MATHEMATICS.md — the theory of record#

Every claim carries an evidence class. RUN means executed this session and the output read. READ means a file and line was opened and quoted. CITED means an external source was resolved and its title matched. DERIVED means it follows from RUN/READ by steps that are written out. Anything that would be GUESS is either labelled or absent.

Numbers carry provenance: which journal, which seed count, and what they were compared against.

0. THE THESIS — WHAT THE MODULE IS SUPPOSED TO PREDICT#

Self-attention is called a next-token predictor, and that description is accurate: it guesses the next best symbol. This module is meant to guess the next best SHAPE — the configuration the sequence settles into — so that the next equilibrium can be reached rather than the next word emitted.

Concretely, and in the author's framing: it does not predict a word. Given the words already present, it predicts several at once, jointly, because their values are determined together. And the way it decides which existing tokens matter is causal: inject an existing token, and observe how the whole shape moves — read through the embedding geometry and the topology of the sequence, not through a similarity score.

0.1 THIS PROJECT HAS NOT BEEN TESTING THAT. `READ`, AND IT IS THE CENTRAL DEFECT.#

scale/negation_scope.py:274, equilibrium_oracle:

"z*_{s-1}, the last coordinate of (I - A)^{-1} b."

The label is one coordinate. z* is the shape; z*_{s-1} is a single point on it. And nrmse at scale/negation_scope.py:672 is a root-mean-square error on that scalar.

So every task in this repository — the static ones and the equilibrium ones — asks for a point prediction.

That is why softmax wins, and the reason is a theorem rather than an accident. Predicting one scalar from a sequence, where the answer is a payload at a latent location times a query-side transform, is exactly the single-location regression problem on which one softmax layer is provably Bayes-optimal (arXiv:2410.01537). The corpus was built as the one task softmax cannot lose.

0.2 WHAT THE THESIS REQUIRES INSTEAD#

Three changes, each of which is a build rather than an argument:

(a) The label becomes the shape. Predict z* — the whole fixed point, a vector of length s — not z*_{s-1}. The oracle already computes the entire vector and then discards all but one entry.

(b) The readout must not be flat — but only where the target is not flat, and that was measured rather than assumed. A root-mean-square error over a configuration weights every coordinate independently and identically, which is the assumption a shape violates: the coordinates of a fixed point are coupled, and a prediction right in aggregate but wrong in arrangement is not a good prediction of a shape. Candidate readouts that respect the coupling are Hilbert projective distance on the positive part, Fisher–Rao or total variation where the object is a distribution, and a Procrustes or optimal-transport cost where only arrangement matters.

A QUALIFICATION THAT SURVIVED A CHECK, AND IT NARROWS THE RULE CORRECTLY. The objection does not bite on the chain family: equilibrium_oracle returns z*_{s-1}, which is exactly N(0, t*) on the real line — the target's own geometry is flat, so a root-mean-square error is the right readout there and every reading already taken under it stands.

It bites on the absorbing-chain corpus, whose label is a probability. The lightweight non-flat readout is one transform and no new scoring code:

    phi(p) = 2 * arcsin( sqrt(p) )

the exact Fisher–Rao geodesic coordinate on the binary simplex, since ds = dp / sqrt(p(1-p)).

RULE 8 was applied to it before it was measured. phi is smooth and strictly monotone, so near the fixed point it acts as a diagonal scaling and cannot move the leading eigenvalue — both readouts must therefore share an asymptotic rate. Confirmed to 6.319e-07 and 4.610e-04.

But the price at the rungs is not zero, which is what makes the choice load-bearing:

`t`	Euclidean	Fisher–Rao	ratio
8 (`_64`)	0.1815582640	0.2576242379	1.418962
32 (`_1024`)	0.0481326124	0.0582150553	1.209472

The cause is measured: _1024's conditional label puts 0.7666 of its mass within one hundredth of a boundary (0.3333 below 0.01, 0.4333 above 0.99), which is exactly where the two metrics diverge.

So the readout is load-bearing where the round reads and inert where it does not, and it must be DECLARED BEFORE the corpus run. A 42 % swing at a rung, chosen after the curve is visible, would be choosing the answer. The assertion in demo() requires both rate agreement below 1e-3 and a rung ratio above 1.15, so a readout that changed nothing would fail the test that says it needs declaring.

(c) The consequence test becomes vector-valued. The existing fidelity metric asks whether sign(Δŷ) matches sign(Δy) for a scalar. Under the thesis it should ask how the whole shape moves under an intervention — a displacement field, compared by direction and magnitude across all coordinates, not a sign on one.

0.3 WHERE SOFTMAX CANNOT FOLLOW — the novelty claim, stated narrowly#

One softmax step computes a convex mixture per query row, and each row is computed independently of the others. There is no mechanism in that step by which the value assigned to position i constrains the value assigned to position j. A jointly determined configuration — where the coordinates must be mutually consistent because they solve a fixed point together — is not what a single independent-per-row mixture computes.

That is the sharpest available statement of what this module could do that softmax cannot, and it is narrow on purpose:

It is not a claim that softmax cannot predict any single coordinate — the theorem says it can, optimally.
It is not a claim about depth — depth composition is separately governed by L = floor(log2 k) + 2 (§2), and stacked softmax layers get that too.
It is a claim about joint determination in one read, which is the only place the fixed point is doing work a mixture is not.

And it is currently UNTESTED, because no vector-valued label exists in the repository. Recorded as the open claim rather than as a result.

0.35 THE USE CASE, AND IT IS WHY JOINT DETERMINATION IS NOT A METAPHOR#

The target application is embodied systems — robots and anything else that must understand causality, consequence, pattern and link rather than emit plausible continuations. That is not a marketing frame; it is the setting in which the thesis of §0.3 stops being a preference and becomes a requirement.

A language model predicting the next token is well served by per-row independence. Each position's continuation can be scored against the context on its own, and softmax's independent-per-query mixture is exactly the right shape for that — which is why it is Bayes-optimal on the single-location problem and why it has been so hard to beat here.

A robot predicting what happens next is not. Joint angles, contact forces and body positions are coupled by constraints, so they are determined together or not at all: moving one joint changes the reachable set of every other, a contact that makes or breaks reconfigures the whole kinematic chain, and a force applied at one point propagates through the linkage. There is no ordering in which these can be read off independently, because the constraint is simultaneous. That is a fixed point in the literal sense, and it is the thing a per-row mixture cannot represent in one read.

The intervention test of §0.2(c) is the robot's own question, stated exactly. "If I move this, what else moves?" is a displacement field over a configuration under a do() operation — which is the consequence-fidelity metric of §6 generalised from a sign on one scalar to a vector over the whole shape. An embodied system needs the vector answer; the scalar answer is not a weaker version of it but a different question.

And the provenance is not a coincidence. This project's own recovered graph corpus comes from a physics engine, where the merged upstream contribution computes connected components over constraint/tree incidence — which is precisely the coupling structure above, expressed as a graph. Bodies joined by constraints form islands that must be solved together; bodies in different islands do not interact. The corpus and the use case are the same object seen twice, which is why an absorbing-chain label on that substrate is the natural test rather than an arbitrary one.

