# R12 WGRQ CPU Preregistration: Delayed-Witness Edge-Parity Ring **Status:** **CLOSED 2026-07-15 before any fit.** The frozen acquisition was generated and independently replay-audited, but the adversarial implementation audit found protocol-breaking defects described in Section 10. No Stage-A fit, Shohin checkpoint fit, H100 job, language-transfer claim, or change to the protected flagship is authorized from this version. **Claim class:** empirical neural optimization under information-identical frozen oracle transcripts. WGRQ is not a new state object, algorithm, oracle-complexity result, recurrent primitive, or general-reasoning mechanism. ## 1. Exact family For even `n >= 4`, the delayed-witness edge-parity ring `DWEPR_n` has physical state `x in GF(2)^n`, initial state `0^n`, and two reversible events: ``` (R x)_i = x_(i+1 mod n) F(x)_0 = x_0 xor 1, with every other coordinate unchanged. ``` The only query is `READ`, with output ``` O(x) = x_0 xor x_1. ``` There is no coordinate-selecting query. A future continuation must rotate a latent difference to the fixed sensor. ### Residual theorem For two states, let `d=x xor y`. Shared flips cancel from their difference and shared rotations only rotate it. Therefore after a continuation containing `r` rotations, ``` O(T_w(x)) xor O(T_w(y)) = d_r xor d_(r+1). ``` The states are future-equivalent exactly when ``` y=x or y=x xor 1^n. ``` Hence there are `2^(n-1)` residual classes. The canonical quotient is the edge vector `e_i=x_i xor x_(i+1)`, whose even parity leaves exactly `n-1` independent bits. `R` rotates the edge vector, `F` toggles edges `e_(n-1)` and `e_0`, and `READ` returns `e_0`. Every exact source-deleted packet therefore needs at least `n-1` history- dependent bits. The canonical edge representation attains the bound. ### Delayed witnesses For inequivalent states, let `b=e(x) xor e(y)`. Their shortest distinguishing continuation is ``` R^k, where k=min{i : b_i=1}. ``` Because a nonzero even-parity `b` cannot have only its final bit set, the maximum shortest-witness depth is `n-2`, and the bound is attained. `n=3` is the smallest physical system with a nonempty worst-case witness; even scales are used in the board for token-count controls. ## 2. Absolute symbolic gates Before any fit, exhaustive enumeration must verify at `n=3` and `n=6`: - physical transitions and reversibility; - `x~y` iff all determining continuations `R^0...R^(n-2)` agree; - exactly `2^(n-1)` quotient classes of size two; - representative-independent quotient transitions; - every shortest-witness depth and the tight `n-2` case; - no collision or over-splitting in the serialized canonical code. The minimum check count is ``` (n-1) * 2^(2n) + 3 * 2^n. ``` Any mismatch rejects the board before training. Cancellation controls include `F F R^n` (identity), `F R F R^(n-1)` (same counts, nonidentity), `F R` versus `R F`, the observationally null global-complement word `G=(F R)^n`, and the equal-count identity `(F F)^(n/2) R^n`. The generator must balance labels within declared length, event-count, endpoint, and gadget strata wherever mathematically possible and report the unavoidable parity obstruction separately. ## 3. Frozen acquisition Training scales are `n in {4,6,8}` with two source-length bands, at most `2n` and `8n`. There are exactly 3,072 episodes in each of the six scale/length cells, or 18,432 episodes total. Each episode contains four source histories: - two distinct histories from one residual class; - two histories from different residual classes; - the non-equivalent pair is stratified over shortest-witness depths `0...n-2`; - histories and pair roles are generated before model initialization. Every history receives eight frozen continuation/read probes. The bank includes all `n-1` determining rotations, adds the redundant final rotation, and repeats the bank deterministically only when needed to reach eight. Thus every episode contains exactly 32 one-bit ordinary oracle answers. Equivalence labels and the first-distinguishing-witness mask are deterministic functions of those public answers and add no oracle channel. All arms receive byte-identical histories, probes, answers, equivalence labels, witness masks, order, and batching. Training acquisition is exactly 589,824 ordinary one-bit answer calls. No