Gravity Analysis Regge Exact Flat Hessian Bloch Symbol4 D Abstract Centered Tend

The abstract limit theorem

In mathematics, a limit describes what a quantity approaches as an input gets arbitrarily close to a target value. The declaration abstract_centered_tendsto_available is a proved theorem about a specific kind of limit: it takes a weighted sum of cosine functions, subtracts its value at zero, divides by the square of the input, and shows that this expression converges to a particular number as the input approaches zero. The theorem is abstract because it applies to any finite collection of weights and phases, not just to the specific 1208 couplings used in the framework's gravity model.

The statement is a recognition ledger, the framework's term for a discrete record of events, of a standard calculus fact. It says that for any weights and phases, the limit of the centered trig polynomial divided by t squared equals the second-order term of its Taylor expansion. This is the kind of result that lets a model builder replace a complicated finite sum with a simpler quadratic approximation when studying behavior near a point. The proof is a direct specialization of a more general theorem in the framework's machine-checked library of formal theorems.

In Recognition Science, the theorem acts as a bridge. The framework's gravity model defines an exact midpoint Bloch symbol as a sum over 1208 couplings, each involving a cosine of a phase. The abstract theorem provides the general limit statement, and a separate specialization applies it to this specific sum. That specialization proves that the exact symbol, after centering and dividing by t squared, converges to a second-order term. This is a technical but necessary step for connecting the discrete model to a continuous, small-scale description.

The declaration does not claim that the discrete model is physically correct, nor that the limit represents a physical observable. It only establishes a mathematical convergence property. The framework's own status flags mark the ledger convergence and the gap action recovery as open, and the normalization gate passes only under a discrete bookkeeping factor. The theorem is a tool, not a physical law.

/-- Re-export: abstract centered Tendsto is available for any weight/phase. -/
theorem abstract_centered_tendsto_available
    {ι : Type*} (w θ : ι → ℝ) (s : Finset ι) :
    Filter.Tendsto (fun t : ℝ => centeredTrigPoly w θ s t / t ^ 2)
      (nhdsWithin 0 {0}ᶜ) (nhds (centeredTrigPolyM2 w θ s)) :=
  tendsto_centeredTrigPoly_div_sq w θ s

/-- **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]

What this page does not claim

The theorem does not prove that the discrete gravity model is physically correct. The theorem does not establish that the limit represents a measurable physical quantity. The theorem does not resolve the open ledger convergence or gap action recovery flags.

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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