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Volume 1: Gravitational Extensions - Waves from the Hidden CircleGravitational waves as messengers from a hidden fifth dimension.In this opening volume of The Echoes of Tau Trilogy, Ernest C. Gatlin III introduces a speculative 5D Kaluza-Klein cosmology where a compact extra dimension-radius τ ≈ 7 h⁻¹ Mpc-leaves detectable imprints on gravitational waves traversing the higher-dimensional ""bulk.""Through the Tau GW Theorem, the book derives heptagonal harmonics, transcendental phase drifts (δθ ≈ 0.0227), fractal damping, and chiral precession, offering falsifiable predictions for: nHz stochastic backgrounds and harmonic spikes (pulsar timing arrays like NANOGrav) Frequency beats and enhanced ringdowns (LIGO/Virgo O5 runs) Multi-messenger correlations with CMB anomalies and cosmic filaments (Euclid, DESI) Complete with equations, detection pipelines, open-source Python simulations, and philosophical reflections on gravity as a projection of deeper geometry, this concise yet rigorous work invites quiet observers to listen for waves from the hidden circle-and glimpse the eternal design echoing through spacetime.

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