D56: Derivation of the Linear Scaling N_comp(k) = k from Axioms C1–C4 — Resolution of Open Problem OP-D41-1-A (Projective Dynamic Logo Framework)

This document resolves Open Problem OP-D41-1-A of the PDL framework: the derivation of the linear scaling N_comp(k) = k for the number of maximal partial closures on a k-particle–k-hole nuclear interface surface, from axioms C1–C4 alone, without any free parameter. The main result (Theorem 1) is an unconditional theorem of C1–C4, proved via three lemmas: (L1) Uniqueness: the unique maximal C1+C2+C3+C4-admissible closure on a single interface unit of surface T = R_surf(p)²/R_sea(n) = 25.26 triangles is K_4 (D16a). Therefore N_comp(1) = 1. (L2) Disjointness: the mixed-triangle sets of k distinct (A)∧(B)-coupled proton–neutron interface units are mutually disjoint. This follows from the vertex-disjointness of the k K_4 blocks (D29). (L3) C3-irreducibility: with no cross-edge between distinct interface units, each unit is a separate connected component; no admissible closure can span two units (axiom C3 directly). Together, L1–L3 give N_comp(k) = k exactly for all k ≥ 1. A corollary establishes R_surf(k) = k·T for the effective active surface of a k-ph nucleus. Combined with D32 (Proposition 3: N_mix ∝ R_surf) and D42 (Indifference Lemma: distinct relations contribute independently), the incoherent-summation argument gives |M_fi|² = k·|m_1|² ∝ k, providing structural support for Conjecture H_B (D41). The remaining formal gap — identification of the E2 multipole operator within the PDL signed-graph formalism — is identified as Open Problem OP-E2-PDL and is explicitly not claimed here. Numerical verification is provided for k = 1..5 via the companion Python script PDL_OP_D41_1_v1.py (deposited alongside this document). The experimental motivation comes from Ha et al. (Nature Communications 16, 10631, 2025) on ⁸⁴Mo/⁸⁶Mo, and from Escudeiro, Recchia, Lenzi et al. (Physical Review C 113, 044304, 2026) on the f₇/₂ mirror nuclei ⁴⁷Cr–⁴⁷V and ⁴⁹Mn–⁴⁹Cr. This document belongs to the core PDL corpus (D01–D56) and resolves OP-D41-1-A. The PDL programme derives fundamental physical constants and nuclear structure from four combinatorial axioms on finite signed graphs, without presupposing spacetime, particles, or fields.