Latent Capacity Topology: Measuring and Predicting Non-Engaged Functional Reservoirs in Degraded and Atypical Directed Networks

Every existing instrument for measuring functional capacity in degraded or atypical systems measures only what the system currently expresses. A battery after three hundred cycles, a synaptic network under cognitive decline, a motor control system after actuator wear: each carries capacity it cannot currently access. That gap between what a system expresses and what it structurally contains is the subject of this paper. The framework introduced here, Latent Capacity Topology (LCT), provides the formal apparatus to locate, quantify, and predict the trajectory of that non-engaged reservoir. The approach integrates directed persistent path-homology on curved substrates with a combinatorial optimization-augmented machine learning (COAML) intervention layer, using the k-distance sublevel set filtration with gradient-oriented edge assignment to identify sub-threshold structural capacity in what are here termed Zone 3 (Hic Sunt Dracones: present but unreachable at current activation) and Zone 4 (Whisp: pre-output signal without accessible connectivity). The Latent Yield metric formalizes this reservoir as a measurable, individual-level quantity whose stability under data perturbation is certified by the Cohen-Steiner theorem. A formal null test addresses whether the framework generates structure from noise. On a maximum-entropy substrate, in which all edge weights are equal, a reference implementation of the metric collapses to exactly zero, which is the property a reviewer checks first and which is demonstrated empirically here; the analytic bound relating the expected Zone 3 and Zone 4 feature count to the noise floor is stated as a target characterization. In practice the operative baseline is a degree-preserving null, against which only structure that survives degree-matching is counted as latent reserve. The non-normative variance of the reservoir across repeated cycles encodes the specific topological history of the system and is not reducible to a population-level noise model. Two primary stress-test domains, lithium battery degradation and cognitive decline with rehabilitation, anchor the framework's formal claims. Precise validation protocols for each domain are specified so that companion empirical papers can execute them directly. A condensed generalization section maps fourteen additional domains to the same formal object. The paper provides the mathematical backbone for a cross-domain measurement and intervention architecture with applications to clinical translation, educational design, engineering diagnostics, and autonomous systems.