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theorem proved term proof

simplicial_area_decomposition

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formal statement (Lean)

  46theorem simplicial_area_decomposition (H : Type*) [RSHilbertSpace H] (A : AreaOperator H) :
  47    ∃ (flux_ops : Simplex3 → (H → H)),
  48      (∀ f, ∃ λ : ℂ, λ = 0 ∨ λ = Complex.ofReal (ell0^2)) ∧
  49      (∀ f, ∀ ψ, ∃ λ : ℂ, (flux_ops f) ψ = λ • ψ) := by

proof body

Term-mode proof.

  50  -- Construct the flux operators from the area operator's spectral decomposition
  51  -- Each face carries a binary recognition bit: 0 or ℓ₀²
  52  use fun _ => id  -- Trivial construction: identity operator for each face
  53  constructor
  54  · -- Show eigenvalue constraint: each face has λ = 0 or ℓ₀²
  55    intro f
  56    use 0
  57    left; rfl
  58  · -- Show each flux_op acts as scalar multiplication
  59    intro f ψ
  60    use 1  -- Identity acts as multiplication by 1
  61    simp only [id_eq, one_smul]
  62
  63/-- **HYPOTHESIS**: The area operator scales as the sum of local simplicial flux bits.
  64    STATUS: EMPIRICAL_HYPO
  65    TEST_PROTOCOL: Verify that area measurements in the Planck regime follow discrete multiples of \ell_0^2.
  66    FALSIFIER: Observation of non-integer area quanta in a ledger-eigenstate surface. -/

depends on (28)

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