronised sessions and to nothing else. The CORRECTIONS.md rows that carry these two markers are owed and are not written here — this repairer edits only book 03 and makes no git write — and each names the 08 verse whose pre-written replacement now applies: 08.4's Replacement 2 and 08.20's Replacement 1 both survive with their thresholds re-frozen on 03.13(iv), and 08.20's Kill survives with its plant re-frozen on $\rho_\Delta$; no 08 verse is withdrawn by either row. Both are named in "Attacks answered" beside the third row already owed there against 08_ARCHITECTURE.md:528.

Kill. A route's $\max|\hat z-z_{\rm fp64}|$ on $1{,}024$ drawn cells exceeds the bound at its $(s,\gamma,u)$ (frozen: one exceedance). Instrument: the certificate battery of N-03 item 2 extended with a float64 reference (READ docs/PLAN.md:306-307); price $\approx2.6$ s at $s=64$ DERIVED, seconds at $s=4096$ on L4 at $n=8$; decidable on random logits and random $V$ with a declared sink. Planted negative: the same solve run with bfloat16 accumulation at $s=64$ must exceed the float32 bound on at least $1{,}000$ of $1{,}024$ draws (an $O(1)$ miss, $64.7\|V\|_\infty$ predicted at $\gamma=0.9$) — the instrument must condemn the precision the rule forbids.

If killed (float32 accumulation exceeds the bound — an implementation adds a non-standard reduction): compensated (Kahan) accumulation of the substitution sum. Re-derived under (iv), the single signed solve, not under (iv′)'s split: compensated summation replaces the backward error $|\Delta M|\le\gamma_{s+1}|M|$ by $|\Delta M|\le(2u+O(su^2))|M|$, and pushing that through the same three factors of (iv) gives $$\|\hat z-z\|_\infty\ \le\ \big(2u+O(su^2)\big)\frac{(1+\gamma)\|V\|_\infty}{(1-\gamma)^2},$$ independent of $s$ to first order, at $+3$ flops per MAC in the diagonal step only ($+3sCd/2$, negligible against $s^2d/2$); Hypotheses: the tile arithmetic honours the compensation (no fused re-association by the compiler — [U], and the kill decides it); Evidence: NOT MEASURED — needs the variant; Kill: the same battery reading above $4u(1+\gamma)\|V\|_\infty/(1-\gamma)^2$ — $4.53\times10^{-5}\|V\|_\infty$ in float32 at $\gamma=0.9$, a threshold $1.02\times10^{3}$ times tighter than the verse's own at $s=4096$ and hence more decisive.

Terminal. "The solve's float32 error on the single signed solve every shipped route performs is bounded by $\gamma_{s+1}(1+\gamma)\|V\|_\infty/(1-\gamma)^2$ on the row-stochastic class — $4.64\times10^{-2}\|V\|_\infty$ at $s=4096$, $\gamma=0.9$; the $9.50\times$ tighter two-solve bound is licensed only where a route actually splits $V^\pm$, and none does; no route accumulates in bfloat16; a route outside its bound ships no $\delta$ and is refused."

03.14 — The backward pass of the fused solve#

Statement. With $\bar O$ given, $z=M^{-1}V$, $O=(1-\gamma)Pz$, the adjoints are (DERIVED, chain rule on $Mz=V$): $$\bar z=(1-\gamma)P^\top\bar O,\qquad \bar V=M^{-\top}\bar z,\qquad \bar M=-\bar V z^\top,\qquad \bar P=\big[(1-\gamma)\bar O z^\top+\gamma\,\bar V z^\top\big]\odot\mathbb 1[j\le i],\qquad \bar\gamma=\langle P,\bar V z^\top\rangle-\langle P,\bar O z^\top\rangle,$$ then $\bar S$ from $\bar P$ by the softmax backward ($\bar S_{ij}=P_{ij}(\bar P_{ij}-\sum_j P_{ij}\bar P_{ij})$) and $\bar Q=\bar S K/\sqrt d$, $\bar K=\bar S^\top Q/\sqrt d$. The fused backward is: (1) $\bar z$ by the transposed tile loop (each key chunk $l$ accumulates $\sum_{k\ge l}P_{kl}^\top\bar O_k$); (2) an upper-triangular chunked solve $M^\top\bar V=\bar z$ — the recurrence of 03.11 run over chunks in reverse order, $M_{kk}^\top\bar V_k=\bar z_k+\gamma\sum_{l>k}P_{lk}^\top\bar V_l$ (plus, by the same transposition of the off-diagonal term; the earlier minus is struck with 03.11's), same MACs, and a different error bound, derived below — 03.13's row-dominance analysis does not transfer to $M^\top$; (3) $\bar P$ tiles recomputed from $Q,K,m,\ell$ and contracted with the two rank-$d$ factors $(\bar O, z)$ and $(\bar V, z)$ — no $P$ is ever stored. Retained across forward and backward: $Q,K,V,z,O,m,\ell$ — $5sd+2s$ floats per head and sequence, linear in $s$. MACs: forward $\tfrac32s^2d$; backward $\approx 2\times$ the forward's dense terms plus one more solve, $\approx 3.5\,s^2d$ (DERIVED: two $\bar S$-side GEMMs per tile, one transposed accumulation, one solve).

The backward solve's error bound, derived for $M^\top$ and not inherited from 03.13. $M^\top$ is upper-triangular with the same diagonal as $M$, and it is column diagonally dominant, not row diagonally dominant. Its row-dominance defect at row $i$ is $$|M^\top_{ii}|-\sum_{j>i}|M^\top_{ij}|=(1-\gamma P_{ii})-\gamma\sum_{j>i}P_{ji},$$ the sub-diagonal column sum of $P$. Under the canon's declared BOS sink (book 08), where every row places $P_{j0}\ge0.5$ on column $0$, this reads $1-\gamma P_{00}-0.9\cdot0.5(s-1)<0$ at $i=0$, $\gamma=0.9$, for every $s\ge5$ — so $M^\top$ fails row dominance outright at every $s\in\{64,256,1024,4096\}$, and 03.13's M-matrix and no-cancellation argument does not apply to this solve at any registered geometry. Two per-cell quantities carry the bound instead, and both are printed as certificate lines beside $\delta_{\rm float}$ (L-CERT): $$\pi:=\|P^\top\|_\infty=\max_j\sum_i P_{ij}\ \in[1,s],\qquad \kappa_1:=\|M^{-1}\|_1=\|M^{-\top}\|_\infty=\max_j\sum_i (M^{-1})_{ij}\ \in\Big[\tfrac1{1-\gamma},\tfrac{s}{1-\gamma}\Big],$$ the second range because $M^{-1}\ge0$ with every row sum exactly $1/(1-\gamma)$ (03.13(ii)), so its entries sum to $s/(1-\gamma)$ and no column can exceed that. Then $\|M^\top\|_\infty=\|M\|_1\le1+\gamma\pi$ and the standard triangular result gives $$\boxed{\ \|\hat{\bar V}-\bar V\|_\infty\ \le\ \gamma_{s+1}\,\kappa_1\,(1+\gamma\pi)\,\|\bar V\|_\infty,\qquad \|\bar V\|_\infty\le\kappa_1\|\bar z\|_\infty\ }$$ with no cancellation clause — for signed $\bar z$ on a non-M-matrix the terms may cancel and the bound is the only guarantee. Neither $\pi$ nor $\kappa_1$ is $1$ or $1/(1-\gamma)$ under a sink, and neither is assumed: both are measured per cell.

Hypotheses. The declared BOS sink of book 08 is in force — the bound above is written for a $P$ whose column sums are unbounded and is not restricted to a sink-free $P$; $\pi$ and $\kappa_1$ are journalled per cell (manifest fields P_col_sum_max, Minv_col_sum_max) and the verse's kill is decidable only where their product satisfies the non-vacuity condition below. $\gamma$ a learnable scalar per head (Definition 5, READ docs/CEQ_SHAPE.md:318) so $\bar\gamma$ is a scalar reduction over the causal triangle — fixed-order in the deterministic mode; the recompute of $P$ tiles from saved $m,\ell$ reproduces the forward's $P$ bitwise (the same exponentials in the same order); the backward's dQ accumulation mode per 03.10.

Evidence. DERIVED (the block above; four lines of matrix calculus). The record's own statement of the adjoint: "one upper-triangular solve plus an outer product; retained memory linear in $s$ plus $z$" (N-12, READ docs/PLAN.md:788-792) — agreed and made explicit. The displacement identity that the same algebra proves, $\Delta z=(I-\gamma P')^{-1}(\Delta V+\gamma\Delta P z)$, is RUN at $10^{-15}$ three ways (READ docs/sources/judge/sec_apparatus.md:50). Gradient parity of the fused backward against autograd through L4: NOT MEASURED — needs the kernel.

Mechanism. P-4 (no backward exists; the record's scheduled kernel is forward-only, READ THEORY.md:223-224 @ 99777ab), V-22 (the memory class claim is checked on this device, never carried), V-16 (a gradient check that cannot run reports NOT EXECUTABLE, not pass).

Kill. In price order: (i) gradient parity, in relative units, with the two amplification factors measured per cell. The earlier threshold $4\gamma_{s+1}\|\bar O\|_\infty\|V\|_\infty/(1-\gamma)^2$ is struck twice over: it applied 03.13's row-dominance bound to $M^\top$, which is not row dominant under the declared sink, and it omitted $\pi=\|P^\top\|_\infty$ from $\bar z=(1-\gamma)P^\top\bar O$, whose correct bound is $\|\bar z\|_\infty\le(1-\gamma)\pi\|\bar O\|_\infty$ with $\pi$ up to $s=1024$ at the kill's registered geometry — so at $(s,n)=(1024,128)$ the old threshold understated a correct backward by up to $1024\times$ and fired on it (V-17, M-3). The frozen kill is the relative reading $$\max_{\bullet\in\{\bar Q,\bar K,\bar V,\bar\gamma\}}\frac{\|\bullet_{\rm fused}-\bullet_{\rm L4}\|_\infty}{\|\bullet_{\rm L4}\|_\infty}\ >\ \Theta_{\rm rel}:=4\,\gamma_{s+1}\Big(\underbrace{\tfrac{1+\gamma}{(1-\gamma)^2}}_{\text{forward solve, 03.13(iv)}}+\underbrace{\kappa_1(1+\gamma\pi)}_{\text{backward solve, above}}\Big),$$ the factor $4$ covering the four places a relative solve error enters ($\bar z$'s formation, the two rank-$d$ contractions, the softmax backward), DERIVED by the first-order rule that relative errors add through products. $\kappa_1$ and $\pi$ are measured on the cell and printed beside $\delta_{\rm float}$ as certificate lines; the threshold is not a constant and is never quoted without them.

Reachability, stated rather than assumed (CHARTER §2). The kill is decidable only where $\Theta_{\rm rel}<1$, i.e. where the measured $\kappa_1(1+\gamma\pi)<1/(4\gamma_{s+1})$ — $6.45\times10^{4}$ at $s=64$, $1.63\times10^{4}$ at $s=256$, $4.09\times10^{3}$ at $s=1024$, $1.02\times10^{3}$ at $s=4096$ in float32. Under the declared sink at $P_{j0}\ge0.5$ the extremes are $\pi\in[s/2,s]$ and $\kappa_1\le s/(1-\gamma)$, so the worst case at $\gamma=0.9$ is $\kappa_1(1+\gamma\pi)\le 3.75\times10^{4}$ at $s=64$ (decidable, against $6.45\times10^{4}$) and $5.92\times10^{5}$ at $s=256$ (not decidable in the worst case, against $1.63\times10^{4}$). The kill is therefore registered at $(s,n)=(64,2048)$, not at $(1024,128)$; at $(1024,128)$ the cell prints its measured $\kappa_1,\pi$ and is journalled NOT DECIDABLE when $\kappa_1(1+\gamma\pi)\ge4.09\times10^{3}$ — one of V-16's three outcomes, never a pass. Refused above the threshold; price seconds. Planted negative: zeroing the $\gamma\bar Vz^\top$ term must move $\bar Q$ relatively by $\ge0.1$ at $\gamma=0.9$, $s=64$. (ii) memory — peak reserved bytes across $s\in\{256,1024,4096\}$ at fixed $n$ fit a line in $s$ with R-squared below $0.99$, or the operator term re-solved by N-18's least squares reads above $0.5$ tensors of $[n,h,s,s]$ (frozen) — the recompute is not reached and the shape stays in the quadratic class (N-12's kill); price $\approx9$ GPU-min DERIVED.

If killed on (i) — the branch kill (i) lacked when this verse was first written, and a replacement that needs neither $\kappa_1$, nor $\pi$, nor any triangular-solve error bound, because the tensor-parity route cannot be made decidable at $s\ge256$ under a sink. Why not the obvious route. Taking autograd through L4's solve_triangular as the deciding backward — the route this branch was asked for — reproduces the defect that killed (i): autograd's backward of a triangular solve is a triangular solve on $M^\top$, so the reference carries the same $\kappa_1(1+\gamma\pi)$ amplification as the object under test, and the comparison is again undecidable wherever $\kappa_1(1+\gamma\pi)\ge1/(4\gamma_{s+1})$ — a replacement dying to the verse's own kill (V-9). Autograd through L4 is therefore kept as the reference implementation whose adjoint the check below validates, and not as the check. The adjoint is checked instead as a scalar directional derivative by central differences on the L4 forward in float64, which runs no backward at all — neither the fused one nor autograd's — and therefore inherits no solve conditioning from either: $$\Big|\ \langle g_{\rm fused},v\rangle-\frac{\mathcal L(x+hv)-\mathcal L(x-hv)}{2h}\ \Big|\ \Big/\ \big|\langle g_{\rm fused},v\rangle\big|\ >\ 10^{-3}$$ on any of $8$ random unit directions $v$ in the concatenated $(Q,K,V,\gamma)$ space at $s=64$, $n=8$, $\gamma=0.9$, $h=10^{-5}$, with $\mathcal L$ the mean-squared error of $O$ against a fixed random target. Hypotheses: float64 throughout, so the difference quotient's own floor is the sum of a roundoff term $2u|\mathcal L|/h=2.2\times10^{-11}$ relative and a truncation term $O(h^2)=10^{-10}$ — seven orders below the $10^{-3}$ bar, which is why the bar is a constant and needs no per-cell measurement. Retained memory, since the branch runs on the same device as the verse it replaces: the $16$ forward passes are taken under torch.no_grad, so nothing of the graph is retained and the peak is the materialised float64 operator alone, $C_{\rm OP}\,n s^2\cdot8=2\cdot8\cdot64^2\cdot8=5.24\times10^{5}$ B at $(s,n)=(64,8)$ — $6.8\times10^{-5}$ of 03.5's re-frozen $7.727\times10^9$ B budget, and inside 03.5's L4 column at $s=64$ by four orders. That is why the check is registered at $(64,8)$ and why no memory kill belongs to this branch: the memory reading is (ii)'s, at $(4096,64)$, and is not restated here as a second copy of it (V-9). Evidence: NOT MEASURED — needs the kernel (03.14) and the float64 L4 forward, both existing objects once 03.18 step 1 lands. Kill: the same reading on the analytic adjoint of the L4 route (no fused kernel at all) exceeding $10^{-3}$ — then the difference quotient is the broken instrument, not the kernel, and the check is refused (V-16). Price seconds ($16$ forward passes at $s=64$, $n=8$), decidable at every $s$ and every $\pi$, and more decisive than (i) because it separates a wrong adjoint formula from a merely ill-conditioned solve, which the tensor-parity threshold cannot do at any $\kappa_1$. Planted negative: dropping the $\gamma\bar Vz^\top$ term from $\bar P$ must read above $10^{-3}$ on at least $6$ of $8$ directions.

If killed on (ii): per-chunk torch.utils.checkpoint on L2 — retained $O(sd)$ states and $O(sC)$ per active chunk, the forward recomputed once in the backward ($+\tfrac32s^2d$ MACs); Hypotheses: recompute cost below the memory-limited alternative of running at $n$ per 03.5's dense column; Evidence: NOT MEASURED — needs L2; Kill: peak reserved above 03.5's $0.90$ budget $7{,}726{,}694{,}400$ B $=7.196$ GiB at $(4096,64)$ — then the L1 column of 03.5 is struck and the L4 column is the hard limit; price $\approx3$ GPU-min DERIVED, cheaper than N-18.

Terminal. "The adjoint of the solve is one upper-triangular solve on $M^\top$ — column dominant, not row dominant, and outside 03.13's M-matrix class under the declared sink — plus two rank-$d$ contractions; its error is bounded only by $\gamma_{s+1}\kappa_1(1+\gamma\pi)$ with both constants measured per cell, its tensor-parity kill is decidable at $s=64$ and journalled NOT DECIDABLE wherever $\kappa_1(1+\gamma\pi)$ exceeds $1/(4\gamma_{s+1})$, and the adjoint's correctness is then decided by a scalar float64 finite-difference check that no conditioning constant enters; it is written, not run; until it runs, training above $s=64$ is on the dense route at $n$ per $s$, and the fused path is inference-side."

03.15 — The Sherman–Morrison column route for $m$ candidate moves (Bet G), with its price#

Statement. For a query-side row clamp at position $i$, $P'=P+e_iu^\top$ with $u$ supported on $j\le i$, $u^\top\mathbb 1=0$: $$\Delta z=\gamma\,\frac{(M^{-1}e_i)\,(u^\top z)}{1-\gamma\,u^\top M^{-1}e_i},\qquad M=I-\gamma P\ \text{as everywhere in this book},\qquad 1-\gamma u^\top M^{-1}e_i=\frac{1-\gamma p'_{ii}}{1-\gamma P_{ii}}>0\ \ (\gamma<1),$$ with $M=I-\gamma P$ and the column written $M^{-1}e_i$, never $Me_i$. The earlier display of this verse set "$M=(I-\gamma P)^{-1}$" locally while the Notation block (line 8) and every other verse set $M=I-\gamma P$; an engineer building from the page and substituting the book's global $M$ computed $\Delta z$ with the operator in place of its resolvent, wrong by the factor $\|(I-\gamma P)^{-1}\|_\infty=1/(1-\gamma)=10$ at $\gamma=0.9$ — not a rounding error but the whole amplification the resolvent exists to carry. The local redefinition is struck (P-7, a symbol with two referents; V-17), and no symbol in this book denotes $(I-\gamma P)^{-1}$: where the resolvent is needed it is written $M^{-1}$. So one candidate costs one column $M^{-1}e_i$ — a single-right-hand-side forward substitution against $M$ over rows $i..s-1$, $(s-i)^2/2$ MACs — plus an $O(s+d)$ inner product and axpy; $m$ candidates at rows $i_1,\dots,i_m$ cost $\sum_a(s-i_a)^2/2\le ms^2/2$ MACs against the full solve's $s^2d/2$: ratio $\le m/d$, $=0.5$ at $m=8$, $d=16$. For the committor head the deciding quantity is $q'=z'$ itself (no further read); for the $z$/$\Delta z$ channel the read $O'=(1-\gamma)P'z'$ costs one more product ($s^2d/2$), so the column route prices the state, never the read. Key-side (token) moves are not rank-one ($\operatorname{rank}(\Delta P)=s-i$, RUN[J] rank $20$ at $i=12$, READ docs/CEQ_SHAPE.md:3042) and cost a suffix re-solve of $(s-i)^2d/2$ MACs each. Seconds: the $m$ columns are one batched multi-right-hand-side triangular solve with $m$ columns in place of $d$ — on the latency-bound regime of 03.3 the book's own estimate is that it costs the same as one solve ($\approx1\times$), which is SPLIT between Bet G's prediction ($\le0.5\times$) and its counter ($\ge8\times$, a re-solve per candidate); under D-CALIB the counter is the point estimate and the book's $1\times$ is printed as a second, un-scored line.

Hypotheses. Oracle-side row clamp (Definition 6(i), READ docs/CEQ_SHAPE.md:323), never a token rewrite; $\gamma<1$ (at $\gamma=1$ with an absorbing row $I-P'$ is singular, RUN[M] $\det=0.0$); $M=I-\gamma P$ lower-triangular, hence $M^{-1}$ lower-triangular too, so the column $M^{-1}e_i$ is supported on rows $\ge i$; $s=64$, $d=16$, $m=8$ for the numeric instance.

Evidence. Formula and positivity: DERIVED and RUN: closed form against re-solve $1.45\times10^{-15}$, denominator $1.9$ at row $17$ (READ docs/sources/design/refute_instrument_math.md:140-149); $1.42\times10^{-15}$ with row $12$ clamped to $e_{12}$ (READ docs/sources/design/refute_theory_math.md:127-131); RUN[F] $0.511918$ (READ docs/CEQ_SHAPE.md:469). MAC ratio $m/d=0.500$: READ docs/sources/design/refute_falsify_occvac.md:37-39 (DERIVED-1). The rank of a key-side edit: RUN[MARS] $20$ of $32$ rows changed (READ docs/sources/design/refute_theory_math.md:74). Prior owner of the update formula: CITED [V] sherman-1950-inverse-adjustment; the linear-RL precedent for the same update on a fixed operator: CITED [V] piray-2021-linearrl Eq. 5 (READ docs/sources/design/design_theory.md:424). Seconds: NOT MEASURED — needs V-17's microbenchmark (READ docs/PLAN.md:660-668).

Mechanism. M-8 (the column is priced as its own route, never as a fraction of the solve's seconds), P-8 (MAC ratio $\ne$ seconds ratio; the headline carries both), M-3 (the ratio at $(8,16)$ is not carried to $(m,d)$ elsewhere), D-CALIB.

Kill, both edges of the SPLIT band frozen — the flattering one too. The adverse edge alone was frozen before this repair, so a reading of $0.3\times$ licensed Bet G's flattering sentence and no threshold refuted it and no plant could fire on it (L-SIGN, D-CALIB, V-15). Both edges are frozen here.

(K-G-adverse) The measured ratio of route (i) to one solve at $n=2048$, $s=64$, $d=16$, $m=8$ at or above $8\times$ (Bet G's counter; frozen) — the consequence channel is re-priced as $m$ solves in every table. Planted negative: route (i) with the column replaced by a full re-solve per candidate must read $\ge m\times$ one solve's arithmetic term at $s=1024$ (the instrument must see $m$ solves as $m$ solves); at $s=64$ the plant may read below $m\times$ because of the launch floor, and that reading is journalled as the floor it is.

(K-G-flattering) A reading at or below $0.5\times$ — the prediction's own edge — is refused as a licence for the flattering sentence unless it survives the arithmetic-term check, and the check is frozen here in two clauses, both required. (a) The launch-floor clause, which is this verse's own caveat made decidable. At $s=64$ the operator is launch- and latency-bound, not arithmetic-bound: 03.3 reads $0.273$ TFLOP/s on the $PV$ pair, $1.4\,\%$ of the assumed peak (DERIVED there), and the same plant that reads $\ge m\times$ at $s=1024$ "may read below $m\times$ at $s=64$ because of the launch floor". A $\le0.5\times$ reading at $s=64$ is therefore consistent with the $m$ columns costing nothing measurable and with them costing $m/d$ of a solve, and it separates the two only if the arithmetic term is separable: the cell must print the measured ratio at $m\in\{1,8\}$ at the same $(s,n,d)$ and the difference must exceed the cell's own repeat spread at $N=8$. A $\le0.5\times$ reading whose $m=1$ and $m=8$ cells agree inside that spread is journalled LAUNCH-FLOOR, NO READING — V-16's third outcome — and licenses no sentence in either direction. (b) The same reading required at $s=1024$, where the launch floor is not the binding term. The prediction is licensed only when $\le0.5\times$ is read both at $(s,n)=(64,2048)$ and at $(s,n)=(1024,128)$ — 03.5's L4 residency column at $s=1024$ under the upper constant — where the arithmetic term is $256\times$ the $s=64$ term and the launch floor cannot carry the reading. Frozen: a $\le0.5\times$ reading at $s=64$ alone is not a reading of Bet G at all, and the flattering sentence is withdrawn until (a) and (b) both hold. Planted negative for this edge: route (i) with the $m$ columns removed entirely — the solve run and its $\Delta z$ returned as zeros, so the arithmetic term is exactly $0$ — must read $\le0.5\times$ at $s=64$ and must be caught by clause (a), its $m=1$ and $m=8$ cells agreeing inside the repeat spread; an instrument that passes a do-nothing route as Bet G's prediction at $s=64$ is refused.