A SECOND SETTING, AND IT SHARES THE STRUCTURE RATHER THAN MERELY THE SPIRIT. Retrieval-augmented generation, where a model must understand the consequence of a false report. A retrieved passage is not a token to be attended to on its own merits; it is an intervention on the context, and the question that matters is what the rest of the answer does when that passage is wrong. That is a do() operation followed by a displacement over the whole output — the same vector-valued consequence measure of §0.2(c), applied to evidence rather than to joints.

And the failure mode there is the one a per-row mixture is worst at. A retrieved falsehood does not corrupt one position; it propagates, because downstream claims are conditioned on it and must be mutually consistent with it. Scoring each output position independently against the context cannot represent that propagation, which is why a system built on independent-per-row attention can produce an answer that is locally plausible at every position and globally wrong. Detecting that requires reading the shape, not the tokens.

What this does NOT license. No claim is made here about robotic performance, about retrieval accuracy, about control, or about any embodied or retrieval benchmark — nothing in this repository has been run on one. This section states why the target shape is what it is, and nothing more.

0.4 THE ARCHITECTURAL POSITION THAT FOLLOWS#

Softmax attention is not the opponent. It is the base case, and the module must contain it.

That is a reversal, and the evidence forced it. Three independent results say the same thing:

Softmax is Bayes-optimal on the single-location task shape. For a target that is a payload at one latent location times a query-side transform, one softmax attention layer provably attains Bayes risk while linear attention provably falls short. CITED: Marion, Berthier, Biau, Boyer, ICLR 2025, arXiv:2410.01537; Duranthon et al., arXiv:2509.21936.
Softmax's single step is already a converged fixed-point step. The attention update is the update rule of a modern Hopfield network, which "converges with one update" with exponentially small retrieval error. CITED: Ramsauer et al., ICLR 2021, arXiv:2008.02217.
Iterating the normalisation to a fixed point is published and buys little. Sinkformer runs Sinkhorn iterations to a doubly-stochastic fixed point in 3–5 steps for small accuracy gains. CITED: Sander, Ablin, Blondel, Peyré, AISTATS 2022, arXiv:2110.11773.

The consequence for design is precise. If softmax lies inside the module's function class, the module is bounded below by softmax and any equilibrium machinery is a strict addition. If softmax lies outside it — which is what was actually built, see §4 — then the comparison measures the exclusion rather than the idea.

Where iteration is known to pay, the iterated object is the representation, not the mixture weights. Weight-tied looped transformers with input injection match standard transformers on algorithmic tasks at under 10 % of the parameters. CITED: Yang, Lee, Nowak, Papailiopoulos, ICLR 2024, arXiv:2311.12424.

1. THE EQUILIBRIUM TASK FAMILY#

1.1 Why the previous corpus could not test anything#

Both originally registered oracles are closed-form functions of their input. READ, scale/negation_scope.py:

negation_scope    x[:, p, CH_PAYLOAD] * x[:, f, CH_FLIP]      a product of two entries
counter_squared   x[:, :, CH_FLIP].sum(dim=1) ** 2            a sum, squared

Neither has a fixed point. An arm that iterates to convergence and then predicts a product of two numbers is doing the same work as an arm that does not iterate. The deciding contrast on that corpus was therefore correct and uninformative.

1.2 The chain family, and the closed form that makes it admissible#

The chain label is the t*-step iterate of an affine recursion z' = a·z + b with a strictly lower triangular, hence nilpotent, so t* is exact rather than a tolerance.

The truncation law. A reading that uses only k hops has normalised error

    ceiling(t*, k) = sqrt( max(0, t* − k) / t* )

RUN, this session:

`t*`	`k=0`	`k=1`	`k=2`
2	1.000000	0.707107	0.000000
8	1.000000	0.935414	0.866025
32	1.000000	0.984251	0.968246

k = 0 is exactly 1.000000, which is the mean-predictor bar. That is what makes the family admissible: a zero-hop reading is worth nothing, so the label cannot be obtained without iterating.

It had to be earned. READ: an earlier encoding gave the query token a driver, so 1/(t*+1) of the label was legible at zero hops and the harness's own 0-step gate aborted INSTRUMENT BROKEN on three of five rungs. Setting b[s-1] = 0 restores the property. A third static task was avoided by a gate firing, not by care.

The formula was checked against drawn batches rather than assumed: max deviation +0.012088 at n = 2048.

1.3 The consequence family#

E2 labels a coordinate of the new fixed point after a one-token do() shock, under a contraction with Lipschitz constant 0.800000 exactly on every draw, enforced at generation with a raise rather than a warning.

Its truncation decays geometrically, bounded by L^k × (k=0 error): 1.000067 / 0.410387 / 0.203305 / 0.070732 / 0.013666 / 0.001017.

Its first design was killed by its own control. The raw new-fixed-point coordinate let a shock-blind reading score 0.194150 — 96 % of that label was the un-intervened game. The mirror-intervention contrast that replaced it has shock-blind error identically sqrt(1 + mean²/var) ≥ 1.0; measured 1.000067.

2. THE DEPTH LAW — WHY THE SHORTFALL IS NOT ABOUT SETTLING#

hop_k composition requires depth logarithmic in k:

    L = floor(log2 k) + 2          width O(1)

CITED: Sanford, Hsu, Telgarsky, ICML 2024, arXiv:2402.09268, Theorem 4.2; Corollary 4.3 gives an Ω(log k) lower bound conditional on the 1-vs-2-cycle conjecture — conditional, and it must be cited as such.

RUN: hop_2 → 3, hop_4 → 4, hop_8 → 5, hop_16 → 6.

The arms in this project are depth 1. At t* = 8 the required depth is about 5.

And the measured behaviour matches. At t* = 8 an arm trains to 0.860972 — sitting on its own two-hop ceiling of 0.866025 — and evaluates at 1.112208. It is not failing to settle; it is memorising, because the term it needs is a three-way product a[s-1]·a[s-2]·b[s-3] that depth-one mixing cannot form.

3. THE CONTRACTION AND ITS CERTIFICATE — WHAT SURVIVED#

Three contraction certificates were attempted and all three died: the sphere metric on an identity (θ = arcsin(√TV) exactly on a one-token mask), Birkhoff on T (the theorem requires a positive linear map and the map is not linear), and Birkhoff on G (failed its own per-seed bind).

What holds is κ = β — chosen, not derived — and it holds in 142/142 journal rows. READ: results/hilbert.jsonl, field bound_respected.

Two numerical facts that were load-bearing and are easy to get wrong:

tanh(Δ/4) reads exactly 1.0 in float64 for Δ ≥ 76.246190. The representable form is 1 − κ = 2/(e^{Δ/2} + 1), which stays finite to Δ ≈ 1400.
d_H(softmax u, softmax v) = osc(u − v) exactly — the log-partition cancels in an oscillation, so the Hilbert distance between two softmax outputs needs no exp.

