[petalas]

Summary

Scoped algebraic proof with exact finite corroboration. Classify every common address permutation for standard cyclic Fourier tensorization on pairwise-coprime axes, including arbitrary nonlinear permutations.

N = product_i s_i; M_i = N/s_i
same permutation on both Fourier indices
    iff every M_i is a square unit modulo s_i
all maps: f(x) = sum_i M_i v_i x_i mod N, with M_i v_i^2 = 1 mod s_i

Axes (13,17) have exactly four maps, including 102x+26y mod 221. Axes (3,5) and (3,7) admit none. The proof forces every admissible bijection to be additive, so nonlinear reindexing cannot escape the criterion. This is an originally posed interface result, not a predecessor audit. No new κ.

Only research/crt-fourier-common-permutation/ is added. The contribution was produced by Nick Petalas. Credit David Ganz / Dugongue's #5564 for the native coprime-grid interface and distinct CRT address conventions, and its retained ToikY, Jose Manuel, Tekalign, Cinna Davis, Chafik Boukhalfa, unpaidmoon and native/tensor lineage. CRT, cyclic characters and Good–Thomas tensorization are classical; no broad literature-priority claim is made.

Evidence

• Before: Construct prime-free native tensor grids with exact CRT replay (no new kappa) #5564 supplies two different CRT address maps with fixed standard local roots. It does not impose or classify a common map.
• After: PROOF.md gives the complete common-map criterion and count. Exact local-root construction agrees with a separate cyclic-generator enumeration on 105 coprime shapes and 7,313 candidates, yielding 147 maps and 887,566 literal phase entries. All 1,592 permutations in the tiny exhaustive domains are checked, including nonlinear maps and translations. The four (13,17) maps receive another 195,364 phase checks. Three distinct-map scope controls cover 1,066 entries; fourteen invalid-input or semantic controls reject.

Local CPython 3.12.15, macOS arm64, standard library only. Both direct warning-strict modes and both registered commands pass. Each registered command takes about one second locally. Output is a generated receipt bound to all six package files and two exact source inputs. Metadata, committed bytes/modes, declared source ancestry, official export integrity and input coverage pass. Normal and optimized replays also pass from a read-only Git-free export with empty PATH; all nine exported files retain their bytes and modes. Actual corruption of each source input rejects in both modes.

From the repository root, reproduce with the exact local interpreter path, or substitute another CPython 3.12 executable:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/crt-fourier-common-permutation/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/crt-fourier-common-permutation/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/crt-fourier-common-permutation
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/crt-fourier-common-permutation --check-source-history --check-modes

Expected markers: PASS_CRT_FOURIER_COMMON_PERMUTATION, PASS_REGISTERED_RESEARCH_PACKAGE, and PASS_RESEARCH_COMMIT_BYTES. A sparse checkout must materialize research/native-coprime-grid first. No predecessor executable is imported or replayed.

Limitations and unrun obligations

The written proof supplies the universal quantifiers; finite checks corroborate them. Standard local roots and the same permutation on both indices are essential. Different maps, changed local roots and general linear transforms remain available. The constructor is a bounded offline reference, without a fast modular-root or physical execution claim.

No movement/storage saving, emitted native program, numerical resampling, coefficient preparation, restoration, stock, precision/tape, full caller bill or supplier admission is supplied. All five selected framework assumptions and κ=.003125804 retain their status. No selected cold replay, Lean or Docker was run. Independent human and formal proof-kernel review are unclaimed.

Open next questions

• Can an actual caller exploit a successful common-permutation grid after its local roots and complete movement bill are fixed? Start with one full input/output ABI and transform.
• At an excluded grid such as (3,5), can paid local root changes beat retaining two CRT maps? Charge phase preparation and the complete transform before claiming a saving.

Related open work

• Research: paid-copy E8 exact-RAM realization with an attributed DFT corollary #647 studies an exact-RAM cyclic DFT construction with supplied roots; it does not classify a common CRT address permutation, and no comparison of its numerical claims is made here.
• Research: close the PR390 heterogeneous physical provider in exact RAM #956 studies an exact-RAM native provider and cyclic DFT transfer; the present result concerns only a fixed-root address identity, with no provider or exponent comparison.
• Audit conditional CRT last-K two-slice streaming tariff #4187 studies CRT sparse-repair streaming tariffs; it neither supplies nor requires this common-permutation Fourier criterion.

Completion refresh inspected this account's open PRs created since the advance claim. No additional PR studies the common CRT Fourier permutation criterion or reports the same map counts. No extra computations of another contribution were run.

Advance claim. Canonical base 2485805a6d47f36ea83ddde6327d453ac82d44dc; priority revision 8. Live overlap checks are bounded and do not establish global novelty.

CI notes

Both hosted checks, plan and replay (research/crt-fourier-common-permutation), passed at head b1895eb132fe9a67798c1878acec0df46b53ea11 in run 41067215161. The completed checks were read once, and the run metadata confirms the same head and successful conclusion. Local registered replay also passes. Hosted CI certifies the registered finite checks; it does not establish the universal proof or supplier admission.

Merge Danger

Door: two-way. Blast Radius: isolated. This adds one opt-in research package and changes no shared code or selected record.