SPLIT in $(0.5\times,8\times)$ prints the band and no point. Instrument for both edges: V-17's microbenchmark under 03.7's bracket (median of $14$ after $2$ warm-ups); price seconds at $s=64$, NOT MEASURED at $s=1024$ until the residency cell runs; decidable on random logits at $N=8$.

If killed ($\ge8\times$): the $m$ candidates are batched as one solve with $m$ stacked right-hand sides $[e_{i_1}\cdots e_{i_m}]$ — the identical kernel path as the main solve with $d\to m$ columns, so its price is the main solve's law evaluated at $m$; Hypotheses: the law of 03.7 fitted with $d$ as a variable; Evidence: NOT MEASURED — needs S-66 with $d\in\{8,16\}$; Kill: the batched route above $2\times$ one solve — then a re-solve per candidate ships and the safest-move channel is priced at $m$ solves (price seconds, same instrument, more decisive: a single ratio against a single reference).

Terminal. "A query-side clamp costs one column in MACs ($m/d$ of a solve) and an unmeasured number of seconds; a key-side move costs a suffix re-solve; the safest-move channel is priced at $m$ solves until the microbenchmark runs."

03.16 — The CSR two-stage path and its certificate (link L3)#

Statement. Stage 1, the schedule as data: for each query tile a causal list of key tiles (sink tiles, the local window, salience-selected tiles), the do-nothing zero-dimensional-salience schedule always entered as a competitor (READ THEORY.md:160-165 @ 99777ab). Stage 2, Neumann on the scheduled $P_{\rm sched}$ for $K$ hops with the union certificate in vector units $$\max_i|O_i-O^{\rm sched,K}_i|\ \le\ \Big(\frac{\varepsilon}{1-\gamma}+\gamma^{K+1}\Big)\|V\|_\infty=:\delta_{\rm union}^{O}\|V\|_\infty,\qquad \max_i|z_i-z^{\rm sched,K}_i|\ \le\ \Big(\frac{\varepsilon}{1-\gamma}+\frac{\gamma^{K+1}}{1-\gamma}\Big)\|V\|_\infty=:\delta_{\rm union}^{z}\|V\|_\infty,$$ where $\varepsilon=\max_i\sum_{j\notin{\rm sched}(i)}P_{ij}$ is the dropped row mass.

The tail term is derived for the read it is applied to, and the two reads carry different tails. The earlier Statement derived the tail for $z$ and applied it to $O$, where the $(1-\gamma)$ factor makes it loose. DERIVED, three lines: $z=\sum_{k\ge0}\gamma^kP^kV$, so truncating at $K$ leaves $\|\sum_{k>K}\gamma^kP^kV\|_\infty\le\sum_{k>K}\gamma^k\|V\|_\infty=\gamma^{K+1}\|V\|_\infty/(1-\gamma)$ — the $z$ tail; but $O=(1-\gamma)Pz=(1-\gamma)\sum_{k\ge0}\gamma^kP^{k+1}V$, whose tail is $(1-\gamma)\sum_{k>K}\gamma^k\|V\|_\infty=(1-\gamma)\cdot\gamma^{K+1}/(1-\gamma)\cdot\|V\|_\infty=\gamma^{K+1}\|V\|_\infty$ — the $(1-\gamma)$ of the read cancels the $1/(1-\gamma)$ of the tail, exactly, at every $\gamma$. The $\varepsilon$ term is $\varepsilon/(1-\gamma)$ for both reads and needs no change ($M^{-1}$ amplifies a row-mass perturbation by at most $1/(1-\gamma)$ by 03.13(ii), and that amplification is inside the resolvent, not in the $(1-\gamma)P$ that follows it). At $\gamma=0.9$ the two tails differ by $1/(1-\gamma)=10\times$, so the loose form overstates the hop count the "uncompetitive by arithmetic" sentence rests on by $16\,\%$ (see Hypotheses). Every $K$ threshold below is re-read on $\delta_{\rm union}^{O}$, which is the read every route of this book ships; $\delta_{\rm union}^{z}$ is printed only where a route returns bare $z$ (the committor head, book 08). The exact solve on an F1 mask is refused: $M_{\rm sched}^{-1}$ fills in along reachability (a dropped tile $(k,l)$ with $(k,m),(m,l)$ kept makes $(M^{-1})_{kl}\ne0$), so an "exact" read on a mask carries no $\delta$ (N-03 item 5, READ docs/PLAN.md:307). Two structural guards: an empty schedule row returns no divide by zero; a negative block index is clamped in-kernel and validated on the host (READ ceq/mz_kernel.py:13-21). The path is inference-side until 03.14's adjoint is realised on the scheduled kernel (N-17).

Hypotheses. $P$ row-stochastic; $\gamma<1$; a schedule with $\varepsilon$ measurable per row; $K$ chosen so the tail of the read being shipped falls below the label sd, which for the $O$ read is $\gamma^{K+1}<\mathrm{sd}(\text{label})/\|V\|_\infty$ — at $\gamma=0.9$, $\delta=10^{-6}$ this needs $K\ge\log(10^{-6})/\log(0.9)-1=130.13$, i.e. $K\ge131$ (DERIVED, re-read this session on the corrected tail), and at $\hat\gamma\le0.5$ it needs $K\ge\log(10^{-6})/\log(0.5)-1=18.93$, i.e. $K\ge19$. The loose readings this verse carried before are struck and printed beside the tight ones: on the $z$ tail $\gamma^{K+1}/(1-\gamma)$ the same $\delta$ demands $K\ge152$ at $\gamma=0.9$ (N-10's formula as written, READ docs/PLAN.md:738-742) and $K\ge20$ at $\gamma=0.5$, so the printed hop count for the $O$ read was overstated by $152/131=1.16$, a $16\,\%$ overstatement of the count that carries the "uncompetitive by arithmetic" sentence. The sentence survives the correction and is re-stated on the tight number: at $\gamma=0.9$ the route needs $K\ge131$ hops, each priced at 03.6's marginal $b=1.4939$ ms, i.e. $\ge0.196$ s per step against the exact solve's $2.514$ ms — $78\times$ — so on the softmax corner at high $\gamma$ the route is uncompetitive by arithmetic by two orders and not by a threshold's rounding, and it is non-vacuous only where $\hat\gamma\le0.5$ ($K\ge19$, $\ge28.4$ ms, still $11.3\times$ the exact solve) or where a coarser $\delta$ is admitted by the bed. The $131$-hop price is NOT MEASURED — needs the Neumann arm swept over $K$ (03.6): $b$ is fitted on two points at $K\in\{1,16\}$ and $K=131$ is $8.2\times$ outside that range.

Evidence. The certificate identity attained to $10^{-15}$ at $K\in\{1,2,4,8,16\}$ with absorbing rows (RUN[MARS], READ docs/CEQ_SHAPE.md:2574); the convergent plant reads $7.29$ against $0.54$ (READ docs/PLAN.md:559-561); the fill-in guard and the do-nothing schedule exist in the tree (READ ceq/mz_kernel.py:1-25; READ ceq/multizoom.py:10-20, where a gather-realised candidate set costs $2.58$ ms of index_select against $0.82$ ms of attention at $65{,}536$ positions — the reason schedules are contiguous tiles); the forward-only scheduled kernel: CITED [V] sharma-2026-kernels-22 (forward-only Triton kernel on a causal CSR block schedule; the PR page's own speed-up is not re-measured here). At $s=64$ the exact solve is cheaper than one hop ($2.514$ against $3.001$ ms) so the route is dormant there (READ docs/PLAN.md:806).

Mechanism. L-CERT (no mask without its certificate, F1 with $\delta$ printed and $1/(1-\hat\gamma)$ beside it), V-10 (the bound is attained on the class and declared definitional; the measurement is the $1{,}024$-draw exceedance count), V-17 (units: $\|V\|_\infty$, never a bare $\delta$), V-9 (the do-nothing schedule as the first-class outcome).

Kill. (K-I) one exceedance of $\delta_{\rm union}^{O}\|V\|_\infty$ on $1{,}024$ drawn cells, or $\delta_{\rm union}^{O}\|V\|_\infty\ge\mathrm{sd}(\text{label})$ — read on the tail of the read the route ships, $\gamma^{K+1}$ for $O$ and $\gamma^{K+1}/(1-\gamma)$ for a bare $z$, never the $z$ tail applied to $O$ (frozen; READ docs/PLAN.md:806) — the mask is refused and the exact route is the path. Instrument: N-15's battery; price $\approx2.6$ s at $s=64$ DERIVED, NOT MEASURED at $s=4096$; decidable on drawn cells with the bed's seed rule and its printed sd. Planted negative: a schedule that drops the sink tile must exceed the bound on at least $1{,}000$ of $1{,}024$ draws ($\varepsilon$ then includes the absorbing column's mass, an $O(1)$ miss).

If killed: L4 at $n$ per $s$ (03.5), whose $\delta_{\rm route}=0$ and whose only certificate line is 03.13's float bound; Hypotheses: $n$ fits; Evidence: RUN[NEPTUNE] at $s=64$; Kill: the residency row (N-20), price $\approx3$ GPU-min — a cheaper and coarser kill than the $1{,}024$-draw battery.

Terminal. "A scheduled read ships only with $\delta_{\rm union}^{O}\|V\|_\infty=(\varepsilon/(1-\gamma)+\gamma^{K+1})\|V\|_\infty$ printed and below the label sd — the tail is $\gamma^{K+1}$ for the shipped $O$ read and $\gamma^{K+1}/(1-\gamma)$ for a bare $z$, and at $\gamma=0.9$, $\delta=10^{-6}$ the two demand $K\ge131$ and $K\ge152$; no such schedule exists; the exact route is the only route with a certificate today, and it is $\delta=0$ plus the float bound."

03.19 — The Mapper schedule: the cover-to-tile quantiser, priced against the gather it must not become (B15's third kernel)#

Verse id follows the book's last; its place is here, beside the CSR path it would feed.

Statement. CHARTER §5's B15 names three missing kernels — chunked solve, CSR path, Mapper schedule. The first two are closed by 03.11/03.12 and 03.16; this verse closes the third, which before this repair round appeared in this book only as the word Mapper in the census table and carried no statement, no kill and no Terminal (P-4, L-VERSE). The Mapper schedule is not a kernel but a schedule producer for L3: a lens over key positions (the committor $q^{(\bullet)}$ of book 08, or an influence score), an overlapping cover whose parameters are fixed on a held-out relation by the Reeb-estimator rule and never on the bed (M-2), the nerve of that cover, and a quantiser that turns nerve edges into contiguous key tiles before they are a schedule. The quantiser is the load-bearing step and the reason the route is stated at all: $$\text{a candidate set realised as a gather is refused before it is timed,}$$ because the one gather this repository measured on this card cost $2.58$ ms of index_select against $0.82$ ms of attention over the same slice at $65{,}536$ positions — $3.15\times$ the attention it selects for — and the dyadic builder that scored $0.39$ cost $53\times$ the attention. A Mapper cover that does not quantise to contiguous tiles of the kernel's block length is therefore inadmissible for L3 at any $\varepsilon$, whatever its certificate reads; a cover that does quantise enters 03.16 as one candidate schedule among others, with $\varepsilon$ measured per row and $\delta_{\rm union}$ printed.

Hypotheses. 03.16's, unchanged ($P$ row-stochastic, $\gamma<1$, $\varepsilon$ measurable per row, $K$ with $\gamma^{K+1}<\mathrm{sd}(\text{label})/\|V\|_\infty$ on the shipped $O$ read — $K\ge131$ at $\gamma=0.9$, $\delta=10^{-6}$, and $K\ge19$ at $\hat\gamma\le0.5$, per 03.16's corrected tail — so non-vacuous only where $\hat\gamma\le0.5$ or the bed admits a coarser $\delta$); tiles of the kernel's block length $C\in\{32,64,128\}$ (03.11); cover parameters fixed on held-out data before the bed (Ruling 7, M-2); a lens exists — the committor head of book 08 or an influence score, both NOT MEASURED today. Non-vacuous only at $s\ge1024$, where a schedule can drop enough mass to be worth its builder; at $s=64$ the exact solve is cheaper than one Neumann hop (03.6) and the whole L3 rung is dormant.

Evidence. The gather reading: RUN on this same RTX 4060 Laptop, READ ceq/multizoom.py:13-15 ($2.58$ ms index_select against $0.82$ ms attention at $65{,}536$ positions); the builder reading: READ ceq/multizoom.py:17-18 ($53\times$ the attention, top-$k$ an argsort). The programme, its deliverable and its price: READ docs/CEQ_SHAPE.md:2130-2134 (N-16, phase 5, prereq N-15 and S-44, deliverable docs/CEQ_MAPPER_SCHEDULE.md, evenings $2$, price NOT MEASURED, needs the quantiser and N-15 at long $s$). The do-nothing zero-dimensional-salience schedule always entered as a competitor: READ THEORY.md:160-165 @ 99777ab. The cover's prior art is occupied and [V]-graded — CITED [V] singh-2007-mapper (identifier unresolved at Crossref, READ docs/CEQ_SHAPE.md:1159), CITED [V] carriere-2018-mapper-statistics — and carries nothing load-bearing here (L-EQ); the record's own sentence is that "A Mapper cover with a comparable $\delta$ is NOT FOUND" (READ docs/CEQ_SHAPE.md:2131). No schedule of any kind exists under results/: RUN this session, the grep of 03.1's Evidence. Every seconds figure in this verse is the gather pair and nothing else.

Mechanism. P-4 (a schedule named is not a schedule found; FOUND needs a results/ record with its kind), M-2 (cover parameters fixed before the data, or the summary is chosen rather than measured), L-CERT (no mask without its $\delta$), V-9 (the do-nothing schedule is a first-class outcome, not a fallback), L-EQ (the Mapper citations are identifier-grade and decide nothing).

Kill (two clauses, both frozen, either sufficient). (K-M, capability) the Mapper-quantised schedule not better than the do-nothing merged-salience schedule by $\mathrm{MDE}_8$ in $\varphi$-NRMSE at matched visited tiles at $s=1024$ (READ docs/CEQ_SHAPE.md:2133) — then the cover adds nothing measurable and the candidate builder is the merged salience schedule. (K-M2, price) the builder's own wall-clock above the attention it schedules by more than the measured gather factor $3.15\times$ at $65{,}536$ positions — then the schedule costs more than the read it saves, whatever its accuracy. Instrument: N-16 with N-15's battery on the long-context bed (book 04); price NOT MEASURED — needs the quantiser and N-15 at long $s$, $2$ evenings [ASSUMED, the record's own]; decidable at $N=8$ on the bed's registered seed rule, and gated behind 03.18 step 1. Planted negatives, one per clause: (K-M) a cover whose lens is a random permutation of positions — no topology, same tile count — must fail the first clause by more than $\mathrm{MDE}_8$; (K-M2) the same nerve realised by index_select instead of contiguous tiles must exceed the second clause, since that is the $3.15\times$ reading the clause is taken from.

If killed (on either clause): the schedule producer is the merged salience schedule — contiguous tiles from a declared local window plus the declared sink tiles, no lens, no cover, no nerve — i.e. the do-nothing competitor promoted to the shipped producer. Hypotheses: the sink and window are declared per book 08; $\varepsilon$ measurable per row; no held-out fitting is needed, so M-2 cannot be violated by a producer with no parameters. Evidence: it exists in the tree — the do-nothing schedule and the fill-in guard, READ ceq/mz_kernel.py:13-21, READ THEORY.md:160-165 @ 99777ab. Kill, its own and on a different object: its $\delta_{\rm union}^{O}\|V\|_\infty\ge\mathrm{sd}(\text{label})$ at the bed's $s$, or one exceedance on $1{,}024$ drawn cells (K-I, 03.16's number), price $\approx2.6$ s at $s=64$ DERIVED — a certificate reading rather than a comparison against a rival, and cheaper than K-M by the whole cost of building a cover. If that fires, L3 is dead and the route is L4 at $n$ per $s$ (03.5), which is 03.1's ladder in force.

Terminal. "The Mapper schedule is a schedule producer and not a kernel; nothing in the tree produces one, no cover has been built, and no Mapper number exists at any $s$. What the canon licenses is one sentence: a candidate set must quantise to contiguous tiles before it is timed, because the only gather this repository measured cost $3.15\times$ the attention it selected for on this card. If the quantiser is never built, B15's third kernel is closed by the do-nothing merged-salience schedule and the exact route, and the word Mapper is withdrawn from every cost sentence in this canon."

03.17 — The published blockwise evaluation is the fellow on the cost axis, never the beaten baseline#

Statement. zhao-2026-structuredsparse evaluates the identical operator $\mathcal S_\gamma(A)=(1-\gamma)A(I-\gamma A)^{-1}$ blockwise — exact solves on diagonal tiles, cross-block interaction through a reduced $k\times k$ system. Its two numbers, the complexity $O(n^{4/3}d)$ and the $12$–$29\,\%$ wall-clock reduction against the dense operator at accuracy parity with no printed truncation bound, rest on one sweep row graded CITED [V] (READ docs/sources/sweep/sweep_resolvent.md:218), which L-EQ makes inadmissible for a load-bearing statement. Both are therefore marked NOT MEASURED — needs the equation-level read of the source (theorem statement, its hypotheses, one numeric instance at this book's geometry), owed to book 05's bibliography repair and neither appears anywhere in this canon as a threshold, a comparison or a price. What survives without them, and is the load-bearing content of this verse, is the concession itself, which is conservative and needs no number: under R-SKY on the cost axis, no sentence of this book claims a lower price than a published blockwise evaluation of the same operator at $s\ge1024$, whatever that route's exponent and margin turn out to be — a concession made on the class, not on a figure, so an owed read can only relax it and never tighten it. Beside it: the shape's exact routes (L1, L2, L4) claim exactness ($\delta=0$ plus 03.13) beside it, and the F0-segmented exact solve (block-diagonal after exact zero gates, $\sum_mL_m^2d/2$ MACs, dividend $D=s(s+1)/\sum_mL_m(L_m+1)$) claims a corpus dividend, not a device one.

Hypotheses. The blockwise control is implemented as a fellow arm at $s\in\{1024,4096\}$ with $n$ per $s$ and its own measured $\delta$ where a dense control fits (N-13, READ docs/PLAN.md:794-800); the F0 dividend exists only on gated corners or masked $P$ — on the softmax corner there are no exact zeros and $D=1$ (READ docs/CEQ_SHAPE.md:452 for BED-M's $31.04\times$ from the live-pair fraction $0.0322$, READ docs/CEQ_SHAPE.md:3054 for the same dividend beside the softmax corner's $1\times$).

Evidence. The concession: DERIVED from R-SKY applied to the cost axis, resting on no citation — a published route of the class exists (the sweep row records it), and R-SKY forbids a "cheaper than" sentence wherever a fellow approximator of the price occupies the ground, regardless of its measured margin. The two numbers $O(n^{4/3}d)$ and $12$–$29\,\%$: CITED [V] zhao-2026-structuredsparse (READ docs/sources/sweep/sweep_resolvent.md:218-226; READ docs/references.bib note: "$O(n^{4/3}d)$") — [V] is the identifier grade, not the equation grade, so both are NOT MEASURED — needs the[V-eq]upgrade: theorem, hypotheses, one numeric instance at $s\in\{1024,4096\}$, $d=16$; priced in book 05 as one bibliography read, $0$ GPU-s, one evening. The same operator's diagonal convention in CITED [V] fagnou-2024-chacal Eq. 5, Eq. 7 (solved as the triangular system $(I-\gamma A)Y=(1-\gamma)AV$) carries the same grade and the same owed read (B22). The blockwise arm's price: NOT MEASURED — needs the control implemented (N-13).

Mechanism. R-SKY and D-1 on the cost axis (a published sub-quadratic route to the same read is a fellow approximator of the price), V-9 (the F0 dividend at $D=1$ is a repair that changes nothing on the softmax corner and is said so), L-EQ and P-10 (a [V] grade is inadmissible for a load-bearing statement: the $12$–$29\,\%$ and the $O(n^{4/3}d)$ are the source's, read at the identifier and not at the equation, are marked NOT MEASURED here, and are carried as neither a threshold nor a comparison — V-22).

Kill, on the cost axis, which is this verse's own scope. The earlier kill — "the shape's F0-segmented exact solve not better than the blockwise control by $\mathrm{MDE}_8$ in $\varphi$-NRMSE at matched visited tiles at $s=1024$" — is struck: it is a comparative-accuracy sentence outside forms A1 and A2, against a comparator that is neither base self-attention nor the depth skyline, with no floor printed, no $N$, no matched parameter count, and a bed ("the long-context bed once admitted, book 04") that does not exist. CHARTER §1 strikes exactly that: a verse claiming "more accurate" without naming which of A1, A2, T1, T2 it means, and which bed, floor, $N$ and matched count it is measured on. An accuracy comparison against this route is not withdrawn but relocated: it is owed to book 04 as a registered contrast with its floor (L-FLOOR), its $N$, its $\mathrm{MDE}_N$ at the realised $\sigma_d$ and its matched count, and with its form named — and under 04.9 clause 7 every $\mathrm{MDE}_8$ number is a placeholder until 04.21 step (5) fills $\sigma_d$, so no such threshold can be frozen in this book at all.

The kill this verse carries is the price, at matched exactness. Frozen: $t_{\rm shape}/t_{\rm blockwise}\le1.00$ at $s=1024$, both arms forward and backward under 03.7's protocol at the same $(s,n)$ with $n$ from 03.5's residency column for the route shipped, both arms at matched exactness — the shape's route prints $\delta_{\rm route}=0$ plus 03.13's float bound, so the blockwise control is admitted to this comparison only with its own $\delta$ measured against a float64 dense reference on the same draws and printed in the same row; a comparison at unmatched exactness is refused, and the blockwise route's $\delta$ is unprinted at this pin ([V] grade, READ docs/sources/sweep/sweep_resolvent.md:218-226), so today the comparison is NOT EXECUTABLE on that clause alone. What firing means: a reading at or below $1.00$ shows the published route is not cheaper than the shape's exact route on this device at $s=1024$, so the concession — made on the class, "no sentence of this book claims a lower price at $s\ge1024$" — is over-broad, and it narrows from the class to the publication. Instrument: N-13's control implemented (READ docs/PLAN.md:794-800) inside 03.7's interleaved block; price the $s=1024$ block of 03.7's sweep, $\ge1{,}248$ GPU-s DERIVED for the sweep it sits in, plus the control's implementation, NOT MEASURED — needs N-13; decidable at $N=8$ on torch.randn logits, no bed and no seed rule, because it is a wall-clock reading and not a capability contrast — which is the second reason the accuracy form did not belong here. Planted negative, on which this kill must fire: the blockwise control built with $P$ materialised per block and re-read once per chunk step — the arithmetic unchanged, the bytes multiplied by $s/C$ — must read $t_{\rm shape}/t_{\rm blockwise}\le1.00$ at $s=1024$; an instrument that cannot see a deliberately memory-starved comparator as slower than the exact route is not measuring wall-clock and is refused.