4. THE CONFOUND — AND IT SUPERSEDES THE ARM COMPARISON#

The pivot family structurally cannot place the deciding token on the value path.

READ, scale/arm_s.py:107: select_pivots(kk, min(k_piv, s - 2), exclude=(s - 1,)), then return piv[piv > 0]. READ, scale/m3_quintuple.py:129: the same set via exclude=(0, s - 1).

So the largest legal pivot is s − 2. Every operator row is tril(-1)-strict, so row s − 2 reads only j ≤ s − 3. Therefore v[s-2] is invisible to every pivot row, while softmax's own row s − 1 reads j ≤ s − 2 and sees it directly.

And the t* = 1 label is a[s-1] · b[s-2] — concentrated on exactly the hidden position. Measured, perturbing that token by +100: the softmax reading moves 101.6983 (direct value pass-through) while the pivot value path moves 3.263746.

Consequence, stated plainly: any arm-versus-softmax number on this corpus measures the exclusion at least as much as it measures the idea. The root cause is a documented rationale rather than a defect — "a pivot reading of the row being settled is not an independent reading of it" — a defensible independence property that became a capability ceiling.

The repair is the same as the architectural position in §0: lift the exclusion so that p = s − 1 is legal, at which point av[s-1] is the softmax row and softmax becomes an interior point of the family.

5. THE SETTLED ARM ITERATES THE WRONG UNKNOWN#

The settled cell iterates α on a simplex over {α @ av} — a reparameterisation inside the unsettled twin's own family. It can reallocate weight; it cannot add a function the twin could not already express.

That is a structural account of the measured result, rather than a post-hoc one:

settled − twin = −0.002959     exact 95% CI [−0.042903, +0.031557]     covers zero

RUN, this session: exact enumeration over all 5**5 = 3125 paired resamples, 126 distinct values — cheaper than B = 10000 Monte-Carlo by 3.2× and with no sampling error at all.

And the variance tells the same story: sd 0.064106 for settled against 0.016547 for the twin, 3.874×, on the same batches, the same initialisation and the same parameter count. Settling widens the distribution without moving the mean.

The one-hot control separates the two claims. argmax − softmax = −0.118456, CI [−0.134115, −0.102786], 0/5 seeds — collapsing the mixture to a lookup is worse than plain softmax. So the mixture is worth something and the fixed point over it is not. Gradient descent on softmax attention converges to the max-margin separator of optimal tokens (CITED: Tarzanagh et al., arXiv:2306.13596), which is why hard selection kills the training signal that soft selection keeps.

6. CONSEQUENCE FIDELITY — THE CAPABILITY METRIC OF RECORD#

For an arm at trained weights, over drawn do() interventions with closed-form oracle effects:

    fidelity  =  fraction of interventions where  sign(Δŷ_model) = sign(Δy_oracle)

reported with a Clopper–Pearson exact interval, alongside the slope of Δŷ on Δy, and compared against softmax by exact McNemar on the same interventions with byte-identical draws.

It is computable for softmax too, which is what makes it a table column rather than a plea. Measured at s=64 d=24 n_train=8192, seed 0 only — the file requests five seeds and contains one:

arm	eval NRMSE	fidelity	CP interval	slope
`softmax`	0.877168	0.807843	[0.754044, 0.854329]	0.319032
`pivot_signed`	0.673762	0.854902	[0.805590, 0.895735]	0.565631
`pivot_unsigned`	0.747528	0.835294	[0.783958, 0.878642]	0.469549
`windowed_signed`	1.075830	0.360784	[0.301809, 0.423038]	0.021058

Paired exact McNemar against softmax: pivot_signed b=28 c=40 p=0.18181 no reject; pivot_unsigned p=0.348889 no reject; windowed_signed p=5.45054e-23 REJECT, in the wrong direction — its fidelity is below chance and it scores flipper 0.000000 (0/72), never once recovering the flipper token's sign.

Two readings, and both matter. Softmax already has consequence fidelity at 0.807843, well clear of chance — causal awareness is not the thing that distinguishes the arms. And the signed arm's apparent advantage does not survive the paired test.

7. THE ABSORBING-CHAIN ORACLE AND ITS RATE LAW#

The label is a coordinate of the absorption probabilities of a Markov chain: N = (I − Q)^{-1}, B = N R, the fixed point of z ← Q z + R.

λ₂ is defined, not assumed. P = [[Q, R], [0, I]] is block-triangular, so the literal second eigenvalue of P is 1 and says nothing. λ₂ := ρ(Q).

The engineered dial is exact. Setting Q := α · P_TT gives ρ(αP_TT) = α·ρ(P_TT), so α = target / ρ(P_TT) lands λ₂ = 0.9250000000 on both graphs, t_rel = 13.3333.

The committor residual has a closed form. With u − u_t = Q^t u and v − v_t = Q^t v:

    q_t − q  =  ( u ⊙ (Q^t s)  −  s ⊙ (Q^t u) )  /  ( s ⊙ s_t )

verified to 1.082467e-15 and 4.583348e-13. Both tails carry the same Perron mode, so the leading term is λ₂^t · w ⊙ (c(s)u − c(u)s) over a converging denominator, and therefore f := decay/λ₂ → 1 exactly: measured 1.00000063 and 1.00158689 at t = 320.

But only the limit is derived. A placement sweep on a fixed graph gives f(160) = 0.99034549 — below one — which no account of late sub-dominant modes permits. The leading amplitude c(s)u − c(u)s is a property of the two boundary columns, so the excess is set by both spectrum and geometry, and geometry can flip its sign.

And the operational consequence is the important part. Every ladder rung t* ∈ {1,2,8,32} sits below the mode time (20.5615, 169.3116). The ladder is entirely pre-asymptotic, so λ₂^{t*} is the wrong predictor at the rungs even though λ₂ is the right asymptotic rate. The closed form above is the right predictor and costs one solve.

A Cheeger obstruction bounds what any one-edge bridge can do. g ≤ 2φ with φ = 1/vol(S) gives t_rel ≥ vol(S)/2 = 1372.50 against a target band of [10, 20]; landing in band would need vol(S) ≤ 40, making every label a 3-hop function. The α-scaling construction relocates the obstruction rather than evading it — a radius-3 ball retains 0.7951152826 of the walk's weight and 0.1090161839 survives past the graph's diameter, so the reach ceiling moves from 3 hops to about 13. A ceiling at 13 is still a ceiling.

