A closed evaluation of tr M_p mod p for periodic continued-fraction operators

A pre-registered computational run (PCF-ORBIT-SCHEMA) asking whether a four-step proof schema — rank-one deflation, an algebraic generating function from Lagrange–Bürmann, a combinatorial interpretation, and a free Z/p action — generalises across an 874-member class of continued-fraction operator pairs with nilpotent mod-p monodromy. It does not. Step (i) is forced for every pair in the search box and carries no information; step (iii) fails for genuine members; step (iv) is shown not to be a well-posed question about a member at all. What the run proves instead is a closed evaluation of the trace: tr M_p = mu * (1 - (Delta|p)) (mod p) for every prime p >= 5 with p not dividing 2*a2, where (Delta|p) is the Kronecker symbol. The membership criterion (mu = 0, or Delta a nonzero perfect square) follows in both directions as a corollary. Verified at 806,760 (pair, prime) cells with zero violations, in exact integer arithmetic throughout. The deposit contains the sealed pre-registration, an append-only ledger of 30 claims with epistemic classes and falsifiers, all verification code, and all data. Seven defects found during the run are logged as append-only corrections rather than edits; one was caught by the automated consistency check and six by external review or by cross-checking a companion run, a ratio the report discusses explicitly. Open item: the second nilpotency clause (M_p not equal to 0) is reduced to a single scalar congruence but not closed.