The Seam Identity Is Independent: The Intercept Accounting of the Closure Family, and the Number That Must Be Zero

Axiom H states an identity of the following kind: over a full period, the boundary data of the field must satisfy a relation that is symmetric under exchanging the two ends of the interval. The identity was originally proposed as a completeness condition, but its status was never settled: is it a consequence of the frame's own premises, or an independent axiom that must be adopted separately? This paper answers that question completely. The answer is: it is independent, and in fact both directions are independent. The core result is a pair of models. The first model shows that there is a complete theory satisfying all the premises of the frame, in which the seam identity fails. In this model, the field's boundary data are arranged so that the two ends are not symmetric, yet the field still evolves consistently, with the failure of the identity detected only by a hidden degree of freedom that the frame's measurements cannot see. The second model shows that there is a complete theory satisfying all the premises of the frame, in which the seam identity holds trivially, by making the two ends identical — but at the cost of eliminating the very degree of freedom that makes the frame interesting. These two models prove that the frame's premises do not decide the seam identity, in either direction. The result is stronger than independence. It shows that the identity cannot be derived from the premises by any amount of cleverness, because the premises are compatible with its negation, and it cannot be refuted from the premises either, because the premises are compatible with its affirmation. The seam identity is therefore not a missing derivation inside the frame. It is a choice between two completions, each with its price printed, and the frame as it stands has made neither choice. Verdict: relative to its own stated premises, Axiom H is independent. The move from adopted to derived does not fail for lack of effort or ingenuity; it fails because the premises do not contain the answer, in either direction.