The Kepler Conjecture 2025, 45 - Scoring

For all saturated packing Λ, v ∈ Λ ∃ D(v, Λ). The negligible function A: Λ → ℝ defines a composite A = A₀○D(_˙, Λ): Λ → DS → ℝ, v → D(v, Λ) → A₀(D(v, Λ)). Each simplex in the Q-system with vertex at v defines a simplex in the Q-system with vertex at 0 attached to D(v, Λ). Q₀(D) is the set of translated simplices at origin, and is determined by D. Let Q be a quarter in Q₀(D). The context of Q is (p, q) if there are p anchors and p - q quarters along the diagonal of Q. Write c(Q, D) for the context of Q ∈ Q₀(D). An orthosimplex is a convex hull of {0, v₁, v₁ + v₂, v₁ + v₂ + v₃} with v₂ a vector orthogonal to v₁, and so on. It's specified up to congruence by the parameters of their absolute values, so that a ≤ b ≤ c. This In this case, the Rogers simplex R(a, b, c) is an orthosimplex of the form S(a, b, c, √(c² - b²), √(b² - a²), √(b² - a²)). A Rogers simplex R has a quoin - a wedge-like solid - above R. It's defined as the solid bounded by the planes through the faces of R, and a sphere of radius c at the origin. quo(R) is the volume of the quoin over R.

Let B(0, t) be a ball of radius t, centered at the origin, v₁, v₂ vertices. |v₁| < 2t, |v₂| < 2t, then when the ball is truncated by cutting away the caps, and the circumradius of the triangle {0, v₁, v₂} is less than t, then the intersection of the caps is the union of four quoins. A measurable subset X of ℝ³, sol(X, v) the area of the radial projection of X\{0} to the unit sphere around origin. The area is the solid angle of X at v. If v = 0, then write sol(X).

A quarter Q with η⁺(Q) the maximum of the circumradii of the two faces of Q along its diagonal. Define A₁(S, c, v) = -vol VC(S, v) + sol(S, v)/3δ_oct - σ(S, c, v)/4δ_oct where σ is a function whose function heavily depends on the definitions of S. Note A₀: DS → ℝ is continuous. A 4-tuple simplex S with context c has Σ⁴_i=1 A₁(S, c, v_i) = 0. The function A as defined above is negligible, in that ∃ C₁ constant: ∀ r ≥ 1 ⇒ Σ_{v∈Λ(x, r)} A(v) ≤ C₁r²

Set the score of the decomposition star σ(D) = -4δ_oct(volΩ(D) + A₀(D)) + 16π/3. For A, A₀ and σ if the maximum of σ on DS is 8pt, then the minimum of the function on DS D→volΩ(D) + A₀(D) is √(32) and vice versa. If the maximum of σ on DS is 8pt, then for all saturated packing Λ, there is a negligible fcc-compatible function A. A standard cluster (R, D) with decomposition star D and R one of its standard regions defines a quad cluster when the standard region is a quadrilateral.