F-star residue (i): mixed anchor-repair-pivot lemma MX, tied-instance certificates, and a tied-domain search (no kappa)

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F-star residue (i): mixed anchor-repair-pivot lemma MX, tied-instance certificates, and a tied-domain search (no kappa)

Scope

Kappa-blocker KB5b (explicit open request; scope changed from the drawn card and recorded in TASK.json): residue class (i) tied-minimum / isotropic lines of open problem F-star of the merged research/root-unit-decoder-lemmas package (section 7): cert(M) >= floor(2v/3) for every exact 3-total rational decoder M = sum_j p_j q_j^T. This is the exact residue that PR 2717 (Aaron Wang) left after proving Lemmas SF, MP, RK and AN: lines the mod-2 anchor cannot anchor (tied-minimum / isotropic lines), where Lemma AN handles only the count of odd diagonals and not their repair or ordering. Package research/fstar-tied-residue-seeddcc97691/, branch research/agent-65-fstar-tied-residue-dcc97691, base main 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2 (priority revision 4). No kappa, certificate, supplier, window or frontier change. F-star itself is not proved and not refuted here.

Agent: Aaron Wang, with help from opencode (substantial AI assistance; no independent human review claimed).

Result 1 — Lemma MX (PROVEN)

A mixed anchor-repair-pivot certificate with exact hypotheses, in two order forms. Anchor labels on a set A (lines with M_tt odd), one-sided repairs a*(t) on a set U (with b fixed at the anchor's b0), matched pivots a(t) = b(t) = e_{c(t)} with c(t) in D(t) on the rest T; per pivot class C_i, MAS(Z[C_i]) is the maximum acyclic induced subset of the zero digraph. Then cert(M) >= |A| + |U| + sum_i MAS(Z[C_i]) under either (a-first) "the repairs kill their A/U and T blockers and no A-to-T conflict remains", or (t-first) "the repairs kill their A/U blockers and no T-to-A/U conflict remains". Written proof in the README. Its U/T-empty special case is Lemma AN of PR 2717; its A/U-empty special case is Lemma MP of PR 2717, so MX subsumes both as sufficient conditions.

Result 2 — the canonical tied instances are closed by structural certificates (CHECKED)

Per-line residue classification (relative to the stored primitive 2-integral normalization: anchor_good = M_tt odd; repairable_1s = full one-sided repair exists with b fixed; tied_isotropic = neither), then exhaustive maximization over all pivot maps as unit masks (4096 on tied_min, 6144 on cex_min_1), every accepted value carrying a witness replayed by an independent checker:

instance	v	floor(2v/3)	ceil(2v/3)	AN	MP (exhaustive)	MX (max)	engine witness	PR 2111 recorded
tied_min	12	8	8	0	9	9	11 (re-found)	11 (recorded)
cex_min_1	19	12	13	8	16	16	18 (replayed)	18 (recorded)

F-star holds on both canonical tied-minimum instances by a structural certificate (9 and 16, at or above both the floor and the ceiling threshold), even though the mod-2 anchor is vacuous on tied_min (0 of 12 lines) and short on cex_min_1 (8 of 19). Both fixtures are entirely tied_isotropic outside the anchor set (0 one-sided repairable), so the residue is total there. The engine witnesses reproduce DreamingOfClouds' recorded one-copy L8 maxima of PR 2111 (their CP-SAT optimality is recorded, not re-verified here).

Result 3 — exact obstruction to the mixed route (CHECKED)

On cex_min_1 no anchor/pivot mixture helps in either block order: the a-first separation fails for 6144/6144 pivot maps (A-to-T conflict) and the t-first separation fails for 6144/6144 (T-to-A/U conflict). The winning value 16 is the pure matched-pivot construction. This is the exact obstruction to extending AN/MP by naive mixing on the canonical hard instance.

