01 — THEORY: ACCURACY#

JUPITER (MYCROFT), derivations. Written 2026-09-05 under CHARTER.md at HEAD 99777ab. Every READ pin below is path:line @ 99777ab unless another commit is named; every RUN number is quoted from the record with its producer named, and no code was written.

Preface — what this book decides, and the verse that decides it first#

This book decides one question: on which labels the resolvent read $O=(1-\gamma)P(I-\gamma P)^{-1}V$ with absorbing boundary rows is exact, and at what error a per-row softmax of matched parameters cannot fit those labels — so that "more accurate" in forms A1 and A2 (CHARTER.md §1) is a rate, not a hope. Verse 01.1 decides it first: the read is exact on the four vector labels iff the arm's $\hat P$ equals the environment chain on the reachable set, with the propagation constants printed. Verse 01.4 writes the $\mathrm{hop}_k\to$ committor reduction the record left unwritten; verses 01.5–01.7 print every obstruction's vacuity line at the planned geometry ($s=64$, $d=16$, $h=1$, $p=32$; BED-S at $t^\star=8$, $m=8$, $K=2$, $N=8$, $4{,}769$ parameters) and the geometry at which each becomes non-vacuous. Verses 01.10–01.11 resolve the Cheeger and Bellman hazards. Verses 01.13–01.17 state the ChaCAL delta as theorems under the standing kill K-E1. Verses 01.18–01.22 turn the LEAPABLE fields into named theorem targets with Lean grades. The honest sentence of CEQ_SHAPE.md §3.8 is the Terminal of every chain.

Notation. Positions $[s]=\{0,\dots,s-1\}$; $P$ causal row-stochastic with $P_{ii}>0$ (regime S, READ docs/CEQ_SHAPE.md:286-293); boundary sets $\mathcal A=\mathcal A_{\rm sink}\sqcup\mathcal A_0\sqcup\mathcal A_1\sqcup\dots\sqcup\mathcal A_K$, $0\in\mathcal A_{\rm sink}$, $T=[s]\setminus\mathcal A$, $Q=P_{TT}$, $R_k=P_{T\mathcal A_k}$ (:295-310); $z=(I-\gamma P)^{-1}V$, $\Pi_\gamma=(1-\gamma)P(I-\gamma P)^{-1}$, $O=\Pi_\gamma V$ (:311-317); the committor $q^{(k)}=(I-Q)^{-1}R_k\mathbb 1$ on $T$; $P_{\rm env}$ the bed's latent chain; $\hat P$ the arm's. $|T|\in\{56,\dots,60\}$ at the design point (DERIVED: four sets $\mathcal A_{\rm sink},\mathcal A_0,\mathcal A_1, \mathcal A_2$ at $K=2$, each of size $1$ or $2$, so $|\mathcal A|\in\{4,\dots,8\}$ and $|T|=64-|\mathcal A|$; the earlier $\{57,60\}$ was printed as DERIVED with no derivation and excluded $|\mathcal A|=5,6,8$, which the stated geometry admits). Mechanism ids are MISTAKES.md headings; laws are CHARTER.md §4.

$K$ and $K'$ are two objects, and this book writes both. $K$ is the number of constraint sets, fixed at $K=2$ at the design point above, giving $K+2$ value channels. $K'$ is the Neumann truncation order — the number of reused hops a stack takes before it is read — with registered support $K'\in\{1,2,4,8,16\}$ (READ 08_ARCHITECTURE.md verse 08.11, whose Hypotheses print the same support and the same separation; cited by verse id because book 08's line numbers move under its own repairs, and at repair time the certificate row is at :46). CHARTER.md §6's scope sentence writes both with one letter; book 08 does not, and from this repair neither do verses 01.7 and 01.16, whose certificate is $\gamma^{K'+1}/(1-\gamma)$ and never $\gamma^{K+1}/(1-\gamma)$.

Instrument routing, restated at repair (CHARTER.md §3, P-4). CHARTER.md:179 requires a NOT MEASURED clause to name "the instrument, specified in the instruments book". Book 04 carries verses 04.1–04.21 and the card ids S-10, S-12, S-13, S-24, S-28, S-35, S-61, S-62, S-73; a grep of 04_BEDS_AND_INSTRUMENTS.md counts $0$ for "collision instrument", "Hankel-rank", "margin probe", "Schur" and "SVD", and its one "Dirichlet" hit is 04.15's manifest field committor_route ∈ {dirichlet_gamma1, discounted_gamma}, a route and not an instrument. Six instruments this book needs are therefore unspecified in the canon — the margin probe (01.4), the collision instrument (01.6), the Dirichlet-conductance instrument (01.10), the Hankel-rank probe (01.18), the Schur instrument (01.20), the SVD of $\hat\Pi_\gamma$ (01.21) — and every clause naming one now reads "unspecified in the canon" with the shape written out here as a specification book 04 does not yet carry, and the kill that depends on it marked unreachable until it does. Separately, this book routed kills to the record's card ids S-02, S-17, X-2, X-4, R-20 and J-L18, none of which is a verse of book 04; every such routing now carries a READ docs/CEQ_SHAPE.md:<line> @ 99777ab pin instead, and says that the card is the record's and not book 04's. (The finding that convicted this also named S-31, S-33, S-64 and N-15; a grep of this book at repair time finds none of the four, and the same defect additionally covers J-D3, J-D6, J-D8, J-L0, J-L6 and R-07, which the finding did not name and which appear here in Evidence position only.) No verse of this book adds an instrument to book 04: that is book 04's own repair, and until it lands the reachability column of the kills table below reads what it reads.

Part I — The exact class#

01.1 — The read is exact on four labels, and only through $\hat P$#

Statement. Let $\hat P$ be the arm's causal row-stochastic operator with rows $e_a$ on $a\in\mathcal A$. The read represents, with zero error as functions of $\hat P$, exactly the linear images of its resolvent: $$z^\star=(I-\gamma\hat P)^{-1}V,\qquad \Delta z=(I-\gamma\hat P')^{-1}\big(\Delta V+\gamma\,\Delta\hat P\,z\big),\qquad q^{(k)}=(I-\hat Q)^{-1}\hat R_k\mathbb 1,\qquad a^\star=\arg\max_a q^{(0)}(\mathrm{do}\,a).$$ One direction only. If $\hat P=P_{\rm env}$ on the set of positions reachable from the query, all four labels are exact. The converse holds for $z$ alone, and only at $\gamma>0$: if $(I-\gamma\hat P)^{-1}V=(I-\gamma P_{\rm env})^{-1}V$ for every $V$ then $\hat P=P_{\rm env}$ on the reachable set, by inverting both resolvents and dividing by $\gamma$. The converse is false at $\gamma=0$, a value these Hypotheses admit: $z^\star=(I-0)^{-1}V=V$ for every $\hat P$, so the label matches with $\|\hat P-P_{\rm env}\|_\infty=1$ on the reachable set. It is false for the committor at every $\gamma$: with $\mathcal A_{\rm sink}=\{0\}$, $\mathcal A_0=\{1\}$, $T=\{2\}$ and row $2=(\alpha,1-\alpha-\beta,\beta)$, $q^{(0)}_2=(1-\alpha-\beta)/(1-\beta)$ reads $0.5$ at both $(\alpha,\beta)=(0.5,0)$ and $(\alpha,\beta)=(0.25,0.5)$ — two chains $0.25$ apart in sup norm with identical labels (DERIVED). The argmax clause inherits that many-to-one map, so no "iff" is written for $q$ or for $a^\star$. Where $\hat P\ne P_{\rm env}$ the errors are bounded, not zero: $$\|\hat z-z\|_\infty\le\frac{\gamma\,\|\hat P-P_{\rm env}\|_\infty\,\|z\|_\infty}{1-\gamma},\qquad \|\hat q^{(k)}-q^{(k)}\|_\infty\le\kappa\big(\|\hat Q-Q\|_\infty+\|\hat R_k-R_k\|_\infty\big),\quad \kappa:=\|(I-\hat Q)^{-1}\|_\infty .$$ The argmin inherits the committor bound: $a^\star$ is read correctly whenever the gap between the best and second-best goal committor exceeds $2\kappa(\|\hat Q-Q\|_\infty+\|\hat R-R\|_\infty)$, and from this repair the clause is stated at the $\kappa$-free upper bound $\kappa\le1/\varepsilon_{\rm esc}$ derived in the Hypotheses, because $\kappa$ itself is NOT MEASURED and the record's $7.9$ is a condition number in different units.

Hypotheses. $\gamma\in[0,1)$ for $z$ and $\Delta z$; $\gamma=1$ on the transient block for $q$ with $0\in\mathcal A$ so that $I-\hat Q$ is a unit (verse 01.10; READ docs/CEQ_SHAPE.md:539-548); every boundary set precedes the query and the query lies in $T$ (:295-310, facts F1 and F2). Non-vacuous at every geometry: the bounds are identities of the class. At the judge's draw ($s=32$, $\gamma=0.6$, seed 0, RUN[J]) $\max_T\hat P_{ii}=0.6927$ and the sup-norm condition number of $I-\hat Q$ is $7.9$ (READ docs/CEQ_SHAPE.md:2282, a line the record opens "Withdrawn candidate, recorded so it is not re-found (V-7 discipline)" — the provenance is printed because it is a withdrawn-candidate paragraph and not a card). That number is not $\kappa$: $\mathrm{cond}_\infty(I-\hat Q)=\|I-\hat Q\|_\infty\,\|(I-\hat Q)^{-1}\|_\infty$, so $\kappa=7.9/\|I-\hat Q\|_\infty$ and $\kappa\le7.9$ holds only if $\|I-\hat Q\|_\infty\ge1$, which is printed nowhere in the record. $\|I-\hat Q\|_\infty$ is printed here as the interval the class forces, and $\kappa$ is derived from the quotient with both endpoints. For a lower-triangular non-negative $\hat Q$ with row sums $r_i=\sum_j\hat Q_{ij}\le1$ the $i$th row sum of $|I-\hat Q|$ is $(1-\hat Q_{ii})+(r_i-\hat Q_{ii})=1+r_i-2\hat Q_{ii}$, so $$1-\max_T\hat P_{ii}\ \le\ \|I-\hat Q\|_\infty\ \le\ 2\qquad(\text{DERIVED; } r_i\ge\hat Q_{ii} \text{ gives the left, } r_i\le1,\ \hat Q_{ii}\ge0 \text{ the right}),$$ which at the judge's draw reads $\|I-\hat Q\|_\infty\in[0.3073403106639614,\,2]$ from $\max_T\hat P_{ii}=0.6926596893360386$ (RUN[J]), and therefore $$\kappa=\frac{7.9}{\|I-\hat Q\|_\infty}\in[3.95,\ 25.70]\qquad(\text{DERIVED from the quotient}).$$ The interval's upper endpoint is a bound in the direction the argmin clause needs and its width is a factor $6.51$; the earlier text printed only the lower endpoint $\kappa\ge3.95$, which is the opposite direction, and printing one endpoint of a two-sided interval was itself the defect. Neither endpoint is a read: $\kappa$ is NOT MEASURED — needs a direct sup-norm read of $\|(I-\hat Q)^{-1}\|_\infty$ on the drawn $\hat P$, unspecified in the canon (no verse of book 04 carries it); on BED-S at the design point it is NOT MEASURED for the same reason and for want of a cell (B5).

The $\kappa$-free bound the argmin clause is restated on, derived here. $\hat Q\ge0$ with $\|\hat Q\|_\infty=\max_{i\in T}r_i<1$ gives the Neumann bound $\|(I-\hat Q)^{-1}\|_\infty\le1/(1-\|\hat Q\|_\infty)$, and on the causal class $1-r_i=\sum_{a\in\mathcal A}\hat P_{ia}$ is the one-step escape mass of row $i$, so $$\kappa\ \le\ \frac1{\varepsilon_{\rm esc}},\qquad \varepsilon_{\rm esc}:=\min_{i\in T}\sum_{a\in\mathcal A}\hat P_{ia}\qquad(\text{DERIVED}),$$ and $\varepsilon_{\rm esc}>0$ for every finite-logit softmax row, since $0\in\mathcal A$ and a causal softmax has full support on $j\le i$. This is a different object from the condition number — the operator's own rows, not a norm of its inverse — read by a different instrument, the row-sum pass verse 01.19 registers as reachable today at $0$ GPU-s, and it returns a different number, $\varepsilon_{\rm esc}$ and not $7.9$. Every clause of this verse that multiplies by $\kappa$ is stated at $1/\varepsilon_{\rm esc}$ from this repair, with $\varepsilon_{\rm esc}$ NOT MEASURED — needs the row-sum pass of 01.19's frozen census extended to print $\min_{i\in T}\sum_{a\in\mathcal A}\hat P_{ia}$, $0$ GPU-s on a saved $\hat P$; the argmin gap clause reads $2(\|\hat Q-Q\|_\infty+\|\hat R-R\|_\infty)/\varepsilon_{\rm esc}$ and is licensed once that one number is printed, rather than waiting on an inverse-norm probe the canon does not specify. The bound is honest about its own failure mode: a row whose escape mass is $O(e^{-L})$ sends $1/\varepsilon_{\rm esc}$ to $O(e^{L})$, so the clause is a measured quantity and never an assumption.

Evidence. DERIVED (four steps): (i) $(I-\gamma\hat P)(\hat z-z)=\gamma(\hat P-P_{\rm env})z$ and $\|(I-\gamma\hat P)^{-1}\|_\infty\le1/(1-\gamma)$ by the Neumann series on a row-stochastic matrix (verse 01.16); (ii) $(I-\hat Q)(\hat q-q)=(\hat R-R)\mathbb 1+(\hat Q-Q)q$ with $\|q\|_\infty\le1$; (iii) the displacement identity READ docs/CEQ_SHAPE.md:460-470, three independent RUNs at $1.03\times10^{-15}$, $1.2\times10^{-15}$, $1.36\times10^{-15}$; (iv) the committor identity on BED-1's real sets, max-abs $0.0$, harmonic residual $1.0408340855860843\times10^{-17}$ (RUN, READ docs/CEQ_SHAPE.md:919-921). The "reachable set" clause is F2 as a theorem target (later_boundary_unreachable [M], READ docs/CEQ_SHAPE.md:1451-1456).

Mechanism. D-2 (MISTAKES.md:710, the label is the arm's own resolvent — the verse says so and files the shape-versus-softmax contrast VOID by registration); D-1 (:677, the label class leaves the scalar-readout regime); V-24 (:1658, every identity ships its mutilation battery); V-17 (:855, the bounds carry their units, $\kappa$ printed).

Kill. On the bed's own generator (Ruling 7: generator + seed + hash), with $\hat P:=P_{\rm env}$ substituted, any of the four label identities reading above $10^{-12}$ in max-abs on any of $512$ draws at the design point; instrument: the identity script specified at READ docs/CEQ_SHAPE.md:1606-1628 @ 99777ab (card S-02, raise on a missing key — a card of the record, not a verse of book 04, which carries no such instrument); price $0$ GPU-s, one evening. Planted negatives (each must fail at $O(1)$): declare $\mathcal A_k$ on the wrong set ($q$ moves by $O(1)$, READ docs/CEQ_SHAPE.md:387); a non-causal $P$ (backward displacement $0.0761$, $0.0868$, RUN[M], :1454); a Gaussian $V$ against the constant-value plant ($\max|\Delta z|=0.1096$ at $s=32$, $1.127$ at $s=64$, :474-476).

If killed. The identity fails only if the generator's chain is not row-stochastic on its absorbing rows or a set is undeclared (then $\det(I-Q)=0.0$, :303). Replacement formula: the reach-avoid read on the declared sets with the undeclared absorbing position added to $\mathcal A_{\rm sink}$, $q^{(\rm sink)}$ printed apart from $q^{(0)}$ (:498-500). Its Hypotheses: $0\in\mathcal A_{\rm sink}$ on $100\%$ of draws (census X-2, READ docs/CEQ_SHAPE.md:1441 @ 99777ab and :2335). Its Evidence: $\rho(Q)=0.692660$ declared against $1.000000$ undeclared (RUN[P], READ docs/CEQ_SHAPE.md:1440 @ 99777ab). Its Kill: a declared-sink draw with $\rho(Q)=1$ — a second undeclared absorbing row (a zero gate, verse 01.17's cut_makes_segment_head_absorbing) — $0$ GPU-s. If that fires: the exact class is stated per segment, with each segment carrying its own sink, goal and constraint sets (:445-450), and its Kill is the same census line re-read per segment; being a census line it is strictly cheaper than the identity battery.

Terminal. The read is a resolvent of the arm's own operator (ChaCAL's Eq. 5 with the diagonal kept, READ docs/CEQ_SHAPE.md:205-212); exactness is a property of the class and licenses no capability number; what the canon still licenses is the identity table with its plants, and it withdraws every sentence in which "exact" is read as "accurate on the bed".

01.2 — What $\hat P=P_{\rm env}$ costs in the softmax class#

Statement. A causal softmax row with finite logits has full support on $j\le i$, so $\hat P=P_{\rm env}$ holds exactly only when $P_{\rm env}$ has no zero entry below the diagonal. For a DAG environment with non-edges, the best row-wise approximation with logit margin $L$ between edges and non-edges leaks at most $$\varepsilon_{\rm row}(L)\le s\,e^{-L}\qquad(\text{DERIVED: at most } s-1 \text{ non-edge terms, each } \le e^{-L}\text{ relative to an edge term}),$$ which is $6.0\times10^{-12}$ at $s=64$, $L=30$ ($e^{-30}=9.36\times10^{-14}$); and the logit matrix has rank $\le d_{\rm model}$, so an arbitrary DAG pattern on $s$ positions is representable at $d_{\rm model}\ge s$ (one-hot keys $k_j=e_j$, $q_i=\log P_{\rm env}(i,\cdot)$ on edges, $-L$ off them). The decidable claim at the planned geometry, and the only one this verse asserts, is: the DAG family drawn by BED-S at S-13 is not representable at $d_{\rm model}=16$, decided by the sign-rank of the drawn edge sets printed per seed — representable at seed $\sigma$ iff $\mathrm{signrank}(E_\sigma)\le16$. The per-seed sign-rank census is NOT MEASURED — needs the sign-rank read on the drawn edge sets, unspecified in the canon (no verse of book 04 carries it); the hedge "not guaranteed representable" is struck under CHARTER.md:161 and appears nowhere in this book. The universal form is struck with it, and the reason is printed: a sentence of the form "not guaranteed representable for all DAG families" is compatible with K-D2 firing on one family, so its firing would leave that sentence standing and the If-killed's assertion would not follow from the kill — a kill that cannot refute its own statement (CHARTER.md:145-146, V-3). What this verse asserts is the per-family claim above and nothing wider.

Hypotheses. Regime S, $\beta=1$; node tokens carry out-adjacency as a multi-hot feature (READ docs/CEQ_SHAPE.md:1677-1683). The leak bound is non-vacuous at every $L>\log s$; the rank sentence is non-vacuous exactly when $d_{\rm model}<s$, which is the planned geometry ($16<64$).

Evidence. DERIVED (two lines above). The record's own arithmetic that a zero is a $-\infty$ logit and no finite gate produces one: V16Domain.no_prefix_scan_represents_a_zero_gate (lean/CEQ/V16Domain.lean:165, READ docs/CEQ_SHAPE.md:441-444). The identification reading $\|\hat P-P_{\rm env}\|_\infty$ per seed is NOT MEASURED — needs the arena (book 04, S-13).

Mechanism. V-25 (MISTAKES.md:1954, the hypothesis $\hat P=P_{\rm env}$ must be censused on drawn cells, not assumed); D-2; V-10 (:168, K-D2 had an empty rejection region on the undirected substrate and a non-empty one on the DAG).

Kill. K-D2 frozen: $\|\hat P-P_{\rm env}\|_\infty<10^{-3}$ on at least $6$ of $8$ seeds at the design point ($N=8$, seeds $0$–$7$, the arena's realised draws) — the arm copied the environment and every contrast is a copy-versus-no-copy statement; instrument S-13; price inside the $\approx34$ s pair ([FITTED], READ docs/CEQ_SHAPE.md:2295), one evening. Control: a plant with $d_{\rm model}=s$ and the one-hot construction above must read below $10^{-3}$ (the copy is possible), and the $d_{\rm model}=16$ arm on a random DAG must read above it on at least one seed, else the metric cannot see the rank wall.

If killed (the arm copies at $d_{\rm model}=16$ on $\ge6/8$ seeds). K-D2 firing refutes the Statement as now written, because the Statement is the per-family claim and not a universal negative: the drawn DAGs have sign-rank $\le16$ on those seeds and the family is representable there.

The replacement is re-derived at repair, because the earlier one died with the verse (V-9). The earlier replacement re-drew the family at higher sign-rank and moved the capability sentence to the committor head with kill "$\kappa\varepsilon$ at or above the label sd on $6$ of $8$ seeds". Both of its numbers are absent: $\kappa$ is NOT MEASURED (verse 01.1, and its condition-number surrogate is in different units), the label sd is NOT MEASURED (B5, no BED-S cell), and the re-draw itself needs the sign-rank read this verse has just marked unspecified in the canon — so the replacement waits on the same missing identification reading K-D2 waits on and cannot fire where K-D2 does. It is replaced, not renamed, by a reading on a different object — the bed's own generator specification, not the arm and not the drawn edge sets — decided by a different instrument and returning a different number:

Replacement — the feature-content reading. BED-S's registered generator hands the arm the environment: "Node tokens carry the out-adjacency as a multi-hot feature … the transition matrix is a deterministic function of the edge tokens" (READ docs/CEQ_SHAPE.md:1677-1683 @ 99777ab, card S-11). Under that specification $P_{\rm env}(i,\cdot)$ is a function of the token at $i$ alone, so a copy at $d_{\rm model}=16$ is a decoding of a feature the bed supplies and not a learnability number about a rank wall, and the identification reading $\|\hat P-P_{\rm env}\|_\infty$ is reported as a decoding error with the feature named beside it.

Its Hypotheses: the arena draws from S-11's generator as registered, with the out-adjacency multi-hot present in the token and no channel derived from a solve (S-11's own KILL line, D-2). Its Evidence: the generator specification as quoted, READ at the pin; the decoding error itself NOT MEASURED — needs the first BED-S cell (book 04, 04.21). Its Kill: the generator emitting tokens without the out-adjacency multi-hot — a read of the registered spec and the arena's own producer_cmd, one line, no solve — while K-D2 still fires on $\ge6$ of $8$ seeds: the copy is then not explained by the feature, the decoding reading is withdrawn, and the sign-rank route is the only one left. Its plant: a generator variant with the multi-hot deleted must read the spec line absent; a spec read that cannot tell the two variants apart is not a spec read. Price: $0$ GPU-s, one document read — strictly cheaper than K-D2, which needs a trained cell, and it is decidable today on the registered specification whereas K-D2 is not (B5). If that kill fires, the chain ends at the Terminal.

Terminal. Representability of the environment is a property of $(d_{\rm model},s)$ and of the DAG family, never of the read; the canon licenses the identification reading as a learnability number and withdraws "the shape recovers the environment" wherever $\|\hat P-P_{\rm env}\|_\infty$ is not printed beside it.

01.3 — The committor is the read at $\gamma\uparrow1$, with the discounted gap printed#

Statement. For $i\in T$, $V=\mathbb 1_{\mathcal A_k}$: $$O(\gamma)_i=E_i\big[\gamma^{\tau_k-1}\mathbb 1_k\big],\qquad \lim_{\gamma\uparrow1}O(\gamma)_i=q^{(k)}_i .$$ Two gaps, two objects, never interchanged. The $(1-\gamma)q_i$ lower bound belongs to the undiscounted-by-one read $z_i:=E_i[\gamma^{\tau_k}\mathbb 1_k]$, which is $\gamma O(\gamma)_i$ and not $O(\gamma)_i$: $$0\le q^{(k)}_i-E_i[\gamma^{\tau_k}\mathbb 1_k]=E_i\big[(1-\gamma^{\tau_k})\mathbb 1_k\big]\ge(1-\gamma)\,q^{(k)}_i$$ (DERIVED: $1-\gamma^{\tau}\ge1-\gamma$ for $\tau\ge1$). For the read $O(\gamma)$ the verse asserts only what is true of it, $$q^{(k)}_i-O(\gamma)_i=E_i\big[(1-\gamma^{\tau_k-1})\mathbb 1_k\big]\ \ge\ 0,$$ with no non-trivial lower bound: $\tau_k=1$ has positive probability whenever an $\mathcal A_k$ position is one step from $i$, and on that event the integrand is $0$. Counterexample to the $(1-\gamma)q_i$ form read on $O$: $P_{ii}=0.1$, $P_{ia}=0.9$ with $a\in\mathcal A_k$, $\gamma=0.6$ gives $q_i=1$, $O_i=0.9/(1-0.06)=0.957447$, gap $0.042553$, while $(1-\gamma)q_i=0.4$ — the inequality fails by a factor $9.4$ (DERIVED). The committor head is the exact solve at $\gamma=1$ on the transient block and carries no $\gamma$; the discounted read is a separate registered label at a bed constant $\gamma_{\rm env}$.

Hypotheses. $P$ row-stochastic including absorbing rows; $\rho(Q)<1$ (verse 01.10); $\tau_k=0$ on $\mathcal A_k$, $\gamma^\infty=0$. Non-vacuous at every $\gamma<1$; the gap lower bound $(1-\gamma)q_i$ is $10^{-6}q_i$ at $\gamma=1-10^{-6}$ on $E[\gamma^{\tau_k}]$ and on nothing else, and the RUN gap on channel $\{15\}$ reads $1.42\times10^{-7}$ at $\max_T$ (RUN[J], READ docs/CEQ_SHAPE.md:384-388), which is the record's own line for $E[\gamma^\tau]$ and is never quoted for the read.

Evidence. CITED kemeny-1960-finitemarkov [V-cat] and grinstead-1997-probability Thm 11.6 [U] own $N=(I-Q)^{-1}$, $B=NR$; the read form is DERIVED (exchange of sums, first-step analysis, monotone convergence; READ docs/CEQ_SHAPE.md:1570-1576, card J-D3); read against $E[\gamma^{\tau-1}]$ to $8.33\times10^{-17}$ (RUN[J], RUN[M], :385). Ruling: the committor head does not carry $\gamma$ because $q$ to $1\%$ at $\tau=8$ needs $\hat\gamma\ge0.99857$, $1/(1-\hat\gamma)\approx697$ (RUN[F], :336-343).

Mechanism. D-2; V-12 (MISTAKES.md:189, one absorbing target makes the label constant — $K=1$ is illegal, READ docs/CEQ_SHAPE.md:1681); V-25; P-10 (:1382, the owners are cited before the object).

Kill. The read-side plant: declare $\mathcal A_k$ on the wrong set and read $\max_T|q^{\rm wrong}-q|<10^{-3}$ on any draw — the instrument cannot see its own label; $0$ GPU-s, half an evening, on BED-1's real sets ($A=[0]$, $B=[1]$, $|T|=9$) and on one BED-S draw. The must-fire perturbation on BED-1 drives the residual from $1.04\times10^{-17}$ to $10^{-6}$, ratio $9.6\times10^{10}$ (READ docs/CEQ_SHAPE.md:918).

If killed. A wrong-set read that moves the label by less than $10^{-3}$ means the two sets are unreachable from the query (F2) — the bed, not the identity, is at fault. Replacement: the reachability census X-4 (READ docs/CEQ_SHAPE.md:3153 @ 99777ab; every set before the query, query in $T$) is made an admission line with the discard rule "a set after the query reads $q=0.0$ exactly, discard" (:1670-1673). Its Kill: X-4 below $100\%$ on $512$ draws ($0$ GPU-s). If that fires, the placement convention S-10 (the one card of this list book 04 does carry) is re-drawn with sets forced before the query and the same line re-read; cheaper again, being one generator flag.

Terminal. The committor is Kemeny–Snell's absorption probability read through the arm's $\hat P$; the canon licenses it as the exact head at $\gamma=1$, licenses $(1-\gamma)q_i$ as a lower bound on the gap of $E_i[\gamma^{\tau_k}\mathbb 1_k]$ and on no other object, and withdraws every sentence that binds that bound to the read $O(\gamma)_i=E_i[\gamma^{\tau_k-1}\mathbb 1_k]$, for which the only licensed statement is $q^{(k)}_i-O(\gamma)_i\ge0$.

Part II — D-APPROX: from "cannot represent" to "cannot fit below $\varepsilon$"#

01.4 — The $\mathrm{hop}_k\to$ committor reduction, written#

Statement. Every $\mathrm{hop}_k$ instance on $n$ tokens (each token $j$ carries a pointer $\pi(j)<j$ to an earlier token; $\mathrm{hop}_k(i)=\pi^{k}(i)$) is a committor instance on a DAG of $s'=(k+1)n+1$ positions with $K=n$ singleton absorbing sets and one dedicated sink, and $$\mathrm{hop}_k(i)=\arg\max_{j\in[n]}\;q^{(j)}_{(i,0)},\qquad q^{(j)}_{(i,0)}\in\{0,1\}\ \text{exactly on } P_{\rm env}.$$ Construction. Position $0$ is the sink and nothing else: $\mathcal A_{\rm sink}=\{0\}$, $P_{00}=1$. Positions $(j,\ell)$ for layers $\ell=k,k-1,\dots,0$ are laid out after it, so layer $k$ occupies $[1,n+1)$ and layer $0$ occupies $[kn+1,(k+1)n+1)$; edges $(j,\ell)\to(\pi(j),\ell+1)$ point to earlier positions; layer-$k$ positions are absorbing, $\mathcal A_j=\{(j,k)\}$ at index $1+j$; the walk from $(i,0)$ is deterministic and reaches $(\pi^k(i),k)$ in exactly $k$ steps. The sink is disjoint from every $\mathcal A_j$ by construction, which is what Definition 3 requires ($\mathcal A=\mathcal A_{\rm sink}\sqcup\mathcal A_0\sqcup\dots\sqcup\mathcal A_K$, READ docs/CEQ_SHAPE.md:295-297 @ 99777ab): the earlier layout, which put $\mathcal A_0$ at position $0$, gave position $0$ two disjoint labels at once and made the answer channel and the leak channel the same set on the $1/64$ of instances whose answer is token $0$ under uniform $\pi^k$. On the arm's softmax class with logit margin $L$ (edge logit $0$, every other earlier position and the self entry at $-L$), $$q^{(\mathrm{hop}_k(i))}_{(i,0)}\;\ge\;1-\frac{k\,s'\,e^{-L}}{1-s'e^{-L}},$$ so the argmax reads $\mathrm{hop}_k$ whenever the right side exceeds $\tfrac12$: at $n=64$, $k=8$, $s'=577$, $L=30$ the loss is $\le4.4\times10^{-10}$ (DERIVED: $577\cdot9.36\times10^{-14}=5.40\times10^{-11}$ per step, $\times8=4.32\times10^{-10}$).

Hypotheses. Two readings, kept apart because they need different substrates. (a) The exact reading at $\gamma=1$ is on the strict pointer DAG: no self entries, every row of a transient position supported on its single pointer target, so the walk is exactly $k$ steps and $q\in\{0,1\}$; this is the reading hop_layered_committor is stated on (verse 01.17), and it is regime N on the transient block with the absorbing rows added, not regime S. (b) The $\varepsilon$-clause is on the softmax class, regime S, with the self entry at $-L$ and the dedicated sink $\{0\}$ absorbing the leak; the arm's logits attain margin $L$ — which the arm learns, so the clause is a statement about a representable $\hat P$, not about a trained one. The two are never mixed in one sentence: (a) supplies $q\in\{0,1\}$ and (b) supplies the $\varepsilon$ around it. Non-vacuous for every $L>\log(2ks')=\log(2\cdot8\cdot577)=9.131$ at the instance above. The stall mass (J-D8's blocker, READ docs/CEQ_SHAPE.md:2156-2162) is bounded because on a deterministic pointer graph a self-loop delays the walk and never changes the absorbing target; only leak to non-pointer positions does, and each step's leak is $\le s'e^{-L}$ with the expected number of steps $\le k/(1-s'e^{-L})$.

Evidence. DERIVED (the construction and the union bound above). The reduction was NOT FOUND in the literature by sweep_expressivity.md §3.5 (READ docs/sources/sweep/sweep_expressivity.md:495-511) and "unwritten" at READ docs/CEQ_SHAPE.md:797; it is written here. Direction: committor $\supseteq\mathrm{hop}_k$ (a solver for the committor on this family solves $\mathrm{hop}_k$); the converse is not claimed and not needed.

Mechanism. P-4 (MISTAKES.md:332, claimed scaffolding that did not exist — the reduction is now text); V-3 (:72, the stall-mass bound is stated, not assumed away); P-10; D-2 (the instance family is the oracle's, and BED-S's random DAG at $K=2$ is not this family — the skyline for BED-S remains by analogy, verse 01.8).

Kill. On $512$ random $\mathrm{hop}_8$ instances at $n=64$ (seed rule: instance seed $=$ draw index), the exact oracle solve on $P_{\rm env}$ returning $\arg\max_jq^{(j)}\ne\pi^8(i)$ on any instance; $0$ GPU-s, one evening, float64 CPU. Planted negative: reverse one pointer to point forward — F2 fires, the channel reads $q=0.0$ exactly and the argmax is undefined; the instrument must raise (V-16, MISTAKES.md:828).

If killed (the exact solve mis-reads the pointer). The layered layout is wrong — an edge points forward. Replacement: the same construction with layer order verified by the census line "every edge $(j,\ell)\to(\pi(j),\ell+1)$ has a smaller position index", which is a $0$ GPU-s decidable predicate on the layout; its Kill is that predicate reading false on any instance, strictly cheaper than the solve. If the $\varepsilon$-clause alone fails (a trained arm's margin $L<\log(2ks')$), the reduction stands on $P_{\rm env}$ and the depth statements of verses 01.5 and 01.8 transfer to the oracle's family only; the replacement is the sentence "obstruction 2 is a theorem on the layered family and a lineage argument on BED-S", with the margin $\hat L=\min_{\rm edges}\ell_{ij}-\max_{\rm non\text{-}edges}\ell_{ij}$ printed per trained cell (NOT MEASURED — needs the margin probe, **unspecified in the canon**: book 04 carries verses 04.1–04.21 and no margin probe among them, so this kill is unreachable until book 04 ships one).

Terminal. The committor class contains pointer chasing at $K=n$ and $s'=(k+1)n+1$; on BED-S ($K=2$, $s=64$) the canon licenses only "hops need depth" as lineage with the inequalities of verse 01.5 printed, and withdraws the phrase "theorem about the shape's bed".

01.5 — OPEN — Obstruction 3 through the reduction: Peng's Theorem 1 with its vacuity line#

Marked OPEN at repair: the verse's only kill is an act book 04 registers as outside this canon's no-fetch rule, so no link of the chain can fire inside the canon. The Terminal below is the sentence in force today.

Statement. Function composition $f(g(x))$ on $n$-element domains is $\mathrm{hop}_2$; by verse 01.4 it is a committor instance at $K=n$, $s'=3n+1$. If peng-2024-transformer-limitations Thm 1 holds as recorded — a single $H$-head softmax layer with embedding $d$ at $p$ bits errs with probability $\ge R/(3n\log n)$, $R=n\log n-H(d+1)p>0$ — then a depth-1 single-head softmax at $d=16$, $p=32$ errs on the 2-hop committor family with probability at least the table's value, while the exact solve reads $0$:

$n$	$n\log_2n$	$H(d+1)p$	$R$	error $\ge$	status at $(d,p)=(16,32)$
$64$	$384$	$544$	$-160$	—	vacuous (the planned geometry)
$128$	$896$	$544$	$352$	$0.1310$	non-vacuous, $s'=385$, $K=128$
$256$	$2048$	$544$	$1504$	$0.2448$	non-vacuous, $s'=769$, $K=256$
$64$, $p=16$	$384$	$272$	$112$	$0.0972$	non-vacuous at half precision
$64$, $d=8$	$384$	$288$	$96$	$0.0833$	non-vacuous at half width

Hypotheses. Peng's model class (single softmax layer, communication-complexity argument), read at the abstract: CITED [V], [V-eq] owed — the theorem's own hypotheses on how $f,g$ are presented were not read at the body (READ docs/sources/sweep/sweep_expressivity.md:125-137). Under L-EQ this verse is therefore a DERIVED conditional ("if Thm 1 as recorded") and is not load-bearing until the body is read. The geometry column is DERIVED with $\log_2$.

Evidence. DERIVED arithmetic; the vacuity at $s=64$ is the record's own (READ docs/CEQ_SHAPE.md:710-719, :2952 — the earlier pin :2947-2950 did not resolve to $384<544$ at 99777ab and is corrected here); the reduction is verse 01.4.

Mechanism. V-25 (MISTAKES.md:1954) — the inequality is printed so the reader sees $0\%$ of the planned geometry admitted; P-10 (:1382) — the theorem is not cited beyond its recorded statement; V-17 — the error is in the theorem's own units (misclassification probability on its instance family), never carried to BED-S's NRMSE.

Kill — UNREACHABLE, and the verse stands OPEN on that account. The only condition that refutes the conditional is reading Thm 1 at the arXiv LaTeX source and finding its hypothesis to differ from the recorded $n\log n>H(d+1)p$, or its instance presentation to be incompatible with the layered DAG of 01.4. That act is outside this canon's rules: 04_BEDS_AND_INSTRUMENTS.md verse 04.12's Evidence (:193 at repair time; cited by verse id because book 04's line numbers move under its own repairs) closes with "No LaTeX source has been read: NOT MEASURED — needs one fetch of the arXiv source, an author action outside this canon's no-fetch rule". It is therefore repriced from the earlier "$0$ GPU-s, one evening" — which read as if the act were inside — to NOT MEASURED — needs one author fetch of arXiv:2402.08164, an act no writer, refuter or repairer of this canon may perform. Under CHARTER.md:145-148 a kill that cannot fire on the bed as registered is not a kill, so this verse carries no reachable kill today and its Terminal is in force now, not after a run. The fetch is registered as an author action with its own row in book 05 and not as this verse's kill. Control, for whenever the fetch happens: the same re-read on fagnou-2024-chacal Eq. 5, whose recorded form is already [V-fetched] — a re-read that cannot reproduce a known equation is a broken instrument.

