[petalas]

Summary

Scoped proof answering Open next question 1 of #4276 negatively. In the two-role right-vector alphabet over F2[t], with binary right vectors and free GL2(F2) maps anywhere:

U = [[1,t],[0,1]], L = [[1,0],[t,1]]
M = (UL)^2 = [[1+t^2+t^4,t^3],[t^3,1+t^2]]
cost(M) = 4
cost((UL)^k) = 2k for every k >= 1; delta((UL)^k) = 2

Every invertible target has a unique reduced expression in constant-free directional transvections followed by one binary free map. Its paid cost equals reduced length. Hence reachable two-role targets have unbounded cost even over a commutative coefficient algebra. F2[t] is not semilocal, so the result is outside #4749's three-factor hypothesis. No new kappa or selected-frontier change.

Produced by Nick Petalas. Wentao Li #4099/#2513 supplies the alphabet and column bound; Nick Petalas #4243/#4276 supplies the free-map normal form, one-factor criterion and explicit open question. ToikY #4334 and Nick Petalas #4601/#4749 supply the field-product, local-product and semilocal boundaries. Hirosi Nagao's 1959 polynomial-matrix structure and degree method retain their classical credit. The restricted-alphabet cost consequence and self-contained binary proof are the contribution here; no new group-theoretic normal-form theorem is claimed.

Evidence

• Before: Prove the exact two-factor right-vector criterion at n = 2; the M2(F2) failure lifts to F2[S3] (no new κ) #4276 asks whether every reachable two-role target over every algebra has cost at most three. Prove the semilocal right-vector cost formula (no new κ) #4749 proves the bound under a semilocal hypothesis.
• After: PROOF.md gives an explicit invertible cost-four target, a quantified lower bound permitting arbitrary polynomial coefficients and all free binary maps, and the direct unbounded-cost corollary.
• The finite verifier enumerates all 1,536 invertible polynomial matrices with maximum entry degree at most four. Exact costs zero through four have histogram [6,270,612,504,144]. All 1,008 reduced degree-four expressions of length at most three, including every free endpoint, miss M. The written degree proof justifies this bounded domain for the finite exclusion; it supplies the unbounded statement separately.
• A separate literal alphabet check covers 3,072 left-vector/right-direction parameter tuples and finds 96 invertible tuples, including repeated identities. Every invertible tuple passes constant splitting and all six free conjugations. Seven controls detect changes to the degree assumptions, coefficients, word and endpoint.
• Both registered warning-strict ordinary/optimized commands passed locally, timed at 0.04 seconds each. The package runner passed. In-memory compilation passed; optional external Python analyzers were unavailable. Every function is below 60 lines and every loop has the documented finite bound. The generated receipt hashes all six package files.
• Committed-byte/source-history preflight passed for seven package/input files at d0158a552352c09608295a7ec47dd31327b360ae. Official committed export creation and integrity check passed. An optimized Git-free export replay produced the same marker and complete generated receipt.

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

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

Any CPython 3.12 installation may replace the local interpreter path. Expected verifier marker:

PASS_RIGHT_VECTOR_POLYNOMIAL_COST targets=1536 cost4=144 target_cost=4 shorter_exclusions=1008 controls=7 no_new_kappa

The package runner ends with PASS_REGISTERED_RESEARCH_PACKAGE. Source pins and the original card are bound to main b58ad9ef4d4e861b0eaa5dba41742bdc072dca51. Pre-submission canonical fetch reached 1f87c3578ae546c8f903f7f4d83dc2e2fb99650b; relevant guidance, selected record, catalogue and proof input were unchanged. New-main README searches found no duplicate. The card's first 945-pairing positive control also passed, before the explicit change to this formal-alphabet question.

