The Graviton on Graph HOG52002

The Graviton on Graph HOG52002 develops a finite mathematical framework for relational geometry, standing-wave coherence, and registered temporal evolution on the Thalean Transducer AT4val[60,6]. The work investigates whether coherent gravitational excitations can be represented through the native symmetry, quotient structure, and relational dynamics of a finite graph.

The construction begins with a 60-vertex, 120-edge graph possessing 40 triangular substructures and an automorphism group of order 480. Its native symmetry includes six cyclic subgroups of order five. From this carrier, the research develops a registered five-frame G9000 construction comprising 9,000 states, with exact finite return relations:

F⁵ = K

F¹⁰ = K²

F²⁰ = I

Across 24 registered constructions, the transport action decomposes into 450 free cycles of length twenty. The fivefold harmonic representation separates into one scalar sector and two real two-dimensional conjugate sectors:

1 ⊕ 2 ⊕ 2

This provides a finite setting for studying coherent recurrence and rank-two transformation laws.

The gravitational interpretation is motivated by the relationship between two G60 carriers and their quotient geometry. In this approach, a candidate gravitational polarization is associated with a relational response of the paired system rather than with an independently assumed gravitational field. The symmetric trace-free rank-two representation supplies a precise transformation criterion, while the existence of a native geometric polarization map remains a separate mathematical requirement.

The proposed polarization must satisfy the equivariance relation:

Qₚₒₗ(g · d) = ρ₂(g) Qₚₒₗ(d)

The fivefold tensor character is related to the geometric vector character by:

χₜₑₙₛₒᵣ = χᵥₑ꜀ₜₒᵣ²

The work also examines normalization-transport defects in the G1800 quotient constructions. These corrections are gauge-valued and admit an ambient section-coboundary description. Their exactness within the admissible gauge coefficient system remains under investigation. This distinction separates presentation-dependent corrections from potentially persistent relational invariants and prevents gauge artifacts from being mistaken for physical curvature.

The accompanying computational evidence includes certified graph data, native transport constructions, Chinese remainder coordinate calculations, quotient and cocycle certificates, and reproducibility material. The graph census and registered G9000 transport calculations have been independently reproduced; the stronger gauge-module exactness claim and the proposed gravitational polarization remain research targets.

The principal contribution is a reproducible finite framework in which graph symmetry, spectral closure, quotient transport, and encounter-relative registration can be studied together. The proposed graviton interpretation is presented as a candidate correspondence, not as an established identification with a physical massless spin-two particle.

The broader research program investigates whether relational geometry and retained encounter history can support a discrete account of gravitational response and emergent temporal structure.

KEYWORDS

Thalean Mechanics; HOG52002; AT4val[60,6]; Graviton; Discrete Gravity; Relational Geometry; Algebraic Graph Theory; Graph Automorphisms; Spectral Graph Theory; Standing Waves; Finite Group Representations; Cyclic Transport; G60; G900; G9000; Rank-Two Tensor Representations; Gauge Cohomology; Relational Time; Computational Mathematical Physics.

RESEARCH CLASSIFICATION

Mathematical Physics; Discrete Geometry; Algebraic Graph Theory; Computational Mathematics; Foundational Physics.

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