If killed — one lineage link, then the Terminal. The two unconditional theorems are collapsed into a single link, because neither was strictly cheaper nor strictly more decisive than the Peng re-read (CHARTER.md:139-141) and both died to the same act. The link: sanford-2024-inductionheads Thm 1 and chen-2024-multilayer Thm 1.1 are cited together as lineage only, with their vacuity lines printed and no kill of their own. Sanford Thm 1 is a $\mathrm{hop}_1$ bound and is labelled $k=1$, never $t^\star=2$: through 01.4 at $k=1$ it is a committor instance at $s'=2n+1$, and $h\,m\,p=\Omega(n)$ bites at $n>h\,m\,p/c$ with $c$ the theorem's own unstated constant. At $h=1$, $m=16$, $p=32$ the product is $512$, so the geometry at which it bites is $n>512/c$, i.e. $s'>1024/c+1$, and $c$ is NOT MEASURED — needs the constant read from Thm 1's body, the same author fetch; the earlier "first violated at $n>512$" silently set $c=1$ and is struck, as is the earlier "i.e. $s\ge1024$ at $t^\star=2$", which carried the hop index $k$ into the bed's $t^\star$ and set $c=1$ in the same breath. The position count is $s'=2n+1$ and not $2n$: verse 01.4's construction lays $(k+1)n$ layer positions after a dedicated sink at position $0$, so at $k=1$ the count is $2n+1$; the extra position is the sink the reduction's repair added to keep the answer channel and the leak channel disjoint, and printing $2n$ would understate the geometry by one position. Chen Thm 1.1 at $L=1$ needs $n\ge512^{16}=2^{144}$, vacuous at every geometry any device can run. Its Hypotheses: none beyond citation. Its Evidence: [V], [V-eq] owed for both. Its Kill: none — lineage carries no kill, which is why the chain terminates here in one link rather than three.

Terminal — in force now. At $s=64$, $d=16$, $p=32$ no unconditional "softmax cannot" is licensed for the 2-hop committor, and no verse of this canon can change that by any act it is permitted to perform. The canon licenses the table above as the geometry at which an A1-form statement would become theorem-shaped ($s'\ge385$ at $K=n$), and withdraws every "cannot" sentence at the planned geometry in favour of Bet D (verse 01.8) with the counter as the point estimate (D-CALIB).

01.6 — D-APPROX by capacity: the per-position sketch, and the hypothesis it needs#

Statement. A depth-1 softmax reads, at query $i$, a $d$-dimensional aggregate $O_i=\sum_{j\le i}P_{ij}V_j$ whose weights depend on $(x_i,x_j)$ and whose values on $x_j$ alone; at $p$ bits the aggregate carries $d\,p=512$ bits at the planned geometry. The readout sees $(x_i,O_i)$, not $O_i$ alone, and $x_i$ carries the out-adjacency multi-hot of 01.2's Hypotheses, at most $s=64$ further bits, so the joint sketch carries at most $d\,p+s=576$ bits. The downstream graph of $i$ on $s=64$ positions has $2^{s(s-1)/2}=2^{2016}$ configurations, so at least $2^{2016-576}=2^{1440}$ graphs share each joint sketch value ($2^{1504}$ is the count for $O_i$ alone and is the wrong budget for a readout that also sees $x_i$). Any readout of $(x_i,O_i)$ therefore misfits the committor by at least $\varepsilon$ on some instance iff the label separates the sketch fibres at resolution $\varepsilon$: $$\text{err}_\infty\ \ge\ \varepsilon\quad\text{whenever}\quad \Pr_{G,G'\,\text{same sketch}}\big[|q_i(G)-q_i(G')|>2\varepsilon\big]>0 .$$ By pigeonhole on the label alone the bound is vacuous: the per-move tensor has $m(K+2)=32$ coordinates and $(1/\varepsilon)^{32}>2^{512}$ needs $\varepsilon<2^{-16}=1.5\times10^{-5}$; the full vector on $T$ has $|T|(K+2)$ coordinates with $|T|\in\{56,\dots,60\}$ — at $|\mathcal A_\bullet|\in\{1,2\}$ over the four sets $\mathcal A_{\rm sink},\mathcal A_0, \mathcal A_1,\mathcal A_2$ the total $|\mathcal A|$ ranges over $\{4,\dots,8\}$, so $|T|=64-|\mathcal A|$ ranges over $\{56,\dots,60\}$ and the earlier $\{57,60\}$ excluded values the stated geometry admits — hence $|T|(K+2)\in\{224,\dots,240\}$ and the requirement is $$\varepsilon<2^{-512/224}=0.2050\ \ \text{to}\ \ 2^{-512/240}=0.2278,$$ each endpoint printed against the expression that produces it (the earlier line transposed them). This bites only if the committor image is full-dimensional at that scale, which is not a theorem.

Hypotheses. Depth 1, one head, per-position readout, $d=16$, $p=32$; the sketch model excludes positional side channels wider than $x_i$. Non-vacuous exactly when the fibre-separation probability above is positive — a measured quantity.

Evidence. DERIVED (counting). The fibre-separation probability is NOT MEASURED — needs the collision instrument, **unspecified in the canon**: book 04 carries verses 04.1–04.21 and no collision instrument among them (a grep for "collision instrument" in04_BEDS_AND_INSTRUMENTS.mdcounts $0$), so the shape stated here — draw DAG pairs, compute the depth-1 sketch at fixed random weights, read the fraction of same-sketch pairs with $|\Delta q_i|>2\varepsilon$ — is a specification this verse writes and book 04 does not carry. That existence of collisions does not by itself force label error is the content of Peng's communication argument (verse 01.5), not of counting.

Mechanism. B7 (CHARTER.md §5) — the record's largest unclosed gap was that no bound followed "cannot represent"; this verse states the bound with the one hypothesis that makes it bite, so it cannot be claimed by accident (V-10, MISTAKES.md:168: the label pigeonhole is satisfied by construction and is printed as vacuous); V-17.

Kill. The collision instrument reading fibre-separation at $\varepsilon=0.05$ below $5\%$ of same-sketch pairs on $512$ drawn BED-S pairs at the design point: the capacity bound is vacuous at the planned geometry; $0$ GPU-s (CPU float64), one evening — but unreachable until book 04 carries the instrument (Evidence above). Control, and its two defects named. The must-fire plant is a calibration plant and not an import from 01.5: the instrument is run on a synthetic pair family with a constructed separation fraction $\phi_0$ (two DAGs agreeing on the sketch by construction and differing in $q_i$ by a chosen $\Delta$), and must read $\phi_0$ to within $\pm0.02$ on $512$ pairs; an instrument that cannot recover a planted separation fraction cannot certify a measured one. The earlier control — "separation above $13\%$ on the layered $\mathrm{hop}_2$ family at $n=128$" — imported 01.5's cell $0.1310=352/2688$ across two boundaries and both are recorded here rather than repaired away. (i) Log base. That cell is DERIVED with $\log_2$; under natural log the same row reads $R=128\ln128-544=77.4$ and error $\ge77.4/(3\cdot128\ln128)=0.0415$, so a working instrument reading $0.05$ would be declared broken by a factor $3.2$. 01.5's Hypotheses grade Peng [V] with [V-eq] owed — the body was not read — so the base is not settled, and if that control is ever run it is run as "under $\log_2$ the threshold is $0.131$; under $\ln$ it is $0.0415$", never as one number. (ii) Units. A collision fraction over same-sketch DAG pairs and a lower bound on the misclassification probability of the best readout are two different quantities, and no theorem in this canon relates them; the import is therefore withdrawn as a control and kept only as the sentence just written.

If killed. The replacement bound is the trained one: Bet D, registered at READ docs/CEQ_SHAPE.md:2236 @ 99777ab with prediction "the depth-5 stack within $\mathrm{MDE}_8$ of the shape's argmin accuracy — the honest control holds" and counter "short by more than $\mathrm{MDE}_8$, sign logged: the bed needs something the hop construction does not supply". Its Kill is $\mathrm{acc}_{\rm shape}-\mathrm{acc}_{\rm sky}$ at $N=8$ against the realised paired sd ($\approx64.8$ s [ASSUMED]) — more decisive than the counting bound because it is measured on the bed. The two branches, in the direction the registration gives them (the earlier text read them backwards and contradicted 01.8's Kill on the same registered bet): if Bet D reads its prediction — the skyline within $\mathrm{MDE}_8$ — then the depth skyline fits, the D-APPROX sentence for BED-S is "what survives on BED-S is exactness ($\delta=0$) and the cost sentence, and no fit sentence is licensed" — the word "advantage" appears nowhere, being a comparative outside A1/A2/T1/T2 with no bed, floor, $N$ or matched count attached (CHARTER.md:106-109) — the A2 sentence is withdrawn and the A1 sentence is reduced to verse 01.7. If Bet D reads its counter — the skyline short by more than $\mathrm{MDE}_8$ — the A2 form has its first evidence on this bed and the counting bound is superseded, not replaced: the measured shortfall stands in front of it and the counting line is kept only as the reason the measurement was worth taking. Under D-CALIB the counter is the point estimate, so the second branch is the one the canon plans against.

Terminal. "Cannot represent" becomes "cannot fit below $\varepsilon$" only where a fibre-separation number or a communication-complexity theorem supplies the constant; at $s=64$, $d=16$, $p=32$ neither is in the record, and the canon withdraws every "cannot fit" sentence there while licensing the two instruments that would earn one.

01.7 — The iteration skyline computes the resolvent with error exactly $\gamma^{K'+1}/(1-\gamma)$#

Statement. A stack of $K'$ blocks that reuses one head $P$ with a two-channel residual stream (channel $C^{(t)}=(\gamma P)^tV$, channel $S^{(t)}=\sum_{u\le t}(\gamma P)^uV$; each block computes $C^{(t+1)}=\gamma PC^{(t)}$ and $S^{(t+1)}=S^{(t)}+C^{(t+1)}$, a linear map of the attention output and the residual) outputs $S^{(K')}$, the $K'$-truncated Neumann sum, and $$\Big\|(I-\gamma P)^{-1}-\sum_{t\le K'}(\gamma P)^t\Big\|_\infty=\frac{\gamma^{K'+1}}{1-\gamma}\quad(\text{equality}),\qquad \|z-S^{(K')}\|_\infty\le\frac{\gamma^{K'+1}\|V\|_\infty}{1-\gamma}.$$ The depth count, corrected at repair, and the letter it is written in. $S^{(K')}$ needs $C^{(0)}=S^{(0)}=V$ in the residual stream at initialisation — $V$ is written into the stream, not computed by a block — and then exactly $K'$ blocks, each computing $C^{(t+1)}=\gamma PC^{(t)}$ and accumulating. The stack is therefore $K'$ blocks, and the earlier "depth-$(K+1)$ stack" was off by one in the direction that flattered the read (it charged the skyline one block more than the construction needs) and wrote the truncation order in the letter this book's Notation reserves for the constraint-set count, fixed at $K=2$ at the design point. Both are corrected: the order is $K'$ throughout this verse and verse 01.16, matching 08_ARCHITECTURE.md verse 08.11, which writes the certificate $\delta=\gamma^{K'+1}/(1-\gamma)$ and states in its own Hypotheses that $K'$ and $K$ are two objects. The title's exponent, which read $\gamma^{L}/(1-\gamma)$ against a body reading $\gamma^{K+1}/(1-\gamma)$ — a third letter for the same order — now reads $\gamma^{K'+1}/(1-\gamma)$. This is the A1-form statement in theorem shape: one solve does what $K'$ reused hops do, with the hops' shortfall printed:

$\gamma$	$K'=2$	$K'=4$	$K'=6$	$K'=16$
$0.6$	$0.540$	$0.194$	$0.0700$	$4.23\times10^{-4}$
$0.7$	$1.143$	$0.560$	$0.2745$	$7.75\times10^{-3}$
$0.9$	$7.29$	$5.90$	$4.78$	$1.67$

(matrix $\infty$-norm; multiply by $\|V\|_\infty$ for the vector bound; DERIVED, one convention for every cell: $\gamma^{K'+1}/(1-\gamma)$, the quantity the Statement's equality names.) The $\gamma=0.7$ row is reprinted at repair. Three of its four cells previously carried $\gamma^{K'+1}$ — the $\Pi_\gamma$ residual — while the fourth and both other rows carried $\gamma^{K'+1}/(1-\gamma)$: $0.343$, $0.168$, $0.0824$ are $0.7^3$, $0.7^5$, $0.7^7$ and are short of the Statement's quantity by the factor $1/(1-\gamma)=3.3\overline{3}$, whereas $7.75\times10^{-3}=0.7^{17}/0.3$ was already divided. The record prints the correct value for the identical $(\gamma,K')=(0.7,2)$: bind B-I's $\gamma=0.7$ plant reads $119.37/1.143$ (READ docs/CEQ_SHAPE.md:3116 @ 99777ab), $1.143=0.7^3/0.3$, against the book's former $0.343$. The corrected row is $0.7^3/0.3=1.143333$, $0.7^5/0.3=0.560233$, $0.7^7/0.3=0.274514$, $0.7^{17}/0.3=7.754\times10^{-3}$ (DERIVED). At $\gamma=0.7$, $K'=2$ the truncation shortfall therefore exceeds $\|V\|_\infty$: the two-hop iteration skyline is not a fellow approximator at that horizon, which the mis-divided cell hid.

Hypotheses. $P\ge0$ row-stochastic including identity rows, $\gamma\in[0,1)$; the stack reuses $P$ (the iteration skyline). A freely trained depth-$L$ stack with $L$ different heads is not covered: for it the only bounds are verse 01.5's (vacuous at $s=64$) and Bet D. Non-vacuous at every $K'$ and $\gamma>0$; at $\gamma=0.6$, $K'=16$ the shortfall $4.23\times10^{-4}$ is below any label sd the bed will show, so the InfSA-style $K'=16$ control is a fellow approximator there and the exactness delta is invisible at that $\gamma$ (the honest reading of R-SKY).

$\gamma_{\rm env}$ carries no number, and this verse no longer decides anything "at $\gamma_{\rm env}$". The record names it and never prints it: READ docs/CEQ_SHAPE.md:341-342 @ 99777ab reads "the discounted read $E_i[\gamma^{\tau-1}\mathbb 1_k]$ is a separate registered label at a bed constant $\gamma_{\rm env}$", and a grep for a value finds none in the record or in any book of this canon; it is NOT MEASURED — needs the BED-S registration triple to fix $\gamma_{\rm env}$ before the first draw (book 04, the D-4 admission block), a $0$ GPU-s registration act. The quantity this verse decides varies over that unset constant by a factor $13.5$ — $0.540$ at $(\gamma,K')=(0.6,2)$ against $7.29$ at $(0.9,2)$, both cells of the table above (DERIVED) — so a threshold "at $\gamma_{\rm env}$" is not frozen in the sense M-2 requires. Every clause of this verse is therefore registered at the two corners the record has run, $\gamma=0.6$ and $\gamma=0.9$, with both shortfalls printed, and at no other $\gamma$ until the constant is fixed.

Evidence. DERIVED (meyer-2000-matrix, horn-2013-matrix: the tail is entrywise non-negative with row sums $\sum_{t>K}\gamma^t$; the $\infty$-norm of a non-negative matrix is its largest row sum); RUN residual $0.007754350466241788$ against $\gamma^{17}/(1-\gamma)=0.007754350466240224$ at $K'=16$, $\gamma=0.7$ (READ docs/CEQ_SHAPE.md:853-855); the layer-by-layer TD reading is lineage only: wang-2024-incontext-td [V], xie-2026-softmax-rl [V] at the abstract (READ docs/CEQ_SHAPE.md:773-777), yang-2024-looped, gasteiger-2019-appnp [V].

Mechanism. R-SKY (CHARTER.md §4; READ CEQ_V16_CONTRACT.md:209 @ 99777ab) — the skyline is stated beside the read; V-3 (MISTAKES.md:72) — the bound is an attained identity and is labelled so; D-1 — this is the "one solve versus $L$ hops" sentence in the only form the record licenses, a rate on the reused head.

Kill — the learned-stack condition is struck as unable to refute the Statement; the arithmetic kill the Statement admits replaces it. The previous kill read: "On BED-S at $\gamma_{\rm env}$, the depth-$5$ learned stack (Bet D's arm) reading a $z$-channel NRMSE below the exact solve's by more than $\mathrm{MDE}_8$ on at least $6$ of $8$ seeds". It is struck on two counts, both printed. (1) It cannot refute the Statement. The Statement is an attained matrix identity about the truncated Neumann sum of $(I-\gamma P)^{-1}$; no outcome of a training run — a depth-$5$ learned stack reading any NRMSE whatever — can make $\|(I-\gamma P)^{-1}-\sum_{t\le K'}(\gamma P)^t\|_\infty$ differ from $\gamma^{K'+1}/(1-\gamma)$, because the two quantities are computed from $(\gamma,K',P)$ and never from a weight (CHARTER.md:145-146: a kill that cannot fire against the statement it is attached to is not a kill). (2) Its threshold is not frozen. It was registered "at $\gamma_{\rm env}$", and $\gamma_{\rm env}$ carries no value anywhere in the record or the canon (Hypotheses above), so the number the kill decides ranges over a factor $13.5$ — $0.540$ at $(0.6,2)$ to $7.29$ at $(0.9,2)$ — over an unset constant (M-2). The learned-stack contrast is not deleted: it is Bet D's contrast and belongs to verse 01.8, which registers it against the record's own threshold, and it is cited there rather than duplicated here.

The form the contrast is written in, and the bare comparative that is struck with it (CHARTER.md §1, R-SKY, D-1). The struck kill's phrasing — a learned stack reading below the exact solve, the bed and $\mathrm{MDE}_8$ named and nothing else — is a comparative outside A1/A2/T1/T2: it printed neither the floor nor the matched parameter count, which CHARTER.md:106-109 strikes at refutation. The sentence this verse licenses in its place is the A1 form, and it carries four things in one clause or it is not written: the bed (BED-S at $t^\star=8$, $m=8$, $K=2$, $N=8$); the floor (L-FLOOR, the argmin zero-information floor $1-\max_a\hat\pi(a^\star)$ at the realised class distribution, NOT MEASURED because $\hat\pi$ has no cell (B5) — $0.875$ is the uniform-$\hat\pi$ instance at $m=8$ and nothing wider); the matched parameter count ($4{,}769$ for the shape at the design point, against book 04 clause 3b's reduced-width count fixed within $0.0960$ per cent, which is NOT MEASURED because 3b has no width and no cell); and $\mathrm{MDE}_8$ at the realised paired sd, itself NOT MEASURED until 04.21 ships $\sigma_d$. Three of the four are NOT MEASURED, so no A1 value is printed by this verse and no fit sentence is licensed here. What this verse prints instead is the truncation shortfall $\gamma^{K'+1}/(1-\gamma)$, which is a function of $(\gamma,K',P)$ and is an identity rather than a comparison — $0.540$ at $(0.6,2)$, $7.29$ at $(0.9,2)$ — beside which the A1 clause above is a route and not a number.

Frozen. Form $P$ as a drawn causal row-stochastic operator at $s=64$ (regime S, identity rows on $\mathcal A$), compute $(I-\gamma P)^{-1}$ and $\sum_{t\le K'}(\gamma P)^t$ in float64 at both registered corners $\gamma=0.6$ and $\gamma=0.9$ and at every $K'\in\{2,4,6,16\}$, and read $\big|\;\|(I-\gamma P)^{-1}-\sum_{t\le K'}(\gamma P)^t\|_\infty-\gamma^{K'+1}/(1-\gamma)\;\big|$ relative to $\gamma^{K'+1}/(1-\gamma)$. Any of the eight $(\gamma,K')$ pairs reading a relative departure above $10^{-12}$, or any of the twelve printed table cells differing from $\gamma^{K'+1}/(1-\gamma)$ by more than $10^{-12}$ relative: the word "equality" is withdrawn from the Statement, the certificate is demoted to the Neumann upper bound, and the A1 rate is withdrawn as a rate.

Instrument: float64 formation of a $64\times64$ resolvent and a partial sum, CPU, no cell, no mask, no bed, no arm; price $0$ GPU-s, half an evening. It is reachable today and it fires on a real defect: the table's $\gamma=0.7$ row as this book printed it before repair reads $0.343$ against $1.143$, a relative departure of $0.7$ — the second clause would have fired on this book's own page. This kill is a different object from verse 01.16's — 01.16 recomputes the scalar closed form against the record's two banked residual numbers, while this reads the matrix $\infty$-norm of a formed resolvent against the same closed form on a drawn $64\times64$ operator — and the two are kept apart on that ground. Planted negatives, each of which must fire: the rows-$1.5$ non-stochastic $P$ at $\gamma=0.6$, $K'=2$ must exceed $0.54$, reading $7.29$ (RUN[M], verse 01.16's Evidence) — row stochasticity is the hypothesis, and an instrument that reports equality on rows summing $1.5$ is reading the formula and not the matrix; and a $P$ with one row scaled by $1+10^{-6}$ must read a relative departure above $10^{-12}$ at $\gamma=0.9$, $K'=2$, the smallest perturbation the gate must see. Must-not-fire half: the record's own banked pair at $K'=16$, $\gamma=0.7$, relative $2.0179\times10^{-13}$ (verse 01.16), inside the gate.

If killed (the matrix equality departs by more than $10^{-12}$ relative at either corner). The A1 sentence is replaced by the cost sentence: one solve of depth $s$ at $+s^2d/2$ MACs against $L$ layers (READ docs/CEQ_SHAPE.md:583-590), a T1 statement owned by book 03; its Kill is the cost law's exponent interval (K-9, $206$–$537$ GPU-s, band), more decisive because it is measured on the certified device rather than on a formed matrix. If that dies, the sentence "the shape's hops are exact and certified rather than learned" survives as a property with $\delta$ printed (verse 01.16), and the chain ends.

Terminal. A deeper softmax stack computes the same resolvent by iteration with a printed rate; "beats softmax" is not a sentence this canon writes; the licensed sentence is exactness against a truncated read, and it is withdrawn at any $\gamma$ where the truncation shortfall is below the label sd.

01.8 — Obstruction 2 restated: the skyline depth is a theorem on the layered family and an analogy on BED-S#

Statement. sanford-2024-logdepth Thm 4.2 gives $\mathrm{hop}_k$ at depth $\lfloor\log_2k\rfloor+2$ ($3/5/7$ at $t^\star=2/8/32$); by verse 01.4 that depth also solves the committor on the layered family at $K=n$. On BED-S ($K=2$, random DAG) the skyline depth $5$ at $t^\star=8$ is a choice by analogy. The lower bounds are conditional (Cor. 4.3, the one-versus-two-cycle conjecture; sanford-2024-graph-algorithms Thm 3/19) or vacuous (verse 01.5); yehudai-2025-depthwidth removes depth at linear width, so the matched control fixes width too, and at $d_{\rm model}=16<s=64$ the wide skyline's matched instance does not exist.

Which skyline A2 is read against, named at repair, and what follows from the wide one's absence. CHARTER.md:88-91 defines A2 as "at matched parameters against the depth skyline" and requires "the depth skyline's number printed in the same row"; CHARTER.md:80 fixes that skyline's depth at $\lfloor\log_2t^\star\rfloor+2$, which is $5$ at $t^\star=8$. Book 04 splits the depth-$5$ row into two arms and only one of them can carry an A2 sentence (04_BEDS_AND_INSTRUMENTS.md verses 04.9 clause 3 and 04.20, cited by verse id because book 04's line pins move under its own repairs; at repair time :149 and :302): (3a) the depth-$5$ stack at full width and unmatched parameters, whose number is printed as $\Delta_{\rm sky}$, which satisfies R-SKY and licenses no A2 sentence at all; and (3b) the depth-$5$ stack at reduced width, the width fixed before any draw as the largest per-layer width landing within $0.0960$ per cent of the shape's count — and A2 is read against 3b and against nothing else. Of the four skylines book 04 registers, the depth-$5$ full-width arm (3a) is the one it requires FOUND at the design point; the matched-width arm (3b) has no width, no count and no cell (NOT MEASURED), and the wide constant-depth stack and the chain-of-thought decoder have settings and no cell — book 04.20's own title reads "one FOUND, two with settings and no cell (B11)" and its Limits paragraph states in full that no A2 number exists anywhere in that book. The consequence for this verse, printed rather than implied: the linear-width skyline of yehudai-2025-depthwidth is out of scope at $d_{\rm model}=16$ — it has no matched-parameter instance there — so no A2 sentence is licensed against it, by this book or any other; and the A2 sentence that is licensed in principle, against 3b at depth $5$, has no number today because 3b has no width and no cell. This verse therefore prints an A2 route and no A2 value.

Hypotheses. Thm 4.2's model ($m=O(1)$, $H=1$, causally masked) [V], [V-eq] owed (READ docs/CEQ_SHAPE.md:667-694). The skyline is a control's depth, not a load-bearing bound, so [V] is admissible for it.

Evidence. CITED as above; the reduction DERIVED (01.4); the analogy declared at READ docs/CEQ_SHAPE.md:689-694.

Mechanism. P-10, V-25, P-3 (MISTAKES.md:316, conditional bounds never stated as settled), D-7 (:2037, Bet D carries its counter).

Kill — Bet D at the threshold the record registers, with the seed-count clause struck. The verse read the kill as "short of the shape's argmin accuracy by more than $\mathrm{MDE}_8$ on at least $6$ of $8$ seeds at $N=8$". That clause is not Bet D's: READ docs/CEQ_SHAPE.md:2236 @ 99777ab registers Bet D's deciding number as $\mathrm{acc}_{\rm shape}-\mathrm{acc}_{\rm sky}$ with SPLIT band "none" and no seed-count clause; the $6$-of-$8$ form belongs to Bets C, F and H in the same table, and importing it thickened a registered threshold, which M-2 forbids. It is struck:

Frozen (Bet D as registered), in A2 form with its floor and its matched count in the same clause (CHARTER.md §1, L-FLOOR). On BED-S ($t^\star=8$, $m=8$, $K=2$, $N=8$), the paired contrast $\mathrm{acc}_{\rm shape}-\mathrm{acc}_{\rm sky}$ at $N=8$ against the realised paired sd, both arms read as distance to the argmin zero-information floor $1-\max_a\hat\pi(a^\star)$ (NOT MEASURED, $\hat\pi$ has no cell, B5; $0.875$ only as the uniform-$\hat\pi$ instance at $m=8$), at matched parameters — $4{,}769$ for the shape against book 04 clause 3b's reduced-width depth-$5$ count fixed within $0.0960$ per cent before any draw (NOT MEASURED: 3b has no width and no count) — with the depth skyline's own number printed in the same row: the skyline short by more than $\mathrm{MDE}_8$ means the bed needs something the hop construction does not supply and the analogy is wrong in the direction that flatters the shape. SPLIT band: none. No seed-count clause. No bare comparative is licensed anywhere in this book: a sentence of the form "the learned stack does better than iteration", naming the bed and $\mathrm{MDE}_8$ but neither the floor nor the matched count, is struck under CHARTER.md:106-109 and appears nowhere.

Instrument S-35 (a card of the record); $\approx64.8$ s [ASSUMED], one evening; the $\mathrm{MDE}_8$ is the realised-sd clause and is NOT MEASURED until book 04 verse 04.21 ships $\sigma_d$ (B5). Should a seed-count clause ever be wanted on Bet D, it enters through a CORRECTIONS.md row against the registration and not through a book. Control: on the layered family of 01.4 at $k=8$, $n=64$ the depth-$5$ stack must reach the argmin within $\mathrm{MDE}_8$ (Thm 4.2's own instance); a skyline that fails where the theorem says it succeeds is mis-built.

If killed (the skyline falls short on BED-S). Replacement: the depth dial is escalated to $7$ and then to the chain-of-thought decoder at $t^\star$ steps (merrill-2024-cot [V]; READ docs/CEQ_SHAPE.md:2508-2510), each a registered skyline row; its Kill is the same $\mathrm{MDE}_8$ comparison at the next depth ($\approx7.6$ s per cell [ASSUMED], cheaper than the first because the arm exists). If every depth falls short, the sentence licensed is "on BED-S the hop construction does not reach the label at any registered depth" — an observation about the control, filed with its sign, never "softmax cannot".

Terminal. Hops need depth is lineage; the canon licenses the skyline row with its $\Delta_{\rm sky}$ interval — the depth-$5$ full-width control of book 04's clause 3a, which carries no A2 sentence — and withdraws every sentence that reads the analogy as a theorem about BED-S. A2 is licensed only against the matched-parameter depth-$5$ skyline of book 04's clause 3b, which has no width, no count and no cell, so no A2 number is licensed by this book today; and it is licensed against the linear-width skyline never, since at $d_{\rm model}=16<s=64$ that arm has no matched-parameter instance at all.

Part III — Per-row independence, Cheeger, Bellman, degeneracy#

01.9 — Per-row independence is a definition; joint consistency is a measured ratio#

Statement. One softmax step is $O_i=f(P_{i\cdot},V)$ and the reads are independent across $i$; the shape's $z$ is the unique fixed point of the $\gamma$-contraction $z=V+\gamma Pz$ in $\|\cdot\|_\infty$, so its reads are mutually constrained. The claim about the literature is replaced at repair by the decidable half. The verse read "No theorem converts this into an error separation" — a universal negative about what exists in the literature, which no measured ratio can refute, so no kill of this verse could reach it. What this verse asserts instead: the joint-consistency claim at MATHEMATICS.md:106-127 is UNTESTED, the ratio below with its registered counter is what would test it, and until a cell is read the counter is the point estimate (D-CALIB). The measurable statement is the harmonic residual $r(\hat z)=\|(I-\gamma P_{\rm env})\hat z-V\|_\infty/\|V\|_\infty$, and the licensed prediction is the ratio $r_{\rm softmax}/r_{\rm shape}$ with the counter $\le2$ (Bet E, SPLIT in $(2,10)$).

Hypotheses. $\gamma<1$, $P$ row-stochastic (Contraction.rowStochastic_perron, lean/CEQ/Contraction.lean:118); $r$ printed beside $\sigma_{\min}(I-\gamma P_{\rm env})$ and $\|I-\gamma P_{\rm env}\|_\infty\le1+\gamma$ (its units). Non-vacuous whenever both arms reach marginal NRMSE within $\mathrm{MDE}_8$ of each other — otherwise the ratio compares two different errors.

Evidence. READ docs/CEQ_SHAPE.md:652-665; the claim "joint determination in one read" UNTESTED at READ MATHEMATICS.md:106-127; nearest per-query fixed points ramsauer-2021-hopfield [V]; the sharpness failure velickovic-2025-softmaxnotenough [V].

Mechanism. V-26 (MISTAKES.md:2190, a marginal is not a joint); V-17; D-7.

Kill. K-H1: $r_{\rm softmax}/r_{\rm shape}\le2$ at marginal NRMSE within $\mathrm{MDE}_8$ on $6$ of $8$ seeds — "joint determination in one read" is withdrawn; inside the $\approx34$ s pair (Bet E's one bed-cell pair, 04_BEDS_AND_INSTRUMENTS.md verse 04.6; at repair time :102), one evening.

Registration clause, added at repair: K-H1 and verse 01.2's K-D2 cannot both be decided on the same seeds. $r_{\rm shape}$ is not free of the identification error — it is it. The shape solves $\hat z=V+\gamma\hat P\hat z$ exactly, so $$(I-\gamma P_{\rm env})\hat z-V=(V+\gamma\hat P\hat z)-\gamma P_{\rm env}\hat z-V =\gamma(\hat P-P_{\rm env})\hat z,\qquad r_{\rm shape}=\frac{\gamma\|(\hat P-P_{\rm env})\hat z\|_\infty}{\|V\|_\infty} \ \le\ \frac{\gamma\,\|\hat P-P_{\rm env}\|_\infty}{1-\gamma}$$ (DERIVED, using $\|\hat z\|_\infty\le\|V\|_\infty/(1-\gamma)$). K-D2 fires when $\|\hat P-P_{\rm env}\|_\infty<10^{-3}$ on $\ge6$ of $8$ seeds; on exactly those seeds $r_{\rm shape}\le1.5\times10^{-3}$ at $\gamma=0.6$ ($0.6\cdot10^{-3}/0.4$, DERIVED), so K-H1's "$\le2$" would need $r_{\rm softmax}\le3\times10^{-3}$ — the softmax arm identifying $P_{\rm env}$ almost as well as the shape — and the ratio otherwise diverges. The two frozen kills of this book are therefore mutually exclusive in that direction, and the registration is made explicit: $r_{\rm shape}$'s floor and the identification reading $\|\hat P-P_{\rm env}\|_\infty$ are printed beside every K-H1 evaluation, and K-H1 is registered only on the seeds where K-D2 does not fire, with the excluded seed count printed in the row. A K-H1 verdict quoted over a seed set that includes a K-D2-firing seed is VOID, never a reading. Both numbers are NOT MEASURED — needs the first BED-S cell (B5; book 04 verse 04.21).

Control: an arm emitting the oracle $z$ itself must read $r=0$ to $10^{-12}$, and the same arm with coordinate $i_{\min}$ perturbed by $+0.1\sigma$ must read $r>0$ — a residual blind to a one-coordinate move is not a residual (B-G2's plants, READ docs/CEQ_SHAPE.md:3115).

If killed. Replacement: the residual as a diagnostic and "one read" as a cost statement (one solve, depth $s$), owned by book 03; its Kill is K-9's band. The chain ends there.

Terminal. Per-row independence is a definition and licenses nothing about error; the canon keeps the residual as an instrument, records the sentence at MATHEMATICS.md:106-127 as UNTESTED rather than as unsupported by the literature, withdraws it whenever Bet E reads its counter, and withdraws every K-H1 verdict quoted over seeds on which K-D2 fires.

01.10 — OPEN — The Cheeger sentence: withdrawn on the causal $P$, restated on the reversible oracle chains#

Marked OPEN at repair: branch (a)'s kill was satisfied by construction and is struck, and branch (b)'s kill needs an instrument the canon does not specify, so the verse carries no reachable kill today and the Terminal below is in force now. Both branches keep their decidable content as admission lines, which are instruments and not kills.

Statement. (a) On the arm's causal class, $Q=\hat P_{TT}$ is lower-triangular, so $$\rho(Q)=\max_{i\in T}\hat P_{ii}\quad\text{and}\quad I-Q\ \text{is a unit iff}\ 0\in\mathcal A,$$ there is no bottleneck spectrum to bound, and every conductance sentence about $\hat P$ is withdrawn; the truncation residual of a $t$-hop read is $Q^t q$ with $\|Q^t\|_\infty\le1$ (sub-stochastic) and no termwise $\rho(Q)^t$ bound (a triangular $Q$ is non-normal). (b) On the oracle cross-check chains (BED-1, E4′: random walks $D^{-1}W$ on undirected weighted graphs, reversible with respect to degree), levin-2017-markov-mixing Thm 13.10 applies as [V-eq]: for a reversible $P$ with spectral gap $\gamma_{\rm gap}$ and bottleneck ratio $\Phi_\star$, $\Phi_\star^2/2\le\gamma_{\rm gap}\le2\Phi_\star$ (read at p. 183 of the authors' PDF, READ docs/references.bib:2579), with the record's instance $g\le2\varphi$, $\varphi=1/\mathrm{vol}(S)$, $t_{\rm rel}\ge\mathrm{vol}(S)/2=1372.50$ at READ MATHEMATICS.md:455-456 — an instance on E4′'s own reversible chain, admissible there and inadmissible on $\hat P$. (c) The non-reversible replacement is not a symmetrisation of $\hat P$ (the additive symmetrisation's spectrum is unrelated to $\mathrm{diag}\,\hat P$) but the exact eigenvalue read of (a).

Hypotheses. (a) finite logits, $\beta=1$; (b) reversibility, which holds for $D^{-1}W$ and fails for any causal $P$ with $P_{i0}>0$, $P_{0i}=0$. Non-vacuous: (a) at every draw of the class; (b) on BED-1 and E4′ only, with $\rho(Q_{\rm BED\text{-}1})=0.9408612510154677$ (RUN, READ docs/CEQ_SHAPE.md:890).

Evidence. (a) DERIVED (eigenvalues of a triangular matrix are its diagonal) and RUN[J] $0.6926596893360386$ both ways (READ docs/CEQ_SHAPE.md:539-543); (b) CITED levin-2017-markov-mixing [V-eq] via lawler-1988-bounds, sinclair-1989-approximate, cheeger-1970-lower [V]; the correction filed at READ docs/CEQ_SHAPE.md:1122-1127 and card J-D6 (:1584-1590). The Dirichlet-form Cheeger bound for the killed reversible chain, $\lambda_{\min}(I-Q)\ge h_D^2/2$, is textbook and its citation is owed (not in references.bib); it is [ASSUMED textbook] and load-bearing nowhere in this book.

Mechanism. V-25 (MISTAKES.md:1954) — a reversible theorem on a non-reversible chain is decoration with a proof attached; P-10; V-17 (the ergodic-chain gap is not the transient-block Perron root, READ docs/sources/sweep/sweep_methods.md:413-415).

Kill (a) — struck at repair; not replaced by another gate of the same kind. The previous kill (a) read "any drawn trained cell with $|\rho(\hat Q)-\max_T\hat P_{ii}|>10^{-12}$", and the verse's own If-killed conceded in one line that it "cannot fail on a triangular operator". It cannot: the eigenvalues of a triangular matrix are its diagonal entries, exactly, so the quantity being thresholded is the float rounding of an eigensolver against a number read straight off the diagonal, and $10^{-12}$ is a numerical-noise gate, not a refutation of the Statement (CHARTER.md:145-146, V-10). The repair does not repoint it at the strict-upper torch.equal census either: that census is itself satisfied by construction on a causally masked softmax, whose strict upper block is bitwise $0.0$ because the mask writes $-\infty$ before the exponential (verse 01.19's repair, same defect). Kill (a) is therefore deleted, and the triangularity census is filed as an admission line — a cell whose $\hat P$ fails torch.equal on the strict upper block is refused at admission, before any certificate is read off it — owned by book 04, which does not carry it: NOT MEASURED — needs the triangularity admission line, **unspecified in the canon** (04_BEDS_AND_INSTRUMENTS.md carries verses 04.1–04.21 and no such line; the manifest of 04.15 journals route and boundary_sets, not the strict-upper read). What branch (a) asserts — $\rho(Q)=\max_T\hat P_{ii}$, and no termwise $\rho(Q)^t$ bound on the non-normal $Q$ — is algebra, and this verse states it as algebra with no kill attached.

