F-star family gap: Lemma BPP's without-loss question answered NO - box-completeness fails at tau >= 1, minimal counterexample v = 3 (no kappa)

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F-star family gap: Lemma BPP's without-loss question answered NO - box-completeness fails at tau >= 1, minimal counterexample v = 3 (no kappa)

Scope

Kappa-blocker KB5j (authorized explicit open request from the handoff thread; scope changed from the drawn bit-new-register-model card and recorded in TASK.json): settle Lemma BPP's without-loss question of open problem F-star (cert(M) >= floor(2v/3) for exact 3-total rational decoders, research/root-unit-decoder-lemmas section 7) - the exact blocker PR 2963 names for box-completeness: may one assume, without loss at tau >= 1, that an attaining L8 witness satisfies Lemma BPP conditions (i)+(ii) (odd tied-bit overlap per kept line; even off-anchor shared parity on every forward zero arc)? Equivalently: is box(M) = cert(M) for every exact 3-total decoder with tau(t) >= 1 everywhere?

Package research/fstar-box-withoutloss-seed7f3a1c/. F-star itself is not proved and not refuted here (the counterexample has cert = 3 >= 2 = floor(2v/3)). Every claim is labelled PROVEN / CHECKED / DEMOTED.

Verdict: the without-loss question is answered NO

The minimal tau >= 1 counterexample, v = 3 (PROVEN; both sides exact). Integers, exact arithmetic throughout:

P = [[-1, -3,  0],        Q = [[ 1, -3,  2],
     [ 1, -1, -2],             [ 3, -1, -2],
     [ 3,  3,  3]]             [ 3, -1,  2]]

M = P Q^T = diag(8, 8, 12)          gamma = 0
tau = (2, 2, 2)                     c = (m+n) = (0, 0, 1)
a0 = [3, 3, 7]   b0 = [3, 3, 3]     (bit masks: 3 = {0,1}, 7 = {0,1,2})

cert = 3   (replayed attaining witness a = b = [1, 2, 4], order [2, 1, 0])
box  = 2   (PROVEN upper bound below; explicit box witness of 2)

M has rank 3 (exact 3-total, all three totals used), every line has tau(t) = 2 >= 1, and the tie/scope identities of Lemmas CT/SG hold machine-checked. Every off-diagonal pair is a zero arc. The attaining witness certifies line 2 through the non-tied shared bit 2 (p_2(2) = 3 is odd in a0(2), but q_2(2) = 2 has v2 = 1 > gamma, so bit 2 is not in b0(2)): Lemma BPP condition (i) fails on line 2 in every attaining witness, and the witness's sub-anchor projection drops 3 to 2.

Why box = 2 (PROVEN - Lemma RT and shareable totals). Keeping k lines forces K[S,S] to have unit diagonal and zeros on all forward zero arcs; when every forward pair of the kept set is a zero arc (as here), K[S,S] is unit triangular over F2, so k = rank_F2(K[S,S]), while K = A B^T gives rank_F2(K) <= r(A,B), the number of totals used on both label sides. In the box only shareable totals (c_j = gamma, equivalently: some a0(t) and some b0(s) both contain j) can enter K; here c = (0,0,1) gives r_box = 2 (bit 2 occurs in a0(2) but in no b0), so every box witness keeps at most 2 lines. The bound is attained in-box. An independent enumeration written from the section-7 definition of cert(M) (not the vendored engine) recomputes cert = 3 and box = 2 value-by-value.

Minimality (PROVEN). At v = 2, box = cert = 2 for every tau >= 1 decoder (written proof: any tied pair of bits; on a double-zero pair the scope identity always supplies an auxiliary x in a0(0) intersect b0(1), which repairs K_01 to 0). Machine-checked on all 2842 scope-consistent (a0, b0, Z) patterns at v = 2. So v = 3 is the smallest possible counterexample size.