model-dependent mining, target-dependent rejection, reseeding, seed search, equivalence oracle, counterexample oracle, or hidden state ID is allowed. Generation uses ``` SHA256(seed || 0x00 || ASCII(domain) || uint64_be(counter)) ``` with rejection sampling only for unbiased finite-bank selection. The generator, auditor, transcript, report, and hashes are frozen before fitting. ## 4. Matched learner Every neural arm uses exactly: - a 15-bit packet, with only the first `n-1` bits active; - a public 15-bit scale mask; - a two-bit event code; - tied transition MLP `32 -> 64 -> 15`; - readout MLP `30 -> 64 -> 1`; - 5,136 trainable fp32 scalars; - straight-through hard bitpacking after every transition; - no source tokens, cache, per-step parameter, oracle handle, or external execution in the committed packet or reader. For hard bitpacking, probabilities are `sigmoid(logits)`, the forward packet is `1[p>=0.5]`, and the backward value is the standard straight-through estimator. At evaluation, exactly 15 bits are serialized; masked bits must be zero. All arms use AdamW with learning rate `3e-4`, betas `(0.9,0.95)`, epsilon `1e-8`, matrix decay `0.01`, gradient clip `1.0`, batch 64, four epochs, exactly 1,152 updates, 64 warmup updates, then fixed cosine decay. There is no dropout, early stopping, checkpoint selection, score-dependent scheduling, or seed replacement. ## 5. Loss arms All loss tensors are computed eagerly in every arm. Only frozen coefficients differ. Let `A` be answer BCE over every frozen probe. Let `E` be behavioral Jensen-Shannon divergence between the equivalent pair over every shared probe. Let `S_short` be a unit-margin separation hinge on the non-equivalent pair at its first distinguishing probe. `S_uniform` uses a deterministic uniform probe from the same bank. `R=(E+S)/2`. `R_sham` applies the same computation after a deterministic wrong-partner permutation within scale, length, event-count, endpoint, and answer-signature strata. Let `C` be the common mean `p*(1-p)` bit-commitment penalty. Primary arms: ``` WGRQ-shortest: 0.75 A + 0.25 R_short + 0.01 C active-answer-only: 1.00 A + 0.01 C uniform-witness: 0.75 A + 0.25 R_uniform + 0.01 C relation-sham: 0.75 A + 0.25 R_sham + 0.01 C ``` A fifth favorable capacity control receives direct canonical edge-bit targets but uses the identical model, optimizer, updates, and data. It is a privileged ceiling, not an information-matched denominator. Exact symbolic partition refinement is the non-neural ceiling. The allowed positive claim concerns optimization only: the relational objective may bias the same finite recurrent program toward the observable quotient. It cannot claim new target information or a better oracle rate. ## 6. Process-level deletion and confirmation Confirmation is generated only after every final checkpoint hash is frozen. It has three untouched strata with 1,024 committed-history episodes each: - length OOD: `n=8`, source length up to `64n`; - scale OOD: `n=16`, source length up to `8n`; - full OOD: `n=16`, source length up to `64n`, including witness depth `n-2`. Each episode contains four histories and 32 continuation/read branches, exactly 128 ordinary one-bit answers. Total confirmation acquisition is 393,216 calls. The writer receives one source history, serializes exactly 15 bits, and exits. A fresh reader process receives only fixed weights, public scale mask, the 15-bit packet, one continuation, and fixed `READ`. It clones the byte-identical packet for all 32 branches. Source events, source IDs, activations, RNG state, cache, paths, simulator, verifier, and cross-branch memory must be absent. The original packet must remain byte-identical after every branch. `episode_exact=1` only when all normal reads, equivalent-history interchanges, non-equivalent donor reads at selected witnesses, process-deletion checks, masked-bit checks, and packet-reuse checks pass. Individual probes are never independent scoring units. ## 7. Seeds and decision rule The paired initialization/order seeds are frozen: ``` 17011, 27103, 38119, 49201, 50311, 61403, 72503, 83609, 94709, 105019, 116027, 127031 ``` All five neural arms run all 12 seeds: 60 fits. No failed seed is replaced. Use 20,000 deterministic two-way paired bootstrap replicates. Resample seed IDs and committed-history