IndisputableMonolith.Physics.PathIntegralFromRS
This module formulates the path integral in Recognition Science by defining classical paths as stationary points with J-cost zero. It supplies the bridge from the RS cost function to quantum path sums via siblings like classical_path and PathIntegralCert. The structure is purely definitional, importing the Cost module and introducing no theorems.
claimClassical paths obey $J=0$, the Euler-Lagrange stationary condition on the recognition cost $J(x)=(x+x^{-1})/2-1$. The path integral is expressed as a weighted sum over fluctuations around these paths, with PathIntegralFormulation and pathIntegralCount supplying the counting measure.
background
The module sits inside the Physics domain and imports IndisputableMonolith.Cost, whose J-cost is the central functional of Recognition Science. It introduces PathIntegralFormulation together with the auxiliary objects classical_path, quantum_fluctuation, PathIntegralCert and pathIntegralCert. The local setting is the translation of the single RS functional equation into the language of path integrals, with the supplied doc-comment identifying the classical path directly with the J=0 condition.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module supplies the definitional layer that later certificates (PathIntegralCert and siblings) rely on when embedding the path integral inside the Recognition framework. It directly encodes the classical-path statement that links the J-cost to the Euler-Lagrange stationary point, preparing the ground for any downstream derivation of the Feynman sum from RS.
scope and limits
- Does not derive the explicit action functional from the RS equation.
- Does not prove equivalence with the standard Feynman path integral.
- Does not treat measure-theoretic convergence or regularization.
- Does not connect to the phi-ladder or mass formula.