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 D₇₎₌ = 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 c (n₃) = 3 from quaternionic minimality; papers O28–O29 establish that the admissible projection space H₄₅₅ = C³ has rank r₄₅₅ = 3 and is the spin-1 symmetric square H₄₅₅ Sym² (V_) of a spin-12 doublet V_ (d_ = 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 H₄₅₅ Sym² (V_) as an su (2) -module (the spin-1 representation), and that the irreducibility of Sym² (V_) 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_) 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 ₂ (L₄₅₅). The isotropy condition AH = Aᵦ has since been resolved by a complementary route: Q8 proves Aᵦ = Cₒₔ (₂) = 2 via Casimir rigidity on Sym² (V_), and Q10 establishes AH 2 via spectral universality, giving AH = Aᵦ = 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 Lₖ₄₈₋ 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 - Aᵦ| decreases monotonically with q in the low-energy sector. The four-point power-law fit gives |AH - Aᵦ| 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 = Aᵦ, reducing the identification problem to a single scalar convergence.