F-star family gap: the sub-anchor-box decomposition and a fixture-density sub-decoder census (no kappa)

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F-star family gap: the sub-anchor-box decomposition and a fixture-density sub-decoder census (no kappa)

Scope

Kappa-blocker KB5g (authorized explicit open request from the handoff thread; scope changed from the drawn card and recorded in TASK.json): close the family gap named(M) < cert(M) of open problem F-star of the merged research/root-unit-decoder-lemmas package (PR 1590, section 7, OPEN: cert(M) >= floor(2v/3) for every exact 3-total rational decoder M = sum_{j<=3} p_j q_j^T) — the precise residual obstruction of the residue as isolated by PR 2901 (its README section 3, open questions 1-2).

Package research/fstar-family-gap-seed1c25f56f/. F-star itself is not proved and not refuted here. Claims are labelled PROVEN / CHECKED / DEMOTED.

CORE: the family-gap verdict

The family gap is harmless on the entire realized domain, and its exact content is the interior of the sub-anchor box.

quantity	tied_min (12 lines)	cex_min_1 (19 lines)
threshold floor(2v/3)	8	12
named_lo = max(MP, anchor reach)	9 (4096 maps, exhaustive)	16 (6144 maps, exhaustive)
named (PR 2901 record, reproduced)	9	16
form_r (named-form labels, bounded)	10	15 (not a bound)
box (sub-anchor labels)	11	18
cert	11 (PR 2111 recorded, witnessed)	18 (PR 2111 recorded, witnessed)
family gap cert - named	2	2
box gap cert - box	0	0

• CHECKED (exhaustive both sides): on all 21 exact sub-decoder records (v = 3..6) cert = box = form_r = named exactly (gap 0) and cert = max(MP, reach) + 1 exactly; every attaining label pair lies in the sub-anchor box.
• CHECKED (given PR 2111's recorded certificate limits 11/18, CP-SAT, not re-verified): the sub-anchor box attains cert exactly on both fixtures (box = 11 = cert, box = 18 = cert), and every attaining label pair lies in the box. The recorded gap 2 against the named families is reproduced; its per-line content is classified (below).
• CHECKED (scoped negative): no threshold counterexample and no sub-threshold record in the searched domain: named_lo >= floor(2v/3) on all 1410 census records and on both fixtures (9 >= 8, 16 >= 12). By the proven chain cost >= cert no falsifier can exist on any searched record; the falsifier criterion was evaluated on every record and its decisive branch was never reached (0 falsifiers, 0 below-threshold, minimum named ratio 2/3).

The exact remaining gap to a proof: prove the sub-anchor box complete (cert = box-max, at least when every line has tau(t) >= 1) and its floor, or prove the matched-pivot floor max_c sum_i MAS(Z[C_i]) >= floor(2v/3) (which would close F-star despite being weaker than cert). Both remain OPEN.

PROVEN: Lemma SB (the sub-anchor box) and Lemma LB (the ladder)

In the canonical reduced normalization (Lemmas SC/SG of PR 2869, Aaron Wang): a0_j(t) = [v_2(p_j(t)) = m_j], b0_j(t) = [v_2(q_j(t)) = gamma - m_j]; tau(t) is the tie number of Lemma CT (PR 2901).

Lemma SB. (i) a0(t) intersect b0(t) is exactly the set of totals counted by tau(t). (ii) A sub-anchor pair (a inside a0(t), b inside b0(t)) certifies line t (K_tt = 1) exactly when |a intersect b| is odd, which needs tau(t) >= 1; for tau(t) = 1 the unique tied bit lies in both labels; for tau(t) = 2 exactly one of the two tied bits lies in a intersect b. (iii) A tau(t) = 0 line has no certifying sub-anchor pair; any witness keeping it must leave the box on that line.

Proof. (i) j lies in both label sets exactly when j is in D(t) with v_2(p_j(t) q_j(t)) = gamma, the definition of being counted by tau(t). (ii) For sub-anchor labels a intersect b lies inside a0(t) intersect b0(t) and K_tt = |a intersect b| mod 2. (iii) is (ii) at tau(t) = 0. QED

Lemma LB. (a) named(M) <= form_r(M): every named-family certificate (Lemma SG anchor, Lemma MX repairs, Lemma MP pivots, Lemma CL3 mixings) uses per-line pairs from the named forms (a0,b0), (a*, b0), (e_c,e_c), and ab_value is the exact certificate value of a fixed label assignment. (b) box(M) <= cert(M) by definition of the label classes. (c) When S_gamma is empty, named = max(MP, MX): the anchor pair has K0_tt = 0 on every line and certifies nothing.

Proof. (a) Each named family assigns each line one Form-R pair; its carried certificate is a certificate of that assignment. (b) is the identity of label spaces. (c) with S_gamma empty every named construction has A empty and the rest are matched pivots and their Lemma MX variants. QED

Part (c) makes tied_min's named value exactly 9 (MP exhaustive over 4096 pivot maps; MX exhaustive 9 per PR 2747, Aaron Wang, reproduced here by a seeded scan).

