Finite Relational Mechanics on AT4val[60,6]
Year: 2026
Type: article
Abstract: Finite Relational Mechanics on AT4val[60,6] ( HOG 52002 )develops a proof-carrying combinatorial mechanics on the 60-vertex tetravalent graph AT4val[60,6], using its intrinsic quotient, covering, incidence, symmetry, holonomy, and closure structure. The construction begins from the Petersen-rooted quotient G₁₅ ≅ L(Petersen), where the transport-induced sector incidence matrix satisfies the exact identity MMᵀ = A³ + 2A² + 2I. An intrinsic complementary pair of six-pentagon factors determines a finite V₄-connection family of 2¹⁰ = 1024 gauge-fixed lifts. Exactly one member has forty triangles, and an explicit graph isomorphism identifies it with L(CDC(Dodecahedron)) ≅ AT4val[60,6]. A derived twelve-section structure factors equivariantly as X₁₂ ≅ X₃ × X₄ With symmetry S₃ × D₈. Its 160 native triangles split into 88 flat and 72 twisted classes with exact binary closure law ω(τ) = 1 ⇔ cₛ(τ) = c_Q(τ), and lifted closure depth d(τ) = 3(1 + ω(τ)). The binary triangle charge admits an edge reconstruction dδ = ω, unique up to the expected 11-dimensional vertex gauge, together with the proved finite normal form δ ∼ 1 + Bₛ + B_Q. The same binary closure class admits exactly two nontrivial symmetry-compatible integer refinements within the complete declared eight-character family. These refinements are inequivalent under overall sign plus integer vertex gauge and induce parent-relative signed polarizations on the twisted sector. Gauge minimization yields exactly two minimum-support representatives of weight 18, each isomorphic to the hexagonal prism C₆ □ K₂. Each prism admits a unique native C₃-equivariant one-factorization and an exact phase-resolved twisted-triangle syndrome 24 = 16 + 8. The paper also develops a constructed F₂³ cube-tetrahedron macrogeometry, a registered five-frame recurrence with exact projected-to-complete split 10 → 20, and a general centered relational magnitude Rₙ on declared finite real-valued registers. All results are finite and combinatorial. No physical time, spacetime, action scale, Planck constant, or empirical correspondence is claimed. The emphasis is on exact finite structure, explicit theorem boundaries, gauge and projection distinctions, and reproducibility through machine-checkable certificates and artifacts.
Author: Cave, Scott Allen
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7203985686