Scoped thresholds: what a coverage-law change must deliver for the selected complex local word (no new kappa)

[navinor]

Status: scoped obstruction + exact threshold table under the optimistic relaxation. No new κ; the selected record is unchanged. Contributor: Navinor, with help from opencode. Apache-2.0.

Adds only research/complex-coverage-law-relief/. upstream/, frozen pins and the selected record are untouched.

Task card

• Card: complex-new-helper-inventory (complex-mechanisms, chooser track, OS-random nonce 6ddf1b6647d8c5131033b16d30f0045b, base bce54fb3f895cbe5cd4c0eb477c3afeed100190f, priority revision 2). Scope is refined to the card's first experiment (the complete optimistic child histogram, rejecting routes without headroom), applied to changed coverage-law interfaces instead of a new contract; reason is recorded in TASK.json/TASK-CARD.md.
• Question: with the selected PR 1094 local word fixed ({1:2412, 2:418, 3:432, 4:142, 5:82, 8:8}, mass 5586 per call; coverage law m = 73, w = 896/9 families per call, centre 64, unpaid 1, idle rank cap 51), which interface changes are necessary and which sufficient for the optimistic moment to reach a = 1/99 (κ = 0.01) or the retained-transfer unbinding saving a = κ_bit/(1−κ_bit) = 0.001736437026484097?

Result (all brackets certified, exact rational intervals, code-disjoint from PR 1240)

Positive control: a* = 0.001690054109088696 reproduces #1240 grid ceiling a < 0.001690054109089 within one 1e-15 step. The incumbent ceiling is below the unbinding target, so pure repacking cannot unbind the complex side; an interface change is required even for the near target.

Lever (others at the selected line)	T2: unbinding	T1: a = 1/99
unpaid coordinates per family u	u* ∈ (1.009826689, 1.009826689), least integer 2	u* ∈ (2.804078009, 2.804078010), least integer 3
centre charge c (from 64)	c* ∈ (63.021698514, 63.021698515)	impossible: sup a* = 0.004711142955614 at c = 0
families per replica F (9 stages)	least met 906 (905 fails)	least met 3093 (3092 fails)
paid-word count scale s	met iff s ≤ 0.969437086	met iff s ≤ 0.049110034

Pair points at F = 896: (u=2, c=0) gives a* = 0.009357801683738 < 1/99 — unpaid relief up to 2 with any centre charge fails T1 (sup over c is at c = 0); (u=3, c=64) gives a* = 0.011001994530886 > 1/99 (optimistic sufficiency). The 4.911% word-scaling threshold independently matches #1240 "about 95% lower fully priced local excess spend". The a* < D/E diagnostic inequality is certified at four points (incumbent D/E = 0.001695287130359 vs a* = 0.001690054109089).

Not covered / limitations

• Necessity holds inside the stated model only; sufficiency is optimistic-relaxation (fractional counts, all idle at rank ≤ 51, no added charges). No physical realizability, owner map, finite invoice, cold replay or new kappa. The card's acceptance gate (a new physically realizable contract) is not met by this PR.
• u, c and the pair points are at F = 896; F, s at u = 1, c = 64. u > 1, c ≠ 64 are what-if coverage laws needing their own admission. κ = a/(1+a) is the retained planning transfer; T2 uses the bit supplier pinned at the base commit.
• Inherited all-size/compiler, common-label/mutable-source, whole-call hosting, precision/tape, analytic-transfer and bit H1–H5 assumptions remain.

Overlap / credit

• PR 1012's coverage identity; Research note: packing ceiling (+0.87%) and 1/99 budget for the selected complex local word (no new kappa) #1240 (Douglas Colkitt) for the moment model and fixed-word ceiling; Review aid: packing ceilings of 14 open complex kappa claims (stacked on #1240; no new kappa) #1264 for the 14-claim census at the unchanged law. The pending levers.py ideal-budget table referenced by Director: priority revision 3 — aim effort at centre path, rank shape and coverage law (stacked on #1240) #1272 proposed centre-path-sharing/coverage-law-change cards is complementary: this PR certifies exact threshold brackets (not first-order multiples), adds the families and word-scaling levers, least-integer thresholds, the pair obstruction and convexity margins, on the base commit.
• Ali Romman's research/heterogeneous-owner-assembly/ bounds the frozen catalogue's 25 cost interfaces and excludes new helper-inventory interfaces (the R489/R480 interfaces of PR 1004 and PR 1005); these thresholds price what such a new interface must deliver.
• Distinct from CLAIM: exact ceiling for within-group repartitions of #1094 idle rank-48 children (#1252 multi-child follow-up), expires 23:30Z #1265 (idle repartitions at a fixed law) and CLAIM: X2 acceptance boundary for the selected owner map (Navinor, expires 2026-10-14 00:40Z) #1315 (X2 owner-map acceptance boundary). Claim thread: CLAIM: coverage-law relief thresholds for the selected complex local word (Navinor, expires 2026-10-14T00:50Z) #1328.

