pathIntegralDeepCert
The definition constructs the PathIntegralDeepCert structure by assigning its three fields directly from prior lemmas on path weight positivity, maximality at the equilibrium point r=1, and normalization to unity there. Researchers deriving the Feynman path integral from the J-cost action would cite this to certify the required properties of the measure. It proceeds as a direct structure constructor with field assignments and no further reasoning.
claimThe path integral deep certificate is the structure satisfying: for all real $r, ħ$, the path weight is positive; for $ħ > 0$ the path weight at any $r$ is at most its value at $r=1$; and the path weight at $r=1$ equals 1 for any $ħ$. The definition supplies this structure by direct assignment of the corresponding lemmas to each field.
background
In this module the path integral is obtained from the J-cost action $S[γ] = ∫ J(γ(t)) dt$, with path weight defined via the exponential of minus the action over ħ. The structure PathIntegralDeepCert packages three properties: positivity of the weight for every path and ħ, maximality at the classical point r=1 where J(1)=0, and normalization to 1 at that point. Upstream results include the native ħ = φ^{-5}, the positivity theorem path_weight_pos via Real.exp_pos, the maximality theorem path_weight_max_at_eq via exp monotonicity, and the normalization theorem gaussian_approx_at_eq via direct substitution of Jcost_unit0.
proof idea
This is a definition that constructs the certificate via the structure constructor PathIntegralDeepCert. It assigns the path_weight_pos field to the theorem path_weight_pos, the path_weight_max field to path_weight_max_at_eq, and the weight_one_at_eq field to gaussian_approx_at_eq. No tactics or reductions are applied beyond the constructor.
why it matters in Recognition Science
The definition supplies the concrete term that populates PathIntegralDeepCert, which is immediately used to prove the structure is nonempty. It completes the master certificate in the path integral module and supports the structural claim that the Feynman path integral follows from the J-action with concentration on the classical path. In the framework it aligns with the classical limit ħ→0, Gaussian fluctuations near equilibrium, and the convexity of J that forces the measure to peak at stationary action.
scope and limits
- Does not derive the explicit closed-form expression for the path weight function.
- Does not prove convergence of the path integral in infinite-dimensional function space.
- Does not extend the construction to interacting fields or curved space-time.
- Does not supply quantitative error bounds beyond the Gaussian approximation.
Lean usage
theorem pathIntegralDeepCert_inhabited : Nonempty PathIntegralDeepCert := ⟨pathIntegralDeepCert⟩formal statement (Lean)
61noncomputable def pathIntegralDeepCert : PathIntegralDeepCert where
62 path_weight_pos := fun r hbar => path_weight_pos r hbar
proof body
Definition body.
63 path_weight_max := path_weight_max_at_eq
64 weight_one_at_eq := gaussian_approx_at_eq
65