[petalas]

Summary

A scoped counting proof answers Open next question 2 of #4745: the exact number of reachable radical two-role right-vector targets over every finite field and every square-zero radical dimension. The predecessor reachability theorem is a cited input; the new contribution is the all-field count.

For a split central algebra R=F_q⊕J with J²=0, d=dim J, z=q^d and A_r(z)=∏_(i<r)(z-q^i):

unreachable = A_4(z) + q(q²-1) A_3(z) + q²(q-1)² A_2(z)/2
reachable   = z⁴ - unreachable

At q=3,d=2 this gives 5,697 reachable and 864 unreachable targets. Every reachable target has formal paid cost two, and every other radical target is unreachable at arbitrary finite word length, by #4745. No new κ or selected-frontier change.

Produced by Nick Petalas. Credit Nick Petalas #4745/#4276/#4243, ToikY #4334, and Wentao Li #4099 following #2513. The matrix-space, irreducible-polynomial and external-line counts are classical; their application to this formal-alphabet count is the scoped contribution. README.md includes a self-contained enumeration proof and a classical geometry reference.

Evidence

• Before: Prove exact square-zero radical reachability and binary count (no new κ) #4745 proves the all-field criterion and binary count, and explicitly leaves the nonbinary count open.
• After: the written proof classifies every rank-one-free annihilator by dimension and counts its spanning coefficient tuples. Literal checks cover 6,832,236 target tuples over F₂,F₃,F₄,F₅,F₇ and 5,580 subspaces, plus irreducible-matrix censuses and seven control groups. The binary d=4 case exercises the full-span term and gives 29,836 reachable and 35,700 unreachable targets.
• Both registered local commands passed on CPython 3.12, separately timed at 0.79 s and 0.81 s. Normal and optimized generated receipts are identical. The package runner passed. Five-file committed-byte/source-history preflight, official export integrity and optimized replay from a read-only Git-free export with PATH=/nonexistent passed on 5af823b1a14b2517235925145e36581bfdfcad41.
• No receipt is handwritten. The verifier writes nothing and generates SHA256 bindings for all five package files. It imports no predecessor executable and needs no unmerged Git object to run.

Exact reproduction from the repository root, using the CPython 3.12 standard library:

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

Another CPython 3.12 path may replace the local path. Markers are PASS_RADICAL_ALL_FIELD_COUNT, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

The package is pinned to main b58ad9ef4d4e861b0eaa5dba41742bdc072dca51. Pre-submission fetch reached 1f87c3578ae546c8f903f7f4d83dc2e2fb99650b; instructions, priorities, catalogue, family status and selected record were unchanged. Live open PR bodies, recent claims, merged PR searches and all-main README searches were rechecked before committing. The overlap helper covered 521 open PRs and 3,679 Discussion records with reported complete pagination; local note coverage was sparse and incomplete. Searches are evidence of coverage, not proof of novelty.

Limitations and unrun obligations

The universal result is the written counting proof, conditional on the cited #4745 reachability theorem. Finite corroboration implements only q=2,3,4,5,7 and the stated bounded dimensions. The algebra must be split, have scalar residue F_q, two roles and J²=0; only targets in M₂(J) are counted. Higher nilpotency, other singular residues and more roles are excluded.

Native preparation, chart selection/emission, routing, restoration, stock, precision and the complete consumer bill remain unknown. No supplier admission or complete multiplication theorem follows. Independent human and formal proof-kernel review are unclaimed. Lean, Docker and selected cold replay were not run. All registered package obligations ran locally. The historical card baseline stopped at omitted sparse input research/fresh-helper-thinned/PROOF.md; no baseline pass is claimed. The card's change to this algebraic side question is explicit in TASK.json.

Open next questions

• Count the distribution of projective common bilinear-zero pairs, beginning with annihilator intersections with the determinant quadric. A restricted chart-choice invoice is still missing.
• Determine second-order obstructions when J³=0, as already proposed by Prove exact square-zero radical reachability and binary count (no new κ) #4745. Prove a cube-zero radical target has exact cost three (no new κ) #4764 addresses one concrete higher-nilpotency target; its ambient algebra differs from this theorem's square-zero hypothesis.

