Formalize the P104 affine-complement endpoint lemma in Lean

[petalas]

Summary

Formalize PR #2920's geometric implication, without changing its mathematics or price claim.

q(x)=1 on D
  -> ker(f) and ker(f+q) partition D
  -> a nontrivial split gives codimension-one kernels
  -> dimension eight gives two genuine dimension-seven subspaces

Adds research/p104-affine-complement-lean/ and one scoped formal-check workflow. Eight named Lean statements include finite cardinalities and counterexamples to weakened premises. A separate source-bound checker reconstructs all 208 actual P104 endpoint hypotheses.

Credit Nick Petalas/#2920/#2909/#2896, Hiroki Kumazaki/#2889/#2353, Arham Tahir/#2675/#2682, ywz65/#2614, Sagar Thakrar, Philippe Van Velzen and the retained geometry/native lineage. The dependency-audit pattern comes from Ali Romman's cyclic-Schur package. Exact source heads, full credit and substantial-assistance disclosure are in TASK.json and README.

Canonical claim and handoff. Publication commit 2291714c9c87de6bec3dd94d7744a0594cb565a8.

Evidence

• Before: Certify P104 separate endpoint geometry and two-history budget #2920 has a written universal affine-complement proof and finite endpoint enumeration. After: the generic partition, nontriviality, dimension and cardinality implications are kernel-checked in Lean 4.21.0 with pinned Mathlib. All thirteen theorem declarations, including five generated helpers, use only standard axioms.
• Five formal controls reject the wrong complement, wrong dimension, incomplete proof, injected custom axiom and missing audit command. Three finite behavioral groups cover the source-bound positive fixture, weakened hypotheses, optimized Git-free execution and source/typed-receipt faults.
• Local Linux/CPython 3.12.15 registered replay and formal runner pass. Ruff, warning-as-error compilation and exact committed-byte preflight pass. The research-workflow suite ran 144 tests successfully with five skips. No independent human or separate-agent review is claimed.

python3.12 -B scripts/research_package.py --package research/p104-affine-complement-lean
python3.12 -I -O -Werror -B research/p104-affine-complement-lean/check_lean.py

The second command requires the existing pinned formal/lean environment. The generic registered replay checks finite premises, not Lean; the dedicated workflow checks the proof and its controls. Hosted CI is separate from local evidence.

Started from canonical main 90a72fb37; refreshed main a7482490b has unchanged relevant guidance, selected record, scientific inputs and formal environment. Read-only merge-tree compatibility passes. Final overlap covers 496 open PRs, 2,119 closed PR bodies and 1,597 Discussion records with complete pagination and no warnings. Arbitrary diffs and unpublished work remain outside coverage. Pending #2920 is pinned at b9ea9def949f9d71d59e21294d0e6cfbe0692a73.

Merge Danger

Door: two-way. One new opt-in package and workflow; frozen evidence and the selected frontier are unchanged.

Blast Radius: scoped. Python extraction and its correspondence to the Lean model are not themselves formalized. Compiled Mathlib caches and the compiler remain trusted. This does not formalize the price optimization, joint source acquisition, coefficients, chronology, dirty restoration, hosting, precision, all-size admission or the full multiplication theorem. No new supplier or kappa is claimed. The family decision is to finish this audit and stop, not expand a construction campaign.