pathWeight
plain-language theorem explainer
Each recognition path receives a positive real weight defined as the exponential of the negative recognition action. Workers deriving the Born rule from classical costs in the Recognition Science measurement interface cite this construction. The definition is a direct one-line assignment of the real exponential to the negated output of the path action integral.
Claim. $w[γ] = e^{-C[γ]}$ where $C[γ]$ is the recognition action obtained by integrating the J-cost along the rate function of the path $γ$ over its positive duration interval.
background
The module supplies a lightweight interface for recognition paths and their associated actions and weights, deliberately omitting measure-theoretic lemmas to keep the build surface stable for paper exports. A RecognitionPath is a structure consisting of a positive real duration T and a strictly positive rate function defined on the closed interval from 0 to T. The sibling pathAction definition computes the recognition action as the integral of the J-cost of the instantaneous rate over that interval.
proof idea
The definition is a direct one-line wrapper that applies the real exponential function to the negation of the path action value returned by the sibling pathAction definition.
why it matters
This definition supplies the weight that feeds the Born rule theorem born_rule_from_C and the C2A bridge theorems weight_bridge and weight_equals_born, which equate the weight to squared amplitudes. It realizes the classical-to-quantum link inside the measurement module and supports the overall Recognition Science correspondence between path costs and probabilities. The construction sits downstream of the J-cost integral and upstream of amplitude-modulus identities.
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