TCFQ as an Effective Matter Model on Synthetic Hyperbolic Networks

We analyse the torsion-coupled field model with constitutive Dirac fermion (TCFQ) on a fixed hyperbolic background H4 and examine to what extent its mass spectrum can be recovered on synthetic hyperbolic networks of type {3, z}. In the continuum, the model admits a unique BPS–RG fixed point with constitutive relations λ = g2/10, κ2 = g2/5, v2 = 5/g2, yielding the mass ratio mA/mH = √5 and purely axial on-shell torsion. We prove that exact supersymmetry is incompatible with this fixed point (λ ̸= g2/2), and that the cosmological constant problem manifests as an unavoidable free parameter Λbare ∼ −3 + 5/(2g2). For the discrete analogue, we formulate the spectral ratio r(z) = q(ℓ−2λ1(A) + ℓ2mA2)/(ℓ−2λ1(H) + ℓ2mH2) using the geometric Laplacian with geodesic edge lengths cosh ℓ(z) = cos(2π/z)/[1−cos(2π/z)] from the hyperbolic triangle {3, z}. For the Klein Map (z = 7, genus 3), √we obtain r(7) ≈ 1.15 in the weak-coupling limit, far from the continuum value 5 ≈ 2.236 (error > 48%). This structural discrepancy is a negative result of publishable significance: the TCFQ mass hierarchy is an emergent property of the H4 continuum that is not directly recoverable in low-genus, finite-volume hyperbolic networks. Recovery would require a network in the large-volume limit ℓ(z)2 ≫ λmax.