8. THE FORMAL RESULTS#

lake build CEQ exits 0. 134 theorems + 41 lemmas across the thirteen files in lean/CEQ/ and lean/CEQ.lean, counted by python scripts/lean_count.py, 2026-09-11. A raw grep -c '^\s*theorem\s' reads 140 + 35 instead: it counts six comment lines that open with the word "theorem" and misses six @[simp] lemma declarations. (The earlier count of 39 covered only the six earliest modules — Contraction, Nilpotent, Occupancy, OracleSeparation, OrbitBound, Refcount — and is still exact for them; V15, V15Fork, V15Kernel, V15Phase, V15Source and V16Domain came later.)

oracle_ne_resolvent. The arm's operator is strictly lower triangular, hence A^n = 0. The oracle's transient block Q is non-negative with symmetric support and at least one positive entry, hence never nilpotent. So the two cannot coincide, and the ladder's creditability does not rest on an assumption.

truncation_never_exact. The partial sum Σ_{k<N} Q^k is not the inverse of (I − Q) at any N — whereas for the arm's own operator it becomes an equality at N = n. Every rung therefore leaves a real residual.

SymmSupport is deliberately weaker than symmetry, because D^{-1}W is not symmetric at unequal degrees.

Both are bound to the shipped object rather than to a paper statement: reachability from the root module, #print axioms on five names with a planted sorryAx seen to fire, the three hypotheses checked entrywise on the actual matrix, and a must-fire in which the arm's own .tril(-1) operator is rejected by the support check.

9. THE NOVELTY POSITION, BOUNDED#

Most of equilibrium-labels-as-attention-capability-bar is occupied: CLRS (arXiv:2205.15659) and CLRS-Text (arXiv:2406.04229) for equilibrium labels as a bar; Sanford et al. NeurIPS 2024 (OpenReview AfzbDw6DSp) for an attention capability hierarchy; deep-equilibrium algorithmic reasoners (arXiv:2402.06445, arXiv:2410.15059) for equilibrium-solving arms.

The nearest contender was checked directly and does not occupy what remains. arXiv:2607.21607, "Spectral Flow Certificates for Depth-Aware Long-Range Propagation in Graph Neural Networks", is real — verified via the arXiv API, title matched, formula SFC(G,k) = 1 − (1 − γ(G))^k and all three R² = 0.910 / 0.881 / 0.863 byte-exact as attributed. Its date/identifier mismatch is block-wide across five neighbouring identifiers and does not impeach it.

But: it is a fixed-graph GCN with Transformer, GAT, GIN and GraphSAGE each appearing zero times; its headline fits are on engineered graphs while its surveyed graphs fall to R² = 0.440; and its label is trained accuracy on a sign-of-sum readout, with equilibrium and fixed point each appearing zero times.

And its own Proposition 1 (iii) states that at fixed depth SFC and γ induce identical rankings — so the headline values are λ₂'s numbers relabelled, and the paper's real gain is pooled-depth R² 0.884 against 0.858.

What remains unoccupied, stated narrowly: attention rather than message-passing, and the equilibrium itself as the label rather than an accuracy. Three further absences are recorded as NOT FOUND rather than assumed: no prior work iterating a routed mixture-weight vector over content-selected, Gram-coupled pivots to a fixed point as the attention read; none on the signed row-L1 normalisation ρ·w/‖w‖₁ as attention; and no unconditional lower bound against one-layer softmax on this product chain.

10. LIMITS#

Collected here rather than scattered.

Every task in the original corpus is static, so no reading taken on it bears on settling. The arm comparison on the equilibrium corpus is confounded by the pivot exclusion of §4 and must not be reported as evidence about settling until that exclusion is lifted. The consequence-fidelity table of §6 is one seed of a requested five, and its companion at d = 256 covers two of four arms. The ladder is pre-asymptotic at every rung, so any curve read against λ₂^{t*} will look wrong for arithmetic reasons. Five seeds cannot decide anything anytime-validly — the attainable evidence ceiling at five seeds is 3.80169140625 against a threshold of 40. No weights have been published. The depth lower bound of §2 is conditional on the 1-vs-2-cycle conjecture. And results/m3_capability.txt:1212 is a stale trap: it records e3_t8 k=2 NRMSE=0.815162 from before b[s-1] = 0 and describes a corpus that no longer exists.

11. THE SECOND EXACT ORACLE — KIRCHHOFF AGAINST THE ABSORBING CHAIN#

§7's label is one computation of the harmonic measure ω_v = P_v(τ_a < τ_b). The matrix-tree theorem gives a second, and the two agreeing is now an instrument law rather than an unstated hope. scale/kirchhoff.py, RUN.

The chain. L = D − A on the merged component, τ(G) its spanning-tree count, F(x | y) the number of spanning 2-forests separating x from y.

Kirchhoff. det(L_g) = τ(G) for L_g the Laplacian with row and column g deleted. Every cofactor is the same number.
All-minors. det(L_{xy}) = F(x | y).
Resistance. Ground at x, so M := (L_x)^{-1} has M_xx = M_xy = 0 and, by Cramer, M_yy = det(L_{xy}) / det(L_x). Since R(x,y) = (e_x − e_y)ᵀ L⁺ (e_x − e_y) and M differs from L⁺ only along the all-ones direction, which cancels in the difference, R(x, y) = F(x | y) / τ(G) — effective resistance is a ratio of forest counts.
Harmonic measure. Unit current a → b gives v = L⁺(e_a − e_b), and polarisation on (e_x − e_b) − (e_a − e_b) = e_x − e_a gives v(x) − v(b) = ( R(x,b) + R(a,b) − R(x,a) ) / 2. Normalising by v(a) − v(b) = R(a,b) yields the harmonic function with boundary values 1 at a and 0 at b, which is ω. Now ground at b: R(x,b) = M_xx, R(a,b) = M_aa, R(x,a) = M_xx + M_aa − 2M_xa, and the three-term expression collapses to

    ω_x  =  M_xa / M_aa  =  F(x, a | b) / F(a | b)

with no cancellation of large resistances, and one solve — L_b z = e_a — rather than an inverse.

It is independent of §7's solve, and that is structural rather than asserted. The chain path inverts I − Q with Q = D^{-1}A restricted to the nodes that are neither endpoint. The Kirchhoff path solves a symmetric unnormalised D − A on the whole component, in which a is an ordinary interior column and only b is removed. The two differ in matrix, normalisation, right-hand side and index map. L_T and I − Q are diagonally related — I − Q = D^{-1} L_T — which is exactly why the Kirchhoff route does not form L_T.

Both ends of the derivation are checked against brute force, not assumed. Steps 1 and 2 are compared to enumeration of spanning trees and separating 2-forests over every edge subset on drawn 7-node graphs; step 4's forest ratio is compared to the grounded solve node by node. tests/jupiter/test_kirchhoff_agreement.py, 11 tests.

The law, and its measured margin. max |ω_kirchhoff − ω_chain| < 1e-10, enforced in e4_harmonic.measure() before any number is read off an instance. Measured gaps, float64, this machine:

    six drawn Erdős–Rényi instances, n = 9              2.220446e-16
    SMALL_CASE,     62 merged nodes,   60 transient     8.992806e-15
    SHIPPED_CASE, 1202 merged nodes, 1200 transient     9.636736e-14

The tolerance sits three orders above the largest clean gap and nine below the smallest defect. A scratch copy of the chain carrying one off-by-one — dividing by len(neighbours) + 1, the bug you get by counting a node among its own neighbours — moves ω by between 1.749951e-01 and 3.156170e-01 over the same six drawn instances, and the check rejects it on 6 of 6. The defect is survivable by construction: every row stays strictly sub-stochastic, so the solve succeeds and returns plausible numbers in [0,1], and nothing downstream would have noticed.

