Joint Omission under Restricted Admissibility: Local Certificates, Fiber Connectivity, and the Exact Condition for Safe Record Compression

This paper gives an exact finite theory of joint record omission under restricted admissibility. Let a finite admitted record domain be embedded in a product of declared role components, and let replay semantics comprise the verdict, reason path, and obligations at a fixed grain. A set of roles is safely omitted exactly when the replay semantics factors through the projection onto the retained roles. The central problem is that independently valid role-wise omission certificates need not compose. To locate the failure precisely, the paper constructs an omission reconfiguration graph on every retained-coordinate fiber. Its edges are the admitted single-role variations on which the local certificates operate. Role-wise certificates force replay semantics to remain constant on each connected component, but establish nothing between disconnected components. Joint omission is therefore safe exactly when all components within each retained fiber carry the same replay semantics. This yields the Joint Omission Decomposition Theorem, a sufficient connectivity-composition corollary, and the canonical obstruction Ωₒ(E) := ker(π_R) \ ker(E) whose vanishing is equivalent to safe joint omission. Every failure admits a complete two-point certificate: two records with the same retained projection and different replay semantics. The paper then constructs the Universal Omission Quotient. The quotient identifies exactly the records connected by licensed single-role variations and is universal among maps invariant under those variations. Replay semantics factors through this quotient whenever the role-wise certificates are valid. The retained projection factors through a second canonical map, and joint omission is safe exactly when the induced semantics factors through that second map. Local certificates therefore determine semantics only up to the canonical component quotient; joint omission additionally requires component identity to be semantically irrelevant. The paper also closes the minimal-record problem. A semantic conflict hypergraph is formed from the coordinate-difference sets of every pair of records with unequal replay semantics. A retained role set is replay-sufficient exactly when it is a transversal of this hypergraph. Inclusion-minimal sufficient records are therefore precisely its minimal transversals, and minimality is certified by one private conflict witness for each retained role. Minimum-cardinality safe retention is exactly minimum hitting set and is consequently NP-hard. The resulting theory closes joint omission at schema level in three senses: correctness, certification, and minimal compression. It conditionally lifts the joint-omission wall in the deployment interface by supplying the exact certificate that the interface already requires. The paper does not instantiate the fifteen-role corpus domain, select concrete omission sets, or claim independent proof verification; those remain declaration-gated or externally verified tasks.