Asymptotic su(2)-Isotropy of the Effective Quadratic Form and the Identification AH = 2

Abstract

Papers Q7 Q9 of the Cosmochrony spectral geometry programme establish that the effective operator $L_{\mathrm{eff}}$ on $\mathbb{R}_\tau \times {\operatorname{Heis}}_3(\mathbb{R})$ has a principal symbol $\sigma_2(L_{\mathrm{eff}})|_{H_{\mathrm{eff}}} = A_H(k_X^2+k_Y^2) + A_Z k_Z^2$ with $A_H, A_Z > 0$ and $A_Z = 2 = {C_{\mathfrak{su}(2)}}$ (Casimir eigenvalue on ${\operatorname{Sym}}^2(V_\rho)$, Q8). The identification $A_H = 2$ would complete the effective metric to $g^{\mu\nu} = \mathrm{diag}(-A_\tau, 2, 2, 2)$ and establish the isotropic Lorentzian geometry of the Cosmochrony framework. The present paper proves $A_H \to 2$ from two structural inputs: (i) the asymptotic character-independence of the O-series spectral observables ($\sigma_c(n) \to \sigma_*(n)$ uniformly in $c$ as $q \to \infty$, supported structurally by the conjugate-pair equality $\sigma_c = \sigma_{q-c}$ (the conjugation identity, O17; recalled in O18 Theorem 3.1) and confirmed numerically by the O25 campaign ); and (ii) the uniqueness of the $\mathfrak{su}(2)$-invariant quadratic form on ${\operatorname{Sym}}^2(V_\rho)$ (Q7 Lemma 4.3 ). The core argument is that character-independence forces the effective form on ${H_{\mathrm{eff}}}$ to be scalar under the $\mathfrak{su}(2)$ action, and the unique such form is the Casimir with value 2. The result is established conditionally on the O-series universality (structurally supported, numerically confirmed for $q \leq 211$) and the bridge non-obstruction hypothesis of Q7 Q9.