episode IDs while retaining every arm, history, probe, and intervention in its cluster. Define ``` G = min over the three OOD strata of: WGRQ_episode_exact - 0.95 WGRQ_episode_exact - AAO_episode_exact - 0.05 WGRQ_episode_exact - uniform_episode_exact - 0.05 WGRQ_episode_exact - sham_episode_exact - 0.05 ``` GO requires all symbolic gates, privileged-edge ceiling accuracy at least 0.99 in every stratum, the simultaneous one-sided 95% bootstrap lower bound of `G` strictly above zero, and at least 10 of 12 paired seeds beating active answer-only by five points on full OOD. Any oracle mismatch, transcript difference, source/cache leak, resource mismatch, symbolic error, missing seed, masked-bit violation, failed stratum, or near miss closes this version. It cannot trigger threshold, seed, board, loss-weight, or hyperparameter changes. ## 8. Prior-art and allowed claim The residual partition is Moore-machine minimization. Distinguishing continuations are active automata-learning suffixes/homing experiments. The committed state is a predictive state. Behavioral swaps are interchange/ bisimulation-style supervision. These boundaries forbid every primitive, algorithm, oracle, and general-intelligence novelty claim. The maximum claim after GO is: > On a frozen delayed-observation reversible-ring family, shortest-witness > relational loss improves exact source-deleted length and scale extrapolation > for a minimal-bit tiny recurrent learner over information-identical neural > controls. No language bridge follows. `R12_CERTIFIED_LANGUAGE_BRIDGE_BOUNDARY.md` remains a separate prerequisite. ## 9. Disjoint implementation namespace Only these new paths are authorized for Stage A: ``` pipeline/wgrq_residual_oracle.py pipeline/generate_wgrq_falsifier_v1.py pipeline/audit_wgrq_falsifier_v1.py pipeline/score_wgrq_falsifier_v1.py pipeline/test_wgrq_residual_oracle.py pipeline/test_generate_wgrq_falsifier_v1.py pipeline/test_audit_wgrq_falsifier_v1.py pipeline/test_score_wgrq_falsifier_v1.py train/wgrq_state_machine.py train/train_wgrq_cpu.py train/eval_wgrq_cpu.py train/test_wgrq_state_machine.py train/test_train_wgrq_cpu.py train/test_eval_wgrq_cpu.py ``` Any implementation need discovered outside this namespace requires a new preregistration revision before editing. ## 10. Post-freeze execution and closure Stokes job `739105` generated exactly 18,432 episodes and 589,824 ordinary one-bit answer calls. Job `739106` independently replayed every history and answer and passed the symbolic/data-admission audit. The immutable artifacts are: ``` artifact bytes SHA-256 train.jsonl 113675439 ae2849db5d57fda36e2e2fd634ce6e1d0f11eaed7fefe8d9ce722f016f28295a ordinary_calls.jsonl 188866874 251d85432d845c31ce64da1adae132fa8df8f6a63b5db744654b519f2413c9e8 generation_report.json 23417 12c1e54f23b27f3a97a86857b723fec3573f5d558b7528e1615c55746899befb audit_report.json 6773 8f5fac80e0c50bdc807287599f8468194431f3612d6d79a1331f51a073fa2dd4 ``` The acquisition is valid, but this version cannot fit or score a claim: 1. The relation-sham implementation rotates a whole sorted batch rather than deranging partners within each frozen stratum. On the exact board this creates 13,045 equivalent-relation and 13,905 non-equivalent-relation stratum mismatches. Thousands of declared strata are singletons, so the preregistered sham is not realizable on this acquisition. 2. The trainer expects obsolete audit fields and can accept the generator report instead of the independent audit, violating the admission barrier. 3. The scorer trusts supplied protocol booleans and `episode_exact` rows after hashing arbitrary checkpoint bytes. Its own positive test uses arbitrary text checkpoints and hand-authored evaluation rows, so the final decision can pass vacuously. 4. The independent auditor proves internal bundle consistency but does not itself require the generator's hard-coded frozen transcript, ledger, and report hashes. Per the locked decision rule, these are version-closing mismatches rather than post-score implementation details. No one of the 60 fits was launched. A future version would require a new board with constructively non-singleton sham strata, strict independent-audit binding, checkpoint/evaluation seals, and end-to-end adversarial negative tests frozen before acquisition.