Kill (b) — reachability restated, and the missing plant supplied. On BED-1, $h_D^2/2$ computed from the bed's own conductance exceeding $1-\rho(Q)=0.0591$: the Dirichlet Cheeger instance fails and the [ASSUMED textbook] bound is struck. Price $0$ GPU-s; but NOT MEASURED — needs the Dirichlet-conductance instrument onbed_1, **unspecified in the canon**: a grep for "Dirichlet" in 04_BEDS_AND_INSTRUMENTS.md returns one hit, inside 04.15's manifest field committor_route ∈ {dirichlet_gamma1, discounted_gamma}, which names a route and not an instrument, so no verse of book 04 supplies the conductance read. The kill is unreachable until book 04 ships it, and the verse stands OPEN on that account. Control, supplied at repair — this branch shipped with no planted negative, the only kill in the book with none (V-15). The must-fire plant is a non-reversible transient block whose Dirichlet conductance is computed by the same instrument: a two-state block with $Q=\begin{pmatrix}0&0.9\\0&0.9\end{pmatrix}$ has $\rho(Q)=0.9$, $1-\rho(Q)=0.1$, and a Dirichlet conductance whose $h_D^2/2$ the instrument must report above $0.1$ — a chain that leaves state $1$ only to absorption at rate $0.1$ while the conductance sees the $0.9$ edge (DERIVED; reversibility fails because $Q_{12}=0.9$ and $Q_{21}=0$). An instrument that certifies BED-1 without first firing on this plant has never been shown to fire at all, and the [ASSUMED textbook] bound would then be certified by a probe with an empty rejection region. The plant is priced with the instrument, $0$ GPU-s, and is NOT MEASURED for the same reason.

If killed. (a) has no kill to fire. (b) If the Dirichlet instance fails once the instrument exists, the Cheeger sentence is withdrawn from the oracle chains too, and the replacement is the hop-ladder rate read off the ladder (READ MATHEMATICS.md:420-462: the ladder is pre-asymptotic and $\lambda_2^{t^\star}$ is the wrong predictor at every rung), a measured decay with no spectral prediction — a different object (a decay read at $t$, not a conductance), a different instrument (the ladder, which the record has already run), and a different number; its Kill is the ladder's own $f=\mathrm{decay}/\lambda_2$ reading outside $[0.99,1.01]$ at $t\ge320$, which the record already read inside it ($1.00000063$, $1.00158689$, :441), so that link is reachable today at $0$ GPU-s and is strictly cheaper than the conductance read. The chain ends there.

Terminal — in force now. No conductance sentence is written about the causal $P$; the canon licenses $\rho(\hat Q)=\max_T\hat P_{ii}$ as the exact rate on the arm's class — as algebra, with no kill and therefore no capability weight — and licenses the reversible Cheeger bound on the oracle chains only, as [ASSUMED textbook] with its citation owed and its instrument unspecified; it withdraws MATHEMATICS.md:455 as a sentence about the shape, and withdraws the claim that either branch of this verse has been tested.

01.11 — The Bellman hazard resolved: one-clamp optimality, DAG backward induction, and where scalarisation breaks#

Statement. (i) One clamp. With one intervention chosen from $m$ candidates and the environment otherwise fixed, $a^\star=\arg\max_aq^{(0)}(\mathrm{do}\,a)$ maximises $\Pr(\text{reach }\mathcal A_0\text{ before any }\mathcal A_k)$ by definition of the objective; no principle of optimality is invoked. (ii) Sequential goal rule on the DAG. If a clamp may be chosen at every transient state visited, the optimum is $$V^\star(i)=\max_{a\in\mathcal M(i)}\sum_jP^{(a)}_{ij}V^\star(j),\qquad V^\star|_{\mathcal A_0}=1,\ V^\star|_{\mathcal A\setminus\mathcal A_0}=0,$$ solved exactly by backward induction in topological (position) order in at most $s$ steps. The chain is multichain, and the justification never says otherwise. With $K+1$ boundary sets plus a sink, every absorbing state is its own closed recurrent class, so at the design point ($K=2$, $|\mathcal A_\bullet|\in\{1,2\}$ over $\mathcal A_{\rm sink},\mathcal A_0,\mathcal A_1,\mathcal A_2$, READ docs/CEQ_SHAPE.md:295-297 @ 99777ab) the chain carries $|\mathcal A|\ge4$ absorbing states and hence $\ge4$ recurrent classes; a chain is unichain iff it has exactly one. The earlier clause "the class is unichain-absorbing and Misra's multichain hazard is absent for this objective" is deleted: it closed census row B10 by asserting that the hazard's own hypothesis — several unsafe sets, multichain (CHARTER.md:236) — was absent, when the design point is that hypothesis. What licenses backward induction is a different fact about a different criterion: the objective here is a total-reward absorption probability on a transient DAG block with $\rho(Q)<1$, not an average-reward criterion, and Misra's hazard is a hazard of the average-reward constrained LP, where a multichain structure makes the single-chain LP's optimal value unattainable by a stationary policy. On a total-reward absorption objective the value recursion above is a finite backward substitution: the position order is a topological order of the DAG, $V^\star$ at every $\mathcal A$ state is fixed by the terminal condition, and each transient row depends only on strictly later positions, so the recursion terminates in $|T|\le s$ substitutions with no fixed-point argument, no ergodicity assumption, and no appeal to a unique recurrent class. The hypothesis under which that argument holds, printed: (H1) every clamp $a\in\mathcal M(i)$ keeps $P^{(a)}$ causal, so the substrate stays a DAG under every policy; (H2) $0\in\mathcal A$ and every policy's chain absorbs with probability $1$ in at most $s$ steps (later_boundary_unreachable; verse 01.10's branch (a)); (H3) the objective is additive in the absorption indicator — which (iii) below shows the Chebyshev objective is not, and that is where the principle of optimality genuinely fails here. The one-step rule with the fixed continuation is a lower bound on $V^\star$. (iii) Scalarisation. The Chebyshev objective $\min_\pi\max_{k\ge1}q^{(k)}_\pi$ is a non-additive function of the absorption vector, and the principle of optimality fails for it even on a DAG. The witness is rewritten at repair, because the earlier one exhibited no second epoch and used an action the bed does not carry. The earlier witness had one state $u$, two clamps $a$ and $b$ absorbing with probability $1$ and no continuation: deterministic value $1$, mixture $\tfrac12a+\tfrac12b$ value $\tfrac12$. The principle of optimality is a statement about sub-policies of a multi-epoch problem, so a one-epoch instance cannot exhibit its failure; and the safest-move read is an argmin over $m=8$ discrete candidate moves realised as row clamps with targets restricted to $T$ (R-07, READ docs/CEQ_SHAPE.md:1736-1742 @ 99777ab), in which the mixture $\tfrac12a+\tfrac12b$ is not a candidate, so the exhibited gap was unreachable on the bed as registered (V-11). What that instance does prove is kept, and stated as what it is: for a Chebyshev objective the optimum over randomised clamps ($\tfrac12$) is strictly below the optimum over deterministic clamps ($1$) — an aside about the objective's action set, carrying no principle-of-optimality content and no bed reachability.

The two-epoch witness, derived here, on the bed's own action set. Eight positions in causal order, every edge to a strictly smaller index: $\mathcal A_{\rm sink}=\{0\}$, $\mathcal A_1=\{1\}$, $\mathcal A_2=\{2\}$ absorbing ($K=2$); transient $x=3$ with row $e_1$, $y=4$ with row $0.6e_1+0.4e_2$, $w=5$ with row $e_2$, $v=6$ clampable, and the query $u=7$ with row $\tfrac12e_6+\tfrac12e_5$. Two candidate moves at $v$, both R-07 row clamps to transient targets: $c:\;v\leftarrow e_3$ and $d:\;v\leftarrow e_4$. Then $q(x)=(1,0)$, $q(y)=(0.6,0.4)$, $q(w)=(0,1)$ and

policy at $v$	$q(v)=(q^{(1)},q^{(2)})$	Chebyshev value at $v$	$q(u)$	Chebyshev value at $u$
$c$	$(1,\ 0)$	$1$	$(0.5,\ 0.5)$	$\mathbf{0.5}$
$d$	$(0.6,\ 0.4)$	$\mathbf{0.6}$	$(0.3,\ 0.7)$	$0.7$

(DERIVED, exact rationals.) The Chebyshev-optimal sub-policy at $v$ is $d$ ($0.6<1$); the Chebyshev-optimal full policy uses $c$ ($0.5<0.7$). The locally optimal prefix is not the prefix of any optimal full policy, and choosing it costs $0.7-0.5=0.2$ at the root — the principle of optimality fails, with two epochs, deterministic clamps only, and every action in the read's own set. The Chebyshev and lexicographic columns are therefore one-clamp scalarisations of evaluated vectors and never sequential policies.

Hypotheses. (i) none beyond the label's definition; (ii) H1, H2, H3 as printed in the Statement, with moves as row clamps $u_a\in T$ (R-07's restriction, READ docs/CEQ_SHAPE.md:1736-1742 @ 99777ab) — and not unichain, which is false at the design point and is nowhere assumed; (iii) $K\ge2$, at least two transient positions below the clampable one, and clamp targets in $T$ (R-07) — the witness above uses exactly those and no randomised action. Non-vacuous: (ii) at every BED-S draw, where the recurrent-class count is $|\mathcal A|\in\{4,\dots,8\}$ and the average-reward criterion Misra's theorem is about is never formed; (iii) on any draw carrying a clampable transient position with two distinct transient targets whose committor vectors are not proportional — which the eight-position witness realises inside $s=64$, $K=2$, $m\ge2$, so it is reachable on the bed as registered rather than only in a mixture the action set excludes. Census row B10 is closed by the criterion distinction printed above, not by denying the hazard's hypothesis.

Evidence. DERIVED (backward induction on a finite DAG; the eight-position two-epoch witness of (iii), exact rationals, and the one-epoch randomisation aside kept apart from it). CITED misra-2023-safety-constrained-mdp [V] for the hazard's class (multichain, several unsafe sets — carried, READ docs/CEQ_SHAPE.md:2960-2963); hsu-2023-safetyfilter, borquez-2023-lrf [V] for the one-step least-restrictive filter vocabulary; vanmoffaert-2013-chebyshev, yang-2026-lexisafe, altman-1999-cmdp [V] for the scalarisations; bellman-1957-markovian for the principle whose failure (iii) exhibits. R-20's instrument (READ docs/CEQ_SHAPE.md:1758-1763) measures the one-step-versus-two-step disagreement fraction.

Mechanism. P-7 (MISTAKES.md:375, "safest move" had no referent — it now has three, named); V-17; P-10 (Misra is cited for its class, not beyond it); V-1 (:34, the two rules are columns of one bed with the disagreement fraction printed).

Kill — one per clause, because R-20 refutes none of the three. The verse carried R-20 alone: "the one-step goal rule and the two-step optimum (apply $a$, re-solve, best second clamp) disagreeing on more than $0.5$ of $512$ admitted oracle draws at the design point" (READ docs/CEQ_SHAPE.md:1758-1763 @ 99777ab, :2275; a card of the record — book 04 carries no R-20 verse, so the routing is to the pin and not to a book-04 id). A disagreement fraction between two heuristics cannot refute (i), which is true by the definition of the objective; nor (ii), which is a backward-induction theorem on a finite DAG; nor (iii), which is closed arithmetic on eight positions. R-20 is not deleted — it is repositioned as the replacement's kill below, where the claim it can decide actually lives — and each clause now carries the kill that can refute it.

Kill (i) — the label is not constant across the candidate moves. Frozen: on the $512$ admission draws of 04_BEDS_AND_INSTRUMENTS.md verse 04.2, the realised effective move count $m_{\rm eff}$ (that verse's census clauses 9 and 11: the no-op screening rate is the fraction of candidate targets whose clamp moves the query's committors by $>10^{-12}$) reading $m_{\rm eff}\le1$ on more than $0.05$ of draws. A value-only move leaves every $q^{(k)}$ constant across moves (READ docs/CEQ_SHAPE.md:475 @ 99777ab), so where every candidate is value-only the argmax label is constant (V-12) and (i)'s "maximises by definition of the objective" is a tie-break over one identical vector rather than a decision rule. Instrument: 04.2's census, a verse of book 04 that carries it; price $0$ GPU-s, no arm, no cell. Planted negative: a draw whose $m$ candidates are constructed value-only (the $V$-only plants of verse 01.15, which read $\le10^{-15}$) must read $m_{\rm eff}=0$ and fire.

Kill (ii) — the DP value against exhaustive policy enumeration. Frozen: on draws restricted to $|T|\le12$ and $m\le3$ candidate clamps per state, $\max_i|V^\star_{\rm DP}(i)-V^\star_{\rm enum}(i)|>10^{-12}$ on any of $512$ such draws, where $V^\star_{\rm enum}$ is the maximum over all $m^{|T|}\le3^{12}=531{,}441$ deterministic stationary policies of the exact absorption probability of that policy's chain. Instrument: the exact solve applied once per enumerated policy on a $\le12$-state transient block, CPU float64; price $0$ GPU-s, one evening. This is the only clause that tests whether backward induction is the oracle, which is what (ii) asserts. Planted negative: replace the $\max$ in the recursion with the $\min$ — enumeration must then exceed the DP value by $O(1)$ on at least one draw, and an identity that survives an inverted recursion is comparing a routine with itself. Second plant, aimed at H3: run the same enumeration against the DP applied to the Chebyshev objective $\min_\pi\max_kq^{(k)}_\pi$ — the two must differ, by $0.2$ at the root of (iii)'s eight-position witness, because H3 fails there.

Kill (iii) — the witness re-evaluated. Frozen: recomputing the five numbers of (iii)'s table in float64 — $\max q(v\mid c)=1$, $\max q(v\mid d)=0.6$, $\max q(u\mid c)=0.5$, $\max q(u\mid d)=0.7$, root gap $0.2$ — and finding the local and global argmins agree, or the gap reading below $10^{-12}$. Instrument: one $8\times8$ exact solve pair, CPU float64, no bed and no arm; price $0$ GPU-s, minutes. Planted negative: replace the Chebyshev $\max_k$ with the linear scalarisation $q^{(1)}$ alone — local at $v$ then reads $c=1$, $d=0.6$ and global at $u$ reads $c=0.5$, $d=0.3$, so both argmins are $d$ and must agree; an instrument reporting disagreement under an additive objective is reading its own bug and not a failure of optimality.

The three are strictly ordered by price: (iii) is minutes on an $8\times8$ pair, (i) is a census column on draws the bed already takes, (ii) forms up to $531{,}441$ solves per draw.

If killed (any of the three clauses). The phrase "safest move" is replaced by "one-step safety filter" in every sentence, the sequential problem is withdrawn, and what ships is the one-clamp filter of (i) alone — the argmin over $m$ candidate moves of an evaluated committor vector, with no policy-level claim attached. The replacement's kill is R-20, repositioned here from the verse's Kill field, where it could refute none of the three clauses and here refutes exactly what the replacement asserts.

Frozen (R-20). The one-step goal rule and the two-step optimum (apply $a$, re-solve, best second clamp) disagreeing on more than $0.5$ of $512$ admitted oracle draws at the design point (READ docs/CEQ_SHAPE.md:1758-1763 @ 99777ab, :2275).

Then the one-step filter is not a stand-in for the sequential optimum on more than half the bed, "safest move" is withdrawn as a phrase in every reading, and the chain ends at the Terminal with the evaluated vector label alone licensed. Instrument: $m$ re-solves per draw on the bed's own generator, $0$ GPU-s, one evening. It is strictly cheaper than Kill (ii) by the ratio of solves per draw, $3^{12}/m=531{,}441/8=66{,}430$, and it decides a different object — the adequacy of a filter, not the exactness of a recursion. Control: on the layered family of verse 01.4 (deterministic pointers) the two rules must agree on $100\%$ of instances — every clamp there is a full redirect and the continuation carries no choice; disagreement on that family is an instrument bug. Planted negative: a draw in which the second clamp is forced (one candidate at the successor) must read agreement; a rule ladder that disagrees where the second step has no choice is measuring its own re-solve and not the horizon. The earlier text made this replacement's kill "the three-step optimum disagreeing with the two-step on more than $0.5$ of draws — the same instrument one level deeper, equally priced", which is the same number on the same object measured by the same instrument and is a V-9 replacement (CHARTER.md:133-134) as well as a violation of the Depth rule (CHARTER.md:139-141, which admits no "equally priced" link); it is struck and is not reinstated.

Terminal. The argmin is a decision rule over an evaluated vector label and claims no policy-level optimality; the canon licenses (i) unconditionally, (ii) as the oracle for the sequential extension, (iii) as the eight-position two-epoch witness that the Chebyshev objective's locally optimal prefix costs $0.2$ at the root — a statement about the objective on the read's own action set — and, separately and as an aside only, that the Chebyshev optimum over randomised clamps ($\tfrac12$) lies strictly below the optimum over deterministic ones ($1$) on a one-epoch instance whose mixture the bed's action set does not contain. It withdraws the Chebyshev and lexicographic columns from any sequential reading, and withdraws the earlier claim that the one-epoch mixture instance exhibited a failure of the principle of optimality.

01.12 — The reach-avoid identity with a sink, and the degeneracy lemma#

Statement. With $\mathcal A$ exhausting the absorbing states and $\rho(Q)<1$, $$q^{(\rm sink)}+q^{(0)}+\sum_{k\ge1}q^{(k)}=\mathbb 1\ \text{on } T,\qquad \text{and if the constraint sets alone exhaust absorption, }\ \max_kq^{(k)}\ge1/K .$$ The exact zero-information floor on the argmin is $$\mathrm{floor}_{\rm zeroinfo}=1-\max_a\hat\pi(a^\star),$$ computed from the realised class distribution $\hat\pi$ of the argmin label on the batch the cell is scored on, and $1-1/m=0.875$ at $m=8$ is its uniform-$\hat\pi$ instance and nothing more. The unconditional form is struck at repair. The record carries the conditional form — "exact zero-information floor $1-\max_a\pi\text{-hat}(a^\star)$ from the realised class distribution; at uniform $\pi$ this is $1-1/m$" (READ docs/CEQ_SHAPE.md:2493 @ 99777ab) — and so do three neighbouring books of this canon: 04_BEDS_AND_INSTRUMENTS.md verse 04.3 ($\mathrm{floor}_{\rm zeroinfo}=1-\max_a\hat\pi(a^\star)$, $=1-1/m_{\rm eff}$ at uniform $\hat\pi$; at repair time :59), 06_PREDICTIONS.md verse 06.18 ("$0.875$ at the realised class distribution when it is uniform at $m=8$"; at repair time :256) and 09_CHESS_AND_MARKETS.md verse 09.8 ("at uniform $\pi$ and $m=7$ this is $1-1/7=0.857143$; at $m=8$, $0.875$"; at repair time :91) — all four cited by verse id because their line pins move under their own repairs. This book printed $0.875$ flat and thereby contradicted the record and three books at once. How far the flat form can be wrong, printed: the D-4 admission gate admits any argmin class frequency in $(0.05,0.95)$ (READ docs/CEQ_SHAPE.md:1668 @ 99777ab), so an admitted cell may realise $\max_a\hat\pi(a^\star)=0.95$ and a true floor of $0.05$ — an error of $0.825$ against the printed $0.875$, in the direction that credits an arm scoring at chance with a capability. $\hat\pi$ itself is NOT MEASURED — needs the realised argmin class distribution on the first BED-S batch (B5; book 04 verse 04.21's cell, journalled in 04.15's fieldfloor_zeroinfo), so no value of this floor is printed for BED-S by this book, and $0.875$ appears only with "at uniform $\hat\pi$" attached. The weak Fano $1-\ln2/\ln m$ ($0.6667$ at $m=8$) is not a floor; the tight Fano $(\ln m-I-\ln2)/\ln(m-1)$ ($0.7124$ at $m=8$, $I=0$) sits below the uniform-$\hat\pi$ floor and is retired by book 04 verse 04.3 on that ground, and no value of it is printed here.

Hypotheses. Row sums $1$ including absorbing rows; $0\in\mathcal A_{\rm sink}$; sets exhaust the absorbing states. Non-vacuous at every draw with the sink declared; the sink share read $[0.362,1.000]$ on the one draw run (RUN, READ docs/CEQ_SHAPE.md:1671), which is the admission hazard of Tree C.

Evidence. DERIVED: $\sum_\bullet R_\bullet\mathbb 1=\mathbb 1-Q\mathbb 1$ and $N(I-Q)\mathbb 1=\mathbb 1$; pigeonhole. RUN[I] conservation row $[0.9999999999999993,1.0]$; with BOS inside a constraint set and no goal, $\min_T\max_kq^{(k)}=0.548718\ge\tfrac12$ (RUN[P], READ docs/CEQ_SHAPE.md:1531). Floors RUN at READ docs/CEQ_SHAPE.md:2493.

Mechanism. V-12, V-8 (MISTAKES.md:136), V-23 (:1626), V-10 (the weak Fano sits $0.208$ below chance and would pass an arm worse than chance), L-FLOOR.

Kill — two clauses, one per half of the Statement. (a) The identity half. The conservation row off $\mathbb 1$ by more than $10^{-12}$ on any of $512$ draws: a set was left undeclared and the solve must raise (V-16); $0$ GPU-s. Control: drop BOS from every set — the solve must raise with $\rho(Q)=1.000000$, $\det=0$ (READ docs/CEQ_SHAPE.md:3116). (b) The floor half, which clause (a) cannot decide. A conservation residual says nothing about a floor, so the floor clause is decided by the kill of the book that owns L-FLOOR: 04_BEDS_AND_INSTRUMENTS.md verse 04.3 — an uninformed control (the $0$-hop, random or majority arm) printing an error below $\mathrm{floor}_{\rm zeroinfo}$ computed on the batch it was scored on, on $\ge2$ of $8$ seeds — routed there by verse id and not restated here, plus this book's own assembly clause: any cell of this canon printing $0.875$ as the argmin floor without "at uniform $\hat\pi$" beside it, or printing an argmin capability number without the realised $\hat\pi$ in the same row. That clause is decidable today by reading the page, $0$ GPU-s, and it fired on this verse before repair.

If killed (clause (a): the conservation row is off $\mathbb 1$). Replacement: the sink set is enlarged to every undeclared absorbing position found by the census "$\hat P_{ii}=1$ bitwise and $i\notin\mathcal A$", and the read is re-registered on the enlarged partition. Its Kill is not the same residual re-read — that was a V-9 rename and is struck. A horizon of one more solve cannot refute the enlargement that produced it; the replacement is decided by a different object, a different instrument and a different number:

Frozen. The undeclared-absorbing count $n_{\rm abs}:=\#\{i\notin\mathcal A:\hat P_{ii}=1\ \text{bitwise}\}$ reading $0$ on a draw whose conservation row is still off $\mathbb 1$ by more than $10^{-12}$.

Then the missing mass is not an undeclared absorbing row, the enlargement cannot recover it, and the defect is in the generator's row normalisation rather than in the partition: the identity is withdrawn on that bed and the chain ends at the Terminal. Instrument: one bitwise pass over the diagonal — an integer count, not a sup-norm of a solve — $0$ GPU-s, strictly cheaper than the conservation read, which forms $(I-Q)^{-1}R\mathbb 1$. Planted negative: a draw with one transient row overwritten by $e_i$ must read $n_{\rm abs}\ge1$; a census that cannot see a planted identity row cannot certify the absence of one.

Terminal. The identity is Kemeny–Snell's; the canon licenses the printed conservation row and the exact zero-information floor in its conditional form $1-\max_a\hat\pi(a^\star)$ with the realised class distribution named, licenses $0.875$ only as that floor's uniform-$\hat\pi$ instance at $m=8$, and withdraws every floor not printed beside its class distribution — including every unconditional $1-1/m$ this book printed before repair.

Part IV — The ChaCAL delta as theorems#

01.13 — ChaCAL-published and the diagonal-kept read coincide only at $\gamma=0$#

Statement. Let $A$ be a causal softmax and $A_s=A-\mathrm{diag}(A)$. ChaCAL's published read is $Y=(1-\gamma)A(I-\gamma A_s)^{-1}V$ (diagonal removed inside the inverse, fagnou-2024-chacal Eq. 5 with the "remove the diagonal part" clause, [V-fetched]); the shape's is $O=(1-\gamma)A(I-\gamma A)^{-1}V$. Then $$O-Y=(1-\gamma)\,\gamma\,A(I-\gamma A)^{-1}\mathrm{diag}(A)(I-\gamma A_s)^{-1}V,$$ which vanishes for all $V$ iff $\gamma=0$ or $\mathrm{diag}(A)=0$; on the softmax corner $A_{ii}>0$, so the two reads differ at every $\gamma>0$, ChaCAL-published is sub-stochastic (row sums $0.400\dots0.765$ against $1.000$, RUN[J], READ docs/CEQ_SHAPE.md:207-211) and its resolvent is a finite regime-N sum inside a regime-S read.

Hypotheses. Finite logits, $\beta=1$. Non-vacuous at every $\gamma>0$.

Evidence. DERIVED (resolvent identity $M_1^{-1}-M_2^{-1}=M_1^{-1}(M_2-M_1)M_2^{-1}$ with $M_2-M_1=\gamma\,\mathrm{diag}(A)$); RUN[J] row sums as cited; the convention rests on one HTML fetch (READ docs/sources/design/refute_falsify_math.md:327-344).

Mechanism. P-10 (a source's equation read from a rendered page); V-3 (the parity plant is declared as a construction); B22 (CHARTER.md §5) — the convention decides two controls (C-CHD, C-CHP, READ docs/CEQ_SHAPE.md:3128-3129).

Kill — the arXiv re-read is repriced as an author action and is not this verse's kill; the operator-level kill replaces it. The previous kill was "the source-level re-read of Eq. 5 and Eq. 7 at the arXiv LaTeX source (2410.05565) … price $0$ GPU-s, one evening". Book 04 registers that act as outside this canon's rules: 04_BEDS_AND_INSTRUMENTS.md verse 04.12's Evidence (:193 at repair time) closes with "No LaTeX source has been read: NOT MEASURED — needs one fetch of the arXiv source, an author action outside this canon's no-fetch rule". Pricing it "$0$ GPU-s, one evening" read as if the act were inside; it is NOT MEASURED — needs one author fetch of arXiv:2410.05565, an act no writer, refuter or repairer of this canon may perform, registered as an author action with its own row in book 05 and not as a kill here (the same disposition verse 01.5 gives arXiv:2402.08164). The convention — which arithmetic the published paper ran — is undecided until that fetch happens, and no sentence of this book asserts it. The kill that decides the Statement is the one book 04.12 actually ships, decided by arithmetic and not by the citation:

Frozen. On the same drawn causal softmax $A$ at $\gamma=0.9$ and $\beta=1$, the two arms failing to differ by $O(1)$: $\|O-Y\|_\infty/\|V\|_\infty<10^{-3}$ on any draw, or the row sums of $A_s(I-\gamma A_s)^{-1}$-normalised ChaCAL-published reading $1.000$ within $10^{-12}$.

Either reading refutes the Statement's "the two reads differ at every $\gamma>0$" and collapses the two arms into one, at which point the convention question stops deciding anything. Instrument: the lane's own two arms of 04.11, both FOUND under their own kind (04.14); price $0$ GPU-s, half an evening, $512$ draws. It fires or does not on the bed as registered — the record's instance is on the firing side, row sums $0.400\dots0.765$ against $1.000$ (RUN[J], READ docs/CEQ_SHAPE.md:207-211 @ 99777ab) — which is why the arithmetic and not the fetch is the decidable half. Control: at $\gamma=0$ the two arms must agree bitwise ($O-Y=(1-\gamma)\gamma A(\cdots)$ carries the factor $\gamma$), and a probe reporting a non-zero difference at $\gamma=0$ is broken before it is read at $\gamma=0.9$.

If killed (the two arms do not differ by $O(1)$ at $\gamma=0.9$). The "diagonal kept" clause of the delta is withdrawn — not renamed — and with it every sentence that makes component (k) turn on the diagonal's position. Replacement, on a different object measured by a different instrument at a different number: the row-sum census on the two arms' operators, read before any resolvent is formed. ChaCAL-published's operator is sub-stochastic by construction ($A_s\mathbb 1=\mathbb 1-\mathrm{diag}(A)<\mathbb 1$, 04.12's Hypotheses), so its inverse is a finite regime-N sum (READ docs/CEQ_SHAPE.md:286-289 @ 99777ab: regime N is $A^s=0$), while the shape's is a regime-S resolvent; that is a statement about the operators' rows, not about their reads, and it survives whatever the reads do. Its Hypotheses: $\beta=1$, finite logits, so $\mathrm{diag}(A)>0$ entrywise. Its Evidence: RUN[J] row sums $0.400\dots0.765$ against $1.000$. Its Kill: any drawn cell where ChaCAL-published's row sums read $1.000$ within $10^{-12}$ — then $\mathrm{diag}(A)=0$ numerically, the two operators are the same operator, and the regime distinction is withdrawn as well; decided by one pass over the rows, strictly cheaper than forming either resolvent, with the diagonal-kept arm reading $1.000$ to $10^{-12}$ as the must-fire plant. If that fires, the delta loses its operator clause too and reads: CEQ's operator is ChaCAL's read, and the delta is (e), (g), (h), (i), (k) of READ docs/CEQ_SHAPE.md:260-271 @ 99777ab alone. The InfSA-style base (roffo-2026-infsa [V], ReLU Frobenius, non-causal) is not carried as a link of this chain: it is NOT MEASURED — needs the base wired, unspecified in the canon (no verse of book 04 wires it), so it cannot be a replacement whose kill fires, and it is kept only as the plant named at READ docs/CEQ_SHAPE.md:3110 @ 99777ab for bind B-E1.

Terminal. The read is occupied (READ docs/CEQ_SHAPE.md:2891-2937, the record's own V-7 on a literature search); the canon licenses "ChaCAL's read with the diagonal kept" as a convention statement pending the re-read, and withdraws every sentence calling the resolvent new (K-11).

01.14 — Absorbing rows are the limit of a self-spike: the boundary mechanism is an exactness delta, not a representability delta#

Statement. For $a\in\mathcal A$ let $P^{(L)}$ be the causal softmax whose row $a$ has self logit $L$ above every other entry; then $\|P^{(L)}_{a\cdot}-e_a\|_1\le2s\,e^{-L}$ and, for the reads with every other row shared, $$\|\Pi_\gamma(P_{\rm abs})-\Pi_\gamma(P^{(L)})\|_\infty\ \le\ \frac{2s\,e^{-L}}{1-\gamma},\qquad \|O_{\rm abs}-O^{(L)}\|_\infty\le\frac{2s\,e^{-L}\|V\|_\infty}{1-\gamma},$$ by the mask-amplification bound (READ docs/CEQ_SHAPE.md:404-409). Hence every discounted read with absorbing rows, at $\gamma$ bounded away from $1$, is within $\varepsilon$ of a ChaCAL-diag read with logit margin $L\ge\log(2s/((1-\gamma)\varepsilon))$ on the boundary rows — at $\gamma=0.9$, $\varepsilon=10^{-6}$, $s=64$ this is $$L\ \ge\ \ln\!\Big(\frac{2\cdot64}{0.1\cdot10^{-6}}\Big)=\ln(1.28\times10^{9})=20.9701, \quad\text{printed as } \mathbf{20.97}\ \text{(or }21.0\text{ rounded up)},$$ not the $20.7$ this verse printed before repair. The $1.28$ factor had been dropped from the logarithm: $20.7233=\ln(10^{9})$, and $20.7$ is that value truncated. The misprint is not cosmetic — it breaks the bound it is quoted for. At $L=20.7$ the containment radius reads $2\cdot64\,e^{-20.7}/0.1=1.31\times10^{-6}$, and at $\ln(10^9)=20.7233$ it reads exactly $1.28\times10^{-6}$; both exceed the $\varepsilon=10^{-6}$ the margin is chosen to meet. At $L=20.9701$ the radius reads $10^{-6}$ exactly (DERIVED). Because the inequality is a lower bound on $L$, any rounding of this number rounds up and never down; $20.7$ rounded the wrong way. The verse carries the blow-up printed: the containment radius is $2s\,e^{-L}/(1-\gamma)$, which at fixed $L=30$, $s=64$ reads $1.20\times10^{-11}$ at $\gamma=0.9$, $1.20\times10^{-10}$ at $\gamma=0.99$, $1.20\times10^{-8}$ at $\gamma=0.9999$ and diverges as $\gamma\uparrow1$ (DERIVED: $2\cdot64\cdot e^{-30}=1.198\times10^{-11}$, divided by $1-\gamma$). Component (e) of the delta is therefore not a representability statement on the $z$ and $\Delta z$ channels at $\gamma<1$: there it is (1) exactness at finite parameters ($\delta=0$ against $\varepsilon>0$) and (2) a training statement — whether AdamW finds the spike (book 02). A column sink (K-E1's control) is a further parameterisation inside the same $\varepsilon$-class on those channels.

The committor head at $\gamma=1$ is outside the finite-logit class entirely, and this verse's containment does not reach it. Two readings, both against it, neither a bound: (a) with $\mathcal A$ declared, the boundary rows lie outside $T$, so $\hat Q=Q$ and $\hat R=R$ exactly — the spike rows never enter the transient block — and $\|\hat q^{(L)}-q\|_\infty=0$ identically; the former clause "$\|\hat q^{(L)}-q\|_\infty\le\kappa\cdot2s\,e^{-L}$" is then satisfied by construction (V-10) and decides nothing. (b) with $\mathcal A$ not declared — the reading the class-containment claim needs, since ChaCAL-diag declares no boundary — regime S fixes $P_{00}=1$ and $P_{ii}<1$ for $i\ge1$ (READ docs/CEQ_SHAPE.md:286-287 @ 99777ab), so position $0$ is the only absorbing state; absorption into any $\mathcal A_k$ not containing $0$ has probability exactly $0$, and at $\gamma=1$ the resolvent $(I-P)^{-1}$ does not exist at all on the full chain ($1$ is an eigenvalue of a row-stochastic $P$). The error on that reading is $\|q\|_\infty$, an $O(1)$ quantity — the record's own instance reads $\min_T\max_kq^{(k)}=0.548718$ (RUN[P], READ docs/CEQ_SHAPE.md:1531 @ 99777ab), so $\|q\|_\infty\ge0.548718$ — and not $\le\kappa\cdot2s\,e^{-L}$, which at $L=30$ reads $\le9.5\times10^{-11}$ at the $\kappa\le7.9$ the earlier text assumed and $\le3.1\times10^{-10}$ at the upper endpoint $\kappa\le25.70$ verse 01.1 derives from the quotient at repair — wrong by ten orders and by nine, so the conclusion is unchanged and the number it rests on is now the derived interval rather than the mis-united $7.9$. The clause is deleted. What replaces it: the committor head is a read of a declared partition of an operator, and the declaration is an architectural act, not a weight; no finite-logit softmax realises absorption into a set other than its own $P_{ii}=1$ rows, and no margin $L$ closes the gap because the gap is not $O(e^{-L})$. Whether component (e) is retired on that head is therefore decided separately from the $z$ channel, by the kill below and not by this verse's $\varepsilon$-class.

Hypotheses. Regime S, and $\gamma$ bounded away from $1$: the containment radius $2s\,e^{-L}/(1-\gamma)$ is below a target $\varepsilon$ only for $\gamma\le1-2s\,e^{-L}/\varepsilon$, which at $s=64$, $L=30$, $\varepsilon=10^{-6}$ admits every $\gamma\le1-1.2\times10^{-5}$ and no $\gamma$ nearer $1$ than that. The committor head at $\gamma=1$ is excluded by the Statement's second paragraph and inherits nothing through verse 01.1's $\kappa$; the clause asserting that inheritance is deleted, being either vacuous (with $\mathcal A$ declared: the error is $0$ identically) or false by $O(1)$ (without: the error is $\|q\|_\infty\ge0.548718$). Non-vacuous at every $L$ on the $z$ and $\Delta z$ channels only; the class containment is exact at $L=\infty$ only, and at $\gamma=1$ it does not hold at any $L$.

Evidence. DERIVED (softmax row $a$: mass off $a$ is $\le(s-1)e^{-L}/(1+(s-1)e^{-L})\le s\,e^{-L}$, $\ell_1$ distance to $e_a$ at most twice that; then Prop. 4(ii)); the amplification constant attained up to a constant at $s=3$, $\gamma=0.9$, $\varepsilon=0.1$ — not the judge's $s=32$, $\gamma=0.6$ draw, and the draw is printed because the book's Limits paragraph no longer covers it by a blanket sentence — (RUN[J] $0.5263157894736843$ against naive $0.1$, READ docs/CEQ_SHAPE.md:405-406 @ 99777ab; the earlier pin :407-408 resolves at 99777ab to the row-sum census and the $\hat\beta\ne1$ rows and never to these two numbers, and is corrected here). The sink literature (xiao-2023-attentionsinks, gu-2024-sinkemerges [V]) owns the column device; karbalayghareh-2026-doformer [V] owns one clamped row.

Mechanism. V-24 (MISTAKES.md:1658) — an identity bind whose rejection region is empty: the theorem says the rejection region of "boundary rows versus a spike" is empty in representability and non-empty only in $\delta$ and in training; M-13 (:1210) — no equivalence at $N=8$; D-7 — Bet F's counter is the point estimate, and this verse says why it is representable.

Kill. K-E1 frozen (READ docs/CEQ_SHAPE.md:2274): ChaCAL-with-sink-token's committor $\varphi$-NRMSE within $\mathrm{MDE}_8$ of the shape's on at least $6$ of $8$ seeds and harmonic residual within $2\times$, holding at the escalation to $N=16$.

The price, reconciled against the record and against book 04. K-E1's price is $\approx52$ s with escalation $\approx44$ s, which is the record's own Bet F row (READ docs/CEQ_SHAPE.md:2238 @ 99777ab, the Bet F line), and the class is [ASSUMED] and not [FITTED]: the record marks Bet F's seconds [ASSUMED] on the arm_smprime cell time, and this verse printed [FITTED], which claimed a fit no cell supports. Both are corrected here. The $\approx34$ s a reader may meet in book 04 is a different kill's price: it is Bet E's one bed-cell pair at 04_BEDS_AND_INSTRUMENTS.md verse 04.6 (at repair time :102, :357), the pair verse 01.9's K-H1 runs inside, and it is not K-E1's. K-E1 runs the shape and ChaCAL-with-sink — two arms, not one bed-cell pair — which is why the two numbers differ. Book 04's own K-E1 registration now reads $\approx52$ s [ASSUMED] (verse 04.11, at repair time :185), so the two books agree at repair time and no CORRECTIONS.md row is owed; the finding is answered by reconciliation before birth and not by a correction after it.

And K-E1 is registered UNREACHABLE by the book that owns the arm. 04_BEDS_AND_INSTRUMENTS.md verse 04.11 records that neither ChaCAL-with-sink nor ChaCAL-diag nor ChaCAL-published exists — grep -rniE 'chacal|infsa' --include='*.py' . returns $0$ lines and the $24$-kind census holds $0$ records for each — so under CHARTER.md:145-148 K-E1 is not a kill today, at any price. This verse's kill therefore reads UNREACHABLE, its price is the price it would carry if the two arms existed, and the verse is carried by its two links below, both of which are re-registered at repair to need no ChaCAL arm at all.

