pathIntegralCert
plain-language theorem explainer
Recognition Science encodes the path integral as a sum over J-cost weighted recognition paths, with this definition certifying that exactly five formulations exist, the classical trajectory has zero J-cost, and all other paths have positive J-cost. A physicist deriving QFT from the recognition equation would cite this to anchor the Euclidean formulation. The definition is a direct bundling of three upstream theorems into the PathIntegralCert record.
Claim. The PathIntegralCert structure holds when the number of formulations is five, the J-cost at unity is zero, and the J-cost is positive for every positive real scale factor other than one: $Fintype.card(PathIntegralFormulation)=5$, $J_{cost}(1)=0$, and $J_{cost}(r)>0$ for all $r>0$ with $r≠1$.
background
In the Recognition Science framework the path integral arises as the sum over recognition paths weighted by exp(-J-cost). The classical trajectory is the stationary point where J-cost vanishes. J-cost is the recognition cost function satisfying J-cost(1)=0 and J-cost(r)>0 for r≠1. The module establishes five canonical formulations corresponding to the five-dimensional configuration space in the RS-native units.
proof idea
This definition constructs the PathIntegralCert record by assigning the five-formulation count from the decide tactic on the enumeration, the classical zero from the Jcost_unit0 lemma, and the quantum positivity from the Jcost_pos_of_ne_one lemma.
why it matters
This certificate anchors the path-integral formulation within the RS derivation of QFT, confirming the five-formulation count that matches the configuration dimension D=5. It directly supports the claim that the dominant contribution is the classical path with weight 1 while fluctuations are suppressed by positive J-cost. No downstream uses are recorded yet, leaving open its integration into explicit computations of correlation functions.
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