Related open work

• Prove exact square-zero radical reachability and binary count (no new κ) #4745, head 3edea13af13b755932852e9306a714b317630dda, supplies the reachability theorem and binary counting law. Our literal binary counts agree numerically: (244,12) at d=2 and (3004,1092) at d=3. This PR extends its count to every q; it does not recheck its arbitrary-word theorem.

• Prove the semilocal right-vector cost formula (no new κ) #4749, head 29e4d3a25225a219f97c3a2815ed3a5a8fd9b891, studies invertible semilocal right-vector targets. Those targets are disjoint from our zero-residue radical targets, so no census number overlaps.

• Prove unbounded two-role right-vector cost over F2[t] (no new κ) #4763, head d0158a552352c09608295a7ec47dd31327b360ae, gives unbounded costs over F₂[t]. The infinite polynomial ring and invertible targets differ from the finite square-zero radical family here; no cost or count overlaps.

• Prove a cube-zero radical target has exact cost three (no new κ) #4764, head b1b163cd5553c75c9bd5ca3ac31a2eab4e2f3511, studies a cost-three target over F_q[x,y]/(x,y)³. Its ambient radical has nonzero square, outside this theorem. At q=2, our direct bilinear masks give zero passing projective pairs out of nine for its three coefficient matrices E11, E12+E21, E22. This agrees numerically with its statement that the same target is unreachable in the smaller square-zero subalgebra K+J²; the larger ambient algebra enables its three-factor word.

• Count rank-one residue lifts over local algebras (no new κ) #4770, head 39ef149ba3ea8adbda18478ea227d3d0826475ab, counts lifts of nonzero rank-one residue matrices over local algebras. Our endpoints have zero residue, so the target sets and census totals are disjoint.

• Prove exact rank-one matrix-algebra cost at every role count (no new κ) #4772, head 116992ef72c8aa9f8994137b04395584d3b350e8, proves exact costs for ordinary rank-one targets over matrix algebras M_m(K), m>=2. This coefficient-algebra family differs from our scalar-residue square-zero algebras; no target-cost or census number overlaps.

• Prove an all-length radical pencil obstruction for any role count #4773, head f5d0f421efec55c5f4b3dbb142eee1e79a4e21db, proves a multi-role necessary obstruction for xI+yA. Its executed 48 excluded three-role pencils are outside our two-role census. For the overlapping two-role specialization, our direct binary enumeration finds exactly two A with no common bilinear zero for I,A. In particular A=[[0,1],[1,1]] gives zero passing pairs, agreeing numerically with its irreducible-pencil obstruction. This compatibility calculation uses only this PR's verifier, not that predecessor's executable.

Related open work refresh

Before DONE, this account's open PRs created since the CLAIM at 2026-10-22 19:17:23 UTC were refreshed. Besides this PR, they are #4761, #4763, #4764, #4767, #4769, #4770, #4771, #4772 and #4773. Related right-vector work and numerical compatibility checks are recorded above. #4761 studies chain retiming; #4767/#4769 count corank-one headers; #4771 studies Schur dimensions. Their quantities do not overlap this radical-target count. No numerical disagreement was found in the shared tiny examples.

CI notes

Both hosted checks passed on 5af823b1a14b2517235925145e36581bfdfcad41 in Registered research packages run 39839900577: plan passed in 2m50s and the registered research/radical-all-field-count replay passed in 1m25s. Hosted CI certifies the registered checks at this exact commit within their stated finite scope. It does not supply independent human or formal proof-kernel acceptance of the universal theorem. All registered local checks, committed-byte/source-history checks and Git-free export replay also passed.

Merge danger

Door: two-way. Blast radius: isolated. Only the five files in research/radical-all-field-count/ are added; no shared implementation or selected record changes.