Prove an all-length balance law for native source-code controllers

[DavidAriasC]

Ali Romman’s PR #1862 provides native identity transfers with an order-six source code, but a controller that restores the full source remains open. This adds a written, all-length conservation law for the fixed-code controller class: the signed count of forward/backward calls must be divisible by six, even with arbitrary invertible whole-bank scalar mixing, signs, scales, extra restored dirty banks and true inverse calls. Consequently every odd-call controller in that class is excluded, including a fixed-code substitution into the inherited nine-call skeleton. Coherently conjugated coordinate changes satisfy the same law.

A complementary theorem allows a different admissible chart at each call of an F-B-F exchange. Its unwanted diagonal block D satisfies R D K = RK - 2I; on the retained source this is 12 I_8, so its rank is at least eight for every choice of those charts. This extends PR #1862's fixed-chart counterexample to all admissible selected rows in this architecture.

The contribution is one scoped proof package under research/native-controller-congruence/. It supplies a construction-level stopping criterion, rather than a bounded search or extra decimal digits. The numerical multiplication bound is unchanged. Port-dependent operations, non-equivariant physical chart transport, new clean workspace and paid repairs remain possible escape routes. The valid PR #1862 local construction is preserved; no native implementation or full supplier is claimed.

Evidence

• The independent source extractor reads the selected PR #1094 archive, whose two local payloads exactly match PR #1862's pinned historical inputs. It checks the actual 112-port source matrix, the 96+16 code decomposition, the exact four-pair nine-stage recipe, and four changed-chart witnesses.
• Twelve separate dense-rational and CLI tests pass, including legal signs/scales, inverses, a six-call algebraic exchange positive control, noncentral matrix fixtures, missing hypotheses and corrupted inputs. No predecessor executable is imported.
• Registered Python 3.12.14 replay and optimized -O replay pass. The research-workflow suite passes (45 tests, three platform skips). Two agents independently reviewed proof scope and source extraction; this is not independent human review or proof-assistant verification.

python3.12 scripts/research_package.py --package research/native-controller-congruence
python3.12 -I -O -B research/native-controller-congruence/verify.py
python3.12 -B -m unittest discover -s tests -p 'test_research_*.py'

README.md contains the proofs and precise reopening conditions; TASK.json, SOURCES.json, ARCHIVE.json, PREDECESSOR.json, RESULT.json and VALIDATION.json record scope, source pins and replay evidence. No full make frontier1094-verify, native routing, hosting, precision or all-size theorem replay is claimed.

Provenance

Started from main ecc75c79c22d909fd42aa68437eb77c4c1361835, then rebased onto ae95f60feec79897b153dba6459bb3f9fd4d7b8e; consumed selected inputs are unchanged. The structural-track choice and change from the centre-path card are recorded. Final overlap search at 2026-10-14 15:33 UTC covered 538 open PRs and 599 discussion records with complete pagination; no remote match for the precise final terms was found, which is not a global novelty guarantee.

Credit Ali Romman's PR #1862 at bb29ea73ad9be9a0c3d53cdcb1d22b2ee9ffe42f for the code and native transfers, Chafik Boukhalfa's PR #1877 at 6d455d2d71eb35ecf8e562711e5f719b994fe3f3 for the related original-logger obstruction, Jeremy Kothe’s PR #1042, and Rohan Arun’s PR #1094, with the earlier gauge44/split-centres contributors retaining their attribution. Exact predecessor proof and notice are retained. Prepared for David Arias with substantial OpenAI Codex assistance; Apache-2.0. No selected frontier, frozen input, shared verifier or review ledger changes.

[Mister-Meeseeks]

Batch review at 165f08a694b26b0f4c521ce5e1132ede0b7362a2.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Fixed-chart determinant argument forces controller winding divisible by six; independent exact source actions and chart-defect checks support the stated obstruction. The restriction is fixed common coefficient algebra and whole-bank scalar mixing; changing charts and native implementation remain separate.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.