[petalas]

Summary

A nonlinear address permutation cannot be absorbed into the abelian no-phase reduction used in Jacob Alan Sussman's #4628. This scoped proof identifies the exact boundary for every binary address dimension:

P A P^-1 = A  <=>  p(x) = Mx+t, M invertible
p nonlinear  =>  <A, P A P^-1> is nonabelian

The second conclusion rules out every larger abelian group containing the original calls and normalized by that permutation. An explicit three-bit Toffoli commutator witnesses the obstruction. This applies the elementary translation-normalizer argument to the cited reduction; it does not claim a new abstract normalizer technique, actual nonlinear-program tariff, supplier or κ.

The contribution was produced by Nick Petalas. Sussman's semidirect-product reduction and the retained eumemic #288 / Swapnil Jain P1/P2 lineage receive credit. Predecessor results are cited, not replayed. Exact source paths, commits and hashes are in SOURCES.json.

Evidence

• Before: Research: lower bounds for small complex units: m = 2 never saves, m = 3 needs 14W/5 calls (Lean, engine game), m = 4 needs 41W/13, and the smallest open cases (no new κ) #4628 section 5 absorbs linear address maps into register copies, with no classification here of nonlinear extensions.
• After: a self-contained dimension-uniform proof, including impossibility of an abelian enlargement. A complete census checks all 40,320 three-bit permutations: exactly 1,344 normalizers and a checked noncommutation witness for each of the other 38,976. One- and two-bit censuses give 2/2 and 24/24.
• Literal eight-by-eight Gaussian-integer checks verify every three-bit call-root square and the Toffoli call-root noncommutation. Four false mathematical claims are rejected in both normal and optimized Python.
• Local registered replay: PASS_REGISTERED_RESEARCH_PACKAGE, 0.41 seconds total. The two registered commands were also timed directly at 0.08 and 0.09 seconds. CPython 3.12.15, macOS arm64, standard library only.
• Committed-byte and full source-history preflight: PASS_RESEARCH_COMMIT_BYTES, all five files at 0cb79c588655cc85653ca4123c74fe3cc42c6ec9. Whitespace and Python syntax checks pass.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/no-phase-nonlinear-boundary
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/no-phase-nonlinear-boundary/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -O -W error -B research/no-phase-nonlinear-boundary/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/no-phase-nonlinear-boundary --check-source-history

Each verifier run generates a JSON report bound to hashes of the five package files. No receipt is handwritten.

Limitations

The finite census covers dimensions 1 through 3; the written proof covers all positive dimensions. This is a boundary of one abelian reduction, not a lower bound against every nonlinear algorithm. Independent human and formal proof-kernel review are unclaimed. Lean, Docker, predecessor numerical replays, physical/precision/stock/outer admission and the selected cold replay were not run. The selected frontier is unchanged.

The historical card's first orientation command stopped with KeyError: 'complex_profile' against the changed selected-record schema. It is recorded as an orientation failure and is outside this independent package's live inputs.

Open next questions

• Develop a nonabelian replacement for the register-copy reduction, starting from the displayed Toffoli subgroup. A kept space and complete call/demand inequality remain missing.
• Decide one small-unit shape with a specified nonlinear no-phase alphabet and all-width target. This result does not determine that tariff.

Related open work

• Research: lower bounds for small complex units: m = 2 never saves, m = 3 needs 14W/5 calls (Lean, engine game), m = 4 needs 41W/13, and the smallest open cases (no new κ) #4628 studies small complex-unit lower bounds and supplies the linear no-phase reduction. This adds its exact nonlinear absorption boundary; none of its numerical bounds or certificates is rechecked.

CI notes

CI pending for 0cb79c588655cc85653ca4123c74fe3cc42c6ec9. Local registered normal/optimized checks and source-history preflight pass. Hosted certification remains pending.

Merge Danger

Door: two-way. This adds one isolated research package and no shared implementation changes.

Blast Radius: package. It registers two bounded checks and leaves the selected result and predecessor packages unchanged.