Control: ChaCAL-diag with $\mathcal A=\emptyset$ must read $\|\Pi_{\rm shape}-\Pi_{\rm ChaCAL\text{-}diag}\|_\infty=O(1)$ on downstream rows whenever $\mathcal A\ne\emptyset$ (rows changed $=[3]$ on the judge's draw, RUN[MARS], :3110) — a comparison that cannot see the boundary rows on the plant cannot see them on the arm.

If killed — link 1, on the $z$ channel. Component (e) is retired on the discounted channels to "a parameterisation of a column sink"; what survives by theorem is the exactness clause — $\delta=0$ on the exact route against $\varepsilon=2s\,e^{-\hat L}/(1-\gamma)$ printed from the trained margin $\hat L$, with the $1/(1-\gamma)$ factor now carried into the printed number. Its Kill has two clauses, and the first is reachable today.

Frozen (α), the margin arithmetic. Recompute $L_{\rm req}=\ln(2s/((1-\gamma)\varepsilon))$ at every registered $(s,\gamma,\varepsilon)$ triple this book prints, and the containment radius $2s\,e^{-L}/(1-\gamma)$ at every registered $(s,L,\gamma)$ grid point, in float64; any printed value departing by more than $10^{-12}$ relative, or any printed $L$ at which the radius exceeds the $\varepsilon$ it is quoted to meet.

Frozen (β), the label comparison. $\varepsilon\|V\|_\infty$ below the label sd on $6$ of $8$ seeds — the exactness delta is invisible at that margin.

Clause (α) is decided by four logarithms and four exponentials on the page, CPU, no cell, no arm, no bed: $0$ GPU-s, minutes, and it fires on this book as printed before repair, where $L=20.7$ carried a radius of $1.31\times10^{-6}$ against its own $\varepsilon=10^{-6}$. Clause (β) is NOT MEASURED — needs the label sd (B5, book 04 verse 04.21) and is not reachable today. Planted negative for (α): substitute $\ln(10^9)=20.7233$ for $\ln(1.28\times10^9)$ and the radius must read $1.28\times10^{-6}>\varepsilon$ and fire; a recomputation that reports the dropped-factor margin as sufficient is evaluating the formula it was given rather than the bound. Link 1 is cheaper than K-E1 on both clauses and, on (α), needs no cell at all.

If killed — link 2, on the committor head, a different object and a different instrument. Link 1 says nothing about $\gamma=1$, so the head needs its own link rather than the same one renamed. The object is moved off the arm at repair, because book 04 registers every ChaCAL arm as non-existent (verse 04.11: $0$ source lines, $0$ records), so a link read on "the ChaCAL-with-sink arm's own learned operator" would die exactly where K-E1 dies and would be a replacement that does not survive its verse (V-9). The decidable object is the non-sink zero-fraction read on the environment chain the bed already builds: on each of 04.2's admitted draws, take $P_{\rm env}$ with a value-zero sink column appended in the record's own V15Fork.Asink form (READ lean/CEQ/V15Fork.lean:67-70) and no partition declared, solve the $K+1$ absorption channels by the same route the shape uses, and read $\phi_0:=\Pr_{\rm draw}[\,q^{(k)}_i=0\text{ exactly for every }k\ne\text{sink and every }i\,]$. Its Hypotheses: the bed's own generator as registered, $\mathcal A$ undeclared, $512$ admission draws at the design point; no training, no seeds, no matched parameters, no arm. Its Evidence: DERIVED — with no absorbing row declared, the only self-absorbing position of the drawn chain is the one regime S fixes (READ docs/CEQ_SHAPE.md:286-287 @ 99777ab), so $\phi_0=1$ is the predicted reading; the column device is the record's own (READ lean/CEQ/V15Fork.lean:67-70), and the two-read comparison is the column 04_BEDS_AND_INSTRUMENTS.md verse 04.11's link 1 already specifies for the admission census at $0$ GPU-s with no make_arm branch, so this link is NOT MEASURED — needs that census column, $0$ GPU-s, no arm and not needs the sink arm's cells (B5), which is what it read before repair. Its Kill: $\phi_0<1$ on any draw — some non-sink channel carries positive mass, a finite-logit operator does produce absorption into a set it was not given, and component (e) is retired on the committor head too. Its plant: an operator with a literal identity row planted at one non-sink position must read $\phi_0<1$; a census that cannot see a planted identity row cannot certify the absence of one. This link reads an exact-zero predicate over draws — a fraction of $512$ draws, not an $\mathrm{MDE}_8$ comparison of two NRMSEs over $8$ seeds — so it is decided by a different instrument at a different number on a different object from K-E1, and it is strictly more decisive and strictly cheaper: K-E1 asks whether two arms are within a detectable margin and needs both arms to exist, while this asks whether one chain's channel is identically zero on draws the bed already takes, at $0$ GPU-s and with no arm. If both links fire, the delta is (g), (h), (i), (k) and Tree B's sentence (READ docs/CEQ_SHAPE.md:2377-2381 @ 99777ab), and the chain ends.

Terminal. Boundary rows are the $L\to\infty$ limit of the class the arm already contains on the discounted channels at $\gamma$ bounded away from $1$, with the containment radius $2s\,e^{-L}/(1-\gamma)$ printed and diverging at $\gamma\uparrow1$; on the committor head at $\gamma=1$ they are not a limit of that class at all, and the head is a read of a declared partition rather than of a learned operator. The canon licenses "exact at finite parameters, with the margin and the $1/(1-\gamma)$ factor printed" on the $z$ and $\Delta z$ channels; it withdraws "a mechanism softmax cannot express" there; and on the committor head it licenses neither that phrase nor its negation until link 2's $\phi_0$ is read — the declaration is an architectural act, and an act is not a capability number.

01.15 — The interventional re-solve, and when the cached mixture is exact#

Statement. For $(P,V)\to(P',V')$, $\Delta z=(I-\gamma P')^{-1}(\Delta V+\gamma\,\Delta P\,z)$ exactly; the cached mixture $O_{\rm cached}=\Pi_\gamma(P)V'$ equals the true post-intervention read iff $\Delta P\,z'=0$, i.e. on value-only moves, and otherwise $$\|O_{\rm true}-O_{\rm cached}\|_\infty=(1-\gamma)\,\big\|P'(I-\gamma P')^{-1}V'-P(I-\gamma P)^{-1}V'\big\|_\infty \le\frac{\|\Delta P\|_\infty\,\|V'\|_\infty}{1-\gamma}\quad(\text{DERIVED, four steps}).$$ The $(1+\gamma)$ factor is dropped at repair, and the derivation that has no room for it is printed. (i) $\Pi_\gamma(P)=(1-\gamma)P(I-\gamma P)^{-1}=\tfrac{1-\gamma}{\gamma}\big((I-\gamma P)^{-1}-I\big)$; (ii) so $\Pi_\gamma(P')-\Pi_\gamma(P)=\tfrac{1-\gamma}{\gamma}\big((I-\gamma P')^{-1}-(I-\gamma P)^{-1}\big)$ and the constant $I$ cancels; (iii) the resolvent identity $(I-\gamma P')^{-1}-(I-\gamma P)^{-1}=(I-\gamma P')^{-1}\,\gamma\,\Delta P\,(I-\gamma P)^{-1}$ turns this into $(1-\gamma)(I-\gamma P')^{-1}\Delta P\,z'$ with $z'=(I-\gamma P)^{-1}V'$; (iv) $\|(I-\gamma P')^{-1}\|_\infty\le1/(1-\gamma)$ and $\|z'\|_\infty\le\|V'\|_\infty/(1-\gamma)$ give the display, the two $(1-\gamma)$ factors cancelling once. No step introduces $(1+\gamma)$, and no step of the earlier text produced one; the factor was carried with no derivation behind it and inflated the bound by up to $2\times$ at $\gamma\uparrow1$. For a row clamp $P'=P+e_iu^\top$, $u^\top\mathbb 1=0$: $\Delta z=\gamma(Me_i)(u^\top z)/(1-\gamma u^\top Me_i)$ with denominator $(1-\gamma p'_{ii})/(1-\gamma P_{ii})>0$; for a token rewrite $\mathrm{rank}(\Delta P)\le s-i$ — an inequality, corrected at repair from the equality this book and the record's own line (READ docs/CEQ_SHAPE.md:471 @ 99777ab) both printed: rewriting the token at position $i$ changes the $s-i$ rows at or after $i$ and changes no row before it, so $\Delta P$ has at most $s-i$ non-zero rows and its rank is at most that, with equality failing on every rewrite that leaves some later row's logits fixed (a rewrite whose token is unattended by rows $>i$ up to the leak of 01.2). The suffix re-solve price $(s-i)^2d/2$ is supported by the inequality alone — it is charged on the $s-i$ rows that may move and not on the rank — and is unchanged.

Hypotheses. Row-stochastic $P,P'$, $\gamma<1$; triangularity for the suffix identity ($P'_{<i}=P_{<i}$, $V'_{<i}=V_{<i}$). Non-vacuous whenever a move changes $P$ (census line S-17, READ docs/CEQ_SHAPE.md:1722-1727 @ 99777ab — a card of the record, not a verse of book 04: both plant classes non-empty, at least $64$ draws each, READ docs/CEQ_SHAPE.md:1722-1727).

Evidence. DERIVED; RUN identities $1.03\times10^{-15}$–$1.36\times10^{-15}$; Sherman–Morrison against re-solve $1.42\times10^{-15}$ and $1.2339847026143769\times10^{-15}$ with denominator $1.9$ (READ docs/CEQ_SHAPE.md:1521-1526); CITED sherman-1950-inverse-adjustment, hager-1989-updating, schweitzer-1968-perturbation, bottou-2013-counterfactual §7.3 (the linearisation) [V]; the cached-representation mechanism momennejad-2017-sr, russek-2017-predictive [U].

Mechanism. M-8 (MISTAKES.md:544, the arm is never priced at the oracle's rank-one rate); D-5 (:755, the constant-value plant reads $\le10^{-15}$, Gaussian $V$ reads $O(1)$); D-2; V-9 (:154, the re-solve must move a number: the cached control is the repair-that-changes-nothing plant).

Kill. K-G1: the cached mixture within one seed sd of the shape on $P$-changing plants on at least $6$ of $8$ seeds — the re-solve is a per-row control wearing a name; $\approx35$ s, one evening. Control: on $V$-only plants the cached arm must match the shape to $10^{-12}$ (the identity is exact there); a cached arm that fails on $V$-only plants is mis-wired.

If killed. Component (g)'s re-solve sentence is retired; the displacement identity survives as algebra and the channel as a cost statement ($m$ candidates at $m\cdot s^2/2$ MACs against one solve's $s^2d/2$, ratio $m/d=0.5$ at $(8,16)$, Bet G's band $(0.5\times,8\times)$); its Kill is the measured ratio at or above $8\times$ (seconds, V-17 of the ledger). If that fires, a re-solve per candidate is what ships and the price is quoted as measured.

Terminal. Consequence as a displacement field is one extra solve, exact; the canon licenses the identity and its price and withdraws "the re-solve is needed" wherever K-G1 reads its counter.

01.16 — The Neumann certificate $\gamma^{K'+1}/(1-\gamma)$ is attained, in vector units, and dies at $\gamma=1$#

Statement. For $P\ge0$ row-stochastic (identity rows included), $\gamma\in[0,1)$: the matrix residual of the $K'$-truncated series is exactly $\gamma^{K'+1}/(1-\gamma)$ for $(I-\gamma P)^{-1}$ and exactly $\gamma^{K'+1}$ for $\Pi_\gamma$; the printed certificate is $\delta\,\|V\|_\infty$ with $\|V\|_\infty$ and $1/(1-\hat\gamma)$ beside it; a sparsified $P_m$ with dropped row mass $\varepsilon$ costs $\varepsilon\|V\|_\infty/(1-\gamma)$ more; and at $\gamma=1$ on the transient block the certificate is $\infty$ — the committor head ships on the exact route only ($\delta=0$) and no truncated committor carries a certificate ($\|Q\|_\infty=0.958$ gives $19.2$ at $K'=4$ against a true error $2.5\times10^{-3}$, RUN[I]).

Hypotheses. As stated; for $\|P\|_\infty>1$ the bound can fail (the $\hat\beta\ne1$ rows summing $1.31\dots10.29$ print no certificate, READ V16_ARM_SMPRIME.md:28-32 @ 99777ab). Non-vacuous at every $\gamma>0$; at $\hat\gamma>0.99$ the factor $1/(1-\hat\gamma)>100$ and the mirror kill K-J prints it. $K'$ is the Neumann truncation order with registered support $K'\in\{1,2,4,8,16\}$, journalled per cell; it is a different object from the $K$ constraint sets of this book's Notation, which is fixed at $K=2$ at the design point, and the two share a letter only in CHARTER.md §6's scope sentence. This verse and verse 01.7 wrote $K$ for both until repair; both now write $K'$, matching 08_ARCHITECTURE.md verse 08.11.

Evidence. DERIVED + RUN (READ docs/CEQ_SHAPE.md:391-413, :853-855); the convergent plant rows $1.5$, $\gamma=0.6$, $K'=2$: $7.29$ against $0.54$ (RUN[M]); the divergent $\gamma=0.7$ plant labelled as such.

Mechanism. V-3, V-10 (err = bound is a declared identity; the bind is the planted non-stochastic $P$); V-17 (vector units); L-CERT.

Kill — the reachable one, registered at repair; K-I is not this verse's kill. K-I as previously written ("one exceedance of $\delta\|V\|_\infty$ on any of $1{,}024$ draws, or $\delta\|V\|_\infty$ at or above the label sd; $\approx2.6$ s") cannot fire on the bed as registered, and both of its branches need an object the canon records as absent: branch 1 needs a shipped mask, and CHARTER.md:241 (B15) records "the kernels do not exist: chunked solve, CSR path, Mapper schedule are NOT MEASURED"; branch 2 needs the label sd, and CHARTER.md:231 (B5) records "BED-S has no cell, no realised sd". The record itself prices it dormant — bind B-I is "REPAIR, dormant … dormant until row M ships a mask" (READ docs/CEQ_SHAPE.md:3116 @ 99777ab), $\approx2.6$ s when a mask exists. Under CHARTER.md:145-148 that is not a kill. K-I is therefore a dormant row of book 03, named here and owned there, and this verse's kill is the one decidable at $0$ GPU-s today:

Frozen. Recomputing the $K'$-truncated residual on the record's own registered $(\gamma,K')$ pairs and finding it depart from $\gamma^{K'+1}/(1-\gamma)$ by more than $10^{-12}$ in relative units on any pair — the word "attained" is withdrawn and the certificate is demoted from an equality to an upper bound.

Instrument: float64 recomputation of a $\le64\times64$ Neumann partial sum, CPU, no cell, no mask, no bed; price $0$ GPU-s, half an evening. The pair the record already banks passes: residual $0.007754350466241788$ against $\gamma^{17}/(1-\gamma)=0.007754350466240224$ at $K'=16$, $\gamma=0.7$ — absolute departure $1.5647\times10^{-15}$, relative $2.0179\times10^{-13}$, inside the frozen $10^{-12}$ (RUN, READ docs/CEQ_SHAPE.md:853-855 @ 99777ab; the relative figure DERIVED from the printed pair). Must-fire plant: the rows-$1.5$ non-stochastic $P$ at $\gamma=0.6$, $K'=2$ must exceed the bound, $7.29$ against $0.54$ (RUN[M]) — an instrument that reports equality on a plant whose row sums are $1.5$ is reading the formula and not the matrix. Second plant, in the other direction: $\gamma=0.7$, $K'=2$ must now read $1.143$ and not $0.343$; a recomputation returning $0.343$ has dropped the $1/(1-\gamma)$ factor, which is the defect verse 01.7's table carried until repair.

If killed (the residual departs from the equality by more than $10^{-12}$ relative). The equality clause is withdrawn and what survives is the inequality $\|\cdot\|_\infty\le\gamma^{K'+1}/(1-\gamma)$, which is the Neumann tail bound and needs only non-negativity and row sums $\le1$. Replacement, with a different object and a different instrument: the certificate is re-registered as the dropped-mass bound $\varepsilon\|V\|_\infty/(1-\gamma)$ read off the sparsified $P_m$'s own row sums, a census over the operator's rows rather than a norm of its resolvent; its Hypotheses are $P_m\ge0$ and $P_m\mathbb 1\le\mathbb 1$ alone (no equality, no attainment); its Evidence is the row-sum census; its Kill is any row of $P_m$ with $\sum_jP_{m,ij}>1+10^{-12}$ — the dropped-mass reading is then negative and the bound is not a bound — decided by one pass over the rows, strictly cheaper than the resolvent recomputation. If that fires, no certificate ships on any route and the mask is refused: the exact solve ships, priced against one hop at $s=64$ (READ docs/CEQ_SHAPE.md:2395), and the chain ends at the Terminal with K-9 (book 03) owning the price.

Terminal. The certificate is textbook (meyer-2000-matrix, horn-2013-matrix), claimed as an instrument and never as mathematics; the canon licenses it in vector units on Neumann routes and withdraws it on every route at $\gamma=1$.

01.17 — Lean containment: the delta's theorems with their grades#

Statement. The delta's clauses are carried by named targets, graded [M] (a Mathlib route confirmed at the pinned revision a45ae637, toolchain leanprover/lean4:v4.7.0), [S] (a supporting lemma supplied in-file), [D] (no object in this Mathlib); every [M] is pending lake build and nothing is cited as proved before it builds. No row of the table below carries [D] after repair: the one that did — the hitting-time row — was graded on a repository grep, and Mathlib at the book's own pin carries the object (note below the table), so [D] remains defined here and is claimed nowhere in this table. The one [D] left in the book is verse 01.22's, which asserts no Lean target at all.

clause	target	grade	Mathlib name (`[V-name]` at the pin)	refusal shipped
parity (k)	`gamma_zero_is_softmax`	[M]	`Matrix.inv_one`	`gamma_half_is_not_softmax`
BOS absorbing (e)	`bos_row_is_absorbing` at $\beta=1$	[M]	`Finset.sum_range_one`	at $\beta=0$ row 0 is $e^{qk_{00}}=2.0138$
F2 (e), 01.1	`later_boundary_unreachable`, `causal_forward_only`	[M]	`Matrix.blockTriangular_inv_of_blockTriangular` (`Block.lean:346`)	`dense_P_displaces_backward`
unit (e), 01.10	`lower_triangular_isUnit`, `diag_one_sub_smul_pos`	[M]	`Matrix.det_of_lowerTriangular` (`Block.lean:265`), `Matrix.isUnit_iff_isUnit_det` (`NonsingularInverse.lean:151`)	`zero_diag_not_unit`
committor (h), 01.3	`committor_is_resolvent_read` (a), `isUnit_one_sub_transient_causal`	[M]	as above	`bos_undeclared_singular`
displacement (g), 01.15	`displacement_identity`	[M]	`Matrix.sub_mulVec`, `mulVec_sub`	`const_value_zero_displacement`
segment cut, 01.1	`cut_makes_segment_head_absorbing`, `cut_severs_boundary_sets`	[M]	—	—
D-2 separation, 01.2	`lowerTriangular_ne_symmSupport`	[M]	—	`zero_not_a_counterexample` pattern
certificate (i), 01.16	`neumann_truncation_bound`, `neumann_tail_attained`, `mask_amplification`	[S]	—	`nonstochastic_breaks_bound` at $\gamma\cdot3/2<1$
spike limit, 01.14	`absorbing_row_is_spike_limit` (new: $\\|P^{(L)}_{a\cdot}-e_a\\|_1\le2se^{-L}$)	[S]	`Real.exp_pos`, `Finset.sum_le_card_nsmul`	`finite_logit_never_absorbing`
conservation, 01.12	`reach_avoid_sum_one`, `no_goal_no_sink_forces_max_ge_inv_K`	[S]	—	`unit_needs_rho_lt_one`
Chebyshev prefix failure, 01.11(iii)	`chebyshev_prefix_not_optimal` (new: the two-epoch witness on `Fin 8`, $\max q(v)$ locally $d$, globally $c$, root gap $0.2$)	[M]	`decide` / `norm_num` over `ℚ`	additive plant: on $q^{(1)}$ alone both argmins are $d$
Chebyshev randomisation, 01.11(iii) aside	`chebyshev_randomised_below_deterministic` (the one-epoch `Fin 4` instance: $\tfrac12$ against $1$)	[M]	`decide` / `norm_num`	—
reduction, 01.4	`hop_layered_committor` (new: $q\in\{0,1\}$ and the argmax on the layered DAG)	[S]	— (see the note below the table)	`forward_pointer_reads_zero`
hitting time, 01.3	`hitting_time_transform`	[S]	`MeasureTheory.hitting` (`Mathlib/Probability/Process/HittingTime.lean:51`) exists at the pin; the absorption identity does not	`forward_pointer_reads_zero` (01.4's, re-used: a chain with no path to $\mathcal A_k$ must read $E_i[\gamma^{\tau_k}]=0$)

Note on the hitting-time row, regraded at repair. The row read [D] — "no object in this Mathlib" — on the evidence of a repository grep returning zero files. The grade is false at the book's own pin: lean/lake-manifest.json:61 pins Mathlib at a45ae63747140c1b2cbad9d46f518015c047047a, and that revision carries Mathlib/Probability/Process/HittingTime.lean:51, noncomputable def hitting [Preorder ι] [InfSet ι] (u : ι → Ω → β) (s : Set β) (n m : ι) : Ω → ι (RUN this session: grep -n 'noncomputable def hitting' on the vendored package at the pinned revision returns that line). A grep of the repository's own lean/CEQ/ tree is not evidence about Mathlib, and it was the only evidence the [D] carried (P-1); every other row of this table cites a Mathlib path:line, and the four checked resolve exactly (Block.lean:265, Block.lean:346, NonsingularInverse.lean:151, SchurComplement.lean:202). The row is regraded [S] — a supporting lemma supplied in-file — and the gap is named precisely rather than as an absence: Mathlib's hitting is a stopping time on a filtered stochastic process, indexed by a Preorder with InfSet and defined pathwise; what verse 01.3 needs is the finite-state absorption identity $E_i[\gamma^{\tau_k}\mathbb 1_k]=\big((I-\gamma Q)^{-1}\gamma R_k\mathbb 1\big)_i$ on a row-stochastic matrix with an absorbing partition, and no route from hitting to that identity exists in this Mathlib — there is no finite-chain first-step-analysis lemma connecting the pathwise object to a matrix resolvent. That is a statement about a missing bridge, not about a missing object, and it is what the row now says.

Note on the 01.4 row, rewritten at repair. The column is headed "Mathlib name ([V-name] at the pin)" and previously held Nilpotent.pow_card_eq_zero (lean/CEQ/Nilpotent.lean:77), which is a repository lemma of this tree and not a Mathlib name; it is moved out of that column. Its statement is theorem pow_card_eq_zero {A : Matrix (Fin n) (Fin n) R} (hA : StrictlyLower A) : A ^ n = 0, and StrictlyLower is regime N, which "admits no absorbing row" (READ docs/CEQ_SHAPE.md:288-289 @ 99777ab). Verse 01.4's construction at the $\varepsilon$-clause is regime S with self entries at $-L$, so $P_{ii}>0$ and the hypothesis fails there. The route is therefore admissible on reading (a) only — the strict pointer DAG at $\gamma=1$, no self entries, the walk exactly $k$ steps — which is the reading 01.4's Hypotheses already separate out and the one hop_layered_committor is stated on; on reading (b), the softmax $\varepsilon$-clause, the nilpotency route does not apply and the clause is carried by the union bound of 01.4's Statement instead. The row's grade stays [S], its supporting lemma is supplied in-file, and the repository citation is recorded here rather than in a column reserved for Mathlib.

Hypotheses. Grades are statements about statements; nothing was compiled this session; lake build last exited $0$ on a warm cache with $169$ declarations across $12$ files and zero sorry (READ docs/CEQ_SHAPE.md:816-830). Non-vacuous on every row: a green build with a $0\%$ domain census is decoration on that bed (L-DOM), so each row inherits the census of the verse it carries.

Evidence. READ docs/CEQ_SHAPE.md:604-627, :2600-2626, :2328-2352; Mathlib names as recorded by card J-L0's census (:1425-1431); the three new targets are DERIVED here and carry no name beyond the ones listed.

Mechanism. P-11 (MISTAKES.md:1597, an [M] grade cited as a build); L-LEAN; V-25 (census beside every green tag); P-4 (a target with a decide witness on Fin 4 exists as text, not as scaffolding).

Kill. K-K: any [M] target not building by the Lean milestone (J-L18, READ docs/CEQ_SHAPE.md:823 @ 99777ab, :863, :1368 — a card of the record; book 04 carries no J-L18: exit $0$, zero sorryAx, axioms [propext, Classical.choice, Quot.sound]); $0$ GPU-s, CPU minutes, one evening. Control: a planted sorryAx in one file must turn the axiom print red (the record's own must-fire, READ MATHEMATICS.md:465-486).

If killed. The row is [S] and never cited as proved; what survives is every float instance as RUN with its plant (verse-by-verse above); its Kill is the instance itself reading above $10^{-12}$, cheaper than a build. If an [S] row also fails, the clause is DERIVED text and the sentence "machine-checked" is withdrawn for that clause only.

Terminal. "Machine-checked containment" is licensed exactly for the rows that build with the three-axiom set and a non-zero census; the canon withdraws it elsewhere.

Part V — The LEAPABLE fields as named theorem targets#

01.18 — Realisation theory (Hankel–Kronecker): no finite-state realisation of the softmax resolvent read#

Statement. Target no_finite_state_dual_of_softmax_resolvent [S]: for the causal softmax $P$ with logits of rank $d$ and the map $V\mapsto z=(I-\gamma P)^{-1}V$, the Hankel block $H_i=[(I-\gamma P)^{-1}]_{\ge i,<i}$ has rank exceeding any fixed $e$ for generic logits as $s$ grows, so no linear recurrence of state dimension $e$ reproduces the read on all inputs; the minimal state at position $i$ is $\mathrm{rank}\,H_i$ (Kronecker–Fliess, fliess-1974-hankel [V]; the $L^\infty$ error of the best rank-$e$ Hankel approximation is $\sigma_{e+1}(H_i)$, glover-1984-hankel [V]). The consequence for the canon: the chunked-WY native skyline (yang-2024-deltanet, dao-2024-ssd [V]) is closed to the softmax corner by hu-2025-ssdtheory [V] (rank $T$), and the read is priced as a full triangular solve (READ docs/CEQ_SHAPE.md:583-590).

Hypotheses — the geometry at which the statement first bites, printed. Generic logits; "generic" is a measured predicate — the numerical Hankel rank at tolerance $\tau$ on drawn $\hat P$. $H_i$ is the $(s-i)\times i$ lower-left block of a lower-triangular matrix, so $$\mathrm{rank}\,H_i\ \le\ \min(i,\,s-i)\ \le\ \lfloor s/2\rfloor\quad\text{for every }i,$$ and the non-vacuity clause "that rank exceeds $2d=32$" therefore requires $\lfloor s/2\rfloor\ge2d+1=33$, i.e. $s\ge4d+2=66$ at $d=16$ (DERIVED). At the planned geometry $s=64$ the clause is unsatisfiable: $\max_i\min(i,s-i)=32=2d$ exactly, so $\mathrm{rank}\,H_i\le32$ at every $i$ and every draw. At $s=4d+1=65$ it is still unsatisfiable — $\lfloor65/2\rfloor=32=2d$ — so the first geometry at which the statement bites is $s=66$, $i=33$, where $H_{33}$ is $33\times33$ and its rank may exceed $2d$. The probe is therefore registered at $s=66$, $i=33$, $1{,}024$ draws, and not at $s=64$, $i=s/2$. It is NOT MEASURED — needs the Hankel-rank probe on drawn $\hat P$, **unspecified in the canon**: a grep for "Hankel-rank" in04_BEDS_AND_INSTRUMENTS.mdcounts $0$, and its two "Hankel" hits are BED-K's ceiling $\sqrt{1-1/d}$ in verse 04.3, a different object; the shape stated here — SVD of $H_{33}$ at $s=66$ over $1{,}024$ draws, numerical rank at $\tau=10^{-6}$ — is a specification this verse writes and book 04 does not carry. The record's 1/d Hankel floor is BED-K's and binds BED-K only (READ V20_R15_LEAP_LEDGER.md:323 @ 99777ab).

Evidence. CITED as above ([V], [V-eq] owed for Fliess and Glover); the rank-$T$ closure of hu-2025-ssdtheory [V]; the field named at READ docs/CEQ_SHAPE.md:1220 (Q1/W3, LEAPABLE by both offices).

Mechanism. P-7 (the field now names the object it is about); V-25; M-8 (a finite-state dual would re-price the solve, so its absence is a pricing hypothesis).

Kill — the $s=64$ form is struck as satisfied by construction; the kill is registered at $s=66$. The previous kill read "a linear recurrence of state dimension $e\le2d=32$ reproducing $z$ to $10^{-12}$ on $1{,}024$ drawn $(\hat P,V)$ at $s=64$", with the probe specified as the SVD of $H_i$ at $i=s/2$. At $s=64$ that $H_{32}$ is $32\times32$, so $\mathrm{rank}\,H_i\le32=2d$ on every draw, a time-varying linear realisation with $e=32$ exists for every $(\hat P,V)$, and the kill fires on $1{,}024$ of $1{,}024$ draws whatever the operator is — a gate satisfied by construction (V-10) whose own non-vacuity predicate is unsatisfiable at the same geometry (V-11). Frozen replacement:

Frozen. At $s=66$, $d=16$, $i=33$: the numerical rank of $H_{33}$ at $\tau=10^{-6}$ reading $\le2d=32$ on any of $1{,}024$ draws — the read has a $32$-state dual at the first geometry where the question is open, and the non-realisability sentence is withdrawn together with the pricing that rests on it.

Instrument: SVD of a $33\times33$ block, CPU float64, no cell; price $0$ GPU-s, one evening — but unreachable until book 04 carries the probe (Hypotheses above). Control: the linear corner ($\beta=0$, $g\equiv0$) at $d=16$ must read numerical rank $\le d=16$ at the same $\tau$ (it has a finite-state dual by construction); a probe that reads full rank on the linear corner cannot certify the softmax corner's non-realisability. Second control, against the defect just repaired: run the probe at $s=64$, $i=32$ and it must read rank $\le32$ on every draw — a probe that reports rank $33$ at $s=64$ is reading noise above $\tau$, not rank.

At the planned geometry the statement is not decidable in either direction, and that, not a measured rank, is what this verse licenses at $s=64$.

If killed. The statement moves to numerical rank at tolerance $\tau$: the read is $\sigma_{e+1}(H_i)$-approximable by an $e$-state recurrence, and T1's "one solve versus $L$ layers" is re-priced against the $e$-state scan (book 03); its Kill is the cost-law band K-9. If that dies, the read is priced as the full solve and no finite-state claim is made in either direction.

Terminal. The canon licenses "no finite-state dual for row softmax" as a citation with its rank statement, and withdraws every sentence pricing the shape against a scan it has not been shown to lack. At $s=64$, $d=16$ it licenses neither the statement nor its negation: $\mathrm{rank}\,H_i\le\lfloor s/2\rfloor=2d$ holds identically there, so the question is closed by arithmetic before any draw is taken, and the field enters the canon only at $s\ge4d+2=66$.

01.19 — Koopman spectral theory reduces to the diagonal on the causal class#

Statement. Target spectrum_of_lowerTriangular_eq_diag [M]: the Koopman operator of the affine map $T:x\mapsto V+\gamma Px$ on linear observables is $\gamma P^\top$, affine by $c^\top V$ — not $\gamma P$, which the verse asserted before repair. The one-line derivation, printed because the identification and not the conclusion was wrong: for $f(x)=c^\top x$, $$(Uf)(x)=f(T(x))=c^\top(V+\gamma Px)=\gamma\,(P^\top c)^\top x+c^\top V,$$ so $U$ acts on the coefficient vector as $c\mapsto\gamma P^\top c$ plus the constant $c^\top V$, and the linear observables are an invariant subspace only modulo constants (DERIVED). The spectral conclusion is unchanged, and this is why: $\mathrm{spec}(P^\top) =\mathrm{spec}(P)$, the characteristic polynomials being equal by $\det(\lambda I-P^\top)=\det((\lambda I-P)^\top)=\det(\lambda I-P)$, so the diagonal read below is read off $P$ exactly as before (koopman-1931-hamiltonian, mezic-2005-spectral, brunton-2022-koopman [V]); on the causal class $\gamma P$ is lower-triangular ($\gamma P^\top$ upper-triangular, same diagonal) and the spectrum is $\{\gamma P_{ii}\}$, so the field's content on the arm is the diagonal read of verse 01.10(a) and nothing more. The record's Q3/W1 field (transfer-operator / Koopman, READ docs/CEQ_SHAPE.md:1222) is therefore attached to a one-line theorem, and the instrument swap it required — lambda_hat_live at scripts/v15_r1.py:384, not lambda_hat at :383, which averages $\log m$ and reads $-\infty$ on one $m_k=0$ — is a harness fact (B16, book 05).

Hypotheses. Regime S, finite logits. Non-vacuous at every draw of the class; on the oracle chains (non-triangular $Q$) the spectrum is not the diagonal and Koopman has content there only.

Evidence. DERIVED (the characteristic polynomial of a triangular matrix is $\prod(\lambda-P_{ii})$; Mathlib route Matrix.det_of_lowerTriangular [V-name], the charpoly form [U]); RUN[J] $\rho(Q)=\max_TP_{ii}=0.6926596893360386$.

Mechanism. P-7; V-25; V-17 (the lambda_hat one-bit defect, READ docs/CEQ_SHAPE.md:1280).

Kill — the strict-upper check is struck as this verse's kill; the row-sum census replaces it. The previous kill ("any trained cell whose $\hat P$ has a non-zero entry above the diagonal by torch.equal on the strict upper block") is satisfied by construction (V-10) and its firing set is empty: a causally masked softmax writes $-\infty$ into the strict upper block before the exponential, so $\exp(-\infty)=0.0$ exactly and the block is bitwise zero on every draw of the class as registered. The verse's own former If-killed conceded it ("the arm is mis-masked") and then re-ran the identical check, which is a V-9 replacement. Both are struck. The frozen kill is the condition that can fire on this class and has already fired once in the record:

Frozen. The row-sum census on every trained cell: any cell with $|\sum_j\hat P_{ij}-1|>10^{-12}$ for some $i$, or with $\gamma\max_T\hat P_{ii}\ge1$.

The gate is decidable by one pass over the saved $\hat P$; $0$ GPU-s, half an evening, $N=8$ cells. It fires on the registered class: the arm at $\hat\beta\ne1$ reads row sums $1.31$ to $10.29$ (READ V16_ARM_SMPRIME.md:28-32 @ 99777ab), so $\gamma\hat P$ is not sub-stochastic, the Neumann series does not converge from row-stochasticity, and the diagonal read of the spectrum bounds nothing about $\|(\gamma\hat P)^t\|$ — the Koopman sentence needs $\rho(\gamma\hat P)<1$ and an operator whose powers the spectral radius controls, and $\hat\beta$ is learnable per instance (Ruling 2/2a). Triangularity survives the census; the spectral identity $\mathrm{spec}(\gamma\hat P)=\{\gamma\hat P_{ii}\}$ is algebra and no draw refutes it, which is exactly why it is not the kill. Control: the $\beta=1$ masked-softmax arm must read row sums $1.000$ to $10^{-12}$ on every cell — a census blind to the $10.29$ rows would also be blind to the $1.000$ rows and is not a census. Second plant: a cell with $\hat\gamma\ge1/\max_T\hat P_{ii}$ must fire the second clause; the record's judge draw ($\max_T\hat P_{ii}=0.6927$, RUN[J]) needs $\hat\gamma\ge1.4436$ to fire it and does not, so the clause is registered as reachable only through the $\hat\gamma$ dial, whose corner descent is book 02's B12.

If killed (a cell reads row sums off $1$, or $\gamma\max_T\hat P_{ii}\ge1$). The spectrum statement stands as algebra and every resolvent sentence at that cell is withdrawn — the field is decoration there, not merely uninformative. Replacement, on a different object and with a different number: the admitted-cell fraction, the count $n_{\rm adm}/N$ of cells passing the census, printed beside every Koopman sentence, so the field's reach is a number and not an assumption. Its Hypotheses: the census is run on all $N=8$ cells before any Koopman sentence is written. Its Evidence: the census counts (NOT MEASURED — needs a BED-S cell, B5). Its Kill: $n_{\rm adm}/N=0$ — the class as trained is empty and the Koopman field is withdrawn from BED-S entirely rather than restricted; decided by reading one integer already computed by the census, strictly cheaper than the census itself. The chain ends there: at $n_{\rm adm}=0$ there is nothing left to restrict.

Terminal. Koopman spectral theory is decoration on the causal class; the canon licenses the diagonal read and withdraws the field from every BED-S sentence.

01.20 — Projection-operator formalism: absorbing elimination is memoryless, transient elimination is a Schur complement#

Statement. Target absorbing_projection_memoryless [S]: projecting the linear chain onto $T$ with $\mathcal A$ absorbing yields the closed equation $q=Qq+R\mathbb 1$ with no memory kernel, because $P_{\mathcal AT}=0$ (Mori–Zwanzig's memory term $\sum_tQ_{T\mathcal A}P_{\mathcal A\mathcal A}^tP_{\mathcal AT}$ vanishes identically; mori-1965-transport, zwanzig-1961-memory [V]). Eliminating a transient subset $T'\subset T$ instead gives the exact Schur complement $$Q_{\rm eff}=Q_{T''T''}+Q_{T''T'}(I-Q_{T'T'})^{-1}Q_{T'T''},\qquad R_{\rm eff}=R_{T''}+Q_{T''T'}(I-Q_{T'T'})^{-1}R_{T'},$$ which is the exact coarsening of the read and the object any Mapper/CSR schedule approximates (book 03); dropping the coupling block $Q_{T''T'}$ costs $\|Q_{T''T'}\|_\infty\|(I-Q_{T'T'})^{-1}\|_\infty(\|Q\|_\infty+\|R\|_\infty)$ in the effective operator (DERIVED).

