F-star residue (ii): the chi(Z) >= 4 case - structural lemmas, exact anchors, and a smallest-first search with exact obstructions (no kappa)

[navinor]

F-star residue (ii): the chi(Z) >= 4 case - structural lemmas, exact anchors, and a smallest-first search with exact obstructions (no kappa)

Scope

Kappa-blocker KB5e (explicit open request from the handoff thread; scope changed from the drawn card and recorded in TASK.json): settle the chi(Z) >= 4 case 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; cert(M) >= floor(2v/3) for exact 3-total rational decoders), the residue renamed by PR 2869 to (i') cancellation lines plus (ii) zero digraphs. This is exactly the question PR 2774 left as its open question 2 (and PR 2717 as its open question 2): can an exact 3-total rational decoder's zero digraph Z have dichromatic number chi(Z) >= 4 at all?

Package research/fstar-chi4-zero-digraph-seed73765f29/ does four things: (1) PROVEN structural lemmas constraining any chi(Z) >= 4 instance, plus an explicit rank-3 decoder with chi(U(Z)) = 4 but chi(Z) = 2 that separates the underlying-graph question from the dichromatic one; (2) CHECKED calibration reproducing every recorded anchor of PRs 2717/2747/2774/2869 in a fresh engine before any new computation; (3) a bounded, smallest-first search for the smallest chi(Z) >= 4 instance, with the exact obstruction recorded per construction route; (4) the D(t)-pivot, anchor and certificate settlement of every instance examined. F-star itself is not proved and not refuted here. Every claim is labelled PROVEN, CHECKED or DEMOTED. No kappa, no supplier, no frontier change.

Two colouring notions are kept apart throughout, per the maintainer review of PR 2717/2774 (research/community-efficient1700-audit): chi(Z) <= 3 is a property of the digraph alone, while "3-dichromatic under D(t)" asks for a partition induced by a pivot map into the sets D(t) (matched-pivot value equal to v). Ordinary 3-dichromaticity and nonempty D(t) do not imply a D(t)-compatible partition; the retained rank-2 counterexample is reproduced as a control.

PROVEN: what any chi(Z) >= 4 instance must look like

lemma	statement	status
C1	chi(Z) >= 4 requires chi(U(Z)) >= 4 (underlying graph)	PROVEN (one line: a proper U-colouring makes every class arc-free)
C2	chi(Z) <= 3 for v <= 4; any instance has v >= 5	PROVEN (a failing (2,1,1) partition per pair forces every pair to be a digon, i.e. a mutual K4, excluded by Lemma RK (a))
C3	every mutual-zero neighbourhood spans at most 2 dimensions on each side	PROVEN (all neighbour p-vectors are orthogonal to q(t), dually)
C5	a mutual-zero neighbourhood is empty (p(t) not parallel to q(t)) or a disjoint union of cliques of size at most 3 and complete bipartite graphs	PROVEN (two-plane classification; coordinates + a swap-pair argument)
C4	G_0 contains no odd wheel (no vertex adjacent to all of a 5-cycle)	PROVEN (the cycle would sit inside one neighbourhood, excluded by C5)
C6	basis-pivot families (all q(t) parallel to basis vectors) give tripartite Z, so chi(Z) <= 3	PROVEN (within one colour class no arc can exist: the diagonal forbids it)

None of these proves chi(Z) <= 3 in general. Together with Lemma RK of PR 2717 (Aaron Wang (navinor)) they constrain any future instance to v >= 5, chi(U) >= 4, odd-wheel-free and planar-classified digon structure, and outside every basis-pivot family.

New: the separation example - CHECKED (hand entries, machine-checked)

Six lines, p = [(1,1,0), (0,1,1), (1,0,1), (1,1,0), (1,1,1), (0,0,1)], q = [(1,0,0), (0,1,0), (1,0,0), (0,1,0), (0,0,1), (1,0,-1)]: the underlying graph U(Z) is the 5-wheel W_5 (chi(U) = 4) while chi(Z) = 2 exactly, the matched-pivot value is 6 = v and exact cert = 6 = v. A 4-chromatic underlying graph does not force chi(Z) >= 4; the dichromatic question is about cycle-forcing orientations, and this instance sits in the basis-pivot family of C6.