Scope. The forest identity is the identity for the undamped walk. At kill > 0 the operator is not a random walk on G and the law raises rather than comparing two different objects and reporting the difference as a defect.

12. THE LADDER VERDICT AS A NAMED STATISTIC — PAGE'S L AND ISOTONIC REGRESSION#

scale/e_ladder.py:217 decides row A against row B by mono = all(b >= a for a, b in zip(deltas, deltas[1:])). That is a monotonicity check on four numbers, and it has no null, so it has no error rate: four independent noisy numbers land in non-decreasing order by chance one time in twenty-four. scale/page_trend.py supplies the null. RUN.

Page's L. Rank the k conditions within each of the n blocks, sum ranks per condition into R_j, take L = Σ_j j · R_j. Under exchangeability within a block every one of the k! rank assignments is equally likely and blocks are independent, so the null distribution of L is the n-fold convolution of the k!-point distribution of Σ_j j·r_j — exact, no table, no normal approximation. At k = 4, n = 5 that enumerates 24⁵ = 7 962 624 assignments; the support is [100, 150], the mean 125, and the distribution is symmetric, which is what makes the one-sidedness a choice rather than an artefact.

The pre-registration, exhaustive before the data.

    RISES := page_p < 0.05  AND  isotonic top rung > 0 with CI excluding zero
    FLAT  := neither clause fires
    SPLIT := exactly one clause fires

SPLIT exists because "both / neither" leaves a hole, and this repo has already paid for one: row H was added to 99777ab:E_LADDER_PREREGISTERED_READING.md because rows A–G all conditioned on settled winning somewhere. Neither clause can pick a rung: L quantifies over every rung by construction and the size clause reads the last rung, fixed by the ladder's definition.

THE READING, on the complete CPU ladder — five seeds, four rungs.

    rank sums R_j       8.0   12.0   13.0   17.0
    L = Σ_j j·R_j       139        support [100,150], null mean 125
    exact p             0.016724
    permutation p       0.016255   (se 0.000283, 200 000 draws, seed 0)
    isotonic fit        −0.036025  −0.017215  −0.004284  +0.016035
    top rung            +0.016035  CI [+0.002826, +0.033167]

The pre-registered branch returns RISES. Two independent p-value paths — an exact convolution and a permutation draw — agree to 4.7e-4, inside 1.7 standard errors.

Two things that verdict rests on, both measured, neither softened.

The size clause is carried by the constraint, not by the data. The same bootstrap without the monotone constraint gives the top rung at +0.016035 with CI [−0.004711, +0.033167], which covers zero — and whose lower bound reproduces results/e_ladder_reading.txt's shipped ci_lo of −0.004711 exactly, so the two bootstraps are the same bootstrap. PAVA pooled the top rung in 13.07% of resamples and lifted the lower bound by +0.007537. Pooling only ever raises a low top, never lowers a high one, so that lift is a property of the estimator under a flat truth.

The trend clause survives 2 of 5 single-seed deletions. Dropping seed 1 leaves p = 0.003864 and seed 2 leaves p = 0.021741; dropping seed 0 or 4 leaves p = 0.050411 and seed 3 leaves p = 0.072401.

But the conjunction is calibrated even though one of its clauses is not, and that is the argument for requiring both. Over 200 drawn tables per arm (5×4, sd = 0.03, seed 11, n_boot = 300):

    flat truth      trend 0.065   size 0.325   RISES 0.040
    rising truth    trend 0.810   size 1.000   RISES 0.810

The isotonic clause alone fires on a third of flat tables — six times nominal. The conjunction reads 0.040 against a nominal 0.05, because the trend clause gates the miscalibrated one. Both arms are drawn from the same generator and the verdict separates them, so the branch is not vacuous.

THE GRANULARITY QUESTION, AND THE TWO CLAUSES ANSWER IT DIFFERENTLY. At five seeds a sign-pattern statistic can express no two-sided p finer than 2/2^5 = 0.0625, which is above the 0.05 this project quotes; measured on the shipped contrast() over 1000 samples, a 5-0 unanimity excludes zero 385/385 times, a 4-1 split 20-44% and a 3-2 split 0-3.7%, so "the CI excludes zero" at N = 5 is very nearly "all five seeds agreed". That floor is real, and it lands on one of the two clauses.

The trend clause clears it. Page's L ranks k conditions within each block instead of reading one sign, so the outcome space is (k!)^n = 24^5 = 7 962 624 rather than 2^n = 32 — a factor of 12^5 = 248 832, or log2 24 = 4.585 bits per block against 1, 22.925 bits against 5. The consequences are exact:

    achievable p-values on the whole support        51
    finest non-zero achievable p                    1 / 24^5 = 1.2558674e-07
    achievable p-values at or below alpha = 0.05    14   (L = 137 .. 150)
    critical value at alpha = 0.05                  L = 137
    TRUE size of the test at that critical value    0.037002877
    next coarser rung, L = 136                      0.052384114  (above alpha)

alpha = 0.05 is reachable, and p = 0.016724386 is an exact atom sum of the discrete null — the cumulative count 133 170 / 7 962 624 at L = 139 — not an interpolation onto it. Being discrete the test is conservative: its true size is 0.037002877, not 0.05. By contrast the sign lattice at N = 5 has six achievable one-sided p-values of which exactly one clears 0.05 (unanimity, 1/32 = 0.03125), and none of their two-sided partners does.

Three routes to the null, sharing no arithmetic: float probabilities through np.convolve, integer polynomials through Python big integers, and literal enumeration of every one of the (k!)^n rank assignments with L rebuilt from whole tables. The first two agree to 1.388e-17 at k=4, n=5; all three agree exactly at (3,3), (4,3) and (3,4), where enumeration is tractable.

The size clause does not clear it, and cannot. It IS a paired percentile bootstrap over five seeds, so 0.0625 is its floor and no interval it prints is a 0.05-level statement at this seed count. The ladder's top rung is a 4-1 split — the regime Venus measured at 20-44% — and its unconstrained interval duly covers zero at [-0.004711, +0.033167]. Only the one-sided PAVA lift carries it across. The granularity floor and the estimator bias are two independent reasons to distrust the same clause, and they push the same way. The pre-registered branch is not refitted after the fact: RISES stands as written, with the trend clause sound at its own level and the size clause now known to be incapable of the level it was written at.

AND ROW G OUTRANKS ALL OF IT. 99777ab:E_LADDER_PREREGISTERED_READING.md credits a rung nothing in either direction when either cell sits at or above predict-the-mean, and it fires on three of the four rungs this statistic is computed on:

    rung      settled       twin    credited
    e3_t1    0.994399   0.958373    yes
    e3_t2    1.013958   0.996743    no
    e3_t8    1.096009   1.091725    no
    e3_t32   1.103711   1.119745    no

A trend in the difference between two arms that both lose to predict-the-mean is an ordering, not a capability. RISES is a statement about the contrast as a number. It is not a claim that settling bought anything at depth, and page_trend.report() prints these cell means beside the verdict so that it cannot be read as one. Replacing an eyeball with a statistic sharpens the reading of a quantity the ladder's own pre-registration had already declined to credit.

