KB5L: tau >= 1 box floor PROVEN at v = 5 - shareable class narrowed - no kappa

[navinor]

KB5L: the tau >= 1 box floor at v = 5 is PROVEN - shareable class narrowed (no kappa)

Scope

Kappa-blocker KB5L (authorized explicit open request from the handoff thread; scope changed from the drawn complex-new-helper-inventory card, recorded in TASK.json): the first open case of the tau >= 1 box floor (v = 5) and the shareable-class remainder - the recorded remainders of PR 3012 (agent-79).

Answer (a): PROVEN. Every exact 3-total rational decoder M = sum_{j<=3} p_j q_j^T on v = 5 lines with M_tt != 0 and tau(t) >= 1 on every line has box(M) >= 3 = floor(2v/3). The tau >= 1 box floor is now PROVEN for v <= 5; the first open case moves to v = 6. The mechanism is a rank obstruction, not a keepable count and not matched pivots: combinatorially the only way to keep box at 2 is a complete zero-arc digraph inside a 2-bit label-cap family, and a complete zero-arc digraph makes M diagonal, which has rank 5 - impossible for a 3-total decoder.

Answer (b): narrowed. PR 2978's 3 realized / 3 unrealizable-column / 21 not-found-in-cap split is replayed exactly (same seed and stages); a strictly larger bounded push (exact kernel q-solver at 10,000 P-restarts per pattern plus the big alphabet at 2,000) realizes none of the 21; and 1 of the 21 is PROVEN unrealizable (the pattern (a0, b0) = (3^3, 3^3) with complete Z: the anchor screen gives p_2 == 0, so rank(M) <= 2, but complete Z forces diagonal rank 3). The other 20 stay CHECKED not-found-in-cap. This also corrects PR 3012's framing: only 20 of the 21 lie in the shareable class (r_box = 3); the settled one forces r_box = 2.

F-star (cert(M) >= floor(2v/3)) is not proved and not refuted. Claims are labelled PROVEN / CHECKED / DEMOTED.

CORE: the v = 5 tau >= 1 verdict

stage	object	result	label
1	5-line type multisets, tau >= 1	749,398 enumerated; envelope box = 2 on exactly 33	PROVEN (exhaustive)
2	the 33 families under the anchor screen	15 unrealizable; 6 shapes survive	PROVEN (written)
3	blocking zero-arc sets per shape	only the complete digraph blocks every triple	PROVEN (exact)
4	complete digraph at rank <= 3	diagonal M has rank 5 > 3	PROVEN (written)
5	realizable replay	120 tau-forced decoders, exact box >= 5, witnesses replayed	CHECKED
6	bounded falsifier hunt	35 instances, 0 hits	CHECKED (bounded)

The reduction (written; engine-anchored definitions). box(M) is the largest kept set over in-box labels a(t) <= a0(t), b(t) <= b0(t) with K_tt = 1 on kept lines and K_ts = 0 at every zero arc (t, s) with t before s in the kept order. (R1) box depends only on (a0, b0, Z) (Lemma SB; labels of non-kept lines never enter a constraint). (R2) box is non-increasing in Z, so box(Z) >= box(Z_even) with Z_even the set of all pairs of even |a0(t) intersect b0(s)|. (R3) Lemma SG (PR 2869, PROVEN; machine-asserted on every realizable instance here) forces every realizable zero arc into Z_even; the engine's scope identity is the exact iff: dot1(a0[t], b0[s]) = 1 if and only if M_ts != 0 and v2(M_ts) = gamma. So a realizable box = 2 decoder must sit inside a box = 2 combinatorial envelope (a0, b0, Z_even), and its actual Z must block every triple in every order.

Stage 1 (PROVEN, exhaustive). All C(41,5) = 749,398 multisets of 5 raw line types ((a0, b0) mask pairs with tau = |a0 intersect b0| >= 1; raw masks preserve the joint bit structure that per-line canonicalization loses) are enumerated with Z = Z_even: memoized singleton witnesses first, the exact kept-set ladder (pinned to an independent definition-level brute enumeration on 400 random configs) as fallback. Result: PASS_KL_V5 configs=749398 cheap=749347 exact=18 envelope-box2-families=33. All 33 families share one shape: every pair-intersection even (Z_even complete) and one label side confined to 2 bits (a rank-2 cap on in-box K = A B^T) - exactly the Lemma RT + shareable-totals route PR 3012 recorded as the only possible falsifier shape.

Stage 2 (PROVEN, written). From the engine's anchor_data: a0_j(t) = [v2(p_j(t)) = m_j] with m_j a minimum over nonzero column entries; b0_j(t) = [v2(q_j(t)) = gamma - m_j]. Hence (A1) bit j in no a0(t) implies p_j == 0 identically (minima are attained), so rank(M) <= 2; (A2) bit j in some b0(t) requires m_j finite and c_j = gamma, so a family with bit j in some b0 but in no a0 is unrealizable. The screen rejects 15 of the 33 and leaves 6 shapes up to total relabeling (5 unshareable-total rank-3 shapes, 1 zero-p-column rank-2 shape): PASS_KL_SHAPES shapes=6 screen-rejected=15.

