The Beating Winding: Biological Rhythm as a Driven Complex Eigenvalue

The Euler Universe series reads structure through a complex winding, e^ ( (λ+i) θ), in which the argument θ carries a phase and the exponent λ carries a rate. This paper applies that reading to biological rhythm, and specifically to its failure. A single excitable cell, linearised about rest, has a Jacobian with a complex-conjugate eigenvalue pair a±bi; as a control parameter crosses threshold the pair crosses the imaginary axis in a Hopf bifurcation and a limit cycle is born, with b the oscillation frequency and a the amplitude rate. Maintenance of the beat is a driven, dissipative winding: the phase direction is marginal and transverse perturbations decay, paid for by metabolic dissipation to a reservoir. Collective rhythm is already written in the literature as a modulus times a phase through the Kuramoto order parameter re^ (iψ). The load-bearing claim concerns tissue: a coherent rotating wave (rotor) about a phase singularity can destabilise and break up into wandering wavelets, and by Pesin's identity hKS = Σλ⁺ this is a passage from coherent rhythm (λKS ≈ 0) to spatiotemporal chaos (λKS > 0). Sudden cardiac death is, in these terms, a decoherence. The i throughout is a classical oscillation frequency — the imaginary part of a Jacobian eigenvalue of a real dynamical system in real time; this paper makes no quantum-coherence claim and is unrelated to quantum biology. The paper imports all mechanism, claims no new cardiology and no new empirical number, and reports in an explicit ledger both what the lens re-describes and where a tempting unification (the Kuramoto coherence collapse and the onset of chaos as 'two faces of one modulus') is declined as a correspondence rather than an identity.