Reduce E8 recursive cost with four coupled frame changes

[aliromman]

Four coupled frame changes lower the recursive cost of Jacob Sussman's E8 certificate at 9c94857. Under the pinned PR352 fallback convention, the conditional complex saving increases from 876248285600677/10^18 to 876412559609473/10^18, approximately 0.01875%.

Two phase-A retimings and two phase-B frame changes reduce the five-stage recursive child count from 20,830 to 20,825. The complete rational scalar operator, including all initial helper values, is unchanged. Helpers (783), copy cost (72), rank mass (56,715), deficit (120), and scalar additions (3,876) are also unchanged.

research/e8-coupled-retiming/PROOF.md expresses the cost change as a positive weighted sum of discrete second differences of x^p. Strict concavity gives a decrease for every 0 < p < 1; the fallback charge also decreases. The proof does not rely on an approximate numerical root. The standalone package includes both certificates, deterministic reconstruction, independent checks, and a dedicated workflow.

Validation

python3 -B research/e8-coupled-retiming/verify.py

• PASS: pinned inputs, exact reconstruction, frame paths and costs, all 1,046,529 entries of the full rational scalar maps, and all initial helper dependencies.
• PASS: exact rational bounds for both savings, rejection of each next 10^-18 grid point, and four damaged-certificate controls, including a forbidden phase-A cancellation. The verifier needs only the Python standard library.
• Separate upstream checks: all three checkers pass from pinned Sussman source 9c94857, including its Lean representation diagnostics. Those scripts are not vendored or run by the package verifier.
• Repository checks before the verifier cleanup: 16 of 17 groups passed, including gen4-bank-verify, 538 tests, and 20 patch checks. The cleanup passed focused and isolated-package verification; the full suite was not rerun. The full suite is not green: verify-joint fails on a stale root Makefile hash in research/joint-dual/SOURCE.json; that manifest also has a stale README.md hash. Both files match the unchanged base, 12a4934d. This PR changes neither file nor the manifest.

Scope and credit

This is a finite circuit and recursive-cost improvement. It does not change the selected integer-multiplication bound. The new certificate has not been compiled in Lean. Frame transfer, helper cleanup, recursion, and multiplication assembly remain inherited dependencies. No global optimum is claimed.

Sussman supplied E8, its checkers, and the proof framework; DaysSky's PR352 supplied the complex connection and fallback convention. Prior retiming includes Sussman's scheduler, PR328, and PR350. The package preserves upstream notices and licenses. Prepared with substantial OpenAI Codex assistance; new contributions are offered under Apache-2.0.

[Mister-Meeseeks]

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

Contribution: Four coupled E8 frame changes.

Review and limits: Fresh exact full scalar-map comparison, rank accounting, moment bounds and mutation checks. No changed-certificate Lean build or blanket upstream gcert acceptance is claimed.

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: a22270515e9c6102dda3b22e75ca2c62e82812ff. Publication: 4040a0f46d67b41c36a472a7cd4af791fd4c36cc.