Construction and geometric properties of the fuzzy $C{P}^3$ onion

We consider the matrix model of a seven-dimensional quantum space built from concentric fuzzy $C{P}^3$’s of increasing matrix size, and show that its continuum limit is the metric cone $C(C{P}^3)$ over $C{P}^3$, not flat $R^7$. Every ingredient is fixed before any comparison, so this geometry is a prediction rather than an input. The two geometries agree in the $s$ wave ($p=0$) but separate in the higher angular sectors (harmonics on $C{P}^3$ of degree $p\ge 1$): in each sector tested, the spectrum of the free Laplacian converges at second order to that of the cone, and at 20480 layers the cone is matched to $1\times {10}^{-9}$ for $p\le 8$, whereas flat space misses by 8-18%. The free propagator converges to that of the cone at the same order, and heat diffusion reproduces the $r^6$ measure and the Bessel spectrum of the cone. The maps between neighbouring layers are obtained in closed form from the polar decomposition of the second-quantized Schwinger-boson lift, using a known operator identity, and coincide with the Kováčik-Tekel coefficient-copying prescription. For $C{P}^1$ the construction reduces, away from the outer boundary, to the Kováčik-Tekel fuzzy onion. Modes along the twistor fibre are exactly degenerate with the harmonics on $S^4$.