Cosmology Lattice Ball Volume Octa Card Eq Sum

The octahedron count

The declaration octa_card_eq_sum establishes a counting identity for the set of integer lattice points inside an octahedron of radius t. The octahedron is defined by the inequality |x| + |y| + |z| ≤ t, where x, y, and z are integers. The theorem states that the number of such points equals a sum over the first coordinate x, from -t to t, of the number of points in a two-dimensional diamond of radius t - |x|. This is a fibered decomposition: each slice of the octahedron at a fixed x is itself a diamond, and the total count is the sum of the counts of those slices.

The identity is proved as a theorem in a machine-checked library of formal theorems, with no gaps and no added axioms. It is a statement about natural numbers, and it holds for every radius t. The proof works by showing that the slice at a fixed x is exactly the set of integer pairs (y, z) with |y| + |z| ≤ t - |x|, which is the definition of a two-dimensional diamond. The theorem then sums the known count of such diamonds over all x. This is a reduction from three dimensions to two, and it is the first step toward the closed-form volume law for the octahedron.

The declaration itself does not give the closed-form polynomial. It only states the sum. The closed form, that three times the count equals 4t³ + 6t² + 8t + 3, is a separate theorem in the same library. The sum identity is the bridge between the geometric definition of the octahedron and the algebraic formula. It is also the basis for the boundary-shell law, which counts the cells added when the radius grows by one. That shell count, 4(t+1)² + 2, is what the framework uses to claim that the cost of recognition localizes to a surface in three dimensions.

In Recognition Science, this counting identity is part of the geometric backbone of the coarsening engine, a simulation that grows its world one radius per cycle. The octahedron is the 3D shape of that world, and the theorem ensures the cell counts the simulation reports are exact. The declaration does not claim anything about physics, about the fine-structure constant, or about the number of spatial dimensions. It is a pure combinatorial fact about integer points, proved for all radii, and its role in the framework is as a tool for later claims about growth and localization.

/-- Fibered form of the octahedron cardinality: each `x`-fiber is a diamond of radius `t - |x|`. -/
theorem octa_card_eq_sum (t : ℕ) :
    (InterfaceComponentBound.Octahedron.ball t).card
      = ∑ x ∈ Finset.Icc (-(t : ℤ)) t,
          (InterfaceComponentBound.Diamond.ball (t - x.natAbs)).card := by
  have hb : InterfaceComponentBound.Octahedron.ball t
      = (Finset.Icc (-(t : ℤ)) t ×ˢ (Finset.Icc (-(t : ℤ)) t ×ˢ Finset.Icc (-(t : ℤ)) t)).filter
          (fun p => p.1.natAbs + p.2.1.natAbs + p.2.2.natAbs ≤ t) := rfl
  rw [hb, Finset.card_filter, Finset.sum_product]
  refine Finset.sum_congr rfl (fun x hx => ?_)
  rw [Finset.mem_Icc] at hx
  have hxnat : x.natAbs ≤ t := by omega
  dsimp only
  -- the (y,z)-fiber over x equals the diamond of radius (t - |x|)
  have hfib : (Finset.Icc (-(t : ℤ)) t ×ˢ Finset.Icc (-(t : ℤ)) t).filter
        (fun q : ℤ × ℤ => x.natAbs + q.1.natAbs + q.2.natAbs ≤ t)
      = InterfaceComponentBound.Diamond.ball (t - x.natAbs) := by
    apply Finset.ext
    rintro ⟨y, z⟩
    rw [InterfaceComponentBound.Diamond.mem_ball_iff]
    simp only [Finset.mem_filter, Finset.mem_Icc, Finset.mem_product]
    omega
  -- the inner (y,z)-sum is exactly the diamond-fiber card
  rw [← hfib, Finset.card_filter]

What this page does not claim

The declaration does not state the closed-form polynomial 3·card = 4t³ + 6t² + 8t + 3. It does not claim any physical or cosmological significance for the octahedron. It does not prove anything about the number of spatial dimensions.

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

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