Log-periodic oscillations in the cosmic expansion from non-commuting projections: a self-consistent emergent framework

We present a self-consistent emergent framework in which two non-commuting orthogonal projections P and Q on a separable Hilbert space naturally generate the observed cosmic energy composition and a logarithmic periodic oscillation in the expansion history. The competition between a unitary flow H and a non-unitary dissipative flow G defines a tension parameter λ = |G|/|H|. Using the Halmos decomposition, non-commutativity induces a complex structure J and spontaneous symmetry breaking along the i-axis, leading to helical trajectories. The resulting spectral layering has weights wₙ = n² e-β n, and the zero-mode exhibits a critical measure divergence as λ→1, which acts as dark energy. By a rigorous calculation of the critical density of states we obtain the dark energy density parameter ΩDE = 1/[1+2(λ-1)], which for λ₀≈1.2 gives ΩDE≈0.71, consistent with current observational constraints (Planck 2018 + Pantheon+ + BAO) without fine-tuning. Linearizing the dynamics around the critical point λ=1 yields a damped oscillator, which translates into a log-periodic modulation of the Hubble parameter and the distance modulus: δμ(z) = A (1+z)-β cos(κ ln(1+z) + φ'), with κ = √λ/(1+λ)≈0.5 and amplitude A~0.01--$0.05$ mag. The model predicts a phantom-like equation of state today (w₀≈-1.1) that asymptotically approaches $w=-1$ at late times, consistent with mild phantom-crossing hints (at low-to-moderate significance) reported in DESI DR2 data. All parameters are derived from the algebraic structure without free functions, providing a falsifiable log-periodic signal testable with Pantheon+ and future surveys. Remarkably, the model constitutes a single-parameter deformation of ΛCDM, with all deviations controlled by the critical exponent α.