Projective Dynamic Logo (PDL) — Global Mapping of Structures, Results, and Open Problems (Version 19)

This document provides a systematic and self-contained guide to the Projective Dynamic Logo (PDL) programme, intended for any scientific colleague who wishes to understand, verify, or continue the research. PDL is a foundational physics programme that derives fundamental physical constants and dynamical laws from four axioms on finite signed graphs, without presupposing spacetime, particles, or fields. The minimal admissible closure is the complete signed graph K₄ on four vertices and six edges. The proton is characterised by the unique integer quintuplet (24, 28, 930, 10087, 11017), from which all results flow without free parameter. Version 19 covers the complete corpus D01–D52 and marks a qualitative threshold in the programme: the causal chain from the four axioms to the cosmological constant ΛPDL is now fully closed. The following advances relative to Version 18 are incorporated. D49 resolves OP-London: the London equation jⁱₛ = − (ncoh e²/mₚ c) Aⁱ is derived from C1–C4 as an unconditional theorem. The central lemma ν = (1/4) ‖ψ₁ − Tᵏ ψ₁‖² is verified exhaustively over 768 configurations in exact integer arithmetic. Axiom C4 forces the London gauge φ = const, and the London equation follows via minimal substitution. The PDL London penetration depth is λL, PDL (N=40) ≈ 7. 25 × 10⁻¹⁵ m. D50 resolves OP12/BH-3: the coefficient 1/4 in the Bekenstein–Hawking formula SBH = kB A/ (4ℓP²) is derived from C1–C4 as the stable fraction 4/16 per surface relation under the (A) ∧ (B) criterion (D29), with statistical independence guaranteed by H3 (D42). The black hole thermodynamics layer of the programme is complete. D51 and D52 together resolve OP1-D35, the PDL analogue of the cosmological constant problem. D51 establishes that β₁ (K₄) = 3 gives exactly three independent leakage cycles (topology), that axioms C1+C3 force each cycle length to be prime (irreducibility), and that the cycle lengths are counted by exact integer identities from the quintuplet: k₁ = nd − nᵤ + Rₑ − 1 = 9, k₂ = nᵤ − Rₑ + 1 = 19, k₃ = Rₑ · nd = 168, with k₁ + k₂ = nd. D52 formally identifies the three leakage bases (1−κ), Rᵥal/Rₜot, and (1−ηL) as the natural and unique representatives of the three cycles, derived respectively from D42 (κ = Rₛurf/Rₜot), D29 (Ωᵥal = 1), and D17/D30 (ηL = εGB). The resulting formula C = (1−κ) ⁹97 × (930/11017) ²3 × (1−ηL) ⁶7 reproduces the observational bound C ≈ 8. 158 × 10⁻⁴⁶ to 0. 17 ppm, compatible with Λₒbs = 1. 089 × 10⁻⁵² m⁻² (Planck 2020) within the 1. 7% measurement uncertainty. The full causal chain C1–C4 → K₄ → β₁ = 3 → (p₊䃑, p₊䃒, p₊䃓) → C → ΛPDL is complete. The document provides: a complete annotated corpus table (D01–D52) ; a nine-layer epistemic architecture table distinguishing unconditional theorems, conjectures, and open problems; a complete list of falsifiable predictions (P1–P9) ; a critical-path dependency map updated to show the closure of all four major layers (quantum dynamics, nuclear stability, black hole thermodynamics, cosmological leakage) ; and a continuation guide for new collaborators.