Exact Distribution Laws and Complete Horizon Classification for Walsh Interaction-Order Energies via Krawtchouk Collision Exclusion

Paper-2 develops an exact computational framework for Walsh interaction-order energies on the Boolean cube. It moves beyond permutation moments to the full distributional problem: exact PMFs and CDFs, structured algorithms, support geometry, multiplicity formulas, and deterministic continuation horizons. The main result is a clear computational picture. Exact evaluation for completely general inputs reaches a #P-hard counting boundary already at the top Walsh order, while important structured classes admit exact and efficient formulas. For sparse fixed-defect fields, the full energy depends only on pairwise Hamming distances, which leads to exact finite-state distribution laws, low-order closed forms, all-order recurrences, finite-alphabet extensions, fixed-parameter support algorithms, Gaussian determinant formulas, and a GKZ/A-hypergeometric representation of exact multiplicities. A second headline result is the complete continuation-horizon classification. The paper proves that the number of additional coordinates required to distinguish exact pair states is H(2) = 1, H(3) = 2, H(k) = 3 for every even k ≥ 4, and H(k) = 4 for every odd k ≥ 5. The even-order result is obtained through a Krawtchouk collision-exclusion argument using recurrence relations, parity-channel factorization, real-rooted polynomials, degree complementation, and coordinate symmetry. Main mathematical formulas For a function f : F₂ⁿ → R, with N = 2ⁿ, the Walsh coefficient and order-k energy are f̂(A) = (1/N) Σₓ f(x)(−1)^(Σi∈A xᵢ) and E_k(f) = Σ|A|=k f̂(A)². The scaled energy used throughout the paper is Q_k(f) = N² E_k(f). The binary Krawtchouk polynomial is K_k⁽ⁿ⁾(d) = Σⱼ₌₀^k (−1)^j C(d,j) C(n−d,k−j), with generating function Σₖ₌₀^n K_k⁽ⁿ⁾(d) z^k = (1−z)^d (1+z)^(n−d), and recurrence (k+1)Kₖ₊₁(x;N) = (N−2x)K_k(x;N) − (N−k+1)Kk−1(x;N). For a fixed-defect field f(v) = b + Σᵢ₌₁^s δᵢ 1v=vᵢ, the exact pair-distance energy kernel is Q_k = C(n,k) Σᵢ δᵢ² + 2 Σi<j δᵢδⱼ K_k⁽ⁿ⁾(dᵢⱼ). If W_n(D) counts the reachable pair-distance states, the exact PMF is Pr(Q_k = q) = 1/(2ⁿ−1){s−1} · Σ{D:Q_k(D)=q} W_n(D). For order 1, Q₁⁽ʳ⁾ = (Σᵢ δᵢ σᵢ,ᵣ)². For order 2, using the pair-Gram matrix G_t, Q₂(t) = ½[tr(ΔG_tΔG_t) − t(Σᵢδᵢ)²], equivalently, Q₂(t) = C(t,2)Σᵢδᵢ² + Σi<jδᵢδⱼ[mᵢⱼ(t)² − t]. For order 3, K₃⁽ᵗ⁾(dᵢⱼ) = [m³ − (3t−2)m]/6, and therefore Q₃(t) = C(t,3)Σᵢδᵢ² + (1/3)Σ_{i