13. THE ATTRIBUTION CLAIM'S SAMPLE-COMPLEXITY LINE#

impact_attribution regresses the query row B[query, :] of length N against n_samples news vectors — the noiseless system y = A b (scale/impact.py:1013-1021, READ). Each row of the planted B carries SUPPLIERS_PER_NODE + COMPETITORS_PER_NODE = 4 nonzeros (scale/impact.py:88-89, 405-410, READ). The fixed sparsity is in the ROW, not the column — a column's nonzero count is however many rows chose it and is unbounded — so the line below is stated for the row the recovery actually reads.

The line. Recovering an s-sparse vector in R^n from m noiseless linear measurements is possible at all only in the RIP regime

    m  ≥  C · s · ln(n / s)

Below it an arm's attribution failure carries no information about the arm: there is provably no procedure separating the planted B from an adversarial one consistent with the same undersampled measurements. scale/rip_line.py:verdict returns the exact string UNDER-SAMPLED there and ADMISSIBLE at or above, so a cell prints a regime rather than a loss.

C is measured, and the oracle transitions across the line. Gaussian A, signed s-sparse b with uniform support and Rademacher signs, drawn not built; recovery by L1 basis pursuit (scipy.optimize.linprog) and, independently, by orthogonal matching pursuit. Every grid reads 0.00 at its bottom and at least 0.96 at its top for basis pursuit — the sweep is non-degenerate at both ends, which is the whole requirement.

    n     s    n/s    ln(n/s)     m_50 (BP)   C (BP)     m_50 (OMP)  C (OMP)
     64    4    16    2.772589       16       1.44270       24       2.16404
    128    8    16    2.772589       32       1.44270       48       2.16404
    256    4    64    4.158883       24       1.44270       28       1.68314
     64   16     4    1.386294       40       1.80337       64       2.88539

C IS NOT BED-INVARIANT, AND THE FIRST TWO BEDS COULD NOT HAVE SHOWN THAT IT WAS. They share n/s = 16, so ln(n/s) is the same number in both and m_50 tracking s forces C to repeat whatever the truth is. Varying the ratio instead: C = 1.44270 holds at n/s = 16 and n/s = 64, and rises to 1.80337 at n/s = 4 — 25% higher. s·ln(n/s) is the asymptotic scaling and its constant creeps as n/s falls toward 1. The operational consequence is that C must be re-swept per bed; reusing 1.44270 elsewhere is a guess. The two solvers likewise disagree by a factor of 1.2 to 1.6 depending on the bed, which is two solvers failing differently and a second reason C is a per-bed measurement.

Where the shipped corpus sits. impact_attribution defaults to n_samples = 256 against N = 1024, s = 4 (scale/impact.py:973, READ), and rip_line(1024, 4, 1.44270) = 32.01, so the default oversamples the floor by roughly 8×. The floor is the deliverable, not a claim that today's default sits on it.

14. THE COHERENCE FLOOR AND THE CALIBRATED SCRAMBLE CONTROL#

Stated as functions of (d, k) so the arithmetic survives any change to either.

The Welch bound. Any k unit vectors in R^d obey

    max_{i<j} |⟨u_i, u_j⟩|  ≥  sqrt( (k − d) / (d (k − 1)) )

which is exactly 0 for k ≤ d, because an orthonormal set of k ≤ d vectors exists and attains it. Above the dimension it is the real constraint and it is tight: at d = 2, k = 3 the bound reads 0.5 and three unit vectors at 120° attain |cos 120°| = 0.5 exactly, agreeing to 1e-12.

M1, at d = 256, k = 16. k ≤ d, so the floor is exactly 0, while random role vectors carry a max coherence of 0.174795. A crosstalk-shaped failure at k = 16 roles in 256 dimensions is therefore a training or design defect, not dimension starvation — the geometry permits perfect separation and nothing in the dimension count obstructs it.

M5, and the author's figure is 15.8% low. The amendment states the random coherence as sqrt(2 ln k / d) = 0.147176. That closed form is a union bound over k events, but the maximum runs over C(k,2) = 120 pairs, not over 16 vectors. Four routes, of which two fail differently from the sampling:

    sqrt(2 ln k / d)              0.147176   author's form, union over k
    sqrt(2 ln(k(k−1)/2) / d)      0.193397   union over C(k,2) pairs — an upper bound
    Monte Carlo, 20 000 trials    0.174795   95% CI [0.174460, 0.175131]
    order-statistic quadrature    0.174499   ∫₀¹ (1 − F(x)¹²⁰) dx, F the exact
                                             |⟨u,v⟩| CDF from (1−t²)^((d−3)/2)

The quadrature lands inside the Monte Carlo interval and both closed forms land outside it. The union bound is above, as a bound must be; the author's form is below, because it counts the wrong number of events. The mean overlap, by contrast, is confirmed on both paths: the exact Γ(d/2) / (√π Γ((d+1)/2)) = 0.049917 matches the Monte Carlo 0.049917 (CI [0.049869, 0.049964]) and its sqrt(2/(πd)) = 0.049868 limit to 4.9e-5.

The consequence, and the error is in the safe direction. Scrambled roles at d = 256, k = 16 carry an expected mean overlap of 0.049917 and an expected worst-case overlap of 0.174795, both by chance and both computable before the control runs. A scramble control calibrated to expect zero residual separation is vacuous before it runs, and it is more vacuous than the amendment's own figure implies, not less. That is the fifteenth pattern, killed pre-birth.

15. A COVERAGE TOOL THAT SCANNED NOTHING#

99777ab:scale/chase_struck_coverage.py walks every .md and .py the shipped struck-constant test does not cover, and reports any struck constant asserted without a strike marker in its paragraph. It printed SCANNING 0 PATHS THE SHIPPED CHECK DOES NOT COVER, uncovered .md: 0, uncovered .py: 0, and exited 0. RUN.

The root cause is a scope error in one expression. The exclusion list (".git", "__pycache__", ".pytest_cache", ".claude", ".benchmarks") was tested against p.parts — the components of the absolute path. Every agent worktree in this project is checked out under <repo>/.claude/worktrees/<name>/, so .claude was a component of the absolute path of every file in the tree and the filter dropped all of them. Measured on this checkout: 366 candidate files, 0 surviving the absolute filter, 365 surviving the same filter applied to p.relative_to(ROOT).parts.

Its own must-fire kept passing throughout, because that control feeds text directly to scan_text and never exercises target selection. A matcher control is not a coverage control, and this is the cleanest available example of the difference: the assertion that fired was true and the instrument was blind.

