[petalas]

Summary

The unchanged rounded native cycle from JoyboyBrian's PR5492 has an explicit orbit with linear drift:

support = (1,10,39,44), all other coordinates zero
x = (3+2i, 1, 3i, 3+i)
delta = (-4+4i, 4+4i, 4+4i, 4-4i)
C^(24k)(x) = x + k*delta, k >= 0

Claim type: scoped proof with an exact finite certificate. The new integral-shift argument proves the all-iteration identity from eight parity checks and a 24-traversal certificate. The ideal identity cycle therefore has infinite order after this rounding, with no path-length-independent absolute error envelope at any fixed dyadic grid.

An arbitrary deterministic correction decoder receiving only the unchanged rounded payload and a tag needs at least K+1 tag states for paths of lengths 0,24,...,24K on unrestricted signed integer inputs. The same final payload arises from K+1 distinct originals. The tag lower bound does not cover a fixed bounded input box or a decoder also given the path length.

Canonical CLAIM. The delivered scope matches the claimed orbit/drift alternative. Main and relevant guidance were refreshed at 2485805a6d47f36ea83ddde6327d453ac82d44dc, priority revision 8. The selected frontier is unchanged.

Evidence

Before this contribution, PR5492 supplied one nonflat traversal. The new result supplies an exact drifting orbit, its universal shift certificate and a correction-state obstruction. No predecessor executable or scientific receipt was audited or replayed.

Local CPython 3.12.15 checks pass:

• All 96 new orbit edges agree between a four-coordinate circuit and a separate full 512-coordinate interpreter. The verifier derives the restricted circuit from the unchanged authenticated signed recipes.
• Eight shift-pair parity checks, 128 signed covariance fixtures, 2,401 complete pair-oracle cases, 769 consecutive iterate formulas and 33 distinct originals with identical final payload pass. Four fault controls and two positive controls pass.
• Warning-strict normal and optimized direct and registered runs produce identical generated receipts. Individually timed commands took 0.741 and 0.747 seconds, respectively. These observations check the replay deadline, not a speedup.
• Exact committed bytes/modes and declared source ancestry pass for all eight package/input files at 6112144851cc13fdd7183d4e909c6dcc841ea8e1.
• Official committed export and integrity checks pass. Both direct modes pass from a read-only Git-free export with empty PATH; all nine exported files, including the export manifest, retain their hashes and modes. Actual source-byte and receipt corruption reject with their intended reasons in both modes.

From the repository root, using the exact local interpreter:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/native-loop-linear-drift/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/native-loop-linear-drift/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/native-loop-linear-drift
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/native-loop-linear-drift --check-source-history

Expected statuses are PASS_NATIVE_LOOP_LINEAR_DRIFT, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. CPython 3.12 and the standard library suffice. Optional generated receipts bind all package file hashes; no receipt is handwritten or committed.

Limitations

This is the fixed-grid width-one frame consistency diagnostic from PR5492, joining occurrences on different supplier fields. It is not a repeated chronological supplier subword. Other tensor widths, dynamic grids, changed edges, a physical implementation and a complete paid caller are unrun or unproved. All five selected framework assumptions remain. No new kappa, supplier admission, complete multiplication theorem, independent human review or formal proof-kernel acceptance is claimed. Every known finite reverse path remains exactly invertible.

Open next questions

1. Bind an equivalent repeated drift segment to an actual chronological caller with a full entry and restored exit.
2. Specify a changed rounded-edge ABI with paid correction storage, erasure and scalar-ADD compatibility; price the complete physical and recursive bill before synthesis.

Related open work

No open PR studying this fixed rounded cycle or its iterates was found in the refreshed coverage. The overlap cache covered 323 open PR bodies and 6,014 Discussion records with complete pagination; only 88 of 8,722 local note records were materialized, supplemented by main-tree formula/object searches. This is bounded coverage, not novelty certification.

• PR6251 shortens the selected P104 scalar word. This result fixes the older signed cycle recipes and changes no scalar program.

• PR6244 pays random-coin delivery at one reached halving site. Its stochastic half interface differs from the deterministic paired lifting studied here.

• PR6249 reconstructs PR6167 numerical compiler conditions. No compiler threshold or invoice is recomputed here.

• PR6257 proves a reader-497 merged-cache tariff on PR6167. It changes the cache contract and does not compute this rounded cycle's iterate formula.

• PR6261 proves a correlated-dirt requirement at a P104 wire meeting. Its local response-space contract differs from repeated rounded frame transport.

The completion refresh checked this account's open PRs created since the CLAIM. No additional open PR studying this fixed rounded cycle or its drift quantity was identified. No new numerical comparison is made for the different contracts above.

CI notes

Both hosted checks, plan and replay (research/native-loop-linear-drift), passed at exact head 6112144851cc13fdd7183d4e909c6dcc841ea8e1 in run 41049121971. The completed checks were read once; run metadata confirms completed/success and the matching head. Hosted replay certifies the registered finite checks. The universal claims rest on the written scoped proof; independent human and formal proof-kernel acceptance remain unclaimed. The registered normal and optimized commands and all listed transport checks also passed locally.

Merge Danger

Door: two-way. Blast radius: isolated. Only the new research package is added; selected records, frozen sources and shared implementation remain unchanged.

This contribution is produced by Nick Petalas. Credit JoyboyBrian / PR5492 for the signed cycle and original defect; Tekalign's reversible lifting; Cao Yifan's word; ywz65's adapters; the coherent-frame-additivity and retained P104/native/controller/geometry lineage. Sricharan Vagvala / PR1631 and Sagar Thakrar / PR1537 supply related orbit-rounding and exact-return precedents under different contracts. Floor translation and the pigeonhole principle are classical. New work is Apache-2.0.