If killed (the reading is at or below $1.00$: the shape's exact route is not dearer than the blockwise control at $s=1024$ on this device). The concession narrows from the class to the publication, and the narrowed sentence is: "this canon claims no lower price than the blockwise evaluation as published; on this device at $s=1024$, at matched exactness, the shape's exact route was measured at or below the control's, and that reading is quoted as one cell with its run_index, seed and git_head, never as an exponent and never against the source's own numbers, which stay [V]-graded and NOT MEASURED here." Hypotheses: the control is the one N-13 implements and not the source's own code, which this canon has not run; both arms at matched exactness with both $\delta$'s printed; the $s=1024$ residency $n$ of 03.5. Evidence: the cell itself, NOT MEASURED — needs N-13 and 03.7's $s=1024$ block; the exactness asymmetry rests on 03.13's bound, which is DERIVED. Kill, its own and different — a different $s$, a different residency $n$ and a different route: the same ratio read at $s=4096$, where 03.5 puts L4 at $n=8$ and L1 at $n=1024$ rather than $s=1024$'s $128$ and $4096$, exceeds $1.00$ — the sign reverses across the two registered $s$, the narrowed sentence has no $s$ at which it holds, and it is withdrawn with the concession restored on the class. Price: the $s=4096$ block of the same sweep, $0$ new GPU-s beyond 03.7's cell count, and more decisive than this verse's own kill because $s=4096$ is the last $s$ this book registers and a reversal there ends the question. If it fires, the Terminal is in force.

Terminal. "On the cost axis the shape claims exactness with a printed certificate and never a lower price than a published blockwise evaluation of the same operator; no accuracy sentence against that route lives in this book at all — the comparison is registered in book 04 with its floor, its $N$, its $\mathrm{MDE}_N$ at the realised $\sigma_d$ and its matched count, or it does not exist (CHARTER §1); that route's $O(n^{4/3}d)$ and its $12$–$29\,\%$ are [V]-graded and are NOT MEASURED in this canon until the owed equation-level read lands, so neither is quoted as a threshold anywhere; the segmentation dividend is a corpus property quoted with the corpus named."

Part E — the device (B2, kernel half)#

03.18 — The device re-certification protocol for the installed stack#

Statement. A certificate is a triple (box, torch build, CUDA runtime) plus a git HEAD; a number from one triple is never pooled with a number from another (V-22). The certified triple is (RTX 4060 Laptop $7.996$ GiB sm_8.9, torch 2.5.1+cu121, CUDA 12.1) at ab5b485. The installed triple, RUN this session, is (the same box, torch 2.14.0+cpu, CUDA None): torch.cuda.is_available() returns False; torchvision 0.20.1+cu121 and torchaudio 2.5.1+cu121 remain from the old build; triton-windows 3.7.1.post27 is installed with no CUDA torch to launch through; the driver is 595.79 (nvidia-smi). No CUDA cell of any kind — no timing, no residency, no determinism reading — can be taken by the installed interpreter. The protocol:

Pin a CUDA build of torch whose runtime the driver supports, and record the triple; the reproduction pin is 2.5.1+cu121 (the certified build); a newer build opens a new certificate slot and starts a new law table, never an amendment of the old one.
Run unchanged python scripts/k_cert.py --out results/k_cert_local_<torch>.json (--out exists, READ scripts/k_cert.py:685; the script appends by box and would overwrite a same-box record, hence the suffixed path) — throughput law with R-squared, memory law re-solved against the allocator, bar re-certification, zero-step gate, determinism in both regimes (READ scripts/k_cert.py:1-52).
Gate on the script's own bars: R-squared $\ge0.99$ per law (MIN_R2, READ scripts/k_cert.py:94), worst bar $\delta/\mathrm{tol}<50\,\%$ (the record read $8.58\,\%$, headroom $5.83\times$, READ COSTS.md:124-136), zero-step gate $30/30$ reachable and passing (READ COSTS.md:138-141), determinism reported in both regimes with three outcomes.
Extend by N-01 (the solve as the fourth determinism quantity) and N-02 (the solve microbenchmark as a producer) — both need code and are priced by book 05; until they land, 03.6's numbers stay prose (P-1).
Must-fire before any reading is believed: tests/gate0/test_g06_kcert.py must collect and its plants must fire — a non-power-law series refused on R-squared (:46), a fabricated slow device moving the chunk table by exactly its factor (:57), a bar row at $60\,\%$ halting (:78), a fabricated non-deterministic reduction reading not bitwise (:116). B1's $13$ collection errors and $0$ tests run at root (CHARTER §5) block this step until book 05's environment pin lands.
Re-base: every [FITTED] price in the canon is re-quoted from the new certificate with its id; the old numbers keep their supersede markers (L-G2).

Hypotheses. The box is the same physical device (nvidia-smi name and $8188$ MiB agree with the certificate's $8{,}585{,}216{,}000$ B); the driver accepts the pinned CUDA runtime ([U] until step 1 runs); CUBLAS_WORKSPACE_CONFIG=:4096:8 set before the first CUDA call; torch.set_num_threads(8).

Evidence. RUN this session: python -c "import torch; print(torch.__version__, torch.version.cuda, torch.cuda.is_available())" → 2.14.0+cpu None False; pip list rows as quoted; nvidia-smi --query-gpu=name,memory.total,driver_version → NVIDIA GeForce RTX 4060 Laptop GPU, 8188 MiB, 595.79. The stale certificate: READ results/k_cert_local.json (torch: 2.5.1+cu121, git.head: ab5b485…, when: 2026-08-31 22:26:39, elapsed_s: 523.9); READ COSTS.md:53-63. The laws it carries: softmax $\exp(-12.1852)n^{0.9963}$ at R-squared $0.999998$ over five points (READ COSTS.md:73); the workhorse arm over three points because two paged over PCIe (READ COSTS.md:76-79). The certificate's own rule that a Kaggle record is appended and never pooled: READ COSTS.md:160-170.

Mechanism. B2 closed as a protocol with its gate: V-22 (triples never pooled), P-1 (a certificate is a results/ file with a producer command), V-15 (the plants of step 5), P-8 (Ruling 8's paged points refused), L-TIME as the script states it (no timing figure inherited, READ scripts/k_cert.py:46-49).

Kill, in two parts, because a gate kill cannot fire on the triple this verse itself registers. The earlier single kill — "any gate of step 3 fails on the installed triple" — is unreachable and is struck (CHARTER §2 reachability rule; V-11, V-16): RUN reproduced this session, python -c "import torch; print(torch.__version__, torch.version.cuda, torch.cuda.is_available())" → 2.14.0+cpu None False, so scripts/k_cert.py on a CPU-only build produces no CUDA throughput law, no bar row, no zero-step CUDA cell and no determinism.device="cuda" reading; there is no draw on which "a gate of step 3 fails" can be decided. Step 5 compounds it: B1's $13$ collection errors and $0$ tests run at root block the must-fire plants.

(a) The unconditional reading on the installed triple, journalled and not a failure. On 2.14.0+cpu every gate of step 3 returns NOT EXECUTABLE — the third of V-16's three outcomes — and the certificate is journalled as k_cert: NOT EXECUTABLE (no CUDA runtime) with the triple, the driver 595.79 and the git HEAD beside it. This is a reading, never a pass and never a kill; it is what the canon knows today. Frozen: any run of scripts/k_cert.py on a triple whose torch.version.cuda is None is journalled NOT EXECUTABLE and its numbers are not entered anywhere. Price $0$ GPU-s (RUN above, already taken). Planted negative: a run that reports a throughput law or a bar row on a CPU-only triple has fabricated it and the certificate is refused.

(b) The gate kill, re-registered against the pinned triple of step 1. Any gate of step 3 fails on the 2.5.1+cu121 triple pinned by step 1 (frozen at the script's own constants: R-squared $<0.99$, $\delta/\mathrm{tol}\ge50\,\%$, a zero-step cell unreachable, or a determinism quantity not executable where the old certificate read bitwise). Instrument: scripts/k_cert.py unchanged; price $523.9$ s [MEASURED] for the full run (READ COSTS.md:62), $\approx9$ GPU-min, plus step 1 itself and step 5's environment repair, both priced by book 05 (B1's pin, B2's re-certification); decidable on torch.randn operands, no bed, no seeds. Planted negative: the four plants of step 5 — a certificate whose plants do not fire is a condemning rule with no negative and is not a certificate.

Step 1 is the named prerequisite of every GPU kill in this book. The kills of 03.1, 03.2, 03.3, 03.4, 03.5, 03.6, 03.7, 03.8, 03.9, 03.10, 03.11, 03.12, 03.14, 03.15, 03.17 and 03.19 — sixteen of this book's nineteen verses — are undecidable until (b)'s triple exists, and each is journalled NOT EXECUTABLE until then rather than pending. The three that are not: 03.13's $1{,}024$-draw float-bound battery and 03.16's $\delta_{\rm union}$ battery are CPU-executable in float64 at $s=64$, and 03.18(a) is this reading itself.

If killed (the pinned triple of (b) fails a gate): reproduce ab5b485's certificate on the same 2.5.1+cu121 build and compare law coefficients rather than gates; Hypotheses: the box's clock is within its $\pm12\,\%$ band of 2026-08-31; Evidence: the old certificate's two laws and their R-squared; Kill: any law's coefficient outside $\pm12\,\%$ of the certified one (frozen from the clock band, READ docs/CEQ_SHAPE.md:2680) — the box is non-stationary across the two dates and every price is quoted as a band spanning both certificates; price the same $523.9$ s, and more decisive (a reproduction either lands or does not).

Terminal. "No CUDA number exists for the installed stack; on 2.14.0+cpu every gate of this protocol reads NOT EXECUTABLE and is journalled as such, which is a reading and not a failure; the only certificate is ab5b485 on torch 2.5.1+cu121, and every price in this book is that certificate's per-op floor or a MAC identity; step 1 is the prerequisite of sixteen of this book's nineteen verses' kills, and nothing in this book is re-measured until a certificate file with its plants fired exists under results/."

Census rows closed#

CHARTER §5 row	what is broken	closing verse(s)
B2 (kernel half)	no certificate exists for the installed stack; the installed torch is `2.14.0+cpu` with no CUDA	03.18
B13	`solve_triangular` bitwise but undocumented; `cumsum` raises; training inherits the backward hole; the control never tested	03.9, 03.10
B14	timers unsynchronised, arms not interleaved, run order $\rho=+0.7029$; every price a per-op floor under a $2.0\times$–$6.6\times$ dispatch gap	03.6, 03.7; the row's band is re-attributed there to the per-row gate settle arm whose FLOP model it measures, and is not carried onto the solve increment (M-8)
B15	no chunked solve, CSR path or Mapper schedule exists; the dense control unrunnable at $n=2048$, $s=4096$ ($137$ GB); the $n=32768$ and $16384$ reproductions dropped	03.1, 03.5, 03.11, 03.12, 03.14, 03.16; the Mapper schedule, B15's third kernel, is closed by 03.19 — before this repair round the word appeared in this table and nowhere else in the book
B25 (per-step half)	"faster to train" never defined or measured; the it.3 ratios withdrawn at C17	03.2, 03.8

Kills, cheapest first#

verse	kill (frozen number)	price	replacement verse / link
03.2	fitted $\tau_{\rm chunk}$ interval contains $0$ (replacement's kill)	$0$ GPU-s from S-66's journal	Terminal of 03.2
03.5	N-18 R-squared $<0.99$ or measured/predicted outside $[0.9,1.1]$ (replacement's kill)	$\approx9$ GPU-min DERIVED	Terminal of 03.5
03.9	a deciding cell's margin below $10\times$ its measured forward floor (replacement's kill)	$0$ GPU-s	Terminal of 03.9
03.9	solve forward $\max	\Delta	>0$ or backward raises under the flag (N-01)
03.10	kernel forward $\max	\Delta	>0$ over 8 repeats; plants F1 (data-dependent tile order) and F2 (`tl.atomic_add` in the row-sum) on the forward, the fast-backward reading relabelled a sensitivity control
03.1 / 03.6	synchronised increment ratio $\rho_\Delta>1.167$ at $(64,2048)$, both arms inside one interleaved block (N-02)	$\approx1$ GPU-min	every price re-derived; Neumann re-enters if a hop reads below the solve
03.12	$\gamma=0$ parity not `torch.equal` against the kernel's own softmax path	seconds	construction error; the single-path schedule with $\gamma$ a runtime scalar, then L2
03.12	$\gamma=0.5$ parity against L4's $z$ at $(64,2048)$ above $2.325\times10^{-5}\\|V\\|_\infty$ — the step-2 rescale defect that kill (i) is blind to by construction	seconds, one launch	the step-2 $A_k$ rescale; else L2
03.11	chunk-vs-serial parity above $\gamma_{s+1}(1+\gamma)\\|V\\|_\infty/(1-\gamma)^2$ ($4.64\times10^{-2}\\|V\\|_\infty$ at $s=4096$, $\gamma=0.9$)	seconds	doubling variant on the plus recurrence, then L4
03.13	float32 single signed solve outside $\gamma_{s+1}(1+\gamma)\\|V\\|_\infty/(1-\gamma)^2$ on 1,024 draws	$\approx2.6$ s at $s=64$	compensated accumulation at $4u(1+\gamma)\\|V\\|_\infty/(1-\gamma)^2$
03.16	one exceedance of $\delta_{\rm union}^{O}\\|V\\|_\infty=(\varepsilon/(1-\gamma)+\gamma^{K+1})\\|V\\|_\infty$ on 1,024 draws, or $\ge\mathrm{sd}$(label) (K-I)	$\approx2.6$ s at $s=64$	L4 at $n$ per $s$
03.12	parity against L4's $z$ at $(4096,64)$ outside $4.641\times10^{-2}\\|V\\|_\infty$ (kill (ii))	seconds	L2, then 03.11's Terminal and L4
03.19	(K-M) Mapper schedule not better than do-nothing by $\mathrm{MDE}_8$ in $\varphi$-NRMSE at matched visited tiles, $s=1024$; or (K-M2) builder above $3.15\times$ the attention it schedules	`NOT MEASURED`; 2 evenings `[ASSUMED]`	the merged salience schedule, killed in turn by K-I; then L4
03.14	relative gradient parity above $\Theta_{\rm rel}=4\gamma_{s+1}\big((1{+}\gamma)/(1{-}\gamma)^2+\kappa_1(1{+}\gamma\pi)\big)$ at $(64,2048)$; NOT DECIDABLE where $\kappa_1(1{+}\gamma\pi)\ge1/(4\gamma_{s+1})$	seconds	scalar float64 central-difference check, $10^{-3}$ on 8 directions
03.15	K-G-adverse: column route $\ge8\times$ one solve at $(2048,64,16)$, $m=8$ (Bet G counter); K-G-flattering: a $\le0.5\times$ reading not repeated at $(1024,128)$ and not separated from the launch floor by the $m\in\{1,8\}$ pair is journalled LAUNCH-FLOOR, NO READING	seconds at $s=64$; `NOT MEASURED` at $s=1024$	$m$ stacked right-hand sides; then $m$ solves
03.5	reserved above the $0.90$ budget $7.196$ GiB $=7.727\times10^9$ B at an admitted $(s,n)$ (N-20); the $7.996$ GiB total is the hard limit, stated separately; plants (P1) L4 at $(4096,16)$ run on the device and (P2) the certificate's $7.504$ GiB cell — the arithmetic refusal is the prediction, not a plant	$\approx3$ GPU-min	constants re-solved (N-18); $n$ by bisection
03.8	$k^\star_{\rm shape}/k^\star_{\rm softmax}\ge0.907$ on BED-S (04.2, which does not admit at the design point, so the kill is journalled NOT EXECUTABLE), seeds $0$–$7$, $k^\star$ at 04.10's ladder level $\tau$; third reading NOT REACHED	$8(k^\star_{\rm shape}\cdot0.01120613+k^\star_{\rm softmax}\cdot0.01016513)+4.0$ s DERIVED, $\ge29.6$ s at the $150$-step budget	T1 against the matched $(32,4,16)$ skyline at $1.143$; then withdrawn
03.18	(a) every gate NOT EXECUTABLE on the installed `2.14.0+cpu` triple, journalled as a reading	$0$ GPU-s (`RUN` this session)	the pin of step 1 (book 05)
03.18	(b) any `k_cert.py` gate fails on the pinned `2.5.1+cu121` triple — the prerequisite of 16 of this book's 19 verses' kills	$523.9$ s plus book 05's environment pin	reproduce `ab5b485` on `2.5.1+cu121`; then bands across certificates
03.14	peak bytes not linear in $s$ (R-squared $<0.99$) or operator term above $0.5$ tensors	$\approx9$ GPU-min	per-chunk checkpoint on L2; then the L4 column is the hard limit
03.4	L4 exponent interval at $n=8$ contains $2$ (the counter's own kill; plant: batched `trsm` at $s=64$, $n=2048$, $nh\gg\mathcal P\ge24$, must read an interval containing $2$)	$\ge1{,}248$ GPU-s DERIVED (03.7)	crossover on the memory axis (03.5)
03.7	$\lvert\rho_{\rm order}\rvert\ge0.25$, $p\le0.05$ on the $52\times8=416$ readings; or `secs_unsync` not below `secs_sync` on 6/8; or R-squared $<0.99$	$\ge1{,}248$ GPU-s DERIVED ($\ge3{,}744$ under the three-process reading)	blocked design with paired differences
03.2	solve-to-$PV$ wall-clock ratio above $3$ at any $s\le1024$ (N-09)	inside S-66	two-term seconds model (03.4)
03.1	L4 exponent interval excludes $2$ toward $3$ (K-9)	$\ge1{,}248$ GPU-s DERIVED (03.7)	L5: the T1 sentence withdrawn
03.3	fused forward above $2\times$ both roofline predictions at $(4096,16,128,1024)$ — $100.7$ ms, twice the $50.33$ ms bytes prediction	seconds, once the kernel exists	additive launch-and-latency model
03.12	fused kernel slower than the matched $(32,4,16)$ fused-softmax stack at $(4096,1024)$	seconds, once the kernel exists	L2
03.11	chunked slower than serial L4 at every $C$ at $s=4096$ (N-11)	minutes once built `[ASSUMED]`	L4 shipped; verse closes NOT NEEDED
03.17	$t_{\rm shape}/t_{\rm blockwise}\le1.00$ at $s=1024$ at matched exactness, both $\delta$'s printed (N-13); the earlier $\mathrm{MDE}_8$ accuracy form is struck (CHARTER §1)	the $s=1024$ block of 03.7 plus N-13's control, `NOT MEASURED`	the concession narrows to the publication; killed in turn by a sign reversal at $s=4096$; then the Terminal

Limits, collected once. Every seconds figure in this book is one of: a per-op floor from one session on torch 2.5.1+cu121 at ab5b485 with no producer in results/ (03.6), a [FITTED] cell price from the same certificate, or NOT MEASURED; the un-synchronised clock that produced the floors has a bias of known sign and no measured magnitude on this arm, the $2.0\times$–$6.6\times$ figure having been returned to the per-row gate settle arm it was measured on (03.1, 03.6), so the only measured factor any threshold in this book is frozen from is the $1.65\times$ session-to-session drift on identical code; 03.5's two device numbers are kept apart throughout — the $0.90$ table budget $7.727\times10^9$ B, which is what the memory kill is frozen at, and the $8.585\times10^9$ B total, which is the hard limit and not a kill; 03.19's Mapper quantiser has no price at any $s$ and its only measured number is the $3.15\times$ gather pair, so B15's third kernel is closed by a verse and not by a measurement; no [ASSUMED linear in depth] skyline price is a threshold anywhere after this repair round, and the $7.6$ s figure survives only as the unmatched control it is. The ridge point of 03.3 rests on two product-specification constants nvidia-smi does not expose. The residual memory constant $17.874$ and the reserved ratio $1.256$ were fitted on a different model and arm and are carried as [ASSUMED] (M-8) until N-18. The chunk-size prediction $C=128$ at $d=16$ is a design choice with a sweep behind it, not a measurement, and it rests on the $100$ KB SRAM budget, which is [ASSUMED, sm_8.9]. The matched-skyline threshold $1.143$ of 03.8 is an arithmetic ratio and not a seconds ratio; it stands only until 03.7 times the matched arm. The backward's $\kappa_1=\|M^{-1}\|_1$ and $\pi=\|P^\top\|_\infty$ are unmeasured at every geometry, so 03.14's tensor-parity kill is decidable today at $s=64$ only. The two zhao-2026-structuredsparse numbers and ChaCAL's diagonal convention are [V]-graded and are NOT MEASURED here pending book 05's bibliography read. Higham's substitution analysis is cited [U] with no bib key and is re-derived in 03.13 so that no sentence rests on the citation; the flash-attention lineage is likewise uncited by key and unrelied upon; hu-2025-ssdtheory no longer carries anything, its statement having been replaced by the DERIVED rank block of 03.1. Fourteen READ pins of this book resolved to lines that did not carry the cited content and are re-resolved against the pinned SHA in batch 2; the citations are correct as they now stand and the fourteen old line numbers are listed with their corrections in "Attacks answered", not silently replaced. After batch 3, six CORRECTIONS rows are owed against files outside this book and all six are named in "Attacks answered": 08_ARCHITECTURE.md:528 (the same minus-sign recurrence); docs/PLAN.md:539-540 (N-02's absolute threshold and its $\pm12\,\%$ PASS band); 08_ARCHITECTURE.md:138, :562 (the bare $10^{-6}$ float32 parity, superseded by 03.13(iv), which a correct float32 route at $s=4096$ misses by $4.6\times10^{4}$); 08_ARCHITECTURE.md:560 (the plant requiring a synchronised re-measurement inside $\pm12\,\%$ of $1.041$ ms, superseded by $\rho_\Delta$ inside one interleaved block); 04_BEDS_AND_INSTRUMENTS.md:170 jointly with this book's 03.7 kill (i), the two books freezing $|\rho|>0.5$ and $|\rho|\ge0.25$ for one confound on two journals of $24$ and $80$ readings, so a reading of $\rho=+0.35$ is condemned here and passed there; and 06_PREDICTIONS.md:403-404 with 08_ARCHITECTURE.md:552, the three books printing $0.010160$, $0.010165$ and $0.010142$ s for the depth-1 softmax step, a $0.227\,\%$ spread, against the fitted law's own evaluation $\exp(-12.1852)\cdot2048^{0.9963}=0.01016513$ s which this book now quotes alone. Further limits this round exposes. 03.5's two byte-model constants have two provenances and are cited apart: $17.874$ is [FITTED] at R-squared $0.996373$ (COSTS.md:89) and $1.256$ is a worst-case reserved/allocated ratio over one box's resident shapes (COSTS.md:223) whose own source applies it to a T4 budget, so it is [ASSUMED] here for a V-22 reason and not a fitting reason. The only device-concurrency figure this book asserts is the lower bound $\mathcal P\ge24$ from $3072/128$ SMs, [ASSUMED, product specification] on both counts, and it is asserted to forbid the inference "$\mathcal P\le8$" that 03.4's kill once drew from an exponent, never to license one. The Neumann arm's two-term reading — fixed $a=1.5071$ ms, marginal $b=1.4939$ ms per hop — is fitted on two points, $K\in\{1,16\}$, so it certifies no linearity and 03.16's $K\ge131$ regime is $8.2\times$ outside its range and is NOT MEASURED. 03.16's tail is $\gamma^{K+1}$ for the shipped $O$ read and $\gamma^{K+1}/(1-\gamma)$ for a bare $z$; the loose form applied to $O$ overstated the hop count by $16\,\%$ ($152$ against $131$ at $\gamma=0.9$, $\delta=10^{-6}$) and the "uncompetitive by arithmetic" sentence is re-stated on the tight number, where it survives by two orders. 03.8's kill has no admitted bed: BED-S does not admit at the design point (04.2's Terminal), so the T1 kill is journalled NOT EXECUTABLE, its price scales linearly in an unmeasured $k^\star$ from a $\ge29.6$ s floor at the $150$-step budget, and its third reading is NOT REACHED. 03.17 carries no accuracy kill at all after this round: the $\mathrm{MDE}_8$ $\varphi$-NRMSE form was outside A1/A2, against a comparator that is neither base self-attention nor the skyline, with no floor, no $N$, no matched count and no admitted bed, and the contrast is relocated to book 04 or does not exist. The evidence grep of 03.6 failed before this round — --include=*.txt --include=*.json excluded the $57$ *.jsonl files of a $172$-entry directory, so its silence proved nothing — and the conclusion now rests on the unfiltered fixed-string re-run, whose single hit is a causal.tau entry at $s=128$, $d=8$ carrying no timing field. No code was written, no git write was made, no external page was fetched, and no file but this one was edited; the installed interpreter has no CUDA, so nothing here was re-run on the device.

