Formalize normal-class rank and complete count/price bounds in Lean

[bitplane]

PR #2338's all-batch P112 argument uses a normal-class rank implication that had not been kernel-checked here. This contribution formalizes that implication, its rectangular fixed-space transfer and its complete count/price consequences, with the exact premises exposed.

For any field and finite term set, distinct endpoint classes and K[i,j] != 0 -> beta[i] = alpha[j] imply 2 dim ker(I-K) <= t. For rectangular factors with AC+DE=I_N, the same hypotheses on CA give 2N <= 2n+t; charging every term yields 64k+3t <= 7t+8n when N=8k, plus the real-power analogue. The proof includes empty term sets. Explicit counterexamples show why equal classes, zero diagonal alone and a one-third replacement fail.

The source mathematics belongs to DreamingOfClouds' #2338, pinned at 82700676e1d42acf98dc1921cb6ef82551620008; exact README/NOTICE copies and hashes are retained. #2342 supplies a narrower independent three-call argument and explicitly excludes auditing the broader all-batch proof. This PR does not verify the geometric/frame premises, run the source SAT/DRAT rigidity check, audit all of #2338 or admit a physical supplier. Selected certificates and kappa are unchanged.

Validation performed locally:

• Fresh Lean 4.33.0-rc1 / pinned Mathlib build: 26 actual axiom probes, only propext, Classical.choice, Quot.sound; three false proof attempts rejected after valid imports.
• Independent fraction and binary rank engines: 39,157 labelled compatible matrices in Q and F2, both orientation minors, nine rectangular fixture cases and an abstract sharp complete factorization.
• Registered Python 3.12 replay and nine fault-control groups passed.
• Full kernel replay under optimized Python 3.14 from a Git-free, read-only 20-file execution snapshot produced an identical kernel receipt; all nine control groups passed and the snapshot was unchanged. This caught and fixed inherited read-only permissions on the new build directory.
• Scoped workflow executes the actual Lean build; hosted CI success is not claimed by the local receipt.

Replay: python3.12 -B scripts/research_package.py --package research/lean-normal-class-rank. That default checks sources/finite evidence and explicitly does not run Lean. README documents the separate full kernel command; VALIDATION.json records the executed checks and their limits.

One fresh random draw selected independent-verification, second suggestion formal-interface-lemma; TASK.json records the scope change and no reroll. Canonical main remained 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2, priority revision 4. Final overlap covered 607 open PRs and 1,137 Discussion records with complete pagination and no warnings.

Coordination: #2380. Open next question: formally bind the actual P112 normal-class/frame evidence to the matrix compatibility premise.

Gareth Davidson (@bitplane), with substantial OpenAI Codex assistance in proof, implementation, verification and writing; no subagents or independent human review. Wrapper/workflow/control structure adapts our #2341/#2295 harnesses, with attribution. New finite engines import no predecessor checker.