CHECKED: what the families miss, per line (PR 2901 open question 2 / PR 2747 open question 3)

The attaining witnesses classify as follows (ANCHOR = (a0,b0), REPAIR = (a*, b0), PIVOT = (e_c,e_c), BOX-INTERIOR = strictly interior box pair):

witness	ANCHOR	REPAIR	PIVOT	BOX-INTERIOR	outside box
tied_min 11-witness (engine re-found)	0	5	1	5	0
tied_min 11-witness (calibration run)	0	5	0	6	0
cex_min_1 18-witness (stored, replayed)	2	10	1	5	0

Every attaining label pair lies in the sub-anchor box on both fixtures and on all 21 exact records. The strictly interior pairs are exactly the pairs Lemma SB (ii) describes: on a tau = 2 line they pick one of the two tied bits in a intersect b, dropping the non-tied bits of a0/b0 — the label space no named family enumerates (named forms fix b = b0(t) on anchor/repair lines or force a = b unit vectors on pivots). On tied_min the Form-R labels alone reached 10 of the recorded 11 (bounded hill-climb over 13,841,287,201 assignments; whether Form-R reaches 11 is OPEN), so at least one line needs strictly interior pairs there.

CHECKED: smallest-first sub-decoder census at fixture density

Every sub-decoder M[S,S] of a fixture is an exact 3-total decoder (restriction preserves rank <= 3 and the nonzero diagonal; control checked) and inherits fixture-density zero digraphs by construction — the exact recorded generator obstruction of PR 2901 (its k-axis generators cannot realize them; Lemma RK (c) forces |pi(t) intersect sigma(s)| >= 2 at every arc).

family	domain	records	deep exact	falsifiers	below threshold
sub-decoders of tied_min	v = 3..6, size 3 complete, 4-6 strided	760	0 flagged	0	0
sub-decoders of cex_min_1	v = 3..6, strided	650	21	0	0

On all 21 exact deep records (each with 1 to 3 cancellation lines): cert = box = form_r = named exactly, cert = named_lo + 1 exactly, all witnesses in the box. The family gap is 0 on every exact record and never reaches 2 below v = 12 in the searched domain; the gap is not a function of the cancellation count (0 at k_cancel up to 3, 2 at k_cancel = 12). The recorded strides per size are in CERTIFICATE.json; sizes v = 7..11 and the unstrided remainder were not enumerated (compute cap) — a scoped statement about the declared domain, not a non-existence claim.

Assignment routes (a), (b), (c) — disposition

• (a) the family certificate meets the threshold (named = MP >= floor(2v/3)): DEMOTED. MP >= floor(2v/3) is witnessed on all 1410 census records and both fixtures (9/12, 16/19) but not proved, and named = cert fails on the fixtures (9 < 11, 16 < 18). A proof of the MP floor would still close F-star (cert >= MP); the exact obstruction recorded here: the floor constrains realizable rank-3 patterns (Lemma RK (c)), fixture-density zeros resist random construction (PR 2901's recorded generator obstruction), and no analytic route was found; the observed feedback-vertex sums v - MP are 3 and 3 against ceilings ceil(v/3) = 4 and 7, with no monotone argument in sight.
• (b) the gap is harmless via a certified lifting: CHECKED on the realized domain, DEMOTED in general. On all 21 exact records the lifting is exact (named = cert); on both fixtures box = cert exactly with the missing labels classified per line. A proved additive bound cert >= named + f is DEMOTED: the realized gaps (0 at v <= 6 with cancellation lines present, 2 at v = 12 with 12 of 12) fit no additive function of the cancellation count or v the analysis produced, and Lemma SB (iii) shows the box certifies nothing on a tau = 0 line, so no uniform per-cancellation-line surplus can hold.
• (c) the first realized threshold counterexample: NOT FOUND. The strict form (cert < floor(2v/3)) and the threshold form (named < floor(2v/3) <= cert) are both unrealized in the searched domain. The minimal realized gap-2 instance remains tied_min at v = 12, and its gap content is exactly the interior-box labels above.

What is proved, what is checked, what is demoted

• PROVEN (README, written proofs): Lemma SB; Lemma LB (named <= form_r, box <= cert, named = max(MP, MX) at empty reach).
• CHECKED (machine, verify.py, stdlib, deterministic, about 230 s, certificate digest 031927ba33ed45a0): controls 5 groups (brute-force-from-definition agreement 3/3 at v = 3, corrupted-witness rejection 2/2, ab_value soundness 10/10, restriction exactness with the Lemma CT and scope identities, diagonal-style decoder sanity); the recorded-anchor calibration (MP 9/16 exhaustive; tied_min hill-climb witness 11; cex_min_1 stored witness 18 replayed; chain 22 on the stored pairing; reach 0/12 and 8/19); the fixture ladder with per-line diffs and box = cert at 11/11 and 18/18; the 1410-record census with 21 exact deep records at named = cert; 0 falsifiers.
• DEMOTED / OPEN: F-star is not proved here and not refuted. The matched-pivot floor (PR 2747 open question 1) is not proved. Sub-anchor-box completeness is CHECKED on the realized domain only; tau = 0 lines are the only structurally excluded case (Lemma SB (iii)) and none occurs in the fixtures. Form-R completeness on tied_min is OPEN. The fixture limits 11/18 rest on PR 2111's recorded CP-SAT optimality, not re-verified here. The CL3 rung on cex_min_1 is a bounded scan (PR 2901's caveat retained). The census thinning is recorded per size.

