PAPER-FBT05A: Effective Berry Geometry on the Relative-Phase Fibre

This paper constructs the projective Berry geometry on the actual relative-phase torus from the positive FBT02B prequantum line. The mixed mother form restricts to zero on a regular fibre, whereas the effective Berry ray line is the dual of the positive prequantum line. At level 24, the complete theta system gives a relative projective embedding, its quantum-geometric tensor (QGT), and the tautological Chern sign. Projective descent does not require a global normalised vector or a trivial scalar Heisenberg cocycle. The canonical positive reference Dolbeault–Dirac operator has a 24-dimensional theta kernel, no opposite-chirality zero modes, and a locally uniform gap on the smooth regular polarised family. Its heat semigroup converges exponentially to the Riesz/Bergman projector. With the specified evaluation data, this gives the heat–Bergman–QGT bridge. The scalar heat trace retains less information and does not reconstruct the projective metric. The first FBT action remains the finite-resolution heat trace of DT 6,E; its Hessian and certified spectral A3 germ are different from the reference QGT. The Serre–Riesz recovery formula is retained as an integrated Hilbert pairing. A separate pointwise pairing defines the density exported to complex periods, with the reference measure and any QGT-volume conversion specified. Full-density extension, convergence and differentiation retain the revised 06-series hypotheses and the same βd. For a supplied basepoint-free two-dimensional theta pencil, the selected ray map has degree 24 to CP1. Its ramification divisor has degree 48, and its QGT degenerates at ramification points. Thus it cannot inherit the everywhere-positive metric of the complete theta embedding. The FBT10C βd = 8 selector remains a fixed numerical benchmark with its stated certification limits. Reference QGT, selected QGT, and coherent PL flow QGT have distinct roles; none alone supplies a four-dimensional mass theorem.