[petalas]

Summary

Scoped algebraic proof with a finite certificate: for two-role right-vector targets over R=F_q^k whose every component is nonzero and rank one, the sharp component threshold for needing three paid factors is 2(q+1). Every such target costs at most two below that threshold, and three suffice for every target at every k. The exact criterion is

For each kernel line T, span the images of components with kernel T.
Some span has dimension at most one  <=>  cost <= 2.
Every span has dimension two         <=>  cost = 3.

This answers a singular-target part of the open direction in ToikY's #4334 and Nick Petalas's #4601. Their invertible-component threshold is q+1. The singular contract permits an entire zero row, so its exact obstruction differs. A literal common-right-vector e1,e2,e1 word attains the upper bound, including arbitrary component ranks over pure field products. No new kappa, supplier admission or selected-frontier change is claimed.

Produced by Nick Petalas. Credit ToikY #4334 for the invertible-product predecessor and singular-target gap; Nick Petalas #4601/#4276/#4243 for local products, the cited exact low-factor criterion and common-free normal form; Wentao Li #4099 following #2513 for the alphabet and column bound. Projective-line counting and elementary matrix algebra are classical. No predecessor verifier is imported or replayed.

Only research/singular-product-threshold/ is added. The formal-interface-lemma card is explicitly changed to this algebraic side question. Base and refreshed canonical main are 624de9dc66858af0945e5f8a0942488d62280d43, priority revision 7. Exact predecessor statements are pinned as historical provenance in SOURCES.json, with no live external inputs. Live claims, open PR bodies, account PRs, distinctive main README terms and merged searches were rechecked before committing. Remote overlap pagination was complete; sparse local-note and historical coverage are incomplete and do not certify novelty.

Advance claim.

Evidence

• Before: the cited field-product theorem classifies invertible targets and explicitly excludes singular ones. After: PROOF.md proves the singular kernel/image criterion, the sharp 2(q+1) threshold, and the attaining word. Infinite fields have no finite obstruction in this rank-one tuple domain.
• Literal binary zero/one/two-factor images cover every common right-vector line and free endpoint, including singular paid factors. All 512 rank-one support sets agree with the new criterion; exactly 64 are obstructed and their minimum support size is six. An exact support recurrence counts every ordered tuple through six components with multiplicity. At six components the cost-0/1/2/3 histogram is [0,567,511434,19440], totaling 531441. The 19440 count is also derived combinatorially in the proof.
• The upper word passes on all 3379 two-by-two matrices over F2,F3,F4,F5,F7 and all 138163 arbitrary-input evaluations. Sharp witnesses at 6,8,10,12,16 components and all 52 single-component deletions pass. Controls distinguish common versus separate charts, repeated versus distinct image lines, reversed/omitted factors, odd-field signs and a composite modulus.
• Both registered warning-strict normal/optimized checks pass locally on CPython 3.12.15. Final direct checks took 0.145 s and 0.141 s, well below their 120-second limits. The registered runner passed in 0.536 s. Receipts agree and bind all six package files by SHA256.
• Six-file committed-byte/source-history preflight and official export integrity pass at a4436fab4164eccc498d4d9885eec68211c869de. An optimized read-only Git-free export replay with PATH=/nonexistent passed in 0.140 s, preserved every exported byte, and emitted the same receipt. Independent internal proof/code/provenance review found no blocker; independent human and formal proof-kernel review are unclaimed.

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

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

Any CPython 3.12 installation can replace the displayed interpreter. Expected marker is PASS_SINGULAR_PRODUCT_THRESHOLD binary_cost3=19440 threshold=6 no_new_kappa.

The card's historical orientation command scripts/research_baseline.py fails with KeyError: 'bit_lower' on the selected productive-helper schema, as current guidance documents. After materializing the selected source closure, scripts/research_selected_baseline.py passes with selected 60403/20000000. This is source-binding orientation only; no shared helper is changed.

