The Internal Geometry of Faster-than-Nyquist Signaling

Faster-than-Nyquist (FTN) signaling is classically understood through the interference it creates. We study instead the internal geometry of the finite constellations it spans: clusters counted by a linear statistic, parallel equispaced runs, an exact self-similar hierarchy with a reversal symmetry, and a within-cluster metric the hierarchy factorizes in closed form. Time compression collapses cluster interiors while leaving the cluster-level geometry intact, and the factorization turns that asymmetry into theorems: a closed-form Mazo-type threshold for the Gaussian pulse, a second rung beneath it, a rate–distance frontier with counted error-event multiplicity, and a pulse dichotomy: a positive floor under the folded spectrum reduces minimum distance to a finite check at every block length and alphabet, a band-edge null voids the reduction, and the strip between the crystallographic angles is closed up to Liouville and rational non-integer cosines. The geometry then yields algorithms: an exact cluster-aware sphere decoder that parallelizes at no critical-path cost; a one-pass trellis decoding every fiber of the cluster statistic exactly; and the junction law, blindness to the binding error event, priced per junction in closed form, whose optimum among sign constraints, the band-edge vector, beats every measured control where the law has slope. Every simulation, benchmark and finite check in the paper is reproduced by the archived verification and benchmark suite, doi: 10.5281/zenodo.21725435.