Phase H — bind#
Phase H asks whether binding a word to an operator, so that composition is a product rather than a sum, buys anything a trained vector model cannot reach.
This page opens with what the phase does not invent, because the fetch row ran before any table was drawn and found that the central move is published.
1. Prior art, stated before any result#
Binding a verb to an operator is not new. Coecke, Sadrzadeh and Clark (2010) already do it, and compose by contraction rather than by addition. From §4.1 of their paper:
the meaning vector of each transitive verb can be thought of as a function that inputs a subject from V and an object from W and outputs a sentence in S
Adjectives-as-matrices is Baroni and Zamparelli (2010), whose abstract "represents nouns as vectors and adjectives as data-induced (linear) functions (encoded as matrices) over nominal vectors". That paper is adjectives only — its §1 states it "addresses instead the combination of nouns and attributive adjectives", and verbs appear only as future work. The extension to verbs as matrices and transitive verbs as rank-3 tensors is Grefenstette et al. (2013), three years later. An earlier draft of this page's source notes attached the verb claim to the 2010 name; that was wrong and is corrected here.
On the sequence-model side the gate landscape is likewise occupied. RetNet
carries a scalar decay, GLA and Mamba-2 a per-token decay, and DeltaNet a matrix
gate I − β_t k_t k_tᵀ — a non-commuting rank-1 reflection, which is to say
DeltaNet already sits inside the family a matrix-valued carry would define
rather than outside it as a rival.
What is left. Not the binding. The claim this phase can still test is about reachability: whether an exact-zero gate yields a second absorbing state where a softmax row cannot, and whether that property is decidable rather than merely observed. DisCoCat and Baroni–Zamparelli say nothing about reachability. Everything below is scoped to that, and any sentence here that reads as a novelty claim about binding is a defect in this page.
2. What the fetch row struck#
Five sources were checked against the sentences this phase wanted to write with them. Three contradicted the intended claim, and the loudest unconditional claim turned out to be in this project's own notes rather than in any paper.
"Constant-depth attention computes iterated commutative products only" does not follow from TC⁰ containment. Containment of log-precision transformers in uniform TC⁰ is an upper bound on capability; it places no restriction to commutative operations, and a search for "commutat" in the source returns nothing. Any statement of the containment must also carry the log-precision qualifier, which the drafted sentence dropped.
The "above TC⁰ via NC¹" leg is unsourced, not merely unconditional. No source fetched in this row mentions NC¹. The background fact is that TC⁰ ⊆ NC¹ and that whether the inclusion is strict is an open problem in circuit complexity. No capability separation is claimed on this page.
"SDPA cannot express a non-commuting recurrence at all" is struck. It was written into this project's own dispatch notes, and a source fetched in the same row refutes it — Liu et al. (2022) §4 report
positive results (> 99% in-distribution accuracy) for every finite-state semiautomaton we considered
including semiautomata generating the non-solvable groups A₅ and S₅. Transformers do simulate non-commuting recurrences in distribution. The failure they exhibit is out of length, and that — not a complexity separation — is the unconditional, measurable claim this phase tests.
Liu et al. is more positive than the drafted sentence implied. Its primary result is that shortcuts are learnable; the out-of-length brittleness is a documented secondary limitation with proposed mitigations, not the paper's headline.
Delétang et al. (2022) does not support the prediction it was cited for. It measures training at length 40 and testing at 41–500, not L against 4L; it reports categorical failure near chance rather than a 0.2 absolute drop; it attributes failure to positional encodings going out of distribution, an architectural cause, not to shortcut learning; and it contains counter-examples of strong transformer generalization on permutation-invariant tasks. The paper does not use the word "shortcut".
