Relational Geometric Mechanics XIV: Graviton Dispersion — α = 1/28 from the Softmax Attention Mechanism over the 13-Dimensional Descriptor Space

This paper derives the graviton dispersion coefficient α introduced in Document 22 and left as "O(1) from the attention-lattice dynamics" in Paper XIII. The result is exact and parameter-free: α = 1/28. The derivation proceeds from a single observation: RGM is background-independent. The foam geometry is not a pre-existing crystal — it is generated dynamically by the attention mechanism. The Gaussian kernel width σ used in Paper XIII cannot be an external parameter; it must emerge from the variance of the softmax attention distribution over the descriptor space. For unit-norm descriptor vectors uniformly distributed on S^(d-1) with d = 13 (the dimension of the SU(3)×SU(2)×U(1) gauge group generators plus the gravitational amplitude component), the softmax variance theorem gives σ² = 1/(d+1) = 1/14 exactly. Therefore α = σ²/2 = 1/28 exactly. The complete graviton dispersion relation is E² = p²c² + (1/28)·p⁴·L_QF²/ℏ². The convergence constant of Paper XIII is simultaneously fixed to C = 1/28. Consistency with GW170817 is confirmed by 31 orders of magnitude. A testable prediction follows: a 251 TeV photon from a 600 Mpc source should arrive approximately 11 days later than a low-energy counterpart, due to the RGM dispersion — measurable by next-generation multi-messenger instruments (Einstein Telescope, LHAASO). The connection to Ng and Steinbring (2025), who confirmed the holographic foam scaling δl ~ l^(1/3)·L_QF^(2/3) using GRB 221009A, is established: their position-space holographic result and the RGM momentum-space dispersion coefficient are complementary descriptions of the same foam structure, related by Fourier transform. With α = 1/28, the RGM quantum gravity framework contains zero free parameters. All fundamental constants of the gravitational sector — G_N, α, and the continuum limit convergence rate — are derived from the five postulates of Paper I. Builds on Papers I–XIII of the RGM series.