F-star residue class (iii): the entry-class scope of Lemma AN/MX - mis-stated and empty (no kappa)

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F-star residue class (iii): the entry-class scope of Lemma AN/MX — mis-stated and empty (no kappa)

Scope

Kappa-blocker KB5d (explicit open request; scope changed from the drawn card and recorded in TASK.json): settle the statement of residue class (iii) "non-dyadic entries" 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 every exact 3-total rational decoder M = sum_j p_j q_j^T. PR 2717 by Aaron Wang (navinor) demoted exactly: "(c) for non-dyadic entries (odd primes in denominators) the mod-2 anchor does not apply at all". PR 2747 by Aaron Wang (navinor) closed class (i) on the fixtures with Lemma MX and left class (iii) untouched. PR 2774 by Aaron Wang (navinor) recorded class (iii) as mathematically murky: odd-denominator rationals are 2-integral, so Lemma AN's mod-2 anchor may already cover them. This PR settles the statement. Package research/fstar-entry-class-scope-seed63d5f999/. F-star itself is not proved and not refuted here; no kappa, supplier or frontier change is claimed.

Agent: Aaron Wang, with help from opencode (substantial AI assistance; no independent human review claimed). Base 544a7e82810e30cab61c297cb2e2ea86127fe944 (frozen assignment pin; canonical main fetched at 66bb142b233f95399922c9e1894a4b5ef850bf40 at the pre-submission refresh, with the pinned inputs research/root-unit-decoder-lemmas/README.md and docs/research/agent-task-catalog.json unchanged between the two; priority revision 5).

The precise entry-class statement (PROVEN)

For a fixed factorization M = sum_{j=1..3} p_j q_j^T, let m_j = min_t v_2(p_j(t)), n_j = min_s v_2(q_j(s)) (nonzero entries), c_j = m_j + n_j, and gamma = min_j c_j. Here v_2 is the 2-adic valuation, Z_(2) is the ring of odd-denominator rationals, and Z[1/2] the dyadic rationals.

Lemma SC (exact scope of Lemma AN's hypothesis). c_j is the 2-adic content of the rank-1 matrix p_j q_j^T, invariant under per-total rescaling; the factorization becomes 2-integral after per-total powers of two if and only if c_j >= 0 for every total. Hence: entries in Z_(2) (odd denominators — this is exactly the "non-dyadic" class of the demoted claim (c)) satisfy Lemma AN's hypothesis verbatim, with no rescaling; entries in Z[1/2] (dyadic) satisfy it exactly when all c_j >= 0 — dyadic entryhood alone does not imply it; general Q is governed by the same per-total criterion.

Lemma SG (scale-invariant anchor; exact reach). For every rational factorization of every entry class, one global rescaling M -> 2^{-gamma} M makes Lemma AN applicable; the reduced labels satisfy K_ts = 2^{-gamma} M_ts mod 2, so K_ts = 0 at every incidence and K_tt = 1 exactly on S_gamma = {t : v_2(M_tt) = gamma}. Therefore cert(M) >= |S_gamma|, and the reach is exact: v_2(M_tt) >= gamma always, and no mod-2 anchor from any power-of-two normalization of the given factorization anchors a line outside S_gamma. Lemma AN's count #{t : M_tt odd} is the special case gamma = 0; PR 2717's corollary sentence about dyadic decoders has a normalization gap as literally stated (control SC1), which Lemma SG repairs for all rational decoders: F-star holds whenever at most floor(v/3) lines have v_2(M_tt) > gamma (the cancellation lines).

Pattern invariance (the collapse). lambda M shares (pi, sigma, Z) with M, hence cert and every chain cost cost(B) are invariant under global rescaling, and every rational pattern has an integer-entry realization (scale each total by the lcm of all denominators). The realizable patterns are identical over Z, Z[1/2], Z_(2) and Q, so F-star holds for all rational decoders if and only if it holds for integer-entry decoders. No residue of F-star can live in entry denominators: class (iii) is empty as an obstruction. Lemmas SF/MP/RK are coefficient-free; Lemma MX's A/U parts use the anchor labels (available everywhere by Lemma SG, with A inside S_gamma) and its T part is coefficient-free, so none of the four lemmas carries an entry-denominator restriction. The only "if and only if" statements are Lemma SC's and Lemma SG's reach, both proved in both directions; no other necessity claim is made.

Fixture settlement and calibration (CHECKED)

Both canonical fixtures are integer-entry decoders, so class (iii) is already empty on the canonical data, and the merge into the cancellation residue is exact on both:

quantity	tied_min (v = 12)	cex_min_1 (v = 19)
entry class of p, q	integer	integer
per-total contents c_j	0, 0, 0	0, 0, 0
gamma	0	0
v_2(M_tt) per line	1 on all 12 lines	0 on 8 lines, 1 on 11 lines
anchor reach S_gamma	0 of 12	8 of 19
PR 2747 anchor_good	0	8
anchor-untouchable v_2(M_tt) > gamma	12 of 12	11 of 19
PR 2747 tied_isotropic	12	11

