F-star residue class (ii): non-3-dichromatic zero digraphs - matched-pivot floor, fixture dichromatic structure, smallest-first falsifier search (no kappa)

[navinor]

F-star residue class (ii): non-3-dichromatic zero digraphs - the matched-pivot floor, the fixtures' dichromatic structure, and a smallest-first falsifier search (no kappa)

Scope

Kappa-blocker KB5c (explicit open request; scope changed from the drawn card and recorded in TASK.json): residue class (ii) non-3-dichromatic zero digraphs of open problem F-star of the merged research/root-unit-decoder-lemmas package (PR 1590 by DreamingOfClouds, with Claude Opus 5.5 (Anthropic) via Claude Code assistance; 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 residue PR 2717 by Aaron Wang (navinor) demoted as "zero digraphs that are not 3-dichromatic (Lemma MP then gives only sum_i MAS(Z[C_i]))" and left untouched by PR 2747 by Aaron Wang (navinor), which closed residue class (i). Package research/fstar-zero-digraph-residue-seede9937f18/, registered replay PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=completed on CPython 3.12.

The package does three things: (1) exact zero-digraph machinery (MAS over all subsets by DP, the exact Lemma MP value max_c sum_i MAS(Z[C_i]) over all pivot maps c : [v] -> D(t), dichromatic numbers of Z with and without the pivot constraint sets D(t)); (2) calibration on DreamingOfClouds' stored fixtures tied_min (12 lines) and cex_min_1 (19 lines) reproducing every recorded anchor, then the exact dichromatic structure of both zero digraphs (the open question 2 of PR 2717); (3) a smallest-first adversarial search for the first instance where the matched-pivot value falls below floor(2v/3) (the open question 1 of PR 2747, which is exactly this class's floor question). F-star itself is not proved and not refuted here. Every claim is labelled PROVEN, CHECKED or DEMOTED.

Calibration - CHECKED (all recorded anchors reproduce)

Machine results (verify.py, stdlib, deterministic; certificate digest 86f48c1bacdbe4a9):

anchor	recorded by	recorded value	this package
tied_min one-copy L8	PR 2111 by DreamingOfClouds (CP-SAT, CHECKED (recorded))	11	11 (hill-climb re-found, witness replayed)
cex_min_1 one-copy L8	PR 2111 by DreamingOfClouds (CP-SAT, CHECKED (recorded))	18	18 (stored witness replayed)
chain engine on cex_min_1 stored pairing	PR 2111 by DreamingOfClouds	22 (ranks 10 + 6 + 6)	22 (admissibility re-checked)
Lemma MP on tied_min	PR 2747 by Aaron Wang (navinor)	9 (4096 pivot maps)	9 (4096 pivot maps, exhaustive)
Lemma MP on cex_min_1	PR 2747 by Aaron Wang (navinor)	16 (6144 pivot maps)	16 (6144 pivot maps, exhaustive)

New: the dichromatic structure of the canonical fixtures - CHECKED

instance	v	floor(2v/3)	chi(Z) unconstrained	MAS(Z)	largest mutual-zero clique	MP (exhaustive)	v - MP	3-dichromatic under D
tied_min	12	8	3 exactly (subset DP, exhaustive)	4	3	9	3	no
cex_min_1	19	12	3 exactly (colouring found; a mutual-zero triangle forces 3)	10	3	16	3	no

Consequences, each CHECKED:

1. Both canonical fixtures have dichromatic number exactly 3 without the pivot constraints. The obstruction that keeps cert(M) below v on cex_min_1 (18 of 19) is therefore not the dichromatic number of the digraph; it lives in the pivot constraint sets D(t) together with the labelling structure. PR 2717's open question 2 ("can Z itself be non-3-dichromatic") is not realized by either fixture; both fixtures are non-3-dichromatic under D, which is the operative sense of class (ii) for Lemma MP.
2. The largest mutual-zero clique is 3 on both fixtures: the Lemma RK (a) bound (no mutual-zero K_4) is attained, so clique bounds alone cannot force a dichromatic number above 3 on these instances.
3. The matched-pivot value loses exactly 3 lines on each fixture (9 below 12, 16 below 19), and both values stay at or above both threshold forms (9 versus 8; 16 versus 12 and 13). F-star holds on both canonical class-(ii) instances by the structural certificate alone.