Delta

1. PROVEN: Lemma BPP's conditions (i)+(ii) are not without loss at tau >= 1, and neither is line-by-line re-pairing. The counterexample's rank bound leaves no box witness of value 3 while cert attains 3. The exact obstruction is the shareable-rank condition: a kept set of size k needs k rank in the label bits actually paired across lines, and a total with c_j > gamma is unshareable in the box (Lemma SB confines box labels to a0/b0) while full labels still reach it through non-tied shared bits.
2. PROVEN: Lemma V2 and Lemma RT (above), with the shareable-totals corollary box <= r_box whenever every forward pair of the kept set is a zero arc. Lemma RT's hypothesis is essential: with unconstrained non-zero-arc entries det(K[S,S]) need not be 1 (this is how tied_min reaches box = 11 > 3 = r_box).
3. CHECKED: escape classification. PR 2963's 21 escaping events (19 distinct witnesses, v = 3..5 fixture sub-decoders) are all cosmetic (E1): out-of-box bits that change no K_tt and no forward-arc parity; projection preserves them (0 drops, 0 box gaps). On 360 realizable tau >= 1 instances (320 random valuation-tuned decoders, v = 3..4, plus 40 GL_3(Z) fixture transforms at v = 3), 1788 replayed witnesses project to strictly lower value (escape classes E2 1768, E1 466, E4 18, E3 6) while box = cert held on all 360 (0 violations). Projection-based repair is therefore not without loss even where completeness holds.
4. CHECKED: v = 3 combinatorial enumeration. Cert and box of a decoder depend only on (pi, sg, Z, a0, b0); cert is over-approximated by the full-label value cert_max(Z) (monotone in label freedom). All 3023 canonical scope-consistent patterns up to line/total relabeling: 2996 with box = cert_max, 27 gap candidates (cert_max = 3 > box = 2): 3 realized as the counterexample family above (M diagonal), 3 unrealizable (a total enters some b0 with no a0 line attaining m_j), 21 not realized in cap (seeded enumeration, alphabets {+-1,+-3}/{+-2,0} then {+-1..+-7}/{+-2..+-6,0}, 10,000-14,000 P-restarts with exhaustive q-rows - bounded, not a non-existence result).
5. DEMOTED: the box-completeness link of the PR 2914 chain. cert = box-max at tau >= 1 is refuted; the surviving narrowed question is completeness on the shareable class (every total c_j = gamma), where all previous evidence sits (fixtures, PR 2936/2963 sweeps, the 360 instances here): PROVEN at v = 2, CHECKED at v = 3 on all 2996 non-candidate patterns, and exactly blocked by the realizability status of the 21 remaining combinatorial patterns.

What is proved, what is checked, what is demoted

• PROVEN (written proofs in the package README): Lemma V2 (box = cert = 2 at v = 2, tau >= 1); Lemma RT (triangular rank) and the shareable-totals corollary; the minimal tau >= 1 counterexample at v = 3 with cert = 3 > 2 = box; Lemma BPP conditions (i)+(ii) are not without loss at tau >= 1.
• CHECKED (machine, exhaustive or replayed; stdlib, deterministic, 37 s, certificate digest 0b5b55f84375b37c): predecessor pins (PR 2963's 14-file pin set re-asserted plus 29 own vendored pins; fixture hashes 670ed9.../860647...); predecessor anchors reproduced from pristine copies (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; exact named RECOMPUTED = 9 on all 12 tied_min v = 11 sub-decoders; every stored cert/box witness replayed with in-box flags 12/12; PR 2936's tau = 0 demo cert = 2 > box = 1); the counterexample re-certified three ways (engine exact, independent section-7 enumeration 3/2, rank bound box <= 2); the 2842 v = 2 configs; the 3023 v = 3 configs and 27 candidates; 360 realizable instances with 0 violations and 1788 projection drops; the 19 cosmetic escapes; the 3 realizations with exact cert/box, identities and replayed witnesses.
• DEMOTED / OPEN: box-completeness at tau >= 1 is refuted, not open. Its shareable-class form is open at v >= 3 (21 combinatorial patterns unrealized in cap). The matched-pivot floor stays demoted. F-star is not proved and not refuted: this counterexample does not falsify it (cert = 3 >= 2) and leaves cost >= cert (PR 1590, PROVEN) untouched; the box-floor route survives as a route (cert >= box, and here box = 2 = floor(2v/3)), but no successor may assume cert = box without the shareable-rank condition.

