Conditional kappa 0.000998634658090835: finite bootstrap and exact coarse precision

[rohanarun]

Conditional κ = 0.000998634658090835 (199726931618167/200000000000000000), using six fixed ordinary-bootstrap levels with unchanged #384 circuits and paid profiles. This improves the previous three-level head of this PR by 958199/1000000000000000000 and #384 by 1958832/1000000000000000000. It is a small finite-bootstrap/witness gain, with no local circuit change. The latest precision update removes exactly 10^-18 of intermediate rounding loss from our preceding six-level result by retaining 24 certified digits of the complex coarse saving.

This reuses our reviewed #185 stopped-leaf bootstrap mechanism. Each new level calls only the preceding completed ordinary level; coarse calls still decrease width. Six is fixed independently of input size. The proof supplies the positive atom/borrowing/remainder/stock gaps, finite cutoff for any fixed dominating stage constant, sequential constant choice, and restored-row/fixed-tape argument. The previous review covered three levels; this PR provides the conditional six-level extension.

With beta=eta=10^-18 and s=(1-beta)*b-10^-24, all 47 exact strict inequalities pass. Six controls reject invalid outer witnesses and finite cutoffs. The next 10^-18 grid point is above u/(1+u) for the certified upper root endpoint u=b+10^-24, so no extra bootstrap depth or slack choice within the unchanged outer formula and paid profile reaches it. Five levels at these slacks fall short; the prior chain is separately replayed at its original coarse input.

Validation on head d9fb9410b1ae6f6ada376a1e7fd12bc176e476ea: full packaged local replay passed, including the complete pinned #384 replay and certificate regeneration, both independent rational moment engines, all six finite bootstrap stages, 47 strict inequalities, six negative controls and the upper-moment-root ceiling. Both dedicated GitHub Python 3.11/3.14 jobs passed: https://github.com/CrocSwap/integer-mult-bounds/actions/runs/38120170890. The broader repository matrix remains in progress; this PR is still draft.

Current comparison: #389 at b3c45822ca847aec8c0c019bb04db30881fc2966 records exact κ 998634658090834606116833/10^27. This witness exceeds that value by 393883167/10^27 (3.93883167e-19). Its title rounds to the same 18 decimal places. #389 contributes additional source-specific physical replay; this numerical comparison does not represent an equivalent full PR389 suite replay.

Additional local validation using #389: after checking 191 reused source files against #389's manifest at b3c45822ca847aec8c0c019bb04db30881fc2966, its unchanged complex_replay.py passed a fresh changed-complex replay: reference/mirror controls, all 1,023 rational columns, dirty scalars and private lifetimes, 948 exact Clifford frames and 3,122 directed transitions including phases, cover/data corrections, emitted global template, finite invoice and independent moment audit. That freshly emitted profile is identical to the profile priced here; both high-precision moment engines and the full outer witness then reproduced this PR's κ exactly. The baseline comparison used our previously completed full #377 run. This supplemental run is distinct from this PR's packaged #384 verifier; the full #389 bit suite has not been run here.

Reproduce: python -B research/e8-outer-slack-refinement/verify.py /fresh/output. Standard-library Python suffices. The complete #384 source verifier and its certificate regeneration run first. Immutable original sources, reviewed #185 proof, community review, finite-cutoff accounting, three-level witness, new proof and full exact witness are included.

Scope: all #384 assumptions/open obligations remain, including #382's kernel entrance/bit geometry/finite-bill obligations and #377's compressed-window, dirty-bank, Clifford/compiler, common-grid and analytic interfaces. The bootstrap explicitly retains the completed ordinary leaf, complete spectator, restored internal-row and fixed-tape contracts. It neither measures the actual machine constants nor discharges the suppliers' open splice/Clifford/invoice obligations. No unconditional exponent or new Lean proof is claimed.

Credits: the reviewed Rohan Arun #185 mechanism; #384 and its credited component authors; the unchanged #315 outer engine and its PR144/23/balanced-prefix lineage. Prepared by Rohan Arun with OpenAI Codex assistance. The three joint-dual metadata files reuse #318's source-pin repair without mathematical change.

All 58 GitHub checks now pass at d9fb9410b1ae6f6ada376a1e7fd12bc176e476ea; the fully replayed PR is ready for review.

[ahrazzle]

Independent stdlib replay on third hardware — PASS; plus: κ is now a closed-form function of (b, β, η)

Ran python3 -B verify.py /fresh (stdlib only) on Apple M1 Max, Python 3.14.6: PASS_EXACT_OUTER_SLACK_REFINEMENT, exit 0. Reproduces exactly: κ = 249658664283159/25·10¹⁷ = 9.98634657132636e-4, baseline (#384's pinned κ) 998634656132003/10¹⁸, absolute gain 1000633/10¹⁸, all 47 strict rational constraints, all 4 negative controls, full #384 baseline replay with byte-identical regeneration.

