Reconstructing AT4val[60,6] from the Intrinsic Pentagon Factors of L(Petersen)

This paper reconstructs the tetravalent graph AT4val[60,6] from intrinsic pentagonal structure in G₁₅ = L(Petersen). An exhaustive finite census shows that the twelve native pentagons of G₁₅ admit a unique unordered partition into two complementary six-pentagon factors, each covering every edge exactly once. Assigning distinct generating V₄ holonomies to the factors produces 1,024 gauge-fixed four-sheeted connections. Their triangle counts assume seven possible values, and exactly one connection attains the maximum of 40 triangles. An explicit sixty-vertex isomorphism identifies this unique maximal lift with L(CDC(Dodecahedron)) ≅ AT4val[60,6], also catalogued as House of Graphs 52002. Its adjacency spectrum and characteristic polynomial are certified independently. A canonical directional duad frame is then recovered, together with the native action on the unordered pentagon-factor pair. Exact finite path censuses yield three dimensionless invariants: the return ratio 6/5, the turn ratio 7/5, and the depth ratio 5/2. The reconstruction theorem and all three ratios are supported by seven machine-readable certificates and reproducible audit scripts. A proposed role-sensitive metric gives the conditional decomposition 13 = 5 + 8, but its existence, canonicity, and uniqueness remain conjectural. No Euclidean, Planck-scale, or physical interpretation is asserted.