New: smallest-first falsifier search in the class-(ii) domain - CHECKED (scoped negative)

Smallest falsifying instance first: v ascending from 3, four generator families per v (entries in {-1,0,1} and {-2..2}), all aimed at the residue: random / random2 (unstructured boxes), sparse (1-2 nonzero coordinates per p(t), q(t), dense zero digraphs), forced (support masks chosen first so at least half the lines have |D(t)| = 1, heavily constrained pivot maps), mutual (rejection sampling for many mutual-zero pairs). 286 instances, v = 3..12, deterministic seeds per instance in CERTIFICATE.json.

quantity	value
falsifiers (explicit admissible B with chain cost below floor(2v/3), or exact cert below threshold)	0
instances with matched-pivot value below floor(2v/3)	0
minimum matched-pivot ratio	2/3 (attained at v = 3, seed 4611, matched-pivot value 2 equal to floor(2v/3))
minimum engine-witness ratio	0.909 (10 at v = 11; every witness above threshold)
instances non-3-dichromatic under D (matched-pivot value below v)	8 (listed below)
instances with chi(Z) >= 4 unconstrained	0 found (286 of 286 backtracking decisions found a 3-colouring; 0 undecided)

The complete class-(ii) population found (every one has chi(Z) at most 3 unconstrained and cert(M) = v exactly by exhaustive a-enumeration):

v	seed	family	MP	floor(2v/3)	exact cert	arcs in Z	mutual-zero pairs
3	4611	random	2	2	3	4	1
4	4641	random	3	2	4	4	1
4	4932	random	3	2	4	3	1
6	6788	random2	5	4	6	5	1
8	8848	sparse	7	5	8	22	5
9	9654	sparse	8	6	9	25	5
10	10932	random	9	6	10	26	3
10	10654	sparse	9	6	10	33	7

Exact obstruction record (early kill). The matched-pivot value never fell below floor(2v/3) anywhere in the domain; its deficit v - MP never exceeded 3 (attained only on the fixtures), while the MP route would need the deficit to exceed ceil(v/3) (4 at v = 12, 7 at v = 19) to fail on an instance. The first instance where the matched-pivot value touches the floor exactly (v = 3, seed 4611) still has cert(M) = v exactly. On all eight class-(ii) instances the general labellings reach cert(M) = v while matched pivots do not: label patterns beyond matched pivots (the same effect PR 2747 documented on tied_min, matched-pivot 9 versus cert 11) carry the certificate there.

What is proved, what is witnessed, what is demoted

• PROVEN: nothing new. This package proves no new lemma; Lemma MP and the chain cost >= sum_j r_j >= cert(M) are cited from PR 2717 by Aaron Wang (navinor) and the merged research/root-unit-decoder-lemmas package (PR 1590 by DreamingOfClouds, with Claude Opus 5.5 (Anthropic) via Claude Code assistance), not restated.
• CHECKED (machine, verify.py, stdlib, deterministic, about 57 s): controls (brute-force-over-all-labellings agreement 3/3 at v = 3, diagonal decoder cert = 3, corrupted-witness rejection 2/2, subset-DP MAS hand values 3/3, dichromatic hand values 4/4, MP 3 on the diagonal decoder); the calibration table above; the dichromatic structure of both fixtures; and the 286-instance search with exact MP values, replayed witnesses, exact cert where a-enumeration terminates, and bounded admissible-B chain scans.
• DEMOTED / OPEN: F-star for general rational decoders is neither proved nor refuted. The matched-pivot floor max_c sum_i MAS(Z[C_i]) >= floor(2v/3) is witnessed on 288 instances (2 fixtures plus 286 searched), not proved. No "exactly when" or necessity claim is made for any hypothesis. Whether some rank-3 zero digraph has chi(Z) >= 4 unconstrained remains open (none found here; the decision is exact when it terminates and no run in this domain was undecided, but the search is bounded in v). Chain costs are bounded-search upper bounds on the minimum, not exact minima. The recorded fixture limits 11 and 18 rest on PR 2111's recorded CP-SAT optimality, not re-verified here. Residue classes (i) and (iii) are not re-attacked. One control's expected value was initially written wrong (the dichromatic number of a directed 3-cycle is 2, not 1; the subset DP returned the correct 2) and the control was corrected after the first run; the correction is recorded in the README rather than hidden.

