Formalize the integral retraction and monomial bridge

[an664]

This closes a scoped formalization obligation from #1497: three concrete integer matrix identities (MP=0, QP=I, I-PQ=WM) imply integer row-image/kernel equality, an actual additive quotient isomorphism, lattice saturation and unique monomial reconstruction. Finite integer products and signed powers are defined and proved; the certificate does not assume those conclusions.

The exact Lean 4.30.0 core/Std replay passes universal and positive-fixture compilation, all 197 queried axiom reports, and three negative fixtures that fail only at the intended false propositions. Observed axiom dependencies are propext and Quot.sound. Fixtures include a signed nonzero-W example, negative powers, a torsion group and missing-identity countermodels. Ten tooling/arithmetic checks also pass. The portable replay pins the exact release/commit, uses fresh isolated compiler outputs, verifies source hashes and rejects incomplete or error-bearing reports.

The generic Python package hook passes but does not run Lean; replay.py supplies the dedicated kernel check with an already installed compiler. The result remains conditional on the displayed identities and ordinary group laws. No actual E8/P112/P120 matrix instance, field nonzero-element adapter, geometric generator/coverage bridge, numeric rank API, local Lemma5 or multiplication theorem is formalized here.

Scoped under Discussion1527, on a fresh branch from main 81e81af67714bdc8d1693b0bf5c29468b92f62ff. The revision-3 checkpoint preserves this completed independent-verification result, with historical revision-2 task/source provenance retained. No selected-frontier or supplier-profile change.

Prepared by Andrey Mas with substantial GPT-6 Pro formalization and diagnostic-repair assistance through ChatGPT; Codex coordinated sources, audited execution, replayed and packaged. The source statement remains credited to #1497 and its inherited Brett Halligan / @burntout76 / Ninefold (#1212), @DreamingOfClouds (#717) and @themilkman216 (#968) lineage. No predecessor Lean implementation is copied. Apache-2.0.

Exact commands, source/body hashes, complete observed axiom receipt and public theorem interfaces are in research/integral-retraction-monomial-bridge.

[Mister-Meeseeks]

Batch review at 985fd79ffea7ae7375fe6aa3ce9940b125c7dca2.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Pinned Lean 4.30 kernel replay checks the conditional integral image/kernel, quotient and monomial reconstruction bridge, all197 reports and intended negative fixtures. Concrete E8 matrices, lawful field-unit adapters and geometric application are outside this package.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.