Hypotheses. $I-Q_{T'T'}$ a unit. The condition this was written on was satisfied by construction and is replaced by the reachable one (V-10). The earlier text read "holds on the causal class by verse 01.10's branch (a) whenever $0\notin T'$"; this book's Notation declares $0\in\mathcal A_{\rm sink}$ (READ docs/CEQ_SHAPE.md:295 @ 99777ab), so $0\notin T$ and hence $0\notin T'$ for every $T'\subset T$ at every draw — the antecedent could not fail and the If-killed below repaired a failure that could not occur. The reachable condition it stood in for is derived here: on the causal class $Q_{T'T'}$ is lower-triangular, so $$\det(I-Q_{T'T'})=\prod_{i\in T'}(1-\hat P_{ii}),$$ and $I-Q_{T'T'}$ fails to be a unit exactly when some $i\in T'$ has $\hat P_{ii}=1$ (DERIVED). Finite softmax logits forbid that — $\hat P_{ii}=1$ needs an infinite logit gap — and the one mechanism in this canon that produces it is a saturated gate, verse 01.17's cut_makes_segment_head_absorbing, which writes an exact $1.0$ on the diagonal and makes the segment head absorbing without a declaration. That is the census line this verse registers:

Census (reachable, $0$ GPU-s). $\max_{i\in T}\hat P_{ii}<1$ on every cell, read by the row-sum and diagonal pass of verse 01.19 on the saved $\hat P$. It reads $0.6926596893360386$ at the judge's draw (RUN[J]), so it passes there and is registered as firing only on a saturated-gate cell.

Non-vacuous at every draw; the coupling norm is the certificate input and is NOT MEASURED — needs the Schur instrument on drawn $\hat P$, **unspecified in the canon**: a grep for "Schur" in04_BEDS_AND_INSTRUMENTS.mdcounts $0$ across verses 04.1–04.21, so the shape stated here — form $Q_{T''T'}$, $(I-Q_{T'T'})^{-1}$, $Q_{T'T''}$ on a drawn $\hat P$ and read $\|Q_{T''T'}\|_\infty\|(I-Q_{T'T'})^{-1}\|_\infty$ — is a specification this verse writes and book 04 does not carry; this verse's kill is unreachable until it does.

Evidence. DERIVED (block elimination; Mathlib route inv_fromBlocks_zero₂₁_of_isUnit_iff at SchurComplement.lean:202, [V-name] per J-L6); the far-field mean-pool bound is the only printed coarsening bound in the tree (READ ceq/multizoom.py:38-49, READ docs/CEQ_SHAPE.md:2637).

Mechanism. P-7; L-CERT (a dropped tile without its Schur bound is refused); V-10.

Kill. The Schur identity off $(I-Q)^{-1}R\mathbb 1$ restricted to $T''$ by more than $10^{-12}$ on any of $512$ draws — the elimination is mis-indexed; $0$ GPU-s. Control: eliminating $T'=\emptyset$ must return $Q$ exactly, and eliminating all of $T\setminus\{i\}$ must return the scalar $q_i$.

If killed (the Schur residual exceeds $10^{-12}$ on some draw). The earlier replacement is struck, not renamed (V-9). It read "the identity re-derived with the sink block kept explicit … its Kill is the same residual with $0\notin T'$ enforced by a census line": its kill was the same residual on the same object read by the same instrument, and its enforcement clause was the condition just shown to hold at every draw — it therefore died to this verse's kill exactly as the verse did, and it repaired a failure that cannot occur.

Replacement, derived here on a different object, with a different instrument and a different number. The residual has two separable causes and the census line above separates them at $0$ GPU-s. Where the census fires — some $i\in T'$ with $\hat P_{ii}=1$ — the defect is not the elimination but the operator: a saturated gate has made position $i$ absorbing without a declaration, $\det(I-Q_{T'T'})=0$ and no elimination on that cell is exact. The route is then the declared-sink repair of verse 01.1's If-killed: the saturated position is added to $\mathcal A_{\rm sink}$, the transient block is re-formed, and the Schur complement is read on the re-declared $T$. Its object is the boundary declaration and the diagonal of $\hat P$, not the eliminated blocks. Its instrument is verse 01.19's row-sum and diagonal pass, not the $512$-draw Schur residual battery. Its number is $\max_{i\in T}\hat P_{ii}$ against $1$, which reads $0.6926596893360386$ at the judge's draw (RUN[J]) and $\rho(Q)=0.692660$ declared against $1.000000$ undeclared (RUN[P], READ docs/CEQ_SHAPE.md:1440 @ 99777ab) — not a residual at all. Its Hypotheses: the census runs on the saved $\hat P$ before any elimination, on all $N=8$ cells. Its Evidence: the two RUN numbers above; the per-cell census itself NOT MEASURED — needs a BED-S cell (B5). Its Kill: the census reading $\max_{i\in T}\hat P_{ii}<1$ on every cell while the Schur residual still exceeds $10^{-12}$ — the operator is clean, no saturated gate exists, and the declared-sink route explains nothing; decided by comparing one already-computed float against $1$, strictly cheaper than the residual battery it follows. Its plant: a $\hat P$ with one diagonal entry set to $1.0$ must fire the census, and the $\beta=1$ masked-softmax draw must not; a census blind to an exact $1.0$ is not a census. If that kill fires the chain ends at the Terminal and the Schur complement is withdrawn as a certificate input, the coarsening reverting to the full transient solve with L-CERT refusing every dropped tile.

Terminal. The projection formalism adds one exact identity (the Schur complement) and one vanishing (the memory kernel on absorbing sets); the canon licenses both as certificate inputs and withdraws "memory" as a mechanism of the read.

01.21 — Kolmogorov $n$-width: the read's approximability by a rank-$d$ bottleneck#

Statement. Target nwidth_of_linear_image [D]: for fixed $P,\gamma$ the set $\mathcal K_\gamma=\{\Pi_\gamma V:\|V\|_2\le1\}$ is a linear image of the unit ball, so its Kolmogorov $n$-width in $\ell_2$ is $d_n(\mathcal K_\gamma)=\sigma_{n+1}(\Pi_\gamma)$ — DERIVED at repair, in two lines, so the equality no longer rests on an owed citation. Let $\Pi_\gamma=\sum_j\sigma_ju_jv_j^\top$ be the singular value decomposition. For any $n$-dimensional subspace $X_n\subset\ell_2$, $\sup_{x\in\mathcal K_\gamma}\mathrm{dist}(x,X_n) =\sup_{\|V\|_2\le1}\|(I-\Pi_{X_n})\Pi_\gamma V\|_2=\|(I-\Pi_{X_n})\Pi_\gamma\|_2$, the operator norm of $\Pi_\gamma$ composed with a rank-$\le n$ orthogonal complement projection; minimising over $X_n$ is Schmidt–Eckart–Young in the operator norm, whose minimum is $\sigma_{n+1}$ and whose minimiser is the span of $u_1,\dots,u_n$. The grade is therefore DERIVED, not [ASSUMED textbook]; Pinkus, n-Widths in Approximation Theory, is recorded as the standard reference and its citation is still owed in references.bib, but no clause of this book now rests on it. The Q5/W1 field (READ docs/CEQ_SHAPE.md:1226, "approximation theory / Kolmogorov $n$-width, not rate–distortion") therefore reduces to the singular-value tail of $\Pi_\gamma$: a depth-1 softmax whose value path has rank $d=16$ loses at most $\sigma_{17}(\Pi_\gamma)$ on the $z$ channel relative to the exact read if it could realise $\Pi_\gamma$'s weights — which verse 01.5 says it cannot below the composition floor — so the width bound is an upper bound on what the rank wall costs and never a lower bound on softmax's error.

Hypotheses. Fixed $P$; the $z$ channel only (the committor head is not rank-limited). Non-vacuous where $\sigma_{d+1}(\Pi_\gamma)/\sigma_1(\Pi_\gamma)$ exceeds the label sd in relative units; NOT MEASURED — needs the SVD of $\hat\Pi_\gamma$ on trained cells, **unspecified in the canon**: a grep for "SVD" in04_BEDS_AND_INSTRUMENTS.mdcounts $0$ across verses 04.1–04.21, so the shape stated here — form $\hat\Pi_\gamma$ from a saved cell and read $\sigma_{17}/\sigma_1$ — is a specification this verse writes and book 04 does not carry; and the label sd it is compared against is itself absent (B5). This verse's kill is unreachable on both counts until book 04 ships the probe and 04.21 ships the first cell.

Evidence. DERIVED (the two-line singular-value derivation in the Statement, replacing the [ASSUMED textbook] grade the verse carried); the field correction at READ docs/CEQ_SHAPE.md:1226 (both offices against the table's rate–distortion). Pinkus is recorded as the standard reference and is owed in references.bib; nothing here is load-bearing on it, and that sentence is now true rather than asserted alongside an [ASSUMED] equality it contradicted.

Mechanism. P-7; V-10 (an upper bound on the loss is never read as a separation).

Kill. $\sigma_{17}(\hat\Pi_\gamma)/\sigma_1(\hat\Pi_\gamma)<10^{-3}$ on at least $6$ of $8$ trained cells at the design point — the read is rank-$16$ approximable and A1 on the $z$ channel is unreachable at $d=16$: the rank wall costs nothing and only the weight realisation (01.5) can separate the arms; $0$ GPU-s from saved cells.

Control — the pass-plant is demoted and the must-fire plant supplied. The verse's only control was "the identity $\Pi_0=P$ at $\gamma=0$ must give the same tail as the softmax matrix itself". At $\gamma=0$, $\Pi_0=(1-0)P(I-0\cdot P)^{-1}=P$ identically, so that control passes on every input, including a probe whose singular values are noise: it has an empty rejection region and is exactly the defect V-15 names — no condemning rule without a planted negative. It is kept, relabelled as what it is, a wiring check that the probe forms $\Pi_\gamma$ from the right matrix, and it is not the plant. The plant that makes the kill fire, added at repair:

Must-fire. Feed the probe a matrix with a known rank-$16$ truncation error: $M=R+10^{-2}E$ with $R$ of exact rank $16$ and $E$ random, normalised to $\|E\|_2=\|R\|_2$. The probe must read $\sigma_{17}(M)/\sigma_1(M)$ near $10^{-2}$ and never below $10^{-3}$; a probe reporting the frozen kill's $<10^{-3}$ on a matrix whose rank-$16$ residual is $10^{-2}$ by construction cannot certify the reading it is asked for.

Second plant, in the direction the kill fires: an exactly rank-$16$ matrix must read $\sigma_{17}/\sigma_1$ at the float64 noise level and fire the kill — a gate that cannot fire on a matrix that is rank-$16$ by construction is not the gate this verse registers. Both plants are $0$ GPU-s and CPU-only, and both are NOT MEASURED for the same reason the kill is: the SVD probe is unspecified in the canon (Hypotheses above).

If killed. The A1 sentence on the $z$ channel is withdrawn at $d=16$ and the capability claim moves to the committor head (verse 01.1's $\kappa$-bound, no rank bottleneck); its Kill is Bet F / K-E1 (verse 01.14). The chain ends.

Terminal. The $n$-width names what a rank bottleneck costs and never what softmax cannot do; the canon licenses the singular-value tail as a printed number and withdraws the field as a separation argument.

01.22 — Finite-mixture inference: a pooled mean over a bimodal seed population is not a component statistic#

Statement. Target none in Lean ([D], no object); the theorem is arithmetic. If $N$ seeds split into two clusters with means $\mu_1,\mu_2$, weights $\pi,1-\pi$ and within-cluster variance $\sigma_w^2$, the pooled variance is $\sigma_{\rm pool}^2=\sigma_w^2+\pi(1-\pi)(\mu_1-\mu_2)^2$, so an $\mathrm{MDE}_N$ computed from $\sigma_{\rm pool}$ overstates each component's resolution by the factor $\sigma_{\rm pool}/\sigma_w$. The record's R1 instance: arm_pl at $N=8$ split $5$ crossed ($0.634002$ to $0.662021$) and $3$ NO READING ($1.113403$, $1.139404$, $1.152430$), nothing between $0.663$ and $1.113$, pooled sd $0.253673$ (READ docs/CEQ_SHAPE.md:927-940). Both cluster bounds are computed by one method at repair. The maximum sample sd (denominator $n-1$) of $n$ points in an interval of width $w$ is attained by splitting them between the endpoints, $\lceil n/2\rceil$ at one and $\lfloor n/2\rfloor$ at the other, and equals $$\sigma_w^{\max}(n,w)=w\,\frac{\sqrt{\lceil n/2\rceil\lfloor n/2\rfloor/n}}{\sqrt{n-1}} =\begin{cases}w/\sqrt3=0.5774\,w,&n=3\\ w\sqrt{0.3}=0.5477\,w,&n=5\end{cases}\qquad(\text{DERIVED}).$$ Three points in width $0.039027$ therefore have sample sd at most $0.039027/\sqrt3=0.0225322$, and five in width $0.028019$ at most $0.5477226\times0.028019=0.0153454$ — not $0.0280$, which is the interval width itself and not a sample sd, and which the earlier line used for the five-point cluster while using the correct $w/\sqrt3$ form for the three-point one: one sentence, two methods. With one method, $$\frac{\sigma_{\rm pool}}{\sigma_w}\ \ge\ \frac{0.253673}{0.0153454}=16.53\ \text{ for the crossed cluster }(n=5),\qquad \ge\ \frac{0.253673}{0.0225322}=11.26\ \text{ for the NO-READING cluster }(n=3),$$ against the earlier $9.06$ and $11.3$. The direction was safe — the claim is a lower bound and the old number was the looser one — and the defect is the split method, which is recorded here rather than repaired away. The pooled resolution $\Delta=0.215326$ is at least eleven times the within-component resolution on either cluster and at least sixteen on the crossed one, and the mean $0.829151$ is a value no seed produced.

Hypotheses. Two clusters, identified by a gap wider than both cluster widths ($0.450$ against $0.039$ and $0.028$). Non-vacuous on R1 and on any BED-S cell whose seeds split.

Evidence. READ as above; DERIVED. The Q5/W3 field (READ docs/CEQ_SHAPE.md:1227, finite-mixture inference against "a mean over a bimodal population is not a statement about either mode") is thereby a scoring rule, which is what MARS ruled it.

Mechanism. V-26 (MISTAKES.md:2190) — the pairing and the split are joint claims; M-9 (:580, $N=8$ verdicts are sign tests); M-3 (:464, the realised spread is the pooled one only if the population is unimodal); D-CALIB.

Kill. On any arena cell, a headline mean printed without $n^+$ (the seed-agreement count) and without the cluster membership beside it; assembly rule, $0$ GPU-s. The decidable statistic: the marginal-preserving permutation of seed labels between the two arms leaving the pairing assertion green (V-26's own check, one line). Control: the R1 cell must read $5/8$ and $3/8$ with the gap $[0.663,1.113]$ empty.

If killed (a cell prints a pooled mean). The number is struck and re-filed as a split with $n^+$; the replacement's Kill is the same permutation check. The chain ends.

Terminal. Every capability number in the canon is a split with $n^+$ beside it; the canon withdraws every pooled mean over a bimodal seed population.

Census rows closed#

row	what is broken (`CHARTER.md` §5)	closed by verse
B4	the label class: one scalar at one position; Q6 F4 for want of a state axis	01.1 (the four vector labels and their exactness conditions), 01.2 (what the environment costs), 01.3 (the committor read); the bed itself is book 04
B7	D-APPROX: "cannot represent" without a bound	01.5 (Peng through the reduction, with its table), 01.6 (the capacity form with its measured hypothesis), 01.7 (the iteration rate, the only theorem-shaped A1 at the planned geometry)
B8	obstruction theorems vacuous at the record's geometry	01.5 (the vacuity table and the geometry at which each bites), 01.8 (Sanford/Chen/Yehudai with their status)
B9	the Cheeger sentence on a non-reversible $P$	01.10 — OPEN: branch (a)'s kill struck as satisfied by construction, branch (b)'s instrument unspecified in book 04; the Terminal is in force
B10	Bellman optimality in multichain constrained MDPs	01.11 — closed by the criterion distinction (total-reward absorption on a transient DAG block versus the average-reward LP Misra's hazard is about), not by denying the hazard's hypothesis: the chain at the design point has $\ge4$ recurrent classes and is multichain
B11	the $\mathrm{hop}_k\to$ committor reduction unwritten; the TD-skyline claim on two abstracts	01.4 (written, with the stall-mass bound), 01.7 (the iteration construction DERIVED, the abstracts demoted to lineage), 01.8
B21	the read is occupied by ChaCAL; the delta and K-E1	01.13, 01.14, 01.15, 01.16, 01.17
B22	the bibliography verified at the identifier; ChaCAL's diagonal on one fetch	01.13 (the theorem that makes the convention decisive, and the re-read as its kill), 01.5 and 01.8 (`[V-eq]` owed, stated as conditional)

Kills, cheapest first#

Reachability is a column, not a footnote: a kill whose instrument the canon does not specify is marked UNREACHABLE and is not walkable this evening, whatever its price reads.

verse	kill	price	reachable today	replacement verse or route
01.19	row-sum census: $\lvert\sum_j\hat P_{ij}-1\rvert>10^{-12}$, or $\gamma\max_T\hat P_{ii}\ge1$	$0$ GPU-s, half an evening	yes — it has already fired (rows $1.31$–$10.29$)	the admitted-cell fraction $n_{\rm adm}/N$, then $n_{\rm adm}=0$
01.10(a)	struck: satisfied by construction on a triangular $\hat P$	—	no kill; the triangularity census is an admission line, UNSPECIFIED in book 04	none — branch (a) is algebra
01.12(a)	conservation row off $\mathbb 1$ by $>10^{-12}$	$0$ GPU-s	yes	sink enlarged, killed by $n_{\rm abs}=0$ on a still-failing draw (a diagonal count, not the residual)
01.12(b)	$0.875$ printed as the argmin floor without "at uniform $\hat\pi$", or a capability number without $\hat\pi$ in the row	$0$ GPU-s (assembly)	yes — it fired on this verse before repair	04.3's floor kill, routed by verse id
01.20	Schur identity off by $>10^{-12}$	$0$ GPU-s	UNREACHABLE — Schur instrument unspecified in book 04	sink block kept explicit
01.22	a pooled mean without $n^+$	$0$ GPU-s (assembly)	yes	the split with $n^+$
01.3	wrong-set plant moving $q$ by $<10^{-3}$	$0$ GPU-s, half an evening	yes	X-4 as an admission line, then S-10 re-drawn
01.16	residual departing from $\gamma^{K'+1}/(1-\gamma)$ by $>10^{-12}$ relative	$0$ GPU-s, half an evening	yes — the banked pair reads $2.0179\times10^{-13}$	the dropped-mass bound by row-sum census; then the exact solve
01.7	the formed matrix identity departing from $\gamma^{K'+1}/(1-\gamma)$ by $>10^{-12}$ relative at $\gamma\in\{0.6,0.9\}$, or any printed table cell doing so	$0$ GPU-s, half an evening	yes — the pre-repair $\gamma=0.7$ row would have fired it at relative $0.7$	the cost sentence (T1, book 03), killed by K-9's band
01.1	any label identity $>10^{-12}$ on $512$ draws	$0$ GPU-s, one evening	yes	per-segment sets (01.17's cut theorem)
01.4	oracle argmax $\ne\pi^8(i)$ on $512$ instances	$0$ GPU-s, one evening	yes	layout predicate; then oracle-family-only transfer
01.13	two arms not differing by $O(1)$ at $\gamma=0.9$: $\\|O-Y\\|_\infty/\\|V\\|_\infty<10^{-3}$	$0$ GPU-s, half an evening	yes (the arXiv re-read is repriced as an author action, book 05)	row-sum census on the two operators; then the delta loses its operator clause
01.11(iii)	the two-epoch witness's local and global argmins agreeing, or the root gap $0.2$ reading $<10^{-12}$	$0$ GPU-s, minutes	yes	— (clause-level)
01.11(i)	$m_{\rm eff}\le1$ on $>0.05$ of the $512$ admission draws (04.2 census clauses 9, 11)	$0$ GPU-s	yes — 04.2 carries the census	— (clause-level)
01.11(ii)	DP against exhaustive enumeration at $\lvert T\rvert\le12$, $m\le3$: $>10^{-12}$ on any of $512$ draws	$0$ GPU-s, one evening	yes	the one-clamp filter, killed by R-20 ($>0.5$ disagreement), $66{,}430\times$ cheaper in solves per draw
01.17	any `[M]` target not building (K-K)	$0$ GPU-s, CPU minutes	yes	`[S]`, then RUN instances
01.6	fibre separation $<5\%$ at $\varepsilon=0.05$	$0$ GPU-s, one evening	UNREACHABLE — collision instrument unspecified in book 04	Bet D (01.8)
01.5	none: the arXiv re-read is an author action outside the canon	`NOT MEASURED`	UNREACHABLE; verse OPEN	Sanford $\to$ Chen as lineage, no kill; Terminal in force
01.18	numerical rank of $H_{33}$ at $s=66$ reading $\le32$ on any of $1{,}024$ draws	$0$ GPU-s, one evening	UNREACHABLE — Hankel-rank probe unspecified; and undecidable at $s=64$	numerical-rank statement; re-price against the scan (book 03)
01.21	$\sigma_{17}/\sigma_1<10^{-3}$ on $6$ of $8$ cells, with the rank-$16$-plus-$10^{-2}$ plant as the must-fire	$0$ GPU-s from saved cells	UNREACHABLE — SVD probe unspecified; no cell (B5)	the committor head; then K-E1 (itself UNREACHABLE, 04.11)
01.10(b)	Dirichlet Cheeger instance failing on BED-1	`NOT MEASURED`, $0$ GPU-s	UNREACHABLE — Dirichlet instrument unspecified; verse OPEN	the ladder's measured rate ($f\in[0.99,1.01]$ at $t\ge320$)
01.9	$r_{\rm softmax}/r_{\rm shape}\le2$ (K-H1), registered only on seeds where K-D2 does not fire	inside the $\approx34$ s pair (Bet E, 04.6)	needs a cell (B5)	the residual as diagnostic; cost sentence (book 03)
01.2	$\\|\hat P-P_{\rm env}\\|_\infty<10^{-3}$ on $6$ of $8$ (K-D2)	inside the pair	needs a cell (B5); S-13 is a card of the record	the feature-content reading on S-11's generator spec, killed by the multi-hot absent while K-D2 fires — $0$ GPU-s, one document read, reachable today
01.15	cached mixture within one seed sd on $P$-plants, $6$ of $8$ (K-G1)	$\approx35$ s	needs a cell (B5)	the identity as algebra; Bet G's band
01.14	K-E1 at $N=8$, escalated to $N=16$	$\approx52$ s $+\approx44$ s `[ASSUMED]` (not the $\approx34$ s of Bet E's pair in 04.6)	UNREACHABLE — 04.11 registers $0$ ChaCAL source lines and $0$ records	link 1($\alpha$): the margin arithmetic, $0$ GPU-s, fires on the pre-repair $L=20.7$; link 1($\beta$): $\varepsilon\\|V\\|_\infty$ against the label sd (B5); link 2: $\phi_0$ on the environment chain with a value-zero sink column, no arm (04.11 link 1's census column); then Tree B
01.8	Bet D as registered: $\mathrm{acc}_{\rm shape}-\mathrm{acc}_{\rm sky}$ at $N=8$ against the realised paired sd, SPLIT band none, no seed-count clause	$\approx64.8$ s `[ASSUMED]`	needs a cell (B5)	deeper skylines, then CoT
01.16 (K-I)	one exceedance of $\delta\\|V\\|_\infty$ on $1{,}024$ draws	$\approx2.6$ s when a mask exists	dormant, owned by book 03; needs a mask (B15) and a label sd (B5)	not a link of 01.16's chain

Limits. Every RUN number in this book is quoted from docs/CEQ_SHAPE.md at the pin or from the round reports at 99777ab; none was re-executed this session, and every RUN[J]/RUN[M] figure carries its own draw. The judge's draw is $s=32$, $\gamma=0.6$, seed 0, and the exceptions this book itself cites are printed here rather than covered by a blanket sentence: the mask-amplification instance of 01.14 at $s=3$, $\gamma=0.9$, $\varepsilon=0.1$ (RUN[J] $0.5263157894736843$ against the naive $0.1$, READ docs/CEQ_SHAPE.md:405-406 @ 99777ab); the BED-1 chain of 01.10(b) at $|T|=9$ (RUN $\rho(Q_{\rm BED\text{-}1})=0.9408612510154677$), which is not $s=32$ at all; and the Gaussian-$V$ plant of 01.1's Kill at $s=64$ (RUN $\max|\Delta z|=1.127$, beside $0.1096$ at $s=32$). The former blanket claim was contradicted by all three on this book's own pages. Verses 01.5, 01.8 and 01.18 rest on [V] citations whose [V-eq] is owed and are stated as conditionals until the source-level re-read of book 04 runs. One citation is owed and not in references.bib: the Dirichlet-form Cheeger inequality (01.10), load-bearing nowhere. The second — the $n$-width of a linear image (01.21) — was [ASSUMED textbook] against a Limits sentence claiming it load-bearing nowhere while the verse's Statement rested on it; at repair the equality is DERIVED in two lines from the singular value decomposition and Schmidt–Eckart–Young, Pinkus is recorded as the standard reference with its citation still owed, and the two sentences no longer contradict each other. One further constant carries no value anywhere in the record or the canon: $\gamma_{\rm env}$ (READ docs/CEQ_SHAPE.md:341-342 @ 99777ab names it and prints no number), over which verse 01.7's shortfall ranges by a factor $13.5$; every clause of that verse is registered at the two run corners $\gamma\in\{0.6,0.9\}$ instead. K-E1 (01.14) is UNREACHABLE, not merely uncelled: 04_BEDS_AND_INSTRUMENTS.md verse 04.11 records $0$ ChaCAL source lines and $0$ records under any ChaCAL kind, so no ChaCAL arm exists to compare; 01.14 is carried by link 1($\alpha$), which is decidable today at $0$ GPU-s, and by link 2, moved at repair off the non-existent arm onto the bed's own environment chain. No A2 number is licensed by this book: A2 is read against the matched-parameter depth-$5$ skyline of book 04's clause 3b, which has no width, no count and no cell, never against the depth-$5$ full-width control and never against the linear-width skyline, which has no matched instance at $d_{\rm model}=16$. No value of the argmin zero-information floor is printed for BED-S: the floor is $1-\max_a\hat\pi(a^\star)$ at the realised class distribution, $\hat\pi$ is NOT MEASURED (B5), and $0.875$ appears only as the uniform-$\hat\pi$ instance at $m=8$ — the unconditional form this book printed before repair contradicted the record and books 04, 06 and 09 at once. The Lean grades are statements about statements; nothing was compiled. One Lean grade was factually wrong at the book's own pin and is corrected: hitting_time_transform read [D] on the evidence of a repository grep, while Mathlib at a45ae637 carries MeasureTheory.hitting (Mathlib/Probability/Process/HittingTime.lean:51, RUN this session); the row is [S] and the gap is the missing bridge from a pathwise stopping time to the finite-state absorption identity, not a missing object. No BED-S cell exists, so every $\mathrm{MDE}_8$ in a kill is the realised-sd clause of CHARTER.md §2 and is a placeholder until S-62 fills it. Six instruments this book's kills depend on are unspecified in the canon — the margin probe (01.4), the collision instrument (01.6), the Dirichlet-conductance instrument (01.10), the Hankel-rank probe (01.18), the Schur instrument (01.20), the SVD of $\hat\Pi_\gamma$ (01.21) — so six kills read UNREACHABLE in the table above for want of an instrument or of a permitted act (01.6, 01.10(b), 01.18, 01.20, 01.21, and 01.5's author fetch), and a seventh, K-E1 (01.14), reads UNREACHABLE because the arm it compares does not exist in the tree at all. Two verses stand OPEN with their Terminals in force: 01.5, whose only kill is an author fetch, and 01.10, whose branch (a) was satisfied by construction and whose branch (b) has no instrument. Verse 01.18's statement is additionally undecidable at the planned geometry: $\mathrm{rank}\,H_i\le\lfloor s/2\rfloor=2d$ at $s=64$, $d=16$, so it enters the canon only at $s\ge4d+2=66$. No code file, no git write, no external fetch.

Attacks answered#

MARS's findings, repair round 1, batch 1 of 3. Every finding appears verbatim — verse, flaw, mechanism, number, severity — beside the repair applied or the sentence now in force. Nothing here reads more gently than when it was filed (CHARTER.md §7); where the number a finding cites has moved (book 04's line pins move under its own repairs), the moved pin is printed and the finding's original is kept.

01.7 — `strike` · V-17, P-1 · repaired#

flaw. The shortfall table's gamma=0.7 row prints gamma^{K+1} where the Statement's equality demands gamma^{K+1}/(1-gamma); three of twelve cells are wrong by a factor 3.333, and the record prints the correct value for the identical (gamma,K).

number. Printed 0.343 / 0.168 / 0.0824 at K=2/4/6, gamma=0.7. Correct: 0.7^3/0.3 = 1.14333, 0.7^5/0.3 = 0.560233, 0.7^7/0.3 = 0.274514. The K=16 cell (7.75e-3 = 0.7^17/0.3) is divided and the other three are not. docs/CEQ_SHAPE.md:3113 (B-I row) prints the gamma=0.7 plant as '119.37 / 1.143' -- bound 1.143 = 0.7^3/0.3 at K=2 -- against the book's 0.343. The gamma=0.6 and gamma=0.9 rows are correct throughout.

replacement_survives. true.

required_repair. Reprint the gamma=0.7 row as 1.143 / 0.560 / 0.2745 / 7.75e-3, or relabel the whole table as the Pi_gamma residual gamma^{K+1} and reprint the gamma=0.6 and gamma=0.9 rows accordingly. One convention per table.

Repair. 01.7's table: the $\gamma=0.7$ row is reprinted $1.143$ / $0.560$ / $0.2745$ / $7.75\times10^{-3}$, the first branch of the required repair, so one convention — $\gamma^{K+1}/(1-\gamma)$, the quantity the Statement's equality names — governs all twelve cells. The paragraph under the table names the defect, the factor $1/(1-\gamma)=3.3\overline{3}$, the three cells that carried $\gamma^{K+1}$ and the fourth that did not, and pins the record's own $1.143$ at READ docs/CEQ_SHAPE.md:3116 @ 99777ab. One consequence the wrong cell hid is now printed: at $\gamma=0.7$, $K=2$ the shortfall exceeds $\|V\|_\infty$, so the two-hop iteration skyline is not a fellow approximator at that horizon. Verse 01.16's new kill cites the same $(\gamma,K)=(0.7,2)$ cell as its second plant, so a recomputation returning $0.343$ now fires a kill.

01.1 — `strike` · V-3, V-25 · repaired (carried from an earlier batch, verified at this repair)#

flaw. The Statement's 'iff' is false in the only-if direction at gamma=0, a value its own Hypotheses admit (gamma in [0,1)), and false for the committor label at every gamma.

number. At gamma=0, z* = (I-0)^{-1}V = V for every P-hat, so the label matches with ||P-hat - P_env||_inf = 1 on the reachable set. For q: take A_sink={0}, A_0={1}, T={2}, row 2 = (alpha, 1-alpha-beta, beta); q^(0)_2 = (1-alpha-beta)/(1-beta) reads 0.5 at both (alpha,beta)=(0.5,0) and (0.25,0.5), two chains differing by 0.25 in sup norm with identical labels.

replacement_survives. false.

required_repair. Weaken to the implication that holds: P-hat = P_env on the reachable set implies the four labels are exact. State the converse only for z as a map on all V at gamma>0, where resolvent equality does force matrix equality, and delete the 'iff' from the committor and argmax clauses.

Repair. 01.1 carries the required repair in full and it is verified at this repair rather than re-derived: the Statement's "One direction only" paragraph asserts the forward implication alone, states the converse for $z$ on every $V$ at $\gamma>0$ by inverting both resolvents, prints the $\gamma=0$ failure with $\|\hat P-P_{\rm env}\|_\infty=1$, and prints the committor counterexample with the two chains $0.25$ apart reading $0.5$ at both $(\alpha,\beta)=(0.5,0)$ and $(0.25,0.5)$; the argmax clause is stated as inheriting that many-to-one map, and no "iff" is written for $q$ or $a^\star$. On replacement_survives: false: 01.1's chain is not the "iff" — its Kill is the four-identity battery with $\hat P:=P_{\rm env}$ substituted, its replacement is the reach-avoid read with the undeclared absorbing position moved into $\mathcal A_{\rm sink}$ (killed by a declared-sink draw reading $\rho(Q)=1$, a census line), and its second link is the per-segment statement (killed by the same census re-read per segment). Three links, three different numbers, and none of them the weakened implication.

01.14 — `strike` · V-10, V-24 · repaired#

flaw. The Hypotheses' clause that the committor head at gamma=1 inherits the 2s e^{-L} bound through 01.1's kappa is either vacuous or false by O(1); it is the clause that carries the verse's conclusion that boundary rows are not a representability delta.

number. With A declared, the boundary rows lie outside T, so Q-hat = Q and R-hat = R exactly and ||q-hat^(L) - q||_inf = 0 identically: the bound kappa2se^{-L} is satisfied by construction and decides nothing. With A not declared -- the reading the class-containment claim needs, since ChaCAL-diag declares no boundary -- regime S fixes P_00 = 1 and P_ii < 1 for i >= 1 (READ docs/CEQ_SHAPE.md:286-287 @ 99777ab), so position 0 is the only absorbing state, absorption into any A_k not containing 0 has probability exactly 0, and the error is ||q||_inf >= 0.548718 (the record's own instance, RUN[P], READ docs/CEQ_SHAPE.md:1531 @ 99777ab), not <= kappa2se^{-L}.

replacement_survives. false.

required_repair. Restrict the epsilon-class containment to the discounted read at gamma bounded away from 1 and print the bound's blow-up 2s e^{-L}/(1-gamma). State that the committor head at gamma=1 is outside the finite-logit class entirely, and re-derive whether component (e) is retired on that head separately from the z channel.

Repair. The clause is deleted. 01.14's Statement restricts the containment to the discounted channels at $\gamma$ bounded away from $1$ and prints the blow-up $2s\,e^{-L}/(1-\gamma)$ with four values at fixed $L=30$, $s=64$: $1.20\times10^{-11}$ at $\gamma=0.9$, $1.20\times10^{-10}$ at $\gamma=0.99$, $1.20\times10^{-8}$ at $\gamma=0.9999$, diverging at $\gamma\uparrow1$. The Hypotheses print the admissible range, $\gamma\le1-2s\,e^{-L}/\varepsilon=1-1.2\times10^{-5}$ at $\varepsilon=10^{-6}$. A second paragraph states that the committor head at $\gamma=1$ is outside the finite-logit class entirely and carries both readings the finding names: with $\mathcal A$ declared the error is $0$ identically and the bound decides nothing (V-10); without, position $0$ is the only absorbing state under regime S, $(I-P)^{-1}$ does not exist at $\gamma=1$ on the full chain, and the error is $\|q\|_\infty\ge0.548718$ against a bound of $\le9.5\times10^{-11}$ at $L=30$, $\kappa\le7.9$ — wrong by ten orders, printed as such. Component (e) is now decided separately per channel. On replacement_survives: false: a second link is derived, not renamed. Link 1 (the $z$ channel) keeps the exactness clause with the $1/(1-\gamma)$ factor carried into the printed $\varepsilon$. Link 2 is a different object measured by a different instrument at a different number: the non-sink zero-fraction $\phi_0$ — solve the $K+1$ absorption channels on the sink arm's own operator with no partition declared and read whether $q^{(k)}_i=0$ exactly for every non-sink $k$ and every $i$ over $512$ draws — killed by $\phi_0<1$ on any draw, planted against a literal identity row at a non-sink position. K-E1 compares two NRMSEs over $8$ seeds against $\mathrm{MDE}_8$; link 2 reads an exact-zero predicate over $512$ draws. The Terminal now licenses the containment on the discounted channels only and, on the committor head, licenses neither "a mechanism softmax cannot express" nor its negation until $\phi_0$ is read.

01.18 — `strike` · V-10, V-11 · repaired#

flaw. The Kill is satisfied by construction at the planned geometry, and the Hypotheses' own non-vacuity predicate is unsatisfiable there.

number. The probe is specified as 'SVD of H_i at i=s/2 over 1,024 draws' at s=64, so H_i is the 32x32 lower-left block of a lower-triangular matrix and rank H_i <= 32 = 2d at every draw. A time-varying linear realisation with e = 32 therefore exists for every (P-hat,V), so 'an e <= 2d = 32 realisation to 1e-12' fires on 1,024 of 1,024 draws. The Hypotheses' clause 'non-vacuous only where that rank exceeds 2d = 32' can never hold at s=64.

replacement_survives. true.

required_repair. Either move the probe to i where min(i, s-i) > 2d -- impossible at s=64, d=16, since max_i min(i,s-i) = 32 -- or raise s to 4d+1 = 65 minimum and state the geometry at which the statement first bites, or drop the Kill and register only the numerical-rank statement at tolerance tau, which is what the If killed already contains.

Repair. The Hypotheses now print $\mathrm{rank}\,H_i\le\min(i,s-i)\le\lfloor s/2\rfloor$ and derive the non-vacuity threshold $\lfloor s/2\rfloor\ge2d+1=33$. The finding's "4d+1 = 65 minimum" is carried and then tightened, not softened: at $s=65$, $\lfloor65/2\rfloor=32=2d$ and the kill still fires by construction, so the first geometry at which the statement bites is $s=4d+2=66$, $i=33$. The probe is re-registered there — SVD of the $33\times33$ block $H_{33}$, numerical rank at $\tau=10^{-6}$, $1{,}024$ draws — and the kill is reprinted as "the numerical rank of $H_{33}$ at $s=66$ reading $\le2d=32$ on any of $1{,}024$ draws", the third branch of the required repair combined with the second. Two controls ship: the linear corner ($\beta=0$, $g\equiv0$) must read numerical rank $\le d=16$, and running the probe at $s=64$, $i=32$ must read rank $\le32$ on every draw, so the repaired defect itself becomes a plant. The Terminal states that at $s=64$, $d=16$ the canon licenses neither the statement nor its negation, the question being closed by arithmetic before any draw is taken.

01.11 — `strike` · P-10, V-25 · repaired#

flaw. Clause (ii) dismisses B10's hazard by classifying the arm's chain as unichain; with K+1 boundary sets plus a sink every absorbing state is its own closed recurrent class, so the chain is multichain by definition -- exactly the class B10 names.

number. At the design point (K=2, |A_bullet| in {1,2}, docs/CEQ_SHAPE.md:295-297 @ 99777ab) the chain has |A| >= 4 absorbing states, hence >= 4 recurrent classes. A chain is unichain iff it has exactly one recurrent class. The census row B10 (CHARTER.md:236) names 'multichain constrained MDPs with several unsafe sets' as the hazard, and 01.11 closes it by asserting the hazard's own hypothesis is absent.

replacement_survives. false.

required_repair. Delete 'the class is unichain-absorbing'. Justify backward induction on the correct grounds -- the objective is a total-reward absorption probability on a transient DAG block with rho(Q) < 1, not an average-reward criterion, and Misra's hazard is a hazard of the average-reward LP -- and print the hypothesis under which that argument holds.

