Prompt-Conditioned Syndrome Dynamics

Status: architecture hypothesis frozen for a minimal matched falsifier.

1. Capability thesis

Shohin's recurring failure is not the absence of local operations. Across DRS, typed-controller, sticky-schedule, integrated-workspace, and product-training experiments, local binding or one-step competence often appears before exact multi-step composition. The failure signature is accumulated state drift: individually plausible edits combine into a globally inconsistent latent state.

Prompt-Conditioned Syndrome Dynamics (PCSD) treats the reasoning state as a problem-specific error-correcting code. The source compiles a sticky set of latent parity checks once. Every recurrent proposal is allowed to change the state only after a differentiable syndrome projection restores those checks. The checks are model-owned, receive no target/query/answer, and remain fixed for the full trajectory.

The intended inductive bias is:

Reasoning steps may change facts, but valid steps preserve task-specific invariants. Enforcing those invariants at every latent commit should prevent locally plausible operations from drifting into an incoherent composition.

This is a falsifiable mechanism claim, not a claim that error correction, recurrence, latent reasoning, or neural code decoding is new. A 2025 recursive latent-reasoning system already combines recurrence, discrete anchoring, intermediate supervision, and random-corruption self-correction. PCSD differs in the proposed object and operation: it compiles a source-conditioned linear constraint geometry and performs an explicit minimum-norm projection after every sparse transaction. A formal novelty claim remains contingent on a broader literature review and a positive matched result.

2. Mechanism

Let the workspace at step t be Z_t in R^(S x D). A prompt encoder produces source features X. A parity compiler emits C normalized check factors:

A(X) in R^(C x S),  B in R^(C x D)
P_c = A_c outer B_c

The sticky reference syndrome is:

r_c = <P_c, Z_0>

A tied recurrent proposer emits a raw update U_t and sparse commit gates g_t. The uncorrected state is:

Z'_t = Z_t + g_t * U_t
e_t  = P Z'_t - r

With G = P P^T + epsilon I, PCSD applies the minimum-norm correction:

Z_(t+1) = Z'_t - P^T G^-1 e_t

Therefore P Z_(t+1) approximately r at every step. The solve is over the small C x C Gram matrix, is differentiable, and uses no host reasoning. The answer head reads only the final corrected state after adaptive halt.

The architecture logs pre/post syndrome norm, correction energy, gate sparsity, Gram conditioning, and halt depth. A zero-syndrome solution with no task gain is not success.

3. What is held constant

All arms use the same:

• tokenizer/backbone or synthetic source encoder;
• workspace slots, width, recurrent steps, late readout, and answer loss;
• train/development/confirmation examples and order;
• optimizer, charged tokens, update count, and random seeds;
• source access and adaptive halt contract;
• parameter and measured training-FLOP budget within 1%;
• model-owned inference with no external host, tool, verifier, or search.

Arms:

1. PCSD: prompt-conditioned sticky checks plus syndrome projection.
2. DENSE: parameter-matched untied dense correction after each proposal.
3. RECURRENT: strongest tied recurrent workspace without correction, padded with live capacity if needed to match parameters/FLOPs.
4. PCSD-SHUFFLED: check geometry from a different prompt in the minibatch.
5. PCSD-ZERO: correction disabled at evaluation with all weights unchanged.

4. Fast discriminating benchmark

The first gate is a generated Conservation-Shift Suite, not a public benchmark. It contains several renderer-randomized register systems whose commands are sparse but whose valid trajectories preserve source-defined affine invariants. The model receives only the rendered world, command stream, and late query.

Training uses command depths 2--8. Development uses new values/renderers at depths 2--8. Confirmation is generated and sealed before training and tests:

• depth 16 and 32;
• unseen entity permutations and group sizes;
• unseen compositions of individually trained operations;
• a one-step latent perturbation at a hidden random trajectory position.

Metrics are exact terminal state, exact queried answer, invariant violation, corruption recovery, and accuracy by depth. Architecture selection never reads the confirmation labels.

5. Pass and kill rules

PCSD passes the minimal gate only if all three fixed seeds satisfy:

1. depth-16 exact answer is at least 70% and depth-32 at least 50%;
2. mean confirmation exact answer is at least 15 absolute points above both DENSE and RECURRENT under matched budgets;
3. latent-perturbation recovery is at least 25 points above both controls;
4. PCSD-ZERO and PCSD-SHUFFLED each lose at least 10 points;
5. post-projection syndrome is at least 100x smaller than pre-projection syndrome without a degenerate zero-update policy;
6. parameter mismatch and measured training-FLOP mismatch are each <=1%.

Kill PCSD after this one cohort if it misses the matched-control margin, if the controls tie it, if the ablations do not remove the gain, or if improvement is restricted to invariant violation without exact-answer transfer. A clean pass advances to language-backed arithmetic/logic and then a scratch small-model integration; it does not by itself establish general reasoning.

6. Immediate implementation sequence

1. Implement the standalone sticky check compiler and minimum-norm projector.
2. Unit-test exact syndrome reduction, gradients, source-only ownership, shuffled checks, and deterministic replay.
3. Implement the Conservation-Shift generator and seal confirmation hashes.
4. Implement the shared trainer with exact parameter/FLOP receipts.
5. Run one smoke seed, then the fixed three-seed matched cohort only if finite.

The frozen Qwen-hosted product route remains a practical control. It is not an arm in this architectural claim and supplies no inference-time component.

7. Result: closed negative

The one-seed full decision used seed 31, data seed 20260806, 4,000 updates, 1,024 examples per update, identical depth-2--8 data, and the frozen 4,096-row development cohorts. PCSD had 68,537 trainable parameters and DENSE had 68,445, a 0.134% mismatch.

PCSD reduced the learned syndrome from roughly 10 RMS to roughly 0.001 RMS and reached 100% decoded invariant validity. That conservation did not produce terminal-state composition. At depth 12, disabling projection loses only 0.976 answer points and shuffled checks lose only 0.659 points, both far below the required 10-point causal ablation. PCSD is also 3.36x slower in measured training throughput.

This seed already violates the required per-seed matched-control gain, exact- state requirement, and ablation effects. Since every seed had to pass, more seeds cannot rescue the preregistered gate. Confirmation depths 16/32 remain unopened. PCSD is closed as a standalone reasoning architecture. Its projector may be reconsidered only as an optional within-particle stabilizer after an independent architecture establishes useful whole-hypothesis reasoning.

Report SHA-256 values:

• PCSD: aee38ad1b52e0c3361d4f0c18d028769b05a72e5746a61f6701c447215b8e1f3
• DENSE: 37360c6ed61dc989c05252b5e53937668be7d19e3efe2e8719db709402927768