Gravity Analysis Regge4 Dtorus Continuum Limit Canonical Finite H4 D Eq Finite T

The finite Hessian identity

In lattice gravity, one often writes a finite, discrete version of the Einstein-Hilbert action on a torus with N sites per side. The declaration canonicalFiniteH4D_eq_finiteTransportedSymbol establishes that this finite action, called the finite torus Hessian, equals a transported symbol for every mode and every edge-length matrix. The equality is a theorem in the framework's machine-checked library of formal theorems, proved by unfolding definitions and reducing to reflexivity.

The content of the identity is a bookkeeping fact. The finite Hessian is built from a fold over distinct hinges, while the transported symbol is a different expression. The theorem says these two expressions coincide once the side length is taken from the family side and the mode is converted to a real mode. The proof is a definitional unfolding, not a deep analytic argument.

The surrounding lemmas give the identity its meaning. The torus has N^4 sites, and the density weight is N^{-4}. A second-difference bookkeeping factor of 2 divided by N^4 multiplies a cell-sum cosine factor of N^4/2, and the product is exactly 1. That cancellation is what makes the surviving dictionary factor equal to 1, and it is what the equality theorem packages.

In Recognition Science, this identity is part of a dictionary between a finite periodic action sequence and a continuum symbol. The framework models the continuum limit as a limit over the family side j of the ratio of the finite Hessian to the momentum norm squared. The identity does not establish that this limit exists; it only identifies the finite object that would appear in the numerator.

What the declaration does not claim is as important as what it proves. It does not prove the four-dimensional cosine cell-sum identity, which remains an open target. It does not resolve residual offsets for non-t11 or t12 orbits on the legacy fold. It does not flip the gap action recovery flag. The identity is a finite algebraic equality, not a statement about the continuum limit itself.

theorem finiteTorusHessian_eq_finiteTransportedSymbol (j : ℕ)
    (m : IntMode4) (E : Mat4) :
    finiteTorusHessian (familySide j) m E =
      finiteTransportedSymbol j m E :=
  canonicalFiniteH4D_eq_finiteTransportedSymbol j m E

theorem canonicalFiniteH4D_eq_finiteTransportedSymbol (j : ℕ)
    (m : IntMode4) (E : Mat4) :
    canonicalFiniteH4D (familySide j) m E =
      finiteTransportedSymbol j m E := by
  unfold canonicalFiniteH4D finiteTransportedSymbol familySide
  rfl

theorem survivingDictionaryFactor4D_eq_cancellation (N : ℕ) [NeZero N] :
    survivingDictionaryFactor4D =
      ttSecondDifferenceDensityWeight N * cellSumCosMulCosFactor N := by
  rw [survivingDictionaryFactor4D_eq_one, density_cellSum_cancellation]

What this page does not claim

The declaration does not prove the four-dimensional cosine cell-sum identity. The declaration does not resolve residual offsets for non-t11 or t12 orbits on the legacy fold. The declaration does not flip the gap action recovery flag.

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/Regge4DTorusContinuumLimit.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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Derived articles

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