BED-H — the belief bed. Pre-registered 2026-09-14, before any code exists.#

Hidden intents, observed statements, exact labels. This document is written before the bed is built and before any arm is run, so that the prediction and the counter-prediction below cannot move afterwards. Every later result on this bed is scored against this page.

Why this bed exists#

Three independent measurements on 2026-09-14 killed the same thing from three directions, and none of them was a bug:

The committor at ply 15 is a function of the ply-15 position, so the walk is Markov and the prefix is conditionally independent of the label. Aggregation costs 0.0810 at the encoder output. Moving the label to ply 19 does not rescue it.
K+Q-vs-K is a fully observed MDP, so conditioning on a move and forcing it are the same operation and the interventional–observational divergence is identically zero. That is the diagnosis of DCM-1's flat move-permutation ablation, not a fault in it.
CausalEncoder PASSED on an i.i.d. EMA bed and its sign flipped on the real target.

On a fully observed Markov chain the future depends only on the present, so an attention operator over a context has provably nothing to do. Every bed this project owned was Markov in the current state. The method also depends on an exact oracle — the absence of one is what killed DCM-1, whose bed carried negative headroom nobody measured first. BED-H is the first bed that makes history necessary and keeps the label exact.

The bed#

Hidden chain x_t over S = 8 states with a non-symmetric P. Emissions y_t at snr ∈ [0.5, 0.7]. Second eigenvalue λ₂ ∈ [0.90, 0.97], giving a memory of roughly 10–30 steps — deliberately 2–3× the baseline's window, so a windowed reader is beaten by the physics rather than by tuning.

Oracle: the forward algorithm. Exact, O(T·S²).

Labels. b_t, the belief, is intensive — the softmax corner. Cumulative expected hits Σ_τ b_τ·h is extensive — the linear corner. Consequences are absorption probabilities under the belief.

This is the first bed on which the corner rule has a referent. The rule beta = 1 − alpha was retired on 2026-09-14 (docs/CORNER_RULE_RETIREMENT.md) on three grounds, the first of which was that pi_jepa's encoder is position-wise, so its scaling exponent is zero by construction and the mask reached its coordinates by index equality alone. Here the intensive and extensive classes are planted by the generator's physics, not by an integer.

Must-fires — gates, not decoration#

Printed per bed instance, and an instance outside the band is REFUSED:

History must be necessary. Window-L oracle error must decay with L. A bed where it does not is a Markov bed wearing a hidden state, and it cannot reward the thing under test.
The label must be alive. Label spread ≥ 0.1.

The trap#

Never hand the model P.

Weak form: P fixed and unknown, learned implicitly.
Strong form: P drawn per instance and encoded in the prefix — in-context system identification. That is the real target.

L-SIMPLE baselines, in order#

Window-L forward algorithm — the oracle with amnesia, exact inside its window. The arm must beat window-4's 0.16–0.22.
An EMA of one-hot observations.
A GRU with S hidden units — the native primitive. This is the SKYLINE, not a rival: a belief update is a first-order recurrence in belief space, so a recurrence is the right tool and beating it is not the claim.

Prior art — REWRITTEN 2026-09-14 after an audit, before any arm ran#

The section this replaces named two references and conceded the wrong half of one of them. An audit resolved 37 identifiers and found that three of the four things this bed would claim are already occupied, with numbers. The corrections are recorded here rather than in a later card, because the whole point of a pre-registration is that it is fixed before the result.

The substrate is published, twice#

arXiv:2512.22471 — The Bayesian Geometry of Transformer Attention. "Bayesian wind tunnels — controlled environments where the true posterior is known in closed form." Its HMM task draws a fresh Dirichlet HMM per sequence, supervises on p(s_t | o_1:t) with forward-recursion ground truth, and scores architectures: Transformer 0.049, Mamba 0.024 ± .009, LSTM 0.411 ± .003 (marked fail), MLP 0.40 in entropy MAE. That is this bed's strong form, already executed.
arXiv:2605.20824 — Markovian Circuit Tracing for Transformer State Dynamic. Calls itself "a controlled benchmark and evaluation framework": six HMM families × three seeds, exact latent states, transition matrices, Bayesian belief vectors, Bayes-optimal predictions and forced-state counterfactual targets, against four controls.

