Gravity Analysis Regge Edge Stencil4 D Finite Ttquadratic Axis Ttplus Ne Zero

A nonzero probe

In numerical analysis, a stencil is a fixed pattern of points used to approximate a derivative or a differential operator on a grid. The declaration finiteTTQuadratic_axisTTPlus_ne_zero concerns a stencil in four dimensions, built from the 15 nonzero displacement vectors whose entries are only 0 or 1, such as (1,0,0,0) or (1,1,0,1). For each such displacement, the framework defines a coefficient from a matrix H, and then sums the squares of these 15 coefficients. The theorem states that when H is a specific matrix called axisTTPlus, this sum of squares is not zero.

The matrix axisTTPlus is a plane-wave perturbation, a sinusoidal disturbance on the grid with a chosen wave vector. The theorem proves that this particular perturbation produces a nonzero total when its squared coefficients are added up. This is a nonvacuity result: it shows the aggregate is not trivially zero for a genuine input. The proof is machine-checked in the framework's library of formal theorems, with no unproved assumptions beyond the standard logical axioms. The same library also records that a pure gauge input, a transformation that should not change physical predictions, gives the distinct value 32 for the same aggregate, and a pure trace input gives 80.

The nonzero value for axisTTPlus matters because the framework is testing whether this aggregate can serve as a finite, computable stand-in for the full four-dimensional Regge action, a discretized form of general relativity. The theorem establishes that the probe is sensitive: it does not collapse to zero on a nontrivial perturbation. However, the declaration does not claim that this aggregate is the correct quantum gravity action. The true weights for the four-dimensional Regge Hessian at flat space remain open; the current construction uses weight 1 on every displacement class, a provisional choice. The theorem also does not prove that this finite sum converges to the Einstein-Hilbert action, and it does not establish gauge invariance, since the pure gauge input gives a nonzero value of 32.

In plain terms, this is a small, exact check in a long campaign. It confirms that a particular test object is alive, not zero, and it records a concrete failure of gauge invariance for the provisional aggregate. The consequence is that the framework now knows this specific probe cannot be dismissed as empty, and it has a precise number, 32, for a gauge transformation that a fully physical theory would need to handle differently. The search for the correct weights and the full four-dimensional theory remains open.

theorem finiteTTQuadratic_axisTTPlus_ne_zero :
    finiteTTQuadratic axisTTPlus ≠ 0 := by
  rw [finiteTTQuadratic_axisTTPlus]
  norm_num

theorem finiteTTQuadratic_axisTTPlus_ne_zero :
    finiteTTQuadratic axisTTPlus ≠ 0 := by
  rw [finiteTTQuadratic_axisTTPlus]
  norm_num

theorem finiteTTQuadratic_gaugePart_axisWave :
    finiteTTQuadratic (gaugePart axisWave axisGaugeVector) = 32 := by
  unfold finiteTTQuadratic
  have hterm :
      ∀ d : Fin 15,
        classCoeff (gaugePart axisWave axisGaugeVector) d ^ 2 =
          (4 : ℝ) * (hasBit0 d : ℝ) := by
    intro d
    rw [classCoeff_gaugePart_axis]
    unfold hasBit0
    cases classBit d 0 <;> norm_num
  simp_rw [hterm, ← Finset.mul_sum, ← Nat.cast_sum, sum_hasBit0]
  norm_num

What this page does not claim

This theorem does not prove that the finite quadratic aggregate equals the full Regge action. This theorem does not establish gauge invariance of the aggregate. This theorem does not derive the fine-structure constant or any other physical constant.

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/ReggeEdgeStencil4D.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: