[petalas]

Summary

Scoped written algebraic proof with exact finite corroboration, produced by Nick Petalas. For a nonempty finite list D of pairs (a,b) with gcd(a,b,m)=1, the exact number of coefficient pairs v modulo any m>=2 with every dot product d.v nonzero is

A_m(D) = m^2 - 1 - sum_{q|m, q>1} phi(q) d_q.

Here d_q is the number of distinct projective demand classes modulo q. The primitive count is J_2(m)-phi(m)d_m. A safe pair exists exactly when d_m<psi(m), and an explicit constructor always supplies a primitive witness when any witness exists. The sharp blocking threshold is psi(m) classes. Empty lists are handled separately.

This contributes one new projection/collision-avoidance lemma to the composite-axis continuation left open by David Ganz / Dugongue's PR5564. Its grid construction is cited context, not an audit or replay target. No new kappa or selected-frontier change is claimed. The supplier-boundary-audit card is explicitly changed to this side interface theorem.

Credit Dugongue, ToikY's guard-safe bands, Jose Manuel's tensor caller, Tekalign's CRT streams and the retained native/CRT lineage. Bezout, CRT, totients and projective ring lines are classical. Havlicek and Saniga's modular projective geometry predates this work; the contribution is the explicit all-divisor count and constructor at this recovery interface, without a broad literature-priority claim. Nick Petalas's PR6273 concerns a separate common-Fourier-permutation contract.

Evidence

PR5564 leaves a complete composite numerical caller open. This result supplies a scoped exact arithmetic condition: modulo 9, the demands (1,0),(1,3) admit exactly 66 safe pairs; omitting the proper-divisor term predicts 68 and a field-style line count predicts 64. Literal dot products reject both replacements. The three points (0,0),(1,0),(2,3) have exactly 60 injective scalar projections out of 81.

• All 12,696 subsets of complete literal-kernel representatives at moduli 2 through 9 agree with the new formula, primitive counts and witness existence.
• Another 2,680 declared comparisons cover sixteen larger composite/prime-power moduli through 64. Empty/full lists, every singleton, every full-list single deletion and pairs among the first twelve representatives are explicit.
• Twenty malformed-domain API calls reject. Three false-formula controls, four arithmetic fixtures, duplicate/signed-demand checks and the literal point-set projection check pass.
• Both warning-strict normal/optimized registered commands pass locally. Direct commands took about two seconds each; the complete registered pair took about four seconds. These are local check-budget measurements, not a performance improvement.
• Eight-file committed-byte/source-history preflight passes at e1af31c. The official export passes integrity checking. An optimized replay from its read-only Git-free tree with an empty PATH passes and preserves all nine exported files and modes. The receipt is generated on stdout and bound to all seven package files and the pinned predecessor context. No receipt is handwritten.

From the repository root, CPython 3.12 and the standard library:

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

Another CPython 3.12 path may replace the displayed interpreter. Expected markers are PASS_COMPOSITE_PROJECTION_DENSITY, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. The counter and literal oracle import no predecessor executable. Discovery stopped at the first verified exact result.

Limitations and unrun obligations

The all-modulus statement rests on PROOF.md. Executable support is bounded to m<=64 and 256 primitive demands; exhaustive subset coverage is only m=2 through 9. The theorem requires two coordinates and every demand primitive modulo m. Its point-set application requires all pairwise differences primitive. The threshold concerns arbitrary primitive demand lists, which are not all asserted to arise as complete difference sets.

No actual selected caller, numerical resampling/recovery bound, native emission, movement, precision/storage, compiler closure or complete paid bill is supplied. All five framework assumptions remain. Selected source-binding orientation passed; selected scientific cold replay, Lean and Docker were not run. Independent human and formal proof-kernel review are unclaimed. No required registered local check was left to CI.

Open next questions

• Extend the count to nonprimitive demands. Start with mixed contents modulo 9; the current primitive-kernel disjointness no longer handles different congruence depths.
• Bind a literal resampling/recovery caller to the primitive-difference hypothesis and price coefficient selection and movement. The gcd proof, complete numerical error contract and physical bill remain missing.

Related open work

No other open PR studying this two-coordinate composite-modulus collision count was identified. The completion refresh inspected all 15 open account PRs created since the CLAIM at 2026-11-03T14:02:06Z, including this PR. The fourteen others are #6623–#6631, #6643–#6647. Their stated scopes contain no competing count or blocking threshold for this cyclic projection problem. No extra computation was run to validate another contribution.

• Prove the sharp two-axis coframe cut threshold (no κ) #6624 studies the fifteen-chart binary Lagrangian rank metric and its cut representation. Its two-axis geometry is a different object from cyclic scalar projections; no shared numerical quantity is computed.
• Certify sharp nonlinear five-cycle recovery probabilities (no κ) #6645 studies nonlinear partial recovery on the binary five-cycle with a shared finite message. Its success curve uses a different demand and observer contract; it does not compute this projection density.

The earlier broad cache covered 575 open PR records and 6,651 Discussion records with complete reported pagination. Sparse local notes, arbitrary diffs and unpublished work remain limits. Source PR5564 and the separate Fourier-permutation PR6273 are retained on main; neither supplies this projection formula.

CI notes

Both hosted checks passed at exact head e1af31cf200339333d2bf21c2134ebeddfcf81b3 in run 41651154282: plan and registered package replay. The completed checks were read once. Run metadata independently confirms completed success and the matching head. The registered replay executes both warning-strict normal and optimized commands.

All registered commands, direct normal/optimized checks, committed-byte/source-history preflight, official export integrity and read-only Git-free exported replay also passed locally. No failed or waived check remains. Hosted CI certifies execution of the bounded registered checks; independent human and formal proof-kernel review of the all-modulus written proof remain unclaimed. The selected frontier is unchanged.

Advance CLAIM.

Merge Danger

Door: two-way

Blast Radius: isolated

Seven files in one opt-in research package. No shared implementation, frozen predecessor or selected-result metadata changes.