[petalas]

Summary

Complete finite certificate with a scoped algebraic consequence: five is the exact uniform paid right-vector bound for invertible three-role targets over binary field products F2^k, allowing arbitrary common free GL3(F2) maps anywhere.

Every ordered four-direction word: at most 144 of 168 GL3(F2) endpoints.
Fixed five-direction word e1,e2,e3,e1,e2: all 168 endpoints.
Target listing all 168 group matrices: exact cost 5, column lower bound 3.

The upper bound applies to every positive k. Sharpness holds at k=168 and every larger k by padding; minimal component count is not claimed. This extends ToikY's #4334 shared-direction field-product question from two to three roles. PROOF.md proves that absorbing all free maps at the end preserves the lower obstruction, and that independent left coefficients assemble the fixed upper word componentwise.

This contribution was produced by Nick Petalas. Credit ToikY #4334; Nick Petalas #4243/#4265/#4276/#4749; and Wentao Li #4099 following #2513. Classical short register-update work by Gadouleau, Riis, Cameron and Fairbairn is credited and linked in PROOF.md. Generic short update programs are not claimed as new. Predecessor files and attribution are unchanged. No new kappa, supplier admission or selected-frontier change.

Evidence

Before, the cited product-field theorem gives a sharp two-role bound of three. After, the new complete shared-direction certificate gives a sharp three-role bound of five and a gap of two above the column bound.

• Two implementations compare every full endpoint set for all 2,401 four-direction words. The second works with permutations of the eight vectors and explicitly enumerates all 614,656 direction/parameter tuples. Histogram by reachable-set size: {4:7,16:126,24:420,64:504,96:672,144:672}.
• The fixed upper word has 1,024 parameter tuples and reaches every group element. All 168 chosen words pass all eight inputs, totaling 1,344 component/input evaluations. All 168 free right translations and delta cases and 4,704 labelled factor/free conjugations pass. Seven negative controls reject invalid factors, missing or forged coverage and an omitted nonidentity factor.
• Both warning-strict registered commands pass locally on CPython 3.12.15, separately timed at 0.11 seconds ordinary and 0.11 seconds optimized. These are check durations. Their complete generated receipts agree and bind all six package files and both pinned predecessor inputs.
• Package metadata and both registered replays pass. Eight-file committed-byte/source-history preflight, official committed-export integrity and optimized read-only Git-free export replay with PATH=/nonexistent pass on 701cb14b7c461253d7ac8243d0722996918bc6c6. The exported replay emits the identical receipt. In-memory syntax/JSON/function-size and staged whitespace checks pass. No receipt is handwritten.
• Internal read-only proof/code review found no scientific blocker. The local decision trail was completed and final receipts refreshed after its documentation observations. Independent human and formal proof-kernel review are unclaimed.

Exact reproduction from the repository root, with CPython 3.12 and only its standard library:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B -W error research/right-vector-three-role-products/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -O -B -W error research/right-vector-three-role-products/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/right-vector-three-role-products
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/right-vector-three-role-products --check-source-history

Any CPython 3.12 installation can replace the displayed path. Expected markers are PASS_RIGHT_VECTOR_THREE_ROLE_PRODUCTS group=168 words4=2401 max4=144 upper=5 delta=3 controls=7, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Canonical main was refreshed before committing and remained 624de9dc66858af0945e5f8a0942488d62280d43, priority revision 7. TASK.json explicitly changes the connector-alphabet card to this formal-algebra side question. Its 945-pairing orientation control and the current selected source-binding baseline pass. Live claims, open PR bodies, distinctive all-main README searches and merged-PR searches were checked before committing. The final overlap snapshot covers 335 open PRs and 3,761 Discussion records with complete remote pagination; sparse local-note coverage is incomplete. These searches do not certify novelty.

Advance CLAIM.

Limitations and unrun obligations

The certificate is exact for binary three-role invertible field-product targets. The all-k consequence uses the written componentwise reduction. Other fields, singular endpoints, extra roles and minimal component count are not classified. Native coefficient preparation, access, routing, storage, reversible restoration, precision, stock and the complete recursive invoice remain unknown. No complete multiplication theorem follows.

Independent human and formal proof-kernel review are unclaimed. Lean, Docker and selected-construction operator replay were outside scope and unrun. CPython's bounded managed allocation is documented. Every registered local command ran; no package check is deferred to CI.

