Mass Scales from the Minimal Quantum State Space

In companion papers — Eigenmode Ratios of the Minimal Quantum State Space and the Standard Model (Paper 1) and Further Consequences of the Minimal Quantum State Space (Paper 2) — we showed that the eigenmode spectra of S² and S³, the state space and symmetry group of the minimal quantum system, produce the Standard Model gauge group, chiral fermion content, Higgs mechanism, Yang–Mills and conformal gravity dynamics, mixing angles, and 4-dimensional Minkowski spacetime, all from two inputs with zero adjustable parameters. Here we address the question those papers left open: can the framework also account for mass scales? We report a zero-parameter formula for the electroweak hierarchy, ln (Mₚₗ/MW) = 8π² Σ (l=1 to 24) 1/bₗ ≈ 39. 57, matching the observed value 39. 56 to 0. 02%. The beta coefficients bₗ use derived matter content at the first two eigenmode levels and pure Yang–Mills at all higher levels. The coupling αGUT = 1/ (4π) follows from Hilbert–Schmidt uniqueness on M₂ (ℂ), giving the prefactor 8π² as the instanton action at unit topological charge with the derived coupling g² = 1. Four structural supports constrain the cutoff L = 24: the factorial L = d! = 4! of the derived spacetime dimension; the identity 2L+1 = b₁² = 49, where b₁ = 7 is the derived QCD beta coefficient; the arithmetic relation b₁² = 2d! + 1 holding only at d₁ = 3, d = 4; and the quasi-periodicity of the octahedral decomposition of spherical harmonics, giving L = χ (S²) × exp (O) = 2 × 12. The effective beta coefficient of the cascade equals the first Laplacian eigenvalue on S²: bₑff ≈ 2 = λ₁, giving ln (Mₚₗ/MW) ≈ 4π² as a leading-order approximation. Supporting results include the first coupling constant from geometry, a second route to the gauge groups via the McKay correspondence, the octahedral l = 2 → 2+3 splitting, and a proof that the eigenmode tower's massive vectors produce the correct sign for Sakharov induced gravity at l = 4 — the same level where the regular representation of the octahedral group first completes. The eigenmode tower also predicts dark matter: the 22 pure Yang–Mills sectors at l ≥ 3 confine into stable, SM-invisible glueballs. The standing wave from the resonance condition, with each dark excitation weighted by √ (λ₁ (S³) ) = √3 and each SM excitation by √ (λ₁ (S²) ) = √2 (reflecting unbroken vs. broken gauge dynamics on the Hopf-connected spaces), gives ΩDM/Ωb = 5. 36, matching the Planck value to 0. 02%. Four negative results establish where the framework stops. The harmonic form is supported by topological independence and eigenspace orthogonality; the cutoff follows from a single resonance condition — Σ 1/bₗ = 1/λ₁, where λ₁ = 2 is the first eigenvalue of the Laplacian on S² — which selects L = 24 as the unique integer solution. A cross-sphere analysis proves that S² is the unique non-degenerate sphere where the resonance sum diverges; for all Sⁿ with n ≥ 3, the eigenmode degeneracies grow too fast and the sum converges below its target, making resonance impossible. Equating the hierarchy exponent with the ground-state eigenvalue of the cascade's standing wave gives (bₑff) ³ = 8, hence bₑff = 2 = λ₁ exactly, with one self-consistency ansatz. However, each confined gauge sector becomes trivially represented in the infrared, so the fully confined tower returns to the same trivial gauge content as l = 0, connecting the cascade to the self-closing conjecture. The hierarchy problem reduces to this ansatz.