η = 49/120 as the Topological Action of T144

Abstract We propose that the spectral invariant η = 49/120 of the Dirac operator on the orbifold T²/Z₁ is not merely a numerical coincidence but constitutes the topological action of the T144 biological timing framework. In the Atiyah-Patodi-Singer formalism, η appears as a boundary term in the index theorem — precisely the role played by an action in variational mechanics. We show that: (i) the decomposition η = 1/3 + 1/5 − 1/8 is the T144 action written in terms of its three symmetry factors Z₃, Z₅, Z₈; (ii) the variational condition δS = 0 on T²/Z₁ quantizes the admissible biological periods to divisors of 120; (iii) Z₈ plays the role of a θ-term — a chiral topological term that breaks time-reversal symmetry and accounts for the irreversibility of biological cycles; (iv) by the generalized Noether theorem, each factor of Z₁₂₀ engenders a conserved biological invariant. This framework provides the mechanistic answer to the question: why does algebraic structure constrain biological periods? The answer is variational: biological periods are eigenvalues of the Dirac operator on T²/Z₁, and they are quantized by the topology of the orbifold. Keywords: topological action, η-invariant, Atiyah-Patodi-Singer, Dirac operator, orbifold, T144, Z₁₂₀, Noether theorem, biological periodicity, θ-term, time-reversal symmetry, Bott periodicity