F-star family gap: the v = 7..11 census, the first gap-2 instance below v = 12, and the box-completeness attack (no kappa)

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F-star family gap: the v = 7..11 census, the first gap-2 instance below v = 12, and the box-completeness attack (no kappa)

Scope

Kappa-blocker KB5h (authorized explicit open request from the handoff thread; scope changed from the drawn supplier-boundary-audit card and recorded in TASK.json): the exact remaining search space of open problem F-star's family-gap question as recorded by PR 2914 (Aaron Wang, agent-72). PR 2914's recorded remaining gap: prove the sub-anchor box complete (cert = box-max, at least for tau(t) >= 1) plus the box floor, or prove the matched-pivot floor, at general v; and its recorded compute obstruction: v = 7..11 unenumerated. This package (1) closes that enumeration with the falsifier criterion armed, (2) attacks box-completeness, (3) reports the matched-pivot floor status as PR 2914 left it. F-star (cert(M) >= floor(2v/3) for exact 3-total rational decoders, research/root-unit-decoder-lemmas section 7) is not proved and not refuted here. Every claim is labelled PROVEN, CHECKED or DEMOTED.

Verdict

1. CHECKED: v = 7..11 is enumerated. 1638 sub-decoder records of the two canonical fixtures (complete at tied_min v = 9..11, deterministic strides elsewhere), 113 deep records, 126 synthetic instances. The falsifier criterion (exact cert below floor(2v/3), or an admissible pairing with chain cost below threshold) was armed on every record, never fired, and is PROVEN inoperative on the domain: named_lo = max(MP, reach) >= floor(2v/3) on all 1638 records (minimum ratio 5/7), each a replayed certificate value, so the proven chain cost >= cert excludes every admissible B below threshold. MP >= floor(2v/3) alone holds on all 1638 records (0 below).
2. CHECKED (exact both sides): the first gap-2-or-more instance below v = 12 is at v = 7. cex_min_1|0,1,2,5,6,8,15: cert = 7 (= v, exact) and named_lo = 5 (MP = 5 exhaustive, reach 5) — gap 2 to the cheap rung. 38 records witness gap 2+ against the cheap rung (all cex_min_1 sub-decoders, v = 7..11). The family gap on both v = 7 gap-2 records is 0 (named = 7 exact): the gap-2 there is to the cheap rung only.
3. CHECKED (exact both sides): the family gap is nonzero from v = 7. tied_min|0,1,2,5,6,8,10 and tied_min|2,3,4,7,9,10,11 (v = 7): named = 6 exact (Lemma MX construction space 823543 exhausted), cert = 7 exact — gap_named = 1. 17 deep records carry an exact gap_named = 1 (v = 7..10). gap_named = 2 below v = 12 is excluded on every enumerated record except the 12 tied_min v = 11 sub-decoders (scan-only named; Lemma MX construction product 1,977,326,743 — the exact obstruction; gap there in 0,1,2). The minimal exact gap_named = 2 instance remains tied_min at v = 12 (PR 2901/PR 2914 record).
4. CHECKED (scoped negative): box-completeness not refuted. box = cert on all 113 deep records (exact, or via a replayed attaining size-v witness in the box), on both fixtures (PR 2914: 11/11, 18/18), on all 21 PR 2914 exact records, and on all 11 tau >= 1 synthetic instances; every recorded attaining witness is box-valued. The one candidate where a box hill-climb sat one below cert (tied_min|0,1,2,3,4,5,6,7,8,10,11, v = 11) dissolved: its attaining cert witness is itself in the box.
5. PROVEN: the tau >= 1 hypothesis is necessary. Explicit exact 3-total rational decoder on v = 2 lines: P = [[1,0,1],[2,1,0]], Q = [[1,0,0],[1,0,1]], M = [[1,2],[2,2]], tau = (1,0) — cert = 2 but box = 1 (the tau = 0 line has no certifying sub-anchor pair, Lemma SB (iii); its witness uses a high total). On all 115 synthetic instances with a tau = 0 line, box = number of lines with tau at least 1 exactly (CHECKED pattern).
6. DEMOTED (retained): the matched-pivot floor. PR 2914 demoted route (a) — MP is strictly weaker than cert on the fixtures (9 below 11, 16 below 18) — and nothing here resurrects it: MP >= floor(2v/3) is witnessed everywhere in the domain (evidence, not proof).

The exact remaining search space: (i) box-completeness for tau >= 1 at general v; (ii) the family gap on the 12 tied_min v = 11 sub-decoders (blocked by the Lemma MX product ~2e9); (iii) the matched-pivot floor (PR 2747 open question 1).

