What question did this study set out to answer?

The aim is to define the Riemannian geometry of nested hyperspheres within a cosmological context.

June 18, 2026Open Access

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

Key Points

The aim is to define the Riemannian geometry of nested hyperspheres within a cosmological context.
Utilized closed-form Riemannian geometry for $S^3$, $S^4$, and $S^5$.
Explored the breathing law and analyzed its empirical metrics.
Mathematical formalization of the cascade mechanism and Pythagorean triad relationships.
Identified the spatial dimension as $n = 3.0$, assigning the observable slice to $S^3$.
Established a fundamental Pythagorean ratio of 3:4:5 for hyperspheres.
Set up frameworks for validating properties through Baryon Acoustic Oscillations (BAO).

Abstract

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 (S³ S⁴ S⁵) 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 V̇/V = nH. When paired with independent metrics from the Spartan Dawn fit (V̇/V = 0. 207 Gyr^-1, H₀ 0. 069 Gyr^-1), it directly measures the spatial dimension as n = 3. 0. This empirically designates the observable spatial slice as the 3-sphere (S³), while framing S⁴ and S⁵ as structural outer shells rather than extra observable dimensions. The + Cascade Nesting Rule: Formally mathematicalizes the cascade-plus-cutoff mechanism introduced in Paper I via the spectrum shift identity _ (Sⁿ) = _ (S^n-1) +. The Fundamental Pythagorean Triad: Demonstrates that the fundamental gaps stand in a rigid 3: 4: 5 ratio (₁ (S³): ₁ (S⁴): ₁ (S⁵) = 3: 4: 5). This links the nested fundamental tones to the shared S¹ 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 = - c²). 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.

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