The concession was inverted#

arXiv:2405.15943 (Shai, Marzen, Teixeira, Gietelink Oldenziel, Riechers, NeurIPS 2024) is real and says what its title says. But it scores no architecture against anything — no baselines, no windowed oracle, no RNN comparison; belief is a probing target recovered by linear regression from residual activations after next-token training. So conceding "the bed does not claim the representation result" gave away ground that paper never held.

And the delta this page kept — "exact oracle labels" — is that paper's own machinery, and is 35 years older than it: Rabiner 1989, 10.1109/5.18626. It is struck from the delta.

Also named, and absent from the original section: arXiv:2502.01954 (Piotrowski, Riechers, Filan, Shai), Constrained belief updates explain geometric structures in transformer representations — attention as the constrained Bayesian update, predicted analytically. That is closer to this project's operator claim than 2405.15943 is.

Frühwirth-Schnatter was miscategorised. Kalman is the continuous twin; Frühwirth-Schnatter (2006) is Markov switching — the same discrete-latent family in a Bayesian idiom, not a continuous analogue.

Occupied elsewhere, with owners#

claim	owner
in-context identification of an HMM	2512.22471, 2506.07298, 2410.16546
⊥ as an output symbol	1907.00208; per-position in a structured output, `garcia18a` ICML 2018
belief entropy as the uncertainty statistic, decomposed per position	1202.6545 (Durand & Guédon, 2012)
the `Σ w·v / (Σ w)^α` family	arXiv:2607.22781, What Softmax Throws Away: Mass-Aware Attention for Evidence Accumulation (Yu & Ha, 24 Jul 2026) — the extensivity argument for attention normalisation, arrived at independently, with a global `p`
softmax normalisation destroys the extensive read	arXiv:2310.08661, with the ablation, 2023
extensive-vs-intensive selects the aggregator	arXiv:2207.13779 — sum for extensive, mean for intensive, chosen by physics

What survives, and it is one sentence#

A protocol that scores a sequence architecture's ⊥ emission against an exactly-known filter posterior, with the abstain threshold anchored at ln S − δ rather than tuned for coverage — on a bed carrying an extensive label alongside the intensive one, against a windowed-forward-algorithm baseline whose headroom is derived from λ₂ in advance.

Every mechanism in that sentence is borrowed and cited. What is new is the evaluation protocol and the maximum-entropy anchor. Nobody scores ⊥ against a filter posterior — four phrasings found nothing. It is a much smaller claim than this page originally made.

The corner rule adds nothing here, and the reason is internal#

Its conjunction — an exponent chosen per output coordinate by dimensional analysis of what that coordinate reads — is genuinely unoccupied: every published exponent is indexed by model, head, layer or query, never by output coordinate. But docs/CORNER_RULE_RETIREMENT.md retired the rule on the same day this bed was pre-registered, on three measured grounds, and closed with "No third reroute is proposed." This bed would be a third reroute. The external gap does not reopen an internally closed question.

Gate correction: must-fire 1 is a theorem, not a gate#

"Window-L oracle error must decay with L" is guaranteed by filter stability — exponential forgetting of the initial condition, Le Gland & Mevel, Math. Control Signals Systems 13:63–93, 2000 — with the rate in this bed's own λ₂ given by arXiv:1710.06078 as B = log(ε)/(λ₂ − λ₁). It fires by construction and discriminates nothing. It is retained as a sanity check on the generator: if it ever fails, the generator is broken, which is the only information it can carry.

It follows that the window-4 band must be derived, not discovered. 0.16–0.22 was written as a bar; it is a function of S, λ₂ and snr fixed at generation time and must be computed before it is measured, with any disagreement reported as the finding.

Skyline correction: the GRU as specified is below a published lower bound#

S hidden units for S = 8 states sits under the Ω(N·|Σ|) bound of arXiv:2310.05161, and arXiv:1805.04908 shows the GRU is strictly weaker than LSTM and ReLU-Elman in finite precision because it cannot count — while the extensive label Σ_τ b_τ·h is a count. The S-unit GRU is therefore a floor, not a skyline; a properly sized recurrent arm and an LSTM must run beside it.

