Minimal scalar completion of a coarse‑grained cross‑cap density (ρ_cap): theory and a tabletop test

We present a minimal, falsifiable framework in which a coarse‑grained cross‑cap/handle density ρ_cap actsas a physical state variable coupled to gravity and to coarse‑grained entropy production. To preserve theBianchi identity and standard tests, we adopt a canonical scalar χ (mass dimension 1) and identify ρ_cap =κχ with [κ] = L⁻² in natural units (c = ħ = 1). The Lagrangian density carries [ℒ] = L⁻⁴ and energy–momentumis covariantly conserved. We provide conservative constraints (c_GW = c; PPN γ,β ≈ GR), a reproducibleKlein³ metamaterial experiment with non‑orientability index τ, simulations, and a preregistered analysis(ANCOVA + Jonckheere–Terpstra). All CAD/data/code are slated for open release.