pathSpace_status
plain-language theorem explainer
pathSpace_status supplies a status string summarizing the Action.PathSpace module components and confirming zero sorries. Framework auditors tracking the Recognition Science mirror would reference it to gauge progress on the variational least-action setup. The declaration is realized by a direct string literal with no lemma applications or computations.
Claim. The status string for the path-space module in the J-action variational principle reports the presence of admissible paths, the J-action integral, interpolation, and fixed-endpoint relations, with zero sorries and axioms.
background
This module sets up the variational stage for the principle of least action derived from the d'Alembert cost functional J. It defines admissible paths as continuous, strictly positive functions on a closed interval [a, b]. The J-action functional computes the integral of Jcost along such a path and serves as the central object of the variational principle. Fixed endpoints record boundary agreement between two paths, while interpolation constructs the convex combination that preserves admissibility and continuity.
proof idea
This is a sorry stub realized by a direct string literal assignment. No lemmas from the upstream results on the J-action, admissible paths, fixed endpoints, or interpolation are invoked; the body simply returns the descriptive status string.
why it matters
This scaffolding declaration tracks the implementation state of the path-space constructions that enable the least-action principle in Recognition Science. It records the key structural fact that admissible paths are closed under convex interpolation, which is required for strict-convexity arguments in related modules. The module companion is the paper RS_Least_Action.tex, linking the J-functional to the broader derivation of physics from a single functional equation.
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