3. Fetch ledger#
| source | reached | verdict on the drafted sentence |
|---|---|---|
| Coecke–Sadrzadeh–Clark 2010 | yes, §4.1 | binding is theirs |
| Baroni–Zamparelli 2010 | yes, PDF | adjectives only; verb claim misattributed |
| Grefenstette et al. 2013 | named | the actual source of verbs-as-tensors |
| Liu et al. 2022 | yes | supports, but tone-inverted in the draft |
| Delétang et al. 2022 | yes | contradicts the L-to-4L prediction |
| Merrill–Sabharwal 2023 | arXiv only | contradicts the commutativity inference |
| Barrington 1989; Krohn–Rhodes 1965 | already in tree, DOIs verified | carried |
| DeltaNet 2406.06484 §2.2; GLA 2312.06635 §4.1 | verified 2026-09-20 | carried forward, not refetched |
Not reached, and nothing may be cited to them. The Merrill–Sabharwal TACL version (ACL server error — the arXiv version covers the content, but TACL page numbers do not exist in this row). Mamba-2 (2405.21060) was never fetched at all; the per-token-decay form quoted for it came from a different paper describing it, a secondary standing in for a primary. Four quotes attributed to arXiv:2210.02671 are lane-only and unverified. The Pāṇini kāraka row is auth-gated and rests on nothing, so the claim that kāraka is the older form of thematic roles is not made on this page.
4. What this phase measures#
Scoped to what survives §1 and §2:
- Order. Vectors add and addition commutes, so "A then B" and "B then A" coincide to machine epsilon. Operators in different planes do not. The question is how much of that untrained separation survives training, since a trained vector model can allocate subspaces, use position, and memorise.
- Length. A scan of operators composes exactly at any length because composition is associative. That is arithmetic and needs no open conjecture. It is the phase's one unconditional claim.
- Forgetting. Two gate primitives and only two: invertible gates (rotations, permutations), which a prefix scan separates by composing with an inverse; and projectors, which are idempotent, rank-deficient and have no inverse. The intended claim was that an invertible-only family cannot forget, and that the separation is structural rather than a matter of optimisation. It is not, and §6 records why — an invertible near-projector solves the task exactly, and the separating property is contraction rather than singularity.
- Decidability. Restricted to a finite semigroup, "does it ever reach this state" is answered by eigenvalue-1 multiplicity and a resolvent residue rather than left undecidable. The claim is that finiteness escapes Rice's preconditions, not that Rice is wrong.
Every model-scoring row carries the standing precondition: a bed whose null scores nothing is void before the model runs.
5. Three beds, three floors, and what that cost#
Nothing in §4 has been measured yet, because every bed built to measure it failed first — and all three failed the same way. The floor each used was chosen for being easy to compute rather than for being the thing a passing arm must beat.
A kernel-carry bed was floored against chance. Its floor was position-blind and sat at chance, 0.125. But a model that sees the context and cannot bind scores 0.3460 on that generator, by guessing uniformly among the values present. The arms read 0.401–0.414, so they cleared the real floor by 0.055–0.068 and recovered about 10% of the 0.346→1.0 headroom. The reported 0.125→0.41 separation was almost entirely "this model mixes tokens at all". Tripling the training budget moved nothing outside one seed spread, and a two-layer variant landed in the same place: the plateau was the task, not the budget.
The Krohn–Rhodes bed shipped no null at all. Its floors_and_null()
contains no shuffle, and its own structured return reports null_scores=false.
Under the standing precondition that is a halt, so the bed is void before any
model runs.
The planted-operator text bed cannot see order, and the reason is geometric.
| baseline | L2 error |
|---|---|
| bag-of-context | 4.9938 |
| last-operator-only | 5.0922 |
| order-free composition | 5.5367 |
| chance | 5.7160 |
Applying the correct operators in the wrong order scores worse than a bag of context and within 3.1% of chance. The planted operators are norm-preserving SO(8) rotations, so a permuted composition lands uniformly on the sphere however wrong the permutation is; the true state lies on that same sphere; and the expected L2 distance between two uniform points on a sphere is a constant. Order-free therefore scores at chance by geometry, not by failing to know the order. Bag-of-context wins only because averaging shrinks a vector toward the origin, a norm artefact unrelated to composition.
Both trained arms on that bed scored worse than chance — 6.3206 and 6.7240 against 5.7160 — which is the signature of a metric that cannot see the property under test, not a result about binding.