The fix is the scope correction plus a refusal. collect_targets filters on the path relative to the scan root, and main returns 1 rather than 0 on an empty target list — a tool that walks zero paths passes every input and must not report success. tests/jupiter/test_struck_coverage_scans.py — deleted from the live tree at c71527a as one of the 53 test files removed with the round-report cleanup that commit describes; readable at git show 99777ab:tests/jupiter/test_struck_coverage_scans.py, not at HEAD — plants a struck constant in a file on disk that target selection must reach, requires it to be found, requires the same text carrying a strike marker not to be found, and requires a file with no struck constant to be silent — three arms, because two would pass for a scanner that flagged everything. 6 of its 8 tests fail against the unfixed scanner; the 8th states the root cause without reference to the refactor so a later rewrite cannot make it vacuous. (This paragraph narrates what that test demonstrated when it ran; the fix and the 27-candidate finding below it are unaffected by the file's later removal.)

What the fix exposed. With 346 paths now scanned the tool exits 1 on 27 candidates. They split into two classes, and the file's own docstring already warns that layer 2 is a text scan and text scans cry wolf. Prose about a strike whose paragraph carries no marker: the scanner's own docstring quoting 1.471448 in its must-fire description, MISTAKES.md:206 discussing 5.4944e-13, 99777ab:PREREGISTRATION_HOLE_AUDIT.md:424 running a git log -S "0.743864" forensic. Live assertions: −1.389 with R² 0.9938 in a 99777ab:RESEARCH.md table row, and the three newly struck U1/N3 constants 0.743864, 0.656532, 0.816955 asserted in tests/cameron/test_harmonic_attribution.py:123-124 and mirrored in tests/deimos/. −1.389 alone accounts for 17 of the 27 and is a short enough string to match inside longer numbers, so the tally is a candidate list and not a finding. Adjudicating it is not this section's business; surfacing it at all required the scanner to look at a file first.

16. LIMITS ADDED BY SECTIONS 11–15#

§10 predates these sections and does not cover them.

The instrument law of §11 is enforced only where e4_harmonic.measure() is the entry point; absorbing_chain and fixed_point remain callable directly and are not guarded, and the law is defined only for the undamped walk. Its agreement margin is measured at three graph sizes on one machine in float64 and no conditioning bound is proved. §12's verdict is RISES under a branch that was fixed before the statistic ran, but its size clause fires on the strength of a one-sided estimator bias AND cannot express a p finer than 2/2^5 = 0.0625 at five seeds, so it is structurally incapable of the 0.05 level it was written at; its trend clause survives only 2 of 5 single-seed deletions, and the percentile bootstrap behind both has 5⁵ = 3125 distinct atoms at five seeds; the calibration sweep is 200 tables per arm under Gaussian noise at one scale and does not establish the error rate under the ladder's real noise, and the trend clause's own TRUE size is 0.037002877 rather than 0.05, so power quoted against a nominal 0.05 overstates it. §13's C is fit at four beds with coarse m grids — m_50 for the n/s = 4 bed is bounded only to (32, 40] — under a Gaussian measurement ensemble, and news_mat in scale/impact.py has not been shown to be one; the line is a necessary condition, never a sufficient one, and nothing here wires UNDER-SAMPLED into an IMPACT cell. §15's fix is verified on this checkout and on synthetic roots; it is not verified on a checkout whose absolute path contains none of the skipped names, where the old and new filters agree and the test falls through to an equality branch. The 27 candidates the fix exposed are unadjudicated, and −1.389 is short enough to match inside longer numerals, so the count is an upper bound on real hits rather than a finding. §14 is arithmetic about independent uniform unit vectors and says nothing about the vectors any trained arm actually holds; the order-statistic quadrature treats the C(k,2) pairwise products as independent, which they are not, and is corroboration for the sampled figure rather than a proof of it.

17. WHAT THE STE RESULT ACTUALLY SHOWS, AND WHAT IT PRICES OUT#

The full candidate catalogue is results/r9_maths_survey.md. This section records only the derivations it rests on. No wall-clock measurement was taken for it.

17.1 Two discrete stages, and only one of them is trained#

	discrete object	site	trained
Stage A	which `k` rows are pivots	`m3_quintuple.py:169` → `pivot_probe.py:80`, `topk(key.norm(dim=-1), k)`	never, in any arm
Stage B	mixture weights `α` over the chosen `k`	`m3_quintuple.py:440`	yes

argmaxste estimates stage B. Its forward is bitwise argmax's: soft − soft.detach() is elementwise exactly +0.0, and soft ∈ [0,1] admits no inf/nan path — checked over 200 drawn gates in float64. Stage A is byte-identical across argmax, softmax, argmaxste, settled and twin. Nothing measured this round bears on stage A.

17.2 The gradient is the effect; the mechanism is not#

Recomputed from results/m3_quintuple_v2.jsonl at ntr8192_nev512, 5 seeds, via m3_synthetic_settled.contrast. Positive = second cell lower.

    argmax    -> argmaxste   +0.225760  CI [+0.212433, +0.245886]  5/5
    softmax   -> argmaxste   +0.107304  CI [+0.082879, +0.140870]  5/5
    argmaxste -> twin        +0.004092  CI [-0.023107, +0.029187]  3/5
    settled   -> twin        +0.002959  CI [-0.031557, +0.048587]  2/5
    argmaxste -> settled     +0.001133  CI [-0.069616, +0.057312]  3/5

Three structurally different stage-B mechanisms — a Neumann-settled fixed point, a straight-through one-hot, and a full softmax mixture — lie within 0.004092 of each other with every CI covering zero. The only contrast excluding zero at 5/5 is the presence of a gradient.

17.3 The pricing rule: what this instrument can and cannot falsify#

Minimum detectable effect, paired, α=0.05 two-sided, power 0.80. Two paths: the normal approximation, and an exact noncentral-t solve. (scipy.stats.nct returns nan at large noncentrality; an unguarded bisection converges upward and returns a non-monotone answer — the guard is load-bearing.)

    paired sd   n=5 normal   n=5 exact   n=10      n=20
    0.022345    0.027996     0.037584    0.022256  0.014758
    0.034451    0.043164     0.057946    0.034313  0.022753
    0.050146    0.062828     0.084345    0.049945  0.033119
    0.082152    0.102929     0.138179    0.081824  0.054257

Seeds required, and their cost at settledrow's 4.560600 s/step (scale/m3_flops.py:107, wall clock, one step including backward and Adam, s=64 n=2048) × 150 steps = 684.09 s:

    contrast                seeds (normal / exact t)   runs    lower-bound clock
    argmax vs argmaxste          0.1 /      3             6         1.1 h
    argmaxste vs twin          556.3 /    559         1 118         8.85 days
    settled vs twin           2254.1 /   2257         4 514        35.74 days
    argmaxste vs settled     41263.5 /  41266        82 532       653.46 days

The seed counts are exact. The day counts are PROVISIONAL: m3_flops.py:117-120 records the same settled unit reading 2.0775 against 3.4372 s/step in two sessions on identical code, 1.65× apart, which is why every clock in this repository carries that label. They are also lower bounds — the compared cells ran at ntr8192, four times the batch the rate was measured at.

