Flavor-Singlet Tetrad Condensation and Monometric Emergent Geometry: A Large-N Functional-RG Framework

We investigate whether a multiflavor fermionic pregeometric framework can dynamicallyselect a single composite tetradic condensate from competing flavor channels. Within aleading projected ladder/large-N functional-renormalization-group truncation, we analyze aJ = 2 channel whose dimensionless coupling obeys βg = 4g − ATg2, with AT > 0 underthe physical transverse-traceless susceptibility prescription adopted in the truncation. Thecorresponding critical separatrix is g∗ = 4/AT. A flavor-dependent splitting permits the window gS > g∗ >gA, so that the flavor singlet reaches the collective instability while a positivesubcritical adjoint branch remains below criticality. In the bosonized branch considered here,singlet criticality is compatible with a nonzero tetradic condensate while the flavor-adjointdirections retain positive quadratic curvature, yielding an operationally monometric phasewithin the truncation. For a Hermitian physical transverse-traceless spin-2 channel withnonzero spectral support and a well-defined subtracted dispersive representation, KällénLehmann positivity implies ZdispTT > 0. This result fixes only the dispersive contribution;it does not determine local matching terms, the complete renormalized Planck coefficient,or the full pole structure of the theory. The construction therefore provides a conditionaldynamical mechanism for selecting a single tetradic background from a multiflavor fermionicsystem rather than a derivation of complete general relativity. Full Ward identities, localmatching, torsional dynamics, ghost tests outside the TT sector, and nonlinear completionremain open problems for subsequent stages of the B-MECG research program