Projective Dynamic Logo (PDL) — Global Mapping of Structures, Results, and Open Problems (Version 14)

This document is the authoritative navigational and epistemic guide to the Projective Dynamic Logo (PDL) programme, updated to cover the complete corpus D01–D40. It is intended for a scientific colleague who wishes to understand, verify, or continue the research independently. PDL derives fundamental physical constants and dynamical laws from four axioms on finite signed graphs, without presupposing spacetime, particles, or fields. The minimal admissible closure under the axioms is the complete graph K₄, identified with the electron prototype. The proton is characterised by the unique integer quintuplet (24, 28, 930, 10087, 11017), from which all results flow without free parameters. Version 13 incorporates document D40 and reflects the following state of the programme. Three critical Gates are resolved: the axiomatic derivation of the amplitude 155/11017 (Gate 1, D29), the structural proof that the QCD correction coefficient equals 2 (Gate 2, D30), and the theorem Gₑff (N) = σ (N) ·GPDL (Gate 3, D31/D36). Four fundamental dynamical equations have been derived from the (A) ∧ (B) coupling criterion without free parameters: the Schrödinger equation (D32), the Dirac equation (D33), Born's rule via Gleason uniqueness (D34), and the Einstein coupling equation (D35). The Hubble tension is resolved at 0. 006% with zero free parameters. Black hole thermodynamics has been derived from PDL axioms: the area law S ∝ Rₛurf is an unconditional theorem (D37), and the Bekenstein–Hawking formula SBH/kB = 4π (Mₑff/MPl) ² follows algebraically from Gate 3 (D38), with three falsifiable parameter-free predictions for primordial black holes and AGN populations. The principal new contribution of v13 is the integration of D40, which derives the complete architecture of nuclear stability and the periodic table from the proton and neutron quintuplets alone: Nₘin (Z) = Z for all Z ≤ 20 as an exact theorem of the neutron survival condition; the conflict saturation C (Z > 20) = 190·Tₚp exactly; and the valley of stability reproduced at 100% accuracy for all 51 elements from Z = 1 to Z = 82 by a rule with filling rates rₑxc ∈ 0, 1, 2, 3 corresponding to the harmonic oscillator sub-shell structure with PDL spin–orbit splitting Δn/ (2nᵤ) = 1/12. D40 also identifies a structural connection between the open problem of deriving these filling rates analytically (OP14) and the open problem of deriving the Bekenstein–Hawking coefficient 1/4 from axioms (OP12/BH-3): both reduce to counting coherent configurations on a collective PDL surface, placing the nuclear and black-hole open problems in the same algebraic category. The epistemic architecture has been extended from six to seven layers to accommodate this new branch. The sole foundational open problem remains the derivation of the Indifference Lemma (H3) from axioms C1–C4 alone (OP1 of D39).