Gravity Analysis Regge4 Dexact Action Symbol Discrete Exact Regge Symbol Smul

The scaling law

The Regge action is a way of doing general relativity without a smooth spacetime: you replace the continuous fabric of space and time with a patchwork of flat, four-dimensional blocks, and you write the gravitational action, the quantity whose minimization gives the equations of motion, as a sum over the blocks' hinges. In this discrete setting, the Recognition Science framework's machine-checked library of formal theorems studies a specific object, the discrete exact Regge symbol, a number computed from a choice of metric and a wave mode that measures how the action responds to that mode. The declaration discreteExactReggeSymbol_smul proves a clean scaling property: if you multiply the metric by a constant factor c, the symbol scales as c squared.

In symbols, the theorem states that for any real number c, any mode m, and any metric E, discreteExactReggeSymbol j m (c • E) = c^2 * discreteExactReggeSymbol j m E. The proof is a short chain of existing results: the same scaling law already holds for the underlying finite exact Regge symbol, and the discrete symbol is defined as exactly twice that finite symbol, so the factor of two cancels out. This is a homogeneity statement, a standard and expected property for a quantity that is quadratic in the metric, and it is proved in the framework's library as a theorem, meaning it is a fully machine-checked result with no unverified assumptions.

The scaling law is a useful sanity check and a building block, but it is deliberately narrow. It does not say that this discrete symbol equals the continuum gravitational action, nor that it is the unique correct discretization. The framework's own documentation marks the full recovery of the continuum limit as an open problem, and it explicitly notes that the geometric fold used here is not equal to a separate algebraic dictionary sequence, differing by a factor of two. The theorem also does not address whether the action's Hessian, the matrix of second derivatives, annihilates gauge modes or has the correct physical spectrum; those are separate properties with their own proofs and open questions.

What the scaling law does give you is a precise, machine-checked fact you can rely on when combining this symbol with other results: the discrete exact Regge symbol is homogeneous of degree two in the metric. That is a concrete anchor for further work, and it is exactly the kind of small, verifiable step that a larger edifice of discrete gravity results can safely stand on.

theorem discreteExactReggeSymbol_smul (c : ℝ) (j : ℕ) (m : Fin 4 → ℤ)
    (E : Mat4) :
    discreteExactReggeSymbol j m (c • E) =
      c ^ 2 * discreteExactReggeSymbol j m E := by
  unfold discreteExactReggeSymbol
  rw [finiteExactReggeSymbol_smul]
  ring

theorem discreteExactReggeSymbol_eq (j : ℕ) (m : Fin 4 → ℤ) (E : Mat4) :
    discreteExactReggeSymbol j m E =
      (2 : ℝ) * finiteExactReggeSymbol j m E := by
  unfold discreteExactReggeSymbol discreteBookkeepingFactor
  ring

What this page does not claim

The discrete exact Regge symbol equals the continuum gravitational action. The discrete exact Regge symbol is the unique correct discretization of the Regge action. The scaling law says anything about the physical spectrum or gauge behavior of the action's Hessian.

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/Regge4DExactActionSymbol.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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