Three-Step Path Fingerprints on LPS Graphs and the Capacity Growth Exponent

The Cosmochrony spectral cascade programme targets a capacity growth exponent δ ∈ [7.4,\,10.6], identified in O7 as the condition under which the cascade exponent β falls within the phenomenological window β^* ∈ (0.09, 0.13) fixed by charged-lepton mass ratios. O7 proved that $k = 2$ adjoint fingerprints on LPS graphs X5,q saturate within the first BFS step, and reformulated the target as: find a fingerprint scale and depth k such that the discrete projective capacity decays as Σ̄ₙ ~ p(n)-δ with δ ∈ [7.4,\,10.6] in a measurable pre-saturation window.The present paper constructs the $k = 3$ path fingerprint using the permutation representation of G = PSL(2, Fq) on P¹(Fq). Each non-backtracking 3-step path v₀ → v₁ → v₂ → v₃ is encoded by the tensor product π₃(v₀ → v₃) = ρₚerm(v₁) ⊗ ρₚerm(v₂) ⊗ ρₚerm(v₃) ∈ R(q+1)³, giving ambient dimension (q+1)³ = O(q³). The saturation threshold is predicted at |S^*| = O(q²) in vertex count, the first growing window in the O-series. On LPS graphs, exponential shell growth (|Sₙ| ≈ pⁿ) compresses this O(q²)-vertex window to only O(log q) BFS steps, which prevents reliable exponent extraction at the group sizes tested.We test the state law Rₙ⁽ᵏ⁾ ≈ Φ(ηₙ) of O7 at this fingerprint level on X5,q for q ∈ \13, 17, 29\, extract the capacity growth exponent δ from the log-log slope of Σ̄ₙ vs.\ $p(n)$, and test whether δ ∈ [7.4,\,10.6]. We also update the O-series status table and identify the remaining open directions toward a first-principles derivation of δ.