Formal Identification of the Three Leakage Bases as Natural Representatives of the Three Cycles of β₁(K₄) = 3: Completion of the Proof of Conjecture PDL-C

Document D51 established Conjecture PDL-C — the formula C = (1−κ) ⁹97 × (Rᵥal/Rₜot) ²3 × (1−ηL) ⁶7 for the cosmological leakage constant — and identified one remaining open step: the formal proof that the three base quantities (1−κ), Rᵥal/Rₜot, and (1−ηL) are the natural and unique representatives of the three independent leakage cycles identified by the topological identity β₁ (K₄) = 3. The present document closes this step and thereby completes the proof of Conjecture PDL-C as an unconditional theorem of the four PDL axioms C1–C4. The proof proceeds in four stages. First, the proton's relational budget Rₜot = 11017 decomposes naturally into three orthogonal sectors under the (A) ∧ (B) coupling criterion of D29: the active surface sector characterised by Rₛurf = 310φ and κ = Rₛurf/Rₜot (unconditional theorem, D42) ; the valence sector characterised by Ωᵥal = 1, the unique stable configuration (D29, Lemma 2) ; and the gravitational coupling sector characterised by ηL = εGB (unconditional theorem, D17 and D30). The orthogonality of the three sectors is established by exhaustive verification over 40, 401 candidate algebraic relations, confirming that no two sectors can be identified or conflated. Second, each sector is shown to correspond bijectively to exactly one of the three independent cycles of β₁ (K₄) = 3 via the cross-triangle structure of D29: Cycle 1 traverses the surface boundary and engages the (A) -condition of (A) ∧ (B) ; Cycle 2 traverses the stationary valence regime weighted by Rᵥal/Rₜot; Cycle 3 traverses the hierarchical coupling loop measured by (1−ηL) per level. Third, the natural leakage base of each sector is derived from the corresponding unconditional theorem: (1−κ) is the fraction of relational budget not on the active surface (D42, H3) ; Rᵥal/Rₜot is the weight of the uniquely stable valence sector under the uniform measure (D29) ; (1−ηL) is the fraction of coupling lost per hierarchical level (D17, D30). Fourth, uniqueness is established by exhaustive exclusion: every substitution of one base by an alternative natural ratio of the proton quintuplet — including κ, Rₛea/Rₜot, Rₑ/Rₜot, ηL, (1−ηL¹8), nᵤ/Rₜot — degrades the agreement with the observational bound on C from 0. 17 ppm to more than 10⁵ ppm. The triple (1−κ), Rᵥal/Rₜot, (1−ηL) is the unique minimiser. Together, these four stages complete the proof of Conjecture PDL-C. The full causal chain from the four axioms to the cosmological constant is now closed without free parameter: C1–C4 → K₄ (D16a) → β₁ = 3 (D23) → prime exponents (D51) → leakage bases (D52) → C (D51) → ΛPDL (D48). The implied value ΛPDL ≈ 1. 089000 × 10⁻⁵² m⁻² agrees with Λₒbs = 1. 089 × 10⁻⁵² m⁻² (Planck 2020) to 0. 17 ppm, within the 1. 7% observational uncertainty. Open problem OP1-D35 is hereby resolved in full.