PathIntegralCert
plain-language theorem explainer
PathIntegralCert defines a structure that certifies the Recognition Science path integral by asserting exactly five formulations exist, the classical path at unit scale has zero J-cost, and all other positive scales have strictly positive J-cost. A physicist deriving QFT from the Recognition Composition Law would cite it to confirm the path sum is dominated by the minimum-cost trajectory plus fluctuations. The definition is a direct structure bundling the inductive enumeration with the J-cost sign properties.
Claim. A structure certifying the RS path integral consists of three fields: the set of path integral formulations has cardinality 5, the J-cost at scale 1 equals 0, and for every positive real $r$ with $r$ not equal to 1 the J-cost at scale $r$ is strictly positive.
background
Recognition Science recasts the Feynman path integral as a sum over recognition paths weighted by J-cost, where J-cost is the functional $J(x) = (x + x^{-1})/2 - 1$ that vanishes only at the self-similar fixed point. The classical trajectory is the unique minimum-J path, which occurs at unit scale with weight 1 in the Euclidean formulation. The module states that five canonical formulations (position, momentum, coherent state, field theory, string) match the configuration dimension in this setting.
proof idea
Direct structure definition that assembles the Fintype cardinality of the PathIntegralFormulation inductive type together with the classical zero J-cost at unity and the strict positivity condition for other positive scales.
why it matters
This definition supplies the certificate object instantiated by pathIntegralCert, which links the Recognition Composition Law to the Euclidean path integral with weights exp(-J). It supports the reduction of QFT to the phi-ladder and eight-tick octave by ensuring the path sum is well-defined over the five formulations and dominated by the classical trajectory. The construction touches the open question of how the RS path measure recovers the standard Feynman measure in the continuum limit.
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