[petalas]

Summary

Scoped counting proof with an exhaustive finite certificate. Exactly 9,216 of 5,308,416 invertible two-role targets over F3^4 need three paid right-vector factors, a fraction of 1/576.

Projective covering support         ordered matrix targets
18 cyclic C4 cosets                 6,912
 6 Klein V4 cosets                  2,304
Total                              9,216

The proof classifies the four projective actions as cosets of regular subgroups of S4, then retains all 24 component orders and sixteen scalar lifts per support. ToikY's #4334 supplies the cited common-unit-pivot criterion and three-factor upper bound. The new deliverable is this exact ternary population and classification, not an audit of the predecessor. The classical projective/Latin-square facts retain their credit.

This contribution was produced by Nick Petalas. Credit ToikY #4334; Nick Petalas #4276/#4243; Wentao Li #4099 following #2513. Exact historical source pins are in SOURCES.json. Only research/ternary-product-threshold-count is added. No new kappa, supplier admission or selected-frontier change is claimed.

Advance CLAIM. Canonical main remains 624de9d at the pre-commit refresh, priority revision 7. TASK.json explicitly changes connector-alphabet-change to this counting side result; its 945-pairing control passed. Live claims, open PRs and distinctive all-main README/merged searches were checked before committing. The overlap snapshot covers 326 open PRs and 3,748 Discussion records with complete remote pagination. Sparse local notes and unpublished work remain outside that coverage; no novelty certificate is implied.

Evidence

• Before: Prove the sharp component threshold for commutative right-vector targets #4334 proves the sharp q+1 component threshold but does not count the first ternary obstruction population. After: PROOF.md derives the exact cost-three count and cyclic/Klein classification.
• One route enumerates all 10,626 four-element subsets of S4 and identifies precisely 24 covering supports. A separate literal F3 bilinear-mask route enumerates all 48^4=5,308,416 ordered matrix tuples, preserving every scalar lift, and finds 9,216 obstructions. It does not call the projective classifier.
• All 2,304 projective multiplication pairs and 147 controls pass, including repeated components, scalar lifts, incomplete input coverage and singular/wrong-size rejection. Explicit checks remain active under optimization.
• Both registered warning-strict commands pass locally, timed separately at 0.181 seconds ordinary and 0.180 seconds optimized, below their 120-second limits. These are check durations, not performance comparisons. Generated receipts agree and bind the six package files.
• Package metadata/replay, in-memory syntax/JSON/function-capacity checks and staged whitespace checks pass. Committed-byte/source-history preflight and official export integrity pass at 6354da6. Optimized read-only Git-free export replay with PATH=/nonexistent emits the identical receipt and leaves all seven exported files unchanged. No receipt is handwritten.

From the repository root, using CPython 3.12 and only its standard library:

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

Any CPython 3.12 installation can replace the local interpreter path. Expected markers are PASS_TERNARY_PRODUCT_THRESHOLD_COUNT with targets=5308416, cost3=9216 and probability=1/576; PASS_REGISTERED_RESEARCH_PACKAGE; and PASS_RESEARCH_COMMIT_BYTES.

Limitations and unrun obligations

The new proof covers F3, two roles, four ordered field components and invertible endpoints. Its exact-cost interpretation cites the predecessor theorem. Other fields, lengths, singular endpoints and local-algebra lifts are outside scope. Native coefficient preparation, routing, control, restoration, precision, stock and the complete supplier invoice remain unknown. No complete multiplication theorem follows. Lean, Docker, a separate static analyzer and selected-construction operator replay were not run. Every registered local check ran; no local package obligation is deferred to CI.

Open next questions

• Count the cost-three population over F3^k for k>4, beginning with inclusion-exclusion over missing entries in the four-by-four permutation coverage board. Repeated projective components require a different count.
• Count the first obstruction population over F4^5 using sharply transitive subsets of PGL2(F4). Its image is a proper subgroup of S5, so unrestricted Latin-square counts do not apply.
• Specify and price one physical producer/consumer before using the formal factor count in a supplier.

