Causal Memory Admissibility (CMAP): A Structural Necessity Classication Discrete-First, Primitives-Grounded, Domain-Independent

What constitutive memory structures are compatible with causality and passivity underfinite resources? Starting from two pre-theoretic primitives -- retardation (causal order prior to geometry)and finite-resource admissibility -- we prove the Causal Memory Admissibility Theorem (CMAP):a bidirectional necessity-and-uniqueness classification of retarded constitutive memoryoperators on finite records. Under a finite-resource admissible class (H1-H4) and five structural gates (P1) retardation,(P2) operator passivity,(F1+F2) genuine fading,(N) non-negativity, and(A) diffusive admissibility, admissible memory is uniquely constrained to be Completely Monotone Finite-resource (CMF):a non-negative-weight superposition of geometric-decay channels, with rates strictly in (0,1). Conversely, any CMF kernel certifies all five gates, yielding a bidirectional ()admissibility certificate. All proof steps use only finite sequences and finite operators; continuum notation appearssolely as optional encoding. Two class-conditional no-go results follow: (NG1) Markovian constitutive laws fail the genuine-fading gate and are structurally excluded.(NG2) the admissible class is open -- both the zero-rate limit (infinite persistence)and the unit-rate limit (Markovian) lie outside the class. CMAP yields a precommitted pass/fail certificate executable on finite records.The result is linear, time-translation-invariant (TTI), and domain-independent within thedeclared class. Keywords:causal retardation; operator passivity; finite-resource genuine fading;complete monotonicity; positive relaxation spectrum; geometric-decay channel;admissibility certificate; discrete-first; structural necessity.