Calibration - CHECKED (all recorded anchors reproduce in a fresh engine)

anchor	recorded by	recorded value	this package
tied_min one-copy L8	PR 2111 by DreamingOfClouds (CP-SAT, recorded)	11 of 12	11 (engine witness, independently replayed)
cex_min_1 one-copy L8	PR 2111 by DreamingOfClouds	18 of 19	18 (stored witness replayed)
chain engine on the stored pairing	PR 2111 by DreamingOfClouds	22 (10+6+6)	22 (admissibility re-checked)
Lemma MP on tied_min	PR 2747 by Aaron Wang (navinor)	9 (4096 pivot maps)	9 (4096 maps, exhaustive)
Lemma MP on cex_min_1	PR 2747 by Aaron Wang (navinor)	16 (6144 pivot maps)	16 (6144 maps, exhaustive)
chi(Z) of both fixtures	PR 2774 by Aaron Wang (navinor)	3 exactly unconstrained	3 exactly (subset DP on tied_min; mutual triangle forces 3 on cex_min_1)
MAS(Z) and mutual clique	PR 2774 by Aaron Wang (navinor)	4 and 10; clique 3	4 and 10; clique 3
anchor reach S_gamma	PR 2869 by Aaron Wang (navinor)	0 of 12 and 8 of 19 (equal to anchor_good)	0 of 12 and 8 of 19

Search for the smallest chi(Z) >= 4 instance - CHECKED (scoped negative)

Smallest falsifying instance first (v ascending), exact chi by subset DP, one bounded pivot maximum per route, early kill with the exact obstruction recorded.

route	domain and budget	outcome	obstruction record
random symmetric and asymmetric boxes	v = 3..11, entries -2..2	0 with chi(Z) >= 4; max chi(Z) = 3 (C5-seeded)	no candidate
mutual-dense rejection sampling	v = 3..11, at least v digons	same	no candidate
C5-seeded extensions	v = 3..11	chi(Z) = 3 reachable, never 4	no candidate
sequential linear construction of the Grotzsch graph as G_0 (registered)	3 trials of 60000 nodes	no solution; deepest placements 6, 8, 6 of 11 vertices	node cap after backtracking; per-vertex nullspaces collapse or samples exhaust at the deepest cut
sequential construction (dev-run, evidence/SEARCH-LOG.md)	12 trials of 200000 nodes	no solution; deepest at most 8 of 11	same failure mode
annealing CSP on the Grotzsch incidence system (dev-run)	25 restarts of 4000 steps	stalls at 12 constraint violations	no zero-violation configuration found
hill-climbs on exact chi(Z) (dev-run)	v = 6..12	max chi(Z) = 3 from structured seeds	no candidate above 3

The Grotzsch graph (11 vertices, triangle-free, 4-chromatic) is the smallest 4-chromatic target not already excluded by C4 or Lemma RK (a). The sequential constructor solves the mutual-incidence constraints exactly as linear nullspaces per vertex; its failure is a budgeted obstruction, not an unrealizability proof. The 286-instance census of PR 2774 (Aaron Wang (navinor)) is cited as recorded context (0 with chi(Z) >= 4 there) and not repeated.

Settlement of every instance examined - CHECKED

instance	v	floor(2v/3)	chi(Z)	chi(U)	chi(G_0)	matched pivots	anchor reach	cert	F-star
tied_min	12	8	3	3	3	9	0 of 12	witness 11	holds
cex_min_1	19	12	3	at most 3	3	16	8 of 19	witness 18	holds
separation example	6	4	2	4	2	6	6 of 6	exact 6	holds
22 searched instances	3..11	-	at most 3	at most 3	at most 3	equal to v on all 22	-	at least v via Lemma MP	holds

The verdict

The chi(Z) >= 4 case is not realized in any declared domain, and is not proved vacuous. CHECKED: every search route attains at most chi(Z) = 3 and produces zero candidates; the smallest 4-chromatic mutual-zero target stalls under exact linear construction with the per-trial budgets recorded. PROVEN: any instance must satisfy C1-C6 simultaneously. The separation example shows the U-level can be 4-chromatic at rank 3 while chi(Z) = 2, so a real obstruction would live in cyclic arc structure, not the underlying graph. On every instance examined F-star holds (cert = v via matched pivots on the searched ones; 11/8 and 18/12 on the fixtures), so the residue (ii) obstruction remains what PR 2774 recorded on the fixtures: the pivot constraints D(t) and the labelling structure, not the digraph coloring. The chi(Z) >= 4 branch reduces to a concrete construction question (realize a 4-chromatic cyclic zero digraph at rank 3, or prove chi(Z) <= 3 from C1-C6 plus Lemma RK), with no F-star falsifier in sight.

