DIVERGE-NFE1: Natural Equation Fault Evidence

Status: frozen before implementation, training, board materialization, or result on 2026-08-06.

1. Purpose

TFS1 proves learned operation alternatives, exact factorized execution, delayed recovery, and coherent abstention in a controlled typed language. Its evidence is still an assessor-issued typed receipt. NFE1 tests the next distinct boundary: can a learned whole-mention interface compile source-bound numeric evidence from independently verified arithmetic reasoning text and drive the same DIVERGE refinement mechanism?

NFE1 does not update TOL3, FTA1, Shohin, or the exact executor. It trains only one equation mention-role model. It is not continuation pretraining or a public reasoning benchmark.

2. Data

The only source corpus is the frozen V10 artifact artifacts/product_reasoning/data/v10_tokenbalanced_35m20c10s10p25t_4m_verified_r1.jsonl at SHA-256 2461d6f70b44a142854d56c24e1fb42d600065e5788a2c4e055ba47b12696549.

Training uses every deduplicated exact integer equation span from rows whose source is reasoning_gym_trace: 2,179 unique equations covering addition, subtraction, and multiplication. A span is admitted only when an independent integer evaluator verifies its RHS.

Confirmation uses source-disjoint augmented_gsm8k rows. A row is eligible only when it contains a complete chain of two to five verified integer equations, its final RHS equals the answer field, every operation's three type-correct alternatives produce distinct immediate values, magnitudes remain below 3,000,000, and none of its exact equations occurs in training. This leaves 109 rows before selection. Hash-sort all eligible rows, retain every depth-three-or-greater row, then take the lowest-hash depth-two rows until the board has exactly 96 episodes. The expected board contains 222 transactions. No model score participates in board selection.

For each confirmation equation, deterministically rotate its visible operator + -> - -> * -> + while preserving the independently verified LHS, argument, and RHS. The corrupted text is the candidate source. It does not reveal the gold operation. The observed RHS remains evidence and is never passed to the operation head after source sealing.

3. Learned interface

The mention parser is a two-layer bidirectional byte GRU with width 128 and no position embeddings. A lexical scanner proposes maximal signed-integer spans; the model assigns each complete mention to one of LHS, ARGUMENT, or RHS. Assignment is a hard one-to-one permutation over whole mentions, never independent byte argmax and never a pooled ten-field query bank. Exact integer parsing occurs only after the model selects a complete span.

Training is one seed (2026080608), 1,000 AdamW updates, batch size 256, learning rate 0.003 with cosine decay, zero dropout, and class-balanced cross-entropy over mention roles. The first checkpoint at update 1,000 is the only checkpoint. No confirmation score may alter width, duration, seed, optimizer, loss, or renderer.

The frozen FTA1 checkpoint 9321b78372d9926930d4de073d70e82c94e8360a69e09be695bab91b2e479f2d provides support scores over the three scalar operation classes. NFE1 retains all three type-correct operations as one coherent categorical fault line; it never mixes operation fields. The visible corrupted operation is expected to make top-1 wrong. Candidate support may rank alternatives but may not remove a type-correct operation before evidence.

4. Execution

Each episode begins from the first learned LHS. At every transaction, the source-sealed packet applies the three candidate operations to each surviving scalar state using the learned ARGUMENT span. The separately sealed learned LHS/RHS evidence packet retains a lineage only when its current state equals the observed LHS and its successor equals the observed RHS. Exact state groups merge only identical complete states. The late query asks for the terminal scalar value.

Raw response text, question text, answer labels, gold operations, model hidden states, and source KV state are unavailable after sealing. The independent assessor may enumerate worlds and check arithmetic but cannot repair a packet.

5. Controls and gates

Matched arms are premature highest-support top-1, equal-memory complete particles, no-evidence factorized support, posterior answer aggregation, and full hard-evidence factorization. Controls shuffle evidence across episodes, swap packet/query commitments, reset the initial state, shift operand semantics, and poison raw source after sealing.

All conditions are conjunctive:

1. at least 220/222 complete confirmation equations have exact learned LHS/ARGUMENT/RHS mention assignment;
2. every accepted packet contains all three distinct type-correct operation candidates and therefore retains the verified gold operation;
3. exact factorized/independent extensional parity on every accepted episode;
4. at least 92/96 exact terminal answers for full NFE1 and at least 90% exact conditional on initially wrong top-1;
5. full NFE1 beats top-1 and equal-memory particles by at least 30 percentage points each;
6. no-evidence support abstains on at least 90/96 terminal queries;
7. shuffled evidence, initial-state reset, and operand-semantic shift each reduce exactness by at least 50 points; every packet/query swap rejects and post-seal source poisoning is bit-invariant;
8. zero evidence receipt accepted with a wrong source, step, mention, or value; zero false commitments, malformed accepted packets, or overflow; and
9. canonical packet/particle bytes, logical/unique execution applications, wall time, source-admission counts, and per-depth/per-operation results are reported even though this low-depth interface gate has no fixed resource promotion threshold.

Failure at the mention gate closes this interface before composition. Failure after a component pass closes NFE1 without a second seed, width, duration, renderer, or loss variant. A pass authorizes a broader natural program/evidence compiler with variables and predicates; it does not authorize continuation pretraining or a general reasoning claim.

6. Pre-implementation control amendment

Frozen on 2026-08-06 before implementation, data materialization, model training, or evaluation. The original operation shift control was specified as the cyclic label permutation + -> - -> * -> +. NFE1 deliberately retains all three scalar operations at every fault line, so that permutation preserves the candidate function set exactly. Full hard-evidence factorization is therefore mathematically invariant to it, making the stated collapse condition impossible rather than discriminating.

Replace only that impossible control with an operand-semantic shift: the executor receives argument + 1 for every candidate operation while the source-sealed packet, operation support, evidence receipts, and expected answer remain unchanged. This tests causal dependence on the compiled argument and arithmetic transition path without removing or relabeling any candidate. Condition 7 consequently reads shuffled evidence, initial-state reset, and operand-semantic shift everywhere. All other data, architecture, budgets, controls, thresholds, and one-run rules remain unchanged.

The same pre-implementation audit re-ran exact equation extraction directly from the frozen corpus. It reproduced 2,256 verified training occurrences and 2,179 exact deduplicated spans, but strict exclusion of any confirmation row containing one of those spans leaves 109 eligible rows with depth counts 85/20/2/2, not the draft's 112 with 87/21/2/2. The deterministic selection therefore retains all 24 depth-three-or-greater rows and 72 hash-first depth-two rows, for 96 episodes and 222 transactions. The component threshold is changed from 221/223 to 220/222, preserving the same greater-than-99% requirement. No source overlap is relaxed to recover a planned count.