Consequence for the kill list. The rule "(a) ties (d) on this bed → operator binding is dead" presupposes the bed measures something. It does not, so that kill does not fire, and no number from those twins carries information about binding either way.
What order-sensitivity does have. The Barrington bed is certified: its oracle was rebuilt by independent array-index permutation composition over 2,000 pairs, both matrix conventions confirmed as exact homomorphisms, and a BFS closure over the two generators reaches exactly 120 elements, so the scan has real depth rather than being a lookup. A scan of 5×5 permutation operators is bitwise exact at length 10,000 in float64, float32 and bfloat16 — every surviving entry of a permutation product is exactly 0 or 1, so no dtype has anything to round. Of the C(120,2) = 7,140 unordered generator pairs, 6,780 do not commute.
S₅ is non-abelian and its word problem is order-defined by construction, so order is demonstrable there. What remains open is whether a text-shaped bed can ever show it, which is a question about the corpus rather than about the operator.
6. Three claims that did not survive their own audit#
The gate class is keyed on the wrong quantity. The intended division is
group (invertible) against reset (projector, rank-deficient, det = 0). An
invertible near-projector R_ε = (1−ε)P + εI solves parity-since-reset with
exactly zero label error for every ε < 1/2, measured down to ε = 1e−15
(det = 1e−15) at length 50,000 with about 5,000 resets. The worst state
entering a reset is (0,1); R_ε sends it to (1−ε, ε); an argmax decode is
correct whenever 1−ε > ε, and the residual is re-multiplied by ε at every reset
so it never accumulates. Meanwhile an orthogonal substitute with |det| = 1
stays at 0.4150 error.
So det = 0 is sufficient but not necessary, and the separating property is
contraction rather than singularity. A load-time gate check that rejects on
"non-idempotent and non-invertible" would admit the wrong objects and reject the
right ones. The earlier structural claim came from measuring an L2 norm on
isolated prefixes and never decoding a task label.
The line gate is a length filter, not a noise filter. It does separate real English from random letters — the fraction of random samples above the threshold is 0.0000 at every length from 50 to 150,000 characters, against 1.63–1.72 for real English. But its reject set is monotone in line length and nothing else: 41.72% of lines under 40 letters are rejected, 4.71% at 80–100, 0.12% at 150–250, and 0.00% above 250. Median reject 51 letters, median keep 126.
The index of coincidence is a function of the letter multiset alone, so the gate is blind to word order, spelling and grammar by construction. Shuffling the letters within each of the first 1,000 lines produces 75 rejects against 75 for the unshuffled lines — identical, not approximately. A control of real corpus words in random order passes the gate 93.6% of the time. A filter whose stated job is to reject noise admits word salad nine times in ten and rejects two in five genuine short sentences.
Separately, the two published index-of-coincidence figures, 1.46 and 1.73, are the same convention — strip to 26 letters, fold case, normalise by 26 — at two sample lengths, and that is the only convention whose random baseline is 1.00. Three alternative conventions were tested on two corpora and all read higher, never lower, so "a different convention reads lower" is ruled out.
A recall bar that measures nothing. Verb-centre recall against a parser oracle reads 1.0000 with a 95% Wilson interval of [0.9922, 1.0] at n = 489, because the system under test and the oracle are the same parser call on the same string, run twice and compared. It is an identity wearing error bars. The replacement is measured and available: agreement against an independent part-of-speech tagger that never sees the parser's output reads 0.8262, 95% CI [0.7901, 0.8572], and a threshold belongs on that distribution.
7. What the Lean corpus gained#
Three results landed in lean/CEQ/V16Domain.lean, and the accounting is stated
rather than implied: the section adds 19 declarations and 3 citable new
facts, and two of the three named targets carry none of their own.
A scalar gate composed over a finite ordered list is invariant under permuting
that list. This is Mathlib's List.Perm.prod_eq specialised to a commutative
monoid; it carries no new mathematics and is present only as the contrast term.
