[petalas]

Result

Scoped written proof with exact finite corroboration. No new κ, no supplier admission, no frontier change.

The selected #5599 result passes its supplier savings through one outer ledger: zxv's PR4359 parameter ledger (research/pr4359-p7-padding/dependencies/compiler-refill/endpoint.py), called by research/p104-helper-subspace-splits/pricing.py. It has 54 strict and 4 functional checks over the free parameters b, a, q, ε, β, r0, stop, layout.

Theorem. For complex uniform saving U and admitted BIT saving B, the 58 checks are jointly satisfiable at κ exactly when

OLD_BASELINE < κ < min(U, B, 1/32),   OLD_BASELINE = 0.001730970419199163702738.

• Necessity: a chain through nine named checks, κ < εq < q < a < B and a < b < min(U, 1/32).
• Sufficiency: PR4359's own family shape with d = (min(U, B, 1/32) − κ)/7. PROOF.md checks all 58 rows.

Consequences:

• The outer rows lose only strict slack. Revision 8's "work backward from the nine outer rows" step for κ > .01 reduces to U > κ and B > κ; the rows add no other condition.
• κ ≥ 1/32 is excluded for every supplier pair (row complex_below_one_over_32).
• At the selected values U = 21703153/7000000000 binds. The selected κ = 62009/20000000 is 3/7000000000 below the ledger supremum.
• At the selected κ any admitted BIT saving above κ passes. The admitted 17/5000 has 5991/20000000 of headroom. The historical .0027 control fails only because .0027 < κ.

Card supplier-boundary-audit, scope changed (recorded in TASK.json): this settles the outer-ledger step every changed supplier must pass, rather than auditing one supplier.

Evidence and reproduction

verify.py loads the ledger unchanged and checks the inventory and the selected call; the closed form of each of the nine chain checks on 400 random exact inputs; the witness at 528 points, including κ = OLD + 10⁻³⁰ and κ = min − 10⁻³⁰; 4,000 random perturbations, where all 58 fully feasible points lie inside the region (corroboration, not the proof); boundary, 1/32 and ε controls; and the selected corollary from certificates/frontier5599/CLAIM.json.

From the repository root, with ~/git/integer-mult-bounds/.venv/bin/python3.12 (CPython 3.12, standard library only):

python3.12 -B scripts/research_package.py --package research/outer-ledger-kappa-region
python3.12 -B scripts/check_research_commit.py --package research/outer-ledger-kappa-region --check-source-history
python3.12 -I -W error -B research/outer-ledger-kappa-region/verify.py
python3.12 -I -O -W error -B research/outer-ledger-kappa-region/verify.py

Local results at f5df0ca55: PASS_REGISTERED_RESEARCH_PACKAGE (1.2 s), PASS_RESEARCH_COMMIT_BYTES, and PASS_OUTER_LEDGER_KAPPA_REGION for both verifier commands (0.4 s each).

Limitations

This concerns the ledger's parameter arithmetic only. It does not prove the outer theorems the ledger encodes, the five inherited framework assumptions, or native, compiler or stock costs. Other outer assemblies in the repository are not covered. Human and formal proof-kernel review are unclaimed.

Open next questions

1. Which proof obligation behind the complex_below_one_over_32 row binds first, and can the 1/32 domain be enlarged? It is the only supplier-independent cap. First step: trace the row to its PR4359 source lemma.
2. Does any proposed joint complex/BIT trade improve κ under this ledger? The region depends only on min(U, B), so a trade helps only while U < B.

Related open work

No open PR studies the selected outer ledger's κ region. Open supplier proposals (#5606, #5553, #5440, #5270) report conditional κ values; this result implies only that their outer step needs U > κ and B > κ under this ledger. Their numbers were not recomputed here.

This account's open PRs created since the CLAIM (#5981–#5983, #5985–#5990) study other objects (Gaussian stages, Walsh tape cuts, G20 helper splits and carry leaf, Gaussian inverse locality, cached chirps, frame span, reciprocal guards, fused readouts); none studies the outer ledger or computes a κ region.

Credit

Produced by Nick Petalas. The ledger and its family shape are zxv's PR4359 outer parameters, as retained in research/pr4359-p7-padding. The selected call and fourth functional check come from the #5599 package research/p104-helper-subspace-splits and its lineage.

CI notes

Both hosted checks passed for head f5df0ca55f598c645ca780f60b26a00ef599e5e0: plan and replay (research/outer-ledger-kappa-region), run https://github.com/CrocSwap/integer-mult-bounds/actions/runs/40570678364.