Repair. The clause "the class is unichain-absorbing and Misra's multichain hazard is absent for this objective" is deleted. 01.11's Statement now opens clause (ii) by conceding the count — $|\mathcal A|\ge4$ absorbing states, hence $\ge4$ recurrent classes at the design point, and a chain is unichain iff it has exactly one — and then justifies backward induction on the criterion: a total-reward absorption probability on a transient DAG block with $\rho(Q)<1$, where the recursion is a finite backward substitution in topological order with no fixed-point argument, against Misra's hazard, which is a hazard of the average-reward constrained LP. Three hypotheses are printed: (H1) every clamp keeps $P^{(a)}$ causal so the substrate stays a DAG; (H2) $0\in\mathcal A$ and every policy absorbs in at most $s$ steps; (H3) the objective is additive in the absorption indicator — and (iii) is now read as the place H3 fails, which is where the principle of optimality genuinely breaks. The census table's B10 row records that the row is closed by the criterion distinction and not by denying the hazard's hypothesis. On replacement_survives: false: the old replacement's kill was "the three-step optimum disagreeing with the two-step on more than $0.5$ of draws — the same instrument one level deeper, equally priced", a V-9 renaming, and it is named as such. It is replaced by an exactness identity against exhaustive policy enumeration: $\max_i|V^\star_{\rm DP}(i)-V^\star_{\rm enum}(i)|>10^{-12}$ on any of $512$ draws restricted to $|T|\le12$, $m\le3$, enumerating all $\le3^{12}=531{,}441$ deterministic stationary policies. Different object (the DP value against the true optimum, not a disagreement fraction between two heuristics), different instrument, different number, strictly more decisive. Two plants: an inverted recursion ($\min$ for $\max$) must be exceeded by enumeration, and a Chebyshev DP must disagree with enumeration by $\tfrac12$ on the two-clamp witness, which is H3's own failure used as a control.

book-level (01.4, 01.6, 01.10b, 01.18, 01.20, 01.21) — `strike` · P-4 · repaired#

flaw. Six verses ship 'NOT MEASURED -- needs (book 04)' for instruments that book 04 does not specify, violating the CHARTER §3 requirement that the instrument be specified in the instruments book.

number. 04_BEDS_AND_INSTRUMENTS.md contains verses 04.1-04.21 and card ids S-10, S-12, S-13, S-24, S-28, S-35, S-61, S-62, S-73. Missing entirely: the collision instrument (01.6), the margin probe (01.4), the Dirichlet-conductance instrument (01.10b), the Hankel-rank probe (01.18), the Schur instrument (01.20), the SVD of Pi-hat_gamma on trained cells (01.21) -- grep counts 0 for 'collision instrument', 'Hankel-rank', 'margin probe', 'Schur instrument', 'Dirichlet' in that file. Book 01 also routes kills to cards S-02, S-17, S-31, S-33, S-64, X-2, X-4, N-15, R-20, J-L18, none of which appear in book 04 (they appear only in docs/CEQ_SHAPE.md).

replacement_survives. false.

required_repair. Either add the six instruments as verses in book 04 with shapes, draws, seed rule and price, or restate each NOT MEASURED clause as 'needs , unspecified in the canon' and mark the six kills unreachable until book 04 carries them. Replace the card-id routing with book 04 verse ids or with a READ pin into docs/CEQ_SHAPE.md at 99777ab.

Repair. The second branch, since a writer of book 01 may not edit book 04 (the task names one file). A new Instrument routing paragraph in the Preface states the rule, the counts, and the six instruments by name and owning verse. Each of the six NOT MEASURED clauses now reads "unspecified in the canon" with the grep count that convicts it and the probe's shape written out as a specification this book supplies and book 04 does not carry: the margin probe (01.4), the collision instrument (01.6), the Dirichlet-conductance instrument (01.10's branch (b) — book 04's one "Dirichlet" hit is 04.15's manifest field committor_route ∈ {dirichlet_gamma1, discounted_gamma}, a route and not an instrument), the Hankel-rank probe (01.18 — book 04's two "Hankel" hits are BED-K's ceiling $\sqrt{1-1/d}$ in 04.3, a different object), the Schur instrument (01.20), the SVD of $\hat\Pi_\gamma$ (01.21). The kills table now carries a reachable today column and marks five UNREACHABLE; the Limits paragraph repeats the six by name and records the two verses that stand OPEN in consequence. Card-id routing: S-02, X-2, X-4, R-20, S-17 and J-L18 each now carry a READ docs/CEQ_SHAPE.md:<line> @ 99777ab pin and say the card is the record's and not book 04's; S-10 is named as the one card of the list book 04 does carry. The finding additionally named S-31, S-33, S-64 and N-15: a grep of this book at repair time finds none of the four, which is recorded in the Preface rather than quietly dropped, together with six further card ids the finding did not name — J-D3, J-D6, J-D8, J-L0, J-L6, R-07 — which carry the same defect in Evidence position.

01.6 — `strike` · D-7, V-17 · repaired (carried from an earlier batch, verified at this repair)#

flaw. The If killed reverses the direction of Bet D as registered, and withdraws the A2 sentence on the outcome that would license it; the same bet is read the other way in 01.8, so two verses of one book contradict each other on one registered bet.

number. docs/CEQ_SHAPE.md:2237 @ 99777ab registers Bet D's prediction as 'the depth-5 stack within MDE_8 of the shape's argmin accuracy -- the honest control holds' and its counter as 'short by more than MDE_8, sign logged'. 01.6 writes 'If Bet D reads the counter, the D-APPROX sentence for BED-S is the depth skyline fits ... the A2 sentence withdrawn'. The counter is the skyline NOT fitting. 01.8's Kill reads the same counter as 'the analogy is wrong in the direction that flatters the shape', i.e. the skyline is short. Under D-CALIB the counter is the point estimate, so the book's stated default outcome and its stated consequence are inverted.

replacement_survives. false.

required_repair. Swap the branches: if Bet D reads its prediction (skyline within MDE_8) the A2 sentence is withdrawn and the D-APPROX sentence for BED-S is 'the depth skyline fits'. If it reads its counter, the A2 form has its first evidence and the counting bound is superseded, not replaced.

Repair. 01.6's If-killed carries the required repair in full and it is verified at this repair: the branches are printed "in the direction the registration gives them", with the registration quoted at READ docs/CEQ_SHAPE.md:2236 @ 99777ab — prediction is the skyline within $\mathrm{MDE}_8$, counter is short by more than $\mathrm{MDE}_8$. On the prediction the depth skyline fits, the A2 sentence is withdrawn and the A1 sentence reduces to 01.7; on the counter the A2 form has its first evidence and the counting bound is superseded, not replaced. The text names the inversion it repairs and the contradiction with 01.8's Kill, which reads the same counter the same way. Under D-CALIB the counter is the point estimate, so the second branch is the one the canon plans against, and that is printed.

01.13 — `strike` · V-11, P-4 · repaired#

flaw. The only Kill is an act book 04 registers as outside the canon's rules, and it is priced as if inside.

number. 01.13 Kill: 'The source-level re-read of Eq. 5 and Eq. 7 at the arXiv LaTeX source (2410.05565) ... price 0 GPU-s, one evening'. 04_BEDS_AND_INSTRUMENTS.md:175 closes 04.12's Evidence with 'No LaTeX source has been read: NOT MEASURED -- needs one fetch of the arXiv source, an author action outside this canon's no-fetch rule'. A kill that cannot fire on the bed as registered is not a kill (CHARTER.md:145-148).

replacement_survives. false.

required_repair. Reprice the kill as NOT MEASURED with the author action named, or replace it with the operator-level kill book 04.12 actually ships -- the two arms must differ by O(1) at gamma=0.9 on the same P, decided by arithmetic and not by the citation -- and state the convention as undecided until the fetch happens.

Repair. Both branches, the two halves being compatible. The re-read is repriced NOT MEASURED — needs one author fetch of arXiv:2410.05565, an act no writer, refuter or repairer of this canon may perform, registered as an author action with its own row in book 05, and the convention — which arithmetic the published paper ran — is stated undecided until that fetch happens, with no sentence of this book asserting it. The verse's kill becomes the operator-level one book 04.12 ships: on the same drawn $A$ at $\gamma=0.9$, $\beta=1$, the two arms failing to differ by $O(1)$ — $\|O-Y\|_\infty/\|V\|_\infty<10^{-3}$ on any draw, or ChaCAL-published's normalised row sums reading $1.000$ within $10^{-12}$ — decided by arithmetic, $0$ GPU-s, $512$ draws, with the record's own instance ($0.400\dots0.765$ against $1.000$, RUN[J]) on the firing side and a $\gamma=0$ bitwise-agreement control. The book-04 pin the finding gives as :175 reads :193 at repair time and is now cited by verse id (04.12) so it survives book 04's own repairs; verse 01.5's copy of the same pin is corrected the same way. On replacement_survives: false: the old replacement routed through the InfSA-style base, which is NOT MEASURED — needs the base wired and therefore cannot fire; it is demoted to the plant it is at READ docs/CEQ_SHAPE.md:3110 @ 99777ab and is not a link of the chain. The new replacement is the row-sum census on the two operators — a statement about the operators' rows, not their reads, which survives whatever the reads do — with its own Hypotheses ($\beta=1$, finite logits, $\mathrm{diag}(A)>0$), its own Evidence (RUN[J] row sums), its own Kill (any cell reading $1.000$ within $10^{-12}$, one pass over the rows, strictly cheaper than forming either resolvent) and the diagonal-kept arm as its must-fire plant.

01.5 — `strike` · V-11, P-3 · repaired as OPEN, Terminal in force (carried from an earlier batch, verified at this repair)#

flaw. Same defect: the Kill is the arXiv-source re-read book 04 registers as outside the canon's no-fetch rule, and both pre-written replacements are killed by 'the same re-read', so the entire chain is unreachable.

number. 01.5 Kill priced '0 GPU-s, one evening'; Replacement 1's Kill: 'the same re-read, on a second source'; Replacement 2's Kill: 'trivial (any finite s)'. 04_BEDS_AND_INSTRUMENTS.md:175: 'an author action outside this canon's no-fetch rule'. Three links, none of which can fire inside the canon.

replacement_survives. false.

required_repair. Mark the chain unreachable and let the Terminal stand as the licensed sentence today: at s=64, d=16, p=32 no unconditional 'softmax cannot' is licensed for the 2-hop committor. Register the re-read as a NOT MEASURED author action with its own row in book 05, not as this verse's kill.

Repair. 01.5 carries the required repair and it is verified at this repair: the verse is headed OPEN, its Kill is headed "UNREACHABLE, and the verse stands OPEN on that account" and repriced NOT MEASURED — needs one author fetch of arXiv:2402.08164, an act no writer, refuter or repairer of this canon may perform, registered as an author action with its own row in book 05. The two unconditional theorems are collapsed into one lineage link with no kill of its own, because neither was strictly cheaper nor strictly more decisive than the re-read and both died to the same act — so the chain terminates in one link rather than three, and the Terminal is headed "in force now": at $s=64$, $d=16$, $p=32$ no unconditional "softmax cannot" is licensed for the 2-hop committor, and no verse of this canon can change that by any act it is permitted to perform. Sanford's constant $c$ is NOT MEASURED and the earlier "first violated at $n>512$", which silently set $c=1$, is struck. The book-04 pin is corrected to verse-id routing at this repair; the kills table's 01.5 row now reads "none: the arXiv re-read is an author action outside the canon" with UNREACHABLE in the reachability column.

01.16 — `strike` · V-10, V-11 · repaired#

flaw. K-I cannot fire on the bed as registered: both of its branches need an object the canon records as absent, and the book's own Kills table labels it '(dormant)'.

number. Branch 1 needs a shipped mask; CHARTER.md:241 (B15) records 'the kernels do not exist: chunked solve, CSR path, Mapper schedule are NOT MEASURED'. Branch 2 needs the label sd; CHARTER.md:231 (B5) records 'BED-S has no cell, no realised sd'. docs/CEQ_SHAPE.md:2280 @ 99777ab prices R-10/K-I at ~2.6 s and Bet J at ':2237' is registered '(dormant)', '~2.6 s when a mask exists'.

replacement_survives. true.

required_repair. Register the kill that is reachable today at 0 GPU-s: the attained-equality check err = gamma^{K+1}/(1-gamma) against the RUN pair 0.007754350466241788 / 0.007754350466240224, with the rows-1.5 plant at gamma=0.6 as the must-fire. Keep K-I as a dormant row of book 03 and say so in the Kill field.

Repair. Exactly as required. 01.16's Kill field now opens by naming K-I unreachable, with both absent objects quoted at CHARTER.md:241 (B15) and CHARTER.md:231 (B5) and the record's own dormancy quoted from bind B-I ("REPAIR, dormant … dormant until row M ships a mask"); it states that K-I is a dormant row of book 03, named here and owned there. The verse's kill is the attained-equality check, frozen at a relative departure above $10^{-12}$ from $\gamma^{K+1}/(1-\gamma)$ on the record's registered $(\gamma,K)$ pairs, priced $0$ GPU-s and half an evening on a float64 recomputation with no cell, no mask and no bed. The banked pair is printed on the passing side with the relative figure derived: $0.007754350466241788$ against $0.007754350466240224$ is an absolute departure of $1.5647\times10^{-15}$ and a relative departure of $2.0179\times10^{-13}$, inside the frozen threshold. The rows-$1.5$ plant at $\gamma=0.6$, $K=2$ is the must-fire ($7.29$ against $0.54$), and a second plant reads the $(0.7,2)$ cell: a recomputation returning $0.343$ has dropped the $1/(1-\gamma)$ factor, which is 01.7's repaired defect turned into a control. The If-killed is re-derived onto a different object — the dropped-mass bound read off $P_m$'s own row sums, killed by any row summing above $1+10^{-12}$ — rather than left on the resolvent. K-I keeps its own row in the kills table, marked dormant and owned by book 03.

01.19 — `strike` · V-10 · repaired#

flaw. The Kill cannot fire on the bed as registered: the arm's P-hat is a causally masked softmax whose strict upper block is exactly zero by construction, and the verse's own If killed concedes it ('the arm is mis-masked').

number. Kill: 'Any trained cell whose P-hat has a non-zero entry above the diagonal by torch.equal on the strict upper block'. A masked softmax writes -inf before the exponential, so the upper block is bitwise 0.0 on every draw; the kill's firing set is empty. The If killed replacement ('the masked corner with the check re-run') is the identical check.

replacement_survives. false.

required_repair. Replace with a kill that can fire on the class as registered: the arm at beta not equal to 1, whose rows sum 1.31 to 10.29 (READ V16_ARM_SMPRIME.md:28-32 @ 99777ab), makes gamma*P non-substochastic and the diagonal read of the spectrum no longer bounds anything -- register the rowsum census on every trained cell as the decidable condition.

Repair. Exactly as required. The strict-upper check is struck by name, with the mechanism printed ($\exp(-\infty)=0.0$ exactly, so the firing set is empty, V-10) and the old If-killed named as the V-9 renaming it was. The frozen kill is the row-sum census on every trained cell: $|\sum_j\hat P_{ij}-1|>10^{-12}$ for some $i$, or $\gamma\max_T\hat P_{ii}\ge1$. It has already fired once on the registered class — rows summing $1.31$ to $10.29$ at $\hat\beta\ne1$, READ V16_ARM_SMPRIME.md:28-32 @ 99777ab — so $\gamma\hat P$ is not sub-stochastic, the Neumann series does not converge from row-stochasticity, and the diagonal read bounds nothing about $\|(\gamma\hat P)^t\|$. The verse records that the spectral identity itself is algebra no draw refutes, which is why it is not the kill. Controls: the $\beta=1$ masked arm must read row sums $1.000$ to $10^{-12}$ on every cell; the second clause needs $\hat\gamma\ge1.4436$ at the judge's draw ($\max_T\hat P_{ii}=0.6927$, RUN[J]) and is registered as reachable through the $\hat\gamma$ dial, book 02's B12. On replacement_survives: false: the replacement is now the admitted-cell fraction $n_{\rm adm}/N$ printed beside every Koopman sentence, killed by $n_{\rm adm}/N=0$ — a different object (a count over cells, not one cell's rows), decided by reading an integer the census has already produced, so strictly cheaper, and terminating, since at $n_{\rm adm}=0$ there is nothing left to restrict.

01.10 (branch a) — `strike` · V-10 · repaired; verse now OPEN, Terminal in force#

flaw. Branch (a)'s Kill cannot fire and the verse states so in its own If killed field.

number. If killed: '(a) cannot fail on a triangular operator'. Kill (a): '|rho(Q-hat) - max_T P-hat_ii| > 1e-12'. Eigenvalues of a triangular matrix are its diagonal exactly; the residual is float rounding on a value read off the diagonal, so the threshold 1e-12 is a numerical-noise gate, not a refutation of the Statement. CHARTER.md:145-146: a kill that cannot fire is not a kill.

replacement_survives. false.

required_repair. Delete kill (a) or repoint it at the reachable condition it actually guards -- a trained cell whose P-hat is not lower-triangular, decided by the strict-upper torch.equal census that the If killed already names -- and file that census as book 04's admission line rather than as this verse's kill.

Repair. Kill (a) is deleted, the first branch of the required repair. The second branch is deliberately not taken and the reason is printed in the verse: repointing at the strict-upper torch.equal census would repoint at another gate satisfied by construction, since a causally masked softmax's strict upper block is bitwise $0.0$ — the same defect 01.19's finding convicts in this same batch, so repairing one verse into the other's defect would be a repair that changes nothing. The triangularity census is instead filed as an admission line — a cell failing it is refused before any certificate is read off it — owned by book 04, which does not carry it: NOT MEASURED — needs the triangularity admission line, unspecified in the canon. Branch (a)'s content is stated as algebra with no kill attached. Consequence, carried honestly rather than hidden: with (a) struck and (b) unreachable (next finding), 01.10 carries no reachable kill today; the verse is headed OPEN, its Terminal is headed "in force now" and now states that neither branch has been tested, and the census table's B9 row records the same.

01.10 (branch b) — `repair` · V-15 · repaired#

flaw. Branch (b)'s Kill ships with no planted negative, the only kill in the book with none; V-15 requires the control that would make the kill fire on a known-bad plant.

number. Kill (b): 'On BED-1, h_D^2/2 computed from the bed's own conductance exceeding 1 - rho(Q) = 0.0591: ... NOT MEASURED -- needs the Dirichlet-conductance instrument on bed_1 (book 04), 0 GPU-s.' No Control clause follows. Every other Kill in the book (01.1-01.9, 01.10a, 01.11-01.22) carries one.

replacement_survives. true.

required_repair. Add the must-fire plant: a non-reversible transient block whose Dirichlet conductance is computed by the same instrument must read a bound that exceeds 1 - rho(Q), so an instrument that never fires is visible before it certifies BED-1.

Repair. The plant is supplied and the omission is named in the verse as "the only kill in the book with none (V-15)". The must-fire plant is a non-reversible transient block $Q=\begin{pmatrix}0&0.9\\0&0.9\end{pmatrix}$: $\rho(Q)=0.9$, $1-\rho(Q)=0.1$, reversibility fails because $Q_{12}=0.9$ and $Q_{21}=0$, and the same instrument must report $h_D^2/2$ above $0.1$ on it. The verse prints the consequence of skipping it: an instrument that certifies BED-1 without first firing on this plant has never been shown to fire at all, and the [ASSUMED textbook] bound would be certified by a probe with an empty rejection region. The plant is priced with the instrument and is NOT MEASURED for the same reason — book 04 carries no Dirichlet-conductance instrument — so the branch is marked UNREACHABLE in the kills table.

01.3 — `repair` · V-17 · repaired (carried from an earlier batch, verified at this repair)#

flaw. The printed gap lower bound belongs to a different object than the read the Statement defines; for the read O(gamma) the inequality is false.

number. The Statement defines O(gamma)_i = E_i[gamma^{tau_k - 1} 1_k] and then asserts 0 <= q - E[gamma^{tau_k} 1_k] = E[(1-gamma^{tau_k}) 1_k] >= (1-gamma) q. The middle object is (1-gamma) z_i, not O_i. Counterexample for O: P_ii = 0.1, P_ia = 0.9 with a in A_k, gamma = 0.6 gives q_i = 1, O_i = 0.9/(1-0.06) = 0.95745, gap 0.04255, while (1-gamma) q_i = 0.4. The record's own line (docs/CEQ_SHAPE.md:384-386 @ 99777ab) states the inequality for E[gamma^tau], never for the read.

replacement_survives. true.

required_repair. State the gap chain for (1-gamma) z_i explicitly, or print the correct bound for O: q_i - O_i = E[(1 - gamma^{tau_k - 1}) 1_k] >= 0, with no non-trivial lower bound, and rewrite the Terminal so it binds the object it names.

Repair. Both branches are taken and verified at this repair. 01.3's Statement carries a "Two gaps, two objects, never interchanged" paragraph: the $(1-\gamma)q_i$ chain is stated explicitly for $z_i:=E_i[\gamma^{\tau_k}\mathbb 1_k]=\gamma O(\gamma)_i$, and for the read $O(\gamma)$ the verse asserts only $q^{(k)}_i-O(\gamma)_i=E_i[(1-\gamma^{\tau_k-1})\mathbb 1_k]\ge0$ with no non-trivial lower bound, with the reason printed ($\tau_k=1$ has positive probability and the integrand is $0$ on that event). The counterexample is printed with the finding's own numbers — $P_{ii}=0.1$, $P_{ia}=0.9$, $\gamma=0.6$, $q_i=1$, $O_i=0.957447$, gap $0.042553$ against $(1-\gamma)q_i=0.4$ — with the failure factor $9.4$. The Hypotheses state that the $10^{-6}q_i$ reading at $\gamma=1-10^{-6}$ holds "on $E[\gamma^{\tau_k}]$ and on nothing else", and the Terminal binds the bound to $E_i[\gamma^{\tau_k}\mathbb 1_k]$ and to no other object, withdrawing every sentence that binds it to the read.

01.4 (construction) — `repair` · V-8, V-12 · repaired (carried from an earlier batch, verified at this repair)#

flaw. The construction gives position 0 two disjoint boundary labels at once, so the answer channel and the leak channel are the same set on every instance whose answer is token 0.

number. The construction declares A_j = {(j,k)} for j in [n] with layer k at indices [0,n), so A_0 = {position 0}; the Hypotheses simultaneously declare 'sink 0 declared as A_sink absorbing the leak'. docs/CEQ_SHAPE.md:295-297 @ 99777ab requires A = disjoint union of A_sink, A_0, A_1, ..., A_K. At n=64 the collision hits 1/64 of instances by uniform pi^k, and on those the epsilon-clause q^(hop_k(i)) >= 1 - k s' e^{-L}/(1 - s' e^{-L}) cannot be separated from the leak the sink is declared to absorb.

replacement_survives. false.

required_repair. Add one dedicated sink position at index 0 and shift the layers, giving s' = (k+1)n + 1 and A_sink = {0}, A_j = {(j,k)} at indices [1, n+1). Reprint the loss bound at s' = 577 and the non-vacuity threshold L > log(2 k s').

Repair. Exactly as required, and verified at this repair. 01.4's Statement reads $s'=(k+1)n+1$ throughout; position $0$ is the sink and nothing else ($\mathcal A_{\rm sink}=\{0\}$, $P_{00}=1$); layer $k$ occupies $[1,n+1)$ with $\mathcal A_j=\{(j,k)\}$ at index $1+j$ and layer $0$ occupies $[kn+1,(k+1)n+1)$; the disjointness is asserted against Definition 3 at READ docs/CEQ_SHAPE.md:295-297 @ 99777ab, and the earlier layout is named as the defect it was, including the $1/64$ collision rate under uniform $\pi^k$. The loss bound is reprinted at $s'=577$: $577\cdot9.36\times10^{-14}=5.40\times10^{-11}$ per step, $\times8=4.32\times10^{-10}$, so $\le4.4\times10^{-10}$; the non-vacuity threshold is reprinted as $L>\log(2ks')=\log(2\cdot8\cdot577)=9.131$.

01.4 (Lean route) — `repair` · P-10, V-25 · repaired#

flaw. The Lean route cited for the reduction target has a hypothesis the construction violates: StrictlyLower is regime N, and the construction is regime S with self entries at -L, i.e. P_ii > 0.

number. 01.17's table routes hop_layered_committor through Nilpotent.pow_card_eq_zero (lean/CEQ/Nilpotent.lean:77), whose statement reads 'theorem pow_card_eq_zero {A : Matrix (Fin n) (Fin n) R} (hA : StrictlyLower A) : A ^ n = 0'. 01.4's Hypotheses: 'regime S with self entries at -L'. docs/CEQ_SHAPE.md:288-289 @ 99777ab: 'regime N admits no absorbing row'. The row also files a repository theorem in a column headed 'Mathlib name ([V-name] at the pin)'.

replacement_survives. true.

required_repair. Either state the reduction on the strict pointer DAG at gamma=1 with no self entries, where pow_card_eq_zero applies and the walk is exactly k steps, and treat the -L self entry only in the softmax-class epsilon-clause; or drop the Nilpotent route and grade the target [S] with its own supporting lemma. Move the repository citation out of the Mathlib-name column.

Repair. The first branch plus the column fix. 01.4's Hypotheses separate the two readings and that separation is verified at this repair: (a) the exact reading at $\gamma=1$ is on the strict pointer DAG with no self entries, the walk exactly $k$ steps and $q\in\{0,1\}$, regime N on the transient block with the absorbing rows added; (b) the $\varepsilon$-clause is on the softmax class, regime S, self entry at $-L$; and the two are never mixed in one sentence. 01.17's table row for the reduction now carries — in the column headed "Mathlib name ([V-name] at the pin)", and a note below the table moves the repository lemma out of that column: its statement is quoted in full, StrictlyLower is identified as regime N which "admits no absorbing row" (READ docs/CEQ_SHAPE.md:288-289 @ 99777ab), the route is declared admissible on reading (a) only, and reading (b) is carried by the union bound of 01.4's Statement instead. The grade stays [S] with its supporting lemma supplied in-file.

Batch 1 disposition, counted exactly. Sixteen findings filed — twelve strike, four repair — and sixteen answered. Eleven were repaired at this batch: 01.7, 01.14, 01.18, 01.11, the book-level P-4 row, 01.13, 01.16, 01.19, 01.10's branch (a), 01.10's branch (b), 01.4's Lean route. Five were verified as already carrying the required repair from an earlier batch and are recorded as verified rather than re-claimed: 01.1, 01.6, 01.5, 01.3, 01.4's construction. No finding is left unanswered and no verse was deleted to escape one. Two verses now stand OPEN with their Terminals in force — 01.5 (its only kill an author fetch) and 01.10 (branch (a) struck as satisfied by construction, branch (b)'s instrument unspecified) — and five kills read UNREACHABLE in the kills table for want of an instrument book 04 does not carry. Four replacements were re-derived onto a different object and a different instrument because the old one died with its verse (V-9): 01.14's link 2 (the non-sink zero-fraction $\phi_0$), 01.11's (DP against exhaustive enumeration), 01.13's (the row-sum census on the two operators), 01.19's (the admitted-cell fraction $n_{\rm adm}/N$); 01.16's was re-derived onto the dropped-mass row census for the same reason.

MARS's findings, repair round 1, batch 2 of 3. Every finding appears verbatim — verse, flaw, mechanism, number, severity — beside the repair applied or the sentence now in force. Nothing here reads more gently than when it was filed (CHARTER.md §7). Where a finding's line pin has moved because the book it cites was itself repaired between filing and answer, the moved pin is printed and the finding's original is kept unaltered.

01.7 — `repair` · M-2, V-3 · repaired#

flaw. The Kill cannot refute the Statement, and the threshold it does decide is not frozen because gamma_env carries no value anywhere in the canon.

mechanism. M-2, V-3.

number. The Statement is an attained matrix identity; a learned depth-5 stack reading a z-channel NRMSE below the exact solve's cannot make ||(I-gamma P)^{-1} - sum_{t<=K}(gamma P)^t||_inf differ from gamma^{K+1}/(1-gamma). The Kill is registered 'On BED-S at gamma_env', which docs/CEQ_SHAPE.md:341-342 @ 99777ab names as 'a bed constant gamma_env' with no number printed anywhere in the record or the canon. The shortfall the verse decides varies from 0.540 (gamma=0.6, K=2) to 7.29 (gamma=0.9, K=2), a factor 13.5, over the unspecified constant. The verse's own Hypotheses concede that at gamma=0.6, K=16 the shortfall 4.23e-4 is 'below any label sd the bed will show'.

replacement_survives. true.

required_repair. Freeze gamma_env with a number and an evidence class, or register the kill at the two corners the record has run (gamma=0.6 and gamma=0.9) with both shortfalls printed. Add the arithmetic kill the Statement actually admits -- any table cell differing from gamma^{K+1}/(1-gamma) by more than 1e-12 -- which fires today.

severity. repair.

Repair. The second branch of the required repair, both halves. (1) $\gamma_{\rm env}$ is not frozen with a number, because none exists to freeze: a new paragraph in 01.7's Hypotheses prints NOT MEASURED — needs the BED-S registration triple to fix $\gamma_{\rm env}$ before the first draw (book 04, the D-4 admission block), a $0$ GPU-s registration act, quotes docs/CEQ_SHAPE.md:341-342 @ 99777ab naming the constant without a value, prints the factor $13.5$ span ($0.540$ at $(0.6,2)$ against $7.29$ at $(0.9,2)$), and registers every clause of the verse at the two corners the record has run, $\gamma=0.6$ and $\gamma=0.9$, and at no other $\gamma$. (2) The learned-stack kill is struck, with both grounds printed in the verse: it cannot refute an attained matrix identity computed from $(\gamma,K',P)$ and never from a weight, and its threshold was hung on the unset constant (M-2). It is not deleted from the canon — it is Bet D's contrast, and it is cited to verse 01.8, which registers it against the record's own threshold. The frozen replacement is the arithmetic kill the finding names, widened to the object the Statement is about: form $(I-\gamma P)^{-1}$ and $\sum_{t\le K'}(\gamma P)^t$ in float64 on a drawn causal row-stochastic $P$ at $s=64$, at both corners and at every $K'\in\{2,4,6,16\}$, and fire on a relative departure above $10^{-12}$ from $\gamma^{K'+1}/(1-\gamma)$ in the formed matrix $\infty$-norm or in any of the twelve printed table cells. It fires today on this book's own page as it stood before batch 1: the $\gamma=0.7$ row read $0.343$ against $1.143$, a relative departure of $0.7$. It is kept distinct from verse 01.16's kill — 01.16 recomputes the scalar closed form against the record's two banked residual numbers, this reads the matrix norm of a formed resolvent — and it carries two must-fire plants (the rows-$1.5$ non-stochastic $P$ at $\gamma=0.6$, $K'=2$ reading $7.29$ against $0.54$; a $P$ with one row scaled by $1+10^{-6}$) and one must-not-fire (the banked pair at relative $2.0179\times10^{-13}$). The chain is unchanged, since replacement_survives was true: the cost sentence (T1, book 03), killed by K-9's band, measured on the certified device.

01.7 — `repair` · V-17, P-7 · repaired#

flaw. The stack depth is off by one and the truncation index K collides with the constraint-set count K that the book's own Notation fixes at 2; book 08 disambiguates the same object as K'.

mechanism. V-17, P-7.

number. S^{(K)} = sum_{u<=K} (gamma P)^u V requires C^{(0)} = S^{(0)} = V in the residual stream and K blocks that each compute C^{(t+1)} = gamma P C^{(t)}; the Statement calls it a 'depth-(K+1) stack'. The title writes gamma^L/(1-gamma) where the body writes gamma^{K+1}/(1-gamma). The Notation fixes 'K' as the constraint count with K=2 at the design point; 08_ARCHITECTURE.md:39 writes the certificate as delta = gamma^{K'+1}/(1-gamma), reserving K for the constraint sets.

replacement_survives. true.

required_repair. Rename the truncation order K' throughout 01.7 and 01.16 to match book 08, fix the title's exponent to gamma^{K'+1}, and state the depth as K' blocks with V written into the stream at initialisation, or K'+1 with the first block named.

severity. repair.

Repair. All three, plus the Notation. The truncation order is renamed $K'$ throughout verses 01.7 and 01.16 — titles, statements, table headers, formulas, evidence lines, kills, plants and the kills-table rows — and the book's Notation now carries a paragraph stating that $K$ is the constraint-set count fixed at $K=2$ at the design point with $K+2$ value channels, that $K'$ is the Neumann truncation order with registered support $K'\in\{1,2,4,8,16\}$, and that CHARTER.md §6's scope sentence writes both with one letter while book 08 does not and, from this repair, neither does book 01. 01.7's title now reads $\gamma^{K'+1}/(1-\gamma)$, replacing a third letter ($\gamma^{L}$) for the same order. The depth is corrected to the first branch offered: a paragraph in the Statement derives that $C^{(0)}=S^{(0)}=V$ is written into the residual stream at initialisation and that exactly $K'$ blocks follow, so the stack is $K'$ blocks and the earlier "depth-$(K+1)$ stack" was off by one in the direction that flattered the read, charging the skyline one block more than the construction needs — which the verse now says in those words. The finding's pin 08_ARCHITECTURE.md:39 has moved under book 08's own repair; the certificate row is at :46 at repair time, and both books are cited by verse id (08.11) from here on so the pin cannot rot again.

01.2 — `repair` · P-7, V-10 · repaired; replacement re-derived#

flaw. The Statement hedges where a number belongs, and the frozen Kill K-D2 does not refute it: 'not guaranteed representable' stays true when the arm copies.

mechanism. P-7, V-10.

number. Statement: 'is not guaranteed representable at d_model = 16 < s = 64'. CHARTER.md:161 forbids 'hedging words in place of a number'. K-D2 fires when ||P-hat - P_env||_inf < 1e-3 on >= 6 of 8 seeds; a universal negative of the form 'not guaranteed for all DAG families' is compatible with that outcome on one family, so the kill's firing leaves the Statement standing and the If killed's assertion ('the rank sentence is false on BED-S's DAG family') does not follow from the kill as written.

replacement_survives. false.

required_repair. Restate as the decidable claim K-D2 refutes: 'the DAG family drawn by BED-S at S-13 is not representable at d_model = 16', with the sign-rank of the drawn edge sets printed per seed as the number that decides it.

severity. repair.

Repair. The Statement's restatement was already carried and is verified, not re-claimed: it asserts "the DAG family drawn by BED-S at S-13 is not representable at $d_{\rm model}=16$", decided per seed by $\mathrm{signrank}(E_\sigma)\le16$, with the per-seed census NOT MEASURED — needs the sign-rank read on the drawn edge sets, unspecified in the canon, and the hedge struck under CHARTER.md:161. Added at this batch: the Statement now also prints why the universal form is struck — a "not guaranteed for all DAG families" sentence survives K-D2 firing on one family, so its firing could not refute the verse, which is a kill that cannot refute its own statement. On replacement_survives: false, the replacement is re-derived onto a different object and is not renamed. The earlier replacement re-drew the family at higher sign-rank and moved the capability sentence to the committor head with kill "$\kappa\varepsilon$ at or above the label sd on $6$ of $8$ seeds": $\kappa$ is NOT MEASURED (01.1) and in different units from the record's condition number, the label sd is NOT MEASURED (B5), and the re-draw needs the same unspecified sign-rank read K-D2 waits on — so it waited on exactly what its verse waited on. The new replacement is the feature-content reading: BED-S's registered generator hands the arm the environment — "Node tokens carry the out-adjacency as a multi-hot feature … the transition matrix is a deterministic function of the edge tokens" (READ docs/CEQ_SHAPE.md:1677-1683 @ 99777ab, card S-11) — so a copy at $d_{\rm model}=16$ is a decoding of a feature the bed supplies and not a learnability number about a rank wall. Different object (the generator specification, not the arm and not the edge sets), different instrument (one read of the registered spec and the arena's producer_cmd), different number (the presence or absence of the multi-hot, not a sup-norm against $10^{-3}$). Its kill: the generator emitting tokens without the multi-hot while K-D2 still fires on $\ge6/8$ seeds, at which the decoding reading is withdrawn and the sign-rank route is the only one left. Price $0$ GPU-s, one document read — strictly cheaper than K-D2, and decidable today, where K-D2 needs a cell that does not exist (B5). The kills-table row is rewritten to match, and the routing "S-13 is in book 04" is corrected to "S-13 is a card of the record".

01.1 — `repair` · V-17, P-2 · repaired; the $\kappa$ clause re-derived#

flaw. kappa is read out of its units: a sup-norm condition number is substituted for a sup-norm inverse norm, and the source line is a paragraph the record marks as a withdrawn candidate.

mechanism. V-17, P-2.

number. The Notation defines kappa := ||(I - Q-hat)^{-1}||_inf. The Hypotheses read 'the sup-norm condition number of I - Q-hat is 7.9 (READ docs/CEQ_SHAPE.md:2282), so kappa <= 7.9'. cond_inf = ||I - Q-hat||_inf * ||(I - Q-hat)^{-1}||_inf, so kappa = 7.9 / ||I - Q-hat||_inf and the inequality holds only if ||I - Q-hat||_inf >= 1, which is nowhere printed. docs/CEQ_SHAPE.md:2282 @ 99777ab opens 'Withdrawn candidate, recorded so it is not re-found (V-7 discipline).'

replacement_survives. false.

required_repair. Print ||I - Q-hat||_inf beside the condition number and derive kappa from the quotient, or read kappa directly and register it as its own RUN. Note the provenance of the 7.9 as a withdrawn-candidate line.

severity. repair.

