Universal Periodic-Orbit Triplet - (UPO)

The RSM propose a reproducible fingerprint for septimal holonomy in polar media: a dominant S-ridge (separatrix-like trace), a geometric ring-comb in the 2-D Fourier magnitude with constant semilog step Δ10, and a quadratic sideband at 14 × the septimal line quantified by C = A14/A7². A simple Frank–Read-style model in a negative-curvature throat predicts Δ10 ≈ π/7 ln 10 ≃ 0.15 after conditioning, a near-degenerate 6↔7 doublet, and a 14-cycle echo. We pilot the pipeline on three disparate images (a Majorana/DNA panel, a Planck CMB map, and a Magellanic Stream map). All three show strong S-ridges; ring-combs are present but shallow in raw sky maps; the 7→14 echo is non-zero in two panels. We preregister a conditioning+bootstrap protocol and release open code to enable independent replication and falsification. A pinned dislocation segment (Frank–Read source) that is periodically driven within a negatively curved throat acts as a prime-selector: it emits twin counter-spirals whose step radii rn lie on a log-geometric comb, log⁡10rn=log⁡10r0+n Δdex with Δdex≈0.15. he source’s emission train carries a septimal line at 7f and a quadratic echo at 14f with A14∼χA72. We derive these invariants from a septimal holonomy in a hyperbolic throat and show that the same ruler/echo recurs in Leidenfrost petal patterns, Faraday waves, and torus-knot choreographies—closing as a (2,7) loop on the time×angle torus. The Frank–Read (FR) source converts shear into loop emission. When the local metric is effectively hyperbolic (negative Gaussian curvature K<0) and the drive carries a seven-phase cadence, the loop-closing condition is quantized by a π/7 holonomy. Consequences are, a prime-filtered set of azimuthal indices m, equal spacing on log⁡r, and a quadratic sideband at twice the septimal line. We argue these are; mechanism-level invariants that explain visually similar “S-separatrices,” log-combs, and 7→14 echoes seen across crystal growth, Leidenfrost/ Faraday experiments, and braided flows.