F-star residue (i-prime): cancellation lines - tie lemma, insertion-finite mixing, fixture settlement, and the corrected dyadic corollary (no kappa)

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F-star residue (i-prime): cancellation lines - tie lemma, insertion-finite mixing, fixture settlement, and the corrected dyadic corollary (no kappa)

Scope

Kappa-blocker KB5f (authorized explicit open request from the handoff thread; scope changed from the drawn card and recorded in TASK.json): the cancellation-line residue (i-prime) 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 case where more than floor(v/3) lines have v_2(M_tt) > gamma ("cancellation lines", PR 2869's rename), so the mod-2 anchor is short of the threshold - plus the small corrective repair it depends on: PR 2717's dyadic-corollary normalization gap, which PR 2869 found and this package reproduces and fixes.

Package research/fstar-cancellation-lines-seed80d69336/. F-star itself is not proved and not refuted here. Claims are labelled PROVEN / CHECKED / DEMOTED.

CORE: the cancellation-line verdict

Verdict: on both canonical fixtures and on every searched instance, F-star holds despite the cancellation lines, certified by a coefficient-free certificate (Lemma MP matched pivots); the true residual obstruction is isolated as the family gap named(M) < cert(M), not a threshold failure.

quantity	tied_min (12 lines)	cex_min_1 (19 lines)
cancellation lines	12 of 12	11 of 19
anchor reach S_gamma	0 of 12	8 of 19
threshold floor(2v/3)	8	12
Lemma MP (exhaustive pivot maps)	9 (4096 maps)	16 (6144 maps)
Lemma CL3 insertion mixing	9 (no anchor lines exist)	16 (5 of 8 anchors placeable at best config; 7 seen)
engine / stored witness	11	18
family gap cert - named	2	2

tied_min is all cancellation lines with anchor reach 0 and cex_min_1 is four anchors short of threshold; in both, the matched-pivot certificate alone reaches 9 >= 8 and 16 >= 12, with no anchor, no repairs and no search. Cancellation lines do not obstruct the threshold in these instances; the pivot structure carries it.

PROVEN - Lemma CT (what a cancellation line is). With tau(t) = #{j : v_2(p_j(t) q_j(t)) = gamma} (the tie number, invariant per total): t is anchored exactly when tau(t) is odd, and t is a cancellation line exactly when tau(t) is even (tau in {0, 2}); every tied total lies in D(t), so a tau = 2 line has |D(t)| >= 2 and a |D(t)| = 1 line is pivot-bit-forced (every K_tt = 1 labelling carries the unique D(t) bit on both sides; the matched pivot attains it). tau = 2 is exactly the tied-minima obstruction named in research/root-unit-decoder-lemmas section 7.

PROVEN - Lemma CL3 (insertion-finite mixing; answers PR 2747's open question 2). Lemma MX mixes anchor lines and matched pivots only in two block orders; its general order has a closed form. A set of anchor lines inserts at valid cuts of one pivot order exactly when the joint demand digraph (pivot conflicts, plus L(x)-before-x and x-before-R(x) for each anchor line x) is acyclic, and then cert(M) >= |A| + sum_i MAS(Z[C_i]). Anchor-anchor conflicts vanish always (scope identity K0 = M' mod 2 = 0 at incidences), so insertions interact only through the shared order. On cex_min_1 (bounded scan of 2^20 mixing configurations, A = S_gamma level exhaustive, exact joint insertion on the top 32): up to 5 of 8 anchor lines placeable at the best configuration and 7 in some configuration, but the mixed value stays 16 - insertion power is real (the two block orders fail completely, 6144/6144, per PR 2747's record) and adds nothing over pure pivots there.