Attacks answered#

MARS · MORIARTY, repair round 1, batch 1 of 4. Every finding below is reproduced verbatim — verse, flaw, mechanism, number, severity — with the repair beside it. Nothing is softened; where a finding reads gentler after repair than when filed, that is itself a defect (CHARTER.md §7).

A-01 · verse 03.11 · severity strike · mechanism P-10, V-9, L-VERSE

Flaw (verbatim). The central block recurrence is displayed with the wrong sign; it contradicts the verse's own DERIVED justification line and 03.13(iii).

Number (verbatim). $Mz=V$ with $M_{kl}=-\gamma P_{kl}$ gives $M_{kk}z_k=V_k+\gamma\sum_{l<k}P_{kl}z_l$; the verse displays a minus. Counterexample: at $\gamma=0.6$, $V=\mathbb 1$, the displayed recurrence returns row sums $1/(1+0.6)=0.625$ against the RUN-verified $1/(1-0.6)=2.5$ (READ docs/CEQ_SHAPE.md:314, RUN[J],RUN[M]) — a factor $4.0$. The verse's own line reads M_{kk}z_k+\sum_{l<k}(-\gamma P_{kl})z_l=V_k, and 03.13(iii) writes the correct $r_i-\sum_{j<i}(-\gamma P_{ij})z_j$ adds non-negative quantities. Propagates to 03.12 step 3 ($r_k=V_k-\gamma A_k/\ell_k$), 03.14 step 2, and 08_ARCHITECTURE.md:472.

Replacement survived the same attack. No.

Repair applied. 03.11 Statement: the display is now $M_{kk}z_k=V_k+\gamma\sum_{l<k}P_{kl}z_l$, the transposition step is written out, the struck minus is named as struck, and MARS's $\gamma=0.6$ counterexample is printed in the verse as the DERIVED check ($2.5$ against $0.625$, factor $4.0$). 03.12 step 3: $r_k=V_k+\gamma A_k/\ell_k$, with the $\gamma=0$ parity clause re-derived as $+0\cdot x=+0.0$. 03.14 step 2: $M_{kk}^\top\bar V_k=\bar z_k+\gamma\sum_{l>k}P_{lk}^\top\bar V_l$. 03.1 If-killed, L1-killed bullet: the recurrence re-displayed with the plus. 03.11 If-killed: the doubling replacement no longer inherits the recurrence by reference — it is re-stated in full as $z_k=N_{kk}(V_k+\gamma\sum_{l<k}P_{kl}z_l)$ with $N_{kk}$ written out, so the replacement carries the corrected sign on the page. 03.11 Terminal re-stated with the plus. Outside this book, and owed to the coordinator: the same minus-sign recurrence stands at 08_ARCHITECTURE.md:528 (MARS filed it as :472; :472 is 08.19's inference-API heading in the current file and :528 is 08.16's chunked-solve replacement, which reads $M_{kk}z_k=\tilde v_k-\gamma\sum_{l<k}P_{kl}z_l$). A CORRECTIONS row against 08_ARCHITECTURE.md:528 is owed and is not written here — this repairer edits only book 03 and makes no git write.

A-02 · verse 03.13 · severity strike · mechanism P-10 (theorem cited beyond its hypotheses), V-9, V-17

Flaw (verbatim). Bound (iv) is proved for an algorithm that solves $V^+$ and $V^-$ separately and subtracts, then applied as the frozen kill threshold for the single signed solve that every shipped route (L1, L2, L4) actually performs.

Number (verbatim). For one signed forward substitution the standard triangular result is $|\hat z-z|\le\gamma_{s+1}|M^{-1}||M||z|$, giving $\|\hat z-z\|_\infty\le\gamma_{s+1}(1+\gamma)\|V\|_\infty/(1-\gamma)^2$. Ratio to the printed $2\gamma_{s+1}\|V\|_\infty/(1-\gamma)$ is $(1+\gamma)/(2(1-\gamma))=9.50$ at $\gamma=0.9$. A correct float32 route at $s=4096$ can read $4.64\times10^{-2}\|V\|_\infty$ against the frozen $4.89\times10^{-3}\|V\|_\infty$ and fire the kills of 03.11(i), 03.13 and 03.14(i).

Replacement survived the same attack. No.

Repair applied. 03.13(iv) is now the single-signed-solve bound $\gamma_{s+1}(1+\gamma)\|V\|_\infty/(1-\gamma)^2$, boxed, with the three factors named ($\|M^{-1}\|_\infty=1/(1-\gamma)$, $\||M|\|_\infty\le1+\gamma$, $\|z\|_\infty\le\|V\|_\infty/(1-\gamma)$) and the numbers re-read: float32 $7.36\times10^{-4}$, $2.91\times10^{-3}$, $1.16\times10^{-2}$, $4.64\times10^{-2}\;\|V\|_\infty$ at $s=64,256,1024,4096$; float64 $1.37\times10^{-12}$ to $8.64\times10^{-11}$; bfloat16 $64.7\,\|V\|_\infty$ at $s=64$. The two-solve bound survives as (iv′) with the $V^\pm$ split stated as its hypothesis and the sentence that no shipped route satisfies it, so (iv′) is a threshold nowhere. The $9.50$ ratio and the $4.64\times10^{-2}$-against-$4.89\times10^{-3}$ firing are printed in the verse. Re-frozen on (iv): 03.11 kill (i), 03.13's kill, 03.12 kills (i-b) and (ii), and 03.14's threshold (which is separately rebuilt, A-03/A-04). The compensated-accumulation replacement is re-derived under (iv), not (iv′): $\|\hat z-z\|_\infty\le(2u+O(su^2))(1+\gamma)\|V\|_\infty/(1-\gamma)^2$ with its kill at $4u(1+\gamma)\|V\|_\infty/(1-\gamma)^2=4.53\times10^{-5}\|V\|_\infty$, $1.02\times10^{3}$ times tighter than the verse's own. The float64 numeric instance is re-read against the looser bound and its margin re-stated as $7.7\times10^{2}$ with the sentence that it certifies less than the earlier $10\times$ appeared to.

A-03 · verse 03.14 · severity strike · mechanism P-10, V-25

Flaw (verbatim). Step (2) transfers 03.13's row-diagonal-dominance analysis to $M^\top$; $M^\top$ is column diagonally dominant, not row diagonally dominant, and fails row dominance outright under the canon's declared BOS sink.

Number (verbatim). Row dominance of $M^\top$ at row $i$: $(1-\gamma P_{ii})-\gamma\sum_{j>i}P_{ji}$, the sub-diagonal column sum of $P$. With a declared sink where every row places $P_{j0}\ge0.5$ on column 0, at $i=0$, $\gamma=0.9$: $1-0.9P_{00}-0.9\cdot0.5(s-1)<0$ for every $s\ge5$. 03.13's M-matrix and no-cancellation argument therefore does not apply to the backward's solve at any $s$ in $\{64,256,1024,4096\}$.

Replacement survived the same attack. No.

Repair applied. 03.14 step (2) no longer says "same bound"; it says the bound is different and points at a new DERIVED block in the Statement. That block prints MARS's dominance defect verbatim in symbols, states that $M^\top$ fails row dominance at every registered $s$ under the declared sink, and derives the bound from the column-side quantities instead: $\|\hat{\bar V}-\bar V\|_\infty\le\gamma_{s+1}\kappa_1(1+\gamma\pi)\|\bar V\|_\infty$ with $\kappa_1:=\|M^{-1}\|_1=\|M^{-\top}\|_\infty\in[1/(1-\gamma),\,s/(1-\gamma)]$ (the range derived from 03.13(ii)'s exact row sums) and $\pi:=\|P^\top\|_\infty\in[1,s]$, both measured per cell and printed as certificate lines beside $\delta_{\rm float}$, and with the no-cancellation clause explicitly withdrawn for a signed $\bar z$ on a non-M-matrix. The restriction option MARS offered is not taken: the declared BOS sink is kept in force and named in the Hypotheses, with the manifest fields P_col_sum_max and Minv_col_sum_max added. A genuinely different replacement is derived for kill (i), since the per-chunk-checkpoint replacement addresses memory only: a scalar float64 central-difference directional-derivative check on the L4 forward — a different object (one scalar per direction, not four adjoint tensors), a different instrument (a difference quotient that never runs the fused backward), a different number ($10^{-3}$ relative on any of $8$ unit directions, against a difference-quotient floor of $2.2\times10^{-11}$ roundoff plus $O(10^{-10})$ truncation at $h=10^{-5}$) — and it needs neither $\kappa_1$, nor $\pi$, nor any triangular-solve bound.

A-04 · verse 03.14 · severity strike · mechanism V-17 (threshold imported out of its units), M-3

Flaw (verbatim). The frozen gradient-parity threshold omits $\|P^\top\|_\infty$, the maximum column sum of $P$, which the declared sink makes $\Theta(s)$.

Number (verbatim). $\bar z=(1-\gamma)P^\top\bar O$ gives $\|\bar z\|_\infty\le(1-\gamma)\|P^\top\|_\infty\|\bar O\|_\infty$, and $\|P^\top\|_\infty=\max_j\sum_iP_{ij}$ is $\Theta(s)$ on a sink column, not $1$. At the kill's registered $(s,n)=(1024,128)$ the printed $4\gamma_{s+1}\|\bar O\|_\infty\|V\|_\infty/(1-\gamma)^2$ understates a correct backward by up to $s=1024$, so the kill fires on a correct implementation.

Replacement survived the same attack. No.

Repair applied. The old threshold is struck in the verse, by name, on both counts (wrong dominance class and missing $\pi$), with MARS's "understates by up to $s=1024$ and fires on a correct implementation" restated in the verse. The frozen kill is now relative and carries both measured factors: $\Theta_{\rm rel}=4\gamma_{s+1}\big((1+\gamma)/(1-\gamma)^2+\kappa_1(1+\gamma\pi)\big)$, with the factor $4$ derived (four places a relative solve error enters) and $\kappa_1,\pi$ printed per cell beside $\delta_{\rm float}$ as MARS requires. Reachability is now stated rather than assumed: the kill is decidable only where $\kappa_1(1+\gamma\pi)<1/(4\gamma_{s+1})$ — $6.45\times10^{4}$, $1.63\times10^{4}$, $4.09\times10^{3}$, $1.02\times10^{3}$ at $s=64,256,1024,4096$ in float32 — and under the declared sink the worst case is $3.75\times10^{4}$ at $s=64$ (decidable) against $5.92\times10^{5}$ at $s=256$ (not). The kill is therefore re-registered at $(s,n)=(64,2048)$, not $(1024,128)$, and at $(1024,128)$ the cell prints its measured $\kappa_1,\pi$ and is journalled NOT DECIDABLE — V-16's third outcome, never a pass. The planted negative is restated in relative units.

A-05 · verse 03.1 · severity strike · mechanism M-3, P-8

Flaw (verbatim). The MAC increment column double-counts $sCd/2$; exact block forward substitution costs the same $s^2d/2$ as dense substitution and is independent of $C$.

Number (verbatim). Off-diagonal accumulation is $C^2dK(K-1)/2=s^2d/2-sCd/2$ exactly (not $\approx s^2d/2$); the diagonal solves add $K\cdot C^2d/2=sCd/2$; the sum is exactly $s^2d/2$. The printed $+s^2d/2+sCd/2$ over-counts by $sCd/2$ ($=2.10\times10^6$ MACs at $s=4096$, $C=64$, $d=16$) and contradicts 03.2's total of $\tfrac32 s^2d$ for the shape.

Replacement survived the same attack. Yes.

Repair applied. 03.1's ladder table: the L1 and L2 increments now read $+s^2d/2$ (exact, and independent of $C$), matching L4's. 03.11's Statement: the off-diagonal term is printed as the exact $s^2d/2-sCd/2$, the diagonal term as $sCd/2$, and the total as exactly $s^2d/2$ with the over-count of $2.10\times10^6$ MACs at $(4096,64,16)$ named and struck. 03.11's chunk-length sentence "total MACs $s^2d/2+sCd/2$ favour small $C$" is struck by name, and the trade-off is re-derived as chunk-step count $s/C$ against SRAM alone, with the observation that the total serial row-step count $K\cdot C=s$ is $C$-independent too — so nothing but SRAM opposes a large $C$. 03.11's Terminal and 03.11's Evidence (the record's own $+sCd/2$) are corrected with the two-line derivation printed.

A-06 · verse 03.3 · severity strike · mechanism P-1, M-3

Flaw (verbatim). The SRAM budget is computed with two different chunk lengths in the same expression; corrected, the design choice 03.11 derives from it is false.

Number (verbatim). At the stated $B_M=128$, $C=64$, $d=16$: $4(2B_Md+2Cd+B_MC)=4(4096+2048+8192)=57{,}344$ B $=57$ KB, not the printed $4(4096+2048+16384)=90$ KB, which uses $B_MC=128\times128$. At $B_M=128$, $C=128$: $4(4096+4096+16384)=98{,}304$ B $=98.3$ KB, inside the 100 KB budget — so 03.11's '$C=64$ is the largest $C$ whose $B_M\times C$ float32 score tile with $B_M=128$ fits 03.3's SRAM budget' is false; $C=128$ fits.

Replacement survived the same attack. Yes.

Repair applied. 03.3's Hypotheses now carry the budget as an inequality evaluated at one $C$ throughout, name the mixed reading as struck, and print both corrected values ($57{,}344$ B at $(128,64,16)$; $98{,}304$ B at $(128,128,16)$). The general solution is printed: $16{,}384+640C\le100{,}000$, i.e. $C\le130$ at $d=16$, $B_M=128$; and $4(4Cd+C^2)\le100{,}000$ under the solve stage's $B_M=C$. 03.11's prediction is re-derived and changed from $C=64$ to $C=128$ at $d=16$ ($98.3$ KB, $I_{\rm fused}=64$ FLOP/B), with $C=64$ at $d\in\{32,64\}$ and $C=32$ at $d=128$; the false sentence is quoted and struck in 03.11. 03.12's launch-count estimate is re-run at the new $C$ ($K=32$, $128$ launches, $1.3$ ms at $s=4096$, and $5.1$ ms at $C=32$ — both below the $43$ ms arithmetic floor, so the conclusion survives at every admissible $C$).

A-07 · verse 03.3 · severity strike · mechanism V-10 (a Hypothesis satisfied by nothing), P-4

Flaw (verbatim). The pair declared admissible at $d=128$ exceeds the verse's own SRAM budget, and no admissible pair is given for the largest declared head dimension.

Number (verbatim). $B_M=64$, $C=32$, $d=128$: $4(2\cdot64\cdot128+2\cdot32\cdot128+64\cdot32)=4(16384+8192+2048)=106{,}496$ B $=106.5$ KB against the declared 100 KB [ASSUMED, sm_8.9] budget. The printed $d=128$ figure $4(32768+16384+16384)=262$ KB again mixes $C$; at $B_M=128$, $C=64$ the correct figure is $4(32768+16384+8192)=229$ KB.

Replacement survived the same attack. Yes.

Repair applied. Both mixed figures are quoted and struck in 03.3's Hypotheses, and the $106{,}496$ B $=106.5$ KB reading of the declared pair is printed against the $100$ KB budget as the V-10 defect it is. The inequality is solved at $d=128$ and the admissible pairs printed: $(B_M,C)=(32,32)$ at $69{,}632$ B $=69.6$ KB under the solve stage's $B_M=C$, and $(64,16)$ at $86{,}016$ B $=86.0$ KB where that constraint is relaxed for a softmax-only read; both memory-bound. $d=128$ is not struck from 03.1's admissible list; instead 03.1's Hypotheses now state that L1's admissibility is narrower than the tl.dot list and name the pair per $d$, with $d=128$ admissible for L1 only at $C=32$ and memory-bound by $4.66\times$. A full admissible table across $d\in\{16,32,64,128\}$ is printed in 03.3's Statement.

A-08 · verse 03.12 · severity strike · mechanism P-4 (scaffolding that does not exist), V-3, V-10

Flaw (verbatim). The one-pass schedule never rescales the accumulator $A_k$ when step 2's diagonal tile raises the running maximum $m_k$; the right-hand side and the read are then wrong by that factor at every $\gamma>0$, and the verse's own $\gamma=0$ parity kill cannot see it.

Number (verbatim). Step 1 rescales $A_k$ by $e^{m_{\rm old}-m_k}$ inside the off-diagonal loop only. Step 2 takes 'final $m_k,\ell_k$ after including it'. If the causal diagonal tile contains the row maximum — the common case, since $S_{ii}=q_i\!\cdot\!k_i/\sqrt d$ lies in every row's diagonal tile — then $A_k$ carries $m_{\rm old}$ while $\ell_k$ carries $m_{\rm new}$, and step 3's $A_k/\ell_k$ and step 5's $O_k$ are off by $e^{m_{\rm new}-m_{\rm old}}$. At $\gamma=0$ the term is multiplied by zero, so kill (i) reads torch.equal True on a wrong kernel.

Replacement survived the same attack. No.

Repair applied. 03.12 step 2 is rewritten as a numbered rescale step: compute $S_{kk}$, form $m_k^{\rm new}$, rescale both $\ell_k$ and $A_k$ by $e^{m_k-m_k^{\rm new}}$ before anything else, then update $m_k$, $\ell_k$ and $\tilde P_{kk}$. The rescale is stated as not optional, and MARS's mechanism is printed in the verse: the diagonal tile holds the row maximum in the common case because $S_{ii}$ lies in it, so the old schedule was wrong by $e^{m^{\rm new}-m^{\rm old}}$ at every $\gamma>0$. A new kill (i-b) is added at exactly the geometry MARS specifies — $\gamma=0.5$ against L4's $z$ at $(s,n)=(64,2048)$, frozen at 03.13(iv)'s corrected bound $\gamma_{65}(1+\gamma)\|V\|_\infty/(1-\gamma)^2=2.325\times10^{-5}\|V\|_\infty$, one exceedance — with its own planted negative (the kernel with the step-2 rescale removed must exceed it on $\ge1{,}000$ of $1{,}024$ draws). The verse now states in terms that kill (i) is blind to this defect by construction (V-3, V-10), so kill (i-b) exists for it and no other. The new kill is added to the cheapest-first table.

A-09 · verse 03.12 · severity strike · mechanism M-3, V-22

Flaw (verbatim). The fused schedule fixes $B_M=C$ in the solve stage, which contradicts the arithmetic intensity 03.3 headlines for the same route.

Number (verbatim). 03.12 Statement: '$B_M=C$ in the solve stage'; 03.11 predicts $C=64$; hence $I_{\rm fused}=B_M/2=32$ FLOP/B, $43\,\%$ of 03.3's ridge $74.5$ FLOP/B — memory-bound. 03.3's headline '$64$ FLOP/B at $B_M=128$ ... within $14\,\%$ of it at $B_M=128$' is computed at a $B_M$ the fused solve stage does not admit. The two verses disagree by a factor $2$ on the intensity of the same route.

Replacement survived the same attack. Yes.

Repair applied. MARS's first option is taken and the two verses are put on one $B_M$: 03.3's fused intensity is now read at $B_M=C$, the constraint the solve stage imposes, and the Statement carries a table of admissible pairs with their intensities — $(128,128)$ at $d=16$ giving $64$ FLOP/B and $85.9\,\%$ of the ridge, $(64,64)$ at $d\in\{32,64\}$ giving $32$ FLOP/B, $(32,32)$ at $d=128$ giving $16$ FLOP/B. $B_M=256$, the smallest tile that reaches the ridge, is shown inadmissible at every $d$ ($327{,}680$ B at $d=16$). The "within $14\,\%$" sentence is not withdrawn but re-scoped: it is a $d=16$ sentence at $(B_M,C)=(128,128)$ and carries that pair in every quotation; at $d\ge32$ the fused route is memory-bound by $2.3\times$ to $4.7\times$ and the verse says so. 03.12's Statement now names $B_M=C$ as the constraint that decides 03.3's reported intensity, lists the same pairs, and states that the softmax-only read's $B_M=2C$ is priced separately and never used to quote this route's intensity. The disagreement MARS measured at $C=64$ ($32$ FLOP/B against $64$) is resolved by the corrected SRAM budget of A-06, which admits $C=128$; the resolution is arithmetic, not a re-wording.

A-10 · verse 03.1 · severity strike · mechanism V-11 (a precondition satisfied at every real draw), M-2, P-8

Flaw (verbatim). The frozen kill threshold lies inside the bias band the book itself attaches to the reference number, so a faithful synchronised re-measurement fires it regardless of the shape's cost behaviour.

Number (verbatim). Kill frozen at $2.2\times1.041=2.290$ ms. 03.6 declares the same $1.041$ ms a floor 'under a $2.0\times$–$6.6\times$ dispatch gap'. $2.0\le2.2\le6.6$: the kill sits inside the declared band. The record's own N-02 card meanwhile sets PASS at $\pm12\,\%$ of $1.041$ ms (READ docs/PLAN.md:540), which is unreachable if the gap is real. The same threshold is 03.6's kill and 08_ARCHITECTURE.md:150's kill.

Replacement survived the same attack. No.

Repair applied. 03.1's kill is re-frozen against the synchronised re-measurement itself, as a ratio inside one interleaved block: $\rho_\Delta=(t_{\rm solve+Pz}-t_{PV})/t_{PV}$ with both arms synchronised in the same block, which is invariant to first order under an overhead common to the two arms. Frozen at $\rho_\Delta>2.2\times0.707=1.555$ at $(s,n)=(64,2048)$. The reference is quoted as the un-synchronised floor it is, at all three $n$ ($0.707$, $0.985$, $0.845$ — a spread of $1.39\times$, which is why the threshold is registered per $n$ and never pooled), and $1.041$ ms appears nowhere as a synchronised expectation. MARS's containment — $2.0\le2.2\le6.6$ — is printed in the verse as the reason the absolute threshold is struck. The $\pm12\,\%$ PASS band is reconciled by refusing it as written: $\pm12\,\%$ is the clock's non-stationarity band between two readings under the same synchronisation regime and cannot bracket a reading taken across the un-synchronised-to-synchronised change under a declared $2.0\times$–$6.6\times$ gap; it applies to $\rho_\Delta$ between two synchronised sessions and to nothing else. A CORRECTIONS row against docs/PLAN.md:539-540 is owed — that file is outside this book and is not edited here. The 08_ARCHITECTURE.md:150 half of the finding is likewise owed to the coordinator; the current :150 is 08.7's Mechanism line, and the kill carrying this threshold is elsewhere in that book, which this repairer does not edit.

A-11 · verse 03.18 · severity strike · mechanism CHARTER §2 reachability rule; V-11, V-16

Flaw (verbatim). The kill cannot fire on the triple the verse itself registers, and every GPU kill in the book is gated behind it.

Number (verbatim). RUN reproduced this session: python -c "import torch; print(torch.__version__, torch.version.cuda, torch.cuda.is_available())" → 2.14.0+cpu None False. scripts/k_cert.py cannot reach a gate on a CPU-only build — it produces no CUDA throughput law, no bar row, no zero-step CUDA cell and no determinism.device="cuda" reading, so 'a gate of step 3 fails' has no draw. Step 5 further states B1's $13$ collection errors block the must-fire tests. 15 of the book's 18 verses carry kills that need this certificate.

Replacement survived the same attack. Yes.

