Functional-Integral Information Update Cost under Dynamic Boundary Fixation

Abstract This paper develops Functional-Integral Information Update Cost under Dynamic Boundary Fixation as the formal path-space completion of information update cost within the single-universe boundary framework. Its central claim is that information update cost is not merely a local correction term, but a structurally enforced payment defined over admissible coherent paths under a fixed ontological boundary. Let D(M) denote the dynamically complete interface on the mother-structure M, let B_0 denote the unique fixed boundary set, and let \mathcal{P}(B_0) denote the admissible path space constrained by that boundary. Within this framework, \Delta I[\Phi] is formalized as a nonnegative functional-integral cost whose payment is required whenever a generated state is bound into boundary-consistent standing. The paper distinguishes the tax itself from the taxation space in which it is levied. This move upgrades information update cost from a descriptive notion to a full mathematical platform with admissible paths, constrained measure, coherence functional, and feedback functional. On this basis, the paper derives a compact theorem chain: only B_0-supported paths retain admissibility; all coherent update costs are nonnegative; settlement occurs at the moment of admissible generation; and apparently distinct experimental structures can be re-read as different morphic realizations of the same cost platform. The result is a sharper boundary-grounded account of why no admissible coherence is free, and why information update, once formalized at the path-space level, becomes a structural tax rather than a metaphor.