The counter-prediction is itself a published result, and its sign flips#

"The GRU skyline wins outright" is arXiv:2406.04089 (June 2024): transformers consistently underperform RNNs across all tested HMMs — in the fixed-HMM regime. In the strong form this bed names as its real target, 2512.22471 reports the opposite with numbers, LSTM failing at 0.411 against Transformer 0.049. Whichever way this bed lands it replicates one published result and contradicts another, and the axis that decides which is exactly the weak/strong form of the trap.

One more gate, from the audit#

A near-singular P makes the bed unlearnable for every arm — learning HMMs without nonsingularity is at least as hard as noisy parity (cs/0502076). A conditioning gate is required, with its refusal fraction reported beside the other two.

Prediction, on the record#

The author's: the read beats window-4 with a confidence interval, and matches the GRU skyline within resolution at horizon ≥ 16.

The counter, point estimate: the GRU skyline wins outright. A belief update is a first-order recurrence in belief space, so the native primitive is a recurrence and attention only approximates it. On that reading the read's honest claim here is parity-with-skyline plus refusal — not a win.

Both are recorded before the bed exists. If the read wins outright, the counter was wrong and says so; if the GRU wins, the prediction was wrong and the claim drops to parity-plus-refusal. Neither may be revised after the first number.

What would make this bed void#

Stated now so it cannot be decided later: if the window-L error does not decay with L, the bed is Markov and every result on it is uninterpretable. If label spread falls under 0.1, the bed is dead. If any arm is handed P, the trap is broken and the run measures nothing. Any of those three and the instance is refused rather than reported.

Corrections from a theory audit, 2026-09-14 — still before any arm ran#

A citation on this page carried the wrong title. arXiv:2607.22781 was written as "Mass-Aware Attention", which is the mechanism name from the subtitle. The actual title is What Softmax Throws Away: Mass-Aware Attention for Evidence Accumulation. Real identifier, wrong title field — the exact pattern a citation audit exists to catch, committed here by the same session that runs the audits. Corrected above.

arXiv:1612.02526 does not transfer to this bed at all, and the earlier note that it "bounds next-observation prediction, not belief estimation" understated it. Full-text: belief appears once, non-technically; posterior appears only inside the proof of Lemma 1 as a submartingale potential, never as an output. Everything is doubly averaged — over a uniformly random t and over the process — and the authors themselves call Proposition 1 "a kind of negative result—that average error is not a good metric". This bed's label is the belief and its metric is per-position TV. log n/ε transfers to nothing here. For a short-memory claim the citation is Atar & Zeitouni, SICON 35(1):36–55, Corollary 2.1 for the rate, and Ye–Ma–Qian eq. (10) for the window length with its constant C flagged as never estimated by its own authors.

Le Gland & Mevel do not own the Lyapunov-gap characterisation. They attribute it to Atar & Zeitouni in their Introduction, p. 66. They remain the correct citation for forgetting.

~~Their own Remark 2.4 attributes it to Atar & Zeitouni; their contribution is a Birkhoff contraction coefficient under primitivity.~~ SUPERSEDED. This page invented a Remark 2.4 that does not exist in the paper, and attributed a Birkhoff contraction coefficient to authors whose text contains zero occurrences of Birkhoff — it is Atar & Zeitouni who use Birkhoff. Real paper, real attribution, fabricated locator and fabricated contribution: the P-16 shape, committed on the page that files P-16. The attribution itself survives at the corrected location.

The concession to arXiv:2405.15943 can be sharpened. Full-text grep of v3 finds zero occurrences of fractal dimension, box-count, Hausdorff, correlation dimension, Lyapunov or contraction. It visualises the fractal and never measures it; its reported R² values are correlations of pairwise distances, on one generator, one trained configuration, no architecture comparison. Measuring the attractor's information dimension concedes nothing to that paper.

The fractal link is an upper bound with no matching lower bound. dim_H(ν) ≤ h/χ holds unconditionally for place-dependent IFS (Jaroszewska–Rams, arXiv:0707.3532 Thm 1), applied to the Blackwell measure by Bárány–Pollicott–Simon (J. Stat. Phys. 148:393–421, Prop. 14). Every equality theorem assumes constant probabilities; the filter's weights are place-dependent by construction, which is exactly the gap. Measured here: the bound holds 4/4 with 11–45% slack, and is vacuous by 23× in the overlap regime, where the attractor collapses to a point while h/χ = 23.276.

