Cube Space as a Named Coordinate Chart: Re-proof of the 15-Embedding, Visualization Fold, and 37D Gap Bridge

Abstract

Cube Space Design asserted a universal 15-dimensional coordinate system for system state, semantic proximity, a 15D→3D visualization at 99.8% preservation and 95.8% efficiency, incremental frames, and a 69.4× combined improvement via a 37-dimensional gap bridge. Those claims are theorems once the maps are written. This revision supplies the maps and re-proves them. C15=ℝ15 is a named dictionary. Five type mappers land in the same ambient. The process mapper admits a global left inverse on (t, id, cpu), has analytic rank 3, satisfies four image equations, and is proper: it is a smooth embedding of those parameters into the 15-slot writing (Theorem 4.5). Same-identity pairs occupy a metric tube strictly inside random pairs. A visualization filling that is a linear 3-chart plus orthogonal remainder of energy ε=0.002 obeys ρ15→3≥0.998 by Eckart–Young; rank-compression η15→3=1−(3/15)2=0.96 and the product ηρ≈0.958 recover the prior efficiency figure. The 37D object is the concatenation C15⊕F22; projection recovers cube identity, so the pair is a gap bridge. Utility Uη=Ubase×49.5=3.95×1012 holds; the four-factor line 25×3×1.8 equals 1.078×1013 and is a different named functional. Stacked work multiplies (counted-pass ratio 69.75 against formula product 69.74). Parallel wall-clock is not that product. Four laboratory runs return existence_ok=true. An isotropic 15-cloud and a random pad fail the visualization and nearness theorems, as they must. Manuscript 2.2.0 (re-proof edition). Zenodo version 2.0.0. PDF-only. Prior deposit: 10.5281/zenodo.18143028.