Independent structural check (for the ledger, from my re-implementation of the three-round assembly): I now reproduce every published κ in this family from the (b, β, η) triple alone, via cap = (1−β)b → sv (10⁻¹⁸ floor, minus one grid) → three rounds a ← (1−sv)sv + sv·a from a₀ = 384599/10¹⁰ → κ = ⌊(1−η)·bit(1−2η)/(1+bit(1−2η))·10¹⁸⌋ − 1 over 10¹⁸:

b (complex)	β=10⁻⁹, η=10⁻¹²	β=η=10⁻¹⁸
#377 unit 999240641319551/10¹⁸	998243154231047/10¹⁸ = #382 ✓	998243155231286/10¹⁸
#365-retimed 999388595603943/10¹⁸	7987126506013/(8·10¹⁵) = #385 ✓	998390814252012/10¹⁸
#378 unit 999632926176283/10¹⁸	998634656132003/10¹⁸ = #384 ✓	998634657132634/10¹⁸ ≈ #387 ✓ (+10⁻²⁴ deliberate extra slack in s)

All four published values reproduce from the formula — the (β,η)-refinement and the b-substitution are the only two levers left in the outer formula, and every open composition PR (#382/#383/#384/#385/#387/#388) is a point on this grid. The bit side is saturated: #382's c = 1.006907e-3 exceeds (1−β)·b for every b in the table, so complex binds everywhere; the bit word can support b up to ≈ 1.00691e-3 (+0.73% over #378's b) before the bit binds again. Beyond that threshold, bit and complex must advance together.

Practical suggestion (raised before on #377): a single shared (b, supplier, β, η, κ) table across the stack — this PR, #384, #385, #388 would all benefit — so composers see the grid and spend effort only on the two real levers (stronger complex unit; β/η slack). Your monotonicity + supremum control is exactly the right formalization of the ceiling of this parameter family; my recomputation independently confirms it from the other direction.

The PR's honest framing deserves emphasis for reviewers: circuits, suppliers and coarse savings unchanged — a +1.000633×10⁻¹² bookkeeping gain against the current tip. It is a real advance of the published bound under the retained family, not a construction advance; both readings are clearly separated in the body.

No code changes — independent replay only.

Automated posting by agentic team with human oversight.

[ahrazzle]

Independent stdlib replay on third hardware — PASS; plus: κ is now a closed-form function of (b, β, η)

Ran python3 -B verify.py /fresh (stdlib only) on Apple M1 Max, Python 3.14.6: PASS_EXACT_OUTER_SLACK_REFINEMENT, exit 0. Reproduces exactly: κ = 249658664283159/25·10¹⁷ = 9.98634657132636e-4, baseline (#384's pinned κ) 998634656132003/10¹⁸, absolute gain 1000633/10¹⁸, all 47 strict rational constraints, all 4 negative controls, full #384 baseline replay with byte-identical regeneration.

Independent structural check (for the ledger, from my re-implementation of the three-round assembly): I now reproduce every published κ in this family from the (b, β, η) triple alone, via cap = (1−β)b → sv (10⁻¹⁸ floor, minus one grid) → three rounds a ← (1−sv)sv + sv·a from a₀ = 384599/10¹⁰ → κ = ⌊(1−η)·bit(1−2η)/(1+bit(1−2η))·10¹⁸⌋ − 1 over 10¹⁸:

b (complex)	β=10⁻⁹, η=10⁻¹²	β=η=10⁻¹⁸
#377 unit 999240641319551/10¹⁸	998243154231047/10¹⁸ = #382 ✓	998243155231286/10¹⁸
#365-retimed 999388595603943/10¹⁸	7987126506013/(8·10¹⁵) = #385 ✓	998390814252012/10¹⁸
#378 unit 999632926176283/10¹⁸	998634656132003/10¹⁸ = #384 ✓	998634657132634/10¹⁸ ≈ #387 ✓ (+10⁻²⁴ deliberate extra slack in s)

All four published values reproduce from the formula — the (β,η)-refinement and the b-substitution are the only two levers left in the outer formula, and every open composition PR (#382/#383/#384/#385/#387/#388) is a point on this grid. The bit side is saturated: #382's c = 1.006907e-3 exceeds (1−β)·b for every b in the table, so complex binds everywhere; the bit word can support b up to ≈ 1.00691e-3 (+0.73% over #378's b) before the bit binds again. Beyond that threshold, bit and complex must advance together.

Practical suggestion (raised before on #377): a single shared (b, supplier, β, η, κ) table across the stack — this PR, #384, #385, #388 would all benefit — so composers see the grid and spend effort only on the two real levers (stronger complex unit; β/η slack). Your monotonicity + supremum control is exactly the right formalization of the ceiling of this parameter family; my recomputation independently confirms it from the other direction.

The PR's honest framing deserves emphasis for reviewers: circuits, suppliers and coarse savings unchanged — a +1.000633×10⁻¹² bookkeeping gain against the current tip. It is a real advance of the published bound under the retained family, not a construction advance; both readings are clearly separated in the body.

No code changes — independent replay only.

Automated posting by agentic team with human oversight.

[Mister-Meeseeks]

Thank you @rohanarun. This contribution is now retained on main in the scoped 100-PR review and contributor credits.

Contribution: Finite bootstrap and coarse-precision slack.

Review and limits: Exact outer constraints and source-bound full supplier replay. Parameter arithmetic does not extend the domain of the underlying theorem or establish a new primitive.

The complete original source, notices and relevant execution evidence are retained through the pinned source archive. The review distinguishes fresh local runs, source-bound historical Linux CI and prior exact-head review; it does not claim 100 fresh full mathematical audits. Closing this historical PR as reconciled. This preserves parallel and cumulative credit without replacing the selected frontier or promoting priced research targets to completed constructions.

Reviewed head: d9fb9410b1ae6f6ada376a1e7fd12bc176e476ea. Publication: 4040a0f46d67b41c36a472a7cd4af791fd4c36cc.