F-star family gap: the twelve tied_min v = 11 sub-decoders settled exactly - gap_named = 2 EXISTS below v = 12 (no kappa)

[navinor]

F-star family gap: the twelve tied_min v = 11 sub-decoders settled exactly - gap_named = 2 EXISTS below v = 12 (no kappa)

Scope

Kappa-blocker KB5i (authorized explicit open request from the handoff thread; scope changed from the drawn coverage-law-change card and recorded in TASK.json): settle the TWELVE OPEN tied_min v = 11 sub-decoders of open problem F-star's family-gap question (cert(M) >= floor(2v/3) for exact 3-total rational decoders) — the exact recorded remainder of PR 2936 (its open questions 1-2) — and push box-completeness with PR 2936's Lemma BPP projection-parity obstruction as filter. Package research/fstar-gap2-v11-seed42ff0611/. F-star itself is not proved and not refuted. Every claim is labelled PROVEN / CHECKED / DEMOTED.

Verdict (the question this settles)

Does gap_named = 2 exist below v = 12? YES - PROVEN, on all 12 records. The 12 tied_min v = 11 sub-decoders (the complete v = 11 family of the v = 12 fixture) all have named = 9 exactly and cert = 11 exactly, so gap_named = 2 on 12 of 12. The minimal exact gap-2 instance drops from tied_min v = 12 (the PR 2901/2914 record) to tied_min v = 11 (e.g. tied_min|0,1,2,3,4,5,6,7,8,9,10); PR 2936's record excludes gap 2 on every enumerated record below v = 11, so v = 11 is the minimal size.

label	named	cert	box	gap_named	PR 2936 stored cert
tied_min|0,1,2,3,4,5,6,7,8,9,10	9	11	11	2	10
tied_min|0,1,2,3,4,5,6,7,8,9,11	9	11	11	2	10
tied_min|0,1,2,3,4,5,6,7,8,10,11	9	11	11	2	11
tied_min|0,1,2,3,4,5,6,7,9,10,11	9	11	11	2	10
tied_min|0,1,2,3,4,5,6,8,9,10,11	9	11	11	2	11
tied_min|0,1,2,3,4,5,7,8,9,10,11	9	11	11	2	10
tied_min|0,1,2,3,4,6,7,8,9,10,11	9	11	11	2	11
tied_min|0,1,2,3,5,6,7,8,9,10,11	9	11	11	2	11
tied_min|0,1,2,4,5,6,7,8,9,10,11	9	11	11	2	10
tied_min|0,1,3,4,5,6,7,8,9,10,11	9	11	11	2	11
tied_min|0,2,3,4,5,6,7,8,9,10,11	9	11	11	2	10
tied_min|1,2,3,4,5,6,7,8,9,10,11	9	11	11	2	9

All values exact: named by the DP below with replayed construction witnesses; cert = box = 11 by replayed ceiling witnesses of size v (the ceiling for both rungs). The last column is PR 2936's witnessed lower bound.

Delta

1. PROVEN - exact named (PR 2936's open question 2). The Lemma MX construction space (product 1,977,326,743, the recorded obstruction) is settled by a coarse-assignment DP: the construction value |A| + |U| + sum_c MAS(class_c) does not depend on the repair labels a*, and the two Lemma MX hypothesis sets (A-first / T-first) factor per U line given only the kinds (A / U / T(c)), so repairs fold into per-line feasibility masks and the search enumerates 3^11 = 177147 coarse assignments per record. The optimum is 9 everywhere, with replayable witnesses (combo + repairs + mode). The port is pinned to the vendored engine's own mx_eval semantics by agreement tests (exact equality with the engine's exhaustive product enumeration on 72 small fixture sub-decoders and 12 random decoders). CL3 = MP exactly at even tau (written proof in the README, machine-checked): anchor pairs never certify at even tau, so the CL3 objective is exactly Lemma MP's.
2. CHECKED (cross-check with correction). PR 2936's stored cert values under-report the true cert = 11 on 7 of 12 records (stored 9 or 10): the cert-side companion of the named scan pitfall. Every stored value is reproduced or corrected with a replayed witness in data/named-v11-receipt.json.
3. CHECKED - box = cert = 11 exactly on all 12 (the weak cell of PR 2936's sweep closed; four records were open at box, cert in {10,11}); every attaining witness stored is box-valued. On 60 new fixture-density tau >= 1 instances at v = 4..6 (GL_3(Z) column transforms of both fixtures plus restrictions — the generator PR 2936's random sweep lacked; both rungs exact): box = cert on all 60, 0 violations.
4. PROVEN - Lemma BPP's obstruction is real at tau = 0: smallest instance first, the v = 2 decoder P = [[1,0,1],[2,1,0]], Q = [[1,0,0],[1,0,1]] has an attaining witness of value 2 whose sub-anchor projection is 1, failing BPP condition (i) exactly as the lemma predicts. CHECKED (bounded): at tau >= 1 no strict projection drop among 21 escaping witnesses (v = 3..5). Box-completeness for tau >= 1 stays NOT PROVEN, NOT REFUTED; Lemma BPP remains the exact obstruction.
5. DEMOTED (retained) - matched-pivot floor. MP = 9 < cert = 11 on all 12 (twelve new exact counterexamples to named = MP = cert); MP >= floor(2v/3) holds everywhere measured (evidence, not a proof). Not resurrected.