The true residual obstruction (precise definition). Let named(M) be the largest certificate value carried by the union of named families: Lemma SG anchor, Lemma MX one-sided repairs, Lemma MP matched pivots, and Lemma CL3 insertion mixings (every value carries a replayed L8 witness). The residual obstruction of residue (i-prime) is an instance with named(M) < floor(2v/3) <= cert(M); the strict form cert(M) < floor(2v/3) is a falsifier to F-star. The threshold form was not realized in 178 records (falsifier criterion armed everywhere, never fired). The realized form is the family gap named(M) < cert(M): 9 vs 11 and 16 vs 18 on the fixtures (exactly 2 on each) and +1 on the hardest re-split. The attaining witnesses use labels outside the anchor/repair/pivot families (PR 2747's open question 3); a proof of F-star must either prove the matched-pivot floor max_c sum_i MAS(Z[C_i]) >= floor(2v/3) for every realizable pattern (PR 2747's open question 1; witnessed on all 178 records and at 9/12, 16/19 on the fixtures) or master those two-bit labels.

CHECKED: calibration, search, and the re-split probe

Calibration reproduces every recorded anchor first: PR 2717 (tied_min 11/12, cex_min_1 18/19, chain 22 on the stored pairing), PR 2747 (MP/MX 9 and 16 exhaustive over 4096/6144 pivot maps), PR 2869 (reach = anchor_good exactly, 0/12 and 8/19), PR 2890 (PASS_FC4_CALIBRATION tied_min=11/12 cex_min_1=18/19 mp=9/16 chain=22 values).

Smallest-first falsifier search across decoders with 1..v cancellation lines (v = 3..12, exact k = 1..v, deterministic seeds):

family	domain	records	falsifiers	min named ratio
parity-engineered	v = 3..12, k = 1..v exact	75	0	1.000
structured (tied_min signature)	v = 3..12, k = 1..v exact	34	0	1.000
adversary (minimizing the pivot floor)	k > floor(v/3), v = 3..11	44	0	1.000
GL_3(Z) re-splits of the fixtures	fixture-density Z	24 + 1	0	0.895

Falsifier criterion (armed on every record): exact cert < floor(2v/3) where exhaustive a-enumeration terminates, or an explicit admissible B with chain cost below threshold (the proven chain of research/root-unit-decoder-lemmas); never fired. Exact generator obstruction, recorded: random and structured factorizations give sparse zero digraphs (arcs need |pi(t) intersect sigma(s)| >= 2 by Lemma RK (c)), so the named families reach cert = v there and the fixture-density hardness resists random construction - a scoped statement about the searched generators, not an unrealizability claim. GL_3(Z) re-splits keep the fixtures' dense Z (MAS 4 of 12, 10 of 19) while moving gamma, supports and S_gamma; there the named floor dips to 17/19 = 0.895 v with engine +1 on the hardest, and the cancellation counts seen are 11 and 12 only (the diagonal valuations are re-split invariant). The CL3 fixture evaluation is a bounded scan (20000 of 2^20 configurations), not an exhaustive maximum.

CORRECTIVE: the dyadic corollary's normalization gap, reproduced and repaired

PR 2717 proved Lemma AN under the hypothesis "after per-total powers of two the factorization is 2-integral: all entries of p, q lie in Z_(2)" and closed with: "Under the added hypothesis that at most floor(v/3) lines have M_tt even, cert(M) >= ceil(2v/3), so F-star holds for dyadic decoders with at most one-third even diagonals." PR 2869 found the gap; this package reproduces it (control SC1) and states the fix.

The gap, exhibited. On two lines with p = q = (1/2)(1, 1) all entries are dyadic, but the total content is c_1 = -2, the per-total rescaling interval [-m_1, n_1] = [1, -1] is empty, and no per-total power of two makes both factors 2-integral - so the mod-2 reduction p mod 2 Lemma AN applies is not even defined on p(t) = 1/2. Dyadic entryhood does not deliver Lemma AN's hypothesis. The count phrase has a second gap (control SC2): M = 2 I_3 has three even diagonals (naive count 0 anchored) while the corrected reach is 3 (gamma = 1); "even diagonals" is not the anchor count when gamma != 0.