A matrix gate is not, witnessed concretely by two nilpotent 2×2 matrices whose products differ at entry (0,0). That is the pair's only new fact, and the pair is the claim — commutativity alone is a library lemma, while commutativity set against non-commutativity is the architectural statement.
Reachability in a finite semigroup is decidable, and now as a def rather
than a theorem, so the Decidable instance carries executable code. The
distinction is not cosmetic: declared as a theorem on a Type-valued goal it
proved a decision procedure exists while being unable to run one. The two
evaluations print true and false during both lake env lean and
lake build, and an instance backed by classical choice would not reduce to
literals at all. Finiteness is load-bearing — the reachability predicate carries
no Fintype binder of its own, and the pigeonhole step is the single place the
cardinality is consumed.
The List-shaped commutation statements are declared as separate combinatorial
facts that do not connect to pathProd, and the reason is worth recording:
pathProd is a Finset.prod, and Finset.prod requires a CommMonoid by
definition. It cannot typecheck over a non-commuting codomain at all, and
permutation-invariance over a Finset is vacuous because there is nothing to
permute. The operator as currently defined in the corpus therefore cannot express
a matrix gate; that is a fact about the definition, not about the mathematics.
8. H.1 — the row that measured something#
A bed rebuilt to demand order finally separated the arms, and the separation survived a matched budget. Discrete-label accuracy over eight fixed anchors, 5 seeds, same instances for every arm.
| arm | params | accuracy | std | headroom over tightest floor |
|---|---|---|---|---|
| (a) vectors + softmax + RoPE | 620 | 0.2238 | 0.0115 | −0.1328 |
(b) scalar gate m·e^{iθ} |
620 | 0.2628 | 0.0148 | −0.0938 |
| (c) operator gate, invertible | 552 | 0.9570 | 0.0048 | +0.6004 |
| (d) operator gate + projector | 654 | 0.9470 | 0.0059 | +0.5904 |
Tightest floor is last_two_ops = 0.3566 on that eval split. Every figure in this table is
produced by the chain in tests/foreman/h1/, which runs with the six commands
listed in its README; until that directory existed these numbers appeared in
this page and nowhere else, and a reader could not re-run them. (c) − (a) = 0.7332
against a pooled per-seed standard deviation of 0.01250 — 59 σ, with every
(c) seed beating every (a) seed by at least 0.708. Complete separation at n = 5.
The arms share instances, not merely a distribution. Both lanes use the same
population offsets, and the per-seed floors recomputed independently in each lane
agree bitwise — 0.3900 / 0.3600 / 0.3190 / 0.3640 / 0.3500. That agreement is
what proves the split is identical. The parameter residue (620 against 552 and
654) favours the losing arms, so it cannot explain the gap.
Ratios are taken on excess over the majority-label baseline of 0.1640, the only oracle-free floor: (c)/(a) = 13.3×, (c)/(b) = 8.0×. Excess over the tightest floor admits no ratio at all, because (c) is +0.6004 and (a) is −0.1328 — opposite signs. That is the honest phrasing, and it is not the same as saying the vector side has no signal.
The 653× figure recorded for the untrained comparison must never appear beside
these. It is in consequence error, an L2 distance; these are accuracies. And that
figure was inflated in any case: its vector leg computed a constant mean
displacement over zero-mean Gaussian input, which is approximately zero for a
rotation, so 4.0717 is √(2d) and the leg is the always-identity null. It
compared a least-squares fit against a do-nothing predictor.
What the gap is actually attributable to#
The four arms do not differ only in the composition primitive. (a) and (b) are a single attention-pooling layer with no running state; (c) and (d) are a three-step recurrence. Matching the budget did not touch that, and an earlier draft of this row claimed otherwise.
That confound was measured rather than conceded. A vector arm was given the identical structural gift — same three-step recurrence, same additive injection, same readout — with the per-verb matrix replaced by a per-verb diagonal gate, which commutes by construction and therefore cannot represent order. A live parameter search put it at 554, a tighter match to (c)'s 552 than the 620 the (a)/(b) lane could reach. Same bed, offsets, seeds, split and optimiser.
