F-star family gap: the box floor FAILS minimally at v = 3 - surviving tau >= 1 form exact (no kappa)

[navinor]

F-star family gap: the box floor FAILS minimally at v = 3 - and its exact surviving tau >= 1 form (no kappa)

Scope

Kappa-blocker KB5k (authorized explicit open request from the handoff thread; scope changed from the drawn portable-replay card and recorded in TASK.json): the surviving box-floor route to open problem F-star (cert(M) >= floor(2v/3) for every exact 3-total rational decoder M = sum_{j<=3} p_j q_j^T on v lines with M_tt != 0, PR 1590 section 7) - the recorded remainder of PR 2978 (agent-78): its surviving box-floor link plus the shareable-class question.

Answer: the box floor, read over all exact 3-total rational decoders, is FALSE, minimally so. The minimal falsifier is a rank-3 decoder at v = 3 with tau = (0,0,0): cert = 3 >= 2 = floor(2v/3) > 0 = box, both sides exact. F-star is not proved and not refuted (every falsifier has cert >= floor(2v/3)). On the tau >= 1 domain the floor is PROVEN for v <= 4 (Lemma LB2) and its first open case is v = 5, where a bounded exact search of 240 realizable decoders found 0 violations. Claims are labelled PROVEN / CHECKED / DEMOTED.

CORE: the box-floor verdict

instance	v	rank M	tau	r_box	threshold	cert	box
F2-wide	2	2	(0,0)	3	1	2	0
F3-tau0	3	3	(0,0,0)	3	2	3	0
F3-tau1	3	3	(1,0,0)	2	2	3	1
F4-tau0	4	3	(0,0,0,0)	3	2	4	0

Threshold is floor(2v/3); r_box = |{j : c_j = gamma}| is the shareable count. The minimal rank-3 falsifier (F3-tau0), all integers, all arithmetic exact:

P = [[1, 1, 0],        Q = [[2, 2, 1],
     [1, 0, 1],             [2, 1, 2],
     [0, 1, 1]]             [1, 2, 2]]

M = P Q^T = I + 3 J = [[4,3,3],[3,4,3],[3,3,4]]    det = 10, rank 3
gamma = 0   c = (0,0,0)   r_box = 3   tau = (0, 0, 0)
a0 = [3, 5, 6]   b0 = [4, 2, 1]   zero arcs: none

1. PROVEN (Lemma K): box = 0 iff every line has tau = 0, and box <= #{t : tau(t) >= 1} always: a kept line needs in-box labels with K_tt = 1, hence a tied bit j in a0(t) intersect b0(t), and a tied singleton keeps a line alone. So the falsifiers live exactly in PR 2901's cancellation residue (Lemma CT: tau even everywhere), where cert keeps its full value (cert = v on all four instances).
2. PROVEN: the minimal falsifier is at v = 3 with box = 0 (the minimum value), in the exact-3-total class; in the wide class (rank at most 3) a rank-2 falsifier exists already at v = 2. Minimal: rank 3 forces v >= 3. Exact both sides three ways: the vendored PR 2914 engine (exhaustive), an independent definition-level enumeration over every label assignment (box side at all four instances, cert side at v <= 3), and written bounds (cert <= v, Lemma K). Witness replayed; tie parity and scope identities machine-checked; rank computed exactly over Q; all three totals used.
3. PROVEN (Lemma LB2, two-line repair): any two lines with tau >= 1 are jointly box-keepable, so box >= min(2, #{t : tau(t) >= 1}). If some ordered pair is a nonzero arc, order so it is forward and tied singletons work; if both are zero arcs and the tied bits differ, singletons kill K_ts; if they coincide, the scope identity (Lemma SG) supplies a repair bit r in a0(t) intersect b0(s) and the labels {j,r}/{j}/{j}/{j,r} give K_tt = K_ss = 1 and K_ts = 0. Construction machine-replayed on 16,761 canonical v = 3 configs with at least two keepable lines (the full Lemma-SG-consistent (a0, b0, Z) space up to line/total relabeling, including tau = 0 lines, which PR 2978 job B excluded) and on 160 random realizable decoders v = 4..6, 0 failures.
4. CHECKED: the tau >= 1 box floor holds exactly at v = 3, 4 (threshold at most 2 by Lemma LB2; PR 2978 job B recomputed from scratch - its vendored receipt ships empty - gives v = 3 minimum exactly 2 = threshold on all 3023 canonical configs, the 27 gap candidates all at exact box = 2) and is bounded at v = 5 (threshold 3): 240 realizable tau >= 1 decoders (80 random valuation-tuned, 80 extensions of PR 2978's counterexample core, 80 unit-role templates), 97 with r_box = 2, every exact box at least 3, 0 below threshold. Bounded, not a non-existence result.
5. PROVEN (Lemma TB, box matched pivots): box >= max_c sum_j MAS(Z[C_j]) over tied-bit colourings c(t) in a0(t) intersect b0(t) - the box form of PR 2717's Lemma MP (keep acyclic subsets per colour class in topological order). Together with PR 2978's Lemma RT + shareable-totals upper bound this fixes the control picture: tau pattern governs the lower side (Lemmas K, LB2, TB), shareable count r_box the rank-capped upper side, and the zero digraph the shape of any remaining falsifier.
6. Recorded exact obstruction to a v = 5 falsifier. Lemma RT blocks one order of a kept set S only when every forward pair of that order is a zero arc; S is blocked in every order by rank only if Z[S,S] is complete (all six ordered pairs zero arcs), and then M[S,S] is diagonal, so |S| <= rank(M) = 3. Hence at v = 5 a falsifier needs every 3-subset blocked in every order: Z-complete triples for the rank route (r_box <= 2 necessary there) and bilinear infeasibility everywhere else. No such instance realized in the bounded search.
7. DEMOTED: the box-floor route as stated. box >= floor(2v/3) is dead on the full class and survives only on tau >= 1, proved for v <= 4, open for v >= 5. The replacement lower-bound shape is cert >= max(named, box): the named/matched-pivot rung carries the tau = 0 region (Lemma MP gives v on every falsifier here), the box rung carries tau >= 1. Shareable-class completeness (PR 2978's 21 unrealized v = 3 patterns) is unchanged and open; this session spent its one bounded pivot on the box floor.