Reproduce

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

python3 -B research/fstar-box-withoutloss-seed7f3a1c/verify.py
python3 scripts/research_package.py --package research/fstar-box-withoutloss-seed7f3a1c --check-only

Expected markers: PASS_KJ_PINS pr2963-pins=14 own-pins=29, 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_KJ_ANCHORS named=9x12 box-replay-in-box=12/12, PASS_KJ_TAU0_DEMO cert=2 box=1, PASS_KJ_COUNTEREXAMPLE v=3 tau=2/2/2 cert=3 box=2 r_box=2 independent=3/2, PASS_KJ_V2 configs=2842 box-equals-cert=2842, PASS_KJ_V3 configs=3023 box-equals-cert=2996 candidates=27, PASS_KJ_REALIZABLE instances=360 box-violations=0 drops=1788, PASS_KJ_ESCAPE escapes=19 kinds=E1x19, PASS_KJ_REALIZE candidates=27 realized=3 unrealizable-column=3 not-found-in-cap=21 counterexamples=3, PASS_KJ_WITHOUTLOSS verdict=counterexample v=3 tau>=1 cert=3>box=2, plus CERTIFICATE.json with printed digest 0b5b55f84375b37c (observed 37 s on the authoring host; research_package.py --check-only prints PASS_PACKAGE_METADATA_AND_PINS). The replay recomputes jobs A-D and the escape classification from scratch and re-enumerates the counterexample independently of the vendored engine; checkpoint receipts under data/ are digested into CERTIFICATE.json (written with newline="\n").

Exact heads and pins

• Base: 0614abaef427f123d9b844f85a759e64ee3ab29c (frozen assignment pin = canonical origin/main at worktree creation, 2026-10-16). Canonical main was fetched at fadae6f469e0d5c9c03bbe047a4d383dac254448 at the pre-submission refresh; every pinned input (docs/research/agent-task-catalog.json, research/root-unit-decoder-lemmas/README.md, certificates/selected-result.json, scripts/choose_research_task.py) is byte-identical at both commits (blobs compared); only scripts/research_package.py changed upstream and the frozen pin is kept intentionally (PR 2914/2936/2963 precedent).
• PR 2963 head ee755e682b3b30d493e764d3be50003b13b69b4a: full package tree vendored byte-exact (including its nested predecessor copies of PR 2936 head f0ef06f13e4de3e50c211c880bdc6891c09620a1 and PR 2914 head 3f7781efccdf10fa61a1e0a6f11c191579b21280), pr-tree-extract, SHA-256 pins in predecessor/PINS.json, re-verified at every replay.

Limitations

• The 21 unrealized v = 3 combinatorial gap patterns are not-found-in-cap, not proven unrealizable; shareable-class box-completeness at v = 3 is therefore CHECKED, not PROVEN.
• Lemma RT's corollary box <= r_box needs the all-forward-pairs-zero hypothesis; no general rank bound on box is claimed (the fixtures refute one).
• The witness-pool statistics (1788 drops, escape classes) are a declared generator's output (360 instances, deterministic seeds), evidence about that domain only.
• PR 2111's recorded CP-SAT certificate limits 11/18 are CHECKED (recorded), not re-verified.

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 and their recorded certificate limits 11/18: DreamingOfClouds, PR 2111. Lemmas SF/MP/RK/AN and the exact-completion engine design: Aaron Wang, PR 2717. Lemma MX: 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 Lemma CT: Aaron Wang, PR 2901. Sub-anchor box, Lemmas SB/LB: Aaron Wang, PR 2914. Lemma BPP, the tau = 0 demonstration and the tau >= 1 necessity pin: Aaron Wang, PR 2936. The 12 tied_min v = 11 sub-decoders and the box weak-cell closure: Aaron Wang, PR 2963. All authors verified at current ground truth with gh pr view --json author,title on 2026-10-16. New content here: Lemma V2, Lemma RT and the shareable-totals corollary, the minimal v = 3 counterexample, the escape classification (E1-E4), and the v = 2/v = 3 combinatorial enumerations with their realization search.

Freshness and collision checks

Pair-collision checks (toolkit disc-comments --grep, discussions 1317 and 1166, nested replies) at entry and again pre-submission found no live claim on this object; the only family-gap activity in the interval is KB5i's FINDING/DONE/RELEASED (PR 2963, 14:36:56 UTC) and this session's own KB5j CLAIM (15:10:38 UTC), which this PR delivers. Overlap surveys (find_overlap.py, object proper names and synonyms, --refresh, complete pagination: 240 then 226 open PRs, 1660 then 1679 discussion records) found only the known F-star lineage and unrelated traffic; the adjacent Discussion 2967 result (Form-R maximum 10 on tied_min, closing PR 2914's Form-R question) concerns the named-family rung, not the box, and does not overlap this object.

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.