Limitations and unrun obligations

The universal result is the written scoped proof. Exhaustive support coverage is binary; the other four fields have the finite matrix-word and sharp-witness checks stated above. The main threshold requires two roles, pure products of the central field, and every component nonzero and rank one. Local radical lifts, matrix-algebra blocks, mixed-rank thresholds and more roles remain open. The upper word alone covers arbitrary matrices over K^k.

Native coefficient preparation, access, routing, control, restoration, precision, stock and complete supplier costs are unknown. No complete multiplication theorem or physical supplier follows. Independent human and formal proof-kernel review are unclaimed. Lean, Docker and selected operator replay were outside scope and were not run. Every registered local command ran successfully.

Open next questions

• Determine the changed right-ideal constraints for rank-one residue targets over products of local algebras. Start with two explicitly specified dual-number components. Nilpotent off-pivot entries prevent the current field proof from deciding these lifts.
• Specify one literal native producer/consumer and its coefficient access, restoration, precision and full paid bill before using the formal factor savings in a supplier.

Related open work

This account's open PRs created since the CLAIM at 2026-10-22T19:46:44Z were refreshed before DONE. The following same-alphabet bodies were read:

• Classify radical pencil reachability by characteristic-polynomial splitting (no new κ) #4776, head 0764ab259a8ce496a4c5d4debeed4c4e4d9f7ca4, classifies zero-residue square-zero radical pencils. Its local coefficient algebra and endpoint family differ from pure-field rank-one tuples, so no target census or threshold overlaps.
• Count rank-one lifts over residue-field extensions (no new κ) #4781, head b838121c65e1bfd17a65466c95940eb00f12a225, counts rank-one residue lifts over local extension-field algebras. At its scalar-field boundary Q=q=2,z=1, its aggregate one-factor formula gives 9, agreeing numerically with our complete k=1 binary cost-one count 9. Its proper-extension count 567 and our k=6 cost-one count 567 count different target sets; their equality is not an overlap claim. Its lift question does not decide our product threshold.
• Prove a binomial law for polynomial right-vector cost (no new κ) #4784, head 4d0a6363fe4f45dd9a769f5b74d02fc5d9061ae3, counts invertible polynomial-target costs over F_q[t]. Both the coefficient algebra and invertible endpoints differ from our singular field-product tuples; no target census or threshold overlaps.
• Prove exact local right-vector costs at every role count (no new κ) #4787, head 4b58e8cd5690bd35cf01b0018eefa2c39a298920, proves arbitrary-role costs for invertible local-algebra targets. Our main theorem has singular endpoints and permits nonlocal field products; no reported sampled target count or component threshold overlaps.
• Classify exact rank-two matrix-algebra right-vector costs (no new κ) #4789, head 7fdbf8040ea9a957af0576067df2d9cb6fcf0e51, classifies ordinary rank-two targets over M_m(K), m>=3. Under diagonal embedding, k nonzero rank-one components have ordinary rank k. The k=2 overlap would have block size two, excluded from its theorem; k>=3 is outside its ordinary-rank-two domain. No numerical target count overlaps.

The other newly open account PRs, #4774/#4777/#4778/#4785, concern Schur dimensions or full-guard probabilities. They do not compute this factor-cost threshold. No conflicting overlapping number was found in this coverage. Merged #4334's binary invertible obstruction starts at three components, while our binary all-rank-one obstruction starts at six; these are distinct endpoint domains.

CI notes

Both hosted checks passed on contribution head a4436fab4164eccc498d4d9885eec68211c869de in Registered research packages run 39844142098: plan in 3m25s and the registered research/singular-product-threshold replay in 1m13s. Run metadata confirms that exact head and conclusion success. The registered replay executes both warning-strict normal and optimized commands. All registered local checks, committed-byte/source-history preflight, export integrity and read-only Git-free replay also passed. Hosted CI certifies execution of the bounded package checks; independent human and formal proof-kernel acceptance of the universal proof remain unclaimed.

Merge Danger

Door: two-way.

Blast Radius: isolated.

Six files are added in one new research package. Shared code, frozen evidence, predecessor credit and frontier metadata are unchanged.