ADDITIONAL CONIC--QUINTIC KEY-VANISHING CERTIFICATE
Date: 2026-08-23 (Asia/Shanghai)

Target:
H^0(P^2,E_{2,8}T^*_{P^2}(log(C_2+C_5)) tensor O(-3))=0.

Proof run: prime 5, full coupled matrices
Replay:    prime 7, full coupled matrices
Blocks:    c0-, c0+, c1-, c1+, c2-, c2+
Columns:   15,849 at each prime
Result:    rank=columns and nullity=0 in all six blocks at both primes

AGGREGATION COMMAND:
.\bin\verify_finite_vanishing_certificates.exe --degrees 2 5 .\certificates\proof\pair_25_p5.jsonl .\certificates\proof\pair_25_m8_t3_p5.jsonl .\certificates\replays\pair_25_p7.jsonl .\certificates\replays\pair_25_m8_t3_p7.jsonl

CERTIFICATE SET VERIFIED: 36 required full-rank block(s), deterministic fingerprints present.
EXIT_CODE=0

The prime-5 run alone is proof-effective. The prime-7 replay is an
independent implementation check and is not an additional mathematical
hypothesis.