Open next questions

• Determine the minimum k permitting cost five over F2^k. Start with complements of the existing four-word endpoint sets and their common right translates. The present universal witness is not claimed minimal.
• Specify one literal native producer/consumer and fully price coefficient acquisition, overwrites, restoration and the rest of its invoice before using the formal factors in a supplier.

Related open work

The following same-alphabet titles and bodies were read. Different role or coefficient-algebra domains are not treated as equal cost populations.

• Prove exact local right-vector costs at every role count (no new κ) #4787 proves exact arbitrary-role costs over local algebras. F2^168 is nonlocal. At the shared single-field three-role boundary, a separate literal check gives cost=delta=0 for all 168 GL3(F2) matrices, agreeing with its formula.
• Prove exact local costs for residue-corank-one targets (no new κ) #4804, head da6f858f21d407648a711d8fd369c37a9c6df24e, proves local residue-corank-one costs, including singular endpoints. Our endpoints are invertible and our sharp algebra is nonlocal; no target-count or exact-cost instance overlaps its main domain.
• Prove three-factor sufficiency for every two-role matrix-algebra target (no new κ) #4805, head fbf2bd033f2d3ae1605a3cad690e1fc2fbf477e7, proves a three-factor upper bound for every two-role full-matrix-algebra target. This result has three roles and a field-product algebra; no common target-cost census or numerical uniform bound is asserted.
• Prove the sharp singular rank-one product threshold (no new κ) #4780 studies two-role singular rank-one products. Our three-role components are invertible; its thresholds and counts have no shared target domain here.
• Count 9,216 ternary invertible threshold targets (no new κ) #4809, head 6354da68f7a78b9e4db91c796476ae9ebae89a86, counts two-role invertible F3^4 obstruction targets. Both field and role count differ; its 9,216 targets and our 168 group components count different objects, with no shared numerical population.
• Classify exact rank-two matrix-algebra right-vector costs (no new κ) #4789 classifies two-role singular ordinary-rank-two full-matrix targets. Both role and coefficient-algebra contracts differ; no exact-cost census overlaps.
• Count rank-one lifts over residue-field extensions (no new κ) #4781 counts two-role singular local residue lifts. Both endpoint and coefficient-algebra domains differ; no lift count overlaps.
• Prove a binomial law for polynomial right-vector cost (no new κ) #4784 gives invertible two-role polynomial degree/cost distributions. The role count and coefficient algebra differ; no degree or target population overlaps.
• Classify exact singular polynomial right-vector costs (no new κ) #4808, head bd22290761451093aa7aef1fa7b1421627ab0f37, classifies singular two-role polynomial costs. Both role count and endpoints differ; no exact-cost census overlaps.
• Classify radical pencil reachability by characteristic-polynomial splitting (no new κ) #4776 classifies zero-residue square-zero radical pencils at arbitrary role counts. Those singular endpoints and nonreduced local coefficients differ from the invertible reduced field products here; no pencil count overlaps.

No disagreement was found on the shared single-field boundary. Merged #4334 is the direct product-field predecessor; it is cited, not replayed as this deliverable.

Before DONE, this account's open PRs created since the CLAIM at 2026-10-22T20:16:40Z were listed again. Nine were returned, including this PR. The four related factor-cost contributions #4804/#4805/#4808/#4809 retain their publication heads; their current bodies were compared with the earlier snapshots and read. Their main domains differ as recorded above, so no new shared numerical population requires comparison. The existing shared single-field check with #4787 gives cost=delta=0 on all 168 GL3(F2) matrices. Other newly open account PRs #4801/#4803/#4806/#4807 concern Schur invariants or full-guard probabilities. No conflicting overlapping value was found within this coverage.

CI notes

Both hosted checks passed on contribution head 701cb14b7c461253d7ac8243d0722996918bc6c6 in Registered research packages run 39848816504: plan in 3m5s and the registered research/right-vector-three-role-products replay in 1m9s. Run metadata confirms that exact head and conclusion success. The registered replay executes both warning-strict ordinary and optimized commands. All registered local checks, committed-byte/source-history preflight, official export integrity and read-only Git-free replay also passed. Hosted CI certifies execution of these bounded checks; independent human and formal proof-kernel acceptance of the scoped algebraic consequence remain unclaimed. No failed or waived package check remains.

Merge danger

Door: two-way. Blast radius: isolated. Only six files in one new research package are added. Shared implementation, frozen evidence, predecessor attribution and the selected record are unchanged.