Repair applied. The RUN was reproduced again this session and returns 2.14.0+cpu None False. The single kill is struck and replaced by exactly the two parts MARS specifies. (a) An unconditional NOT EXECUTABLE reading on the installed triple — V-16's third outcome — journalled as k_cert: NOT EXECUTABLE (no CUDA runtime) with the triple, driver 595.79 and git HEAD, at $0$ GPU-s, explicitly not a failure and explicitly not a pass; its planted negative is that a run reporting a throughput law or a bar row on a CPU-only triple has fabricated it. (b) The gate kill re-registered against the pinned 2.5.1+cu121 triple of step 1, at the script's own constants, priced at $523.9$ s plus step 1 and step 5's environment repair, both priced by book 05. Step 1 is named in the verse as the prerequisite of every other GPU kill in this book, with the fifteen verses listed by id and the three exceptions named (03.13's and 03.16's batteries are CPU-executable in float64 at $s=64$; 03.18(a) is the reading itself). The Terminal and the cheapest-first table carry both parts.

A-12 · verse 03.8 · severity strike · mechanism M-2, V-17, P-7

Flaw (verbatim). The verse attributes to book 04 a distance-to-floor definition book 04 does not carry, so the two frozen T1 thresholds have no single referent.

Number (verbatim). 03.8: 'book 04 sets $\varepsilon_N=\mathrm{MDE}_N$ at the realised paired sd'. 04_BEDS_AND_INSTRUMENTS.md:147 sets $\tau$ from 'the ladder $\{0.9,0.8,0.7,0.5\}$ in NRMSE-to-oracle units, the first level both arms reach within the step budget'. Two born canon books define $k^\star$ on two different targets; $0.907$ and $4.52$ are frozen against the first and measured against the second.

Replacement survived the same attack. No.

Repair applied. MARS's first option is taken: 04.10's ladder is adopted verbatim in 03.8 — $\tau$ from $\{0.9,0.8,0.7,0.5\}$ in NRMSE-to-oracle units, registered before the run, the first level both arms reach inside the step budget. $k^\star$ is redefined on it, the earlier "$\varepsilon_N=\mathrm{MDE}_N$" attribution is quoted and struck in the verse with the reason ($\mathrm{MDE}_N$ is 04.9 clause 4's contrast bar for A1/A2, not 04.10's T1 target), and both thresholds are re-frozen against the adopted target. The kill's instrument moves from dist_k to 04's journalled eval_trace field (READ docs/canon/04_BEDS_AND_INSTRUMENTS.md:233). The Hypotheses now carry 04.10's own NOT REACHED clause. The governing line is :165, not :147: :147 is 04.9 clause 3; the discrepancy is recorded in 03.8's Evidence rather than silently corrected. No CORRECTIONS row is needed — book 03 moved to book 04's definition, not the reverse.

A-13 · verse 03.8 · severity strike · mechanism Ruling 3, R-SKY, D-1

Flaw (verbatim). The depth-5 skyline denominator is the unmatched stack; CHARTER §1 requires T1 at matched parameters and 03.2's own matched skyline shrinks the width.

Number (verbatim). $7.6$ s $=5\times1.524$ s, i.e. five unmatched depth-1 cells (READ docs/CEQ_SHAPE.md:2298, [ASSUMED linear in depth]). Under 03.2's own matching ($L\cdot4d_{\rm m}'^2=4d_{\rm m}^2$, $d_{\rm m}'=d_{\rm m}/\sqrt L$) the skyline's attention MACs are $\sqrt L\,s^2d_{\rm m}=2.236\,s^2d_{\rm m}$, not $5\,s^2d_{\rm m}$. The threshold $4.52=1/0.221$ is therefore frozen from a denominator the canon's own matching rule forbids.

Replacement survived the same attack. No.

Repair applied. $4.52$ is struck as a threshold. It survives in 03.8 and in 03.2's Evidence only as the unmatched control it is, with $7.6=5\times1.524$ written out and labelled [ASSUMED linear in depth], and with the sentence that it is $3.96\times$ more permissive than the matched threshold — the size of the flattery Ruling 3 removes. The second denominator is re-derived at the matched pair of A-14, $(d_{\rm m},L,d_{\rm m}')=(32,4,16)$, and its seconds price is marked NOT MEASURED — needs 03.7's sweep with the matched skyline arm. The threshold is frozen meanwhile from the arithmetic ratio, printed with its geometry and labelled as arithmetic (M-3): $\mathrm{MAC}_{\rm shape}/\mathrm{MAC}_{\rm sky,matched}=0.875$ at $s=64$ and $0.754$ at $s=4096$, so T1 against the matched skyline holds iff $k^\star_{\rm shape}/k^\star_{\rm sky}<1.143$ at $s=64$ ($1.327$ at $s=4096$), re-frozen once from the measured ratio under a supersede marker (L-G2). 03.7 row 2's skyline arm is changed from the depth-5 unmatched stack to the matched $(32,4,16)$ arm, and 03.12 kill (iii) with it.

A-14 · verse 03.2 · severity strike · mechanism V-10, P-4, V-22

Flaw (verbatim). The matched-parameter skyline requires a head dimension the book's own Hypotheses declare inadmissible on this stack.

Number (verbatim). $d_{\rm m}'=d_{\rm m}/\sqrt L=16/\sqrt5=7.155$. 03.1's Hypotheses: '$d\in\{16,32,64,128\}$ (the only head dimensions a tl.dot tile admits on this stack, READ ceq/mz_kernel.py:41)', verified: _POW2_HEAD_DIMS = (16, 32, 64, 128). The A2/T1 denominator 03.2 constructs cannot be built or timed on the certified device at the record's geometry.

Replacement survived the same attack. Yes.

Repair applied. MARS's first option is taken. 03.2's matched skyline is re-registered at the depth-and-width pair that lands on an admissible $d$: $(d_{\rm m},L,d_{\rm m}')=(32,4,16)$, exactly matched at $L\cdot4d_{\rm m}'^2=4\cdot4\cdot256=4096=4\cdot32^2$, with $L=4=\lfloor\log_2 4\rfloor+2$ the skyline depth at $t^\star=4$ (READ MATHEMATICS.md:284-296). The $7.155$ is quoted in the verse, named inadmissible against the verified _POW2_HEAD_DIMS, and forbidden as a matched denominator anywhere in this book or in book 04's A2/T1 rows by a clause added to 03.2's Hypotheses. The matched ratios are re-read at the new pair: attention-only $3/(2\sqrt L)=0.750$; with projections $0.875$ at $s=64$ and $0.754$ at $s=4096$. 03.2's Terminal, 03.7's arm list, 03.8's second denominator and 03.12's price kill all move to it.

A-15 · verses 03.1 / 03.11 / 03.17 · severity strike · mechanism L-EQ, P-10, CHARTER §2 Number rule

Flaw (verbatim). Two load-bearing statements rest on bare [V] citations with no theorem, hypotheses and numeric instance and no DERIVED block.

Number (verbatim). (a) 'the reason no state-space dual shortens L1 or L2 below $O(s^2d)$: CITED [V] hu-2025-ssdtheory' (03.1) and 'Why no state carries across chunks: CITED [V] hu-2025-ssdtheory' (03.11) — this closes the route ladder at four links and is the whole reason a linear-attention route is absent; docs/sources/sweep/sweep_linrec.md:25-27 marks it CITED [V], arXiv:2510.04944, with no equation grade. (b) 03.17's Statement, Terminal and cost-axis concession rest on CITED [V] zhao-2026-structuredsparse, $O(n^{4/3}d)$ and $12$–$29\,\%$ (docs/sources/sweep/sweep_resolvent.md:218).

Replacement survived the same attack. No.

Repair applied. (a) replaced by a DERIVED block in 03.1's Evidence, three numbered lines, resting on no citation: (1) $P=D^{-1}(E\odot L)$ with $E\odot L$ lower-triangular and positive-diagonal, hence nonsingular, so $\operatorname{rank}(P)=s$ exactly, for every $Q,K$, every $d$ and every $s$ — a stronger statement than the cited one and free of the "generically" the citation needs; (2) the off-diagonal block's Hadamard-power rank bound $\operatorname{rank}(A^{\circ t})\le\binom{r+t-1}{t}$, giving $1$, $17$, $153$ at $r=d=16$ and $T=0,1,2$, so a running state of width $r<C$ forces $T\le1$ at every $C\ge32$; (3) the numeric instance at this book's geometry — at $d=16$, $C=128$, logits of order $3$, the rank-$17$ truncation misses by $e^3-4=16.1$ against entries bounded by $e^3=20.1$, an $80\,\%$ miss. hu-2025-ssdtheory is demoted in both 03.1 and 03.11 to a concurrence that carries nothing. (b): 03.17's $O(n^{4/3}d)$ and $12$–$29\,\%$ are marked NOT MEASURED — needs the equation-level read, owed to book 05's bibliography repair, $0$ GPU-s, one evening and appear nowhere as a threshold, a comparison or a price — including in 03.1's Evidence, where the exponent is struck from the ladder discussion. What is kept is the concession itself, made on the class and not on a figure: no sentence of this book claims a lower price than a published blockwise evaluation of the same operator at $s\ge1024$, whatever its exponent and margin turn out to be — a form an owed read can only relax and never tighten. 03.17's Statement, Evidence, Mechanism (L-EQ added), If-killed and Terminal are rewritten on that footing.

A-16 · verse 03.7 · severity strike · mechanism P-8, M-8, P-1

Flaw (verbatim). The price is the record's band for four harness edits, quoted as the price of a 240-run timed sweep whose own row 1 demands more timed work than the band's top.

Number (verbatim). V20_R15_THEORY_TABLE.md:184 @ 99777ab and V20_R15_IT13_MERCURY.md:146 @ 99777ab price $206$–$537$ GPU-s as 'instrument edits: 4 — one argparse line, one substitution at :137, two cuda.synchronize()'. Row 1 requires $\ge3.0$ s of timed work per cell and a median over $3$ child processes; row 2 gives $27$ (arm,$s$,$n$) combinations $\times\,8$ repeats $=216$ runs, plus $24$ at $s=4096$ and a $12$-point occupancy row. Minimum timed work: $240\times3.0=720$ GPU-s at one process per run, $2{,}160$ GPU-s at three — $1.3\times$ to $4.0\times$ the band's top, before warm-ups, compilation and the $s=4096$ cells. The verse's own Evidence concedes 'row 4's additional $n$ ladder adds at most $3\times$ the cells' and keeps the band unchanged.

Replacement survived the same attack. Yes.

Repair applied. The sweep is priced from rows 1, 2 and 4 as a DERIVED lower bound with the cell count printed in a table, and the $206$–$537$ GPU-s band is quoted only for the four harness edits it prices, with its source's own words reproduced. MARS's $27$-combination count came from this verse's own three-arm count line; row 2 lists four arms, so the count line is corrected upward to four, which makes the price larger, not smaller: $36$ cells at $s\le1024$, $4$ at $s=4096$, $12$ occupancy cells, $52$ cells total; at $N=8$ repeats and $\ge3.0$ s of timed work per cell, the lower bound on timed work alone is $\ge1{,}248$ GPU-s at one process per repeat and $\ge3{,}744$ GPU-s under row 1's three-child-process median — $2.32\times$ to $6.97\times$ the band's top of $537$, before warm-ups, process start-up, Triton compilation and the $s=4096$ allocator pressure. Both numbers stand in the same line wherever the sweep is priced, and the conceding "at most $3\times$ the cells" sentence is deleted. The re-price propagates to 03.7's Kill and If-killed, to 03.4's kill price, to 03.1's L4 bullet and K-9 row, and to the cheapest-first table. The pooled-cell figure in 03.7 kill (i) is corrected from $216$ to $52\times8=416$ readings.

MARS · MORIARTY, repair round 1, batch 2 of 4. Same rule: every finding verbatim, the repair beside it, nothing softened.

B-01 · verse 03.5 · severity strike · mechanism V-11, V-10, M-2

Flaw (verbatim). The table is built against the $0.90$ budget but the kill fires only above total VRAM, so every row in the gap between them is refused by the table and unreachable by the kill.

Number (verbatim). Table budget $0.90\times7.996$ GiB $=7.196$ GiB $=7.727\times10^9$ B; kill threshold $7.996$ GiB $=8.585\times10^9$ B. An $11.1\,\%$ band exists in which the table refuses a row and the kill cannot fire. The record's own certificate holds such a cell: results/k_cert_local.json memory.cells, arm_smprime, $n=16384$, reserved $8{,}057{,}257{,}984$ B $=7.504$ GiB, resident: true, fits: true — above the budget, below the kill.

Replacement survived the same attack. Yes.

Repair applied. MARS's first option is taken. 03.5's kill is re-frozen at the same $0.90$ budget the table is built against — $7{,}726{,}694{,}400$ B $=7.196$ GiB — and the $8{,}585{,}216{,}000$ B $=7.996$ GiB total is stated separately, in the Statement and in the kill, as the hard device limit whose exceedance produces no cell at all. The $858{,}521{,}600$ B band ($11.1\,\%$ of the budget) is printed in the verse as the V-11/V-10 defect it was. MARS's certificate cell is adopted as the kill's reachability witness and as a second planted negative — arm_smprime, $s=64$, $n=16384$, reserved $8{,}057{,}257{,}984$ B $=7.504$ GiB, resident: true, fits: true — a real draw at which the re-frozen kill fires and the old one could not. The threshold moves with it in three other places: 03.4's If-killed (N-20's reading), 03.14's If-killed on (ii), and the cheapest-first table, whose row now names both numbers.

B-02 · verse 03.15 · severity strike · mechanism P-7 (vocabulary with no fixed referent), V-17

Flaw (verbatim). $M$ is redefined as the resolvent inside the verse while the book's Notation and every other verse set $M=I-\gamma P$; an engineer building from the page computes $\Delta z$ with the wrong matrix.

Number (verbatim). Line 8: '$M=I-\gamma P$, $z=M^{-1}V$'. 03.15: '$\Delta z=\gamma\,\frac{(Me_i)(u^\top z)}{1-\gamma\,u^\top Me_i},\ M=(I-\gamma P)^{-1}$', and 'the column $Me_i$ — a single-right-hand-side substitution'. Substituting the book's global $M$ gives $\Delta z$ wrong by the factor $\|(I-\gamma P)^{-1}\|_\infty=1/(1-\gamma)=10$ at $\gamma=0.9$, not a rounding error.

Replacement survived the same attack. Yes.

Repair applied. MARS's first option is taken: the column is written $M^{-1}e_i$ throughout 03.15 and the denominator is $1-\gamma u^\top M^{-1}e_i$, with $M=I-\gamma P$ as in the Notation block and in every other verse. The local redefinition "$M=(I-\gamma P)^{-1}$" is quoted in the verse and struck, with MARS's factor $\|(I-\gamma P)^{-1}\|_\infty=1/(1-\gamma)=10$ at $\gamma=0.9$ printed as the size of the error an engineer building from the page would have made. No new symbol is introduced: the canon writes the resolvent as $M^{-1}$, and nothing in this book denotes $(I-\gamma P)^{-1}$ by a bare letter. The Hypotheses line is corrected with it ($M$ lower-triangular, hence $M^{-1}$ lower-triangular, hence $M^{-1}e_i$ supported on rows $\ge i$).

B-03 · verse 03.16 · severity strike · mechanism P-4, L-VERSE

Flaw (verbatim). The verse is named as a closer of census row B15, which includes the Mapper schedule; the word Mapper appears nowhere in the book outside the closing table.

Number (verbatim). grep -nic mapper docs/canon/03_KERNEL.md returns $1$, at line 435 — the census table row itself. B15 as written in CHARTER §5 names three missing kernels: chunked solve, CSR path, Mapper schedule. The first two are closed by 03.11/03.12 and 03.16; the third has no verse, no kill and no Terminal.

Replacement survived the same attack. Yes.

Repair applied. MARS's first option is taken rather than the withdrawal, because a withdrawal needs a CORRECTIONS.md row this repairer may not write and would leave B15 open until someone else wrote it. A verse for the Mapper schedule is written: 03.19, placed beside 03.16 because it produces the schedule 03.16 consumes, with all seven fields. Statement: the Mapper schedule is a schedule producer and not a kernel — lens, cover, nerve, and the quantiser to contiguous tiles, which is the load-bearing step. Evidence: the gather it must not become, RUN on this card, $2.58$ ms of index_select against $0.82$ ms of attention at $65{,}536$ positions, i.e. $3.15\times$ the attention it selects for (READ ceq/multizoom.py:13-15), and the $53\times$ builder (READ ceq/multizoom.py:17-18); the programme, its deliverable and its price (READ docs/CEQ_SHAPE.md:2130-2134, N-16, $2$ evenings, price NOT MEASURED); "A Mapper cover with a comparable $\delta$ is NOT FOUND" (READ docs/CEQ_SHAPE.md:2131). Kill: two frozen clauses — (K-M) not better than the do-nothing merged-salience schedule by $\mathrm{MDE}_8$ in $\varphi$-NRMSE at matched visited tiles at $s=1024$, and (K-M2) the builder above $3.15\times$ the attention it schedules — each with its own planted negative. If killed: the merged salience schedule with no lens and no parameters, killed in turn by K-I on its own certificate, then L4. Terminal: the word Mapper is withdrawn from every cost sentence in this canon if the quantiser is never built. The census table's B15 row now names 03.19 and records that the word appeared in this book only in that table before this repair round. 03.18's GPU-kill count moves from fifteen of eighteen to sixteen of nineteen.

B-04 · verse whole book · severity strike · mechanism P-1, P-2, CHARTER §3

Flaw (verbatim). Roughly fourteen READ pins resolve to lines that do not carry the cited content; the pin is the canon's provenance mechanism and these citations do not resolve.

Number (verbatim). Verified identical at the pin and at HEAD (git diff --stat 99777ab HEAD -- docs/CEQ_SHAPE.md docs/PLAN.md COSTS.md is empty; HEAD c71527a). docs/CEQ_SHAPE.md — :2297-2298 cited for $1.681/1.524$, actual :2292-2293 (:2297-2298 hold the Neumann and skyline cells); :2300-2301 cited for $7.6$ s and $1.681$ s, actual :2298 and :2293; :2299 cited for the $\approx34$ s bed-cell pair, actual :2295; :2290 cited twice (03.7, 03.18) for the $\pm12\,\%$ clock band, actual :2680 (:2290 is a table header); :2681 cited for the thread-count floor $2.345\times10^{-3}$, actual :2679 (:2681 is blank); :614/:615 cited for lower_triangular_isUnit/resolvent_fromBlocks, actual :612/:613; :326-329 cited for the bare row sum $2.5$ at $\gamma=0.6$, actual :314; :331-335 cited for Definition 5, actual :318; :334-340 cited for Definition 6(i), actual :323; :337-340 cited for the key-side rank RUN, actual :3042; :456-459 cited for BED-M's $31.04\times$, actual :452 and :3054. docs/PLAN.md — :920 cited for Bet L, actual :923; :912 cited for Bet D, actual :915; :552 cited for 'a run reporting pass on an operator that raised is refused', actual docs/CEQ_SHAPE.md:1872. COSTS.md — :61 cited for $523.9$ s, actual :62.

Replacement survived the same attack. Yes.

Repair applied. Every pin MARS names was re-resolved this session by reading the line itself (RUN: sed -n '<L>p' on each file; the three files are byte-identical to 99777ab by MARS's own git diff --stat, reproduced), and corrected in place — sixteen edits over fourteen citations. $1.681/1.524$ → :2292-2293, split into :2293 and :2292 where the two numbers are quoted apart (03.2, 03.8); $7.6$ s → :2298 with $1.524$ s at :2292 beside it; the $\approx34$ s bed-cell pair → :2295; the $\pm12\,\%$ clock band → :2680 in all three places it is quoted (03.1's kill, which inherited it, 03.7's Hypotheses, 03.18's If-killed); the thread-count floor → :2679; the two Lean rows → :612/:613; the row sum $2.5$ → :314 in both 03.11 and 03.13; Definition 5 → :318; Definition 6(i) → :323; the key-side rank RUN → :3042, now quoted with the reading it carries (RUN[J] rank $20$ at $i=12$); BED-M's $31.04\times$ → :452 for the reading and :3054 for the dividend beside the softmax corner's $1\times$; Bet L → docs/PLAN.md:923; Bet D → docs/PLAN.md:915; the cumsum-must-raise sentence → docs/CEQ_SHAPE.md:1872, with docs/PLAN.md:552 named in the verse as cited for it and not carrying it; $523.9$ s → COSTS.md:62. The sources/ and code pins MARS verified are unchanged. The fourteen old line numbers are not silently replaced: they stand in this finding, and the Limits paragraph now records that this book's provenance mechanism failed on fourteen citations before this round.

B-05 · verse 03.6 · severity repair · mechanism M-8 (a constant carried across arms), P-8

Flaw (verbatim). The $2.0\times$–$6.6\times$ figure headlined as the solve increment's dispatch gap is read from a different arm's FLOP-model-versus-wall-clock comparison; carrying it is the mechanism the verse invokes against others.

Number (verbatim). scale/m3_flops.py:101-121 reads: 'THIS ENTRY IS A FLOOR ON THE ROW CELLS' COST', with $s{=}16$ ratio $3.48$, $s{=}64$ ratios $11.19$ and $7.43$ against a flat FLOP ratio $1.700$, concluding 'the model is optimistic by 2.0x at the pilot geometry and by 4.4x to 6.6x at the shipped one'. That is the optimism of a FLOP model for the per-row gate settle arm — a Python loop over t_max steps with a 268 MB Gram copy — not a timer bias measured on solve_triangular, which the same file never touches.

Replacement survived the same attack. Yes.

Repair applied. The band is returned to the arm it was measured on, in 03.6's Statement and in 03.1's Kill, in MARS's own terms: it is the optimism of a FLOP model for the per-row gate settle arm, whose file never calls solve_triangular, and carrying it here is M-8 — the mechanism this book invokes against others. It is struck from every threshold in this book and survives only where that arm is named beside it (03.3's and 03.4's Evidence, where it already was). The two hazards MARS licenses are named (h1) the absent synchronize() (READ V20_R15_JOURNAL.md:53 @ 99777ab), sign known and magnitude unmeasured, and (h2) the $1.65\times$ session-to-session drift on identical code and geometry ($2.0775$ against $3.4372$ s/step, READ scale/m3_flops.py:117-121). The threshold of 03.1 and 03.6 is re-derived from them and moves from $\rho_\Delta>1.555$ to $\rho_\Delta>1.167$: (h1) is neutralised to first order by taking the ratio inside one interleaved block, so the only measured factor left is (h2), and $1.65\times0.707=1.167$. The verse states that (h1)'s residual is second order and bounded by no number in this repository, so this kill's consequence is always "the record's cost number is void" and never "the shape is slow" (M-3). The $2.2$ that produced $1.555$ was N-02's card constant with no derivation on this arm and is struck with the band. The change makes the kill stricter, not weaker: the firing region grows from $\rho_\Delta>1.555$ to $\rho_\Delta>1.167$. The census table's B14 row carries the re-attribution, and 03.1's Terminal drops the band for "an un-synchronised clock whose bias has a known sign and no measured magnitude".

B-06 · verse 03.2 · severity repair · mechanism V-17, M-3

Flaw (verbatim). The Kill names two incompatible MAC referents for the ratio it freezes at $3$.

Number (verbatim). Evidence: 'solve$+Pz$ $2.514$ ms against $PV$ $1.473$ ms, ratio $1.707$ against the MAC ratio $2.0$ for that pair'. Kill: 'the solve-to-$PV$ wall-clock ratio above $3$ ... (MAC prediction $1.0$ for that pair; the floor at $s=64$ reads $1.707$)'. The solve-alone-to-$PV$ reading is $1.041/1.473=0.707$, not $1.707$. The threshold $3$ is $1.76\times$ the reading under one definition and $4.24\times$ under the other.

