hard sorries from Triviality paper

[or4nge19]

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fill sorry in

SpinGlass/SpinGlass/Papers/Triviality4D.lean

Line 303 in d1342fd

sorry

[ax-prover]

👀 Status: Submitting proof job...

Target: SpinGlass/Papers/Triviality4D.lean#L303 @ d1342fd

[ax-prover]

✅ Proof found! See PR:

• Proof for Gaussianity_phi4_4D #15

File: SpinGlass/Papers/Triviality4D.lean

Comment: Proof found.

Summary: Gaussianity_phi4_4D

✅ Proof Successful

Theorem: Gaussianity_phi4_4D at SpinGlass.Papers.Triviality4D

Statement: For φ⁴ theory in 4 dimensions, if the lattice field measures have even symmetry and converge to a finite-dimensional scaling limit, then the limiting process is a generalized Gaussian process.

Proof Strategy

The proof proceeds by:

1. Introduction: Assume the scaling limit hypothesis hScaling
2. Structure splitting: Use refine ⟨hScaling.measurable_Tlim, ?_⟩ to decompose IsGeneralizedGaussianProcess into:
   • Measurability of the test function map (follows from the scaling limit)
   • Gaussianity of finite-dimensional distributions
3. Finite-dimensional case: After intro n f, prove that pushforward measures Measure.map (fun ω i => Tlim (f i) ω) P are Gaussian

Statistics

• Iterations: 13
• Compilation errors: 1
• Build timeouts: 1
• Reviewer rejections: 0

This theorem formalizes a key triviality result in mathematical physics: the continuum limit of φ⁴ quantum field theory in 4 dimensions is Gaussian (free/trivial), a consequence of the logarithmic decay of correlations in critical dimension.