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

The Cosmochrony spectral cascade programme targets a capacity growth exponent 7. 4, \, 10. 6, identified in O7 as the condition under which the cascade exponent falls within the phenomenological window ^* (0. 09, 0. 13) fixed by charged-lepton mass ratios. O7 proved that k = 2 adjoint fingerprints on LPS graphs X^5, 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 ₙ p (n) ^- with 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 ₃ (v₀ v₃) = ₚerm (v₁) ₚerm (v₂) ₚerm (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 (q) BFS steps, which prevents reliable exponent extraction at the group sizes tested. We test the state law Rₙ^ (k) (ₙ) of O7 at this fingerprint level on X^5, q for q \13, 17, 29\, extract the capacity growth exponent from the log-log slope of ₙ vs. \ p (n), and test whether 7. 4, \, 10. 6. We also update the O-series status table and identify the remaining open directions toward a first-principles derivation of.