Replacement survived the same attack. Yes.

Repair applied. MARS's second option is taken and the pair is named once: the kill is read on $t_{\rm solve+Pz}/t_{PV}$ and on no other pair, MAC prediction $2.0$ ($\tfrac32s^2d$ against $\tfrac12s^2d$), floor reading $1.707$, so the frozen $3$ is $3/1.707=1.757\times$ the floor and $1.5\times$ the MAC prediction — the arithmetic is printed in the kill. The parenthesis "(MAC prediction $1.0$ for that pair)" named the solve-alone pair, whose reading is $0.707$ and against which the same $3$ would be $4.24\times$; it is quoted in the verse and struck, and $0.707$ now appears in this book only as 03.1's $\rho_\Delta$ reference, where it belongs.

B-07 · verse 03.2 · severity repair · mechanism M-3, Ruling 3

Flaw (verbatim). The matched-parameter ratio drops the projection term the same paragraph's unmatched instances include, so the printed number is not the ratio at the geometry the verse is asserted at.

Number (verbatim). $3/(2\sqrt L)=0.671$ is attention-only. With projections at $s=64$, $d_{\rm m}=16$, $L=5$: $(1.5\cdot65536+65536)/(\sqrt5\cdot65536+65536)=163840/212074=0.773$. The two preceding instances ($0.250$ at $s=64$, $0.298$ at $s=4096$) include projections; the matched line does not, without saying so.

Replacement survived the same attack. Yes.

Repair applied. The matched paragraph now prints the ratio at both limits, each labelled, at the pair A-14 registered: with projections $\mathbf{0.875}$ at $s=64$, $d_{\rm m}=32$, and $\mathbf{0.754}$ at $s=4096$; and $3/(2\sqrt L)=\mathbf{0.750}$ only as the $s\gg d_{\rm m}$ attention-only asymptote, which is the ratio at neither registered geometry. MARS's own two numbers are printed in the verse as the arithmetic of the pair this book forbids: $(1.5\cdot65536+65536)/(\sqrt5\cdot65536+65536)=163840/212074=0.773$ with projections against $3/(2\sqrt5)=0.671$ attention-only at $(d_{\rm m},L)=(16,5)$ — a gap of $0.102$, the size of the error a bare asymptote makes when it is quoted at $s=64$ — and neither is a threshold anywhere in this canon.

B-08 · verse 03.3 · severity repair · mechanism P-1, M-3

Flaw (verbatim). The bytes-predicted time is printed as $51.5$ ms but the printed expression evaluates to $50.33$ ms; the frozen kill and its plant inherit the error.

Number (verbatim). $6\cdot4096^2\cdot16\cdot1024/128=1.28849\times10^{10}$ B; $/2.56\times10^{11}$ B/s $=50.33$ ms, not $51.5$ ms ($51.5$ ms corresponds to $\mathrm{BW}=250.2$ GB/s, not the verse's $256$ GB/s). The frozen kill '$2\times$ the larger, $103$ ms' becomes $100.7$ ms; the plant's $412$ ms at $B_M=16$ becomes $402.6$ ms.

Replacement survived the same attack. Yes.

Repair applied. MARS's first option is taken: the expression is re-evaluated at the verse's own $\mathrm{BW}=16\cdot16\times10^9=256$ GB/s and printed in two steps ($1.28849\times10^{10}$ B, then $50.33$ ms), with the FLOP prediction beside it in its own two steps ($8.246\times10^{11}/1.908\times10^{13}=43.2$ ms). The kill is re-frozen at $2\times$ the larger $=100.7$ ms and the $B_M=16$ reading at $402.6$ ms; $51.5$ ms is quoted in the verse and struck together with the bandwidth it implies ($250.2$ GB/s). The two dependants move with it: 03.12's kill (iii) now reads "frozen $100.7$ ms forward", and the cheapest-first table's 03.3 row names both $100.7$ ms and the $50.33$ ms it is twice of.

B-09 · verse 03.3 · severity repair · mechanism V-15

Flaw (verbatim). The planted negative requires the kill NOT to fire; it is a positive control, not the known-bad plant V-15 asks for.

Number (verbatim). Kill fires when the fused forward exceeds $2\times$ both predictions. The plant reads: 'the same kernel with $B_M=16$ must read within $2\times$ of its bytes prediction at $B_M=16$ ($412$ ms)' — i.e. the model must track a memory-bound configuration, which is the kill not firing. No plant is given on which the kill fires.

Replacement survived the same attack. Yes.

Repair applied. MARS's plant is adopted in mechanism: the same kernel with the $K$, $V$, $z$ tile reload forced per query row instead of per query tile — a deliberate $B_M=1$ traffic pattern with the arithmetic unchanged — must read above the frozen $100.7$ ms, its bytes prediction being $128\times$ the $B_M=128$ figure, $6.44$ s, i.e. $64\times$ the threshold (DERIVED from $I_{\rm fused}=B_M/2$); a kernel that reads inside $100.7$ ms on that plant is not being measured and the instrument is refused. The $B_M=16$ reading is kept and relabelled as the separate tracking control it is, at the corrected $402.6$ ms, with the sentence that a control on which the kill cannot fire is a positive control and that the verse now carries both, labelled.

B-10 · verse 03.1 · severity repair · mechanism V-15

Flaw (verbatim). The planted negative is a timer-sanity control and does not make the verse's kill fire.

Number (verbatim). Kill: 'L4's increment at $n=2048$, $s=64$ re-measured above $2.290$ ms'. Plant: 'the same microbenchmark with synchronize() removed must read a smaller number than with it ... within $1\,\%$ the timer is not measuring the device'. A smaller un-synchronised reading does not exceed $2.290$ ms; it condemns the timer, not the number.

Replacement survived the same attack. Yes.

Repair applied. A plant on which the kill fires is added to 03.1 and to 03.6, needing no new instrument and no assumed latency: the same interleaved block with the exact solve replaced by the $K=16$ Neumann iteration on the same operator — a known-slow route whose increment the record has already measured at $25.409-1.473=23.936$ ms against the same $PV$ control at $n=2048$, i.e. $\rho_\Delta=16.25$ against the frozen $1.167$, $13.9\times$ over (DERIVED from 03.6's own RUN pair). An instrument that reads $\rho_\Delta\le1.167$ there is refused. The synchronise/un-synchronise comparison is kept and relabelled as a timer sanity control (the un-synchronised reading smaller on at least $6$ of $8$ repeats; equality on all $8$ refuses the instrument), with the sentence that as a kill it lives in 03.7 (ii), where MARS says it already does. MARS's quoted absolute form, $2.290$ ms, had already been struck in batch 1 and stays struck; batch 2 replaces the ground on which it is struck, since the containment argument A-10 used rested on the band B-05 removes.

B-11 · verse 03.7 · severity repair · mechanism P-1, M-3

Flaw (verbatim). The cell count and the arm list disagree, and the Kill's pooled sample size inherits the wrong count.

Number (verbatim). Row 2 lists four arms: softmax control, shape, depth-$L$ skyline at $L=5$, one Neumann hop. Evidence: 'DERIVED count: $3$ arms $\times3$ $n$ $\times3$ $s$ $\times8$ repeats $=216$ cells plus $24$ at $s=4096$'. With four arms: $4\times3\times3\times8=288$ plus $32$. Kill (i) is stated 'on the pooled $216$ cells'.

Replacement survived the same attack. Yes.

Repair applied. MARS's first option — recount at four arms — was applied in batch 1 under A-16, and is verified here against MARS's own arithmetic rather than restated: the count line reads four arms; the table reads $36$ cells at $s\le1024$, $4$ at $s=4096$ and $12$ occupancy cells, $52$ in all; kill (i) reads $52\times8=416$ timed readings. MARS's $288+32=320$ are exactly the first two blocks' readings ($36\times8=288$, $4\times8=32$), and the occupancy row adds $12\times8=96$, giving $416$. The three-way arithmetic is now printed inside kill (i) so the count cannot drift from the arm list again. The skyline arm of row 2 is the matched $(32,4,16)$ arm and not the depth-$L$ stack at $L=5$ MARS quotes — that substitution is A-13/A-14's, and it leaves the cell count unchanged.

B-12 · verse 03.7 · severity repair · mechanism M-2, V-17

Flaw (verbatim). The stated Spearman critical value does not correspond to the sample size it is attributed to.

Number (verbatim). 'the critical value $2/\sqrt{66}=0.246$ at $n=64$'. $2/\sqrt{66}=0.2462$ is the rule $2/\sqrt{n-1}$ at $n=67$; at $n=64$ the same rule reads $2/\sqrt{63}=0.2520$. The frozen threshold $0.25$ sits between the two.

Replacement survived the same attack. Yes.

Repair applied. The per-arm pools are computed from the experiment's own geometry and printed rather than asserted: $9$ cells $\times\,8=72$ readings at $s\le1024$ plus $1\times8=8$ at $s=4096$, i.e. $80$ readings for each of the softmax control, the matched skyline and the Neumann arm, and $80+96=176$ for the shape, which carries the occupancy row alone. The smallest per-arm pool is $80$, where $2/\sqrt{n-1}=2/\sqrt{79}=0.225$; pooled, $2/\sqrt{415}=0.098$. The threshold stays frozen at $0.25$, now shown to be at or above both ($0.25>0.225>0.098$), so it fires on no arm below its own critical value. The struck line is quoted with MARS's arithmetic beside it: $2/\sqrt{66}=0.2462$ is the rule at $n=67$, at $n=64$ it reads $2/\sqrt{63}=0.2520$, and no pool of either size exists in this experiment.

B-13 · verse 03.7 · severity repair · mechanism L-VERSE, CHARTER §2 depth rule

Flaw (verbatim). Kill (iii) has no If-killed branch.

Number (verbatim). Three kills are frozen: (i) $|\rho_{\rm order}|\ge0.25$, (ii) secs_unsync not below secs_sync on $\ge6/8$, (iii) R-squared below $0.99$ per arm. The If-killed opens 'the instrument condemned by (i) or (ii)'. Kill (iii) fires into no replacement and no Terminal sentence of its own.

Replacement survived the same attack. No.

Repair applied. MARS's first option is taken and the branch is written in 03.7 rather than moved into 03.3, because (iii) condemns the law while (i) and (ii) condemn the instrument, and 03.3's If-killed is a roofline replacement, not a sweep's. If killed on (iii): the power law $t=a\,n^bs^c$ is abandoned per arm for the two-term additive model $t=\max(\mathrm{FLOP}/R_{\rm eff},\text{bytes}/\mathrm{BW})+n_{\rm launch}\tau_{\rm launch}+(s/C)\tau_{\rm chunk}\lceil nh/\mathcal P\rceil$, refitted on the same journal, with every point quoted individually beside the fit and never summarised by an exponent. Hypotheses: the three terms add; $\tau_{\rm launch}$ and $\tau_{\rm chunk}$ independent of $s$ and $n$ over the swept range; the same $52$ cells, no new run. Evidence: NOT MEASURED — needsresults/cost/s_sweep.jsonlrefitted, with 03.3's $0.273$ TFLOP/s at $s=64$ — $1.4\,\%$ of the assumed peak — as the DERIVED reason to expect a launch term. Kill: the additive fit's R-squared also below $0.99$ on any arm, or both the $\tau_{\rm chunk}$ and the $\tau_{\rm launch}$ intervals containing $0$ — then no cost law of any form exists in this canon and every price is a single measured cell carrying its run_index, seed and git_head, which is 03.7's Terminal in force. Price $0$ GPU-s (a refit of a journal already paid for), cheaper than the verse's $\ge1{,}248$ GPU-s and more decisive, since it ends the search for a law rather than re-running it. The (i)/(ii) branch is retained unchanged and is now labelled with the clauses it answers.

B-14 · verse 03.12 · severity repair · mechanism L-VERSE, V-9

Flaw (verbatim). Kills (i) and (ii) have no If-killed branch in the verse body; only (iii) is answered.

Number (verbatim). Three kills are frozen: (i) $\gamma=0$ parity not torch.equal, (ii) parity against L4's $z$ outside 03.13's bound at $(4096,64)$, (iii) price at $(4096,1024)$. The If-killed opens 'If killed on (iii)'. The closing table at line 448 supplies 'construction error; L2' for (i), which the verse does not derive.

Replacement survived the same attack. No.

Repair applied. Three branches are written into the verse body, one for each kill the verse now carries — (i), (i-b) and (ii) — each with its own Hypotheses, Evidence and Kill, and each ending in a Terminal inside the depth rule. (i) is derived as the construction error it is: at $\gamma=0$ step 3 returns $r_k=V_k$ by the exactness of $+0.0$, so a mismatch means the two programs do not share an accumulation order; the replacement is the single-path schedule — one program with $\gamma$ a runtime scalar and no $\gamma=0$ specialisation anywhere — whose own kill is against the torch route's $\gamma=0$ output up to the record's printed fused-versus-naive gap ($\max|\Delta|>1.110223\times10^{-16}$ on more than $19/64$ entries at $s=64$, READ docs/sources/judge/sec_record.md:80), a different object that can fire where (i) cannot, a single-path program having no second program to disagree with; if it fires, 03.12's Terminal is in force and L2 decides. (i-b) routes to the two-pass schedule — a statistics pass over $l=0,\dots,k$ including the diagonal tile, then a recompute pass against the final $m_k$ — where no running rescale of $A_k$ exists to omit, at $+s^2d/2$ MACs, killed by a price threshold of $\tfrac43\times$ the one-pass forward rather than by a parity bound. (ii) routes to L2, which the closing table asserted and the verse did not derive, with L2's own chunk-versus-serial parity at $(4096,64)$ against the same 03.13(iv) bound as its kill — convicting the recurrence rather than the kernel, the tile loop being gone — and 03.11's Terminal and L4 beyond it. The closing table's 03.12 rows are updated and a row for kill (ii), which the table did not carry, is added.

B-15 · verse 03.14 · severity repair · mechanism L-VERSE, V-9

Flaw (verbatim). Kill (i) has no If-killed branch in the verse body.

Number (verbatim). Kills: (i) gradient parity above $4\gamma_{s+1}\|\bar O\|_\infty\|V\|_\infty/(1-\gamma)^2$ at $(1024,128)$, marked 'refused'; (ii) memory linearity. The If-killed opens 'If killed on (ii)'. The closing table at line 452 supplies 'refused; L2 with per-chunk checkpoint'.

Replacement survived the same attack. No.

Repair applied. The branch was written in batch 1 under A-03 and is completed here against MARS's specification, which asked for autograd through L4's solve_triangular as the deciding backward, with its own retained-memory Hypotheses and a peak-reserved kill per 03.5's L4 column. That route is examined in the verse and refused, with its reason printed: autograd's backward of a triangular solve is a triangular solve on $M^\top$, so the reference carries the same $\kappa_1(1+\gamma\pi)$ amplification as the object under test and the comparison is undecidable wherever $\kappa_1(1+\gamma\pi)\ge1/(4\gamma_{s+1})$ — a replacement dying to the verse's own kill (V-9). Autograd through L4 is kept as the reference implementation whose adjoint the branch validates, not as the check; the check is the scalar float64 central-difference directional derivative on the L4 forward, which runs no backward at all. MARS's retained-memory clause is written in regardless, the branch running on the same device: under torch.no_grad the peak is the float64 operator alone, $2\cdot8\cdot64^2\cdot8=5.24\times10^{5}$ B at $(s,n)=(64,8)$ — $6.8\times10^{-5}$ of 03.5's re-frozen $7.727\times10^9$ B budget and four orders inside its L4 column at $s=64$ — which is why the check is registered at $(64,8)$ and why no memory kill belongs to this branch: the memory reading is (ii)'s, at $(4096,64)$, and is not restated as a second copy of it (V-9).

B-16 · verse 03.11 · severity repair · mechanism M-3, DERIVED arithmetic

Flaw (verbatim). The doubling variant's GEMM count omits the squarings that produce the powers it multiplies.

Number (verbatim). $(I-\tilde N)^{-1}=\prod_{j=0}^{k-1}(I+\tilde N^{2^j})$ with $k=\lceil\log_2C\rceil$ needs $\tilde N^{2^{j+1}}=(\tilde N^{2^j})^2$ for each $j$, i.e. $k-1$ squarings plus $k$ products $\approx2\log_2C$ tile GEMMs of $C^3$ MACs, so $2sC^2\log_2C$, not $sC^2\log_2C$. The crossover condition becomes $4C\log_2C<d$; at $(C,d)=(16,128)$ it reads $256<128$, failing by $2\times$ rather than at equality.

Replacement survived the same attack. Yes.

Repair applied. The count is corrected in the four places it appears. The Statement's bullet reads $k^\dagger=\lceil\log_2C\rceil$ with $k^\dagger-1$ squarings plus $k^\dagger$ products, $2k^\dagger-1\approx2\log_2C$ tile GEMMs of $C^3$ MACs each and $\mathbf{2sC^2\log_2C}$ in total, naming the halved figure it replaces. The variant-choice sentence reads the condition $\mathbf{4C\log_2C<d}$, failing at the most favourable admissible pair $(C,d)=(16,128)$ as $256<128$ — by a factor $2$, not at the equality $128<128$. The If-killed's displayed $N_{kk}$ carries $k^\dagger$ and the same GEMM count, so the replacement does not inherit the halved one. The Evidence line comparing the record's $sC^2/3$ reads $2sC^2\log_2C$. The conclusion strengthens, as MARS says: the substitution variant is the specification at $d\le128$ and the marginal case disappears.

MARS · MORIARTY, repair round 1, batch 3 of 4. Same rule: every finding verbatim — verse, flaw, mechanism, number, severity — the repair beside it, nothing softened.

C-01 · verse 03.5 · severity repair · mechanism P-1, M-8

Flaw (verbatim). The reserved-over-allocated ratio is attributed to a fit it is not part of, at a citation range that does not contain it.

Number (verbatim). Hypotheses: 'the residual constant $17.874$ and the reserved-over-allocated ratio $1.256$ were fitted on ceq/lm.py::TinyLM at $B=4$, $d=256$, $L=4$, $H=4$ (READ COSTS.md:84-99)'. COSTS.md:84-99 contains $17.874$ (:89) and $3.823$ (:90) from the seq-sweep least squares; $1.256$ is at COSTS.md:223 and is described there as 'the worst reserved/allocated ratio measured over this box's resident shapes' — a different object with a different provenance.

Replacement survived the same attack. Yes.

Repair applied. MARS's pins are re-read this session (RUN: grep -n "1.256" COSTS.md returns exactly 223:reserved/allocated ratio measured over this box's **resident** shapes (**1.256**); sed -n '84,99p' COSTS.md returns the two-parameter least-squares table with C_RESIDUAL (fp32) | 18 | 17.874 | 0.996373 at :89 and C_OPERATOR (fp32) | 3.9 | 3.823 | 0.996373 at :90, and no 1.256). 03.5's Hypotheses now carry the two constants apart, each with its own provenance and its own [ASSUMED] reason. (a) $17.874$ is [FITTED] at R-squared $0.996373$ on the seq-sweep least squares, READ COSTS.md:89, with $C_{\rm OP}=3.823$ named as the same fit's other coefficient at :90. (b) $1.256$ is READ COSTS.md:223 and is quoted in the source's own words — "the worst reserved/allocated ratio measured over this box's resident shapes" — with the three ways it differs printed: a different object (allocator reserve against allocation, not a byte-model coefficient), a different provenance (a maximum over measured shapes, not a least squares), and no R-squared. The earlier attribution of $1.256$ to the COSTS.md:84-99 fit is quoted in the verse and struck. Carrying either constant to the shape at $d_{\rm m}=16$ is stated as M-8 and [ASSUMED] until N-18 (READ docs/PLAN.md:751-755), and the two [ASSUMED] tags now carry different reasons: (a) fitted on another model at another width; (b) a worst-case ratio over one box's resident shapes is not a law, and its own source applies it to a T4 budget (READ COSTS.md:224-225) that V-22 forbids pooling with this device's. The Limits paragraph records both provenances.

C-02 · verse 03.8 · severity repair · mechanism P-8, CHARTER §2 reachability rule

Flaw (verbatim). The kill's price is the price of one 150-step cell pair, not of the to-floor training runs the kill requires, and no bed is named.

Number (verbatim). 'price inside the $\approx34$ s bed-cell pair at $N=8$'. docs/CEQ_SHAPE.md:2295 prices $34$ s as 'one bed-cell pair, 8 seeds each ... two invocations'; that is $150$ steps per cell. The kill needs $k^\star$ for both arms, which the verse's own Evidence marks NOT MEASURED — needs the arena on an admitted bed (book 04, S-62) and the training curve (book 02), and whose Hypotheses admit $T1=\infty$ when an arm never reaches the level. The kill names a geometry ($s=64$, $n=2048$, $d=16$) but no bed, no draw and no seed rule.

Replacement survived the same attack. Yes.

Repair applied. All three of MARS's required repairs are applied and the kill is rewritten under the heading it now carries, "with its bed, its draw, its seed rule and its price from the step budget the ladder level requires". The bed is named: BED-S as registered in 04.2 (READ docs/canon/04_BEDS_AND_INSTRUMENTS.md:27) at $s=64$, $d_{\rm m}=16$, $n=2048$, $t^\star=8$, $m=8$, $K=2$ — and the verse prints that BED-S does not admit at the design point (READ docs/canon/04_BEDS_AND_INSTRUMENTS.md:53, 04.2's Terminal verbatim: "BED-S at the design point does not admit"), so this kill is journalled NOT EXECUTABLE (no admitted bed) today, V-16's third outcome, and becomes decidable only on whichever object 04.2's own three-link chain produces. The draw and seed rule are named: seeds $0$–$7$ of 04.9 clause 1 (READ docs/canon/04_BEDS_AND_INSTRUMENTS.md:147), deduplicated, one thread lane read from the journal, both arms on byte-identical draws (train at seed, eval at seed+12345), $N=8$ arm-cells per arm. The $\approx34$ s price is struck by name, with MARS's reading of docs/CEQ_SHAPE.md:2295 printed in the verse — two arms $\times$ eight seeds $\times$ $150$ steps, two invocations — and replaced by the step-budget formula $\text{GPU-s}=8(k^\star_{\rm shape}\cdot0.01120613+k^\star_{\rm softmax}\cdot0.01016513)+4.0$ s, DERIVED from the two step prices and the arena's $4.0$ s fixed cost (READ docs/CEQ_SHAPE.md:2294), which at the record's registered $150$-step cell length (READ scale/e_ladder.py:140, steps=150) with both arms running the budget out reads $\ge\mathbf{29.6}$ s as a lower bound and not a price, scaling linearly in an unmeasured $k^\star$ ($260$ s at $1{,}500$ steps, $\approx29$ GPU-min at $10^4$). Book 04's own re-priced $150$-step pair, $\approx30.9$ s (READ docs/canon/04_BEDS_AND_INSTRUMENTS.md:183), is named as the same object and the same kind of lower bound. The NOT REACHED outcome is written as the third reading of the kill, beside fired and not-fired: at least one arm failing to reach any level of $\{0.9,0.8,0.7,0.5\}$ inside the budget gives $T1=\infty$, an undefined ratio and 04.10's VOID clause, the row reading NOT REACHED with the step budget printed; a journal reporting (a) or (b) on such a cell is refused. The cheapest-first table's 03.8 row carries the bed, the formula, the $\ge29.6$ s floor and the third reading.

C-03 · verse 03.4 · severity repair · mechanism V-22, M-3

Flaw (verbatim). The kill's stated inference about device concurrency contradicts the book's own device constants.

Number (verbatim). Kill: 'then the chain is hidden even at $nh=8$, $\mathcal P\le8$ on this device'. 03.4 defines $\mathcal P=$ SMs $\times$ resident programs; 03.3 asserts $3072$ CUDA cores, which at $128$ cores per SM on sm_8.9 is $24$ SMs, so $\mathcal P\ge24$ under the verse's own definition. The two verses cannot both hold.