PRICING RULE. At 5 seeds this instrument resolves 0.057946 or larger. The stage-B mechanism differences are 14×, 20× and 51× below that floor; the gradient effect is 3.9× above it. Any candidate whose contribution is a better relaxation of stage B is unfalsifiable here, and an unfalsifiable improvement is inadmissible under this project's own evidence rule.

17.4 Why the one stage-A ablation on record could not have said anything#

99777ab:DONE_ARCHIVE_ROUND1.md:4707 records K4: randpivot_signed, k=8 content-blind pivots, slope +0.081, CP intervals overlapping the content-selected arm, and the pre-registered consequence that content selection is "not load-bearing for M2".

That null was forced. aa82df7:scale/recall_probe.py:3-7 states the identity: for any content-blind schedule of size k,

    P(c reachable) × (share | reachable)  =  (k/s) × (1/k)  =  1/s

bit-for-bit the dense rate, independent of k. M2 draws c from P, so it measures the second factor and conditions the first away. Verified on two paths that fail differently — the symbolic factorisation, and a 200 000-draw Monte Carlo over 12 (s,k) pairs (at s=1024, k=8: symbolic 0.00097656, sampled 0.00099187). A content-blind schedule reproduces the dense rate exactly, so K4 was structurally incapable of reading anything else.

recall_probe.py:9-11 names the quantity that is not an artefact: whether P(c selected) stays Θ(1) as s grows under content-conditional selection. That file is imported by zero Python files, has no results/ artifact, and 99777ab:DONE_ARCHIVE_ROUND1.md:5833 calls it "already sitting unrun". Stage A's search space is C(62,8) = 3 381 098 545 sets, of which topk explores exactly one.

17.5 A correction to §C of `FINDINGS.md`#

arXiv:2410.01537 (Marion, Berthier, Biau, Boyer, ICLR 2025) is cited there as "softmax is provably Bayes-optimal" against "linear attention". Equations fetched from ar5iv.labs.arxiv.org/html/2410.01537: the predictor is T_λ^{k,v}(𝕏) = erf(λ𝕏k)ᵀ𝕏v — erf, not softmax; Corollary 2 gives asymptotic optimality under "d→∞ and L=o(d)"; and Proposition 3 refutes linear regression, ℛ(β⋆) → ε²+γ², not linear attention. The label shape does match (Y = X_{J₀}ᵀv⋆ + ξ, scalar, latent informative position), so the worry is sound — but this repo runs L = s = 64 against D_MODEL = 16 (m3_capability.py:79; the d24 in journal keys is make_batch's flipper offset, negation_scope.py:84-86). L/d = 4.00, the opposite of L = o(d). "Provably" is not earned at this geometry. Single-source: the ICLR proceedings PDF returned compressed streams and the second path failed.

17.6 The `nash.py` shared-`tau` defect is real and too small to be the cause#

ceq/nash.py:64 returns margin * float(matrix_norm(m, ord=2).max()) / 4.0 — one Python float for the batch, from the worst-conditioned instance, applied at :146. Since stance = 2σ(z/τ) − 1 ≈ z/(2τ) for small z/τ, an instance run at τ_batch is attenuated by τ_i/τ_batch. On games built exactly as nash.py:138-144 builds them, Gaussian q,k: median shrink 0.877526 at n=64, 0.851143 at 256, 0.789701 at 2048 (min 0.690213). Two paths — the small-signal ratio and the exact sigmoid amplitude (0.50 → 0.547752, 0.25 → 0.281624, 0.10 → 0.113593).

A 1.27× attenuation cannot produce the recorded OOD NRMSE of 2.6151 to 5.8198, which is 2.6× to 5.8× worse than predicting the mean. Before the fix and the rerun, one line settles it: measure matrix_norm(game, ord=2).max()/median on a real batch. Near 1.27 and the bug cannot be the cause. The Gaussian draw is a surrogate, so this is a lower bound on the attenuation, not the corpus's value.

17.7 The pivot objective's set structure, and where the greedy bound dies#

Derived here, not cited. pivot_probe.pivot_hop2 uses the pivot set as B_P = A[:,P]A[P,:] = Σ_{p∈P} u_p v_pᵀ with u_p = A[:,p], v_p = A[p,:] — a sum of rank-1 outer products, one per pivot, each independent of the rest of P. That independence is the whole reason set-function structure exists here. With M_pq := (u_p·u_q)(v_p·v_q), captured mass is a quadratic form in the indicator,

    f(P) := ‖B_P‖_F²  =  1_Pᵀ M 1_P ,
    f(S∪{a,b}) − f(S∪{a}) − f(S∪{b}) + f(S)  =  2 M_ab   — constant in S

Captured mass is SUPERMODULAR on a nonnegative operator. A softmax attention matrix is nonnegative, so every inner product is ≥ 0, M_ab ≥ 0, and the second difference is ≥ 0: increasing returns. Greedy has no 1−1/e on it. The intuition that pivot selection is obviously a diminishing-returns problem is wrong on the obvious objective.

Reconstruction is monotone submodular, and there the bound is real. h(P) := ‖A²‖_F² − ‖A² − B_P‖_F² has second difference −2M_ab ≤ 0, and monotonicity is derived rather than assumed: with R = A² − B_S and a ∉ S,

    h(S∪{a}) − h(S) = 2⟨R, u_a v_aᵀ⟩ − ‖u_a v_aᵀ‖²  ≥  ‖u_a v_aᵀ‖²  >  0

since R ≥ u_a v_aᵀ elementwise for A ≥ 0, a ∉ S. Checked over 21 936 (S,a) pairs: min(gain − ‖u_a v_aᵀ‖²) = −3.55e−15, zero to float precision and tight, the bound being attained when S holds every other pivot. With h(∅)=0 and h ≥ 0, Nemhauser–Wolsey–Fisher gives greedy 1 − 1/e = 0.632121. (NWF 1978 is classical, cited from standard knowledge and not fetched — flagged.)

The signed arm voids it. M_ab ≥ 0 needs A ≥ 0. On signed operators M carried a negative entry in 297/300 draws at n ∈ {4,5,6} and 300/300 at fixed n = 6, so neither objective is submodular there.

A prediction of mine that was wrong, recorded. I expected h to be non-monotone, reasoning that B_P is a fixed sum rather than a projection so an added pivot could raise the residual. It cannot, for A ≥ 0: 0 of 21 936 checked pairs decreased h. The derivation above is the corrected version.

Two paths throughout: the closed form ±2M_ab, and direct evaluation of f and h on all four sets, agreeing on every checked subset. scale/pivot_selection_theory.py holds both, with demo() asserting them.

What it is worth. It bounds greedy against the best pivot set under a reconstruction objective. It says nothing about whether a better pivot set lowers NRMSE — that is §17.4's question, and aa82df7:scale/recall_probe.py was written to answer it and never run. A provable guarantee on an objective nobody has shown to matter is still a guarantee about nothing.