Gravity Analysis Regge Exact Flat Hessian Bloch Symbol4 D Exact Midpoint Bloch S

The zero-waveform identity

The declaration exactMidpointBlochSymbol_zero_eq is a theorem in the framework's machine-checked library of formal theorems. It states a simple algebraic fact: for any 4x4 real matrix H, the value of the exact midpoint Bloch symbol at the zero wavefunction is exactly the sum of all coupling weights. In plain words, when the wavefunction is identically zero, the symbol's value equals the sum of the weights associated with each of the 1208 couplings in the unit cell. The proof is a direct computation: each coupling phase is zero, the cosine of zero is one, and the sum simplifies.

This identity is a building block, not a physical law. It is a definitional equality that follows from how the symbol and the weights are defined. The theorem does not claim that this zero value has any physical significance, nor does it assert that the symbol at zero momentum represents an energy or a force. It is a technical lemma that supports a larger limit statement: the centered symbol divided by t squared tends to a specific quadratic form as t approaches zero. That limit, not the zero value itself, is what connects the discrete lattice model to a continuum description.

The theorem is tagged as a MODEL in the framework's own accounting. It establishes a fact about the chosen definitions, not a derived physical consequence. The framework's status flags confirm this: the symbol is named, the abstract limit is proved, and the normalization gate passes, but the ledger convergence and gap action recovery remain open. The identity is a precise, narrow statement: at zero wavefunction, the symbol equals the sum of weights. It does not say what happens at nonzero wavefunctions, it does not say the sum has a physical meaning, and it does not claim that the limit statement is a physical law.

theorem exactMidpointBlochSymbol_zero_eq (H : Mat4) :
    exactMidpointBlochSymbol H (fun _ => 0) = exactMidpointBlochSymbolZero H := by
  unfold exactMidpointBlochSymbol exactMidpointBlochSymbolZero
  refine Finset.sum_congr rfl fun i _ => ?_
  rw [couplingPhaseIdx_zero, Real.cos_zero, mul_one]

theorem exactMidpointBlochSymbol_zero_eq (H : Mat4) :
    exactMidpointBlochSymbol H (fun _ => 0) = exactMidpointBlochSymbolZero H := by
  unfold exactMidpointBlochSymbol exactMidpointBlochSymbolZero
  refine Finset.sum_congr rfl fun i _ => ?_
  rw [couplingPhaseIdx_zero, Real.cos_zero, mul_one]

What this page does not claim

The zero value of the symbol has no physical significance asserted by this theorem. The theorem does not describe the behavior of the symbol at nonzero wavefunctions. The limit statement is not claimed to be a physical law by this identity alone.

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/ReggeExactFlatHessianBlochSymbol4D.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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