Gravity Analysis Regge4 Dtorus Continuum Limit Finite Torus Hessian Eq Finite Tr

The finite torus identity

In numerical gravity, the same physical action can be written in different mathematical dialects. One dialect writes the action as a sum over the edges of a fine grid, the other as a sum over the hinges where grid cells meet. The declaration finiteTorusHessian_eq_finiteTransportedSymbol is a machine-checked theorem stating that, on a four-dimensional torus with side length N, these two dialects produce exactly the same number for every configuration of the metric field. It proves that the finite torus Hessian, the second variation of the discrete action, equals the finite transported symbol, the object obtained by carrying the continuum symbol onto the grid. The proof is a definitional equality: unfolding the definitions shows both sides reduce to the same expression, so the identity holds by computation rather than by a long argument.

The identity is part of a larger bookkeeping story. The discrete action carries a density weight of N to the minus fourth power, one factor of N for each of the four torus directions. The cell-sum identity for the four-dimensional cosine fold contributes a factor of N to the fourth over two. These two factors cancel exactly, leaving a surviving dictionary factor of one. That cancellation is what makes the finite torus Hessian equal the raw cosine fold, and it is the same bookkeeping that worked in three dimensions with N cubed. The theorem finiteTorusHessian_eq_finiteTransportedSymbol is the four-dimensional version of that cancellation, stated as an equality between the two finite objects before any continuum limit is taken.

What the theorem does not claim is just as important. It does not assert that the continuum limit of these finite objects exists, nor that the limit equals the Einstein-Hilbert action. Those are separate statements, and the pack marks them as open targets. The theorem does not claim that the four-dimensional cosine cell-sum identity holds for all modes; that identity is named as an open problem in the same file. It does not claim that the discrete action recovers the correct gravitational dynamics in the large-N limit, and it does not claim anything about the physical significance of the finite transported symbol beyond its equality with the finite torus Hessian. The theorem is a precise algebraic statement about two finite expressions, and it earns its place by being exactly that and nothing more.

In the framework's library, this identity is a small but load-bearing step. It shows that the two dialects of the discrete action agree on every finite torus, which is a necessary condition for any continuum limit to be well-defined. The identity also pins down the correct density weight: the pack includes a theorem that the wrong power, N to the minus second, is provably not equal to the correct weight for N at least two. That discrimination matters because a wrong weight would break the cancellation and the identity would fail. The finite torus identity is thus a check that the bookkeeping is consistent before the harder question of the continuum limit is addressed.

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]

theorem torusDensityWeight_ne_wrong {N : ℕ} (hN : 2 ≤ N) :
    torusDensityWeight N ≠ wrongMeshPowerWeight N := by
  rw [torusDensityWeight_eq_correct]
  exact (decoy_wrong_mesh_power hN).symm

What this page does not claim

The continuum limit of the finite torus Hessian exists or equals the Einstein-Hilbert action. The four-dimensional cosine cell-sum identity holds for all modes. The discrete action recovers correct gravitational dynamics in the large-N limit.

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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