Gravity Analysis Regge Exact Flat Hessian Bloch Torus Bridge4 D Tendsto Exact Mi

The torus bridge

In the Recognition Science framework's machine-checked library of formal theorems, a new result connects a discrete grid of wave modes to a continuous limit. The theorem, named tendsto_exactMidpointBloch_torus_family_centered, concerns a family of wave modes arranged on a torus, a shape like the surface of a donut. As the grid spacing shrinks toward zero, the behavior of these discrete modes approaches the behavior described by a continuous formula. The statement is a limit statement: the discrete family converges to a continuous value as the grid becomes infinitely fine.

The theorem applies to any nonzero wave mode and any polarization matrix. It states that the ratio of a certain symbol difference to the squared momentum norm converges to a specific continuous expression. This expression is the second derivative of the symbol at the zero point, divided by the squared norm of the wave mode. The proof composes a previously established continuous limit with a scaling factor that tends to zero, ensuring the discrete family tracks the continuous behavior.

In Recognition Science, this result is part of a bridge between discrete and continuous descriptions. The framework models physical reality as a ledger, a discrete record of recognition events. This theorem shows that a particular discrete structure, the torus family, has a well-defined continuous limit. This is a step toward reconciling the discrete ledger with continuous geometric descriptions used in classical physics.

The theorem is a formal statement about limits. It does not claim that the discrete and continuous descriptions are identical, only that they agree in the limit of infinitely fine grids. It does not establish any physical law or measurement. It is a mathematical result about the convergence of a specific construction within the framework's library.

/-- 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]

/-- 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]

theorem tendsto_torusScale_nhdsWithin_ne_zero :
    Tendsto torusScale atTop (𝓝[≠] (0 : ℝ)) := by
  refine tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within _ tendsto_torusScale_nhds_zero ?_
  filter_upwards [eventually_torusScale_ne_zero] with j hj
  simp only [Set.mem_compl_iff, Set.mem_singleton_iff]
  exact hj

What this page does not claim

The theorem does not establish any physical law or measurement. The theorem does not claim the discrete and continuous descriptions are identical, only that they agree in the limit. The theorem does not apply to the zero wave mode.

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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