Replacement survived the same attack. Yes.

Repair applied. MARS's arithmetic is adopted and printed. The inference "$\mathcal P\le8$ on this device" is struck in the kill, by name, with the contradiction stated: $3072/128=24$ SMs on sm_8.9, at least one resident program each, so $\mathcal P\ge24$ under this verse's own definition, and $\mathcal P\le8$ cannot hold beside the $3072$-core reading (V-22, M-3). The kill now claims only what the reading licenses — the chain is not visible in seconds at $nh=8$, a statement about the sweep and not about $\mathcal P$, consistent with $\mathcal P\ge24$ and needing no concurrency count — and states that $\mathcal P$ is measured by the occupancy row of 03.7 (row 4: the shape at $s=64$, $n\in\{1,2,4,\dots,2048\}$, $12$ cells, the $n$ at which seconds stop being flat in $n$) and is never inferred from an exponent. The SM count is printed in 03.4's Evidence as [ASSUMED, product specification] alongside 03.3's core count, with the derivation shown ($3072/128=24$) and both inputs tagged, since nvidia-smi --query-gpu= exposes neither (RUN this session); the lower bound $\mathcal P\ge24$ is stated to be "the only concurrency figure this book asserts, and it is asserted to forbid an inference, never to license one", and the number of resident programs — hence $\mathcal P$ itself — stays NOT MEASURED — needs the occupancy sweep.

C-04 · verse 03.17 · severity repair · mechanism CHARTER §1, L-FLOOR, D-1

Flaw (verbatim). The kill is a comparative-accuracy sentence outside forms A1 and A2, against a comparator that is neither base self-attention nor the depth skyline, with no floor, $N$ or matched count in the row.

Number (verbatim). 'The shape's F0-segmented exact solve not better than the blockwise control by $\mathrm{MDE}_8$ in $\varphi$-NRMSE at matched visited tiles at $s=1024$'. CHARTER §1: 'A verse that says "more accurate" ... without naming which of A1, A2, T1, T2 it means, and which bed, floor, $N$ and matched count it is measured on, is struck.' No floor is printed and no matched parameter count is given; the bed is 'the long-context bed once admitted (book 04)', which does not exist.

Replacement survived the same attack. Yes.

Repair applied. MARS's first option is taken in full and the second is executed as a relocation rather than a re-registration, because this repairer edits only book 03. The $\mathrm{MDE}_8$ $\varphi$-NRMSE kill is struck in the verse by name, with CHARTER §1's sentence quoted against it and the four missing items listed: no form named, no floor printed, no $N$, no matched parameter count, and a bed that does not exist. The accuracy comparison is not withdrawn but relocated: it is owed to book 04 as a registered contrast with its floor (L-FLOOR), its $N$, its $\mathrm{MDE}_N$ at the realised $\sigma_d$, its matched count and its form named — and the verse records that under 04.9 clause 7 every $\mathrm{MDE}_8$ number is a placeholder until 04.21 step (5) fills $\sigma_d$, so no such threshold could have been frozen in this book at all. The kill is restated on the cost axis, which is this verse's own scope: $t_{\rm shape}/t_{\rm blockwise}\le1.00$ at $s=1024$, forward and backward under 03.7's protocol at the same $(s,n)$ with $n$ from 03.5's residency column, at matched exactness — the shape's $\delta_{\rm route}=0$ plus 03.13's float bound printed against the blockwise route's own $\delta$ measured on the same draws, whose absence at this pin ([V] grade, READ docs/sources/sweep/sweep_resolvent.md:218-226) makes the comparison NOT EXECUTABLE on that clause alone today. Firing narrows the concession from the class to the publication. Instrument N-13 inside 03.7's block; decidable at $N=8$ on torch.randn logits with no bed and no seed rule, which is the second reason the accuracy form did not belong here. Planted negative on which the cost kill fires: the blockwise control with $P$ materialised per block and re-read once per chunk step — arithmetic unchanged, bytes $\times s/C$ — must read $\le1.00$; an instrument blind to a memory-starved comparator is refused. The cheapest-first table's 03.17 row and the Terminal both carry the change.

C-05 · verse 03.17 · severity repair · mechanism L-VERSE ('nothing in a verse without its Kill'), P-4

Flaw (verbatim). The If-killed declares itself to need no kill.

Number (verbatim). 'Kill: none needed — this is already the Terminal's sentence, reached in one link.'

Replacement survived the same attack. No.

Repair applied. MARS's second option is taken: the If-killed is kept and given a kill, because deleting it would leave 03.17 with a kill and no link, and the narrowed sentence is a genuinely different statement from the Terminal's — the Terminal concedes on the class, the replacement concedes on the publication and asserts one measured cell. The replacement is now a full link: Statement — the concession narrows to "no lower price than the blockwise evaluation as published", with the $s=1024$ reading quoted as one cell carrying its run_index, seed and git_head, never as an exponent and never against the source's [V]-graded numbers. Hypotheses — the control is N-13's and not the source's own code, which this canon has not run; both arms at matched exactness with both $\delta$'s printed; the $s=1024$ residency $n$ of 03.5. Evidence — NOT MEASURED — needs N-13 and 03.7's $s=1024$ block; the exactness asymmetry rests on 03.13's DERIVED bound. Kill, its own and different — the same ratio at $s=4096$, where 03.5 puts L4 at $n=8$ and L1 at $n=1024$ rather than $s=1024$'s $128$ and $4096$, exceeds $1.00$: the sign reverses across the two registered $s$, the narrowed sentence has no $s$ at which it holds, and it is withdrawn with the concession restored on the class. A different number ($>1.00$ against $\le1.00$), on a different object (a different $s$, a different residency $n$, a different route), so it is not the verse's kill renamed (V-9). Price: the $s=4096$ block of the same sweep, $0$ new GPU-s beyond 03.7's cell count, and more decisive because $s=4096$ is the last $s$ this book registers. The chain is Statement → If-killed → Terminal, two links, inside the depth rule. MARS's alternative kill for the replacement — the cited $O(n^{4/3}d)$ not reproducing at $s\in\{1024,4096\}$ — is refused with the reason printed: that exponent is [V]-graded and marked NOT MEASURED throughout this verse, so a kill frozen against it would import a number the verse itself refuses to quote (L-EQ, V-17), and the sign-reversal kill is frozen on this canon's own measured ratio instead.

C-06 · verse 03.15 · severity repair · mechanism L-SIGN, D-CALIB, V-15

Flaw (verbatim). The prediction's flattering direction carries no kill; only the adverse edge of the SPLIT band is frozen.

Number (verbatim). SPLIT is $(0.5\times,8\times)$; Bet G's prediction is $\le0.5\times$, the counter $\ge8\times$; the Kill freezes only $\ge8\times$. A reading of $0.3\times$ — which would license the flattering sentence — has no threshold that refutes it and no plant.

Replacement survived the same attack. Yes.

Repair applied. MARS's required repair is applied and the kill is split into two named clauses, (K-G-adverse) and (K-G-flattering), with the defect stated in the verse in MARS's own terms: the adverse edge alone was frozen, so a $0.3\times$ reading licensed the flattering sentence with no threshold refuting it and no plant firing on it. (K-G-flattering) freezes the lower edge in two clauses, both required. (a) The arithmetic-term check, which is this verse's own $s=64$ caveat made decidable: at $s=64$ the operator is launch- and latency-bound (03.3's $0.273$ TFLOP/s, $1.4\,\%$ of the assumed peak), and this verse's own adverse plant already concedes it "may read below $m\times$ at $s=64$ because of the launch floor", so a $\le0.5\times$ reading there is consistent both with the $m$ columns costing $m/d$ of a solve and with their costing nothing measurable. The cell must print the measured ratio at $m\in\{1,8\}$ at the same $(s,n,d)$ and the difference must exceed the cell's own repeat spread at $N=8$; agreement inside that spread is journalled LAUNCH-FLOOR, NO READING (V-16's third outcome) and licenses no sentence in either direction. (b) The same reading required at $s=1024$, at 03.5's L4 residency $n=128$ under the upper constant, where the arithmetic term is $256\times$ the $s=64$ term and the launch floor cannot carry the reading. Frozen: a $\le0.5\times$ reading at $s=64$ alone is not a reading of Bet G at all, and the flattering sentence is withdrawn until (a) and (b) both hold. The flattering edge carries its own planted negative: route (i) with the $m$ columns removed entirely — the solve run and $\Delta z$ returned as zeros, arithmetic term exactly $0$ — must read $\le0.5\times$ at $s=64$ and must be caught by clause (a); an instrument that passes a do-nothing route as Bet G's prediction is refused. The cheapest-first table's 03.15 row carries both clauses.

C-07 · verse 03.6 · severity repair · mechanism M-3, M-8

Flaw (verbatim). Two Neumann cost numbers are printed side by side that no constant-per-hop model reconciles, and different canon books price from different ones of them.

Number (verbatim). 'one Neumann hop at $3.001$ ms ... ($K=16$: $25.409$ ms)'. $16\times3.001=48.02$ ms against $25.409$ ms — a factor $1.89$. The marginal per-hop cost implied by the $K=16$ figure is $(25.409-1.473)/16=1.496$ ms. 03.6 licenses its sentence from $3.001$; docs/canon/04_BEDS_AND_INSTRUMENTS.md:164 and docs/canon/06_PREDICTIONS.md:349 derive cell prices from $25.409$.

Replacement survived the same attack. Yes.

Repair applied. MARS's required repair is applied and the arithmetic is printed in the verse. The Neumann arm is stated as a two-term reading, both terms DERIVED from the same RUN pair and from nothing else: $t_{\rm Neumann}(K)=a+bK$ with $b=(25.409-3.001)/(16-1)=\mathbf{1.4939}$ ms per hop and $a=3.001-b=\mathbf{1.5071}$ ms fixed; MARS's $PV$-referenced marginal $(25.409-1.473)/16=\mathbf{1.496}$ ms is printed beside $b$, agreeing to $0.15\,\%$, and MARS's $16\times3.001=48.016$ ms against $25.409$ ms, factor $1.890$, is printed as the reason no constant-per-hop model reconciles the pair. Which term each sentence uses is now explicit and no sentence uses both: (i) this verse's licensed comparison, "at $s=64$ the exact solve is cheaper than one Neumann hop", is a comparison of totals at $K=1$ — $2.514$ ms against $3.001$ ms, margin $0.487$ ms — and uses neither $a$ nor $b$ alone; (ii) a $K=16$ cell is priced from the $K=16$ total $25.409$ ms, so a $150$-step cell reads $3.811$ s; (iii) reading $25.409$ ms as a per-hop figure and multiplying is a units error and appears nowhere in this book. MARS's two cross-book pins are re-resolved and the correction is printed: 04:164 is 04.10's Statement and 06:349 is 06's C-20 row, neither carrying $25.409$; the figure lives at 04:183, 04:781, 06:847 and 06:849, and at all four the neighbouring books read it as a per-hop cost and record that $16\times25.409\ \mathrm{ms}=0.407$ s fails to reproduce their $5.12$ s InfSA-16 cell by a factor $12.6$ — so MARS's attribution of a cell price derived from $25.409$ to those books does not hold as filed, while the underlying defect it names does. Under reading (ii) the $150$-step cell is $3.811$ s and not $5.12$ s either, so $5.12$ s is not derivable from this pair under any of the three readings, which is the sentence both neighbouring books already carry and which 03.6 now supplies the arithmetic for. The extrapolation beyond the measured range is marked: two points fix a line and certify no linearity, so $K$ outside $\{1,16\}$ — 03.16's $K\ge131$ regime is $8.2\times$ outside it — is NOT MEASURED — needs the Neumann arm swept over $K$ inside 03.7's interleaved block, and the Limits paragraph records it.

C-08 · verse 03.6 · severity repair · mechanism P-1, V-16

Flaw (verbatim). The grep offered as the evidence that no producer row exists excludes the file extension most of results/ uses.

Number (verbatim). 'RUN this session: grep -rl 2.514 --include=*.txt --include=*.json results/ returns nothing'. results/ holds $172$ entries of which $57$ are *.jsonl, excluded by both --include filters; grep -rl 2.514 results/ (RUN this session) returns results/cameron_aggregators.jsonl. The conclusion may still hold but the cited command does not establish it.

Replacement survived the same attack. Yes.

Repair applied. MARS's second option is taken and the filtered command is struck. The earlier evidence line is quoted in the verse and struck, with MARS's counts reproduced RUN this session (ls results/ | wc -l $=172$; ls results/*.jsonl | wc -l $=57$) and the reason printed: both --include filters excluded every .jsonl file, so the command searched a third of the directory and its silence proved nothing (V-16, silence read as a pass). The re-run is quoted with its hit and the reason the hit is not the producer: RUN this session, grep -rlF 2.514 results/ returns exactly one file, results/cameron_aggregators.jsonl, and the hit is 2.5148398876190186, an entry of the causal.tau array of a record whose fields are {agent: "cameron", tag: "smoke", s: 128, d: 8, k: 4, draws: 12, ...} (RUN this session, first line of the file) — a threshold statistic at $s=128$, $d=8$ with no timing field of any kind, no secs, no n=2048 and no ms. The verse also records that MARS's unescaped grep -rl 2.514 results/ returns $13$ files because . matches any character, and that the fixed-string form is the one this verse runs and quotes. The conclusion — no results/ record produces $2.514$, $4.659$ or $9.916$ ms, so under P-1 they are prose until N-02 runs — now rests on the re-run command. The Limits paragraph records that the book's own producer grep failed before this round.

C-09 · verse 03.7 vs 04.10 · severity repair · mechanism M-2 (thresholds frozen), D-1

Flaw (verbatim). Two born canon books freeze two different run-order thresholds for the same confound on the same interleaved timing instrument.

Number (verbatim). 03.7 Kill (i): $|\rho_{\rm order}|\ge0.25$ with $p\le0.05$ on a pooled cell set. 04_BEDS_AND_INSTRUMENTS.md:151: 'Spearman $|\rho|>0.5$ with $p<0.05$ ... on the $N\ge16$ interleaved timings'. A journal reading $\rho=+0.35$ is condemned by book 03 and passed by book 04.

Replacement survived the same attack. Yes.

Repair applied. MARS's second option is taken — this repairer edits only book 03 and cannot adopt one threshold across two born books — and 03.7's kill (i) now carries a paragraph, "Why this threshold is not book 04's, and what the disagreement costs", with both critical values computed from their own $n$ and printed. Book 04's journal is $8$ seeds $\times\ 3$ interleaved arms $=24$ arm-cells (04.10's own arm-cell reading of $N$), where $2/\sqrt{n-1}=2/\sqrt{23}=0.417$ and the $p=0.05$ boundary of $t=\rho\sqrt{(n-2)/(1-\rho^2)}$ sits at $\rho=0.404$, so its $0.5$ is above both at its own $n$; this book's smallest per-arm pool is $80$ readings, where $2/\sqrt{79}=0.225$ and the $p=0.05$ boundary sits at $\rho=0.220$, so the frozen $0.25$ is above both at its own $n$. MARS's number is printed in the verse and not softened: a journal reading $\rho=+0.35$ is condemned by this book and passed by book 04. What separates them is stated as power and not as disagreement about the confound — at $n=80$, $\rho=0.35$ gives $t=3.300$, $p=0.0014$ and both of this verse's clauses fire; at $n=24$ the same $\rho$ gives $t=1.752$, $p=0.094$ and both of book 04's clauses spare it — so the two books deliver opposite verdicts because the same number carries $67\times$ different evidence on the two journals. What is not defensible is recorded as owed: a single canon-wide $\rho$ bar cannot exist while the two instruments differ in $n$ by $3.3\times$, so a CORRECTIONS.md row is owed against 04_BEDS_AND_INSTRUMENTS.md:170 and this verse's kill (i) jointly, adopting "$|\rho|$ above the larger of $2/\sqrt{n-1}$ and the $p=0.05$ boundary at that journal's own $n$, both printed in the cell" in place of two frozen constants. The two constants stand until the coordinator writes it. MARS filed book 04's kill at :151, which is 04.9 clause 6 and carries no $\rho$; the pin is re-resolved to :170 in the verse and the old one is printed here.

C-10 · verse 03.11 / 03.13 vs 08_ARCHITECTURE.md · severity repair · mechanism M-2, L-G2, V-17

Flaw (verbatim). A neighbouring born book still carries the parity threshold this book declares repaired, and the plant this book's kill contradicts, with no CORRECTIONS row reconciling them.

Number (verbatim). 03.13 Mechanism: "N-11's '$10^{-6}$ in float32' is repaired here, since at $s=4096$ a correct float32 kernel can legitimately miss $10^{-6}$ by $10^2$." 08_ARCHITECTURE.md:472 still freezes 'parity $10^{-6}$ against the serial solve' for the same chunked route. 08_ARCHITECTURE.md:470's plant requires the microbenchmark to 'reproduce $1.041$ ms within $\pm12\,\%$' while 03.1/03.6 freeze the kill at $2.2\times$ that number under a declared $2.0\times$–$6.6\times$ gap.

Replacement survived the same attack. Yes.

Repair applied. MARS's second option is taken, because a CORRECTIONS.md row is a file this repairer may not write. 03.13's Mechanism now carries a paragraph, "Supersession of book 08's two constants, marked as L-G2 requires", with both markers written out. (1) 08_ARCHITECTURE.md:562 (08.20's Replacement 1, the chunked block-triangular solve) and 08_ARCHITECTURE.md:138 (08.4's Replacement 2, the dense inverse per chunk) both freeze parity at $10^{-6}$ in float32 against the serial solve; both are superseded by 03.13(iv), since a correct float32 single signed forward substitution at $s=4096$, $\gamma=0.9$ reads up to $4.64\times10^{-2}\|V\|_\infty$ and misses $10^{-6}$ by $4.6\times10^{4}$ — printed here at the corrected magnitude, MARS's "$10^{2}$" being the miss at the two-solve constant that A-02 struck. Marker: supersedes: 08_ARCHITECTURE.md:138, :562 (parity 1e-6, float32) — superseded by 03_KERNEL.md 03.13(iv), both 08 cells keeping their text beside it (L-G2). (2) 08_ARCHITECTURE.md:560 (08.20's Kill) plants "the dense per-op microbenchmark must reproduce $1.041$ ms within $\pm12\,\%$", which demands that a synchronised re-measurement land inside $\pm12\,\%$ of a number this book quotes only as an un-synchronised floor with a known-sign, unmeasured-magnitude bias. Marker: supersedes: 08_ARCHITECTURE.md:560 — superseded by 03_KERNEL.md 03.1/03.6, $\rho_\Delta$ inside one interleaved block. MARS's threshold reading of 03.1/03.6 is stale by one batch and the current one is printed: the kill is frozen at $\rho_\Delta>1.167$, derived from (h2)'s measured $1.65\times$ drift and not at $2.2\times$, and the $2.0\times$–$6.6\times$ gap was returned to the per-row gate settle arm in batch 2 (B-05) — the finding's mechanism stands unchanged and the supersession is filed against the current threshold. The verse names the 08 verses whose pre-written replacements now apply and states that no 08 verse is withdrawn by either row: 08.4's Replacement 2 and 08.20's Replacement 1 survive with their thresholds re-frozen on 03.13(iv), and 08.20's Kill survives with its plant re-frozen on $\rho_\Delta$. MARS's pins :472 and :470 are re-resolved to :562/:138 and :560 and the old two are printed here. Both CORRECTIONS.md rows are owed and are not written; they join those already owed, and the Limits paragraph now lists six.

C-11 · verse 03.8 vs 06.x / 08.x · severity note · mechanism M-8, M-2

Flaw (verbatim). Three born books print three values for the depth-1 softmax step price that the T1 thresholds are frozen from.

Number (verbatim). 03.8: $1.524$ s per $150$-step cell $=0.010160$ s/step. 06_PREDICTIONS.md:349: $0.010165$ s. 08_ARCHITECTURE.md:462: $0.010142$ s [MEASURED]. The law itself, $\exp(-12.1852)\cdot2048^{0.9963}$, evaluates to $0.010163$ s. The spread $0.010142$–$0.010165$ is $0.23\,\%$ and feeds $0.907$ (03.8) versus $1/1.1024=0.9071$ (06).

Replacement survived the same attack. Yes.

Repair applied. Applied though filed note. 03.8's denominator 1 now quotes one value with one class and one pin: the evaluation of the fitted law, $\exp(-12.1852)\cdot2048^{0.9963}=\mathbf{0.01016513}$ s per step, [FITTED] at R-squared $0.999998$ over five points (READ COSTS.md:73), re-evaluated RUN this session. The other three are named and placed: $0.010165$ s is that law to six figures; $0.010160$ s is the $150$-step cell price divided back out, a rounding of the same law; $0.010142$ s is the single measured point of results/k_cert_local.json throughput.points, a datum the law was fitted through and not a competing law, and is quoted as such wherever it appears. The ratio is re-derived from the law — $(0.01016513+0.001041)/0.01016513=1.10241$ — with $1.681/1.524=1.10302$ printed beside it as the cell reading it is, and the threshold is frozen at $1/1.10241=\mathbf{0.9071}$, unchanged at three figures under either reading. MARS's evaluation of the law, $0.010163$, is corrected in the verse: recomputed this session the law reads $0.01016513$, which is 06's figure to six figures, so the spread MARS reports is $0.010142$–$0.010165$ $=0.227\,\%$ and the law sits at its top, not between. MARS's pins 06:349 and 08:462 are re-resolved to 06:403-404 and 08:552. A CORRECTIONS.md row is owed against 06_PREDICTIONS.md:403-404 and 08_ARCHITECTURE.md:552 so all three books quote the law's evaluation with the measured point labelled; it is named in the Limits paragraph and not written here.

C-12 · verse 03.16 · severity note · mechanism V-17, M-3

Flaw (verbatim). The union certificate's tail term is derived for $z$ and applied to $O$, where the $(1-\gamma)$ factor makes it loose; the vacuity line is quoted at the loose constant.

Number (verbatim). For $O=(1-\gamma)Pz$ the Neumann tail is $\gamma^{K+1}\|V\|_\infty$, not $\gamma^{K+1}\|V\|_\infty/(1-\gamma)$. At $\gamma=0.9$, $\delta=10^{-6}$: the tight bound needs $K\ge131$, the printed one $K\ge152$ — a $16\,\%$ overstatement of the hop count that carries the 'uncompetitive by arithmetic' sentence. The $\varepsilon$ term $\varepsilon/(1-\gamma)$ is correct for $O$ and needs no change.

Replacement survived the same attack. Yes.

Repair applied. Applied though filed note. 03.16's Statement now carries two certificates, one per read: $\delta_{\rm union}^{O}=\varepsilon/(1-\gamma)+\gamma^{K+1}$ for the shipped $O$ read and $\delta_{\rm union}^{z}=\varepsilon/(1-\gamma)+\gamma^{K+1}/(1-\gamma)$ for a bare $z$ read, each times $\|V\|_\infty$, with the cancellation derived in three lines — $O=(1-\gamma)\sum_{k\ge0}\gamma^kP^{k+1}V$, whose tail is $(1-\gamma)\cdot\gamma^{K+1}/(1-\gamma)=\gamma^{K+1}$ exactly, at every $\gamma$ — and MARS's clause that the $\varepsilon$ term needs no change stated with its reason ($M^{-1}$'s amplification is inside the resolvent, not in the $(1-\gamma)P$ that follows it). The $K$ thresholds are re-derived on the corrected tail and the loose ones printed beside them as struck: $K\ge131$ at $\gamma=0.9$, $\delta=10^{-6}$ against the loose $K\ge152$, an overstatement of $152/131=1.16$; $K\ge19$ at $\hat\gamma\le0.5$ against the loose $K\ge20$. The "uncompetitive by arithmetic" sentence survives the correction and is re-stated on the tight number, with the price attached: $131$ hops at 03.6's marginal $b=1.4939$ ms is $\ge0.196$ s per step against the exact solve's $2.514$ ms, $78\times$, and at $\hat\gamma\le0.5$ nineteen hops is $28.4$ ms, still $11.3\times$ — two orders, not a rounding. The $131$-hop price is marked NOT MEASURED because $b$ is fitted on two points and $K=131$ is $8.2\times$ outside their range. The kill (K-I), the Terminal, 03.1's L2-killed bullet, 03.19's Hypotheses and If-killed and the cheapest-first table all carry $\delta_{\rm union}^{O}$.

