Gravity Analysis Regge Bloch Local Incidence M2 Eval4 D Continuum Face Mean Loca

The continuum face value

In the framework's discrete model of gravity, space-time is built from elementary building blocks, and the model tracks how a quantity called the mean-local moment behaves along different directions. The declaration in question computes this moment for a specific configuration: a normalized direction called the plus axis, measured against a fixed reference direction called the symbol direction. The result is a single rational number, negative one sixteenth, written as -1/16.

The computation is not a numerical simulation but a theorem in a machine-checked library of formal theorems. The library proves that the mean-local moment for this configuration equals -1/16, and it proves this value is not equal to the Einstein-Hilbert coefficient, the constant that appears in the classical continuum theory of gravity. The theorem also shows that the moment for the plus axis differs from the moment for the cross axis in one of the tested directions, meaning the model retains a directional asymmetry even after averaging.

In Recognition Science, the declaration is part of a larger evaluation of how the discrete model relates to the continuum limit. The docstring explains that the mean-local Path B agrees with a different construction called distinct-hinge on every ray, but the position-resolved version does not close the Einstein-Hilbert gap and does not restore isotropy. The status flags confirm this: mean-local equals distinct-hinge on witnesses, anisotropy persists, the factor-4 residual persists, position-resolved does not close Einstein-Hilbert, and gap-action recovery is false.

What the declaration does not claim is just as important. It does not claim that the discrete model reproduces the Einstein-Hilbert action, the standard continuum action of general relativity. It does not claim that isotropy, the property of being the same in every direction, is restored by this averaging procedure. It does not claim that the gap between the discrete model and the continuum theory is closed. These are all open targets, and the declaration is a precise measurement of one point in that larger search.

theorem continuumFace_meanLocal_normalizedPlus_symbolDir :
    m2MeanLocalAllOrbitMoment ((Real.sqrt 2)⁻¹ • axisTTPlus) symbolDir /
      (∑ i : Fin 4, symbolDir i * symbolDir i) =
      (-1 / 16 : ℝ) := by
  rw [m2MeanLocalAllOrbitMoment_smul,
    m2PathB_meanLocal_axisTTPlus_symbolDir, symbolDir_normSq,
    inv_pow, Real.sq_sqrt (by norm_num : (0 : ℝ) ≤ 2)]
  norm_num

theorem meanLocal_pinned_face_ne_eh :
    m2MeanLocalAllOrbitMoment ((Real.sqrt 2)⁻¹ • axisTTPlus) symbolDir /
      (∑ i : Fin 4, symbolDir i * symbolDir i) ≠
      einsteinHilbertTTCoefficient4D := by
  rw [continuumFace_meanLocal_normalizedPlus_symbolDir,
    einsteinHilbertTTCoefficient4D_eq]
  norm_num

theorem m2PathB_meanLocal_plus_cross_disagree_e0Dir :
    m2MeanLocalAllOrbitMoment axisTTPlus e0Dir ≠
      m2MeanLocalAllOrbitMoment axisTTCross e0Dir := by
  rw [m2PathB_meanLocal_axisTTPlus_e0Dir, m2PathB_meanLocal_axisTTCross_e0Dir]
  norm_num

What this page does not claim

The discrete model reproduces the Einstein-Hilbert action of general relativity. Isotropy is restored by the mean-local averaging procedure. The gap between the discrete model and the continuum theory is closed.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Gravity/Analysis/ReggeBlochLocalIncidenceM2Eval4D.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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Derived articles

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