[Lean] Connect complete-bank endpoints to PR516 realization

[jamesyc]

Change and mathematical scope

One new theorem, CurrentMain.PR516CompleteBankEndpoint.full_state_endpoint, connects the full physical helper/rest endpoint from #516 to the sequential completed-bank interface in #513. It directly reuses LemmaA.family_endpoint, LemmaA.endpoint_indep, #509's event semantics and Mathlib's Equiv.funSplitAt; it does not reprove their inductions.

Author-checked conditional source context:

• main/base: 12a4934d15006e8c10f9b2264f13f8f341be802b
• κ = 90446448750933/125000000000000000 = 0.000723571590007464
• This is the unchanged selected construction context, not a new exponent saving derived by this adapter.

Coverage

• Finite: no new selected finite witness, census, matrix computation or numerical instance checker is claimed.
• Conditional interface: for arbitrary initial helper and rest values, the completed word gives the specified helper operator and rest output independent of the initial helper. Full unit row and zero off-diagonal helper column are separate hypotheses. The reference zero in the rest definition imposes no clean-state assumption.
• Full theorem: actual physical execution, injective owners, exact recipes, well-labelled events, full rows/columns including the work register, common-chart partial swap, all-60-replica partition/order, boundaries and physical costs remain explicit. No full multiplication/runtime theorem is proved.

Reuse and exact pins

• Lemma A of #484: the realization of paid wrappers and spectators, proved in Lean 4 #516: 1c6b6ff9366a94e1f86a6c5d661a7df75c6ff814, Realize only, with a reversible import alias and explicit modification notice.
• Lemma G of #484: clean proof and Lean 4 formalisation of its core #509: 7b9c720ecc9f483c383e7cd9c2ab81f06ee5df59, Stream bytes unchanged.
• [Lean] Add sequential completed-bank source bindings #513: 005dd3f4e3cba8d1761abc4ab391aaf6d14d6f73, six support sources unchanged.
• Lean 4.21.0; Mathlib 308445d7985027f538e281e18df29ca16ede2ba3.
  DreamingOfClouds and predecessor authors retain attribution and Apache-2.0 notices. PR516's numerical/window checkers are outside this closure, including the fixed-layout checker discussed in its latest comment.

Existing #509/#513/#516 were compared before publication; generic realization is credited upstream. The new contribution is the complete pair-valued interface adapter. No superseded PR is closed.

Validation and reproduction

Fresh isolated nine-source package replay passed: 70 complete axiom probes, each using only propext, Classical.choice and Quot.sound; 19.3728 compiler seconds, 0.00412 FIFO seconds, 3,301,359,616 bytes peak VM. Eight object hashes match the research check; the notice-modified Realize object is separately recorded. Twenty changed files, all under research/lean-complete-bank-endpoint; no root or scientific CI changes.

Offline command (with existing exact source worktrees and built pinned runtime):

python3 research/lean-complete-bank-endpoint/check.py --lean /path/to/lean-4.21.0-linux/bin/lean --mathlib-root /path/to/mathlib4 --pr509-root /path/to/PR509/research/e8-lemma-g-lean --pr516-root /path/to/PR516/research/e8-lemma-a-lean --pr513-root /path/to/PR513/research/lean-sequential-bank-bindings --build-dir /path/to/new-empty-reproduction-directory

Exact hashes, full receipts, resources and replay boundaries are packaged. Local kernel results are separate from fork/upstream CI, which has not been reported as passed. The inherited joint-certificate root-file provenance issue is outside this chunk.

Prepared by James Chang with OpenAI dot and GPT Astra assistance; upstream assistance disclosures are preserved.

[Mister-Meeseeks]

Integrated on main in 9449300, with original commit history and individual credit retained.

Your nine exact Lean modules compiled under Lean 4.21.0, and all 70 complete axiom reports passed. All 2,213 external sources and objects match the pinned closure. The maintainer used the native macOS executable, not the author’s Linux runtime/memory reproduction. The full-state adapter and its precise execution, row/column, ownership and operator hypotheses are documented; no current finite instance or complete multiplication theorem is inferred automatically.

Maintainer review and exact scope · Contributor record. The multiplication claim remains conditional on the inherited all-size/realization/analytic interfaces; these checks do not establish a complete unconditional theorem.

Thank you for contributing this work.