(1) The v = 7..11 census (CHECKED)

Sub-decoders M[S,S] of both canonical fixtures (restriction is exact), smallest first, deterministic strides recorded per cell in the receipts:

fixture	v = 7	v = 8	v = 9	v = 10	v = 11
tied_min	240 of 792	240 of 495	220 of 220	66 of 66	12 of 12
cex_min_1	240 of 50388	200 of 75582	160 of 92378	140 of 92378	120 of 75582

Totals (replayed by verify.py):

quantity	value
records	1638
trivial gap-0 closures (named_lo = v forces cert = v)	946
witnesses independently replayed	1384
records witnessing gap 2+ vs named_lo	38
gap-2-potential records (v - named_lo >= 2)	70
records with named_lo below threshold	0
records with MP below threshold	0
falsifiers	0
minimum named_lo / v	5/7

Family gap by v (deep tier, 113 records; exact named where the Lemma MX + CL3 construction products are exhausted, exact cert via enumeration or a replayed size-v witness):

v	exact gap_named	gap-2 status
7	1 first (2 tied_min records), 0 elsewhere	excluded (both sides exact)
8	1 (2 cex_min_1 records)	excluded
9	1 (3 cex_min_1 records)	excluded
10	1 (2 cex_min_1 records)	excluded (candidate settled: named = 10 exact)
11	none exact	open exactly on the 12 tied_min v = 11 sub-decoders

The scan pitfall, quantified. PR 2914's lineage caution is real: on 3 of the 10 records settled exhaustively (raised 3M cap), the seeded 20,000-iteration Lemma MX scan under-reported named by 1 to 2 (the v = 10 gap-2 candidate's scan value 8 is exactly 10). Scanned named values are lower bounds; every exact gap_named above uses an exhausted construction space.

(2) Box-completeness: NOT PROVEN, NOT REFUTED, hypothesis pinned

Evidence for completeness (CHECKED): box = cert on all 113 deep records, both fixtures (given PR 2111's recorded CP-SAT limits), all 21 PR 2914 exact records, and all 11 tau >= 1 synthetic instances; every recorded attaining witness is box-valued.

The hypothesis tau >= 1 is necessary (PROVEN): the explicit v = 2 decoder P = [[1,0,1],[2,1,0]], Q = [[1,0,0],[1,0,1]] has tau = (1,0), cert = 2 (keep both lines with unit pairs on total 0; no zero arcs) but box = 1 (line 1 has a0 = 1 in bit notation 2, b0 = bits 0 and 2, so no sub-anchor pair certifies it). Unrestricted box-completeness fails at the smallest possible size.

Lemma BPP (box projection parity) — PROVEN. For any labelling and its sub-anchor projection (a'(t) = a(t) intersect a0(t), b'(t) = b(t) intersect b0(t)), for every pair of lines t, s:

K'_ts = K_ts + |(a(t) intersect b(s)) \ (a0(t) intersect b0(s))|  (mod 2)
K'_tt = |a(t) intersect b(t) intersect (a0(t) intersect b0(t))|   (mod 2)

by set algebra mod 2. So a witness projects to a box witness of the same value only if each kept line's shared bits contain an odd number of tied bits and every forward zero arc has even shared-high-bit parity. A counterexample must violate both conditions on every attaining witness: certify some kept line purely through non-tied shared bits, or land odd shared-high-bit parity on some forward zero arc. No realized instance needs that.

Synthetic sweep (CHECKED): 126 deterministic random exact 3-total decoders at v = 4..7, exact cert vs exact box on each: 11 with tau >= 1 everywhere, all box = cert; the 115 with a tau = 0 line follow box = number of tau-positive lines; 0 counterexamples. Random decoders never produced a hard (cert < v) tau >= 1 instance — fixture-density zeros are needed (PR 2901's generator obstruction), and those are the census records.

(3) Matched-pivot floor: DEMOTED (retained)

No new evidence to resurrect PR 2914's demoted route (a): MP is strictly weaker than cert on the fixtures (9 below 11, 16 below 18). New data in favor of the statement, none of it a proof: MP >= floor(2v/3) on all 1638 census and 113 deep records.

What is proved, what is checked, what is demoted

• PROVEN (written proofs in the package README): tau >= 1 necessary for box-completeness (explicit v = 2 decoder, argument in prose); Lemma BPP; the falsifier criterion inoperative on the enumerated domain (chain plus replayed named_lo witnesses); named_lo = v forces cert = named_lo = v.
• CHECKED (exhaustive machine computation, both sides exact): first gap-2-or-more instance below v = 12 at v = 7; family gap 1 from v = 7; gap_named = 2 excluded below v = 12 outside the 12 tied_min v = 11 sub-decoders.
• CHECKED (machine, verify.py, stdlib, deterministic, 50.9 s, certificate digest a4525ee97862ba34): predecessor pins 11/11 byte-exact; PR 2914's controls and every recorded anchor reproduced (PASS_FG_CALIBRATION tied_min=11/12 cex_min_1=18/19 mp=9/16 chain=22 reach=0/12,8/19; PR 2914's own certificate digest 031927ba33ed45a0 reproduced first from a pristine copy); all 1638 cheap records (identities, 1384 replayed witnesses, gap bookkeeping, MP recomputed on a sample plus every gap-2 record); all 113 deep records (products recomputed, exact-rung reproduction on a bounded sample, gap-2 residual classification asserted); the box-attack core (the v = 2 demonstration and 100 sweep instances re-run value-by-value).
• CHECKED (receipts, not recomputed in the registered run): the sweep's v = 6,7 cells; deep values outside the reproduction sample (each has a replayed witness or a recorded exact rung; the heavy census is a resumable checkpoint run with receipts under data/census_v711/).
• DEMOTED / OPEN: F-star not proved, not refuted. Box-completeness for tau >= 1 not proved (Lemma BPP pins what a counterexample must do). gap_named = 2 not excluded on the 12 tied_min v = 11 sub-decoders. Matched-pivot floor not proved. Fixture limits 11 and 18 rest on PR 2111's recorded CP-SAT optimality (CHECKED (recorded)).

