DONE: actual stage-zero Gaussian commutation, with its required translation (#4464)

[mihir-s-05]

DONE / released — 2026-10-21, priority revision 7. PR #4464, commit 82c7db427100033aa1275ecdb16ebc2e2ef10b91, answers #4457 open question 1 for the source-specified complex caller. The proof and exact checker establish D0 Q = Q D0 = F9 on all 112 ports. The port Gaussians are polynomials in XOR translations, so commutation does not require q to be a standard axis. The recorded rank-eight normalization retains a nonzero translation at every port; dropping it is wrong. The written tensor-composition argument applies to all positive column multiplicities, conditional on the retained native realization. This is an independently implemented check, not maintainer approval or a new kappa. Full native/adapter implementation, precision, BIT and outer admission remain separate. No active claim remains.

Public artifact: package and proof, exact receipt.

Evidence: 114,688 exact commutator coefficients, 57,344 Fourier-character comparisons, 448 initial caller maps and 338 semantic/binding rejection cases. Registered Python 3.12 replay and source pins pass. Git-free optimized replay matches the receipt and rejects changed source/receipt data. Committed-byte/source-history preflight passes all 15 files. Hosted CI is pending.

Concrete normalization witness: port 0 has q=11, complement [13,14,7,16,32,64,128,256], and translation t=15. The actual identity is D0=T_t product C_b; dropping T_t gives error -1/16 on input e_0, output 0. Inserting a noncommuting character before F9 instead gives commutator coefficient (1+i)/16. The first is a required existing phase correction, not a defect in the source's retained recipe.

The initial transport idea was not pursued after fresh overlap found #4457. This follows its explicit commutation question and remains complementary to #4455 and #4448. Final refresh: canonical main 2a13eb1827784206202db90e5dc8327cf8a250db, #4457 dd40f88a4d45a0ae2487eb24f935516f749cefef, 317 open PRs, 3,318 Discussion records and complete tracked top-level note coverage.

Next useful task: separately audit one full native invocation and its whole-call phase/coordinate adapters at arbitrary column multiplicity, retaining spectator and return conditions. The current result closes only the specified commutation implication.

Credit: Mark/Matarener for #4457; zxv for the source caller and phase utilities; Cinna Davis/1337isnot1337 for #4440 and the native successors; Jacob Alan Sussman for the mixed-target compiler; QQ16/native phase predecessors including ywz65. Prepared for Mihir Srivastava / mihir-s-05 with substantial OpenAI Codex assistance, no subagents. Apache-2.0.

{
  "schema": "integer-mult-coordination/v1",
  "kind": "DONE",
  "task": "independent-moment",
  "scope": "PR4457 open question 1: actual complex stage-zero Gaussian commutation at all 112 ports",
  "base": "2a13eb1827784206202db90e5dc8327cf8a250db",
  "expiry": null,
  "artifact": "https://github.com/CrocSwap/integer-mult-bounds/pull/4464"
}