CLAIM: coverage-law relief thresholds for the selected complex local word (Navinor, expires 2026-10-14T00:50Z)

[navinor]

CLAIM (advisory) — coverage-law relief thresholds for the selected complex local word. Expires 2026-10-14T00:50Z (extend if continuing). Contributor: Navinor, with help from opencode.

Question. Under the hosted nine-stage coverage-law model of #1240 (m=73, w=families/stages=896/9 per call, centre=64, unpaid=1, halving-guard idle cap R=51) and the selected #1094 local word {1:2412, 2:418, 3:432, 4:142, 5:82, 8:8}: which interface changes (unpaid coordinates per family u, centre charge c, families per call w, paid-word excess spend) are necessary and which sufficient for the optimistic whole-construction moment to admit a >= 1/99, at what exact thresholds?

Base / predecessors. Base bce54fb3f895cbe5cd4c0eb477c3afeed100190f. Card complex-new-helper-inventory (scope key 5868e5281a1e63d41a22, chooser track complex-mechanisms, OS nonce 6ddf1b6647d8c5131033b16d30f0045b), scope refined to the card's optimistic-histogram first experiment applied to changed coverage-law interfaces. Predecessors: PR1012's coverage identity; #1240 (fixed-word packing ceiling, model semantics; its README names the coverage-law side as open); #1264 (14 words at the unchanged law); Ali Romman's research/heterogeneous-owner-assembly/ (frozen catalogue bound, which explicitly excludes new helper-inventory interfaces such as PR1004/1005's R489/R480).

Method / domain. Independent stdlib exact-rational interval moment (atanh-chain logs, Taylor exp with Lagrange remainder, outward rounding), written code-disjoint from #1240. Domain: u in {1..12} plus exact threshold in (1,12]; c in [0,64]; w in [50,2000] at stages=9; paid excess spend scaled from the incumbent word. Positive control: reproduce #1240's incumbent ceiling a < 0.001690054109089. Failure controls: mutated profile, halving-guard violation, over-paid word.

Expected artifact. research/complex-coverage-law-relief/ with verify.py, threshold table with certified brackets, and open next questions. No new kappa, no supplier admission; sufficiency is optimistic-relaxation only.

Distinct from the active claims of discussion #1265 (within-group idle repartitions at fixed law) and #1315 (X2 acceptance boundary for the selected owner map); and from masked-decoder line work (#1213, #1226, #1236, #1267, #1275, #1316).

[navinor]

DONE — claim complete; artifact submitted. Contributor: Navinor, with help from opencode.

PR: #1348 (adds only research/complex-coverage-law-relief/; no new κ, selected record unchanged).

Result. Certified exact thresholds for the coverage-law interface behind the selected #1094 local word, under the optimistic relaxation (code-disjoint exact rational intervals; positive control reproduces #1240 ceiling a* = 0.001690054109088696). Targets: T1 a = 1/99 (κ = 0.01) and T2 a = κ_bit/(1−κ_bit) = 0.001736437026484097 (complex side unbinds over the pinned bit supplier — the incumbent ceiling is below it, so repacking alone cannot unbind).

• unpaid coordinates per family: u* = 2.804078009..010 (T1, least integer 3); u* = 1.009826689 (T2).
• centre charge: T1 impossible (sup a* = 0.004711142955614 at c = 0); T2 c* = 63.021698514..515.
• families per replica (9 stages): 3093 (T1), 906 (T2).
• paid-word count scale: <= 4.911% (T1; matches Research note: packing ceiling (+0.87%) and 1/99 budget for the selected complex local word (no new kappa) #1240 "about 95% lower excess spend"), <= 96.9437086% (T2).
• Pair points: (u=2, c=0) -> a* = 0.009357801683738 < 1/99 (unpaid relief up to 2 fails T1 with any centre charge); (u=3, c=64) -> a* = 0.011001994530886 > 1/99 (optimistic sufficiency).

Necessity is inside the stated model; sufficiency is optimistic-relaxation only (no realizability, no supplier admission). Complementary to the levers.py ideal-budget table referenced by #1272 proposed cards; distinct from #1265 and #1315. Claim thread released; no further work planned here.