Repair. The provenance note and the units correction were already carried and are verified: the Hypotheses quote the record's own opening "Withdrawn candidate, recorded so it is not re-found (V-7 discipline)", state that $7.9$ is not $\kappa$, and give $\kappa=7.9/\|I-\hat Q\|_\infty$ with the inequality $\kappa\le7.9$ holding only if $\|I-\hat Q\|_\infty\ge1$, which is printed nowhere. What was missing and is supplied at this batch: $\|I-\hat Q\|_\infty$ is now printed as the interval the class forces, and $\kappa$ is derived from the quotient with both endpoints. For lower-triangular non-negative $\hat Q$ the $i$th row sum of $|I-\hat Q|$ is $1+r_i-2\hat Q_{ii}$, so $1-\max_T\hat P_{ii}\le\|I-\hat Q\|_\infty\le2$, which at the judge's draw reads $[0.3073403106639614,\,2]$ from $\max_T\hat P_{ii}=0.6926596893360386$ and gives $\kappa\in[3.95,\,25.70]$ — a two-sided interval of width factor $6.51$, where the earlier text printed the lower endpoint alone, which is the direction the argmin clause cannot use, and printing one endpoint of a two-sided interval was itself the defect. On replacement_survives: false, the clause that dies is re-derived rather than renamed. The $\kappa$-multiplying clauses are restated on a $\kappa$-free upper bound derived here: $\kappa\le1/\varepsilon_{\rm esc}$ with $\varepsilon_{\rm esc}=\min_{i\in T}\sum_{a\in\mathcal A}\hat P_{ia}$, from the Neumann bound $\|(I-\hat Q)^{-1}\|_\infty\le1/(1-\|\hat Q\|_\infty)$ on a sub-stochastic $\hat Q$. That is a different object (the operator's own rows, not a norm of its inverse), read by a different instrument (verse 01.19's row-sum pass, which this book already registers as reachable today at $0$ GPU-s), returning a different number ($\varepsilon_{\rm esc}$, not $7.9$), and it bounds $\kappa$ from above, which is the direction the argmin gap clause needs. The verse prints the bound's own failure mode — a row with escape mass $O(e^{-L})$ sends $1/\varepsilon_{\rm esc}$ to $O(e^{L})$ — so it stays a measured quantity. $\kappa$ itself remains NOT MEASURED. Verse 01.14's paragraph that quoted "$\kappa\le7.9$" is corrected to carry both endpoints ($\le9.5\times10^{-11}$ and $\le3.1\times10^{-10}$ against $\|q\|_\infty\ge0.548718$ — wrong by ten orders and by nine), so its conclusion no longer rests on the mis-united number.

01.12 — `repair` · V-17, L-FLOOR · repaired; replacement re-derived#

flaw. The argmin floor is printed without the uniform-distribution hypothesis that makes it true, contradicting the record and three neighbouring canon books.

mechanism. V-17, L-FLOOR.

number. 01_THEORY_ACCURACY.md:658 prints 'the zero-information floor on the argmin is chance, 1-1/m (0.875 at m=8)' flat. docs/CEQ_SHAPE.md:2493 @ 99777ab: 'exact zero-information floor 1 - max_a pi-hat(a*) from the realised class distribution; at uniform pi this is 1 - 1/m'. 04_BEDS_AND_INSTRUMENTS.md:43, 06_PREDICTIONS.md:246 and 09_CHESS_AND_MARKETS.md:75 all carry the conditional form. The admission gate admits any argmin class frequency in (0.05, 0.95) (docs/CEQ_SHAPE.md:1668 @ 99777ab), under which the true floor can be 0.05 -- an error of 0.825 against the printed 0.875, and no BED-S cell exists to measure pi-hat (B5).

replacement_survives. false.

required_repair. Print the floor as 1 - max_a pi-hat(a*) with the realised class distribution named, and 0.875 only as its uniform-pi instance, matching books 04, 06 and 09.

severity. repair.

Repair. The Statement now prints $\mathrm{floor}_{\rm zeroinfo}=1-\max_a\hat\pi(a^\star)$ from the realised class distribution of the argmin label on the batch the cell is scored on, with $1-1/m=0.875$ named as its uniform-$\hat\pi$ instance at $m=8$ and nothing more; the unconditional form is struck in those words. The record's conditional line is quoted at READ docs/CEQ_SHAPE.md:2493 @ 99777ab, and the three neighbouring books are cited by verse id with their moved pins printed beside the finding's originals: 04.3 (filed at :43, at repair time :59), 06.18 (filed at :246, at repair time :256), 09.8 (filed at :75, at repair time :91). The admission gate's band is printed with the arithmetic the finding gives: $(0.05,0.95)$ admitted (READ docs/CEQ_SHAPE.md:1668 @ 99777ab), so an admitted cell may realise a true floor of $0.05$, an error of $0.825$ against the printed $0.875$, in the direction that credits an arm scoring at chance with a capability; $\hat\pi$ is NOT MEASURED (B5), so no value of this floor is printed for BED-S by this book. The retired Fano forms are stated as retired by book 04 verse 04.3 with no value printed here. On replacement_survives: false, two repairs. First, the verse's Kill is split so the floor clause has a kill at all: clause (a) is the conservation residual, clause (b) routes the floor to 04_BEDS_AND_INSTRUMENTS.md verse 04.3's own kill by verse id and adds this book's assembly clause — any cell printing $0.875$ as the argmin floor without "at uniform $\hat\pi$", or an argmin capability number without the realised $\hat\pi$ in the same row — which is decidable today by reading the page and fired on this verse before repair. Second, the identity half's replacement no longer re-reads the same residual: that was a V-9 rename and is struck; it is killed instead by the undeclared-absorbing count $n_{\rm abs}=\#\{i\notin\mathcal A:\hat P_{ii}=1\ \text{bitwise}\}$ reading $0$ on a draw whose conservation row is still off $\mathbb 1$ — an integer count over the diagonal, not a sup-norm of a solve, strictly cheaper than the read it replaces, with a planted identity row as its must-fire. The Terminal carries the conditional form.

01.5 — `repair` · P-3, P-10 · already repaired at batch 1; verified and tightened here#

flaw. The chain's second link is neither strictly cheaper nor strictly more decisive than the first, violating the Depth rule, and the theorem it invokes is a hop_1 bound registered against t* = 2.

mechanism. P-3, P-10.

number. 01_THEORY_ACCURACY.md:315: 'equally priced and decisive only for t* = 2'. CHARTER.md:139-141: 'each link's kill must be strictly cheaper or strictly more decisive than the one before it'. Sanford Thm 1 is cited in the same sentence as 'hop_1 over a three-symbol alphabet needs h m p = Omega(n)'; through 01.4's construction k=1 gives s' = 2n, so 'i.e. s

= 1024 at t = 2' conflates the hop index k with the bed's t. Omega(n) also carries an unstated constant that 'first violated at n > 512' silently sets to 1.

replacement_survives. false.

required_repair. Reorder the chain so each link is cheaper or more decisive, or collapse Sanford and Chen into one lineage link with a single kill. Label the Sanford row k=1 and print the geometry at which it bites as s' = 2n with n > h m p / c and c named or NOT MEASURED.

severity. repair.

Repair. The second branch was taken at batch 1 and is verified at this repair, not re-claimed: Sanford and Chen are collapsed into one lineage link carrying no kill at all, the phrase "equally priced and decisive only for $t^\star=2$" appears nowhere in the book (grep, this session, zero hits outside this verbatim finding), Sanford Thm 1 is labelled $k=1$ and never $t^\star=2$, and the constant is named: $h\,m\,p=\Omega(n)$ bites at $n>h\,m\,p/c$, at $h=1$, $m=16$, $p=32$ the product is $512$, the geometry is $n>512/c$, and $c$ is NOT MEASURED — needs the constant read from Thm 1's body, the same author fetch, with "first violated at $n>512$" struck for silently setting $c=1$. Added at this batch, and it makes the finding's number larger, not smaller: the phrase "i.e. $s\ge1024$ at $t^\star=2$" is now named and struck in the verse itself, and the position count is printed as $s'=2n+1$ and not $2n$ — verse 01.4's repaired construction lays $(k+1)n$ layer positions after a dedicated sink at position $0$, so at $k=1$ the count is $2n+1$; the finding's $2n$ predates that repair, and printing $2n$ would understate the geometry by one position. On replacement_survives: false: there is no replacement, and the verse says so. 01.5 stands OPEN, its only conceivable kill is an author fetch of arXiv:2402.08164 that no writer, refuter or repairer of this canon may perform, the lineage link carries no kill by construction, and the Terminal is in force now: at $s=64$, $d=16$, $p=32$ no unconditional "softmax cannot" is licensed for the 2-hop committor, and no verse of this canon can change that by any act it is permitted to perform.

01.11 — `repair` · P-3, V-3 · repaired; kill split three ways, R-20 repositioned#

flaw. The If killed link is declared 'equally priced', violating the Depth rule, and the Kill refutes none of the three clauses of the Statement.

mechanism. P-3, V-3.

number. 01_THEORY_ACCURACY.md:643: 'the same instrument one level deeper, equally priced'. CHARTER.md:139-141 requires each link strictly cheaper or strictly more decisive. R-20's disagreement fraction above 0.5 on 512 draws cannot refute (i), which is true by definition of the objective; nor (ii), which is a backward-induction theorem; nor (iii), which is closed arithmetic on Fin 4.

replacement_survives. false.

required_repair. Give each clause its own kill: (i) the census that the m candidate moves are not all value-only, since a value-only move leaves every q^(k) constant (docs/CEQ_SHAPE.md:475 @ 99777ab) and the argmax label is then constant (V-12); (ii) the backward-induction residual against the exact V* on 512 draws at 1e-12; (iii) the Fin 4 witness re-evaluated. Reprice the deeper link so it is strictly cheaper.

severity. repair.

Repair. All four parts. The verse's Kill field now carries three frozen kills, one per clause, and R-20 is removed from it with the reason printed: a disagreement fraction between two heuristics refutes none of (i), (ii) or (iii). Kill (i) is the value-only census the finding names: on the $512$ admission draws of 04_BEDS_AND_INSTRUMENTS.md verse 04.2, the realised $m_{\rm eff}$ (that verse's census clauses 9 and 11, the no-op screening rate) reading $m_{\rm eff}\le1$ on more than $0.05$ of draws, since a value-only move leaves every $q^{(k)}$ constant (READ docs/CEQ_SHAPE.md:475 @ 99777ab) and the argmax label is then constant (V-12); instrument in book 04, price $0$ GPU-s, no arm, planted against $V$-only moves that must read $m_{\rm eff}=0$. Kill (ii) is the exactness identity against exhaustive enumeration — $\max_i|V^\star_{\rm DP}-V^\star_{\rm enum}|>10^{-12}$ over all $m^{|T|}\le3^{12}=531{,}441$ deterministic stationary policies at $|T|\le12$, $m\le3$ — promoted from the If-killed, where batch 1 had placed it, into clause (ii)'s own Kill, because it is the only statement that tests whether backward induction is the oracle, which is what (ii) asserts; both plants (the inverted $\max\to\min$ recursion, and the Chebyshev DP that must differ) are carried. Kill (iii) is the witness re-evaluated: the five numbers of the new two-epoch table ($1$, $0.6$, $0.5$, $0.7$, gap $0.2$) recomputed in float64, firing if the local and global argmins agree or the gap reads below $10^{-12}$, with the linear scalarisation $q^{(1)}$ as the plant under which both argmins must agree. The deeper link is repriced by inverting the chain. R-20 is repositioned as the replacement's kill, where it refutes exactly what the replacement asserts — that the one-clamp filter is a stand-in for the sequential optimum — and it is strictly cheaper than Kill (ii) by the ratio of solves per draw, $3^{12}/m=531{,}441/8=66{,}430$. The phrase "the same instrument one level deeper, equally priced" is struck and named as both a V-9 rename and a violation of CHARTER.md:139-141, which admits no "equally priced" link, and is not reinstated. The three kills are printed in strict price order (iii) $<$ (i) $<$ (ii), and the kills-table rows are split to match.

01.11 — `repair` · V-3, V-11 · repaired; a two-epoch witness derived#

flaw. Clause (iii)'s witness shows that randomised clamps beat deterministic ones at a single decision epoch; it exhibits no second epoch, so it does not exhibit a failure of the principle of optimality, and the mixture it uses is not in the read's action set.

mechanism. V-3, V-11.

number. The witness has one state u, two clamps a and b absorbing with probability 1, and no continuation: deterministic value 1, mixture (1/2, 1/2) value 1/2. The principle of optimality is a statement about sub-policies of a multi-epoch problem. The safest-move read is an argmin over m = 8 discrete candidate moves (docs/CEQ_SHAPE.md:1736-1742 @ 99777ab, R-07's row-clamp restriction), and the 1/2 a + 1/2 b mixture is not a candidate, so the exhibited failure is unreachable on the bed as registered.

replacement_survives. false.

required_repair. Replace with a two-epoch witness on a DAG of depth 2 in which the Chebyshev-optimal prefix is not the prefix of any Chebyshev-optimal full policy, or restate (iii) as what the witness proves -- for a Chebyshev objective the optimum over randomised clamps is strictly below the optimum over deterministic clamps -- and regrade the Lean target chebyshev_no_bellman to match.

severity. repair.

Repair. The first branch — the stronger of the two — is taken, and the second is carried beside it. A two-epoch witness is derived here on eight positions in causal order: $\mathcal A_{\rm sink}=\{0\}$, $\mathcal A_1=\{1\}$, $\mathcal A_2=\{2\}$ absorbing ($K=2$); transient $x=3$ with row $e_1$, $y=4$ with row $0.6e_1+0.4e_2$, $w=5$ with row $e_2$, $v=6$ clampable, query $u=7$ with row $\tfrac12e_6+\tfrac12e_5$. The two candidate moves at $v$ are R-07 row clamps to transient targets — $c:\,v\leftarrow e_3$ and $d:\,v\leftarrow e_4$ — so both are in the read's own action set and no mixture is used. Then $q(v\mid c)=(1,0)$ with Chebyshev value $1$ and $q(v\mid d)=(0.6,0.4)$ with value $0.6$, so locally $d$ is optimal; while $q(u\mid c)=(0.5,0.5)$ with value $0.5$ and $q(u\mid d)=(0.3,0.7)$ with value $0.7$, so globally $c$ is optimal. The Chebyshev-optimal prefix is not the prefix of any Chebyshev-optimal full policy, and taking the locally optimal one costs $0.7-0.5=0.2$ at the root — a failure of the principle of optimality with two epochs, deterministic clamps only, and every action reachable on the bed as registered. The old one-epoch instance is kept and restated as what it proves, the second branch of the required repair: for a Chebyshev objective the optimum over randomised clamps ($\tfrac12$) is strictly below the optimum over deterministic clamps ($1$) — an aside about the objective's action set, with the verse printing in full that a one-epoch instance cannot exhibit a failure of the principle of optimality and that the mixture is not a candidate under R-07, so that gap was unreachable on the bed (V-11). The Lean target is regraded to match: 01.17's table row is renamed chebyshev_prefix_not_optimal on Fin 8 with decide/norm_num over ℚ and the additive $q^{(1)}$ plant in its refusal column, and a second row carries chebyshev_randomised_below_deterministic for the one-epoch Fin 4 aside; chebyshev_no_bellman appears nowhere. On replacement_survives: false: the clause's kill is now Kill (iii) above, an arithmetic re-evaluation of the new witness at $0$ GPU-s, which the old witness could not carry because the quantity it computed was not the quantity clause (iii) claimed. 01.11's Hypotheses and Terminal are rewritten to carry the two objects apart.

01.6 — `repair` · V-17, P-10 · already repaired at batch 1; verified here#

flaw. The Control's must-fire threshold is imported from a log base the verse's own Hypotheses declare unread, and it compares a collision fraction with a misclassification-probability lower bound, two different quantities.

mechanism. V-17, P-10.

number. Control: 'separation above 13% on the layered hop_2 family at n=128' comes from 01.5's table cell 0.1310 = 352/2688, computed 'DERIVED with log_2'. Under natural log the same row reads R = 128ln128 - 544 = 77.4 and error >= 77.4/(3128*ln128) = 0.0415, so a working instrument reading 0.05 would be declared broken by a factor 3.2. 01.5's Hypotheses grade Peng '[V], [V-eq] owed -- the theorem's own hypotheses ... were not read at the body'. Separately, no theorem relates the fraction of same-sketch DAG pairs at fixed random weights to a lower bound on the error of the best readout.

replacement_survives. true.

required_repair. State the log base as a condition of the control ('under log_2; under ln the threshold is 0.0415') until the [V-eq] is owed, and derive the control threshold from the collision instrument's own calibration -- a plant with a known separation fraction -- not from Peng's error probability.

severity. repair.

Repair. Both halves were applied at batch 1 and are verified at this repair, not re-claimed. 01.6's Kill carries a section headed "Control, and its two defects named": the must-fire plant is a calibration plant — a synthetic pair family with a constructed separation fraction $\phi_0$, which the instrument must recover to within $\pm0.02$ on $512$ pairs, on the ground that an instrument that cannot recover a planted separation fraction cannot certify a measured one — and the imported threshold is recorded with both of its defects rather than repaired away. (i) Log base: the verse prints that $0.1310=352/2688$ is DERIVED with $\log_2$, that under natural log the same row reads $R=128\ln128-544=77.4$ and error $\ge77.4/(3\cdot128\ln128)=0.0415$, that a working instrument reading $0.05$ would therefore be declared broken by a factor $3.2$, and that because 01.5 grades Peng [V] with [V-eq] owed the base is not settled — so if that control is ever run it is run as "under $\log_2$ the threshold is $0.131$; under $\ln$ it is $0.0415$", never as one number. (ii) Units: the verse states that a collision fraction over same-sketch DAG pairs and a lower bound on the misclassification probability of the best readout are two different quantities, that no theorem in this canon relates them, and that the import is therefore withdrawn as a control and kept only as that sentence.

01.8 — `repair` · R-SKY, D-1 · repaired#

flaw. The verse declares that the wide-skyline matched instance does not exist at the planned geometry, which leaves A2 -- the form that requires the depth skyline's number in the same row -- unmeasurable there, and the book never says which skyline A2 uses.

mechanism. R-SKY, D-1.

number. Statement: 'yehudai-2025-depthwidth removes depth at linear width, so the matched control fixes width too, and at d_model = 16 < s = 64 the wide skyline's matched instance does not exist.' CHARTER.md:88-91 defines A2 as 'at matched parameters against the depth skyline, where the skyline's learned hops fall short of the exact solve by more than MDE_N' and requires 'the depth skyline's number printed in the same row'. The book's only skyline cells are the depth-5 and depth-7 rows; 04_BEDS_AND_INSTRUMENTS.md:274 records 'two with settings and no cell' (B11).

replacement_survives. false.

required_repair. Name the skyline A2 is measured against -- depth 5 at t* = 8 by CHARTER.md:80 -- and state explicitly that the linear-width skyline is out of scope at d_model = 16, so no A2 sentence is licensed against it. Print which of the two skyline rows has a cell.

severity. repair.

Repair. All three, and the naming is narrower than the required repair rather than wider, because CHARTER.md:88-91 requires matched parameters and book 04 splits the depth-$5$ row on exactly that. A new paragraph in 01.8's Statement names: CHARTER.md:80 fixes the skyline depth at $\lfloor\log_2t^\star\rfloor+2=5$ at $t^\star=8$; book 04 verse 04.9 clause 3 splits that depth into (3a) the full-width, unmatched-parameter arm whose number is printed as $\Delta_{\rm sky}$ and which licenses no A2 sentence at all, and (3b) the reduced-width, matched-parameter arm, against which A2 is read and against nothing else (the finding's pin :274 has moved; at repair time 04.9 clause 3 is :149 and 04.20 is :302, and both are cited by verse id). The linear-width skyline of yehudai-2025-depthwidth is stated to be out of scope at $d_{\rm model}=16$ — no matched-parameter instance exists there — so no A2 sentence is licensed against it, by this book or any other. Which rows have cells is printed: of book 04's four skylines, the depth-$5$ full-width arm (3a) is the one 04.20 requires FOUND at the design point (its title reads "one FOUND, two with settings and no cell (B11)"), while 3b has no width, no count and no cell, and the wide and chain-of-thought arms have settings and no cell — and book 04's own Limits paragraph states that no A2 number exists anywhere in that book. The consequence is printed in the verse and again in its Terminal and in this book's Limits: this book prints an A2 route and no A2 value. On replacement_survives: false: the chain is untouched by this finding — its links are the depth dial escalated to $7$ and then the chain-of-thought decoder, each a registered skyline row killed by the same $\mathrm{MDE}_8$ comparison at the next depth — but the A2 licence those links would carry is now stated as absent for all of them, since every one is a depth row and A2 lives only on 3b's matched-width arm.

01.8 — `repair` · M-2 · repaired#

flaw. The Kill thickens Bet D's registered threshold with a seed-count clause the record does not carry.

mechanism. M-2.

number. 01.8 Kill: 'the depth-5 stack short of the shape's argmin accuracy by more than MDE_8 on at least 6 of 8 seeds'. docs/CEQ_SHAPE.md:2237 @ 99777ab registers Bet D's deciding number as 'acc_shape - acc_sky' with SPLIT band 'none' and no seed-count clause; the 6-of-8 form belongs to Bets C, F and H in the same table. M-2 freezes thresholds where they are registered.

replacement_survives. true.

required_repair. Restate the kill as the registered paired contrast acc_shape - acc_sky at N = 8 against the realised paired sd, or file a corrections row adding the 6-of-8 clause to Bet D before using it.

severity. repair.

Repair. The first branch. 01.8's Kill is restated as the registered paired contrast $\mathrm{acc}_{\rm shape}-\mathrm{acc}_{\rm sky}$ at $N=8$ against the realised paired sd, SPLIT band none, and no seed-count clause, with the struck clause quoted and the reason printed: the $6$-of-$8$ form belongs to Bets C, F and H in the same table (READ docs/CEQ_SHAPE.md:2236 @ 99777ab), and importing it thickened a registered threshold, which M-2 forbids. The verse states that a seed-count clause on Bet D, should one ever be wanted, enters through a CORRECTIONS.md row against the registration and not through a book. The kills-table row is rewritten with the same three qualifiers, and the row that previously paired 01.7 with 01.8 on Bet D is split, since 01.7's kill is no longer Bet D at all (the 01.7 · M-2, V-3 finding above). Verse 01.6's If-killed already cited Bet D without the seed clause and is unchanged.

01.9 — `repair` · V-10, V-3 · repaired#

flaw. K-H1 and 01.2's K-D2 are mutually exclusive in one direction, so the book's two measured kills cannot both be decidable on the same draws; and the Statement is a universal negative its own kill cannot refute.

mechanism. V-10, V-3.

number. r_shape = ||(I - gamma P_env) z-hat - V||_inf / ||V||_inf and the shape solves z = V + gamma P-hat z exactly, so r_shape = gamma ||(P-hat - P_env) z-hat||_inf / ||V||_inf. K-D2 (01.2) fires when ||P-hat - P_env||_inf < 1e-3 on >= 6 of 8 seeds; on exactly those seeds r_shape is driven toward 0 and r_softmax/r_shape diverges, so K-H1 ('ratio <= 2') cannot fire there. The Statement 'No theorem converts this into an error separation' is a claim about the literature that no measured ratio can refute.

replacement_survives. true.

required_repair. Print r_shape's floor and the identification reading ||P-hat - P_env||_inf beside every K-H1 evaluation, and register K-H1 only on seeds where K-D2 does not fire. Restate the Statement as the decidable half: the joint-consistency claim at MATHEMATICS.md:106-127 is UNTESTED, and the ratio with its counter is what would test it.

severity. repair.

Repair. Both halves. The Statement's universal negative is struck and replaced by the decidable half the finding names: the joint-consistency claim at MATHEMATICS.md:106-127 is UNTESTED, the ratio with its registered counter is what would test it, and until a cell is read the counter is the point estimate (D-CALIB). A new registration clause in the Kill carries the finding's derivation in the verse: $(I-\gamma P_{\rm env})\hat z-V=\gamma(\hat P-P_{\rm env})\hat z$ from $\hat z=V+\gamma\hat P\hat z$, so $r_{\rm shape}=\gamma\|(\hat P-P_{\rm env})\hat z\|_\infty/\|V\|_\infty \le\gamma\|\hat P-P_{\rm env}\|_\infty/(1-\gamma)$, and on a K-D2-firing seed ($\|\hat P-P_{\rm env}\|_\infty<10^{-3}$) that caps $r_{\rm shape}$ at $1.5\times10^{-3}$ at $\gamma=0.6$ — so K-H1's "$\le2$" would need $r_{\rm softmax}\le3\times10^{-3}$ and the ratio otherwise diverges. It follows, and the verse says so: $r_{\rm shape}$'s floor and the identification reading $\|\hat P-P_{\rm env}\|_\infty$ are printed beside every K-H1 evaluation, K-H1 is registered only on the seeds where K-D2 does not fire, the excluded seed count is printed in the row, and a K-H1 verdict quoted over a seed set including a K-D2-firing seed is VOID. The Terminal records the sentence as UNTESTED rather than as unsupported by the literature and withdraws every K-H1 verdict quoted over K-D2-firing seeds; the kills-table row carries the restriction and names the $\approx34$ s pair as Bet E's (04.6).

01.14 — `repair` · M-2, V-17 · repaired; link 2 re-derived off the non-existent arm#

flaw. The verse prices K-E1 at a number that contradicts book 04's price for the identical frozen kill.

mechanism. M-2, V-17.

number. 01.14 Kill: 'approx 52 s, escalation approx 44 s ([FITTED])'. 04_BEDS_AND_INSTRUMENTS.md:169 (Kill (K-E1)): 'Instrument: the pair of 04.9, approx 34 s'. docs/CEQ_SHAPE.md Bet F @ 99777ab: 'approx 52 s, escalation approx 44 s'. Two canon books give two prices for one kill with one frozen threshold.

replacement_survives. false.

required_repair. Reconcile against the record: K-E1 runs the shape and ChaCAL-with-sink, not one bed-cell pair, so 52 s is the price and book 04's 34 s is the pair's. A corrections row must name which book is wrong before birth.

severity. repair.

Repair. Reconciled exactly as the finding directs, and one further error of class is corrected beside it. 01.14's Kill now prints: K-E1's price is $\approx52$ s with escalation $\approx44$ s, the record's own Bet F row (READ docs/CEQ_SHAPE.md:2238 @ 99777ab), because K-E1 runs the shape and ChaCAL-with-sink — two arms, not one bed-cell pair; the $\approx34$ s a reader meets in book 04 is Bet E's one bed-cell pair at verse 04.6 (at repair time :102, :357), the pair verse 01.9's K-H1 runs inside, and it is not K-E1's. The class is corrected from [FITTED] to [ASSUMED], which is what the record marks Bet F's seconds and what this verse claimed a fit for. No corrections row is owed: book 04's K-E1 registration was itself repaired between the filing of this finding and its answer and now reads $\approx52$ s [ASSUMED] (verse 04.11, at repair time :185; the finding's pin :169 has moved), so the two books agree before birth and the reconciliation is a book repair rather than a post-birth correction. On replacement_survives: false, one further fact the finding's pin exposes and this repair carries: book 04 verse 04.11 registers K-E1 UNREACHABLE — grep -rniE 'chacal|infsa' --include='*.py' . returns $0$ lines and the $24$-kind census holds $0$ records for every ChaCAL arm — so under CHARTER.md:145-148 K-E1 is not a kill today at any price, and this verse now says so. Link 2, which batch 1 had read "on the ChaCAL-with-sink arm's own learned operator", would have died exactly where K-E1 dies and is moved off the arm entirely: $\phi_0$ is now read on the environment chain the bed already builds — $P_{\rm env}$ on 04.2's admitted draws with a value-zero sink column appended in the record's own V15Fork.Asink form (READ lean/CEQ/V15Fork.lean:67-70) and no partition declared — which needs no arm, no training, no seeds and no matched parameters, is the census column book 04 verse 04.11's own link 1 already specifies at $0$ GPU-s, and reads NOT MEASURED — needs that census column instead of needs the sink arm's cells (B5). Link 2 is therefore now both strictly more decisive and strictly cheaper than K-E1, and the kills-table row carries the price contrast, the UNREACHABLE mark and the moved object.

01.14 — `repair` · P-1 · repaired#

flaw. The required logit margin is misprinted: the 1.28 factor is dropped from the logarithm, and at the printed margin the bound exceeds the epsilon it is meant to meet.

mechanism. P-1.

number. L >= ln(2s / ((1-gamma) epsilon)) = ln(128 / 1e-7) = ln(1.28e9) = 20.9701, not the printed 20.7 (= ln(1e9) = 20.7233). At L = 20.7 the bound reads 264e^{-20.7}/0.1 = 1.28e-6 > epsilon = 1e-6.

replacement_survives. false.

required_repair. Print 20.97 (or 21.0 rounded up, never down, since the bound is a lower bound on L).

severity. repair.

Repair. $\mathbf{20.97}$ is printed, with $21.0$ named as the rounded-up form and the rule stated — the inequality is a lower bound on $L$, so any rounding rounds up and $20.7$ rounded the wrong way. The verse now prints $L\ge\ln(2\cdot64/(0.1\cdot10^{-6}))=\ln(1.28\times10^{9})=20.9701$, names the dropped $1.28$ factor, and prints the failure the misprint causes at both readings of the old number: at $L=20.7$ the containment radius reads $2\cdot64\,e^{-20.7}/0.1=1.31\times10^{-6}$ and at $\ln(10^{9})=20.7233$ it reads exactly the finding's $1.28\times10^{-6}$ — both exceeding the $\varepsilon=10^{-6}$ the margin is chosen to meet — while at $20.9701$ the radius reads $10^{-6}$ exactly. On replacement_survives: false: link 1's kill read "$\varepsilon\|V\|_\infty$ below the label sd on $6$ of $8$ seeds", which computes $\varepsilon$ from the same formula and needs a label sd that does not exist (B5), so it could neither catch this defect nor fire at all. Link 1's Kill now carries two frozen clauses, and the first is reachable today: clause $(\alpha)$ recomputes $L_{\rm req}=\ln(2s/((1-\gamma)\varepsilon))$ at every registered triple and the radius $2s\,e^{-L}/(1-\gamma)$ at every registered grid point, firing on a relative departure above $10^{-12}$ or on any printed $L$ at which the radius exceeds its own $\varepsilon$ — four logarithms and four exponentials, $0$ GPU-s, no cell, no arm, and it fires on this book as printed before repair; clause $(\beta)$ is the label-sd comparison, marked NOT MEASURED and not reachable today. The plant for $(\alpha)$ is the misprint itself: substituting $\ln(10^9)$ must produce $1.28\times10^{-6}>\varepsilon$ and fire.

01.21 — `repair` · V-15, L-EQ · repaired#

flaw. The Control is a pass-plant that holds by construction and cannot make the kill fire, and the Statement's own equality rests on a citation the book declares owed while the Limits paragraph asserts it is load-bearing nowhere.

mechanism. V-15, L-EQ.

number. Control: 'the identity Pi_0 = P at gamma = 0 must give the same tail as the softmax matrix itself'. At gamma = 0, Pi_0 = (1-0) P (I)^{-1} = P identically, so the control passes on every input including a broken SVD probe. V-15 requires the plant that makes the kill fire. The Statement is 'd_n(K_gamma) = sigma_{n+1}(Pi_gamma)' graded '[ASSUMED textbook]; the citation -- Pinkus, n-Widths in Approximation Theory -- is owed, not in references.bib', while the Limits paragraph says both owed citations are 'load-bearing nowhere'.

replacement_survives. true.

required_repair. Add a must-fire plant: a matrix with a known rank-16 truncation error (a rank-16 matrix plus 1e-2 times a random perturbation) must read sigma_17/sigma_1 near 1e-2, not below 1e-3. Either add Pinkus to references.bib or grade the Statement DERIVED from the singular-value decomposition of a linear image of a ball, which is a two-line derivation.

severity. repair.

Repair. Both, and the second by the branch that does not require touching another file — no file but this book is edited at this repair. Control: the $\gamma=0$ identity is demoted in place, with the arithmetic printed — $\Pi_0=(1-0)P(I-0\cdot P)^{-1}=P$ identically, so it passes on every input including a probe whose singular values are noise, an empty rejection region, the exact defect V-15 names — and is relabelled a wiring check that the probe forms $\Pi_\gamma$ from the right matrix. The must-fire plant the finding specifies is added verbatim in substance: $M=R+10^{-2}E$ with $R$ of exact rank $16$ and $E$ normalised to $\|E\|_2=\|R\|_2$ must read $\sigma_{17}(M)/\sigma_1(M)$ near $10^{-2}$ and never below $10^{-3}$. A second plant is added in the firing direction: an exactly rank-$16$ matrix must read the ratio at the float64 noise level and fire the kill. Grade: the Statement is regraded DERIVED by the two-line derivation the finding names — $\sup_{x\in\mathcal K_\gamma}\mathrm{dist}(x,X_n)=\|(I-\Pi_{X_n})\Pi_\gamma\|_2$, minimised over $n$-dimensional $X_n$ by Schmidt–Eckart–Young at $\sigma_{n+1}$ with minimiser $\mathrm{span}(u_1,\dots,u_n)$ — so [ASSUMED textbook] is gone, nothing in this book rests on Pinkus, and the Limits paragraph's "load-bearing nowhere" is now true of what remains rather than contradicting a Statement that leaned on the owed citation. Pinkus stays recorded as the standard reference with its citation owed in references.bib, and the Limits paragraph is rewritten to say one citation is owed and load-bearing nowhere (01.10's Dirichlet-form Cheeger) and to name the contradiction that has been removed.

01.17 — `repair` · P-1, P-10 · repaired#

flaw. The [D] grade for hitting_time_transform is evidenced by a repository grep, and the object it claims absent is present in the vendored Mathlib at the book's own pinned revision.

mechanism. P-1, P-10.

number. Table row: 'hitting time, 01.3 | hitting_time_transform | [D] | no finite-chain hitting-time object (RUN[P] grep, zero files)'. lean/.lake/packages/mathlib at rev a45ae63747140c1b2cbad9d46f518015c047047a (the book's pin, lean/lake-manifest.json:61) carries Mathlib/Probability/Process/HittingTime.lean:51, 'noncomputable def hitting [Preorder iota] [InfSet iota] (u : iota -> Omega -> beta) (s : Set beta) (n m : iota) : Omega -> iota'. Every other row of the table cites a Mathlib file:line and all four checked resolve exactly (Block.lean:265, Block.lean:346, NonsingularInverse.lean:151, SchurComplement.lean:202).

replacement_survives. true.

required_repair. Regrade the row to [S] with the gap named precisely -- Mathlib's hitting is a stopping time on a filtered process, and no finite-state absorption identity E_i[gamma^{tau}] = ((I - gamma Q)^{-1} R 1)_i is derived from it -- and cite HittingTime.lean:51 as the object that exists.

severity. repair.

Repair. The row is regraded [S], its Mathlib column now cites MeasureTheory.hitting at Mathlib/Probability/Process/HittingTime.lean:51 as the object that exists, and its refusal column carries forward_pointer_reads_zero re-used from 01.4 (a chain with no path to $\mathcal A_k$ must read $E_i[\gamma^{\tau_k}]=0$). A note below the table records the whole finding in the verse: the pin is confirmed by RUN this session — the rev at lean/lake-manifest.json:61 is a45ae63747140c1b2cbad9d46f518015c047047a and grep -n 'noncomputable def hitting' on the vendored package at that revision returns line $51$ — and the struck evidence is named for what it was, a grep of the repository's own lean/CEQ/ tree offered as evidence about Mathlib (P-1), which is the only evidence the [D] carried, in a table where every other row cites a Mathlib path:line and all four checked resolve exactly. The gap is stated precisely rather than as an absence: Mathlib's hitting is a pathwise stopping time on a filtered process, indexed by a Preorder with InfSet, and what verse 01.3 needs is the finite-state absorption identity $E_i[\gamma^{\tau_k}\mathbb 1_k]=((I-\gamma Q)^{-1}\gamma R_k\mathbb 1)_i$ on a row-stochastic matrix with an absorbing partition; no route from the first to the second exists in this Mathlib, which is a missing bridge and not a missing object. This book's Limits paragraph carries the correction as one Lean grade that was factually wrong at the book's own pin.

Batch 2 disposition, counted exactly. Sixteen findings filed — sixteen repair, zero strike — and sixteen answered. Fourteen were repaired at this batch: 01.7 twice (the unfrozen $\gamma_{\rm env}$ kill; the $K\to K'$ rename with the depth off-by-one), 01.2, 01.1, 01.12, 01.11 twice (the three-way kill split with R-20 repositioned; the two-epoch witness), 01.8 twice (the A2 skyline naming; Bet D's thickened threshold), 01.9, 01.14 twice (the K-E1 price with link 2 moved off the non-existent arm; the logit margin), 01.21, 01.17. Two were verified as already carrying the required repair from batch 1 and are recorded as verified rather than re-claimed — 01.5 and 01.6 — with 01.5 additionally tightened at this batch ($s'=2n+1$ and not $2n$, and the struck "$s\ge1024$ at $t^\star=2$" named in the verse). No finding is left unanswered, no verse was deleted to escape one, and nothing in this section reads more gently than when it was filed.

Six replacements were re-derived onto a different object and a different instrument because the old one died with its verse (V-9). 01.2's, from a re-drawn family killed by $\kappa\varepsilon$ against an absent label sd, to the feature-content reading on S-11's generator specification, killed by one document read at $0$ GPU-s. 01.1's $\kappa$-multiplying clauses, from an unread inverse-norm probe to the escape-mass bound $\kappa\le1/\varepsilon_{\rm esc}$ read by verse 01.19's row-sum pass. 01.12's, from the same conservation residual re-read to the undeclared-absorbing count $n_{\rm abs}$, a bitwise diagonal pass. 01.11's, from R-20 one level deeper to a chain in which the enumeration identity is clause (ii)'s own kill and R-20 becomes the replacement's kill at $66{,}430\times$ fewer solves per draw. 01.14's link 2, from the ChaCAL-with-sink arm — which book 04 registers as $0$ source lines and $0$ records — to the bed's own environment chain with a value-zero sink column, needing no arm at all. 01.14's link 1, from a label-sd comparison that cannot fire (B5) to the margin arithmetic, which fires today on this book's own printed $L=20.7$.

What this batch withdrew and did not replace. No A2 number is licensed by this book: A2 lives only on book 04's matched-width depth-$5$ skyline, which has no width, no count and no cell, and never on the linear-width arm, which has no matched instance at $d_{\rm model}=16$. No value of the argmin zero-information floor is printed for BED-S: the floor is $1-\max_a\hat\pi(a^\star)$ and $\hat\pi$ is NOT MEASURED. No clause is registered at $\gamma_{\rm env}$: the constant carries no value in the record or the canon, and verse 01.7 is registered at $\gamma\in\{0.6,0.9\}$ instead. K-E1 is UNREACHABLE, not merely uncelled — the arm it compares does not exist in the tree. Verse 01.5 stands OPEN with no replacement at all, its Terminal in force now: at $s=64$, $d=16$, $p=32$ no unconditional "softmax cannot" is licensed for the 2-hop committor, and no verse of this canon can change that by any act it is permitted to perform.

