Exact Nested-Sphere Geometry and the Breathing Dimension-Counter: S 3 , S 4 , S 5 , the +ℓ Cascade, and the 3:4:5 Triad

Title: Exact Nested-Sphere Geometry and the Breathing Dimension-Counter: S³, S⁴, S⁵ the +ℓ Cascade, and the 3:4:5 Triad Series: TZPID Gold Spine Series II, Paper XIII of XX Abstract This paper supplies the exact closed-form Riemannian geometry for the nested hyperspherical enclosures ($S^3 \subset S^4 \subset S^5$) required by the cosmological framework established in Paper XII. Using standard differential geometry, it treats these hyperspheres as physical cosmological enclosures, uncovering key exact structural and empirical metrics: The Dimension Anchor: Explores the exact breathing law $\dot{V}/V = nH$. When paired with independent metrics from the Spartan Dawn fit ($\dot{V}/V = 0.207\text{ Gyr}^{-1}$, $H_0 \approx 0.069\text{ Gyr}^{-1}$), it directly measures the spatial dimension as $n = 3.0$. This empirically designates the observable spatial slice as the 3-sphere ($S^3$), while framing $S^4$ and $S^5$ as structural outer shells rather than extra observable dimensions. The $+\ell$ Cascade Nesting Rule: Formally mathematicalizes the cascade-plus-cutoff mechanism introduced in Paper I via the spectrum shift identity $\lambda_\ell(S^n) = \lambda_\ell(S^{n-1}) + \ell$. The Fundamental Pythagorean Triad: Demonstrates that the fundamental gaps stand in a rigid $3:4:5$ ratio ($\lambda_1(S^3) : \lambda_1(S^4) : \lambda_1(S^5) = 3:4:5$). This links the nested fundamental tones to the shared $S^1$ Hopf fiber carrying the comma holonomy detailed in Paper III. Higher-Dimensional Friedmann Scaling: Analyzes invariants across dimensions under the generalized breathing law, evaluating matter, radiation, and constant vacuum densities ($p = -\rho c^2$). The paper cleanly demarcates established Riemannian mathematics from the framework’s physical synthesis, setting up acoustic validation via Baryon Acoustic Oscillations (BAO) carried out in Paper XIV.