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

The limit of the symbol

In the Recognition Science framework, gravity is studied through a discrete ledger, a record of events on a lattice, rather than through a continuous field. The declaration tendsto_exactMidpointBloch_centered_div_sq is a theorem about a specific quantity called the exact midpoint Bloch symbol. This symbol is a finite sum of cosine terms, each weighted by a coupling constant, that encodes the curvature of the lattice at a midpoint. The theorem states that as a small parameter t approaches zero, the difference between the symbol at a scaled point and its value at zero, divided by t squared, converges to a limit. That limit is the second-order term of the symbol's Taylor expansion, which is precisely the quadratic form known as the mass matrix.

This is a statement about the local behavior of the discrete model. It says that for very small perturbations, the complicated finite sum of cosines behaves like a simple quadratic polynomial. The theorem is proved in the framework's machine-checked library of formal theorems, meaning the proof is verified by a computer. The proof works by specializing a more general abstract result about trigonometric polynomials to this specific 1208-coupling case, avoiding a direct computation over the large finite set. The theorem also confirms a normalization identity: a discrete bookkeeping factor of 2 recovers the frozen coefficient of -1/4, which is a known gate for the model's consistency.

What the theorem does not claim is just as important. The declaration does not establish that the discrete ledger converges to the continuous Einstein-Hilbert action in any global sense. The framework's own status flags mark two related goals as open or false: the ledger S_RS_converges_EH_4d and the gap_action_recovery are not proved. The theorem is a local, pointwise limit result about the symbol's second-order term, not a proof that the whole discrete theory reproduces general relativity. It is a building block, a necessary but not sufficient step toward a continuum limit.

In plain terms, this theorem is a precise statement about the smoothness of the discrete model at small scales. It shows that the model's curvature symbol has the correct local quadratic behavior, which is what one would expect from a well-behaved discrete approximation. It does not, by itself, show that the model is equivalent to general relativity, nor does it validate the model's global predictions. It is a piece of the machinery, verified rigorously, that will be needed for any future claim about the continuum limit.

/-- **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]

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

What this page does not claim

The theorem does not prove that the discrete ledger converges to the continuous Einstein-Hilbert action. The theorem does not establish the ledger S_RS_converges_EH_4d or the gap_action_recovery. The theorem does not validate the model's global predictions about gravity.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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