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 $S^2$ and $S^3$, 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, $\ln(M_{\mathrm{Pl}}/M_W) = 8\pi^2 \sum_{l=1}^{24} 1/b_l \approx 39.57$, matching the observed value $39.56$ to $0.02%$. The beta coefficients $b_l$ use derived matter content at the first two eigenmode levels and pure Yang--Mills at all higher levels. The coupling $\alpha_{\mathrm{GUT}} = 1/(4\pi)$ follows from Hilbert--Schmidt uniqueness on $M_2(\mathbb{C})$, giving the prefactor $8\pi^2$ as the instanton action at unit topological charge with the derived coupling $g^2 = 1$. Four structural supports constrain the cutoff $L = 24$: the factorial $L = d! = 4!$ of the derived spacetime dimension; the identity $2L+1 = b_1^2 = 49$, where $b_1 = 7$ is the derived QCD beta coefficient; the arithmetic relation $b_1^2 = 2d! + 1$ holding only at $d_1 = 3$, $d = 4$; and the quasi-periodicity of the octahedral decomposition of spherical harmonics, giving $L = \chi(S^2) \times \mathrm{exp}(O) = 2 \times 12$. The effective beta coefficient of the cascade equals the first Laplacian eigenvalue on $S^2$: $b_{\mathrm{eff}} \approx 2 = \lambda_1$, giving $\ln(M_{\mathrm{Pl}}/M_W) \approx 4\pi^2$ 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 $l = 2 \to 2+3$ 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 $l \geq 3$ confine into stable, SM-invisible glueballs. The standing wave from the resonance condition, with each dark excitation weighted by $\sqrt{\lambda_1(S^3)} = \sqrt{3}$ and each SM excitation by $\sqrt{\lambda_1(S^2)} = \sqrt{2}$ (reflecting unbroken vs.\ broken gauge dynamics on the Hopf-connected spaces), gives $\Omega_{\mathrm{DM}}/\Omega_b = 5.36$, 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 --- $\sum 1/b_l = 1/\lambda_1$, where $\lambda_1 = 2$ is the first eigenvalue of the Laplacian on $S^2$ --- which selects $L = 24$ as the unique integer solution. Equating the hierarchy exponent with the ground-state eigenvalue of the cascade's standing wave gives $b_{\mathrm{eff}}^3 = 8$, hence $b_{\mathrm{eff}} = 2 = \lambda_1$ 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.