Every recorded calibration anchor reproduces (machine-checked, witnesses replayed by the independent checker):

anchor	recorded by	recorded	here
tied_min one-copy L8	PR 2111 by DreamingOfClouds (CP-SAT, CHECKED (recorded))	11	11 (hill-climb re-found; upper bound 12)
cex_min_1 one-copy L8	PR 2111 by DreamingOfClouds (CP-SAT, CHECKED (recorded))	18	18 (stored witness replayed)
chain 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 (exhaustive)
Lemma MP on cex_min_1	PR 2747 by Aaron Wang (navinor)	16 (6144 pivot maps)	16 (exhaustive)
Lemma MX on the fixtures	PR 2747 by Aaron Wang (navinor)	9 and 16	9 and 16 (all-T wins; 2048 mixed maps evaluated on cex_min_1, matching its recorded mixing obstruction)
chi(Z) unconstrained	PR 2774 by Aaron Wang (navinor)	3 and 3	3 (exhaustive) and 3 (decision plus mutual-zero triangle)
non-3-dichromatic under D(t)	PR 2774 by Aaron Wang (navinor)	no and no	no and no (MP 9 < 12, 16 < 19)

Smallest-first falsifier search (CHECKED, scoped negative)

Smallest falsifying instance first, v ascending from 3, three families aimed at the audited statement; falsifier criterion an explicit admissible pairing B with chain cost below floor(2v/3) or exact cert below threshold; an instance is certified once a machine-checked L8 witness at or above threshold is replayed.

family	entries	instances	class	Lemma AN hypothesis	falsifiers	certified
evenden	halves, quarters	28	dyadic	fails on 28 (gamma < 0)	0	28
oddden	thirds, fifths	28	2-integral, odd denominators	holds on 28	0	28
mixed	halves and thirds	28	general Q	fails on 28 (gamma < 0)	0	28

84 instances, v = 3..10, zero falsifiers, 84 of 84 certified (minimum ratio 1.0; exact cert = v on all 36 decided instances, v = 3..5). 56 instances lie outside Lemma AN's literal hypothesis and were covered by Lemma SG's scaling; 66 of 84 have anchor reach below threshold and were carried by the coefficient-free matched-pivot or general-labelling certificate. The scope identity K_ts = M_ts mod 2 was machine-checked on every instance. Early kill with the exact obstruction recorded: by pattern invariance the entry-class axis cannot produce a falsifier beyond the shared combinatorial (pi, sigma, Z) space, so the search was a regression test of the scope machinery; the falsifier criterion (admissible-B chain cost) was armed but never needed to fire. The true residual obstruction is exactly (i-prime) the cancellation lines {t : v_2(M_tt) > gamma} — now containing PR 2747's tied-minimum repair/ordering problem and Lemma AN's anchor-count axis — together with (ii) the zero digraphs of PR 2774. Residue class (iii) is DEMOTED: mis-stated (odd primes in denominators are 2-integral and anchor-covered verbatim) and empty (entry-class-invariance). The residue classification is renamed to (i-prime) and (ii).

Claims ledger

• PROVEN: Lemma SC (with the Z[1/2] / Z_(2) / Q coverage and both directions of the criterion); Lemma SG (scale-invariant anchor, exact reach S_gamma, repaired AN corollary); pattern invariance (integer realization included); the Lemma MX / SF / MP / RK scope remark.
• CHECKED: controls 3/3 brute-force agreement, diagonal cert = 3, corrupted-witness rejection 2/2, MAS 3/3, chi 4/4, MP = 3 diagonal control, and 4 scope controls (dyadic negative-content instance where AN's hypothesis fails and SG anchors everything; scaled M = 2 I_3 with naive count 0 and reach 3; odd-denominator anchor verbatim; reach-witness replay); the fixture settlement and every calibration anchor above; the 84-instance search. Certificate digest 39f5c167c495f457.
• DEMOTED / OPEN: class (iii) as stated; F-star for general rational decoders remains open; the fixture limits 11/18 rest on PR 2111's recorded CP-SAT optimality, not re-verified here; chi(Z) >= 4 above v = 13 remains budget-limited.

Reproduce

python3 -B research/fstar-entry-class-scope-seed63d5f999/verify.py
python3 scripts/research_package.py --package research/fstar-entry-class-scope-seed63d5f999 --check-only
python3.12 scripts/research_package.py --package research/fstar-entry-class-scope-seed63d5f999

Markers: PASS_FSC_CONTROLS, PASS_FSC_FIXTURES, PASS_FSC_SEARCH, PASS_FSTAR_ENTRY_CLASS_SCOPE, and the registered replay PASS_REGISTERED_RESEARCH_PACKAGE research/fstar-entry-class-scope-seed63d5f999 outcome=completed. 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 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 model, the fixtures tied_min and cex_min_1, their recorded limits 11 and 18 (CP-SAT, CHECKED (recorded)) and the stored cost-22 pairing are DreamingOfClouds' work in PR 2111 and PR 2162; the fixture data files are byte-identical copies (sha256 670ed90924b601ade22f0d4adaff46d1667e5449326a0f7ba7b88dd8f5906f93 and 86064701dde94b4f35427bc3e19275eaf1e40e9e90d4fea28e7857f3564d4dd7). The residue statement, Lemmas SF/MP/RK/AN and the exact-completion engine design adapted here are Aaron Wang's PR 2717; Lemma MX and the class-(i) closure are Aaron Wang's PR 2747; the class-(ii) machinery and the murkiness record are Aaron Wang's PR 2774; the falsifier census is Aaron Wang's PR 2335; exact cost 12 of tied_min is Aaron Wang's PR 2379; exact cost 21 of cex_min_1 is Nick Petalas' PR 2206; the stored 18-witness is from PR 2717's package. PR 2613 by thakrarsagar is a same-motif, different-object neighbor (a mod-2 2-integral rank floor for a bit register model). New content here: Lemmas SC and SG with proofs, pattern invariance, the Lemma MX scope remark, the fixture entry-class settlement and the smallest-first entry-class search.