Derivation of the Sub-Shell Filling Rates and the Periodic Table from the PDL Axioms: Resolution of OP13 and OP14

This document resolves two open problems of the Projective Dynamic Logo (PDL) programme: OP13 (derivation of the spin-orbit splitting from axioms C1–C4) and OP14 (derivation of the sub-shell filling rates governing the valley of nuclear stability). The central result is a number-theoretic uniqueness theorem: Δn = 4 is the unique value in 0, 4, 8, 12, … for which the quasi-completeness equation admits a positive integer solution satisfying constraint C2. The discriminant Δ = 22201 = 149² is a perfect square for Δn = 4 and for no other multiple of 4. This forces the spin-orbit splitting s = Δn/ (2nᵤ) = 1/12 as an unconditional theorem of C1–C4, without invoking any experimental input. Combined with the Mirror Lemma — the theorem of D22 that the proton–neutron coexistence condition forces nᵤ (n) = nᵤ (p) = 24 and nd (n) = nd (p) = 28 — the isomorphism of the proton and neutron HO-PDL levels follows. From this isomorphism, the filling rates are derived: rₑxc (Z) = 1 for all even Z in (20, 82) with Z ∉ 28, 50, 82 (mirror filling, ratio 2/2 = 1 independent of sub-shell degeneracy), and rₑxc (Z) = 0 for Z ∈ 28, 50, 82 (shell closure). Verified exhaustively: 31 entries, zero exceptions, zero free parameters. As a consequence, the complete valley of stability Nₘin (Z) for Z = 1 to 82 is an unconditional theorem of axioms C1–C4. The gross structure of the periodic table — which elements exist as stable nuclei and which neutron numbers they prefer — follows from four combinatorial axioms on finite signed graphs, without any free parameter from nuclear physics, atomic physics, or quantum field theory. This document depends on D16a, D22, D25, D29, D40, and D46, and resolves OP13 and OP14 of the global mapping document DM v17 (DOI: 10. 5281/zenodo. 19811860).