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

We propose a conservative and falsifiable framework in which a coarse‑grained cross‑cap/handle density(ρ_cap) acts as a physical state variable. We make no claims of modified gravity or superluminal effects;all predictions are laboratory‑scale and pre‑registered. To keep Bianchi/conservation and standard testsintact, we use a canonical scalar χ (mass dimension 1) with ρ_cap = κχ (κ has dimension L⁻² in c = ħ = 1), andthe Lagrangian density has units L⁻⁴.Motivation. Recent evidence shows that high‑level visual cortex representations are aligned with largelanguage‑model (LLM) embeddings, supporting a shared representational format that our state‑variablehypothesis can connect to engineered non‑orientability (Nature Machine Intelligence, 2025).Decisive test (pre‑registered). In a three‑level non‑orientability design (τ ∈ {0,1,2,3}) crossed with threemicro‑lattice pitches a ∈ {0.6, 0.8, 1.2} mm, we predict to first order that Δ(1/Q) ∝ τ/a³ (measured via twoindependent resonator methods), and that the lateral heat‑flow anisotropy A_L increases monotonicallywith τ. We further require cross‑channel ranking consistency between the Q and A_L trends (Spearman ρ≥ 0.9). SOPs, controls, and preregistration are specified for blinded replication.