Stage 3 (PROVEN, exact). Per surviving shape, exact per-triple blocking tables (kept-3 infeasibility of the 3-line ladder over all 2^6 arc patterns) are combined with a full 2^20 arc-subset enumeration. In every shape the only subset blocking all ten triples is the complete digraph (blocking=1 minimal=1 each): with one non-arc (x, y), the order (x, y, z) leaves K_xy free and the 2-bit cap can always be met with det(K[S,S]) = 0.

Stage 4 (PROVEN, written). Complete Z makes M diagonal with nonzero diagonal, so rank(M) = 5 > 3; no exact 3-total decoder exists with v = 5, tau >= 1 everywhere and box < 3. The rank-2 shapes fail a fortiori. This is the same obstruction PR 2978's realized v = 3 counterexample family sidesteps only because a 3x3 diagonal has rank 3.

Corroboration (CHECKED). PASS_KL_REALIZABLE_V5 instances=120 violations=0 min-box=5 (tau-forced generator; identities asserted; exact box by the vendored engine; in-box witnesses replayed). PASS_KL_HUNT tested=35 hits=0 (tau-targeted, PR 2978-core extensions, nullspace-dense; ladder-exact criterion; bounded, not a non-existence result). PASS_KL_PROBE model-ok=142 tau>=1-all=0: untargeted random entries make all-5-lines tau >= 1 rare, which is why the combinatorial envelope route is the decisive one.

Shareable class: the 21-pattern remainder

shareable_v3.py recomputes PR 2978 job B from scratch (the vendored receipt ships empty): PASS_KJ_V3 configs=3023 box-equals-cert=2996 candidates=27. Pass 1 replays PR 2978's realize_kj.py pipeline exactly (seed 20261017, same stages and caps) and reproduces its recorded split, PASS_KL_SHAREABLE candidates=27 replay={"not-found-in-cap": 21, "realized": 3, "unrealizable-column": 3}. Pass 2 pushes the cap with an exact kernel q-solver: for fixed P (anchor recipe), q(s) lies in the integer kernel of its zero-arc predecessor rows, avoids the off-arc hyperplanes, and matches the parity box; a solution exists iff p(t) is outside span{p(u) : (u, s) in Z} for every non-arc t. 10,000 deterministic P-restarts per pattern plus the vendored big alphabet at 2,000: no new realization (push-final equals replay). One pattern is PROVEN unrealizable (candidate 7; A1 + diagonal rank). The remaining 20 are shareable-class (r_box = 3) and stay CHECKED not-found-in-cap.

Methodological caveat (CHECKED observation, not a proof). PR 2978 job B's canonicalization applies total relabeling per line (canon_total on each line's (a0, b0) separately) and computes pair parities from the chosen representatives. Total relabeling is a global operation on the shared bit indices, so two joint structures with the same multiset of per-line canonical types can have different cross-line parities and therefore different Z spaces. The recorded 3023-pattern census may under-count the joint (a0, b0, Z) space, so "the 21" (or 20) may not be the complete gap-candidate set at v = 3. The v = 5 enumeration in this package is unaffected (raw joint type multisets preserve cross-line structure). A joint-sound v = 3 census is the natural follow-up check.

What this does and does not say about F-star

Nothing here lowers cert on any instance. The box rung of cert >= max(named, box) now carries the threshold on the entire tau >= 1 domain for v <= 5, and the named/matched-pivot rung carries the tau = 0 region (PR 3012's falsifiers all have cert = v). The shareable-class question is about box = cert completeness, not about the threshold: even a realized gap pattern would not falsify F-star, since its cert would still have to drop below floor(2v/3) - which no falsifier test in the lineage has produced.

What is proved, checked, demoted

• PROVEN: the tau >= 1 box floor at v = 5 (four-stage chain above); the envelope classification (33 families); the anchor-attainment screen (15 rejected); the blocking-Z result (complete digraph only); unrealizability of one v = 3 shareable pattern.
• CHECKED (machine, deterministic): ladder pinned to definition-level brute on 400 configs; 120 realizable tau-forced decoders with replayed witnesses; the 35-instance bounded hunt; the exact PR 2978 classification replay and the cap push.
• DEMOTED / OPEN: v = 6 (threshold 4) is the new first open case of the tau >= 1 floor; the 20 remaining shareable-class patterns are not-found-in-cap, not proven unrealizable; the matched-pivot floor stays demoted.

