Emergent Spin–2 Gravity from an E₈ Contact Lattice: Rank–Four Carrier Selection, Channel Reconstruction, and Exact Finite-Spacing Gauge Symmetry

Abstract

A self-contained finite-resolution construction of massless spin–2 gravity from the full E₈ root shell. Instead of assuming a D₄ ⊕ D₄ carrier, the construction begins with a reverse selection problem over root-generated orthogonal rank-four adapters. Exact enumeration leaves two saturated classes, D₄ ⊕ D₄ and 4A₁ ⊕ 4A₁. The latter is retained as an adversarial control: it survives metric-rank, Lorentz-feasibility, quartic and one-loop Newton-sign tests. Two explicitly stated physical selectors—maximal saturated contact capacity and connected single-sector local channel dynamics—independently select D₄ ⊕ D₄. The internal gauge algebra is reconstructed from oriented microscopic channel semantics rather than postulated Lie generators. Root-supported composition, opposite-channel toral closure, crystallographic Cartan action, root-string normalization and the Chevalley–Serre relations yield the unique complex algebra associated with the selected D₄ root system, namely so(8,C); positivity and norm-preserving transport select the compact so(8) real form. Downstream, the E₈ contact map B : R²⁴⁰ → Sym²(R⁴) has rank 10 and a 230-dimensional kernel, producing the metric sector as an exact quotient of the microscopic state space. The projected lattice fixes the native cochain spacing, canonical bond weights are derived from the contact quotient, and the finite-spacing spin–2 kernel possesses exact linear Ward and BRST symmetry. This protects the massless graviton branch at finite lattice spacing. The functional-determinant sign convention is derived directly from bosonic and Grassmann path integrals, and a representation-resolved heat-kernel ledger yields the induced Einstein–Hilbert term and the correct one-loop Newton sign. The construction further includes a finite microscopic measure and gauge quotient, a van Hove thermodynamic limit under stated hypotheses, an infrared Symanzik continuum expansion, quantitative lattice-anisotropy bounds, an interacting Adler sign certificate in the explicitly evaluated equal-gap branch, and hidden/visible gauge-sector consistency tests. Several negative results are kept as explicit no-go theorems. Contact capacity alone does not select Lorentz signature; exact quartic Lorentz invariance is impossible for the nearest-neighbour carrier; a naive fixed-ledger a → 0 limit sends the induced Newton constant to zero; the spectral-current identity ρ|v_g| = 1/π does not imply a residue 1/π for each finite spectral mode or a π⁻²⁴⁷ vacuum factorization; and the observed cosmological constant is not claimed to be predicted. The accompanying reproducibility package contains the complete LaTeX source, independent verification programs, machine-readable claim and primitive registries, certificates, release audits, and SHA-256 integrity manifests.