The Cosmological Leakage Constant from the PDL Axioms: A Structural Derivation of C and Resolution of OP1-D35

Open problem OP1-D35 asks for the derivation of the dimensionless constant C in the PDL cosmological leakage term C^ (leak) _μν = −C · ηL¹8 · (mₚ c/ħ) ² · g_μν (D48) from the four PDL axioms C1–C4, without invoking observational input. This document presents a structural derivation that reduces OP1-D35 to a single remaining step, supported by three exact algebraic identities and a five-step logical argument rooted in the axioms. The derivation proceeds in four stages. First, the topology of K₄ — the unique minimal admissible closure of C1–C4 — imposes β₁ (K₄) = 3: the first Betti number of its 1-skeleton equals three, giving exactly three independent leakage cycles. The universe leaks in exactly three ways, for the same reason it has exactly three spatial dimensions: the topology of K₄ compels it. Second, the conjunction of C1 (binary pulsation: existence is change) and C3 (non-decomposability: existence cannot be factored into independent sub-existences) imposes that each leakage cycle of length N is irreducible if and only if N is prime. A composite leakage cycle would decompose into independent sub-cycles, violating C3. The residual incoherence of the universe — the cosmological constant — is irreducible. Third, the three cycle lengths are counted exactly by the proton quintuplet through the coupling structure between K₄ and the quark sectors: k₁ = nd − nᵤ + Rₑ − 1 = 9 (the asymmetric deficit of the d-sector), k₂ = nᵤ − Rₑ + 1 = 19 (the surface of relational possibility of K₄ for the u-sector), and k₃ = Rₑ · nd = 168 (the total K₄-to-d coupling surface — the rule of the game). These are exact integer identities, verified by exhaustive arithmetic with zero exception, satisfying the completeness condition k₁ + k₂ = nd. Fourth, the resulting formula C = (1−κ) ^p₁₆₈ × (Rᵥal/Rₜot) ^p₉ × (1−ηL) ^p₁₉ = (1−κ) ⁹97 × (930/11017) ²3 × (1−ηL) ⁶7 where pₖ denotes the k-th prime, reproduces the observational bound C ≈ 8. 158 × 10⁻⁴⁶ to 0. 17 ppm — within the 1. 7% uncertainty on Λₒbs (Planck 2020). Replacing any prime exponent by its nearest composite neighbour degrades the agreement by at least four orders of magnitude, confirming that the prime structure is structurally necessary, not incidental. No free parameter enters the formula. This result is presented as Conjecture PDL-C: a strong conjecture with full structural justification from C1–C4. The unique remaining step — the formal identification of the three leakage base quantities (1−κ), Rᵥal/Rₜot, and (1−ηL) as the natural representatives of the three independent cycles — is identified precisely and constitutes the content of a future document. The smallness of the cosmological constant is not a fine-tuning accident: it is the product of 997 + 23 + 67 = 1087 elementary irreducible leakage events across three orthogonal sectors, each governed by the topology of K₄, the quark asymmetry of the proton, and the non-decomposability of existence.