Related open work

• Prove the sharp singular rank-one product threshold (no new κ) #4780, head a4436fa, studies singular rank-one tuples over the same field-product algebras. Every component here is invertible, so endpoint sets and numerical censuses are disjoint; its binary cost-three count 19,440 is not this ternary count 9,216.

• Count rank-one lifts over residue-field extensions (no new κ) #4781, head b838121, counts singular rank-one residue lifts over local extension-field algebras. Its algebra and endpoints differ; no counted target or numerical cost overlaps.

• Prove a binomial law for polynomial right-vector cost (no new κ) #4784, head 4d0a636, counts invertible polynomial targets. F_q[t] differs from the four-component field product, so no degree-conditioned count overlaps this population.

• Prove exact local right-vector costs at every role count (no new κ) #4787, head 4b58e8c, proves invertible costs over local algebras at arbitrary role counts. F3^4 is nonlocal, so its theorem does not determine these targets and no sampled target count overlaps.

• Classify exact rank-two matrix-algebra right-vector costs (no new κ) #4789, head 7fdbf80, classifies singular ordinary-rank-two targets over full matrix algebras. Under a diagonal block embedding these four invertible components have ordinary rank eight, outside its rank-two domain; no target-cost count overlaps.

• Classify radical pencil reachability by characteristic-polynomial splitting (no new κ) #4776, head 0764ab2, classifies zero-residue square-zero radical pencils. Both the coefficient algebra and endpoints differ, so no reachability count overlaps.

• Prove exact local costs for residue-corank-one targets (no new κ) #4804, head da6f858, proves local residue-corank-one costs. Our invertible four-component algebra is nonlocal, so no endpoint or target count overlaps.

• Prove three-factor sufficiency for every two-role matrix-algebra target (no new κ) #4805, head fbf2bd0, proves a universal three-factor upper bound over full matrix algebras. Under diagonal embedding, our cyclic fixture (I, [[0,1],[1,1]], [[1,1],[1,2]], [[1,2],[2,0]]) is an invertible target over M4(F3). A separate literal computation multiplies its three common-direction factors to the full target, agreeing numerically with that upper bound of three. The broader algebra permits additional left coefficients; this compatibility check does not assert equal population counts across algebras.

• Classify exact singular polynomial right-vector costs (no new κ) #4808, head bd22290, classifies singular two-role polynomial targets over K[t]. Our endpoints are invertible over F3^4; both the coefficient algebra and endpoint domains differ, so no target-cost census or population count overlaps.

• Prove sharp five-factor cost for binary three-role products (no new κ) #4810, head 701cb14, proves a five-factor uniform bound for invertible three-role binary field products. The shared field-product setting has a different base field and role count. Its 168 group components and our 9,216 ternary two-role targets count different objects; no numerical population or uniform bound overlaps.

Before DONE, this account's open PRs created since the CLAIM at 2026-10-22T20:15:16Z were listed again. The newly related #4808 and #4810 titles and full bodies were read and added above. Existing related #4804/#4805 retain the same heads as the publication comparison. Other new account PRs concern Schur invariants or guard probabilities and have no shared target-cost population.

These titles and bodies were read. No shared population count is asserted or disagrees. The #4805 three-factor compatibility check was run without predecessor executable code. The account's open PRs created after the CLAIM were also listed before publication. Merged #4334 is the direct predecessor, cited rather than replayed.

CI notes

Both hosted checks passed on contribution head 6354da68f7a78b9e4db91c796476ae9ebae89a86 in Registered research packages run 39848405189: plan in 2m21s and the registered research/ternary-product-threshold-count replay in 1m20s. Run metadata confirms that exact head and conclusion success. The registered replay executes both warning-strict ordinary and optimized commands. All registered local commands, committed-byte/source-history preflight, official export integrity and read-only Git-free replay also passed. Hosted replay certifies the bounded registered checks; independent human and formal proof-kernel acceptance are unclaimed.

Merge Danger

Door: two-way.

Blast Radius: isolated.

Six files in one research package are added. Shared implementation, frozen sources and selected metadata are unchanged.