A trap for anyone reproducing this. Box-counting the support gives d_box = 2.57–3.31, which appears to violate h/χ. It does not: the theorem bounds the dimension of the measure, and dim_H(μ) ≤ dim_B(supp μ) always. Only the information dimension D₁ is the comparable quantity. Box-counting the support here produces a false refutation.

One exact identity, worth asserting as a free correctness check. The top Lyapunov exponent of the filter's random product equals minus the entropy rate of the observation process, λ₁ = −h(Y), identically — because the product's norm is the sequence likelihood. Measured residual 1.26e-05 on the weak draw and 1.41e-13 on a closed-form control.

A second correction pass, 2026-09-14 — after the bed ran, before any arm is published#

Three citations on this page were misaddressed, and all three are corrected above or here. They were written by the same session that runs the citation audits, which is the point of recording them.

Le Gland & Mevel "Remark 2.4" does not exist. Corrected in place above.
arXiv:2102.10487 Eq. (15) is the dimension formula, not the normaliser identity. The identity this page needs is the prequel, arXiv:2008.12886 Eq. (16). Cited to the wrong paper in the same series — an off-by-one-paper error, which resolves cleanly and is therefore invisible to an identifier check.
Ye-Ma-Qian eq. (10) gives a slope and no level. Its constant C is never estimated by its own authors, so no absolute window length can be derived from it. The derived-level figures this page carried were produced by multiplying a quenched (geometric) rate onto an annealed (arithmetic) level, and read 2.5-17x too low on every draw. Both derived levels are dropped. The slope is kept and verified: fitted on L >= 4 against the geometric error, quenched slope over exp(Lyapunov gap) reads 1.0073 weak and 1.0142 strong (1.65% / 2.75% mean error), while the annealed form misses by 40.6% / 52.5%. Jensen holds 48/48 in both forms. lambda_2 of P over-predicts the decay factor by 2.1x.

The window-4 band was pre-registered wrong. Measured 0.1423 against the pre-registered 0.16-0.22, with 26.6% of draws inside. The replacement is measured rather than preferred: the admissible region is a diagonal ridge, snr = 1.181 * lambda_2 - 0.575, not an interval in either coordinate alone.

The free identity holds. lambda_1 = -h(Y) verified to 2.2e-04 on the live draw — a correctness check on both the filter and the QR code, costing nothing.

The counter, resolved — 2026-09-14#

The counter holds and the author's delta shrinks to what the counter said it would. A trained GRU-64, given nothing but its own predicted entropy and the same ln S - delta anchor every other arm gets, reaches recall 0.8959 at precision 0.8903 at delta = 0.10, and 0.9101 / 0.9028 at delta = 0.20, on the same 64 draws with delta unchanged. Every untrained consumer reaches recall 0.0000. Inside a planted noise stretch the GRU's mean entropy is 1.9035 against the oracle's 2.0400, while window-4 sits at 1.1206 and EMA at 1.2340.

The refusal channel is therefore real and it is not the read's. The surviving claim is exactly the pre-registered counter's sentence: abstention scored against an exact filter posterior — a metric, not an architecture. No architectural advantage is claimed for the resolvent read on this bed.

The one unborrowed finding: the two metrics come apart in opposite directions. The GRU skyline reads recall 0.90 with selective-risk dominance 0.00 — its entropy tracks what is knowable rather than its own error, so abstaining by it does not lower its risk on what remains. The EMA reads recall 0.00 with dominance 0.86 — it ranks its own errors well while never crossing the absolute anchor. Neither metric alone characterises a refusal channel, which is why both were demanded in advance. A scorer reporting only one of the two would have called each of these arms a success.

The real bar for any arm round is the ordering arm_TV < prior_mean_TV, not a bare magnitude: the prior-mean-P baseline (ceqjepa/beds/bed_h.py:622-635) costs 0.0802 per-position total variation on 32 instances of the strong form, against 0.1118 for a random other draw, so 0.0802 is the floor an arm must clear — not window-4 at 0.1440. No baseline built here reaches it, so in-context identification on this bed is open rather than settled.