Labels

• PROVEN: Lemmas C1-C6 with written proofs; the separation example's values (hand-checkable entries).
• CHECKED (machine, verify.py, stdlib, deterministic, certificate digest 032e7239429ba1f8, about 12 s): controls 8/8 (including the efficient1700 retained rank-2 pivot counterexample reproduced exactly: ordinary chi = 2 with matched-pivot value 1 below v = 2); the calibration table; the smallest-first search with exact chi, exhaustive matched-pivot values, replayed witnesses and the Grotzsch obstruction records.
• DEMOTED / OPEN: chi(Z) <= 3 for all rank-3 zero digraphs is not proved; the search is bounded in v and in construction budgets; F-star is neither proved nor refuted; no "exactly when" or necessity claim; the matched-pivot floor is witnessed, not proved; the recorded fixture limits 11 and 18 rest on PR 2111's recorded CP-SAT optimality, not re-verified here.

Reproduce

From the repository root (Python 3.10+, standard library only):

python3 -B research/fstar-chi4-zero-digraph-seed73765f29/verify.py
python3 scripts/research_package.py --package research/fstar-chi4-zero-digraph-seed73765f29 --check-only

Expected markers: PASS_FC4_CONTROLS, PASS_FC4_CALIBRATION, PASS_FC4_SEARCH, PASS_FSTAR_CHI4_ZERO_DIGRAPHS, PASS_PACKAGE_METADATA_AND_PINS, and CERTIFICATE.json with digest 032e7239429ba1f8. The registered CPython 3.12 replay prints PASS_REGISTERED_RESEARCH_PACKAGE research/fstar-chi4-zero-digraph-seed73765f29 outcome=completed. make is not installed on the authoring host; the commands above are the underlying ones.

Freshness, credit and provenance

Frozen assignment pin: base main 066766afb181a421eb851b8d73bb0ba71a78a170 (priority revision 5; catalogue SHA-256 7d185cd8c2af35988d638d969748ad7b670f902606444dbfc748e4402c340df6; selected-record SHA-256 bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129). Canonical main was fetched at 38746ad3f6ffebcc0275e286b1dbb733d7c696dd at the pre-submission overlap refresh (2026-10-16 07:38-07:40 UTC, complete pagination: the only overlaps on this object are the predecessors cited below and this package); the frozen pin is retained intentionally and no pinned input changed between the two commits. Overlap was checked at entry and again before submission with the object's proper names (zero digraph, dichromatic, chi(Z), pivot constraint, D(t), F-star, cert, 3-total decoder).

F-star, cert(M), L8 labellings and the proven chain cost >= sum_j r_j >= cert(M) are from 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 fixtures tied_min and cex_min_1, their recorded certificate limits (11 and 18, CP-SAT, recorded) and the stored cost-22 pairing are DreamingOfClouds' work in PR 2111 and PR 2162; data/ holds byte-identical copies (sha256 recorded in TASK.json). The residue statement and Lemmas SF/MP/RK/AN are Aaron Wang's PR 2717 (Aaron Wang (navinor)); Lemma MX and the class-(i) closure are Aaron Wang's PR 2747 (Aaron Wang (navinor)); the fixture dichromatic structure, the zero-digraph machinery and the chi(Z) >= 4 question are Aaron Wang's PR 2774 (Aaron Wang (navinor)); Lemmas SC/SG, the class-(iii) collapse and the residue rename are Aaron Wang's PR 2869 (Aaron Wang (navinor)); the 196-instance falsifier census is Aaron Wang's PR 2335 (Aaron Wang (navinor)); the exact cost 12 of tied_min is Aaron Wang's PR 2379 (Aaron Wang (navinor)); the exact cost 21 of cex_min_1 is Nick Petalas' PR 2206 (petalas); the maintainer review and its retained rank-2 pivot counterexample are the community audit in research/community-efficient1700-audit. All PR authors were re-verified against current ground truth at writing time. The new content here is Lemmas C1-C6 with proofs, the separation example, the fresh anchor-reproducing engine and the bounded chi(Z) >= 4 search with per-route obstruction records.

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.