L(6+7): Exact 1⊕6 Spectral Reduction, Discrete Vacuum Topology, and Transverse Response in a Seven-Field Scalar–Tensor Effective Field Theory

This preprint presents a self-contained theorem-level formulation of the local seven-field scalar–tensor core of the L(6+7) research program. The paper derives an exact radial–angular 1⊕6 decomposition on the stationary all-components-nonzero branch, explicit Einstein-frame kinetic and Hessian projectors, closed canonical masses, the rational symmetry-breaking point F*=100/121, a disconnected 2^7=128 vacuum set, and a complete classification of the physical inverse roots in the light-radial regime. A separately specified transverse response sector is constructed from a covariantly conserved coexact current and a positive spectral representation. Within the stated local microscopic class, the pair-response carrier reduces to a conserved Dirac vector-current representation. The same field-space geometry determines the local tree-level r+a→r+a amplitude and its P-wave projection. At the reference point the dimensionless finite-energy tree zero is sqrt(s_zero)/sqrt(s0) = 1.141899082719035. This number is deposited as a dimensionless theoretical result; no identification with a named experimental resonance is claimed without an independently fixed physical scale and channel-specific matching. The record includes the full paper source and an independent machine-readable reproduction package. The present work consolidates and sharpens the local mathematical core developed across the earlier published L(6+7) corpus, which is linked through the Related works metadata and cited in the paper.