It scores 0.3080 at 400 steps and 0.3342 ± 0.0161 at 4000.
So the recurrence, handed free to a vector arm at matched parameters, buys +0.1104 — 15.1% of the 0.7332 gap — and still lands below the tightest floor. The remaining 84.9% requires the non-commutative matrix.
The limit this bed cannot remove#
The operator arms' forward pass is state = ops[verb] @ (state + emb[patient]),
which is character-for-character the bed's own generative recursion. So 0.9570
measures that the generator's own form fits its own generator. What this bed
cannot separate is non-commutative operator from the generator's exact operator
family, because on this bed they are the same thing by construction.
The confound was removed, and the result survived it#
The limit above was tested rather than left standing. The same five arms — the four above plus the commuting-diagonal control — were re-raced on S5's certified 120-state automaton presented only as integer symbol ids: no permutation matrices, no transition table, no group structure reaches any arm, so no arm's forward pass can be the generator by construction. Krohn–Rhodes guarantees the target is reachable at some dimension; nothing tells an arm the parameterisation.
The kill was pre-registered before the numbers: if the operator arm does not beat the diagonal control at matched parameters on a generator it was never handed, the 84.9% attribution was the arm being the generator. It did not fire.
| arm | params | mean | std |
|---|---|---|---|
| a vectors + softmax | 460 | 0.2285 | 0.0176 |
| b scalar gate | 460 | 0.2305 | 0.0190 |
| c operator, invertible | 404 | 0.8620 | 0.0556 |
| d operator + projector | 398 | 0.6975 | 0.1034 |
| DIAG commuting control | 404 | 0.2860 | 0.0150 |
An independent re-run from a clean directory returned c = 0.8817 ± 0.0869
against DIAG = 0.2803 ± 0.0104, with (c) ahead on 5 of 5 seeds pairwise and
no overlap — worst case 0.7300 against 0.2937.
The floor that binds a commuting arm is not the one the row shipped. A commuting
gate sees only the multiset, which for a length-14 binary word is the count of
symbol 1 — fifteen values. That ceiling measures 0.2898 fitted honestly and
0.3110 as an oracle fit on the eval split itself. DIAG scored 0.2803, within
0.031 of its absolute upper bound, so the control is saturated rather than
undertrained and the separation is not a training artefact. Against the real
tightest floor of 0.3110, the operator arm's worst seed is still 2.35×.
Arms (a) and (b) sit at 0.23, below that ceiling, so they are underfit
against a baseline this row did not measure; their scores bound the vector side
from above rather than measuring it. One parameter budget and one word length
were swept, with no length-generalisation check.
Producer and full caveats: tests/foreman/iaut/, one command.
An earlier draft of this row called the floors "trivial closed-form" baselines.
That is wrong and understates them: last_op_only, last_two_ops, the
order-free replay and the bag predictor are all handed the true entities and the
true operators. They are oracle-assisted, which makes clearing them harder rather
than easier.
9. H.1 — what the census extension turned out to be#
The stability zone reproduces exactly, and it says nothing checkable about the head.
The zone. The recursion c ← e^{ac} has fixed point c = −W₀(−a)/a and is
attracting exactly when |W₀(−a)| < 1. The boundary of the attracting region in
the parameter plane is the analytic curve a(t) = −e^{it}·exp(e^{it}), with
|a(t)| = exp(cos t) and arg a(t) = t + sin t + π. Its ray radii are
| φ | 0 | π/4 | π/2 | 3π/4 | π |
|---|---|---|---|---|---|
| r(φ) | 0.3678794411714423 | 1.214461 | 1.961309 | 2.513997 | 2.718281828459045 |
with both endpoints exact to machine precision at 1/e and e. An earlier
empirical ray scan read 0.3669 and 2.7178; the gaps of 0.000979 and
0.000482 are both at or under that scan's own grid step of 0.0009970, so the
discrepancy is resolution and not a convention error. The substitution z = a·c
turns the recursion into z ← a·e^z with a maximum floating-point difference of
exactly 0.0 over twenty trajectories, so the census map is the exponential
family, bit for bit.
