Gravity Analysis Regge Ttlocal Symbol Existence Sqrt Edge Plane Wave Cont Diff A

The smooth hinge factor

A recognition ledger, a discrete record of events, can carry a model of gravity as a lattice of tetrahedra. The Regge action, a standard way to write gravity on such a lattice, sums over the edges of these tetrahedra, and each edge contributes a term proportional to the square root of its squared length. The declaration sqrtEdge_planeWave_contDiffAt concerns what happens to one of these square-root factors when a plane wave, a sinusoidal disturbance, passes through the lattice.

The theorem states that for any edge in the lattice, the square root of its squared length, viewed as a function of the wave's amplitude, is smooth at zero amplitude. Smooth here means it has derivatives of every finite order at that point. This is not a statement about a special edge: it holds for every edge, for every polarization of the wave, and for every wave vector. The proof works because at zero amplitude the edge has a positive length, and the square root function is smooth away from zero.

This smoothness is the hinge that lets the framework take a second derivative of the entire action profile. The action's second derivative at zero amplitude is the object the framework wants to study: it is the fixed-size Bloch symbol, a quantity that encodes how the lattice responds to the wave. The declaration sqrtEdge_planeWave_contDiffAt is one of several smoothness results that together establish that this second derivative exists. It is a local result, holding at the point of zero amplitude, not a global statement about the whole action.

In Recognition Science, this smoothness is a proved theorem in the framework's machine-checked library of formal theorems. It is a necessary step in a larger program, but it is not the destination. The declaration does not compute the value of the Bloch symbol, and it does not say anything about the continuum limit as the lattice spacing shrinks. It only guarantees that a certain derivative exists, which is the foundation for asking what that derivative equals.

/-- The square-root hinge factor of each edge is `ContDiffAt` at `0`: the
edge value at `t = 0` is `periodicDispSqEdge ∈ {1,2,3} > 0`, so `√` is
smooth there. -/
theorem sqrtEdge_planeWave_contDiffAt (E : Fin 3 → Fin 3 → ℝ) (k : Fin 3 → ℝ)
    (e : PeriodicEdge N N N) (n : ℕ∞) :
    ContDiffAt ℝ n
      (fun t : ℝ => Real.sqrt (planeWaveEdgeField N E k t e)) 0 := by
  refine ContDiffAt.sqrt ((planeWaveEdgeValue_contDiff N E k e n).contDiffAt) ?_
  simp only [planeWaveEdgeField, zero_mul, add_zero]
  exact ne_of_gt (periodicDispSqEdge_pos e.disp)

/-- Squared displacement class values are strictly positive (they are
`1, 1, 1, 2, 2, 2, 3`). -/
theorem periodicDispSqEdge_pos (d : Fin 7) : 0 < periodicDispSqEdge d := by
  fin_cases d <;> norm_num [periodicDispSqEdge]

What this page does not claim

The declaration does not compute the value of the Bloch symbol. The declaration does not address the continuum limit as the lattice spacing shrinks. The declaration does not claim smoothness for all amplitudes, only at zero amplitude.

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

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