Gravity Analysis Regge Exact Flat Hessian Bloch Symbol Zero4 D Exact Midpoint Bl

The zero-momentum identity

The declaration exactMidpointBlochSymbolZero_eq_quartic is a theorem in the framework's machine-checked library of formal theorems. It concerns a specific quantity called the exact midpoint Bloch symbol, evaluated at zero momentum. In plain terms, the theorem states that this quantity can be rewritten as a sum of terms, each involving a product of four entries of a 4x4 matrix H, multiplied by a rational coefficient. This is a quartic form: a polynomial of degree four in the matrix entries.

The proof proceeds by expanding the zero-momentum expression and showing it equals this quartic form. The coefficients of the quartic are then shown to be zero, using a kernel certificate that clears denominators and scales the computation. This establishes the main result: the exact midpoint Bloch symbol at zero momentum is identically zero for any matrix H. The theorem inhabits a typed residual, meaning it satisfies a specific formal specification.

What this theorem does not claim is broader physical significance. It does not assert that the exact midpoint Bloch symbol vanishes at all momenta, only at zero. It does not claim anything about the behavior of gravity in general, or about the physical interpretation of the Bloch symbol. It is a purely algebraic identity, verified by formal proof, that a particular expression simplifies to zero under a specific condition.

/-- **THEOREM:** the exact midpoint Bloch symbol vanishes at zero momentum. -/
theorem exactMidpointBlochSymbolZero_eq_zero (H : Mat4) :
    exactMidpointBlochSymbolZero H = 0 := by
  rw [exactMidpointBlochSymbolZero_eq_quartic]
  refine Finset.sum_eq_zero fun a _ => Finset.sum_eq_zero fun b _ =>
    Finset.sum_eq_zero fun c _ => Finset.sum_eq_zero fun d _ => ?_
  simp [qCoeff_eq_zero a b c d]

/-- Zero-momentum symbol equals the quartic form with coefficients `qCoeff`. -/
theorem exactMidpointBlochSymbolZero_eq_quartic (H : Mat4) :
    exactMidpointBlochSymbolZero H =
      ∑ a : Fin 4, ∑ b : Fin 4, ∑ c : Fin 4, ∑ d : Fin 4,
        (qCoeff a b c d : ℝ) * H a b * H c d := by
  unfold exactMidpointBlochSymbolZero couplingWeightIdx
  exact sum_weight_eq_sum_quartic_terms H

What this page does not claim

The theorem does not claim the Bloch symbol vanishes at nonzero momenta. The theorem does not claim any physical consequence about gravity. The theorem does not claim the quartic form is unique.

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/ReggeExactFlatHessianBlochSymbolZero4D.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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