The Future-Relevance Spectrum From Dynamical Influence to Protection Value in Physical Computation

Physical systems typically represent more distinctions than a specified future task values equally. We introduce a Future-Influence Map from present perturbations to variations in specified terminal-task statistics and define its singular values as the Future-Relevance Spectrum. Zero singular values recover exact future-null directions, while the nonzero spectrum resolves future influence by degree. Combining this geometry with disturbance exposure, protection efficacy, recoverability, and physical cost separates future influence from future risk and marginal protection value. An exact three-direction model proves that complete exact relevance can coexist with sparse optimal protection at finite cost, while a matched-tolerance formulation yields a spectral water-filling law in which the protected set expands as tolerance tightens and recovers the complete relevance support in the exact-performance limit. Under stated convexity assumptions, decision margins further provide sufficient conditions for protection-set stability under bounded model uncertainty. We distinguish continuous relevance evolution under a fixed future specification from exact reclosure caused by a change in that specification, and we specify a falsifiable many-mode benchmark against uniform, noise-based, coherence-based, local-sensitivity, direct optimal-control, and full-state protection strategies. The framework does not replace observability, balanced truncation, goal-oriented reduction, or optimal control; rather, it organizes these neighboring tools around the physical question of which represented distinctions are worth preserving for a specified future. The central result is that exact future relevance and optimal physical protection are distinct objects: what can matter to the future need not be worth protecting equally. Keywords: future sufficiency; future-relevance spectrum; partial closure; observability; goal-oriented reduction; optimal control; physical computation; protection scheduling; quantum coherence; decision sufficiency; dynamic resource allocation.