Gravity Analysis Regge Exact Flat Hessian Bloch Torus Bridge4 D Discrete Torus F

Torus bridge theorem

A discrete torus is a finite grid of points arranged in a loop, used to approximate a continuous shape. In the Recognition Science framework, the theorem discrete_torus_family_bridge_of_symbolZero (a machine-checked result in the framework's library of formal theorems) states that as the grid grows finer, a certain ratio involving the midpoint Bloch symbol, a quantity from wave analysis, converges to a specific continuum value. This convergence holds for all nonzero wave modes and all polarizations, provided a symmetry condition is met: the midpoint Bloch symbol at zero is zero for all matrices.

The theorem is a bridge. It connects a discrete, grid-based description of a wave to a continuous, analytic one. The ratio in question is the exact midpoint Bloch symbol divided by the momentum norm squared. As the torus side length grows, this ratio approaches the ratio of the second derivative of the midpoint Bloch symbol to the wave norm squared. This is a precise statement about how discrete approximations behave in the limit of infinite refinement.

The bridge is built in two steps. First, a continuous-scale version of the convergence is established. Second, this is composed with a scaling that sends the torus scale to zero, the punctured neighborhood of zero. The theorem discrete_torus_family_bridge_of_symbolZero packages these steps into a single statement, requiring the symmetry condition as a hypothesis. A separate theorem, discrete_torus_family_bridge, states the same convergence without the symmetry condition, using a different proof path.

The theorem does not claim that the symmetry condition holds. It does not claim that the convergence is uniform in the wave mode or the matrix. It does not claim that the discrete torus family is the only way to approximate the continuum limit. It does not claim that the midpoint Bloch symbol at zero is zero for any particular matrix. It only establishes the convergence under the stated hypothesis.

/-- Package: R2 for all polarizations inhabits the discrete torus bridge Prop
shape (same binders as `TypedResidual_discrete_torus_family_bridge`). -/
theorem discrete_torus_family_bridge_of_symbolZero
    (hZ : ∀ H : Mat4, exactMidpointBlochSymbolZero H = 0) :
    ∀ (m : IntMode4) (E : Mat4),
      m ≠ 0 →
        Tendsto
          (fun j : ℕ =>
            exactMidpointBlochSymbol E (realMode (torusSide j) m) /
              momentumNormSq (torusSide j) m)
          atTop
          (nhds
            (exactMidpointBlochM2 E (fun i => (m i : ℝ)) /
              waveNormSq (fun i => (m i : ℝ)))) :=
  fun m E hm => tendsto_exactMidpointBloch_torus_family m E hm (hZ E)

/-- Package: R2 for all polarizations inhabits the discrete torus bridge Prop
shape (same binders as `TypedResidual_discrete_torus_family_bridge`). -/
theorem discrete_torus_family_bridge_of_symbolZero
    (hZ : ∀ H : Mat4, exactMidpointBlochSymbolZero H = 0) :
    ∀ (m : IntMode4) (E : Mat4),
      m ≠ 0 →
        Tendsto
          (fun j : ℕ =>
            exactMidpointBlochSymbol E (realMode (torusSide j) m) /
              momentumNormSq (torusSide j) m)
          atTop
          (nhds
            (exactMidpointBlochM2 E (fun i => (m i : ℝ)) /
              waveNormSq (fun i => (m i : ℝ)))) :=
  fun m E hm => tendsto_exactMidpointBloch_torus_family m E hm (hZ E)

/-- Centered discrete torus family: banked continuous Tendsto along `t(j)`. -/
theorem tendsto_exactMidpointBloch_torus_family_centered
    (m : IntMode4) (E : Mat4) (hm : m ≠ 0) :
    Tendsto
      (fun j : ℕ =>
        (exactMidpointBlochSymbol E (realMode (torusSide j) m) -
            exactMidpointBlochSymbolZero E) /
          momentumNormSq (torusSide j) m)
      atTop
      (nhds
        (exactMidpointBlochM2 E (fun i => (m i : ℝ)) /
          waveNormSq (fun i => (m i : ℝ)))) := by
  let dir : Wave4 := fun i => (m i : ℝ)
  let n : ℝ := waveNormSq dir
  have hn : n ≠ 0 := waveNormSq_intMode_ne_zero m hm
  have hcont := tendsto_exactMidpointBloch_m2_div E dir n hn
  have hcomp := hcont.comp tendsto_torusScale_nhdsWithin_ne_zero
  refine hcomp.congr' ?_
  filter_upwards with j
  dsimp only [Function.comp_apply]
  rw [realMode_eq_scale, momentumNormSq_eq_scale_sq]

What this page does not claim

The symmetry condition exactMidpointBlochSymbolZero H = 0 is not proven to hold for any specific matrix. The convergence is not claimed to be uniform across all wave modes or matrices. The discrete torus family is not claimed to be the unique approximation scheme for the continuum limit.

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/ReggeExactFlatHessianBlochTorusBridge4D.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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