Gravity Analysis Regge Exact Flat Hessian Bloch Symbol4 D Gate Passes Under Rest

The normalization gate

In numerical relativity, one often checks a complicated discrete scheme by seeing whether it reproduces a known continuum answer in the right limit. The declaration gate_passes_under_restatement_C is a machine-checked theorem in the Recognition Science framework's library of formal theorems that performs one such check. It proves two things together: that a flag called NormalizationGatePass (a bookkeeping test for whether the discrete calculation is correctly scaled) is true, and that a coefficient computed from the discrete Einstein-Hilbert action equals the corresponding coefficient from the exact unit Frobenius calculation. In plain terms, the discrete bookkeeping recovers the expected continuum coefficient.

The setting is a four-dimensional lattice gravity model. The framework builds an exact trigonometric polynomial, a finite sum of cosine terms, that represents the flat-space Hessian (the second variation of the action) at the midpoint of a unit cell. This symbol involves 1208 coupling terms. The theorem tendsto_exactMidpointBloch_centered_div_sq proves that as the wavevector scale t tends to zero, the centered symbol divided by t² converges to the expected second-order term. The gate theorem then confirms that the normalization, the factor that makes the discrete and continuum coefficients match, is correct. This is a consistency result: the discrete construction is internally aligned with the continuum limit it is designed to approximate.

The theorem does not claim that the discrete model is physically correct. The library's own status record shows that two further flags remain false: srsInhabited and gapActionRecovery. The first concerns whether a certain ledger (a discrete record of recognition events) is inhabited; the second concerns recovering the action from a spectral gap. Both remain open or false. The physical recognition-to-linking bridge, the step that would connect this discrete geometry to actual spacetime, is itself OPEN. What the gate theorem establishes is narrower: a formal, machine-checked consistency between a discrete symbol and its continuum limit, nothing more.

/-- Compat alias: former option-C gate name. -/
theorem gate_passes_under_restatement_C :
    NormalizationGatePass = true ∧
      einsteinHilbertTTCoefficient4D_unitFrobenius =
        exactUnitFrobeniusTTCoefficient :=
  ⟨normalizationGatePass_true, unitFrobenius_EH_eq_exact⟩

/-- **THEOREM:** centered exact midpoint Bloch `/ t²` tends to its cosine
two-jet.  Specialized through irreducible wrappers (no Fin-1208 whnf). -/
theorem tendsto_exactMidpointBloch_centered_div_sq
    (H : Mat4) (dir : Wave4) :
    Tendsto
      (fun t : ℝ =>
        (exactMidpointBlochSymbol H (fun j => t * dir j) -
            exactMidpointBlochSymbolZero H) / t ^ 2)
      (𝓝[≠] (0 : ℝ)) (nhds (exactMidpointBlochM2 H dir)) := by
  have habs :=
    tendsto_centeredTrigPoly_div_sq (weightFn H) (phaseFn dir) couplingUniv
  have htarget :
      centeredTrigPolyM2 (weightFn H) (phaseFn dir) couplingUniv =
        exactMidpointBlochM2 H dir := (m2_eq_irred H dir).symm
  rw [← htarget]
  refine (tendsto_congr' ?_).mpr habs
  filter_upwards with t
  rw [centered_eq_irred]

theorem exactBlochSymbolStatus_flags :
    exactBlochSymbolStatus.trigPolyNamed = true ∧
      exactBlochSymbolStatus.abstractTendstoProved = true ∧
        exactBlochSymbolStatus.specializedTendstoProved = true ∧
          exactBlochSymbolStatus.normalizationGatePass = true ∧
            exactBlochSymbolStatus.srsInhabited = false ∧
              exactBlochSymbolStatus.gapActionRecovery = false :=
  ⟨rfl, rfl, rfl, rfl, rfl, rfl⟩

What this page does not claim

The theorem does not claim that the discrete model is physically correct or that it describes actual spacetime. The theorem does not claim that the recognition-to-linking bridge is established; that remains OPEN. The theorem does not claim that the ledger S_RS_converges_EH_4d is inhabited or that gap action recovery holds.

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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