[petalas]

Result

Counterexample plus finite certificate. No new κ. This answers open question 1 of #4243: is the paid-factor cost of every target M in the right-vector alphabet (I + u y^T, u in R^n, y in K^n, free GL_n(K) maps anywhere) equal to delta(M) = min_T dim_K (M T - I) K^n?

• No in general. Over the noncommutative algebra M2(F2) at n = 2, 72 of 65,536 targets cost 3 while delta = 2. One witness is [[E11, E01], [E10, E00]], which in M_4(F_2) is the transposition of coordinates 0 and 3. A separate M_4(F_2) enumeration confirms that no two-factor word reaches it and a three-factor word does.
• Exact on ten commutative domains (278,274 targets): F2, F3 (n=2), F2 (n=3), F4, F2[C2], F2 x F2, F3 x F3, F2[C3], F2[x]/(x^3) (n=2) and F2[C2] (n=3).
• The minimum over T is necessary. With no free map, cost exceeds dim_K (M - I) K^n on 2 targets over F2 x F2 ((A, A^{-1}), A of order three), 6 over F2[C3] and 684 over M2(F2).

Claim types: the M2(F2) result is a counterexample to the proposed formula; the commutative agreement and the no-free-map gaps are finite certificates on the listed domains. No general theorem is claimed. Package: research/right-vector-general-target.

Evidence and reproduction

verify.py builds each algebra from a structure table, checks associativity and that K is central, computes exact minima by breadth-first search over #4243's normal form Q S, and computes delta over every T in GL_n(K). It asserts the #4243 lower bound on every target and the exact failure and gap counts.

~/git/integer-mult-bounds/.venv/bin/python3.12 -B -W error research/right-vector-general-target/verify.py --stage main
~/git/integer-mult-bounds/.venv/bin/python3.12 -B -W error research/right-vector-general-target/verify.py --stage large
~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/right-vector-general-target
~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/right-vector-general-target --check-source-history

Local results (CPython 3.12, NumPy 2.5.4, macOS arm64): --stage main prints PASS_RIGHT_VECTOR_GENERAL_TARGET ... formula_failures=M2(F2):72 no_free_map_gaps=F2xF2:2,F2[C3]:6,M2(F2):684 no_new_kappa in about 74 s; --stage large prints PASS_RIGHT_VECTOR_GENERAL_TARGET_LARGE F2[C2]:n3:262144 formula_failures=0 no_free_map_gaps=0 no_new_kappa in about 110 s. research_package.py prints PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=completed, and check_research_commit.py prints PASS_RESEARCH_COMMIT_BYTES.

Limits

An exact count in a formal alphabet. Native frames, hosting, routing, precision and recursive cost are not priced; no supplier changes. Commutative agreement is checked only on the listed domains. Transfer of the M2(F2) failure to a group algebra such as F2[S3] is not checked. No proof obligation registered here is left unrun. Independent human and formal proof-kernel review are unclaimed.

Open next questions

1. Is delta exact for every commutative K-algebra, or every local one with residue field K? The README states the one-factor peeling condition that a proof for a minimizing T would need.
2. Does the failure lift to F2[S3], which has M2(F2) as a block?
3. What replaces delta over M2(K)? Compare cost with ranks in M_2n(K).

Related open work

• Prove free basis changes keep the right-vector diagonal count (no new κ) #4243 (head f0cde9eef1808d5b540d2aaa2cc3adddb1bf7197) asked this question and proves the diagonal case. Over M2(F2) at n = 2, all 256 diagonal targets have cost equal to delta, and all 72 failures have a nonzero off-diagonal entry, checked numerically. This is consistent with its diagonal theorem. Its F2[C2 x C2], F2[S3] and F3[x]/(x^2) domains are not rerun here.
• No other open PR studies this alphabet (checked the open PR list, a right-vector search over all PRs, merged package READMEs and the latest handoff comments).

CI notes

Hosted CI is green on head 665bfe79d726a0cb3494326d2f63ae81d313302f: run 39506725266. plan passed in 2m51s and the package replay in 7m56s. No check failed.

Credit

Produced by Nick Petalas. The alphabet, normal form and lower bound come from #4243; the column bound and right-vector alphabet are Wentao Li's research/f2-right-vector-diagonal-minimum (#4099), building on research/bit-cyclic-root-barrier (#2513). Apache-2.0.