Nuclear Stability and the Periodic Table as Combinatorial Closure Hierarchies in the Projective Dynamic Logo Framework

Within the Projective Dynamic Logo (PDL) framework, physical reality is reconstructed from relational axioms on finite signed graphs, without presupposing space, time, or particles. The proton is characterised by a unique integer quintuplet (24, 28, 930, 10087, 11017), established in earlier work. The present paper extends this programme to the nuclear scale and establishes the following results, all without free parameters. First, the neutron quintuplet (24, 28, 1032, 9960, 10992) is derived for the first time from structural principles, via the exact relation Rₜot (n) = Rₜot (p) − (Δn+1) ² = 10992, where Δn = nd − nᵤ = 4. The gap of 40 between the natural budget 6μₙ and the actual Rₜot (n) is decomposed into a structural permutation cost (25) and a mass fatigue contribution (≈15). A new relation Rₜot (p) ≈ 6μₚ (accurate to 0. 001%) is identified, linking the total relational budget to the proton-to-electron mass ratio via the six edges of the elementary (4, 6) closure. Second, the nuclear binding mechanism is formalised: the quantity T = Rₛurf (p) ²/Rₛea (n) ≈ 25 simultaneously measures the neutron's coherence deficit and its capacity to form mixed coherent triangles with a neighbouring proton. This balanced exchange — what the neutron lacks equals what it provides — is the relational foundation of nuclear binding. The stability condition N·T ≥ Z (Z−1) /2 · Tₚp generates the valley of stability, with a saturation threshold Zₛat = Rₛea (n) /Rₛurf (p) ≈ 20 (calcium) marking the structural origin of the frontier between symmetric and neutron-rich nuclei. Third, the sea-exhaustion condition yields Ncrit, max = Rₛea (p) / (2 × gap) = 126. 1, reproducing the largest nuclear magic number without adjustment. Bismuth-209 is identified as the last stable nucleus through structural impossibility rather than probabilistic decay. Finally, the asymmetry Δn = 4 is identified as the structural source of the nuclear spin-orbit coupling introduced phenomenologically by Goeppert-Mayer and Jensen in 1949. The splitting energy Δn/ (2nᵤ) ≈ 0. 083 ħω₀ is consistent with empirical values, and the full sequence of magic numbers 2, 8, 20, 28, 50, 82, 126 is recovered as a structurally motivated conjecture. The principal open problem — the explicit combinatorial derivation of the shell splitting from the signed-graph structure of the collective surface — is formulated precisely.