Systematic Search for Gravitational Wave Echoes in Non-Orientable Extra Dimensions: A Comprehensive Multi-Topology Analysis

We present the first systematic theoretical and observational study of gravitational wave echoes
from non-orientable extra-dimensional topologies. Building on our previous Klein bottle analysis
(Di Bacco, 2025), which established 2.80σ evidence for gravitational wave echoes, we investigate
whether other non-orientable surfaces can produce similar phenomena.
We derive rigorous theoretical frameworks for five distinct topologies: Klein Bottle, Real Projective
Plane (RP2), Möbius Band, Twisted Torus, and String Orientifolds. Using geometric factors
derived from fundamental topological properties—not fitted parameters—we predict specific observational
signatures for each model. Application to 65 LIGO-Virgo events with cosmological corrections
yields definitive results: Klein Bottle achieves 9.25σ combined significance with 87.5%
detection rate, while Twisted Torus shows 5.71σ with 64.1% rate. Crucially, harmonic mode analysis
confirms Klein bottle’s key prediction: strong odd-harmonic signals (11.9σ) with suppressed
even modes (0.5σ), providing a 22:1 suppression ratio exactly as predicted by non-orientable
topology.
These findings establish non-orientable surfaces as a viable class for extra-dimensional physics
and provide a robust theoretical foundation for gravitational wave astronomy as a probe of fundamental
geometry.