Gravity Analysis Regge Edge Stencil4 D Finite Ttquadratic Gauge Part Axis Wave N

The gauge obstruction

Gauge invariance is the requirement that a physical theory's predictions do not change under a transformation that merely relabels its mathematical description. In gravity, the analogous freedom is the ability to change coordinates without changing the geometry. A formula that is not gauge invariant cannot be a physical observable, because its value would depend on an arbitrary choice.

The declaration finiteTTQuadratic_gaugePart_axisWave_ne_zero is a theorem in the framework's machine-checked library of formal theorems. It establishes that a particular provisional formula, called the finite TT quadratic, evaluates to a nonzero number when its input is a pure gauge perturbation. The formula is a sum over fifteen edge classes of a four-dimensional cube, where each term is the square of a coefficient that measures how the perturbation loads that edge. The theorem states that this sum equals 32 for the specific gauge perturbation defined by the axis wave and axis gauge vector.

This result is significant because it demonstrates a failure of exact gauge invariance for this provisional formula. If the formula were gauge invariant, a pure gauge perturbation, which represents no physical change, would produce a value of zero. Instead, the formula yields 32, showing that it is sensitive to the arbitrary gauge choice. The framework's documentation explicitly records this failure and provides a counterexample, rather than claiming the formula is complete.

The theorem is part of a larger effort to construct a discrete version of the Einstein-Hilbert action from a recognition ledger. The ledger is a discrete record of events from which the framework derives physical structure. This particular calculation is a checkpoint that identifies a specific obstruction that a future, fully correct formula must overcome. The theorem does not prove that the full theory converges to general relativity, nor does it establish the correct weights for the four-dimensional Regge action. It is a precise, negative result that narrows the space of possible valid formulas.

theorem finiteTTQuadratic_gaugePart_axisWave_ne_zero :
    finiteTTQuadratic (gaugePart axisWave axisGaugeVector) ≠ 0 := by
  rw [finiteTTQuadratic_gaugePart_axisWave]
  norm_num

/-- Provisional finite TT quadratic form on edge classes:
`Q(H) = Σ_d w_d c_d(H)²` with provisional weights `w_d = 1` for every
nonzero 0/1 class.  **OPEN:** replace `w_d` by the true 4D Regge
flat-Hessian class weights when derived; do not fit them to EH. -/
def finiteTTQuadratic (H : Mat4) : ℝ :=
  ∑ d : Fin 15, classCoeff H d ^ 2

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 full recognition science action converges to the Einstein-Hilbert action in four dimensions. This theorem does not establish the correct weights for the true four-dimensional Regge Hessian. This theorem does not show that the finite TT quadratic is a valid physical observable.

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)

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