Finite assembly theorem and source-bound PR799 application

[lydakis]

Fixed backoffs and a fixed bootstrap depth leave a gap below the retained balanced assembly's supplied-cap ceiling. This PR adds a constructive finite-parameter theorem, an exact verifier, and source-bound applications to PR433 and PR799. The source-basis, contract-checking and E8 contributions below remain included.

Review coverage (2026-10-13)

The published upstream review and review comment record that the two primary finite-assembly packages passed, including the pinned PR799 application, at this exact head. PR275 remains open with partial coverage: the other research/tooling packages require individual review and execution. Inherited generic CI does not cover the whole expanded submission.

Additional contributor-side validation at this unchanged head:

• research/e8-exact-bank-optimizer: all four existing tests passed, including the independent 16,875-inventory oracle, pinned witness cases, large compressed counts, architecture constraints and rejection controls. The proof and implementation were reviewed; all three public pattern fixtures matched their source commits. This establishes the stated finite bank result under its fixed-inventory premises.

• research/e8-dirty-chart-audit: all six existing tests passed. The PR416 control preserves its normalized dirty inputs; the original and changed PR419 fixtures produce the expected endpoint counterexamples in both orientations. Tests also cover a witness first appearing after address zero and exact operator cancellations. The two vendored implementations matched their immutable public sources, and all four fixtures matched their retained source bytes. This validates the diagnostic, without admitting PR419 or excluding a different coupled-bank construction.

• research/e8-hosted-bit-contract: all four existing tests and the exact two-line in-memory audit integration passed on the authentic pinned PR442 bit fixture. The guard accepts the intact schedule and rejects the deliberate terminal-exchange and bridge-coefficient regressions, which the original audit accepts with unchanged profiles. The verifier checked retained reconstruction receipts; the seven reconstruction stages were not rerun. The five public source/notice/license/receipt files matched their immutable revision. This strengthens verification and does not claim that the actual PR442 construction is invalid.

• research/e8-emitter-digest-cache: the unchanged offline verifier passed all 2,742,552 record-byte comparisons across 144 calls, matching complete digests, event counts and paid instruction profiles, plus both focused encoding/cache-lifetime tests. The proof and source-extracted loop were reviewed against the immutable PR447 emitter. Original source, license, notice and manifest matched their public revision.

• research/e8-scalar-address-cache: the unchanged offline verifier passed two complete 9,417-column, eleven-stage scalar replays, 648,692 operand comparisons, all 24 bridge omission controls per fixture and both focused address-boundary/cache-lifetime tests. Exact patch scope and composition with the digest cache passed. Both packages use the retained completed PR438/PR442 workloads; no benchmark or complete supplier stage was rerun. Their documented speedups remain historical microbenchmark measurements, not new end-to-end claims.

• research/e8-bit-basis-order: the unchanged offline audit and all three existing postprocessor tests passed. It checks all 998 PR433 and 987 PR438 literal source/target/helper columns in both orientations, equal copied forms, inverse/dirty-compensator rejection, and exact prefix/finite-coefficient arithmetic. The coefficient sizes reproduce 223→192 and 226→193 bits; the two-level sizes reproduce 483→420 and 488→423 bits, with all 32 extra local additions charged in each case. Tests cover input preservation, idempotence, six unsafe interfaces and CLI overwrite refusal. Four public source files matched their immutable revisions. The five/nine supplier-stage and complex-component receipts are retained evidence, not fresh supplier or phase replays; the PR438 baseline coefficient is reconstructed algebraically. Caps and κ are unchanged.

These were bounded CPython 3.14.8 runs. Submitted package bytes and manifests are unchanged. The bank and dirty-chart verifiers used a disclosed startup shim that adds -E -S to their single unittest child; the hosted-bit and both cache verifiers ran unchanged under -E -S -B with in-process tests. The basis default audit and its three tests ran under -E -S -B, with those startup flags also added to the two exact postprocessor CLI calls, including the expected overwrite refusal. The coordinating Codex session separately reviewed source and rehashed runtime/input pins and execution receipts. This is AI-assisted contributor validation. Upstream acceptance of these additional packages remains pending; no new supplier, native-cost or κ improvement follows.

Finite assembly theorem

For admitted rational bit and complex caps c,b, put a0 = 384599/10^10 and u = min(b,c). Under a0 < u, b < 1/32, an admitted finite coefficient charge, and the inherited weighted-bootstrap and bridge premises, every rational

0 < kappa < u/(1+u)

has a finite constructible witness. The constructor selects positive backoffs, finitely many bootstrap levels and explicit integer cutoffs, then checks all 47 retained strict inequalities. The proof establishes termination and the greatest admissible point on any supplied rational grid. This ceiling is relative to the supplied caps and this assembly family; the endpoint is never attained.

Start with the proof, constructor and historical package. The complete 51-file historical package is retained unchanged from e819106, including its PR433 fixture, five behavioral tests, source notices and receipts. It remains self-contained for default offline verification.

Source-bound PR799 application

The PR799 adapter applies that same theorem to the maintainer-admitted native-complex PR799 and unchanged PR589 bit supplier at upstream f567884aeb8851b6e794af1184119a3bfbe1e631.

