Projective Dynamic Logo (PDL) — Global Mapping of Structures, Results, and Open Problems (Version 20)

This document is the twentieth version of the global mapping of the Projective Dynamic Logo (PDL) programme — a foundational physics framework that derives fundamental physical constants and dynamical laws from four axioms on finite signed graphs, without presupposing spacetime, particles, or fields. Version 20 covers the complete corpus D01–D53 and incorporates document D53, which provides a self-contained synthesis and independent numerical verification of the complete causal chain from axioms C1–C4 to the cosmological constant Λ ≈ 1. 089 × 10⁻⁵² m⁻². The chain traverses fifty-seven orders of magnitude — from combinatorial graph theory to the rate of expansion of the universe — without free parameter. The independent verification confirms the leakage constant C to 0. 41 ppm, and establishes that the apparent 8. 17 ppm offset relative to Planck 2020 is entirely metrological in origin. The programme has reached the following resolved state: three critical Gates (derivations of 155/11017, a=2, and Gₑff (N) = σ (N) ·GPDL from C1–C4 alone) ; complete quantum dynamics layer (Schrödinger, Dirac, Born Level 1 and Level 2 with U (1) phase freedom, London equation) ; complete nuclear stability layer (spin-orbit splitting Δn=4, sub-shell filling rates, valley of stability for Z=1–82, all as unconditional theorems) ; complete black hole thermodynamics layer (area law, Bekenstein–Hawking formula, one-quarter coefficient) ; and complete cosmological leakage layer (C, ΛPDL). The Hubble tension is resolved at 0. 006% with zero free parameters. Nine open problems remain, classified by priority, along with nine falsifiable parameter-free predictions testable at current experimental facilities (Fermi-LAT, FRIB, RIKEN). The document includes the complete corpus inventory (D01–D53), a nine-layer epistemic architecture table, a critical-path dependency map, and a guided reading pathway for new collaborators. This document is part of the PDL open-access corpus available at https: //zenodo. org/communities/pdl-framework and https: //github. com/laubscher-lab/PDL-framework.