From Four Axioms to the Cosmological Constant: Causal Closure of the Chain C1–C4 to Λ — A Synthesis and Computational Verification (PDL Framework — D53)

This document presents a self-contained synthesis of the causal chain leading from the four combinatorial axioms C1–C4 of the Projective Dynamic Logo (PDL) programme to the cosmological constant Λ ≈ 1.089 × 10⁻⁵² m⁻², together with an independent numerical verification at 50-decimal precision. No new mathematical result is established: all theorems are proved in the cited documents D51 and D52. The contribution of D53 is threefold. First, it presents the complete derivation chain in a single linear text, tracing each step from the topology of K₄ (β₁ = 3, giving three independent leakage cycles) through the prime structure of the exponents (23, 67, 997, forced by the non-decomposability axiom C3 and the proton quintuplet) to the leakage constant C = (1−κ)⁹⁹⁷ × (R_val/R_tot)²³ × (1−η_L)⁶⁷ and the cosmological constant Λ_PDL = C · η_L¹⁸ · (m_p c/ħ)². Second, it provides three fully self-contained Python cells (executable in Google Colab) that verify C_PDL against the value published in D51 to 0.41 ppm, and confirm that replacing any prime exponent by its nearest composite neighbour degrades the agreement by at least four orders of magnitude. Third, it situates the result epistemically: the deviation of 8.17 ppm apparent when comparing against the Planck 2020 value of Λ_obs is shown to be entirely metrological in origin (an 8.6 ppm offset in (m_p c/ħ)² between the CODATA versions used), carrying no structural significance. The chain spans fifty-seven orders of magnitude in physical scale, from four axioms on finite signed graphs — without presupposing spacetime, particles, or fields — to the rate of expansion of the universe. No free parameter enters the cosmological derivation. The smallness of Λ is not a fine-tuning: it is the arithmetic consequence of three independent suppressions, each raised to a prime power imposed by the structure of the axioms. This document is part of the PDL corpus (D01–D53), openly available at https://zenodo.org/communities/pdl-framework and https://github.com/laubscher-lab/PDL-framework.