Reproduce

python3 -I -B research/complex-coverage-law-relief/verify.py
python3 scripts/research_package.py --package research/complex-coverage-law-relief --check-only
python3.12 scripts/research_package.py --package research/complex-coverage-law-relief

Expected final line PASS_COMPLEX_COVERAGE_LAW_RELIEF (23 PASS checks: coverage identity, the PR 1240 ceiling reproduction, threshold brackets, pair points, four D/E margins, six failure/monotonicity controls). Also ran python3 -B -m unittest discover -s tests -p 'test_research_*.py': 14 pass; the 6 test_research_onboarding.PackageTests errors are pre-existing on this Windows machine at the pristine base commit (verified with my changes stashed) and unrelated to this package.

Credit: all construction credit for the selected local word stays with its authors (see certificates/selected-result.json and research/gauge44-slots-t16/README.md). Prepared by Navinor with substantial opencode assistance. No human review or full formal verification is claimed.

[Mister-Meeseeks]

100-PR batch review at 6840924bf22523a9f0a965238a22cc3bcfc7297b.

This remains open after scoped triage. Next review step: The baseline exports 0.001733427… as bit_saving, not final kappa. T2 applies a second inverse outer conversion a=kappa/(1-kappa), then calls this the complex-unbinding threshold. Re-derive T2 from the actual coupled assembly (or label it only an artificial planning target); the T1/coverage arithmetic is a separate claim.

This is a specific continuation point, not a rejection or a request to redo already recorded checks. Completed local evidence is retained separately from outstanding whole-head obligations.

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

[navinor]

Follow-up on the review note in #1348 (comment) (batch review: https://github.com/CrocSwap/integer-mult-bounds/blob/main/docs/research/community-efficient500-review.md). Taking option (A): T2 is re-derived from the actual coupled assembly. The T1/coverage arithmetic is untouched.

Old claim (reviewed head 6840924b):

T2 (unbinding): a = κ_bit/(1 − κ_bit) with κ_bit = 1733427039589844234749/10^24, the bit saving pinned by scripts/research_baseline.py at the base commit. This is where the complex side would stop binding over that bit supplier.

with target a = 0.001736437026484097.

New claim (head 74768ab4):

T2 (complex-unbinding, re-derived from the coupled assembly): a = c_bit/(1 − β) = 1733427039589844234749/(10^24 − 1), decimal 0.001733427039589844234749001733… (period-24 repeat), where c_bit = 1733427039589844234749/10^24 = 0.001733427039589844234749 is the pinned #589 bit supplier's moment endpoint in a-units (bit_saving in scripts/research_baseline.py, bit_lower in the pinned assembly certificate — a supplier saving, not a kappa) and β = 10^-24 is the assembly's transfer factor on the complex endpoint.

The retired value a = κ_bit/(1 − κ_bit) = 0.001736437026484097 is kept in the certificate as retired_a_T2, explicitly labelled defined-not-derived.

Exact conversion chain (from the pinned sources). Sources: the selected record's assembly certificate research/community-frontier1094-audit/ASSEMBLY.json and the selected outer reduction research/pending-claims-independent-pricing/arith.py (kappa_from), whose kappa_equals_record gate in research/community-batch1099-audit/check_frontier1094.py binds that assembly to certificates/selected-result.json. The package replays the reduction code-disjoint and reproduces the record kappa exactly.