Open next questions

1. Sub-anchor-box completeness. Is cert(M) = box-max(M) whenever every line has tau(t) >= 1? It would reduce F-star to a finite per-line label space (at most 24 certifying pairs per line). First step: search for the smallest instance where an attaining certificate must leave the box (all 21 exact records and both fixtures are box-valued). Blocker: none.
2. The matched-pivot floor (PR 2747 open question 1). max_c sum_i MAS(Z[C_i]) >= floor(2v/3) for every realizable rank-3 pattern? First step: a SAT/CP construction of rank-3 patterns minimizing the floor. Blocker: needs a constraint solver, not random search.

Reproduce

python3 -B research/fstar-family-gap-seed1c25f56f/verify.py
python3 scripts/research_package.py --package research/fstar-family-gap-seed1c25f56f --check-only

Expected markers: 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_FG_CENSUS records=1410 exact=21 exact-named=21 gap1=49 gap2=0 below-threshold=0 falsifiers=0 max-gap-named=0, PASS_FSTAR_FAMILY_GAP no-falsifier-in-domain, and CERTIFICATE.json with digest 031927ba33ed45a0 (observed 217-229 s across runs on the authoring host). make is not installed on the authoring host; the commands above are the underlying ones.

Freshness and overlap

Base ba3de47290c3882cbbe406db1f657578f7ecda69 is the frozen assignment pin (kept intentionally). Canonical main was fetched at 475fec190d5ad746bd4690574dca30d6901acdaf on 2026-10-16 10:35 UTC; the interval from the base pin contains unrelated batch integrations only, and research/root-unit-decoder-lemmas/README.md, certificates/selected-result.json, docs/research/agent-task-catalog.json and scripts/research_package.py are byte-identical at both commits (pins verified). Pair-collision checks (discussions 1317 + 1166, nested replies, 2026-10-16 08:58-09:00 UTC and again 10:33 UTC) found no live claim on this object; the last F-star activity is KB5f's FINDING/DONE/RELEASED at 2026-10-16 08:53:52 UTC. Overlap surveys (scripts/find_overlap.py, explicit object terms, complete pagination: 492 open PRs, 1585 discussions at 10:29-10:30 UTC) found only the known F-star lineage (PRs 2717/2747/2774/2869/2890/2901, research/fstar-entry-class-scope-seed63d5f999, research/fstar-cert-census-seed298df375) and unrelated traffic.

Credit

F-star, cert(M), L8 labellings and the proven chain cost >= sum_j r_j >= cert(M) are from research/root-unit-decoder-lemmas (PR 1590 by DreamingOfClouds (Claude Opus 5.5 (Anthropic) via Claude Code assistance)). The rank-3 configuration model, tied_min and cex_min_1, their recorded one-copy certificate limits (11 and 18, CP-SAT, CHECKED (recorded)) and the stored cost-22 pairing are DreamingOfClouds' work in PR 2111 and PR 2162; data/tied_min.json, data/cex_min_1.json, data/cex_min_1.B22.json and data/cex_min_1.witness.json are byte-identical copies of their data files (sha256 recorded in the package README). The residue statement, Lemmas SF/MP/RK/AN and the exact-completion engine design this package adapts are Aaron Wang's work in PR 2717. Lemma MX and the exhaustive MP/MX fixture values (9 and 16) are Aaron Wang's PR 2747. The zero-digraph machinery is Aaron Wang's PR 2774; the chi(Z) machinery is Aaron Wang's PR 2890. Lemmas SC/SG and the canonical reduced normalization are Aaron Wang's PR 2869. The family-gap definition and record are Aaron Wang's PR 2901. The exact cost 12 of tied_min is Aaron Wang's PR 2379; the exact cost 21 of cex_min_1 is Nick Petalas' PR 2206. All PR authors were verified at current ground truth with gh pr view --json author,title on 2026-10-16. This package does not claim their results; the new content is Lemma SB, Lemma LB, the sub-anchor-box machinery and decomposition, the exact named/Form-R/box/cert ladder on small instances, the per-line classification of the attaining witnesses, and the sub-decoder census at fixture density.

Prepared by Aaron Wang with help from opencode (substantial AI assistance; no model named; no independent human review claimed). No kappa claim.