Delay-Sum Invariance, Hopf Bifurcation, and Composite Recursive Complexity in Cyclic Delay Networks

LOGOS Framework version 10: an incremental release combining the v9 theoretical closure with two post-closure numerical findings on information transmission in cyclic delay-differential networks. Contents of this version: (1) LOGOS_v9_Final.pdf — the v9 theoretical closure (April 2026, unchanged), establishing the unified variational structure J = ∫ TW·[L − αE² − βR] dt, the stochastic Hamilton–Jacobi–Bellman equation, and the principal result V₄ = −l₁/(2ωc) ≈ +0.448·TW̄ identifying the first Lyapunov coefficient with the quartic value-function curvature at the Hopf threshold. (2) LOGOS_Information_Channels_EN.pdf — a new post-v9 extension paper reporting two numerical experiments: (i) A sharp threshold ε* ≈ 0.20–0.30 separates the linear-information regime from the structural-shock regime for state-space pulse perturbations. This is the numerical signature of the limit-cycle basin boundary defined analytically in v9. (ii) An information-theoretic asymmetry between two parameter modulation channels. A modulation of the nonlinearity gain g(t) is recoverable at the readout with BER ≈ 0.12, whereas equal-amplitude modulation of a delay τ₂(t) is unrecoverable (BER ≈ 0.50, indistinguishable from noise). This asymmetry is a direct information-theoretic consequence of the delay-sum invariance theorem proven analytically in v9: perturbations preserving T = Στᵢ define channels of zero Shannon capacity. The extension formalises the "critical-point injection" intuition geometrically — informationally observable parameters are those that span the normal direction of the bifurcation manifold; perturbations tangent to it are unobservable regardless of amplitude (within the linear regime). The v9 theoretical core is unchanged; this release extends it. Source code (Python) reproducing all simulations and figures of the extension paper is included.