MARS's findings, repair round 1, batch 3 of 3. Every finding appears verbatim — verse, flaw, mechanism, number, severity — beside the repair applied or the sentence now in force. Nothing here reads more gently than when it was filed (CHARTER.md §7).

book-level — `repair` · P-1, V-17 · repaired#

verse. book-level

flaw. The Limits paragraph's claim about every RUN[J]/RUN[M] figure is contradicted by three figures the book itself cites.

mechanism. P-1, V-17

number. Limits: 'every RUN[J]/RUN[M] figure is one float64 draw at s = 32, gamma = 0.6, seed 0'. 01.14 cites RUN[J] 0.5263157894736843 which the source records at s = 3, gamma = 0.9 (docs/CEQ_SHAPE.md:405 @ 99777ab). 01.10(b) cites rho(Q_BED-1) = 0.9408612510154677 on BED-1 with |T| = 9, not s = 32. 01.15 cites RUN 1.127 at s = 64.

replacement_survives. false.

required_repair. Restate as 'every RUN[J]/RUN[M] figure carries its own draw; the judge's draw is s = 32, gamma = 0.6, seed 0, and the exceptions are the mask-amplification instance at s = 3, gamma = 0.9, the BED-1 chain at |T| = 9, and the Gaussian-V plant at s = 64.'

severity. repair.

Repair. The Limits paragraph now reads that every RUN[J]/RUN[M] figure carries its own draw, names the judge's draw as $s=32$, $\gamma=0.6$, seed 0, and prints all three exceptions with their pins: the mask-amplification instance at $s=3$, $\gamma=0.9$, $\varepsilon=0.1$ (READ docs/CEQ_SHAPE.md:405-406 @ 99777ab), the BED-1 chain at $|T|=9$ ($\rho(Q_{\rm BED\text{-}1})=0.9408612510154677$), and the Gaussian-$V$ plant at $s=64$ ($\max|\Delta z|=1.127$, beside $0.1096$ at $s=32$). Verse 01.14's own Evidence line now prints its draw ($s=3$, $\gamma=0.9$, $\varepsilon=0.1$) beside the number rather than relying on a blanket paragraph. On replacement_survives: false: the Limits paragraph is prose and carries no chain, so there was nothing to replace; the defect was a universal quantifier over three known counterexamples on this book's own pages, and the repair is the quantifier's removal, not a substitute claim.

book-level (01.6, 01.7) — `repair` · D-1, R-SKY · repaired#

verse. book-level (01.6, 01.7)

flaw. Two comparative sentences appear outside forms A1/A2/T1/T2 without the bed, floor, N and matched count CHARTER §1 requires.

mechanism. D-1, R-SKY

number. 01_THEORY_ACCURACY.md:429 'the learned stack does better than iteration' -- names the bed and MDE_8 but no floor and no matched count. 01_THEORY_ACCURACY.md:375 'the shape's advantage on BED-S is exactness and cost, not fit' -- 'advantage' with no form named. CHARTER.md:106-109 strikes such a sentence at refutation.

replacement_survives. false.

required_repair. Rewrite 429 as the A1-form comparison with the floor (L-FLOOR) and the matched parameter count printed in the same clause, and 375 as 'what survives on BED-S is exactness (delta = 0) and the cost sentence, and no fit sentence is licensed'.

severity. repair.

Repair, in three places, because the contrast moved between verses during the round. The line the finding numbers 429 is verse 01.7's learned-stack kill, which an earlier batch of this round struck on a different ground — it cannot refute an attained matrix identity, and its threshold was registered at the valueless $\gamma_{\rm env}$ — and whose contrast it forwarded to verse 01.8. That strike removed the kill and left the bare comparative unaddressed, which is this finding's charge and is now answered where the sentence lives:

01.7 gains the paragraph "The form the contrast is written in, and the bare comparative that is struck with it", which names the struck phrasing as a comparative outside A1/A2/T1/T2, states the A1 form with all four required items in one clause — the bed (BED-S at $t^\star=8$, $m=8$, $K=2$, $N=8$), the floor (L-FLOOR: the argmin zero-information floor $1-\max_a\hat\pi(a^\star)$, NOT MEASURED, $0.875$ only as the uniform-$\hat\pi$ instance at $m=8$), the matched parameter count ($4{,}769$ against book 04 clause 3b's count within $0.0960$ per cent, NOT MEASURED) and $\mathrm{MDE}_8$ at the realised paired sd (NOT MEASURED) — and then states that three of the four are NOT MEASURED, so no A1 value is printed and no fit sentence is licensed there.
01.8's Frozen Bet D block, which is where the contrast now lives, is rewritten in A2 form with the floor and the matched count inside the same clause and the depth skyline's own number required in the same row, and closes with the flat prohibition: a sentence of the form "the learned stack does better than iteration", naming the bed and $\mathrm{MDE}_8$ but neither the floor nor the matched count, appears nowhere in this book.
01.6's Bet-D-prediction branch is restated exactly as the finding requires — "what survives on BED-S is exactness ($\delta=0$) and the cost sentence, and no fit sentence is licensed" — with the word "advantage" named as the comparative it is and struck.

On replacement_survives: false: neither sentence was a verse chain, so neither carried a replacement; what stood behind them was a licensing claim, and the repair withdraws the claim rather than re-siting it. No A1 and no A2 value is added: both remain routes whose floor and matched count are NOT MEASURED.

book-level (01.5, 01.14) — `note` · P-2 · repaired, and twelve further pins corrected#

verse. book-level (01.5, 01.14)

flaw. Two READ pins do not resolve to the number they carry at the book's own commit.

mechanism. P-2

number. 01.14 cites ':407-408' for 'RUN[J] 0.5263 against naive 0.1'; at 99777ab those numbers sit at docs/CEQ_SHAPE.md:405-406 and lines 407-408 carry the rowsum census and the beta-not-1 rows. 01.5 cites ':2947-2950' for the vacuity inequality; '384 < 544' appears at docs/CEQ_SHAPE.md lines 128, 1421, 1580, 1614, 2559, 2593, 2952, 3071, 3072 and 3164, and in none of 2947-2950.

replacement_survives. false.

required_repair. Repin to :405-406 and :2952. Re-verify every READ range in the book by grepping the cited string at 99777ab; docs/CEQ_SHAPE.md is byte-identical between 99777ab and HEAD, so the check is mechanical.

severity. note.

Repair, and the wider census the required check produced. 01.14 is repinned to :405-406 with the failed pin named in the text: :407-408 resolves at 99777ab to the row-sum census and the $\hat\beta\ne1$ rows, and to neither number. 01.5's :2947-2950 was already repinned to :2952 at an earlier batch of this round and is re-verified here — git show 99777ab:docs/CEQ_SHAPE.md | sed -n '2952p' (RUN, this session) prints "this reads $384 < 544$ (DERIVED, $\log_2$), so the inequality is vacuous". The mechanical sweep the finding asks for was then run over every docs/CEQ_SHAPE.md pin in this book against the $3{,}311$-line file at 99777ab (RUN: git show 99777ab:docs/CEQ_SHAPE.md, one sed -n per cited range). Twelve further pins did not resolve and are corrected, each to the line that carries the cited string:

claim	pin as printed	pin at `99777ab`
$\mathcal A_k$ on the wrong set moves $q$ by $O(1)$ (01.1)	`:388`	`:387`
$\rho(Q)=0.692660$ declared against $1.000000$ undeclared (01.1)	`:1436`, a blank line	`:1440`
the must-fire perturbation, ratio $9.6\times10^{10}$ (01.3)	`:921`	`:918`
Bet D's registration (01.6, 01.8)	`:2237`, Bet E's row	`:2236`
Bet F's price row (01.14)	`:2237`, Bet E's row	`:2238`
bind B-I, the $\gamma=0.7$ plant $119.37/1.143$ (01.7, 01.11, 01.17)	`:3113`, B-G2's row	`:3116`
K-E1 as frozen (01.14)	`:3180`, a table header	`:2274`
$\mathrm{rank}(\Delta P)$ for a token rewrite (01.15)	`:473`	`:471`
Q3/W1, the Koopman field (01.19)	`:1199`	`:1222`
the `lambda_hat` one-bit defect (01.19)	`:1275-1278`	`:1280`
Q5/W3, finite-mixture inference (01.22)	`:1204`	`:1227`
Q1/W3, realisation theory (01.18)	`:1197`	`:1220`
Q5/W1, the $n$-width field (01.21)	`:1203`	`:1226`

(Thirteen rows for twelve pins: :2237 was wrong twice, once for Bet D and once for Bet F, and resolves to two different lines.) The two occurrences of :2237 and the one of :3113 that sit inside MARS's own quoted findings above are left exactly as MARS wrote them; only this book's own prose is repinned.

What the sweep did not do, stated so the check is not overclaimed. Ranges naming a card (S-02, S-11, S-17, R-07, R-20, J-D3, J-D8, J-L4, J-L14) were verified at the line carrying the card header and, where the range spans a card body, at the line carrying the quoted string — :1677-1683 was checked to the quoted multi-hot clause at :1679 — but not string by string through every card body; a pin that lands inside the right card on the wrong line of it would survive this sweep. docs/CEQ_SHAPE.md is byte-identical between 99777ab and HEAD, so the sweep is repeatable exactly as filed. On replacement_survives: false: a pin carries no chain; the defect is a false citation and the repair is the true one, with the false pin kept in the text beside it so the correction is legible.

01.19 — `note` · P-10 · repaired#

verse. 01.19

flaw. The Koopman operator on linear observables is gamma P transpose plus an affine shift, not gamma P; the spectral conclusion survives, the stated identification does not.

mechanism. P-10

number. For T(x) = V + gamma P x and f(x) = c^T x, (Uf)(x) = f(T(x)) = gamma (P^T c)^T x + c^T V. The Statement reads 'the Koopman operator ... on linear observables is gamma P itself'. spec(P^T) = spec(P), so 'the spectrum is {gamma P_ii}' stands.

replacement_survives. true.

required_repair. Write 'is gamma P^T on linear observables, affine by c^T V', and note spec(P^T) = spec(P) so the diagonal read is unchanged.

severity. note.

Repair. The Statement now reads that the Koopman operator of $T:x\mapsto V+\gamma Px$ is $\gamma P^\top$ on linear observables, affine by $c^\top V$, and prints MARS's own computation as the derivation: $(Uf)(x)=c^\top(V+\gamma Px)=\gamma(P^\top c)^\top x+c^\top V$, so $U$ acts on the coefficient vector as $c\mapsto\gamma P^\top c$ and the linear observables are invariant only modulo constants. The spectral conclusion is kept with the reason printed: $\det(\lambda I-P^\top)=\det((\lambda I-P)^\top)=\det(\lambda I-P)$, so $\mathrm{spec}(P^\top)=\mathrm{spec}(P)$ and $\{\gamma P_{ii}\}$ is unchanged; the clause "$\gamma P$ is lower-triangular" now carries "$\gamma P^\top$ upper-triangular, same diagonal" beside it. The verse's Kill, its row-sum census, its replacement and its Terminal are untouched — the defect was the identification of the operator, not the spectrum the verse reads off it, and the verse says which of the two was wrong rather than calling the error immaterial.

01.6 — `note` · V-17 · already repaired at an earlier batch; verified here#

verse. 01.6

flaw. The pigeonhole count omits the bits the readout carries in x_i, which 01.2's Hypotheses give the out-adjacency multi-hot.

mechanism. V-17

number. The readout is of (x_i, O_i); 01.2's Hypotheses: 'node tokens carry out-adjacency as a multi-hot feature'. That adds up to s = 64 bits, so the joint sketch carries at most d p + s = 576 bits and the fibre size is 2^{2016-576} = 2^{1440}, not the printed 2^{2016-512} = 2^{1504}. The conclusion is unaffected; the exponent is not.

replacement_survives. true.

required_repair. Print 2^{1440} with the x_i budget named, or state the count for O_i alone and say the readout also sees x_i.

severity. note.

Repair, verified at this batch. 01.6's Statement carries the joint budget $d\,p+s=576$ bits with the $x_i$ term named as 01.2's out-adjacency multi-hot, prints $2^{2016-576}=2^{1440}$, and keeps $2^{1504}$ only in the parenthesis saying it is the count for $O_i$ alone and the wrong budget for a readout that also sees $x_i$. The finding stands as filed; being answered before this batch opened does not soften it.

01.6 — `note` · P-1 · already repaired at an earlier batch; verified here#

verse. 01.6

flaw. The interval of admissible epsilon is printed with its endpoints transposed, and |T| in {57,60} is declared DERIVED with no derivation and excludes values the stated geometry admits.

mechanism. P-1

number. 2^{-512/240} = 0.2278 and 2^{-512/228} = 0.2107, so the text 'epsilon < 2^{-512/240} ... 2^{-512/228} = 0.211 ... 0.228' maps each endpoint to the wrong expression. The Notation asserts |T| in {57,60} at |A_bullet| in {1,2}, K=2, s=64; four sets of size 1 or 2 give |A| in {4,...,8} and |T| in {56,...,60}.

replacement_survives. true.

required_repair. Print '2^{-512/228} = 0.2107 to 2^{-512/240} = 0.2278'. Either derive |T| in {57,60} by naming the set-size convention that forbids |A| = 5, 6 and 8, or print |T| in {56,...,60} and the corresponding coordinate range {224,...,240}.

severity. note.

Repair, verified at this batch, on the finding's second branch. The book took the branch the finding offers second: the Notation and 01.6 both print $|\mathcal A|\in\{4,\dots,8\}$ and $|T|=64-|\mathcal A|\in\{56,\dots,60\}$ with the derivation in the Notation, and both state that the earlier $\{57,60\}$ was DERIVED with no derivation and excluded values the stated geometry admits. The coordinate range is therefore $|T|(K+2)\in\{224,\dots,240\}$ and not $\{228,240\}$, so the interval the finding computes for $\{57,60\}$ is superseded by the wider one: the book prints $\varepsilon<2^{-512/224}=0.2050$ to $2^{-512/240}=0.2278$, each endpoint against the expression that produces it, with the transposition named. The finding's own endpoints stand as the interval for the geometry it was filed against, and the book does not present the two as the same number.

01.15 — `note` · V-3 · repaired#

verse. 01.15

flaw. The printed bound carries an unexplained (1+gamma) factor absent from its own derivation, and a rank equality is asserted where only an inequality follows.

mechanism. V-3

number. O_true - O_cached = (1-gamma)(I - gamma P)^{-1} Delta P z', giving ||O_true - O_cached||_inf <= ||Delta P||_inf ||V'||_inf/(1-gamma) with no (1+gamma). The display multiplies by (1+gamma). 'For a token rewrite rank(Delta P) = s - i' -- rewriting token i changes s-i rows, so rank(Delta P) <= s - i; the record's own line (docs/CEQ_SHAPE.md:473 @ 99777ab) carries the same overclaim.

replacement_survives. true.

required_repair. Drop the (1+gamma) or show the step that introduces it. Write rank(Delta P) <= s - i and keep the suffix re-solve price (s-i)^2 d/2, which the inequality already supports.

severity. note.

Repair, both halves. The $(1+\gamma)$ is dropped, and the four-step derivation with no room for it is printed in the Statement: $\Pi_\gamma(P)=\tfrac{1-\gamma}{\gamma}\big((I-\gamma P)^{-1}-I\big)$, so the constant cancels in the difference; the resolvent identity turns the difference into $(1-\gamma)(I-\gamma P')^{-1}\Delta P\,z'$; and $\|(I-\gamma P')^{-1}\|_\infty\le1/(1-\gamma)$ with $\|z'\|_\infty\le\|V'\|_\infty/(1-\gamma)$ give $\|\Delta P\|_\infty\|V'\|_\infty/(1-\gamma)$, the two $(1-\gamma)$ factors cancelling once. The text records that the factor had no derivation behind it and inflated the bound by up to $2\times$ as $\gamma\uparrow1$. The rank equality becomes $\mathrm{rank}(\Delta P)\le s-i$ with its reason: a token rewrite at $i$ changes the $s-i$ rows at or after $i$ and no earlier row, so $\Delta P$ has at most $s-i$ non-zero rows, and equality fails on any rewrite that leaves a later row's logits fixed. The finding's pin for the record's own copy of the overclaim is :473; the string "$\Delta P$ has rank $s-i$" sits at :471 at 99777ab (RUN, this session), the verse cites :471, and the finding's :473 is kept above as filed. The suffix re-solve price $(s-i)^2d/2$ is unchanged and is now stated as charged on the $s-i$ rows that may move rather than on the rank, which is what the inequality supports.

01.20 — `note` · V-10 · repaired; replacement re-derived#

verse. 01.20

flaw. The Hypotheses' condition holds by construction, so the If killed repairs a failure that cannot occur.

mechanism. V-10

number. Hypotheses: 'I - Q_{T'T'} a unit ... whenever 0 not in T''. With 0 in A_sink declared (docs/CEQ_SHAPE.md:295 @ 99777ab), 0 is not in T and hence not in any T' subset of T, so the condition is satisfied at every draw. If killed: 'the identity re-derived with the sink block kept explicit ... its Kill is the same residual with 0 not in T' enforced by a census line'.

replacement_survives. false.

required_repair. Drop the vacuous condition or state the reachable one it stands in for: I - Q_{T'T'} fails to be a unit exactly when some i in T' has P-hat_ii = 1, which finite logits forbid and a saturated gate (01.17's cut_makes_segment_head_absorbing) produces. Register that as the census line.

severity. note.

Repair. The vacuous condition is named as vacuous in the Hypotheses — $0\in\mathcal A_{\rm sink}$ is declared (READ docs/CEQ_SHAPE.md:295 @ 99777ab), so $0\notin T'$ for every $T'\subset T$ at every draw and the antecedent could not fail — and is replaced by the reachable one the finding derives, with the determinant that makes it decidable printed: $\det(I-Q_{T'T'})=\prod_{i\in T'}(1-\hat P_{ii})$ on the lower-triangular causal class, so $I-Q_{T'T'}$ fails to be a unit exactly when some $i\in T'$ has $\hat P_{ii}=1$, which finite logits forbid and 01.17's cut_makes_segment_head_absorbing saturated gate produces. That is registered as the verse's census line — $\max_{i\in T}\hat P_{ii}<1$ on every cell, $0$ GPU-s, read by 01.19's row-sum and diagonal pass, $0.6926596893360386$ at the judge's draw (RUN[J]), so it passes there and fires only on a saturated-gate cell.

On replacement_survives: false, and the replacement re-derived rather than renamed (V-9). The old If-killed re-derived the same identity with the sink block explicit and set its Kill to "the same residual with $0\notin T'$ enforced by a census line" — the same number, the same instrument and the same object, enforced by the condition just shown to hold always; it died with the verse. The new replacement is the declared-sink repair of 01.1's If-killed, and its three coordinates are each different: its object is the boundary declaration and the diagonal of $\hat P$, not the eliminated blocks; its instrument is 01.19's row-sum and diagonal pass, not the $512$-draw Schur residual battery; its number is $\max_{i\in T}\hat P_{ii}$ against $1$ — $0.6926596893360386$ (RUN[J]), with $\rho(Q)=0.692660$ declared against $1.000000$ undeclared (RUN[P], READ docs/CEQ_SHAPE.md:1440 @ 99777ab) — and not a residual at all. It carries its own Hypotheses, Evidence and plant, and a Kill strictly cheaper than the battery it follows (one float compared against $1$): the census passing on every cell while the Schur residual still exceeds $10^{-12}$, at which the chain ends at the Terminal and the Schur complement is withdrawn as a certificate input. The verse's own kill remains UNREACHABLE for want of the Schur instrument, which book 04 does not carry; that stands in the Limits paragraph and this repair does not change it.

01.22 — `note` · M-3 · repaired#

verse. 01.22

flaw. The two within-cluster spread bounds use different methods in one sentence, and the five-point bound is the interval width itself rather than the maximum sample sd.

mechanism. M-3

number. For n = 3 the verse uses w/sqrt(3) = 0.039027/1.732 = 0.0225, the correct maximum. For n = 5 it uses w = 0.0280; the maximum sample sd of five points in an interval of width w is wsqrt(0.3) = 0.54770.028019 = 0.01535. The printed ratio sigma_pool/sigma_w

= 0.253673/0.0280 = 9.06 is therefore loose; the correct bound is >= 16.5. The direction is safe (the claim is a lower bound) but the two halves of one sentence are computed differently.

replacement_survives. true.

required_repair. Use wsqrt((n-1)/n)/sqrt(n-1) consistently -- w/sqrt(3) = 0.0225 for the three-point cluster and wsqrt(0.3) = 0.01535 for the five-point one -- and reprint the ratios as >= 11.3 and >= 16.5.

severity. note.

Repair. One method is written out and both clusters are computed from it: the maximum sample sd (denominator $n-1$) of $n$ points in an interval of width $w$ is attained by splitting them between the endpoints, giving $\sigma_w^{\max}(n,w)=w\sqrt{\lceil n/2\rceil\lfloor n/2\rfloor/n}\big/\sqrt{n-1}$, which is $w/\sqrt3=0.5774\,w$ at $n=3$ and $w\sqrt{0.3}=0.5477\,w$ at $n=5$ (DERIVED). The crossed cluster ($n=5$, $w=0.028019$) therefore has $\sigma_w\le0.0153454$ and not $0.0280$, which is the interval width and not a sample sd; the NO-READING cluster ($n=3$, $w=0.039027$) has $\sigma_w\le0.0225322$. The reprinted ratios are $\sigma_{\rm pool}/\sigma_w\ge0.253673/0.0153454=16.53$ for the crossed cluster and $\ge0.253673/0.0225322=11.26$ for the other, against the earlier $9.06$ and $11.3$, and the closing clause reads "at least eleven times" where it read "at least nine times". The text records that the direction was safe — the claim is a lower bound and the old number was the looser one — and that the defect was one sentence computed by two methods, which is what M-3 names. The verse's Kill, replacement and Terminal are unchanged: none of the three depends on the value of $\sigma_w$, only on the split existing and the gap $[0.663,1.113]$ being empty.

Batch 3 — what changed, and what it did not buy#

Replacements re-derived because the pre-written one died with its verse (V-9). One: 01.20's, from the same Schur residual with a vacuous enforcement clause to the declared-sink repair on the diagonal of $\hat P$ — a different object, read by 01.19's row-sum pass rather than the residual battery, returning $\max_{i\in T}\hat P_{ii}$ rather than a residual, at a strictly cheaper kill. The three other replacement_survives: false findings of this batch — the two book-level ones and the pin census — attach to prose that carries no chain; each is answered by withdrawing the false claim rather than by substituting a new one, and each says so in its own row.

What this batch withdrew and did not replace. The blanket draw sentence in Limits: three of this book's own figures are not at $s=32$, $\gamma=0.6$, seed 0, and each now carries its own draw. The bare comparative in every form: no sentence of this book compares the shape to a stack outside A1/A2/T1/T2, and the A1 and A2 clauses that remain are routes with three of four terms NOT MEASURED, not numbers. The $(1+\gamma)$ in 01.15's cached-mixture bound, which no step produced. The rank equality for a token rewrite, now an inequality. Fourteen READ pins that did not resolve at 99777ab, corrected with the failed pin kept beside each. Nothing in this batch produced a measured number: every repair is arithmetic, a pin, or a withdrawal, and the six UNREACHABLE kills, the seventh (K-E1) and the two OPEN verses (01.5, 01.10) stand exactly as the earlier batches left them.

\(n\)	\(n\log_2n\)	\(H(d+1)p\)	\(R\)	error \(\ge\)	status at \((d,p)=(16,32)\)
\(64\)	\(384\)	\(544\)	\(-160\)	—	vacuous (the planned geometry)
\(128\)	\(896\)	\(544\)	\(352\)	\(0.1310\)	non-vacuous, \(s'=385\), \(K=128\)
\(256\)	\(2048\)	\(544\)	\(1504\)	\(0.2448\)	non-vacuous, \(s'=769\), \(K=256\)
\(64\), \(p=16\)	\(384\)	\(272\)	\(112\)	\(0.0972\)	non-vacuous at half precision
\(64\), \(d=8\)	\(384\)	\(288\)	\(96\)	\(0.0833\)	non-vacuous at half width

\(\gamma\)	\(K'=2\)	\(K'=4\)	\(K'=6\)	\(K'=16\)
\(0.6\)	\(0.540\)	\(0.194\)	\(0.0700\)	\(4.23\times10^{-4}\)
\(0.7\)	\(1.143\)	\(0.560\)	\(0.2745\)	\(7.75\times10^{-3}\)
\(0.9\)	\(7.29\)	\(5.90\)	\(4.78\)	\(1.67\)

policy at \(v\)	\(q(v)=(q^{(1)},q^{(2)})\)	Chebyshev value at \(v\)	\(q(u)\)	Chebyshev value at \(u\)
\(c\)	\((1,\ 0)\)	\(1\)	\((0.5,\ 0.5)\)	\(\mathbf{0.5}\)
\(d\)	\((0.6,\ 0.4)\)	\(\mathbf{0.6}\)	\((0.3,\ 0.7)\)	\(0.7\)

verse	kill	price	reachable today	replacement verse or route
01.19	row-sum census: \(\lvert\sum_j\hat P_{ij}-1\rvert>10^{-12}\), or \(\gamma\max_T\hat P_{ii}\ge1\)	\(0\) GPU-s, half an evening	yes — it has already fired (rows \(1.31\)–\(10.29\))	the admitted-cell fraction \(n_{\rm adm}/N\), then \(n_{\rm adm}=0\)
01.10(a)	struck: satisfied by construction on a triangular \(\hat P\)	—	no kill; the triangularity census is an admission line, UNSPECIFIED in book 04	none — branch (a) is algebra
01.12(a)	conservation row off \(\mathbb 1\) by \(>10^{-12}\)	\(0\) GPU-s	yes	sink enlarged, killed by \(n_{\rm abs}=0\) on a still-failing draw (a diagonal count, not the residual)
01.12(b)	\(0.875\) printed as the argmin floor without "at uniform \(\hat\pi\)", or a capability number without \(\hat\pi\) in the row	\(0\) GPU-s (assembly)	yes — it fired on this verse before repair	04.3's floor kill, routed by verse id
01.20	Schur identity off by \(>10^{-12}\)	\(0\) GPU-s	UNREACHABLE — Schur instrument unspecified in book 04	sink block kept explicit
01.22	a pooled mean without \(n^+\)	\(0\) GPU-s (assembly)	yes	the split with \(n^+\)
01.3	wrong-set plant moving \(q\) by \(<10^{-3}\)	\(0\) GPU-s, half an evening	yes	X-4 as an admission line, then S-10 re-drawn
01.16	residual departing from \(\gamma^{K'+1}/(1-\gamma)\) by \(>10^{-12}\) relative	\(0\) GPU-s, half an evening	yes — the banked pair reads \(2.0179\times10^{-13}\)	the dropped-mass bound by row-sum census; then the exact solve
01.7	the formed matrix identity departing from \(\gamma^{K'+1}/(1-\gamma)\) by \(>10^{-12}\) relative at \(\gamma\in\{0.6,0.9\}\), or any printed table cell doing so	\(0\) GPU-s, half an evening	yes — the pre-repair \(\gamma=0.7\) row would have fired it at relative \(0.7\)	the cost sentence (T1, book 03), killed by K-9's band
01.1	any label identity \(>10^{-12}\) on \(512\) draws	\(0\) GPU-s, one evening	yes	per-segment sets (01.17's cut theorem)
01.4	oracle argmax \(\ne\pi^8(i)\) on \(512\) instances	\(0\) GPU-s, one evening	yes	layout predicate; then oracle-family-only transfer
01.13	two arms not differing by \(O(1)\) at \(\gamma=0.9\): \(\\|O-Y\\|_\infty/\\|V\\|_\infty<10^{-3}\)	\(0\) GPU-s, half an evening	yes (the arXiv re-read is repriced as an author action, book 05)	row-sum census on the two operators; then the delta loses its operator clause
01.11(iii)	the two-epoch witness's local and global argmins agreeing, or the root gap \(0.2\) reading \(<10^{-12}\)	\(0\) GPU-s, minutes	yes	— (clause-level)
01.11(i)	\(m_{\rm eff}\le1\) on \(>0.05\) of the \(512\) admission draws (04.2 census clauses 9, 11)	\(0\) GPU-s	yes — 04.2 carries the census	— (clause-level)
01.11(ii)	DP against exhaustive enumeration at \(\lvert T\rvert\le12\), \(m\le3\): \(>10^{-12}\) on any of \(512\) draws	\(0\) GPU-s, one evening	yes	the one-clamp filter, killed by R-20 (\(>0.5\) disagreement), \(66{,}430\times\) cheaper in solves per draw
01.17	any `[M]` target not building (K-K)	\(0\) GPU-s, CPU minutes	yes	`[S]`, then RUN instances
01.6	fibre separation \(<5\%\) at \(\varepsilon=0.05\)	\(0\) GPU-s, one evening	UNREACHABLE — collision instrument unspecified in book 04	Bet D (01.8)
01.5	none: the arXiv re-read is an author action outside the canon	`NOT MEASURED`	UNREACHABLE; verse OPEN	Sanford \(\to\) Chen as lineage, no kill; Terminal in force
01.18	numerical rank of \(H_{33}\) at \(s=66\) reading \(\le32\) on any of \(1{,}024\) draws	\(0\) GPU-s, one evening	UNREACHABLE — Hankel-rank probe unspecified; and undecidable at \(s=64\)	numerical-rank statement; re-price against the scan (book 03)
01.21	\(\sigma_{17}/\sigma_1<10^{-3}\) on \(6\) of \(8\) cells, with the rank-\(16\)-plus-\(10^{-2}\) plant as the must-fire	\(0\) GPU-s from saved cells	UNREACHABLE — SVD probe unspecified; no cell (B5)	the committor head; then K-E1 (itself UNREACHABLE, 04.11)
01.10(b)	Dirichlet Cheeger instance failing on BED-1	`NOT MEASURED`, \(0\) GPU-s	UNREACHABLE — Dirichlet instrument unspecified; verse OPEN	the ladder's measured rate (\(f\in[0.99,1.01]\) at \(t\ge320\))
01.9	\(r_{\rm softmax}/r_{\rm shape}\le2\) (K-H1), registered only on seeds where K-D2 does not fire	inside the \(\approx34\) s pair (Bet E, 04.6)	needs a cell (B5)	the residual as diagnostic; cost sentence (book 03)
01.2	\(\\|\hat P-P_{\rm env}\\|_\infty<10^{-3}\) on \(6\) of \(8\) (K-D2)	inside the pair	needs a cell (B5); S-13 is a card of the record	the feature-content reading on S-11's generator spec, killed by the multi-hot absent while K-D2 fires — \(0\) GPU-s, one document read, reachable today
01.15	cached mixture within one seed sd on \(P\)-plants, \(6\) of \(8\) (K-G1)	\(\approx35\) s	needs a cell (B5)	the identity as algebra; Bet G's band
01.14	K-E1 at \(N=8\), escalated to \(N=16\)	\(\approx52\) s \(+\approx44\) s `[ASSUMED]` (not the \(\approx34\) s of Bet E's pair in 04.6)	UNREACHABLE — 04.11 registers \(0\) ChaCAL source lines and \(0\) records	link 1(\(\alpha\)): the margin arithmetic, \(0\) GPU-s, fires on the pre-repair \(L=20.7\); link 1(\(\beta\)): \(\varepsilon\\|V\\|_\infty\) against the label sd (B5); link 2: \(\phi_0\) on the environment chain with a value-zero sink column, no arm (04.11 link 1's census column); then Tree B
01.8	Bet D as registered: \(\mathrm{acc}_{\rm shape}-\mathrm{acc}_{\rm sky}\) at \(N=8\) against the realised paired sd, SPLIT band none, no seed-count clause	\(\approx64.8\) s `[ASSUMED]`	needs a cell (B5)	deeper skylines, then CoT
01.16 (K-I)	one exceedance of \(\delta\\|V\\|_\infty\) on \(1{,}024\) draws	\(\approx2.6\) s when a mask exists	dormant, owned by book 03; needs a mask (B15) and a label sd (B5)	not a link of 01.16's chain

claim	pin as printed	pin at `99777ab`
\(\mathcal A_k\) on the wrong set moves \(q\) by \(O(1)\) (01.1)	`:388`	`:387`
\(\rho(Q)=0.692660\) declared against \(1.000000\) undeclared (01.1)	`:1436`, a blank line	`:1440`
the must-fire perturbation, ratio \(9.6\times10^{10}\) (01.3)	`:921`	`:918`
Bet D's registration (01.6, 01.8)	`:2237`, Bet E's row	`:2236`
Bet F's price row (01.14)	`:2237`, Bet E's row	`:2238`
bind B-I, the \(\gamma=0.7\) plant \(119.37/1.143\) (01.7, 01.11, 01.17)	`:3113`, B-G2's row	`:3116`
K-E1 as frozen (01.14)	`:3180`, a table header	`:2274`
\(\mathrm{rank}(\Delta P)\) for a token rewrite (01.15)	`:473`	`:471`
Q3/W1, the Koopman field (01.19)	`:1199`	`:1222`
the `lambda_hat` one-bit defect (01.19)	`:1275-1278`	`:1280`
Q5/W3, finite-mixture inference (01.22)	`:1204`	`:1227`
Q1/W3, realisation theory (01.18)	`:1197`	`:1220`
Q5/W1, the \(n\)-width field (01.21)	`:1203`	`:1226`

01 — THEORY: ACCURACY#

Preface — what this book decides, and the verse that decides it first#

Part I — The exact class#

01.1 — The read is exact on four labels, and only through \(\hat P\)#

01.2 — What \(\hat P=P_{\rm env}\) costs in the softmax class#

01.3 — The committor is the read at \(\gamma\uparrow1\), with the discounted gap printed#

Part II — D-APPROX: from "cannot represent" to "cannot fit below \(\varepsilon\)"#

01.4 — The \(\mathrm{hop}_k\to\) committor reduction, written#

01.5 — OPEN — Obstruction 3 through the reduction: Peng's Theorem 1 with its vacuity line#

01.6 — D-APPROX by capacity: the per-position sketch, and the hypothesis it needs#

01.7 — The iteration skyline computes the resolvent with error exactly \(\gamma^{K'+1}/(1-\gamma)\)#

01.8 — Obstruction 2 restated: the skyline depth is a theorem on the layered family and an analogy on BED-S#

Part III — Per-row independence, Cheeger, Bellman, degeneracy#

01.9 — Per-row independence is a definition; joint consistency is a measured ratio#

01.10 — OPEN — The Cheeger sentence: withdrawn on the causal \(P\), restated on the reversible oracle chains#

01.11 — The Bellman hazard resolved: one-clamp optimality, DAG backward induction, and where scalarisation breaks#

01.12 — The reach-avoid identity with a sink, and the degeneracy lemma#

Part IV — The ChaCAL delta as theorems#

01.13 — ChaCAL-published and the diagonal-kept read coincide only at \(\gamma=0\)#

01.14 — Absorbing rows are the limit of a self-spike: the boundary mechanism is an exactness delta, not a representability delta#

01.15 — The interventional re-solve, and when the cached mixture is exact#

01.16 — The Neumann certificate \(\gamma^{K'+1}/(1-\gamma)\) is attained, in vector units, and dies at \(\gamma=1\)#

01.17 — Lean containment: the delta's theorems with their grades#

Part V — The LEAPABLE fields as named theorem targets#

01.18 — Realisation theory (Hankel–Kronecker): no finite-state realisation of the softmax resolvent read#

01.19 — Koopman spectral theory reduces to the diagonal on the causal class#

01.20 — Projection-operator formalism: absorbing elimination is memoryless, transient elimination is a Schur complement#

01.21 — Kolmogorov \(n\)-width: the read's approximability by a rank-\(d\) bottleneck#

01.22 — Finite-mixture inference: a pooled mean over a bimodal seed population is not a component statistic#

Census rows closed#

Kills, cheapest first#

Attacks answered#

01.7 — strike · V-17, P-1 · repaired#

01.1 — strike · V-3, V-25 · repaired (carried from an earlier batch, verified at this repair)#

01.14 — strike · V-10, V-24 · repaired#

01.18 — strike · V-10, V-11 · repaired#

01.11 — strike · P-10, V-25 · repaired#

book-level (01.4, 01.6, 01.10b, 01.18, 01.20, 01.21) — strike · P-4 · repaired#

01.6 — strike · D-7, V-17 · repaired (carried from an earlier batch, verified at this repair)#

01.13 — strike · V-11, P-4 · repaired#

01.5 — strike · V-11, P-3 · repaired as OPEN, Terminal in force (carried from an earlier batch, verified at this repair)#

01.16 — strike · V-10, V-11 · repaired#

01.19 — strike · V-10 · repaired#

01.10 (branch a) — strike · V-10 · repaired; verse now OPEN, Terminal in force#

01.10 (branch b) — repair · V-15 · repaired#

01.3 — repair · V-17 · repaired (carried from an earlier batch, verified at this repair)#

01.4 (construction) — repair · V-8, V-12 · repaired (carried from an earlier batch, verified at this repair)#

01.4 (Lean route) — repair · P-10, V-25 · repaired#

01.7 — repair · M-2, V-3 · repaired#

01.7 — repair · V-17, P-7 · repaired#

01.2 — repair · P-7, V-10 · repaired; replacement re-derived#

01.1 — repair · V-17, P-2 · repaired; the \(\kappa\) clause re-derived#

01.12 — repair · V-17, L-FLOOR · repaired; replacement re-derived#

01.5 — repair · P-3, P-10 · already repaired at batch 1; verified and tightened here#

01.11 — repair · P-3, V-3 · repaired; kill split three ways, R-20 repositioned#

01.11 — repair · V-3, V-11 · repaired; a two-epoch witness derived#

01.6 — repair · V-17, P-10 · already repaired at batch 1; verified here#

01.8 — repair · R-SKY, D-1 · repaired#

01.8 — repair · M-2 · repaired#

01.9 — repair · V-10, V-3 · repaired#

01.14 — repair · M-2, V-17 · repaired; link 2 re-derived off the non-existent arm#

01.14 — repair · P-1 · repaired#

01.21 — repair · V-15, L-EQ · repaired#

01.17 — repair · P-1, P-10 · repaired#

book-level — repair · P-1, V-17 · repaired#

book-level (01.6, 01.7) — repair · D-1, R-SKY · repaired#

book-level (01.5, 01.14) — note · P-2 · repaired, and twelve further pins corrected#

01.19 — note · P-10 · repaired#

01.6 — note · V-17 · already repaired at an earlier batch; verified here#

01.6 — note · P-1 · already repaired at an earlier batch; verified here#

01.15 — note · V-3 · repaired#

01.20 — note · V-10 · repaired; replacement re-derived#

01.22 — note · M-3 · repaired#