Gravity Analysis Regge Exact Flat Hessian Bloch Symbol Zero4 D Typed Residual Mi

A vanishing check

In the Recognition Science framework's machine-checked library of formal theorems, a declaration named typedResidual_midpointBloch_symbolZero proves a specific identity. The statement is concise: for any matrix H representing a configuration, the quantity exactMidpointBlochSymbolZero H equals zero. This is a theorem, meaning it has been derived through the framework's logic and verified by a computer, with no unproven assumptions beyond the standard axioms of the underlying type theory.

The proof works by expanding the quantity into a quartic form, a sum of terms like Q_abcd * H_ab * H_cd. The coefficients Q_abcd are rational numbers computed from a table of coupling weights. The theorem shows that every one of these coefficients is zero, and therefore the entire sum vanishes. The computation is done with exact rational arithmetic, not floating-point approximations, and the result is certified by the kernel of the proof assistant.

In plain language, this establishes a constraint on the framework's model of gravity at a specific point: the zero-momentum midpoint Bloch symbol is exactly zero. This is a technical consistency check, not a derivation of a new physical law. It confirms that a particular mathematical expression, built from the framework's definitions, satisfies a required condition. The declaration does not claim that gravity is generally quantized, that the framework's model of gravity is physically correct, or that this identity holds at non-zero momentum.

/-- Inhabits `TypedResidual_midpointBloch_symbolZero`
(`∀ H, exactMidpointBlochSymbolZero H = 0`). -/
theorem typedResidual_midpointBloch_symbolZero :
    ∀ H : Mat4, exactMidpointBlochSymbolZero H = 0 :=
  exactMidpointBlochSymbolZero_eq_zero

/-- **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

/-- **THEOREM:** every rational quartic coefficient vanishes. -/
theorem qCoeff_eq_zero : ∀ (a b c d : Fin 4), qCoeff a b c d = 0 := by
  intro a b c d
  rw [qCoeff_eq_kernelGlue]
  exact KernelGlue.qCoeff_eq_zero a b c d

What this page does not claim

This theorem does not claim that the framework's gravity model is physically correct. This theorem does not claim that the identity holds at non-zero momentum. This theorem does not derive a new physical law or a value for a 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/ReggeExactFlatHessianBlochSymbolZero4D.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

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