The Spatial Integration Coefficient Σ(t): Place, Spatial Entropy, and the Geometry of Lived Space A Structural Extension of the Unified Dynamic Model of Mind

Abstract

The Unified Dynamic Model of Mind (UDMM) has established the temporal topology of the organism via the Temporal Openness Coefficient Θ(t) = B_future/B_past. Yet the spatial dimension of the organism's existence — how it constructs, navigates, and loses the lived space that makes action possible — has remained formally undeveloped. This paper introduces the Spatial Integration Coefficient Σ(t) as the spatial analogue of Θ(t), defined as the ratio of friction-grounded known space to available but unexplored space. Σ is not a geometric or navigational variable — it is a topological-thermodynamic property of the organism's spatial state-space, derived from the same UDMM Master Equation that grounds all prior constructs. We derive six formal results: (1) the Spatial Gap Field R(x,t) = |I_bio(x,t) − Ω_env(x)| as the spatially-indexed generalization of the vital gap r; (2) Place formation as a spatial attractor in the R-field, governed by accumulated biological friction Γ_spatial(x) = ∫F_ric(x,τ)dτ · (1/ζ); (3) Σ(t) = dim(Ω_known)/[dim(Ω_available) + ε] derived from the spatial diffusion term D∇²φ of the Master Equation; (4) the Axiom of Embodied Space — the body is the irreducible primary spatial attractor, so Σ → 0 requires BAC → 0 (dissociation), not merely external confinement; (5) Spatial entropy S_spatial = −Σᵢ p(x_i)log p(x_i) such that Σ ≈ exp(ΔS_spatial), making Σ an entropic quantity; and (6) the Unified Existence Equation I(t) = G_A · Θ_B · Σ · (1 + λ·Θ_R·Awareness·BAC), establishing that behavioral intent is jointly determined by temporal openness and spatial integration. A friction criterion resolves the digital-space problem: virtual environments expand Ω_known via virtual friction (η_virt << 1) but do not expand the physically-reachable Ω_available. We propose a three-type spatial psychopathology taxonomy and derive four falsifiable predictions, including a novel friction-weighted spatial entropy test using GPS + EDA wearable data.