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

We analyse a minimal N-node cyclic delay-differential network withbounded nonlinear coupling. Linear stability analysis yields acharacteristic equation that depends only on the total loop delayτ_total, establishing delay-sum invariance as a topological propertyof the cyclic architecture. This structural result leads directly to closed-form Hopf bifurcationconditions: the onset frequency ω_c = sqrt(g²−1) is universal andindependent of both the node count N and the delay distribution.Center manifold reduction yields first Lyapunov coefficientl₁ ≈ −3.47 < 0, confirming a supercritical Hopf bifurcation and astable limit cycle for Λ > Λ_crit. Part II of this paper shows that the three free parameters — gain g,noise amplitude σ, and delay ratio r — are structurally orthogonalwith respect to the characteristic equation. This orthogonalitymotivates a three-component decomposition of dynamical complexity: • C_int (integrative complexity): governed by g, controls bifurcation proximity and inter-node coherence. • C_expl (exploratory complexity): governed by σ, controls state-space coverage and trajectory diversity. • C_temp (temporal complexity): governed by r, controls spectral bandwidth and memory depth. A mechanism isolation analysis demonstrates that state-space expansiondriven by noise alone does not produce increased integration ororganised dynamics — variability and organisation are operationallydistinct properties within this framework. This version (v8) consolidates the full two-part paper with allanalytic proofs, 9 figures, mechanism isolation experiments, andreproducible simulation code. Also included: logos_agape_v3.py —a 175-year (2026–2200) LOGOS+Agape simulation modelling long-rangesystem stability under technological growth and ethical dynamics. v8 also includes logos_agape_v3.py: a 175-year (2026–2200) LOGOS+Agape simulation modelling long-range system stability under technological growth and ethical dynamics. Three-phase structure: Fragility (2026–2075), Agape Activation (2075–2125), Stable Oscillation (2125–2200). 2100+ mean S = 0.295, CV = 0.542.