Three Admissible Directions and Three Spatial Dimensions: A Structural Bridge Candidate between Heff ≃ C3 and the Horizontal Geometry of Heis3(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 houses a spin-1/2 trajectory subspace V_ρ with 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\"{o}dinger 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) that simultaneously resolves 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\ confirms the vanishing of cross terms for all tested pairs, and the residual isotropy gap |AH - Az| is observed to decrease monotonically with q in the low-energy sector. All available analytical and numerical evidence is consistent with asymptotic isotropy AH = Az, reducing the identification problem to a single scalar convergence.