The LOGOS Framework (v11): Z₂ Symmetry Breaking and Time Warping as Structural Boundary Conditions

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.