1. Supplier endpoints in a-units: complex b = 67018507072324140917/(4·10^22) = 0.001675462676808103522925 (moment certified without surcharge) and bit c_bit = 1733427039589844234749/10^24 = 0.001733427039589844234749 (bad-residue surcharge inside the endpoint). c_bit is the baseline's bit_saving export: a supplier saving, not a final kappa.
2. Binding step: s = min(c_bit, (1−β)·b) − 10^−27, β = η = 10^−24; the binding label is bit iff c_bit ≤ (1−β)·b. The selected record takes the complex branch: s = (1−β)·b − 10^−27.
3. Bootstrap: a ← s(1−s+a) seven times from 384599/10^10; the fixed point is exactly s (error contracts by the factor s per step).
4. Outer reduction: q = a₇(1−2η); κ is the largest 10^-18 grid point strictly below (1−η)·q/(1+q), giving 836330098538459/(5·10^17) = 0.001672660197076918 — the selected record's kappa.

The complex side unbinds exactly when the min in step 2 flips to the bit branch: (1−β)·b ≥ c_bit, i.e. b ≥ c_bit/(1−β). The 10^−27 shave, the bootstrap, the (1−2η)/(1−η) factors and the grid floor all act strictly after the min, so none of them moves the flip point. The old value instead applied the inverse outer conversion a = κ/(1−κ) to c_bit, which is already an a-unit endpoint — a second conversion — and overshoots the flip by (c_bit − β)/(1−c_bit) ≈ 0.1736%. No κ = a/(1+a) transfer enters the corrected threshold at all.

Threshold table updated (T2 column, old → new; T1 column and pair points unchanged):

• unpaid u*: (1.009826689, 1.009826689) → (1.009188952, 1.009188953); least integer 2 in both
• centre c*: (63.021698514, 63.021698515) → (63.085188751, 63.085188752)
• families per replica: least met 906 (905 fails) → 905 (904 fails)
• paid-count scale: s ≤ 0.969437086 → s ≤ 0.971370799

The headline obstruction is unchanged: the incumbent packing ceiling a* = 0.001690054109088696 stays below T2, so pure repacking of the selected word cannot unbind the complex side.

T1 / coverage arithmetic: unchanged and separate, as the review states. T1 a = 1/99 remains a chosen planning target; its κ = 0.01 caption is now explicitly labelled as defined under the retained planning transfer κ = a/(1+a), not derived. Every T1 bracket, the (u=2, c=0) and (u=3, c=64) pair points, and the D/E diagnostics are identical to the reviewed head.

Verification on the corrected tree (files updated: research/complex-coverage-law-relief/{README.md,TASK.json,VERIFY.json,SOURCES.json,verify.py}; SOURCES.json now pins ASSEMBLY.json as an input and arith.py as provenance):

• python3.12 scripts/research_package.py --package research/complex-coverage-law-relief --check-only → PASS_PACKAGE_METADATA_AND_PINS
• python3.12 scripts/research_package.py --package research/complex-coverage-law-relief → PASS_REGISTERED_RESEARCH_PACKAGE research/complex-coverage-law-relief outcome=completed
• python3.12 -I -B research/complex-coverage-law-relief/verify.py → PASS_COMPLEX_COVERAGE_LAW_RELIEF, including the new checks ASSEMBLY_BIT_PIN, ASSEMBLY_CHAIN_REPLAY (record kappa 0.001672660197076918 and binding label reproduced), T2_BINDING_FLIP_IDENTITY, T2_BINDING_FLIP_POINT, T2_RETIRED_DOUBLE_CONVERSION, and the unchanged failure controls (unpaid = 0, halving-guard R = 52, negative centre and over-paid word rejected; one extra rank-1 child lowers the ceiling; F+9 raises it).

Citations: the review note #1348 (comment) and the batch review https://github.com/CrocSwap/integer-mult-bounds/blob/main/docs/research/community-efficient500-review.md, on PR #1348.

Aaron Wang, with help from opencode.

[woklerx]

Independent corroboration of corrected PR1348 head 74768ab408b1ae2ce2c7094b49457a86fcb06e94. A closed-form bootstrap oracle agrees with all 315 boundary/grid cases; six fixtures pass all 47 retained outer constraints. The exact switch is c_bit/(1-beta). A rounded-kappa plateau below the switch does not establish bit binding. Omitted-bootstrap, zero-beta and omitted-shave controls reject. Python 3.12.14 registered replay passed.

Source-pinned audit artifact. No new threshold claim, coverage-table audit, supplier replay, or new kappa. Credit Navinor/opencode for the correction; independent audit by woklerx with substantial OpenAI Codex assistance.

Base: 0b1cccf86e2f5132ed286a26ef0a681a495740ea. Coordinated research batch; round 16. The selected record is unchanged.