Audit the P104 index-to-tensor bridge without replaying events.

[unpaidmoon]

Change and mathematical scope

Audit of one published convention. No new κ. Selected κ stays 0.001672660197076918.

Under the row-major index in PR #3996,

index(bank, i, j) = (bank*104 + i)*104 + j

the eight stated local sweeps and shears equal the published tensor factors, and each omitted identity side has rational rank 10816.

• Axis 0 (vary i, fixed j) is K ⊗ I: left multiplication.
• Axis 1 (vary j, fixed i) is I ⊗ K: right multiplication by K^t. Right multiplication by K fails that identity.
• The four sweeps U, U^{-1}, U^t, and U^{-t} match that convention. The four shears are the block maps with A and M = U U^t.
• A = U^t P U satisfies A^2 = 13 A. I - A and M have exhibited rational inverses. M is not U^t U.
• The bank block P ⊗ (M - U^t U) has Frobenius square 18432.
• N - P = U^{-t}(I - A)U^{-1} is invertible, so (N - P) ⊗ M and M ⊗ (N - P) each have rational rank 10816.

The 763,392 literal events were not replayed. No fiber of PR #3979 was localized, and no physical consumer or supplier is admitted. Algebraic rank is not a native-call price. Outcome: partial.

Task and overlap

Base commit: bb4bccbf42844d2801487450727c2f63e10d1d0f (upstream CrocSwap/integer-mult-bounds main).

Exact dependency heads:

• PR Audit every scalar event in the actual P104 tensor code #3996 head b8dd4e04bf8490aca121abc7ba0e503651db6c97 (proof under audit; not merged here).
• PR Construct a P104 dyadic tensor code and price repeated Gram stages #3979 head a8a1d744811cd7d1f9c43ec7ccbae85ef9695b61 (scalar code whose index sentence is audited).
• Fixture research/p104-native-channel/inputs/P104-FIXTURE.json, SHA256 516c06f6ec3cb84be588c546108de897650288576dd5fbf5fe44b9c5546ad8a4.

Task card: p104-side-correction, priority revision 6, from autoresearch.md. Scope: changed. The card asks for a chronological correction program. Discussion #1166 comment 19035716 asks for an independent inspection of the literal-index-to-tensor bridge. This package answers that audit. It does not construct a correction word. The bridge decides whether the scalar maps are the tensors with M = U U^t. Replacing M by U^t U changes the coded shear. Rank 10816 says the Gram-only saving is not an identity-shear completion. No κ change.

This session took item 6 of the chronological open-handoff list on #1166, starting at comment 19016211 (2026-10-18T09:48Z), after dropping later DONE work, this account's already delivered items, and next tasks that already have an open PR:

1. Typed actual-width physical H — comment 19016699.
2. Single-hub two-bearing factorization of the PR Classify all thirty two-port support transfers #3573 certificate — comment 19021149.
3. Integer-shear supplier/recurrence inequality left by Prove minimum binary carrier and native XOR bill for the integer-shear target #3947 — comment 19031016.
4. Adjacent completed-call pair — comments 19034066 and 19034676.
5. Coupled source-normalization — comment 19035513 (updated on PR research: derive P104 two-axis endpoint intertwiners and cost gate #3954).
6. This audit — comment 19035716. The physical-consumer half of that handoff is left open.

Nearest overlap is PR #3996 itself and its predecessor PR #3979. This package does not import either checker and does not repeat the event census. The added result is the executed index bridge, the exact transpose-order gap, and the exhibited inverse of N - P.

Validation

Python 3.12, standard library, from the repository root:

python3.12 -B scripts/research_package.py --package research/p104-index-tensor-bridge

Observed on Python 3.12.3:

PASS_P104_INDEX_TENSOR_BRIDGE {"A_square_is_13A": true, "I_minus_A_invertible_over_Q": true, "M_invertible": true, "b2_nonzero": 384, "contraction_samples": [2224, 12798], "index": "(bank*104+i)*104+j", "literal_events_replayed": false, "naive_kronecker_positions": 4112, "new_kappa": false, "omitted_side_rank": 10816, "wrong_factor_block_frobenius_square": "18432/1"}
PASS_P104_INDEX_TENSOR_BRIDGE_TESTS {"groups": 3, "new_kappa": false}
PASS_REGISTERED_RESEARCH_PACKAGE research/p104-index-tensor-bridge outcome=partial

Failure controls in test_verify.py: a column convention that drops the transpose has a positive gap, and a one-entry mutation of B2 exits FAIL.

These are finite checks. The contraction identity was checked against the dense Kronecker matrix at widths 2 and 3. The identity (A ⊗ B)(A^{-1} ⊗ B^{-1}) = I was checked on one 2×2 pair. At width 104 the rank step is that same entrywise rule together with exhibited inverses of the factors; no 10816-square product is stored.

Not run: the 763,392-event replay, encoder-fiber localization, a physical consumer, Lean, and any selected-frontier update.

Credit

The code, the index sentence, and the rank claim under audit are Chris R / Clear1nsight's PR #3979, audited by ToikY in PR #3996. The fixture and root normalization come from the retained P104 native channel (Rohan Arun, Chafik Boukhalfa, Adam Gall, and the native/E8 lineage named there). This package does not import either checker's source.

Prepared by unpaidmoon with substantial Cursor Grok 4.7 assistance, including the verifier and tests. Apache-2.0. No independent human review is claimed.

Open next questions

From research/p104-index-tensor-bridge/TASK.json:

1. Does one frozen fiber of PR Construct a P104 dyadic tensor code and price repeated Gram stages #3979's event list implement these eight broadcasts? Compare one encoder fiber from the frozen PR Audit every scalar event in the actual P104 tensor code #3996 word bytes with axis 0 of U, without calling that package's checker. Blocker: those bytes live on head b8dd4e04bf8490aca121abc7ba0e503651db6c97 and are not on this main checkout.
2. Is there one repeated physical consumer whose complete bill keeps the k=2 Gram saving? Pin one chronological producer/consumer pair and price (N - P) ⊗ M with owners, frames, and dirty return. Blocker: PR Audit every scalar event in the actual P104 tensor code #3996 open question 1 is still open. Rank 10816 says the Gram shear is not that payment.

Workflow feedback (optional)

Chronological handoffs on #1166, after dropping items already covered by an open PR, selected this audit without rerunning the randomized catalogue picker. A short pointer on comment 19035716 that the index bridge was still unexecuted would have made the same cut.