Result 4 — tied-domain falsifier search, smallest first (CHECKED, scoped negative)

40 instances at v = 3..12 (two per family and v), families tied_isotropic (every line constructed with two tied minimal valuations at level 0 and even diagonal) and mixed (half unique-minimum anchorable, half tied). Zero falsifiers. cert = v on all 40: 16 decided by exhaustive a-enumeration at v <= 6, 24 pinned by full-size replayed witnesses (a witness of size v pins cert = v exactly). Minimum witness ratio 1.0. The falsifier criterion (explicit admissible pairing B with block-rank chain cost below floor(2v/3), which by the proven chain would refute F-star) was tested by an exhaustive B-scan up to 12 free pairs and a deterministic local search above: no B below threshold on any instance (bounded search, not an exhaustive non-falsification proof). Lemma MP reaches v on all 40; Lemma MX is never below MP and never strictly above it on this domain.

What is proved, witnessed, demoted

• PROVEN: Lemma MX, both order forms, exact hypotheses, written proof.
• CHECKED (stdlib verify.py, deterministic, certificate digest 439d75261e9a2b9e): controls (brute/engine agreement 3/3 at v = 3, diagonal decoder cert = 3, corrupted-witness rejection 2/2, MX all-anchor reproducing AN's count, MP control with replayed witness); the fixture classification; the exhaustive MP/MX maxima 9/16 with replayed witnesses; the 6144/6144 obstruction counts in both forms; engine witnesses 11/18; the 40-instance search (cert = v on all, zero falsifiers).
• DEMOTED: F-star for general rational decoders is not proved. That matched pivots always reach floor(2v/3) is witnessed on 42 instances, not proved. No hypothesis is claimed necessary. Residue classes (ii) (non-3-dichromatic zero digraphs) and (iii) (non-dyadic entries) are untouched. B-scan values are bounded-search upper bounds on the minimum chain cost. The fixture certificates 11/18 rest on PR 2111's recorded CP-SAT optimality. Exact pairing costs elsewhere (12 on tied_min, PR 2379 by Aaron Wang; 21 on cex_min_1, PR 2206 by Nick Petalas) are cited, not recomputed.

Reproduce

python3 -B research/fstar-tied-residue-seeddcc97691/verify.py
python3 scripts/research_package.py --package research/fstar-tied-residue-seeddcc97691 --check-only
python3.12 scripts/research_package.py --package research/fstar-tied-residue-seeddcc97691

Expected: PASS_FSTARTIED_CONTROLS, PASS_FSTARTIED_FIXTURES, PASS_FSTARTIED_MX, PASS_FSTARTIED_SEARCH, PASS_FSTAR_TIED_RESIDUE, PASS_REGISTERED_RESEARCH_PACKAGE research/fstar-tied-residue-seeddcc97691 outcome=completed (registered replay: 378.81 s, under the 600 s manifest cap). make is not installed on the authoring host.

Credit

F-star, cert(M), L8 labellings and the proven chain: merged research/root-unit-decoder-lemmas (PR 1590 by DreamingOfClouds, with Claude Opus 5.5 (Anthropic) via Claude Code assistance). Fixtures tied_min/cex_min_1, their recorded certificate limits and the stored cost-22 pairing: DreamingOfClouds, PR 2111 and PR 2162. The residue statement, Lemmas SF/MP/RK/AN and the engine design this adapts: Aaron Wang, PR 2717. The falsifier-census frontier: Aaron Wang, PR 2335. Exact cost 12 of tied_min: Aaron Wang, PR 2379. Exact cost 21 of cex_min_1: Nick Petalas, PR 2206. New work here is Lemma MX with its proof, the residue classification, the mixed-route obstruction and the tied-domain search.

Open next questions are in the package README: (1) does the matched-pivot family always reach floor(2v/3) (which would prove F-star)? (2) does insertion-finite mixing (per-anchor-line cuts of the pivot order) survive the 6144/6144 obstruction? (3) which label patterns do the attaining certificates 11 and 18 actually use?