Dimension 1. Not fractal. A prediction of a non-integer box dimension was
withdrawn before measurement because it contradicts a theorem: the Julia set of
λe^z has Hausdorff dimension 2 for every λ. That set lives in the dynamical
plane; the census lives in the parameter plane, and the two had been conflated.
Basin boundaries of periodic attractors can indeed have dimension between 1 and 2,
which is a different object again.
One stated behaviour of the zone is false off the real axis. "Converges at 0.95r and diverges at 1.05r on every ray" holds only at φ = 0. At π/4, π/2, 3π/4 and π the fixed point is locally unstable at 1.05r exactly as theory predicts, with multiplier ≈ 1.0245 on every ray — but the orbit is then captured by a bounded attracting cycle of period 7, 16, 15 and 2 respectively, verified to 200,000 iterations without escape. Any test asserting literal divergence at 1.05r will misfire on non-real phases.
And the reduction to the head fails, established twice by different routes.
The first route found there is no quantity to reduce. Under the gate route the
modulus row is bitwise phase-independent — the gate phase reaches the complex gate
and never the real score, which extends an earlier finding from a constant phase
to a positional ramp. Under the logit route the row does move with phase, but the
whole operator output is then exactly real, with maximum imaginary part 0.0
across all twenty cells, so no complex quantity exists there at all. And the
normalizer is computed before β is ever applied, so the (1−β) factor in the
mapping has no counterpart in the per-position dynamics. By Cauchy–Schwarz the
normalizer is bounded by (i+1)·e^{w_max} — linear in position, never
exponential in itself.
The second route measured the closest candidate directly. The row-sum tracks
c* = −W₀(−a)/a within 5% only at a = 0.10, and misses by +7.6% to +8.8% at
0.20, +19.6% to +22.4% at 0.30, and +37.6% to +49.1% at 0.36 — 15 of 20 cells
fail. The cause is derived rather than observed: a single static application of the
operator has no self-referential feedback loop, so its row-sum tracks e^a,
smooth and singularity-free, while c* has an essential branch point at a = 1/e
only because the census recursion feeds c back into itself. The two agree to
first order in a — both approximately 1 + a — which is exactly why the error is
small at 0.10 and diverges toward the fold.
So the Lambert bifurcation is exactly right about the scalar map and unconnected
to this operator. What survives is the real-axis statement: the fold line restated
as |W(−a)| < 1 is exact and is a genuine sharpening, whether or not the complex
part reaches anything.
10. H.1 — what the transformer autopsy returned#
Identification first. "JEV" does not resolve to a JEPA derivative or to a decision-making module. It resolves to a separate commercial product launched in September 2026: a non-autoregressive, schema-constrained classifier emitting one structured output and a calibrated confidence per query. It contains no state-transition model, no reward, no rollout and no value function, so it is not prior art for the resolvent read, and it should not be named beside JEPA.
Nothing in it is liftable here. Its schema-constrained decoding solves invalid tool-call output, a problem this project does not have, and is a weaker relative of the exact-zero gate already killed as a masking trick.
But it names the sharpest absence in the project. Calibration as an
objective — training predicted confidence so that P(correct | confidence = c) ≈ c
against the real hit rate — is a property this project has no mechanism to produce
and has never checked. Exactness and calibration are orthogonal: being bit-identical
to a reference says nothing about whether a stated confidence tracks an empirical
rate. Only the former is covered anywhere here. It is not attachable today, because
there is no labelled decision stream to train against, and that is precisely why it
is worth recording.
JEPA, organ by organ. One organ is already resident: the EMA target encoder
with a stop-gradient is implemented and tested in ceqjepa/pi_jepa.py, at
tau = 0.99, and costs nothing to adopt because it is adopted. The spatial
multi-block masking is not applicable — it solves occlusion of 2D and 3D regions,
while these beds are 1D sequences whose context/target split is a temporal horizon
shift.
