Certifiable Semantic Agreement Among LLM Agents: What the Admissibility Instrument Decides
About
Can a committee of LLM agents reach agreement that is certifiable at the level of meaning, not only at the level of a label? We build a protocol to find out. H-CSC emits one of three typed outcomes per round -- semantic commit, verdict commit, or typed abort -- under a common 2f+1 distinct-signer certificate, and we use it to measure what such agreement costs and buys. The answer is conditional, and the condition is not the protocol. We prove a containment lemma: whenever the semantic core is large enough to make the committed verdict deterministically valid, the verdict margin already exceeds f, so at matched deterministic guarantees no coverage separation from certificate-wrapped majority is possible. Measurement agrees: the two rules commit the identical task set. What matters instead is the admissibility instrument. Against adversaries that preserve the verdict and corrupt only the reasoning, a 442 MB fine-tuned encoder reaches AUROC 0.621-0.744 at 0-8% TPR (5% honest FPR), while a training-free lexical predicate reaches 0.865-0.982 at 38-80% (50 tasks, 200 attacks, 400 honest) and is exactly deterministic, discharging an assumption the digest proofs rely on. Honest agents disperse further than attacks displace (90th-percentile honest angular distance 0.992 rad against a 0.65 radius), so on free-text rationales no embedding filter is viable -- a property of the output schema, which when tightened collapses dispersion twenty-fold. The committed digest is not inert: under non-disclosure a similarity-preserving sketch separates faithful from substituted derived statements at AUROC 0.741 (n=350), where a cryptographic hash and a verdict-only digest score exactly 0.500. We also report a tie-break capture vulnerability found in our own protocol, and its four-line fix. We claim no safety or coverage advantage over verdict-only certification; the lemma explains why there is none to claim.