HCSM-47 Graviton Self-Coupling from D4 The Derivative Structure of the Substrate and the Pure Polynomial and Derivative Included Counts
Year: 2026
Type: preprint
Abstract: The Hodge Complex Standard Model identifies the graviton as the self paired curvature mode of the Hodge complex on the periodic 8 × 8 torus: the symmetrized bilinear h = ω ⊗ ω in the substrate 1-form, projected onto the B1 ⊕ B2 isotypic piece of Sym2(Ω1|λ=4), with po- larization space V = B1 ⊕ B2 (HCSM-45). This paper establishes the self-coupling structure of the graviton by computing the D4-invariant content of the graviton vertex spaces. We prove: (i) the pure-polynomial A1 counts of Symn(V ) for n = 1, 2, 3, 4 are 0, 2, 0, 3, forbidding pure- polynomial kinetic and cubic graviton self-couplings; (ii) the derivative-included A1 counts with two derivatives are 5,12,11 at orders 2,3,4, establishing the representation theoretic content required for the kinetic, cubic, and quartic vertices; (iii) the odd-derivative selection rule: the A1 content vanishes for odd numbers of derivatives, as a consequence of the faithful action of D4 on the derivative irrep E; (iv) the D4 action on ker D ∼= 2A1 ⊕ E embeds as a finite subgroup of O(3,1), so D4-invariant structures are a subset of Lorentz-invariant structures. The results re- produce the derivative structure of the Einstein–Hilbert expansion: the pure-polynomial kinetic and cubic vertices vanish, and the physical vertices carry exactly two derivatives. The graviton is a composite bilinear, and its nonlinearity arises from the composite structure of h = ω⊗ω, not from the Faddeev–Popov measure, which is field-independent on the discrete torus. All claims are HCSM-native and verified at machine precision.
Author: Preschutti, Stanley
Institution: Information Physics Institute
Language: English
Subfield: Astronomy and Astrophysics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214729783