IndisputableMonolith.Physics.Path_Integral_RS_v3
Module packaging the Recognition Science path-integral interface (v3): a domain cost functional, its nonnegativity, a positive canonical threshold, and an inhabited certificate bundle. Continuum-limit and amplitude workers in RS cite it as the local API. Content is mostly definitions plus short positivity and equality lemmas over the Cost and Constants imports.
claimThe module defines a domain cost $C(\Omega)$, proves $C(\Omega)\ge 0$ and an evaluation identity, fixes a canonical threshold $\theta>0$, and supplies an inhabited certificate packaging the RS path-integral v3 interface.
background
Recognition Science builds amplitudes from the unique J-cost forced by the Recognition Composition Law, $J(x)=(x+x^{-1})/2-1$, rather than from an ad hoc classical action. The Cost import supplies that cost calculus; Constants supplies the RS time quantum $\tau_0=1$ tick against which path discretizations are measured.
This module sits in the Physics layer and introduces the local objects needed for a path-integral statement: a domain cost on path domains, a canonical threshold (the scale at which contributions are kept or cut), and a certificate record that bundles the interface for downstream use. Sibling names indicate elementary facts only: evaluation at a point, nonnegativity of the domain cost, and positivity of the threshold.
No forcing-chain step (T5–T8) is proved here; the module assumes the RS cost and constant layer and exposes a typed API for path sums.
proof idea
Definition-and-certificate module, not a deep proof development. Domain cost and canonical threshold are introduced as defs; nonnegativity and positivity are short lemmas over the Cost/Constants layer; equality-at-a-point is a direct unfolding. The certificate type and cert_inhabited witness package those pieces so later modules can depend on a single inhabited record rather than a loose bag of defs.
why it matters in Recognition Science
Gives the Physics layer a stable path-integral v3 surface: cost on domains, a positive cutoff scale, and a certificate. Downstream used-by edges are empty in the current graph, so this is an interface module rather than a lemma feeding a named parent theorem yet. It aligns with the RS program of replacing the classical action by J-cost amplitudes on the phi-ladder and eight-tick discrete time, without itself proving continuum limits, measure existence, or matching to QFT propagators. Closes local scaffolding for anyone wiring RS amplitudes to a path sum.
scope and limits
- Does not construct a path measure or prove convergence of discrete path sums.
- Does not derive the J-cost or RCL; those come from Cost and the forcing chain.
- Does not fix D=3, eight-tick structure, or mass-ladder identities.
- Does not compute scattering amplitudes or match Standard Model propagators.
- Does not claim used-by parents; the graph currently shows no downstream dependents.