Prediction in representation space is a gap for the kernel, and this
repository's own code is the proof: pi_jepa.py already implements that objective
around a plain three-layer MLP encoder, never around the operator, which appears
only as the predictor. A kernel cannot contain an objective, and no setting of
β, g or qk reaches it.
One correction to this project's own objection. The anisotropy argument — that clustering is anisotropy, so an effective-rank read prefers a frozen-random encoder at every width — was aimed at JEPA's collapse machinery. Nothing in the fetched papers shows them validating non-collapse by a bare post-hoc geometric read, so the objection targets a validation method they may not use. The argument stands against the instrument this project built; its reach into that literature is unestablished.
The encoder route is probably not where this belongs#
pi_jepa.py's encoder is a pointwise MLP with no sequence mixing at all,
despite its own docstring describing otherwise. A prior round already built the
repair and benchmarked it — a causal attention block lifts R² from −0.19 to
0.986 on this project's own bar — but it uses stock scaled dot-product attention
and never the operator.
Two structural obstacles stand between the operator and an encoder slot. Its causal mask is not a setting but a structure: no bidirectional path exists, and a JEPA encoder is bidirectional. And its positional mechanism is content-conditioned rather than index-conditioned, so it is unverified to break permutation symmetry the way a sinusoid does.
The sceptical finding is the one that matters, and it is stated rather than buried: none of the three surviving certificates — exact refusal, decidable NEVER, a certified resolvent — is a property a representation-learning loss or a downstream probe can query. If that holds, the encoder slot is not where this contribution belongs, and knowing that is worth more than an interface nobody would use.
11. H.1 — three repairs, one of which is progress#
Three rows struck earlier in the phase were rebuilt. One became a real row; two became struck rows with better prose, and are recorded as that.
The Krohn–Rhodes bed is alive#
It had shipped no null at all. It now carries two, both scored on the task's own argmax label metric rather than on finiteness, at n = 2000 per seed:
| null | per-seed error | correct construction |
|---|---|---|
| shuffled operator assignment | 0.5045 / 0.4970 / 0.5175 / 0.5170 / 0.5160 | 0.0000 |
orthogonal reset substitute, det = 1 |
0.5445 / 0.5465 / 0.5445 / 0.5420 / 0.4645 | 0.0000 |
The second null is the one that matters, because the property under test turned
out to be contraction rather than singularity, and a near-singular invertible
matrix would pass the task and prove nothing. To rule out a lucky draw the entire
2×2 orthogonal family was swept — 721 angles, both determinant signs, singular
values exactly [1, 1] — and the best member still errs between 0.393 and
0.544 on every seed.
The contraction reading is confirmed from the other side too: R_ε = (1−ε)P + εI
scores exactly zero label error at every ε tested, from 0.4999 down to
1e−15. Against the same sequences, an orthogonal substitute errs 0.4175–0.572
while the contracting one errs 0.0.
One sentence of this project's own correction was itself incomplete. The failing pairs under the pseudo-inverse prefix form require both indices past the projector, not one — a pseudo-inverse cannot undo a rank-one projection at either end.
The line gate is still a length filter#
The replacement is a two-level model — a character trigram and a word bigram, each with its own placebo — and it passes both placebo tests that the original failed. Letter-shuffle passes the character level 0.00% of the time; word-salad passes the word level 2.40%, both under the 5% void line.
It also reproduces the expected division of labour at real corpus size: the character level kills letter-shuffle but is 68.5% blind to salad, while the word level kills salad and kills letter-shuffle as well through unknown tokens.
But the confound that struck the original is worse, not better:
| line length | rejected |
|---|---|
| under 40 chars | 60.0% |
| 40–100 | 18.78% |
| 100–250 | 2.7% |
| over 250 | 0.0% |
A spread of 0.60 against the struck gate's 0.4172. The mechanism is genuinely different — this gate is order-sensitive where an index of coincidence is a letter-multiset statistic — but the symptom is identical and larger. It is a better-behaved length filter. The within-bucket control that would settle whether anything else is being measured was not run.