Limitations and unrun obligations

The infinite claim is a written scoped proof, with finite corroboration. Two roles, base field F2, coefficient algebra F2[t], binary free maps and no scratch are essential to the stated classification. Singular targets, other base fields and multivariate algebras are not classified. Native preparation, conversion, routing, control, precision, restoration, stock and complete supplier costs remain unknown. No native supplier, complete multiplication theorem, independent human acceptance or formal proof-kernel review is supplied. Lean, Docker and the selected operator replay were not run; they are outside the stated result. No registered package check remains unrun locally.

Open next questions

• Classify costs over F2[s,t], starting with one named reachable word. General reachability and a multivariate normal form are separate obligations.
• Identify one literal native producer/consumer coefficient algebra, then price left-coefficient preparation and the complete physical bill before using the abstract factor count in a supplier.

Related open work

• Prove the semilocal right-vector cost formula (no new κ) #4749, head 29e4d3a25225a219f97c3a2815ed3a5a8fd9b891, proves a three-factor bound for invertible semilocal targets. F2[t] lies outside that hypothesis. Our six constant binary free targets agree numerically with the six cost-zero targets in its reported binary dual-number compatibility control; no other target census overlaps.
• Prove exact square-zero radical reachability and binary count (no new κ) #4745, head 3edea13af13b755932852e9306a714b317630dda, classifies radical targets over split square-zero local algebras. Its singular radical endpoints are disjoint from our invertible polynomial endpoints; no shared target count is claimed.

Merged #4601/#4613 concern invertible local algebras and finite controls. Their local target domains are disjoint from the polynomial domain here. Main/merged searches used the alphabet, polynomial algebra, explicit matrix, unbounded cost and Nagao terms. The overlap helper reported 521 open PR bodies and 3,679 Discussion records with complete pagination at refresh. Its local-note coverage is sparse, and arbitrary diffs and unpublished work are not exhaustively covered. Live open PRs and recent handoff claims were rechecked immediately before committing.

Claim: #1166 (comment)

Related open work refresh before DONE

This account's open PRs created since the CLAIM at 2026-10-22 19:18:43 UTC were listed, and every returned title and body was read. Besides this PR, they are #4761, #4764, #4766, #4767, #4769, #4770 and #4771. The three studying the same right-vector alphabet are:

• Prove a cube-zero radical target has exact cost three (no new κ) #4764, head b1b163cd5553c75c9bd5ca3ac31a2eab4e2f3511, proves exact cost three for a singular radical target over K[x,y]/(x,y)^3. Its finite local algebra and singular target differ from our polynomial algebra and invertible targets. No target-cost or census number overlaps.
• Count square-zero radical targets over every finite field (no new κ) #4766, head 5af823b1a14b2517235925145e36581bfdfcad41, counts radical targets over finite split square-zero algebras. Its zero-residue singular endpoints are disjoint from our invertible polynomial endpoints. No cost or census number overlaps.
• Count rank-one residue lifts over local algebras (no new κ) #4770, head 39ef149ba3ea8adbda18478ea227d3d0826475ab, counts lifts of rank-one residues over finite-dimensional local algebras. Those singular targets and coefficient algebras are outside the present invertible F2[t] domain. No target-cost or census number overlaps.

#4761 concerns chain retiming, #4767 and #4769 concern full-guard header probabilities, and #4771 concerns scalar-pivot Schur dimensions. They study different objects and quantities. No conflicting overlapping number was found. The earlier six-constant-target compatibility agreement with #4749 remains recorded above.

CI notes

Both hosted checks passed on d0158a552352c09608295a7ec47dd31327b360ae in Registered research packages run 39839662719: plan passed in 2m53s and the registered research/right-vector-polynomial-cost replay passed in 1m6s. The run metadata confirms that exact head and conclusion success. The hosted replay certifies the registered finite checks within their stated scope. It does not provide independent human acceptance or formal proof-kernel verification of the written infinite theorem. Local checks and limits remain recorded above.

Merge Danger

Door: two-way.

Blast Radius: isolated.

Only the six new package files are added. Shared code, frozen sources, predecessor credit and the selected record are unchanged.