The spectral shift function and the moment sequence carry the same information

A theory that cannot compute something usually says so in prose, and prose of that kind is unfalsifiable. The reader cannot tell a genuine obstruction from a gap in our effort. This paper turns one such admission into a theorem. The pair of operators is the free and the sealed Jacobi operator of the Reacher Endogenous Theory (RET), a framework whose physical structure follows from one sealed action. Their coefficient deviation is _n-116=-16364n^2+O(n^-3), whose leading term is summable. The difference is therefore trace-class, that is, the singular values of the difference operator sum to a finite number, and it is trace-class structurally rather than by assumption. For that pair the Krein spectral-shift function exists and is unique, and the trace identity tr(f(H)-f(H_0))=\,f'\,dlambda holds. Applied to the powers, the identity yields the equivalence tr(H^k-H_0^k)=k^k-1\,dlambda. The moment sequence and the shift function therefore carry equal information. The shift contraction is an identity rewriting rather than an information compression, and uniqueness of the measure from its moments closes the converse. Freezing the numerical couplings is then equivalent to pinning the spectral measure. A separate theorem, proved in the framework's archive and stated here with its record, shows that this pinning is undecidable within this framework. The constraint set is not a curve. The archived dimension count for the drift is withdrawn by the errata to the archived record, and is not repeated here. The numerical layer is therefore a provable information-conservation boundary rather than a technical gap. The paper states which parts of that statement are proved here and which are carried from the record. –-