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 Nₘin (Z) = Z for all Z ≤ Zₛat = 20 as an exact consequence of the neutron survival condition; (ii) the conflict saturation C (Z > 20) = 190·Tₚp is exact, with Zₛat = 20 recovered without free parameters; (iii) the valley of stability Nₘin (Z) is reproduced exactly for all 51 elements from Z = 1 to Z = 82 by the rule Nₘin (Z) = Z + Σ rₑxc (Z'), where the excess tally rₑxc ∈ 0, 1, 2, 3 corresponds to the sub-shell structure of the harmonic oscillator with PDL spin-orbit splitting Δn/ (2nᵤ) = 1/12; (iv) the upper boundary Nₘax (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.