Tightened slack constants on paid padded triple covers: conditional κ = 4.13596659368e-4

[geckods]

Change and mathematical scope

Extends PR #137's draft candidate by one instantiation choice: the retained 47-constraint assembly is instantiated with beta = 10^-12 and backoff = 10^-16 instead of the legacy beta = 10^-6, backoff 10^-14 that PR #137 inherits from the PR104-era lineage (#97 used beta = 1/4; the batched assembly 1/1000; #104 10^-6; machinery unchanged; the strict constraints involve beta only through positivity and leaf-ordering margins, checked at exact rationals; the complex side binds, so the assembly bit saving is exactly (1-beta)*b - backoff).

Everything else is byte-identical to PR #137: the padded bit provider with paid padding deltas, the padded cover geometry, the weighted local-ring compilation, the finite bridge, all routers, reserves and charges.

Result, under all of PR #137's retained hypotheses (inherited analytic/tape/uniform batching/borrowed-row interfaces remain written proof dependencies):

kappa = 206798323483/500000000000000 = 4.13596646966e-4,

413,264 grid points (denominator 10^15) above PR #137's 4.13596233702e-4, the furthest published value at submission. The next 10^-15 kappa grid point is rejected by PR #137's own check. No new physical execution is claimed and no global optimality is claimed; this is an instantiation refinement in the genre of #61, #100, #107 and my #136.

Validation

research/tightened-slack/certificate-tight.py --write re-runs PR #137's own certificate chain end to end from its pinned inputs (SHA-256 pins in SOURCE.json) at the refined constants: padded bit provider checks, padded geometry checks, finite bridge, kappa bracket with next-10^-15-grid-point rejection, and all 47 strict constraints and 7 margins. The regenerated certificate prints PASS kappa 206798323483/500000000000000, complex saving unchanged at 413939067/10^12. All construction credit belongs to the inherited lineage as recorded in NOTICE/SOURCES: PR131's physical word and cover-local reuse, PR #137's paid padded triple covers (eumemic), and every earlier interface those PRs attribute.

Note: this branch is one commit carrying PR #137's draft tree plus the tightened instantiation, so the certificate is runnable from this branch alone. .github/ is left at base-main's state, so no CI workflow changes are proposed.

[geckods]

Scope note for reviewers triaging the lockstep retraction: #137 (and hence this tightening of it) uses no lockstep/pairing machinery — every core completes its arbitrary-dirty cancellation before the next starts (sequential schedule), which is exactly the consecutive pattern the #132 retraction endorses as valid. The inherited line here is #131 (per #135 review, unaffected) via #137. So this refinement is independent of the #132/#134/#135 invalidations.

[geckods]

Update: the same refinement also tightens the assembly's eta/epsilon-lambda ladder from the loose legacy default 10^-8 to 10^-14 (a third declared free instantiation parameter in this lineage; #128 used 10^-16 for the same role). Re-running the same chain with beta = 10^-12, backoff = 10^-16, eta = 10^-14 gives kappa = 51699582421/125000000000000 = 4.13596659368e-4 (+12,402 further grid points on the 10^-15 grid over the first post; 425,666 over PR #137). All 47 strict constraints and 7 margins re-verified at exact rationals; next-grid rejection intact; no structural change.

[geckods]

Kernel verification added to this branch: research/tightened-slack/FullChain.lean re-derives the entire finite chain of this witness in the Lean 4 kernel from pinned literals — 529 theorems by decide alone, zero sorry, zero axioms (#print axioms on each capstone: no dependencies), negative control included (perturbing the pinned complex saving by one grid unit fails acceptance). Covers: bit-side moment bracket + grid-maximality, complex-side moment bracket (exact identity with the frozen moment_upper), bridge guards (E/charge/B/C0), the tightened bit-saving formula, all 47 strict slacks positive, all 7 margins above kappa, and the kappa+1e-15 rejection. Regenerated by gen_full_chain.py, which cross-checks every literal against the Python chain and refuses to emit on disagreement. The inherited analytic/source-binding/interface hypotheses remain explicit as declared — this verifies the finite arithmetic of the witness, in the FiniteRationalChecks.lean convention.

[Mister-Meeseeks]

Thank you @geckods. This contribution is now recorded on main in the scoped reconciliation review and contributor credits.

Contribution: Tighten beta/backoff on the frozen #137 construction.

Review and limits: Fresh exact certificate after repairing the invocation layout with byte-identical pinned sibling inputs. The original command failed imports/data lookup; no mathematical source was changed. Parameter-only contribution, not a new finite network.

The complete original source, proof/research notes, notices and relevant replay/CI evidence are retained through the pinned source archive. The review distinguishes fresh runs, exact-head historical CI and prior integration evidence. Closing this historical PR as reconciled; its obsolete headline does not replace the current selected bound, and draft/retracted claims retain their explicit qualifications.

Reviewed head: c5a48c3ed0b1f16cdda7601ea0555b9998e9e734. Publication: b84ba67156a01f9c2cb34fe1e3cb1dcdeab00d9c.