The recall bar clears its floors and still measures the wrong thing#
Cross-tagger agreement reproduces exactly: 0.8262 (404/489), Wilson 95% interval [0.7901, 0.8572], between a dependency parser's root verb and an independent part-of-speech tagger's first verb-tagged token.
It clears four no-parsing floors without interval overlap — the strongest being a
first -ing/-ed heuristic at 0.3579 [0.3166, 0.4013], then longest word at
0.0716, first alphabetic token at 0.0102, sentence-final token at 0.0041. The
system beats the strongest by 0.4683, and 0.4013 < 0.7901.
The old bar of ≥ 0.95 is unreachable against an independent tagger and was only ever met by the tautology it replaced. A defensible bar is the measurement's own lower bound, 0.7901.
What it cannot do is the thing a bar is for. Two imperfect taggers disagree on 17.4% of lines when nothing is broken at all, driven by participles used as adjectives, copulas and fronted adjuncts. So a below-bar reading cannot separate an extraction bug from ordinary tagger ambiguity: it measures concordance, not correctness. It is an honest number and a weak bar.
12. H.1 — two certificates, and the end of the exactness claim#
Both certificates pass, and neither is a result. They are theorems with checks, and the page says so rather than counting them.
The dilation identity holds with four orders of slack. Across r ∈ {2, 4, 8} at
five seeds, with σ = 0 and σ = 1 included deliberately rather than left to a
random draw, the doubled-space operator is orthogonal to a maximum of
2.220446049250313e-16 against a bar of 1e−12, its top-left block equals Σ
bitwise in every run, and the reconstruction error is 0.0 exactly.
Two things the hunt for a failure turned up. An earlier draft claimed the
angle route and the direct-product route diverge at σ = 1; they diverge at
σ = 0, because arccos(0) = π/2 and cos(π/2) = 6.123233995736766e-17, not
zero — σ = 1 round-trips exactly. The implementation this repository ships is the
direct-product route, which never calls either function and is therefore exact at
both corners. And the naive √(1−σ²) loses against the stable
√((1−σ)(1+σ)) by a relative 2.756e−10 at σ = 0.99999999, but orthogonality
never degrades past 1.1e−16 there: the precision loss lives in a scalar erasure
formula, not in the defining identity.
The attribution is Halmos 1950 for the block form and Sz.-Nagy 1953 for power dilations. Stinespring 1955 concerns completely positive maps on C*-algebras and is not the citation. The parallel with a groups-and-flip-flops decomposition over finite semigroups is a resemblance between different mathematical settings and is not claimed as a theorem.
And the last named structural difference is dead on the bed built to show it.
The resolvent read is the successor representation applied to a reward, which is
prior art, and it agrees with the temporal-difference fixed point to
8.882e−15 at worst across five seeds against a 1e−12 bar, with successor-
representation row sums exactly 10.000000.
The only difference this project still claimed for it was exactness at absorbing states — a gate reaching exactly zero where a sigmoid only approaches it. On a six-state chain with one absorbing state:
| route | value at the absorbing state |
|---|---|
| resolvent read | 0.0 exact |
| TD(0) after 20,000 sweeps | 2.470e−323 |
| sigmoid gate after 20,000 steps | 0.03817905137506084 |
The gap between the exact read and TD(0) is a subnormal float64 value. No tolerance anyone would set can register it.
The sigmoid's 0.038 is not a rescue, and the measurement says why: it comes from
gradient starvation as the argument approaches zero, which is a fact about
gradient descent on a sigmoid rather than about value estimation. It is not the
mechanism being contrasted.
So exactness at absorbing states buys nothing measurable against the method it was supposed to beat. That is consistent with this project's own earlier concession that the closed-versus-open distinction is a parameterisation default rather than a novelty; it is now measured rather than conceded.
Limits#
No result tables appear on this page yet; the rows that would fill them were running when it was written, and it exists so that the prior-art concession is fixed before any number can tempt an adjustment to it. The unreached sources in §3 are unreached, and their absence is not evidence either way. The reachability claim that §1 leaves standing carries this project's own existing concession that the exact zero is proved for a constructed gate and remains untested for a trained one.