Reproduce

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

python3 -B research/fstar-boxfloor-v5-seedab80f07c/verify.py
python3 -B research/fstar-boxfloor-v5-seedab80f07c/predecessor/pr3012/anchor_replay.py
python3 scripts/research_package.py --package research/fstar-boxfloor-v5-seedab80f07c --check-only

Observed: PASS_KL_PINS vendored-pr3012=57; PASS_KL_PIN configs=400 brute-equals-ladder=400; PASS_KL_V5 configs=749398 cheap=749347 exact=18 envelope-box2-families=33; PASS_KL_REALIZABLE_V5 instances=120 violations=0 min-box=5; PASS_KL_ALL v5-configs=749398 envelope-families=33 realizable=120; PASS_KL_PROBE model-ok=142 tau>=1-all=0; PASS_KL_SHAPES shapes=6 screen-rejected=15; PASS_KL_HUNT tested=35 hits=0; PASS_KL_FAMILY_ALL shapes=6 hunt-tested=35 hunt-hits=0; PASS_KJ_V3 configs=3023 box-equals-cert=2996 candidates=27 certmax-cache=64; PASS_KL_SHAREABLE candidates=27 replay={"not-found-in-cap": 21, "realized": 3, "unrealizable-column": 3} push-final={"not-found-in-cap": 21, "realized": 3, "unrealizable-column": 3} counterexamples=3; PASS_KL_VERIFY v5-configs=749398 families=33 shapes=6 realizable=120 hunt=35 shareable=27 pins=57 digest=0d393ada94fd3809. The vendored PR 3012 anchor replay prints PASS_KM_ANCHORS markers=12 digest=0b5b55f84375b37c vendored-verify=exit0, re-running the vendored PR 2978 verify.py unchanged (including PASS_FG_CALIBRATION tied_min=11/12 cex_min_1=18/19 mp=9/16 chain=22 reach=0/12,8/19 and PASS_KJ_COUNTEREXAMPLE v=3 tau=2/2/2 cert=3 box=2 r_box=2 independent=3/2) and the PR 3012 markers PASS_KM_VERIFY falsifiers=4 lb2=16761 mixed=240 v3-configs=3023 pivot=240 digest=fb29a75bde520732. research_package.py --check-only prints PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check).

Exact heads and pins

• Base: d8e0f83f00191df62174b641e9fff055c357f641 (frozen assignment pin = canonical origin/main at worktree creation, 2026-10-16); branch research/agent-80-kb5l-boxfloor-f0, package head e75906fc4.
• PR 3012 head 63c6525120c20543c4a7acae1a1e1ea36e099b81, package research/fstar-boxfloor-minimal-seed758ae8fe vendored byte-exact under predecessor/pr3012/ (57 files, pr-tree-extract), SHA-256 pins in predecessor/PINS.json, re-asserted at every replay. Its nested vendored chain (PR 2978 head 12eb7b832de9239a314a7cddf6c6033d618cd8f2, PR 2963 head ee755e682b3b30d493e764d3be50003b13b69b4a, PR 2936 head f0ef06f13e4de3e50c211c880bdc6891c09620a1, PR 2914 head 3f7781efccdf10fa61a1e0a6f11c191579b21280) carries its own pin sets (14 + 29 files), re-asserted by the vendored replays.

Open next questions

1. The tau >= 1 box floor at v = 6 (threshold 4): the same envelope classification (6-line multisets) and blocking-Z enumeration with constraint propagation; smallest falsifying instance first.
2. The 20 remaining shareable-class patterns: the kernel feasibility condition settles the Z side exactly; the anchor side needs a valuation-level realizability characterization.
3. F-star at general v: the box rung covers the whole tau >= 1 domain for v <= 5; the tau = 0 region lives in the named rung. A joint lower bound in the shareable-class vocabulary is the natural next lemma.

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, Lemma MX, zero-digraph machinery, Lemmas SC/SG (the scope identity used as R3), chi(Z) machinery, Lemma CT, the sub-anchor box and Lemmas SB/LB (used as R1), Lemma BPP with the tau >= 1 necessity pin, the twelve tied_min v = 11 sub-decoders, Lemma V2, Lemma RT + shareable totals with the minimal box = cert counterexample and the v = 3 combinatorial census with realize_kj (the 27 patterns replayed here), and Lemma K / Lemma LB2 / Lemma TB with the four tau = 0 falsifiers and the recorded v = 5 obstruction shape: Aaron Wang (navinor), PR 2717 / 2747 / 2774 / 2869 / 2890 / 2901 / 2914 / 2936 / 2963 / 2978 / 3012. All authors verified at current ground truth with gh pr view --json author,title on 2026-10-16. New here: the exhaustive v = 5 envelope classification (33 families), the anchor-attainment screen (A1/A2), the exact blocking-Z enumeration (complete-digraph-only), the rank obstruction closing the tau >= 1 floor at v = 5, the exact kernel realization solver, and one shareable-class pattern proved unrealizable.

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.