Reproduce

Standard library only, Python 3.10+:

python3 -B research/fstar-family-gap-v711-seed3e3478d8/verify.py
python3 scripts/research_package.py --package research/fstar-family-gap-v711-seed3e3478d8 --check-only

Observed output (50.9 s, Python 3.12):

PASS_KB5H_PINS files=11
PASS_FG_CONTROLS brute-force=3/3 corrupted=2/2 ab-sound=10 restriction=1/1 diagonal=1/1
PASS_FG_CALIBRATION tied_min=11/12 cex_min_1=18/19 mp=9/16 chain=22 reach=0/12,8/19
PASS_KB5H_CENSUS records=1638 trivial-gap0=946 gap1-witnessed=452 gap2-witnessed=38 gap2-potential=70 falsifiers=0 below-threshold=0 mp-below-threshold=0 min-named-lo-ratio=0.7143 witnesses-replayed=1384
PASS_KB5H_DEEP records=113 gap-named-proven>=17 gap-named-proven-max=1 gap-named-ge2-stored=5 gap2-open-scan-only=12 box-violations=0 reproduced=24
PASS_BOX_ATTACK tau0-demo-cert2-box1 sweep-replayed=100/100 sweep-total=126 tau-ge1=11 box-violations=0 tau0-pattern=115/115
PASS_FSTAR_FAMILY_GAP_V711 no-falsifier-in-domain
certificate digest a4525ee97862ba34

The heavy runs (10 cheap cells, 10 deep cells, the 3M named-settlement pass, the 126-instance sweep; about 25 minutes total) are resumable checkpoint jobs with receipts and are documented in the README; their evidence is the receipt set digested into CERTIFICATE.json.

Credit and provenance

F-star, cert(M), L8 labellings and the proven chain cost >= cert(M): research/root-unit-decoder-lemmas, PR 1590 by DreamingOfClouds (Claude Opus 5.5 (Anthropic) via Claude Code assistance). The rank-3 configuration model, the fixtures tied_min and cex_min_1, and their recorded certificate limits 11 and 18 (CP-SAT): DreamingOfClouds, PR 2111. Lemmas SF/MP/RK/AN and the exact-completion engine design: Aaron Wang, PR 2717. Lemma MX and the exhaustive MP/MX fixture values: Aaron Wang, PR 2747. Zero-digraph machinery: Aaron Wang, PR 2774. Lemmas SC/SG and the canonical reduced normalization: Aaron Wang, PR 2869. chi(Z) machinery: Aaron Wang, PR 2890. Family-gap definition and record: Aaron Wang, PR 2901. Sub-anchor box, Lemmas SB/LB, the v = 3..6 census and the recorded remaining gap: Aaron Wang, PR 2914. All authors verified at current ground truth with gh pr view --json author,title on 2026-10-16. The engine is PR 2914's package extracted byte-exactly with the shared toolkit (pr-tree-extract, head 3f7781efccdf10fa61a1e0a6f11c191579b21280 at extraction) and vendored with SHA-256 pins verified at every replay. New content here: the v = 7..11 census and its verdicts, the exact-residual classification, the scan-pitfall quantification, the v = 2 tau = 0 demonstration, Lemma BPP, and the synthetic sweep.

Freshness and collision checks

Base 5a0dc2c0f4685b02bef7776bc196c7c1231cbac1 is canonical origin/main at entry (fetched and pinned 2026-10-16 10:45 UTC; the pre-submission fetch saw d3add2cc28e958bedec197ab32b5526c5a222628, which touches no pinned input — the frozen assignment pin is kept intentionally, PR 2914's precedent). Pair-collision checks (toolkit disc-comments --grep, discussions 1317 and 1166, nested replies, at entry 10:45-10:47 UTC and the refined pre-computation re-run 11:21 UTC) found no live claim on this object; the only F-star family activity in the last hours is KB5g's FINDING/DONE/RELEASED (PR 2914, 10:40:51 UTC), which this assignment explicitly extends. Overlap surveys (find_overlap.py with the object's proper names and synonyms, complete pagination: 494 then 496 open PRs, 1588 then 1597 discussions) found only the known F-star lineage and unrelated board traffic.

Attribution: Aaron Wang, with help from opencode (substantial AI assistance; no model named; no independent human review claimed). No kappa, supplier, word or moment claim.