Overlap and coordination

Pair-collision check before computation (toolkit disc-comments --grep, discussions 1317 and 1166, nested replies included, 2026-10-15 20:05-20:12 UTC): no live claim on F-star residue (ii) or (iii); PR 2717 (KB5) is DONE/RELEASED 18:20 UTC and PR 2747 (KB5b, residue (i)) is DONE/RELEASED 19:50 UTC. The refined overlap pass on the object's proper names (F-star, cert(M), 3-total decoder, zero digraph, 3-dichromatic, dichromatic number, matched pivot, L8 labelling, MAS, residue (ii), root-unit decoder) at 2026-10-15 21:10-21:14 UTC with complete pagination (716 open PRs, 1407 discussion records) returned only the known predecessors and unrelated objects. The chooser draw (OS seed e9937f18f68bb83ac4e780cc2833deb6, track complex-mechanisms) landed on the saturated cards complex-new-helper-inventory, complex-rank-shape-decoder and p104-side-correction; the assignment's explicit open request KB5c is taken instead, recorded in TASK.json.

Reproduce

Python 3.10+, standard library only, from the repository root:

python3 -B research/fstar-zero-digraph-residue-seede9937f18/verify.py
python3 scripts/research_package.py --package research/fstar-zero-digraph-residue-seede9937f18 --check-only
python3.12 scripts/research_package.py --package research/fstar-zero-digraph-residue-seede9937f18

Expected markers: PASS_FZD_CONTROLS, PASS_FZD_FIXTURES, PASS_FZD_SEARCH, PASS_FSTAR_ZERO_DIGRAPHS, PASS_REGISTERED_RESEARCH_PACKAGE research/fstar-zero-digraph-residue-seede9937f18 outcome=completed, and CERTIFICATE.json with digest 86f48c1bacdbe4a9. make is not installed on the authoring host; the commands above are the underlying ones.

Credit and provenance

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). The rank-3 configuration model, the named instances tied_min and cex_min_1, their recorded one-copy certificate limits (11 and 18, CP-SAT, CHECKED (recorded)), the stored cost-22 pairing and the structural lemmas L5/SCC: DreamingOfClouds in PR 2111 and PR 2162; the two data files are byte-identical copies of their data (sha256 670ed90924b601ade22f0d4adaff46d1667e5449326a0f7ba7b88dd8f5906f93 and 86064701dde94b4f35427bc3e19275eaf1e40e9e90d4fea28e7857f3564d4dd7). The residue statement, Lemmas SF/MP/RK/AN and the engine design this package adapts: Aaron Wang in PR 2717. Lemma MX and the class-(i) closure: Aaron Wang in PR 2747. The falsifier-census frontier (196 instances): Aaron Wang in PR 2335. Exact cost 12 of tied_min: Aaron Wang in PR 2379. Exact cost 21 of cex_min_1: Nick Petalas in PR 2206. This package does not claim their results; the new content is the exact zero-digraph machinery, the dichromatic structure of both fixtures' zero digraphs, and the class-(ii) smallest-first search with its exact obstruction record.

Prepared by Aaron Wang with help from opencode (substantial AI assistance; no independent human review claimed). New work in this package is Apache-2.0. No kappa, supplier, or frontier claim.