Three Admissible Directions and Three Spatial Dimensions: A Structural Bridge Candidate between H_eff C^3 and the Horizontal Geometry of Heis_3(R)

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 Dₕₒₘ = 4 of Heis₃(Z/qZ) (Bass–Guivarch), with three spatial directions: the two horizontal generators X, Y and the central element Z = [ X, Y] (the last entering via sub-principal corrections, open problem Q5b-O2). Paper O23 derives Σc(n₃) = 3 from quaternionic minimality; papers O28–O29 establish that the admissible projection space Heff = C³ has rank reff = 3 and is the spin-$1$ symmetric square Heff Sym²(V_ρ) of a spin-1/2 doublet V_ρ (d_ρ = 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: su(2) Im\,H is semisimple, while heis₃ is nilpotent. Second, we show that Heff Sym²(V_ρ) as an su(2)-module (the spin-$1$ representation), and that the irreducibility of Sym²(V_ρ) 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 su(2) Casimir restricted to Sym²(V_ρ) 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 (± 1, 0) matching the Carnot grading $(1, 2)$ of heis₃ 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 σ₂(Leff). The isotropy condition AH = Az has since been resolved by a complementary route: Q8 proves Az = Cₛᵤ₍₂₎ = 2 via Casimir rigidity on Sym²(V_ρ), and Q10 establishes AH → 2 via spectral universality, giving AH = Az = 2 independently of Q5b-O2. A new analytical result completes the picture: we prove that the chirped discrete Fourier transform Fc (the metaplectic representative of the U(1) rotation) commutes with LWeil for all $(q,c)$, which implies that the cross-term part of the criterion holds structurally. Numerical verification on O25 checkpoints for q ∈ \61, 101, 151, 211\ confirms the vanishing of cross terms for all tested pairs, and the residual isotropy gap |AH - Az| decreases monotonically with q in the low-energy sector. The four-point power-law fit gives |AH - Az| ≈ 0.364\,q-0.52 (R² = 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 AH = Az, reducing the identification problem to a single scalar convergence.