Environmental and Internal Gating of Memory: ARCH Multiplicative Threshold Framework for Encoding and Retrieval

Memory encoding depends on the joint convergence of substrate readiness, internal drive, environmental input, and a permissive temporal-physiological state. The same logic recurs across vertebrate learning systems: hippocampal-dependent spatial and schema-modulated memory (the Navigia substrate in the Systema Behavorum taxonomy), filial imprinting in the chick intermediate medial mesopallium (Lorenz's prägung), vocal learning in the songbird auditory pallium, and Pavlovian conditioning in the basolateral amygdala and ventral striatum. Despite differences in substrate, timescale, and output, writing events are gated by the multiplicative convergence of substrate (A), drive (D), content/context (C), and a permissive state (Φ); retrieval is gated by a parallel conjunction on the same substrate; and behavioral output depends on the match between current input and the stored trace. The framework specifies f in Lewin's B = f(P, E) as the multiplicative threshold A × D × C × Φ ≥ T, decomposing person and environment into computationally separable terms with empirical signatures.Three contributions advance beyond descriptive integration. First, retrieval is formalized as ARCH × Φ-gated in the same form as encoding, generating dissociation predictions between trace-degradation and Φ-mediated retrieval failures that current accounts do not formalize. Second, inherited priors (ethology) and acquired schemas (Tse, Morris, and colleagues) are unified under the same conjunctive logic, both biasing C through Bayesian-like prior weighting. Third, the multiplicative form generates a quantitatively precise prediction - supra-additive failure under combined partial perturbations - that distinguishes the framework from interactionist accounts; a tractable 2 × 2 factorial in chick imprinting is developed in §4.7, with detection power calculated in Appendix A.The framework yields four testable predictions. Single-domain perturbation should cause categorical encoding failure regardless of other domains. Combined partial perturbations should produce supra-additive failure distinguishable from additive null models in factorial designs not yet performed in any of the four systems. Reading-event Φ-suppression should produce retrieval failures dissociable from trace-degradation failures. Pharmacological Φ-modulators (ketamine, psilocybin, MDMA) are reframed as substrate-state modulators that widen the retrieval envelope, converting reading events into writing events, and supply a clinical-translational test. The framework is offered as a falsifiable account of common computational structure across vertebrate memory systems.