Reproduce

Standard library only, Python 3.10+ (registered replay: 3.12):

python3 -B research/fstar-boxfloor-minimal-seed758ae8fe/verify.py
python3 -B research/fstar-boxfloor-minimal-seed758ae8fe/anchor_replay.py
python3 scripts/research_package.py --package research/fstar-boxfloor-minimal-seed758ae8fe --check-only

Observed: PASS_KM_PINS vendored-pr2963=14 vendored-own=29 and PASS_KM_VERIFY falsifiers=4 lb2=16761 mixed=240 v3-configs=3023 pivot=240 digest=fb29a75bde520732; PASS_KM_ANCHORS markers=12 digest=0b5b55f84375b37c vendored-verify=exit0; PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check). The anchor replay runs the vendored PR 2978 verify.py unchanged on a byte-copy and asserts every recorded anchor, including PASS_FG_CALIBRATION tied_min=11/12 cex_min_1=18/19 mp=9/16 chain=22 reach=0/12,8/19, PASS_KJ_COUNTEREXAMPLE v=3 tau=2/2/2 cert=3 box=2 r_box=2 independent=3/2, PASS_KJ_V2 configs=2842, PASS_KJ_V3 configs=3023 box-equals-cert=2996 candidates=27, and the certificate digest 0b5b55f84375b37c.

Exact heads and pins

• Base: 066f57ff795dbe6d0a0ecc11fe742eadcec00c39 (frozen assignment pin = canonical origin/main at worktree creation, 2026-10-16); branch research/agent-79-box-floor-kb5k-7f3a, package head 63c652512.
• PR 2978 head 12eb7b832de9239a314a7cddf6c6033d618cd8f2, vendored byte-exact under predecessor/pr2978/ with its pin sets (14 + 29 files) re-asserted at every replay; PR 2963 head ee755e682b3b30d493e764d3be50003b13b69b4a, PR 2936 head f0ef06f13e4de3e50c211c880bdc6891c09620a1, PR 2914 head 3f7781efccdf10fa61a1e0a6f11c191579b21280 through the vendored copies.

Credit

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). Fixtures tied_min/cex_min_1 with recorded limits 11/18: DreamingOfClouds, PR 2111. Lemmas SF/MP/RK/AN and the exact-completion engine, Lemma MX, zero digraph machinery, Lemmas SC/SG, chi(Z) machinery, the family-gap definition and Lemma CT, the sub-anchor box and Lemmas SB/LB, Lemma BPP with the tau >= 1 necessity pin, the twelve tied_min v = 11 sub-decoders, and Lemma V2 / Lemma RT + shareable totals with the minimal box = cert counterexample: Aaron Wang (navinor), PR 2717 / 2747 / 2774 / 2869 / 2890 / 2901 / 2914 / 2936 / 2963 / 2978. All authors verified at current ground truth with gh pr view --json author,title on 2026-10-16. New here: Lemma K, Lemma LB2, Lemma TB's box form, the four minimal falsifiers with their exact gradient, the mixed-tau characterization, and the v = 5 tau >= 1 obstruction shape.

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