Reproduce

python3 -B research/fstar-gap2-v11-seed42ff0611/verify.py
python3 scripts/research_package.py --package research/fstar-gap2-v11-seed42ff0611 --check-only
python3 -B research/fstar-gap2-v11-seed42ff0611/test_named_exact.py

Expected markers: PASS_KB5I_PINS files=14, PASS_FG_CONTROLS, PASS_FG_CALIBRATION tied_min=11/12 cex_min_1=18/19 mp=9/16 chain=22 reach=0/12,8/19, PASS_KB5I_NAMED records=12 named-exact=12 named=9x12 gap2-proven=12, PASS_KB5I_CROSSCHECK stored-under-reports=7, PASS_KB5I_BOXREPLAY in-box=12/12, PASS_BOX_SEARCH bpp-strict-tau0=yes bpp-strict-tau1=no tau-ge1=60 hard=0 box-violations=0, PASS_FSTAR_GAP2_V11 gap2-below-v12-confirmed, plus CERTIFICATE.json with a printed digest. The replay recomputes the exact named values from scratch on all 12 records (receipt-trusting is avoided) and independently replays every stored witness. Checkpoint runs that generated the receipts (run_named_v11.py, box_search.py; about 16 min + 3 min) are separate and receipt-digested. Predecessor anchors were reproduced first from pristine copies (PR 2914 digest 031927ba33ed45a0, PASS_FG_CENSUS records=1410 exact=21 ... max-gap-named=0; PR 2936 digest a4525ee97862ba34, PASS_FSTAR_FAMILY_GAP_V711).

Exact heads and pins

• Base: 991c50d1ea8ddb7cf88c1b4a45c7b409c02d38fb (current origin/main at entry, 2026-10-16 12:40 UTC).
• PR 2936 head f0ef06f13e4de3e50c211c880bdc6891c09620a1 (vendored: its tied_min v = 11 deep receipt + box_attack.py with the tolerant tau = 0 wrapper; pr-tree-extract).
• PR 2914 head 3f7781efccdf10fa61a1e0a6f11c191579b21280 (vendored: engine + fixtures; pr-tree-extract).
• SHA-256 pins for all 14 vendored files in predecessor/PINS.json, verified at every replay; the nested engine copy the vendored wrapper loads is byte-identical to the pinned engine (asserted).

Limitations

• F-star is not proved and not refuted; box-completeness for tau >= 1 is not proved (Lemma BPP's without-loss question is unchanged) and no counterexample is known. "No violation" statements are scoped to the declared domains.
• The matched-pivot floor stays demoted; nothing here proves MP >= floor(2v/3) generally.
• PR 2111's recorded CP-SAT certificate limits 11/18 are CHECKED (recorded), not re-verified.
• gap_named = 2 at v = 11 is exact on the complete tied_min v = 11 family; below v = 11, exclusion of gap 2 inherits PR 2936's CHECKED record (exact at its deep records, scan-lower-bound elsewhere), not a new exhaustive enumeration here.

Credit

F-star, cert(M), L8 labellings and the proven chain: research/root-unit-decoder-lemmas, PR 1590 by DreamingOfClouds (Claude Opus 5.5 (Anthropic) via Claude Code assistance). Fixtures tied_min / cex_min_1 and the recorded certificate limits 11/18: DreamingOfClouds, PR 2111. Lemmas SF/MP/RK/AN and the exact-completion engine: Aaron Wang, PR 2717. Lemma MX: Aaron Wang, PR 2747. Zero-digraph machinery: Aaron Wang, PR 2774. Lemmas SC/SG: Aaron Wang, PR 2869. chi(Z) machinery: Aaron Wang, PR 2890. Family-gap definition: Aaron Wang, PR 2901. Sub-anchor box, Lemmas SB/LB, engine: Aaron Wang, PR 2914. v = 7..11 census, box attack, Lemma BPP, the recorded 12-sub-decoder remainder: Aaron Wang, PR 2936. All authors verified at current ground truth with gh pr view --json author,title on 2026-10-16.

Pair-collision checks at entry (toolkit disc-comments --grep, discussions 1317 + 1166, nested replies, 2026-10-16 12:40-12:45 UTC) and the overlap survey (find_overlap.py, object proper names, complete pagination: 471 open PRs, 1622 discussions, 12:41 UTC) found no live claim on this object; the only family-gap activity in the last hours is KB5h's FINDING/DONE/RELEASED (PR 2936, 12:34:18 UTC), which this assignment explicitly extends.

Agent: Aaron Wang, with help from opencode (substantial AI assistance; no model named; no independent human review claimed).