Corrected dyadic corollary (PROVEN, from PR 2869's Lemmas SC/SG). For every exact 3-total rational decoder in any entry class: cert(M) >= |S_gamma| = v - #{t : v_2(M_tt) > gamma} with gamma = min_j c_j and c_j the 2-adic content of total j; F-star holds on M whenever at most floor(v/3) lines are cancellation lines. Lemma AN's literal hypothesis holds exactly when c_j >= 0 for every total (Lemma SC (b)) - a condition on the factorization, not a consequence of dyadic entryhood - and when it fails the single global rescaling M -> 2^{-gamma} M restores it (Lemma SG (a)). The anchored set is S_gamma = {t : v_2(M_tt) = gamma}; it equals the odd-diagonal set exactly when gamma = 0.

A corrective comment with this content is posted on PR 2717 (the authorized remedy; its branch is not edited and no follow-up PR is opened for it).

Reproduce

python3 -B research/fstar-cancellation-lines-seed80d69336/verify.py
python3 scripts/research_package.py --package research/fstar-cancellation-lines-seed80d69336 --check-only

Markers: PASS_CL_CONTROLS, PASS_CL_CALIBRATION tied_min=11/12 cex_min_1=18/19 mp=9/16 chain=22 reach=0/12,8/19, PASS_CL_RESPLIT, PASS_CL_GAP dyadic-corollary-normalization-gap=reproduced SC1 SC2, PASS_CL_SEARCH, PASS_FSTAR_CANCELLATION_LINES; certificate digest 7f27c1e761b34c98; observed about 67 s, stdlib only, deterministic. Predecessor engines were also rerun for reproduction receipts (PR 2717 digest 134665ac6a0800e4, PR 2747 digest 439d75261e9a2b9e, PR 2869 digest 39f5c167c495f457, PR 2890 digest 032e7239429ba1f8, each printing its recorded PASS markers).

Freshness: base 38746ad3f6ffebcc0275e286b1dbb733d7c696dd is the frozen assignment pin (kept intentionally); canonical main was fetched at ba3de47290c3882cbbe406db1f657578f7ecda69 on 2026-10-16 08:45 UTC - the interval contains unrelated batch integrations only; research/root-unit-decoder-lemmas/README.md, certificates/selected-result.json and the task catalogue are byte-identical at both commits (pins verified), and the current scripts/research_package.py differs only in CI matrix batching. Refined overlap re-run 2026-10-16 08:50-08:51 UTC (complete pagination: 489 open PRs, 1565 discussions) found no other work on this object; pair-collision check (discussions 1317 + 1166, nested replies) found no live claim on it at entry.

Credit

F-star, cert(M), L8 labellings and the proven chain cost >= sum_j r_j >= 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, tied_min and cex_min_1, their recorded one-copy limits (11 and 18, CP-SAT, CHECKED (recorded)), the stored cost-22 pairing and the tied-minima obstruction: DreamingOfClouds, PR 2111 and PR 2162. Lemmas SF/MP/RK/AN, the residue statement and the engine design this package adapts: Aaron Wang (navinor, with help from opencode), PR 2717. Lemma MX and the class-(i) closure: Aaron Wang (navinor, with help from opencode), PR 2747. The zero-digraph machinery: Aaron Wang (navinor, with help from opencode), PR 2774. Lemmas SC/SG, the entry-class collapse and the residue rename: Aaron Wang (navinor, with help from opencode), PR 2869. The chi(Z) machinery and calibrated engine variant: Aaron Wang (navinor, with help from opencode), PR 2890. Falsifier census frontier: Aaron Wang (navinor), PR 2335. Exact cost 12 of tied_min: Aaron Wang (navinor), PR 2379. Exact cost 21 of cex_min_1: Nick Petalas, PR 2206. All PR authors verified at current ground truth with gh pr view --json author,title on 2026-10-16. New content here: Lemma CT, Lemma CL3 with its exact insertion criterion, the fixture settlement of the insertion route, the family-gap definition and record, the cancellation-line-axis falsifier search with its exact generator obstruction, the GL_3(Z) re-split probe (also PR 2869's open question 2), and the corrected dyadic corollary with its reproduced normalization gap.

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