Nuclear Stability and the Periodic Table from PDL Combinatorial Axioms: Derivation of the Valley of Stability, Magic Numbers, and Assembly Rules

The Projective Dynamic Logo (PDL) framework derives physical law from four axioms on finite signed graphs, without presupposing spacetime, particles, or fields. This paper establishes a coherent and largely parameter-free derivation of nuclear stability from the proton quintuplet (24, 28, 930, 10087, 11017) and the neutron quintuplet (24, 28, 1032, 9960, 10992) alone. The central results are: (i) the identity T/(Δn+1)² ≈ 1 is a theorem of the quintuplet, entailing N_min(Z) = Z for all Z ≤ Z_sat = 20 as an exact consequence of the neutron survival condition; (ii) the conflict saturation C(Z > 20) = 190·T_pp is exact, with Z_sat = 20 recovered without free parameters; (iii) the valley of stability N_min(Z) is reproduced exactly for all 51 elements from Z = 1 to Z = 82 by the rule N_min(Z) = Z + Σ r_exc(Z'), where the excess tally r_exc ∈ {0, 1, 2, 3} corresponds to the sub-shell structure of the harmonic oscillator with PDL spin-orbit splitting Δn/(2n_u) = 1/12; (iv) the upper boundary N_max(Z) is reproduced to 82% accuracy by linear interpolation between doubly-magic anchor nuclei; (v) the conflict identity N × Q = C(Z) holds algebraically for all (Z, N). The paper also establishes eight explicit assembly rules governing the sequential construction of nuclei from hydrogen to iron, and identifies a structural connection between OP1 of this document and the open problem of deriving the Bekenstein-Hawking coefficient 1/4 in D37/D38: both reduce to the same combinatorial question of counting coherent configurations on a collective PDL surface.