path_weight
plain-language theorem explainer
The definition assigns to each real r the contribution exp(-J(r)/ℏ) in the path integral constructed from the J-cost action. Workers deriving the Feynman integral from the Recognition Composition Law cite this when building the partition function or proving concentration on the classical path. It is introduced by direct application of the real exponential to the scaled negative J-cost.
Claim. The weight of a configuration with J-cost r is given by $w(r, ħ) = exp(-J(r)/ħ)$.
background
The module derives the path integral Z = ∫ exp(-J(γ)/ℏ) Dγ from the J-cost action S[γ] = ∫ J(γ(t)) dt, with the measure concentrating near the stationary point because J is convex. J-cost denotes the function J(x) = (x + x^{-1})/2 - 1 that satisfies the Recognition Composition Law. The scale ħ is the RS-native reduced Planck constant ħ = φ^{-5}.
proof idea
Direct definition that applies Real.exp to the term -(Jcost r)/hbar.
why it matters
This supplies the basic weight used by PathIntegralDeepCert (positivity, maximum at r = 1, unit value at equilibrium) and by the Gaussian approximation and classical-limit statements in the same module. It realizes the structural claim that the path integral follows from the J-action, linking to T5 J-uniqueness and the ħ → 0 limit.
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