C-13 · verse 03.5 · severity note · mechanism V-10, V-15

Flaw (verbatim). The planted negative is arithmetic the table performs by construction, not a device reading.

Number (verbatim). 'Planted negative: the table must refuse $(4096,16)$ on L4 at the upper constant ($7.96$ GB predicted against $7.73$ GB budget)'. Verified: $1.256[5.823\cdot16\cdot4096^2\cdot4+2\cdot16\cdot4096\cdot16\cdot4+17.874\cdot16\cdot4096\cdot16\cdot4]=7.958\times10^9$ B against $7.727\times10^9$ B. The formula refuses the row before any device is touched; no plant makes the kill (reserved bytes above the threshold on the device) fire.

Replacement survived the same attack. Yes.

Repair applied. Applied though filed note. The arithmetic refusal is struck as a plant and relabelled as the prediction it is, with MARS's evaluation printed in the verse ($7.958\times10^{9}$ B against $7.727\times10^{9}$ B, DERIVED and re-verified this session) and the reason stated: it is satisfied by construction before any device is touched, so it makes a kill on reserved bytes on the device fire on nothing (V-10, V-15). Two device plants replace it. (P1) L4 at $(s,n)=(4096,16)$ — the row the table refuses — run on the certified triple under a training loop with synchronize(), must read torch.cuda.max_memory_reserved above $7{,}726{,}694{,}400$ B or raise torch.cuda.OutOfMemoryError; a formula that refuses a row the device then runs resident inside the budget is a formula and not a ceiling, and the kill is refused as insensitive. (P2) the certificate's arm_smprime $n=16384$ cell, re-run as a row this table admits, must fire the kill at $7.504$ GiB — the plant batch 2 took from the record, kept. The cheapest-first table's 03.5 row names both plants and records that the arithmetic refusal is the prediction, not a plant.

C-14 · verse 03.10 · severity note · mechanism V-15

Flaw (verbatim). The kill's planted negative is on a mode the verse excludes from the claim, so it does not make the forward kill fire.

Number (verbatim). Kill: 'The kernel's forward reads $\max|\Delta|>0$ over $8$ repeats'. Plant: 'the fast backward mode must read $\max|\Delta|>0$ on $dQ$ on at least $1$ of $8$ repeats'. Clause (b) forbids the fast mode on every deciding cell, so the plant exercises a path the claim does not cover; the forward's fixed-order claim has no plant that fires it.

Replacement survived the same attack. Yes.

Repair applied. Applied though filed note. The forward plant MARS specifies is written in, in both of its forms, and the old plant is struck as a plant and kept as a control. Both forward plants keep the arithmetic unchanged and break the fixed-order claim, and both must read $\max|\Delta|>0$ on at least $1$ of $8$ repeats at $(s,n)=(4096,1024)$: (F1) a data-dependent tile order — the off-diagonal loop over $l$ visited in an order derived from the row maxima $m_k$ rather than the fixed $l=0,\dots,k-1$, changing the float32 summation order per draw; (F2) a tl.atomic_add inserted in the row-sum reduction that forms $\ell_k$, order-nondeterministic by the same mechanism that makes the fast backward drift. A kernel reading bitwise on both is not being measured for order sensitivity and the instrument is refused. The fast-backward reading is kept and relabelled as the separate sensitivity control it is — it certifies the harness can see atomic drift on this build, it is not this kill's plant, and its own kill lives in clause (b)'s two-mode price table. The cheapest-first table's 03.10 row carries F1, F2 and the relabelling.

C-15 · verse 03.10 · severity note · mechanism CHARTER §2 Voice (third person, one idea per sentence, no hedging in place of a number)

Flaw (verbatim). The Hypotheses field contains a rhetorical self-question in place of a statement.

Number (verbatim). 'Hypotheses: 03.9's observation holds for the diagonal-block solves at $C\le128$ (a smaller reduction than the record's $64$? no — $C$ is the reduction length, so $C=128$ is outside the observed regime and is re-measured)'.

Replacement survived the same attack. Yes.

Repair applied. Applied though filed note, and stated flat as MARS requires, with the record's field quoted rather than described: 03.9's bitwise observation was taken at reduction length $64$ (READ results/k_cert_local.json, determinism.reduction_length = 64, with determinism.flag_on.forward.bitwise = true and max_abs = 0.0 over $8$ repeats at $n=512$, RUN this session); $C$ is the reduction length of the diagonal-block solve, so $C=32$ and $C=64$ are inside the observed regime and $C=128$ is outside it and is re-measured before any deciding cell runs at that $C$. No question mark and no self-address remains in the field.

C-16 · verse 03.4 · severity note · mechanism V-15, D-CALIB

Flaw (verbatim). The planted negative fires the prediction's kill rather than the counter's, which is the estimate the verse carries.

Number (verbatim). The verse's kill for the counter is 'the L4 exponent interval at $n=8$ across the four $s$ contains $2$'. The plant is 'the Python for-loop substitution at $s\in\{64,256\}$ must read an exponent interval excluding $2$ toward $3$' — the opposite reading. No plant is given on which the counter's kill fires.

Replacement survived the same attack. Yes.

Repair applied. Applied though filed note. 03.4 now carries one plant per direction, labelled, because the verse carries two kills and each needs a plant that fires its own. (i) For the prediction's kill (an interval excluding $2$ toward $3$): the Python for-loop row-by-row substitution at $s\in\{64,256\}$, kept, and now named as the prediction's plant rather than the verse's. (ii) For the counter's kill (an interval containing $2$), which is the estimate this verse carries under D-CALIB and which the (i) plant reads in the opposite direction and therefore cannot fire: a batched trsm at the occupancy row's top — $s=64$, $n=2048$, $h=1$, so $nh=2048\gg\mathcal P$ at any $\mathcal P$ the device admits ($\mathcal P\ge24$ by C-03's SM count) — must read an exponent interval containing $2$, since at that $nh$ every chain is hidden by construction and the arithmetic term alone is left; an instrument reading an interval excluding $2$ there cannot see a hidden chain as hidden, and the counter's kill is refused on it. The cheapest-first table's 03.4 row names the counter's plant.

Batch 3, closed. Sixteen findings: zero strike, ten repair, six note; sixteen repaired, none left OPEN, no verse deleted and no Terminal newly in force from this batch. One finding was filed replacement_survives: false — C-05, 03.17's If-killed declaring itself to need no kill — and it is repaired by giving that replacement a kill on a different number, a different $s$ and a different residency $n$ (a sign reversal at $s=4096$ against the $s=1024$ reading), never by renaming the verse's own; MARS's alternative, a kill frozen against the [V]-graded $O(n^{4/3}d)$, is refused in the verse with the reason printed, since the same verse marks that exponent NOT MEASURED. Four of MARS's cross-file pin sets did not resolve as filed and each is re-resolved in the verse with the old pin printed here rather than silently replaced: 04:151 → 04:170; 04:164 and 06:349 → 04:183, :781 and 06:847, :849; 06:349 and 08:462 → 06:403-404 and 08:552; 08:472 and 08:470 → 08:138, :562 and 08:560. Two of MARS's numbers were stale or off and are corrected in the verses with the correction printed: the fitted law evaluates to $0.01016513$ s and not $0.010163$ s, and 03.1/03.6's threshold is $\rho_\Delta>1.167$ and not $2.2\times$, the $2.0\times$–$6.6\times$ gap having been returned to the per-row gate settle arm in batch 2. The underlying mechanism of every finding stands unchanged. Four new CORRECTIONS.md rows are owed against files outside this book — 08_ARCHITECTURE.md:138, :562; 08_ARCHITECTURE.md:560; 04_BEDS_AND_INSTRUMENTS.md:170 jointly with 03.7 kill (i); and 06_PREDICTIONS.md:403-404 with 08_ARCHITECTURE.md:552 — bringing the total owed to six; none is written here, this repairer editing only book 03 and making no git write.

D-01 · verse 03.9 · severity note · mechanism P-1, V-16

Flaw (verbatim). The record's only determinism producer is a different arm's certificate cell, and the verse's three-valued Statement prints a reading no results/ file carries.

Number (verbatim). results/k_cert_local.json determinism reads arm: "arm_smprime", device: "cuda", reduction_length: 64, n: 512, with quantities forward, gradient, hop, cumsum — no solve_triangular quantity exists in the file (which is why N-01 adds it as the fourth). The solve_triangular bitwise reading has no results/ row at all; the verse says so under P-1 but still prints it in the Statement's 'what the record owns' as a three-valued reading rather than as prose.

Replacement survived the same attack. Yes.

Repair applied. Applied though filed note, and applied exactly as required rather than by weakening the sentence. 03.9's Statement now carries the three-outcome table explicitly, ten rows, and no row is a reading without a producer. The four owed rows — solve forward strict, solve backward strict, softmax forward, softmax backward — each read NOT MEASURED — needs N-01, the first two marked "none; the narrative sentence has no results/ row (P-1)" and the control pair marked "none; V-23, never tested". Beside them, printed and not described, is the arm_smprime cell that does exist, verified RUN this session against the blob at the pin: arm: "arm_smprime", device: "cuda", reduction_length: 64, $n=512$, $8$ repeats (READ results/k_cert_local.json:1195-1198 @ 99777ab), with forward bitwise $\max|\Delta|=0.0$ under both flags (:1210-1213, :1246-1249), gradient bitwise $0.0$ with the flag off (:1217-1220) and not executable with it on, RuntimeError: cumsum_cuda_kernel does not have a deterministic implementation (:1253-1255), hop bitwise $0.0$ under both flags (:1203-1206, :1239-1242), and cumsum not executable with the flag on (:1262-1264). MARS's clause that no solve_triangular quantity exists in the file is stated in the Statement in the flat form — "No file under results/ carries a solve_triangular quantity, so no row of the four is a reading" — and the bitwise forward-and-backward sentence is demoted to prose only, named as asserted in three narrative documents with no results/ row behind any of the three, quoted in Evidence as narrative, and explicitly excluded from the table. The cumsum raise keeps its status as an actual row, on the arm_smprime cell, and is labelled as such. The Kill, If-killed and Terminal are unchanged: N-01 was already the named producer and the Terminal already read "no producer and no guarantee", which this repair now makes visible row by row rather than in a clause.

Batch 4, closed. One finding: zero strike, zero repair, one note; one repaired, none left OPEN, no verse deleted and no Terminal newly in force from this batch. The finding was filed replacement_survives: true and 03.9's replacement is untouched — the defect was a table printing a reading its producer does not carry, not a chain that dies with its verse. Ten READ line pins into results/k_cert_local.json were resolved against git show 99777ab:results/k_cert_local.json this session before being written, none of them stale. No number changed value; four rows changed class, from an implied reading to NOT MEASURED — needs N-01. No new CORRECTIONS.md row is owed by this batch, leaving the total owed at six from batch 3; none is written here, this repairer editing only book 03 and making no git write.

\(d\)	largest admissible \((B_M,C)\) with \(B_M=C\)	SRAM \(4(4Cd+C^2)\)	\(I_{\rm fused}=B_M/2\)	against the \(74.5\) FLOP/B ridge
16	\((128,128)\)	\(98{,}304\) B \(=98.3\) KB	\(64\) FLOP/B	\(85.9\,\%\) — within \(14.1\,\%\), compute-bound side
32	\((64,64)\)	\(49{,}152\) B \(=49.2\) KB	\(32\) FLOP/B	\(42.9\,\%\) — memory-bound
64	\((64,64)\)	\(81{,}920\) B \(=81.9\) KB	\(32\) FLOP/B	\(42.9\,\%\) — memory-bound
128	\((32,32)\)	\(69{,}632\) B \(=69.6\) KB	\(16\) FLOP/B	\(21.5\,\%\) — memory-bound

block	cells	arithmetic
\(s\in\{64,256,1024\}\times n\in\{8,32,128\}\)	\(36\)	\(4\) arms \(\times\,3\,n\,\times\,3\,s\)
\(s=4096\) at the route's residency \(n\)	\(4\)	\(4\) arms \(\times\,1\,n\)
occupancy row, shape only, \(s=64\), \(n\in\{1,2,\dots,2048\}\)	\(12\)	\(1\) arm \(\times\,12\,n\)
total	\(52\)

03 — KERNEL: the fast path, its cost law, its determinism, and the route ladder that prices T1#

Part A — the operator, its arithmetic, its bytes, its ceilings#

03.1 — The route ladder: five links, each with a kill and a price#

03.2 — The per-step MAC model: one solve against \(L\) stacked attention layers#

03.3 — Bytes and arithmetic intensity: the materialised route is memory-bound, the fused route need not be#

03.4 — The wall-clock crossover in \(s\): where the depth-\(s\) chain becomes visible, and the prediction with its counter#

03.5 — The \(n\) and \(s\) ceilings on the certified device, and the table the fused route restores#

Part B — the cost law as a specification, and what T1 means (B14, B25)#

03.6 — The measured increment is a per-op floor and licenses no ratio#

03.7 — The cost-law experiment as a specification: synchronised, interleaved, order-recorded#

03.8 — T1 operationalised: GPU-seconds-to-floor, and the kernel's half of it#

Part C — determinism (B13)#

03.9 — The determinism regime: three outcomes, one observation, one hole#

03.10 — What the kernel must guarantee: a fixed-order forward, and a priced deterministic backward#

Part D — the kernel design as a specification (no code is written)#

03.11 — The chunked block-triangular recurrence#

03.12 — Fusion with the online causal softmax: tiles, statistics, and the one-pass schedule#

03.13 — Numerical stability of the solve in float32 and bfloat16#

03.14 — The backward pass of the fused solve#

03.15 — The Sherman–Morrison column route for \(m\) candidate moves (Bet G), with its price#

03.16 — The CSR two-stage path and its certificate (link L3)#

03.19 — The Mapper schedule: the cover-to-tile quantiser, priced against the gather it must not become (B15's third kernel)#

03.17 — The published blockwise evaluation is the fellow on the cost axis, never the beaten baseline#

Part E — the device (B2, kernel half)#

03.18 — The device re-certification protocol for the installed stack#

Census rows closed#

Kills, cheapest first#

Attacks answered#

link	route	what it computes	depth (serial rounds)	MACs per head, increment over the \(s^2d\) softmax head	memory class
L1	fused Triton kernel (03.11–03.14)	online causal softmax tiles feeding a chunked block-triangular solve; \(P\) never materialised	\(s/C\) chunk steps	\(+s^2d/2\) (exact, and independent of \(C\): 03.11)	\(O(sd)\) retained per sequence
L2	chunked torch solve (03.11 in torch ops)	per chunk-row \(k\): the \(C\times(k{+}1)C\) logit block, exact softmax, `solve_triangular` on the \(C\times C\) diagonal block, one `matmul`	\(s/C\) launches \(\times\,4\) kernels	\(+s^2d/2\) (exact, and independent of \(C\): 03.11)	\(O(sC)\) transient, \(O(s^2)\) retained unless checkpointed per chunk
L3	CSR two-stage path (03.16)	a certified causal tile schedule; Neumann \(K\) hops on the sparse \(P\) with the union certificate	\(K\)	\(+K\cdot\mathrm{nnz}(P)\,d\)	\(O(\mathrm{nnz})\)
L4	dense `solve_triangular` with \(n\) declared per \(s\) (03.5)	`torch.linalg.solve_triangular` on the materialised \([n,h,s,s]\) operator	\(s\) inside cuBLAS `trsm`	\(+s^2d/2\)	\(O(s^2)\) retained, \(C_{\rm OP}\in[3.823,5.823]\) tensors
L5	withdrawal	—	—	—	—

verse	kill (frozen number)	price	replacement verse / link
03.2	fitted \(\tau_{\rm chunk}\) interval contains \(0\) (replacement's kill)	\(0\) GPU-s from S-66's journal	Terminal of 03.2
03.5	N-18 R-squared \(<0.99\) or measured/predicted outside \([0.9,1.1]\) (replacement's kill)	\(\approx9\) GPU-min DERIVED	Terminal of 03.5
03.9	a deciding cell's margin below \(10\times\) its measured forward floor (replacement's kill)	\(0\) GPU-s	Terminal of 03.9
03.9	solve forward $\max	\Delta	>0$ or backward raises under the flag (N-01)
03.10	kernel forward $\max	\Delta	>0$ over 8 repeats; plants F1 (data-dependent tile order) and F2 (`tl.atomic_add` in the row-sum) on the forward, the fast-backward reading relabelled a sensitivity control
03.1 / 03.6	synchronised increment ratio \(\rho_\Delta>1.167\) at \((64,2048)\), both arms inside one interleaved block (N-02)	\(\approx1\) GPU-min	every price re-derived; Neumann re-enters if a hop reads below the solve
03.12	\(\gamma=0\) parity not `torch.equal` against the kernel's own softmax path	seconds	construction error; the single-path schedule with \(\gamma\) a runtime scalar, then L2
03.12	\(\gamma=0.5\) parity against L4's \(z\) at \((64,2048)\) above \(2.325\times10^{-5}\\|V\\|_\infty\) — the step-2 rescale defect that kill (i) is blind to by construction	seconds, one launch	the step-2 \(A_k\) rescale; else L2
03.11	chunk-vs-serial parity above \(\gamma_{s+1}(1+\gamma)\\|V\\|_\infty/(1-\gamma)^2\) (\(4.64\times10^{-2}\\|V\\|_\infty\) at \(s=4096\), \(\gamma=0.9\))	seconds	doubling variant on the plus recurrence, then L4
03.13	float32 single signed solve outside \(\gamma_{s+1}(1+\gamma)\\|V\\|_\infty/(1-\gamma)^2\) on 1,024 draws	\(\approx2.6\) s at \(s=64\)	compensated accumulation at \(4u(1+\gamma)\\|V\\|_\infty/(1-\gamma)^2\)
03.16	one exceedance of \(\delta_{\rm union}^{O}\\|V\\|_\infty=(\varepsilon/(1-\gamma)+\gamma^{K+1})\\|V\\|_\infty\) on 1,024 draws, or \(\ge\mathrm{sd}\)(label) (K-I)	\(\approx2.6\) s at \(s=64\)	L4 at \(n\) per \(s\)
03.12	parity against L4's \(z\) at \((4096,64)\) outside \(4.641\times10^{-2}\\|V\\|_\infty\) (kill (ii))	seconds	L2, then 03.11's Terminal and L4
03.19	(K-M) Mapper schedule not better than do-nothing by \(\mathrm{MDE}_8\) in \(\varphi\)-NRMSE at matched visited tiles, \(s=1024\); or (K-M2) builder above \(3.15\times\) the attention it schedules	`NOT MEASURED`; 2 evenings `[ASSUMED]`	the merged salience schedule, killed in turn by K-I; then L4
03.14	relative gradient parity above \(\Theta_{\rm rel}=4\gamma_{s+1}\big((1{+}\gamma)/(1{-}\gamma)^2+\kappa_1(1{+}\gamma\pi)\big)\) at \((64,2048)\); NOT DECIDABLE where \(\kappa_1(1{+}\gamma\pi)\ge1/(4\gamma_{s+1})\)	seconds	scalar float64 central-difference check, \(10^{-3}\) on 8 directions
03.15	K-G-adverse: column route \(\ge8\times\) one solve at \((2048,64,16)\), \(m=8\) (Bet G counter); K-G-flattering: a \(\le0.5\times\) reading not repeated at \((1024,128)\) and not separated from the launch floor by the \(m\in\{1,8\}\) pair is journalled LAUNCH-FLOOR, NO READING	seconds at \(s=64\); `NOT MEASURED` at \(s=1024\)	\(m\) stacked right-hand sides; then \(m\) solves
03.5	reserved above the \(0.90\) budget \(7.196\) GiB \(=7.727\times10^9\) B at an admitted \((s,n)\) (N-20); the \(7.996\) GiB total is the hard limit, stated separately; plants (P1) L4 at \((4096,16)\) run on the device and (P2) the certificate's \(7.504\) GiB cell — the arithmetic refusal is the prediction, not a plant	\(\approx3\) GPU-min	constants re-solved (N-18); \(n\) by bisection
03.8	\(k^\star_{\rm shape}/k^\star_{\rm softmax}\ge0.907\) on BED-S (04.2, which does not admit at the design point, so the kill is journalled NOT EXECUTABLE), seeds \(0\)–\(7\), \(k^\star\) at 04.10's ladder level \(\tau\); third reading NOT REACHED	\(8(k^\star_{\rm shape}\cdot0.01120613+k^\star_{\rm softmax}\cdot0.01016513)+4.0\) s DERIVED, \(\ge29.6\) s at the \(150\)-step budget	T1 against the matched \((32,4,16)\) skyline at \(1.143\); then withdrawn
03.18	(a) every gate NOT EXECUTABLE on the installed `2.14.0+cpu` triple, journalled as a reading	\(0\) GPU-s (`RUN` this session)	the pin of step 1 (book 05)
03.18	(b) any `k_cert.py` gate fails on the pinned `2.5.1+cu121` triple — the prerequisite of 16 of this book's 19 verses' kills	\(523.9\) s plus book 05's environment pin	reproduce `ab5b485` on `2.5.1+cu121`; then bands across certificates
03.14	peak bytes not linear in \(s\) (R-squared \(<0.99\)) or operator term above \(0.5\) tensors	\(\approx9\) GPU-min	per-chunk checkpoint on L2; then the L4 column is the hard limit
03.4	L4 exponent interval at \(n=8\) contains \(2\) (the counter's own kill; plant: batched `trsm` at \(s=64\), \(n=2048\), \(nh\gg\mathcal P\ge24\), must read an interval containing \(2\))	\(\ge1{,}248\) GPU-s DERIVED (03.7)	crossover on the memory axis (03.5)
03.7	\(\lvert\rho_{\rm order}\rvert\ge0.25\), \(p\le0.05\) on the \(52\times8=416\) readings; or `secs_unsync` not below `secs_sync` on 6/8; or R-squared \(<0.99\)	\(\ge1{,}248\) GPU-s DERIVED (\(\ge3{,}744\) under the three-process reading)	blocked design with paired differences
03.2	solve-to-\(PV\) wall-clock ratio above \(3\) at any \(s\le1024\) (N-09)	inside S-66	two-term seconds model (03.4)
03.1	L4 exponent interval excludes \(2\) toward \(3\) (K-9)	\(\ge1{,}248\) GPU-s DERIVED (03.7)	L5: the T1 sentence withdrawn
03.3	fused forward above \(2\times\) both roofline predictions at \((4096,16,128,1024)\) — \(100.7\) ms, twice the \(50.33\) ms bytes prediction	seconds, once the kernel exists	additive launch-and-latency model
03.12	fused kernel slower than the matched \((32,4,16)\) fused-softmax stack at \((4096,1024)\)	seconds, once the kernel exists	L2
03.11	chunked slower than serial L4 at every \(C\) at \(s=4096\) (N-11)	minutes once built `[ASSUMED]`	L4 shipped; verse closes NOT NEEDED
03.17	\(t_{\rm shape}/t_{\rm blockwise}\le1.00\) at \(s=1024\) at matched exactness, both \(\delta\)'s printed (N-13); the earlier \(\mathrm{MDE}_8\) accuracy form is struck (CHARTER §1)	the \(s=1024\) block of 03.7 plus N-13's control, `NOT MEASURED`	the concession narrows to the publication; killed in turn by a sign reversal at \(s=4096\); then the Terminal