[petalas]

Summary

Scoped written arithmetic proof with exact finite corroboration. For every radix r=3^a, every irreducible factor of PR621's integral kernel over K=Q_3(u) has even degree. In Newton block k its degree is divisible by 2*3/gcd(3,k-1). The first block is an unramified irreducible quadratic; the second, when present, is an irreducible sextic with ramification index three and residue degree two. There is no odd-degree factor over the global cyclotomic field either. The two local blocks are not asserted to be global factors.

new contact units -> irreducible quadratic over F3
                 -> even residue degree
cited slope denominator + [L:K]=e*f -> factor-degree obstruction

The contribution was produced by Nick Petalas. Credit Hao Yan / FanPUYen PR796, exact head 22902155c7875139e43e4fdf15723142d6709bdd, for the coefficient polygon, contacts and root valuations, and Ali Romman / PR621 for the kernel. Those are cited premises, not audit or replay targets. Cyclotomic, Newton-polygon and local-field theory are classical; Milne's precise theorem locators are in PROOF.md. The new step is the contact-unit formula and residual quadratic obstruction. No literature-wide priority is claimed.

Advance CLAIM. Canonical main was refreshed before publication at 7c49fe32c55de02c2c873655c804b16e4c3fef0a, priority revision 8, selected PR6030. TASK.json explicitly changes the card to this self-posed arithmetic side result. No new kappa or selected-frontier change.

Evidence

Before, PR796 gives the root valuations and contact locations. After, PROOF.md proves the new factor-degree divisibility and identifies the first two local blocks. No predecessor scientific executable is imported or run.

• Exact cyclic coefficient multiplication and uniformizer conversion pass 45 normalized contact checks at r=3,9,27,81,243. A separate dense uniformizer multiplication agrees on all 15 distinct contact coefficients at r=3,9,27.
• Literal unit products through k=9, both residual quadratics, all 64 checked slope denominators, 10,368 admissible ramification/residue pairs and 15 rejection controls pass. The two coefficient algorithms are separately implemented here, not independently authored. Finite checks corroborate the written proof.
• Local CPython 3.12.15, macOS arm64, standard library only: both warning-strict registered modes pass and were timed below 0.1 seconds each. Their generated hash-bound receipts agree exactly. Each registered timeout is 60 seconds.
• Registered metadata/source pins, committed bytes/modes and full declared source history pass for all six package files at db97db5e3f1e39700e13dab52ca082773e938ae7. Official export/integrity and both read-only Git-free, empty-PATH modes pass with identical receipts; all seven exported files retain their hashes.
• In-memory source compilation, function-size inspection and staged whitespace checks pass. The largest function has 29 lines. No separate static analyzer is claimed. The managed-allocation deviation and explicit finite capacities are documented.

Exact reproduction from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/ternary-kernel-factor-degrees/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/ternary-kernel-factor-degrees/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/ternary-kernel-factor-degrees
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/ternary-kernel-factor-degrees --check-source-history --check-modes

Other machines may substitute their CPython 3.12 executable. Expected markers are PASS_TERNARY_KERNEL_FACTOR_DEGREES, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Direct replay is read-only and generates a receipt bound to all six source files. No receipt is handwritten.

Limitations and unrun obligations

The universal field statements rest on the written proof and its cited Newton/local-field premises. Actual p-adic factors or roots are not produced. Higher slope blocks remain unclassified; their degree divisors are not asserted sharp. The result gives no improved complex circle gap, coefficient/root preparation, rounded inverse, native hosting, owned restoration, stock/compiler/tape/precision or complete supplier/outer invoice. All five selected assumptions remain unchanged. No complete multiplication theorem, independent human or formal proof-kernel acceptance is claimed. No predecessor scientific replay, selected cold replay, Lean or Docker was run. No required finite check was deferred to CI.

Open next questions

1. Are the divisors attained by every irreducible factor in higher blocks? Start with higher residual lifting for the degree-18 third block; its wild factorization is missing.
2. Can a declared finite-precision local factor producer help an actual paid consumer? Specify coefficient, conversion and restoration costs before claiming an inverse or supplier saving.

Related open work

No other open PR on this local factor-degree obstruction was found in the covered searches. The direct predecessor #796 and related valuation corroboration #804 and complex radix-nine audit #820 are merged. Their reported results are cited or distinguished, not rechecked.

• #6426 studies generic complex cyclic-inverse norm growth, not these local irreducible factor degrees; no common numerical quantity is compared.
• #6468 studies dyadic balancing of weighted cyclic recurrences; this proof supplies no scaling or grid-cost comparison.
• #6450 studies weighted factor multiplicities in a static dirty-echo tree; no tree tariff is computed here.
• #6486 optimizes a complete dirty-phase replay in the cyclic-inverse lineage; its scientific consumer and runtime quantity differ from this local arithmetic proof.

The completion refresh inspected all twelve account open PRs created since the advance CLAIM, #6574 through #6585 including this PR. The eleven others concern Gaussian rounding, shared-power and ADDRESS tariffs, parity fanout, algebra automorphisms, E8 geometry, phase-chart isolation and separate replay implementations. None studies this ternary kernel local factor-degree quantity or reports a competing value computed here. No additional scientific computations audited or compared those contributions.

Overlap coverage includes 523 open PR bodies and 6,538 Discussion records with complete cached pagination, a newer live open/claim refresh and focused all-state/main/merged searches. Only 123 of 8,917 tracked local notes were materialized for the helper; main README searches supplement that sparse view. Arbitrary historical diffs and unpublished work remain outside coverage. No extra scientific computations checked another contribution.

CI notes

Both hosted checks passed at exact head db97db5e3f1e39700e13dab52ca082773e938ae7 in Registered research packages run 41618035575: plan in 4m34s and registered replay in 1m25s. The completed checks were read once; separate run metadata confirms the same head, completed status and success.

Hosted replay executes both registered warning-strict normal and optimized finite checks. All registered commands, metadata, committed-byte/source-history and read-only export checks also passed locally. No failed, waived or locally unrun package check remains. CI certifies execution of the bounded checks and source bindings; independent human or formal proof-kernel acceptance of the universal written theorem remains unclaimed.

Merge Danger

Door: two-way. Blast radius: isolated. Six new files in one opt-in package; shared code, predecessor packages, pinned evidence and selected-frontier records are unchanged.