Research: masked root-star trimmed-star census on P112; exact covering bound 784/13, 64 falsified for beta-7 atoms (no new kappa)

[navinor]

Status

Scoped research result / method-obstruction on the masked decoder model; no new kappa, no supplier. Only research/masked-root-star-floor/ is added. The selected record (kappa 836330098538459/5e17), certificates/selected-result.json, upstream/ and all frozen pins are unchanged.

Task and scope

Task selector draw: track complex-mechanisms, OS-random seed b27e6ad9fd261c4b4134d5d02523a8a1, card complex-masked-decoder (scope key 7acfe4d9cac04f7ba4cb), refined to the trimmed-star mixed-beta slice of PR 717's D1 open problem. Base e208d5e77a204252f92b395aae271679cc3a56f0; the full task card is research/masked-root-star-floor/CLAIM.md. Coordination claim: discussion CrocSwap/integer-mult-bounds#1210.

Object: on P112 in #717 B14 model, masked E-graph identity (exactness only on the 6720 even-pair entries, x_i.y_j = 0 off-diagonal, x_i.y_i = 4; the 5824 odd entries fixed by cost-0 singleton terms), root-star decoders with trimmed supports N(alpha) — the pairing-rank-7 mixed-beta atoms excluded from PR 717 D1(b)'s flat-support census — with unrestricted real coefficients on both factors and any number of terms.

What this establishes

1. Complete census. All 240 root-star supports (120 normals x {untrimmed C(alpha), self-trimmed N(alpha)}) and all 57,600 ordered support pairs with exact F2 pairing ranks and exact frame-hit counts over the 1260 E-graph 8-cliques. Rank histogram: beta 7: 7,616 atoms; beta 8: 37,440; beta 9: 12,544.
2. Exact covering bound for the family. Every frame is an 8x8 rank problem, so >= 8 terms meet every frame; cst >= 10080 * min(beta/hit). The complete minimum is beta/hit = 7/1170, so the frame-substitution bound over the whole root-star family is exactly 784/13 ≈ 60.31 (vs the ~43.8 of PR 717 D1(e)'s sampled general-support bound).
3. The hoped-for 64 bound is falsified. Beta-7 trimmed atoms meet 1170 of the 1260 frames (the card's threshold for a 64 bound was 1102), e.g. (N(f), N(beta)) with f a frame normal and <f,beta> odd; the eight best atoms are recorded in RESULTS.json as leads. So nine or more cheap trimmed stars at total cost 63-64 are not excluded by frame counting; the family's assembly question is pinned to the exact window [784/13, 64].
4. All the slack lives in the trimmed atoms. Over atoms of pairing rank >= 8 the bound is exactly 64, sharp: the eight coordinate stars d_k = t_k = (<r_i,f_k>)_i are trimmed root-star atoms of the census, satisfy sum_k <r_i,f_k><r_j,f_k> = 2<r_i,r_j> on every entry, and cost 8 per term (cst 64). Any root-star decoder whose terms all have rank >= 8 therefore costs >= 64 at any term count, masked or not.

The written argument (covering lemma, reduction lemma, falsification record, scope) is research/masked-root-star-floor/PROOF.md.

Reproduce

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

python research/masked-root-star-floor/verify.py

Expected: 28 PASS lines and FAILS: none, including

PASS C.bound-beta-ge-8-is-64 | bound over beta>=8 atoms = 64
PASS C.census-complete | max hit/beta = 1170/7 at (a0,tr=False,b2,tr=True) beta=7 hit=1170; bound = 784/13
PASS E.coordinate-cst-64 | cst = 64

The run rebuilds the 120-line E8 model and the full 57,600-pair census from scratch and compares every field of the recomputed receipt with the committed RESULTS.json. A corrupted receipt (altered hit count) is rejected with a field diff and exit 1 (checked); the run also passes under python -O. Exact integer and F2 arithmetic throughout; no network, no third-party packages.

Overlap and credit

• DreamingOfClouds Format limits part 2 (no κ claim): erratum to #595, exact invoice prices, frontier-word censuses, decoder floors, spec requirements, centre and fragmentation walls #717 Format limits (no κ claim): certified ceiling κ ≤ 5.1595e-3, spec census, structural lemmas, helper bounds and open problems #595 Conditional κ = 0.00162128508944984: cold factory-scratch helpers end at 4-dimensional frames on the PR601 witness (+0.439%) #604: the B14 model, the masked reduction D1(a), the mixed-beta open problem, and the sampled covering bound D1(e) whose exact family value this PR computes.
• themilkman216 PR 968 (with Anthropic Claude assistance): the eight-term all-pairs floor, and woklerx PR 1077 (with OpenAI Codex assistance, the seed44/seed48 packages): its complex extension; both explicitly defer masked E-graph identities, which this PR addresses in one bounded family. Joel Pulikkan / GamingPuzzled Research notes (no κ claim): E8 decoder floors are tight; extension-ring, triple and Lorentzian units lose after helpers; cold-end, dispatch and fusion-search negatives #620: star floors and the complex frame decoder of cost 64. Mihir Srivastava Scoped negative result: one added P112 total cannot reduce fixed-reader copy cost #845: fixed-reader ninth total (scope excludes masked completions).
• Concurrent masked results found in the final overlap refresh, all complementary. Research: rank-8 complex decoders are gauge-rigid; E-graph eight-term floor (no new kappa) #1212 (E-graph eight-term floor via rank-8 gauge rigidity) and Research: classify the P112 masked decoder coordinate component (no new kappa) #1213 (masked coordinate component, eight terms; it cites discussion CrocSwap/integer-mult-bounds#1210 as the announced scope of this census) both leave nine-or-more terms open. Research: one-sided E8 mask rigidity on 1,373 punctured units (no new kappa) #1191 closes masked identity at arbitrary term count when one factor is root-linear; the trimmed, non-root-linear star atoms censused here are outside that hypothesis. Bit-side star label-code results (Prove E8 star-decoder puncture thresholds through 35 deleted lines #631 Research: exact star-decoder hierarchies, target budgets, and COPY bounds #669 Scoped negative result: sharp global-star punctures and 67 attaining patterns #725 Research note: sharp off-target proper-star puncture threshold #732 Research: all-puncture E8 star decoder exclusion and p=56 classification (seed 49, no new kappa) #1171) are a different model and are not imported.

Remaining obligations and limits

No construction below 64 is claimed or attempted; the assembly of the recorded beta-7 leads (nine or more terms, every frame met by a rank-8 restricted block) is the open step, in the exact window [784/13, 64]. Not covered: deeper trims, unions of stars, non-star supports (PR 717 D3's P111/P104 hits are non-star), general supports, punctures other than P112. The favorable-cost accounting is unchanged (one cst unit is +2.81% of b; complex-side changes are capped by the PR 589 bit supplier at kappa 0.001730427469823602); nothing here lowers cst, and realization charges (centre reads, frame legality, dirty restoration) are not priced because no construction is offered. Finite replay is not a formal proof of the multiplication theorem; the all-size/compiler, common-label/mutable-source, hosting, precision/tape and analytic obligations of the selected construction remain inherited and unchecked here. The repository-wide suite and the selected cold replay are not rerun for this isolated additive package.

Prepared by Navinor with help from opencode. Apache-2.0.

[Mister-Meeseeks]

100-PR batch review at 8b10989e4e45f0f9ce41004d7898a21a32f87e11.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Complete57600 trimmed root-star support-pair census: frame-cover bound784/13, not64 for beta7 atoms; sharp64 within beta≥8 subclass. Records a limit of the counting method, not a below64 construction.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.