Conditional value	Kappa
Pinned PR799 baseline	0.001661634653669567
Finite witness on the 10^-27 grid	0.001661634653669567075843432

The difference is only 7.5843432e-20. The contribution is a reusable theorem and reproducible admission interface. This parameter improvement is not a substantive frontier advance, new supplier construction or practical speedup. The upstream reconciliation at 50d7688 retains the selected conditional frontier 0.001672660197076918, above this historical PR799 application. No current numerical lead is claimed here.

The adapter verifies 144 immutable upstream file bindings and the unchanged core package. It uses the original PR741/PR799 receipt checkers and admission code, binds the exact integrated replay's owner and invoice, reproduces the complete published assembly, and then constructs the new witness with nine finite levels. Both rational engines check the selected seed; all finite cutoffs and 47 strict inequalities pass. The next grid point and ceiling endpoint are rejected, as are zero eta, zero beta and unit beta.

Recurrence policy is explicit: native complex omits the bad-residue surcharge; bit retains it. Conservative admission runs first. The complex finite invoice is recomputed. The unchanged bit coefficient bound 393 < 512 and row bound 18355 are reused through the authenticated PR589 records; the underlying bit coefficient integer and physical suppliers are not rebuilt.

Reproduction and validation

Historical example:

python3 -B research/finite-assembly-frontier/verify.py

Pinned PR799 application, after obtaining the pinned upstream snapshot using the README commands:

python3 -E -S -B research/finite-assembly-pr799/verify.py \
  --upstream ../integer-mult-pr799-upstream \
  --output ../integer-mult-pr799-result

The verifier runs offline with Python 3.11+ and the standard library. It records exact parameters, cutoffs, moment intervals, constraints and completion hashes. A local bounded validation reproduced the complete portable result and rejected three isolated source mutations before scientific imports: sibling README, upstream input and core input. Original inputs remained unchanged. Root independently checked the source bindings, execution receipts and exact rational result; Ody performed the bounded runs and independent workflow review.

Current head: 524375c. All 60 jobs passed across the five existing fork push workflows on this exact commit: research/formal verification, graded box convolution, terminal elimination, recycled bit registers and early reset/kernel composition. These existing workflows do not include a dedicated PR799 adapter job; the new adapter is covered by the bounded local validation described above. Upstream pull-request workflows separately report action_required without running jobs. Fork CI does not establish upstream approval.

Other contributions retained

These packages retain their source bindings, proofs, validation receipts and attribution. Recorded performance figures refer to the original scoped benchmarks.

Package	Contribution and limit
Source-basis finite-cost reduction	Guarded postprocessor reduces complete coefficient sizes from 223 to 192 bits on PR433 and 226 to 193 bits on PR438. The added operations are charged; supplier caps and kappa stay unchanged.
Hosted-bit parent-contract guard	Enforces the inherited scalar schedule and rejects deliberate mutations accepted by the original profile audit. The authentic PR442 construction passes; no flaw in its published word is alleged.
Emitter digest caching	Byte-identical hashing across 2,742,552 records and 144 calls; recorded 5.47x hashing-loop speedup, not a whole-emitter benchmark.
Scalar address caching	Exact replay equivalence with private restoration and all omission controls; recorded 1.736x/1.730x replay-section speedups. These separate speedups must not be multiplied into an overall claim.
Exact bank optimizer	Packing conditions, optimal patterns and minimum replication for the width-73 rank family, checked against 16,875 inventories.
Dirty-endpoint audit	Exact all-address Gaussian checks with a counterexample invisible at address zero and a positive control; scoped to the stated independent-bank interface.
Retimed component verification	Retained emitter, phase, lifetime and finite-accounting evidence for the pinned PR384 complex component.
Weighted donor-rank matching	Earlier reproducible construction reused by subsequent contributions; its numerical bound is historical.

Scope and attribution

Supplier compiler, common-label/mutable-source realization, whole-call hosting, restored work, fixed tape/common grid, precision, weighted bootstrap and analytic/H1-H5 interfaces remain assumptions. This PR does not establish unconditional multiplication complexity, global optimality or a new Lean proof.

George Lydakis contributed the finite-assembly theorem and adapter with OpenAI Codex assistance, including Ody. Original licenses and cumulative notices remain authoritative. The application credits Jeremy Kothe's PR799, Chafik Boukhalfa's PR741 frames, DreamingOfClouds' PR589 on eumemic's PR555, Rohan Arun's PR559 correction, icekylinx and the earlier supplier, engine, owner and analytic-interface contributors. The historical package retains PR433 and its full upstream attribution.

[lydakis]

@Mister-Meeseeks, could you review this for research/tooling integration? Head a65780b passes all 60 CI jobs. A focused starting point is the finite-assembly-frontier theorem and verifier; the PR body links the proof, pinned inputs and reproduction commands. The result remains conditional on the inherited interfaces. Would splitting this package from the other optimizations make review easier?

[Mister-Meeseeks]

Thank you @lydakis. Your work and attribution are retained in the published 100-PR review, credits and complete original source.

Covered: Finite assembly theorem, PR799 application and a broad accumulated research/tooling stack. The updated head contains many separately scoped packages. Keep open until all new packages and the source-bound PR799 application are reviewed; inherited generic CI does not cover this expanded submission.

Keeping this PR open: The two primary finite-assembly packages passed, including the pinned PR799 application. Eleven additional research/tooling packages at this expanded head still need individual review and execution coverage.

The source archive preserves the entire submission; it does not substitute for the remaining execution or review. The batch's machine-readable disposition records this as partial coverage. Reviewed head: 524375c61b75dc10cef68a71d504e79e7e7f138c.