Question, contribution and prior scope

Question, contribution and prior scope

This paper gives finite computational realizations of target sufficiency within Shadow Theory’s pre-existing source–readout programme. Its abstract descent and minimal-completion foundations are established in [10, 11]; the contributions here concern their constructive realization, constrained readout design and exact resource analysis. Let a finite source set have readout and target . Use the reachable readout space), includingwhen. A readout-only correct answer exists exactly when is constant on each fibre of: necessity follows by evaluating a factorization ̄ on equal readouts; sufficiency defines ̄) as the common target value of the fibre. This is the existing target-relative descent criterion[10]. Restrict every extension space to its reachable image); inactive extra points are not source distinctions. An extension is a map through which the original readout factors,; it is target-sufficient when also . The joint image is then the coarsest target-sufficient extension[11]: the map)) is well-defined because both coordinates factor through Neither abstract statement alone determines which permitted measurements realize the completion or what they cost. When is empty the criterion is vacuous, but no conditional prediction is thereby assigned to an impossible readout. This paper adds a circuit compiler, witness-producing separation, certified constrained synthesis and exact resource examples under declared finite menus. Generic implicit hitting-set methods are established[2, 5]; Boolean Fourier and sum–product methods are established[4, 6, 13]. Our claims concern the explicit source/target interfaces, proofs, cost formulas and implementations, not invention of those methods or a universal physical efficiency theorem. The source model in the

main constructions is classical and finitely described. Its menu can impose incompatibilities, but it is not a simultaneous-value model for incompatible quantum observables. Resource prices, available ports and prediction deadlines are premises. A returned compatible preimage does not identify which source actually occurred. Probabilities require a nominated measure; no Born-selection law follows from counting. The finite model does not identify Boolean gate workspace with reusable physical memory. A source assignment here, a quantum symbol map generating a word law, a single-copy detector outcome, and a retained validation trajectory are different mathematical objects. To use a physical intervention as a menu item one must first specify its admitted preparations, joint protocol, disturbance, accessible records and finite representation; deterministic values for incompatible observables are not supplied by the compiler. The appendices preserve related exact weighted and tensor reductions without making them dependencies of the finite synthesis theorem. Their factorization/context hypotheses and costs are explicit. This separates the unconditional finite algorithms from external asymptotic counting claims.