Gravity Analysis Regge Ttbloch Assembly Raw Cosine Evaluator Bucket Key Of

The bucket key identity

In the framework's machine-checked library of formal theorems, the declaration rawCosineEvaluator_bucketKeyOf proves a compact identity about how a certain cosine value is computed. The theorem states that the raw cosine evaluator, applied to the bucket key of a triple of edge slots, equals the cosine of the difference of two slot phases. In plainer terms: the cosine of an angle that appears in a Regge calculus gravity calculation can be rewritten as the cosine of a difference of two other angles, where those angles come from the slot phases of the two edges involved.

The identity matters because it shows a structural simplification. The bucket key, which is an integer triple derived from the edge slot data, carries exactly the information needed to evaluate the cosine. The theorem proves this for any lattice size N and any commensurate integer wave vector m, under the condition that N is nonzero. The proof is grounded in the framework's library, which is a machine-checked collection of formal theorems, and it relies on the cosine periodicity and an explicit integer-turn phase decomposition for periodic wrapping.

What the declaration does not claim is broader. It does not claim that the bucket key itself is a physical observable or that it has any direct gravitational meaning. It is a bookkeeping device, a way to index and organize the calculation. The theorem does not establish any property of the full gravity amplitude; it only establishes an identity about the cosine evaluator. It also does not claim that the cosine evaluator is the only way to compute the relevant quantity, nor does it claim anything about the convergence or physical interpretation of the Regge TT Bloch assembly as a whole. Those are separate questions, addressed by other declarations in the library.

/-- The bucket cosine for a raw triple is the cosine of the difference of
its two cell-independent slot phases. -/
theorem rawCosineEvaluator_bucketKeyOf (N : ℕ) [NeZero N]
    (m : Fin 3 → ℤ) (p : Fin 6 × Fin 6 × Fin 6) :
    rawCosineEvaluator N m (bucketKeyOf p) =
      Real.cos (slotPhase N m p.1 p.2.1 - slotPhase N m p.1 p.2.2) := by
  unfold rawCosineEvaluator bucketKeyOf slotPhase
  have harg :
      (∑ i : Fin 3,
        commensurateMomentum N m i *
          (((slotMidTwice p.1 p.2.2 i - slotMidTwice p.1 p.2.1 i : ℤ) : ℝ) / 2)) =
        -(∑ i : Fin 3,
            commensurateMomentum N m i * (((slotMidTwice p.1 p.2.1 i : ℤ) : ℝ) / 2) -
          ∑ i : Fin 3,
            commensurateMomentum N m i *
              (((slotMidTwice p.1 p.2.2 i : ℤ) : ℝ) / 2)) := by
    simp only [Fin.sum_univ_three]
    push_cast
    ring
  rw [harg, Real.cos_neg]

/-- Bucket key of one raw stencil triple.  Its phase is the doubled
midpoint displacement from the left slot to the right slot. -/
def bucketKeyOf (p : Fin 6 × Fin 6 × Fin 6) : Bucket :=
  ⟨p.2.1, p.2.2,
    fun i => slotMidTwice p.1 p.2.2 i - slotMidTwice p.1 p.2.1 i⟩

/-- Cosine form of the phase decomposition.  Integral seam corrections
disappear by `2*pi` periodicity. -/
theorem cos_localEdge_eq_cell_slot (N : ℕ) [NeZero N]
    (m : Fin 3 → ℤ) (cell : Vertex N N N) (t f : Fin 6) :
    Real.cos
        (edgeMidpointPhase N (commensurateMomentum N m)
          (localEdgeOf cell t f)) =
      Real.cos (BlochCellSum.theta N m cell + slotPhase N m t f) := by
  rw [localEdge_phase_decomposition N m cell t f]
  rw [show 2 * Real.pi * (slotWrapTurns N m cell t f : ℝ) =
      (slotWrapTurns N m cell t f : ℝ) * (2 * Real.pi) by ring]
  exact Real.cos_sub_int_mul_two_pi _ _

What this page does not claim

The bucket key is a physical observable with direct gravitational meaning. The theorem establishes any property of the full gravity amplitude beyond the cosine evaluator identity. The cosine evaluator is the only way to compute the relevant quantity.

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

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