Three Admissible Directions and Three Spatial Dimensions: A Structural Bridge Candidate between Heff ≃ C³ and the Horizontal Geometry of Heis3(R)

Abstract

Two independent sub-programmes of the Cosmochrony framework independently produce the integer three as a structural output. Paper Q5b derives a four-dimensional effective Lorentzian geometry from the homogeneous dimension ${D_{\mathrm{hom}}} = 4$ of ${\mathrm{Heis}_3}(\mathbb{Z}/q\mathbb{Z})$ (Bass–Guivarch), with three spatial directions: the two horizontal generators $\tilde X, \tilde Y$ and the central element $\tilde Z = [\tilde X, \tilde Y]$ (the last entering via sub-principal corrections, open problem Q5b-O2). Paper O23 proves that the neutral sector of a supplied spinor carrier $V_\rho \cong \mathbb{C}^2$ is ${\mathfrak{su}}(2) \simeq {\mathrm{Im}\,\mathbb{H}}$, of real dimension exactly three (Theorem 3.1); the threshold $\Sigma_c(n_3) = 3$ is a supplied selection rule, with the carrier selection and the identification of $\Sigma_c$ with that dimension open; papers O28–O29 establish that the admissible projection space ${H_{\mathrm{eff}}} = \mathbb{C}^3$ has rank $r_{\mathrm{eff}} = 3$ and is the spin-$1$ symmetric square ${H_{\mathrm{eff}}} \simeq {\mathrm{Sym}}^2({V_\rho})$ of a spin-$\tfrac{1}{2}$ doublet ${V_\rho}$ ($d_\rho = 2$). We identify a representation-theoretic mechanism that constrains any possible bridge between these two "3"s. First, we prove that the bridge cannot be a Lie algebra isomorphism: ${\mathfrak{su}}(2) \simeq {\mathrm{Im}\,\mathbb{H}}$ is semisimple, while ${\mathfrak{heis}_3}$ is nilpotent. Second, we show that ${H_{\mathrm{eff}}} \simeq {\mathrm{Sym}}^2({V_\rho})$ as an ${\mathfrak{su}}(2)$-module (the spin-$1$ representation), and that the irreducibility of ${\mathrm{Sym}}^2({V_\rho})$ forces any equivariant map to the spatial sector to be unique up to positive scalar (Rigidity Lemma via Schur). Third, we prove a Schr\"odinger Quadratic Form Lemma: the ${\mathfrak{su}}(2)$ Casimir restricted to ${\mathrm{Sym}}^2({V_\rho})$ pulls back, under the bridge map, to a positive-definite rank-$3$ quadratic form on the spatial block of the effective symbol, with the spin-weight grading $(\pm 1, 0)$ matching the Carnot grading $(1, 2)$ of ${\mathfrak{heis}_3}$ in a precise sense. These results stop short of identifying the two "3"s but they uniquely constrain the form any identification must take, and reduce the open question to a single computable criterion on $\sigma_2(L_{\mathrm{eff}})$. The isotropy condition $A_H = A_z$ has since been resolved by a complementary route: Q8 proves $A_z = C_{\mathfrak{su}(2)} = 2$ via Casimir rigidity on ${\mathrm{Sym}}^2({V_\rho})$, and Q10 establishes $A_H \to 2$ via spectral universality, giving $A_H = A_z = 2$ independently of Q5b-O2. A new analytical result completes the picture: we prove that the chirped discrete Fourier transform $F_c$ (the metaplectic representative of the $\mathrm{U}(1)$ rotation) commutes with $L_{\mathrm{Weil}}$ for all $(q,c)$, which implies that the cross-term part of the criterion holds structurally. Numerical verification on O25 checkpoints for $q \in \{61, 101, 151, 211\}$ confirms the vanishing of cross terms for all tested pairs, and the residual isotropy gap $|A_H - A_z|$ decreases monotonically with $q$ in the low-energy sector. The four-point power-law fit gives $|A_H - A_z| \approx 0.364\,q^{-0.52}$ ($R^2 = 0.98$), consistent with $O(q^{-1/2})$ convergence as predicted by Q5b Theorem 3.2. All available analytical and numerical evidence is consistent with asymptotic isotropy $A_H = A_z$, reducing the identification problem to a single scalar convergence.