Version 11 (May 2026) This version adds two new results that were completed prior to v10 but not yet formally incorporated into the main document: 1. Z₂ Symmetry Breaking (Section 6) The oscillatory regime of the LOGOS system depends structurally on the odd symmetry of the nonlinearity f (x) = gx/ (1+x²). Introducing a controlled even-order perturbation f_ε (x) = gx/ (1+x²) + εx² reveals a critical threshold εc ≈ 0. 12–0. 20 at which the limit cycle undergoes a fold-of-cycles bifurcation and collapses. No hysteresis is observed. This result indicates that the oscillatory regime occupies a codimension-one set in function space: the Z₂ symmetry is load-bearing, not merely a mathematical convenience. 2. Time Warping — A Null Result (Section 7) Delay modulation (time warping) at fixed τₜotal yields a bit error rate BER_τ ≈ 0. 50, indistinguishable from random guessing. This is a direct algebraic consequence of the delay-sum invariance theorem (Proposition 1): since delay redistribution at fixed τₜotal does not alter the characteristic equation, no recoverable state change is produced at the output. Gain modulation remains the sole viable information channel (BERg ≈ 0. 12). Time-varying delays act as stabilisers on the attractor — a structural buffer, not an information carrier. No changes were made to Sections 1–5 or the previously established results.
Mustafa Serkan Taşkoyan (Thu,) studied this question.