The Spiral Model of Time: Direction, Span, and Self-Similarity

Within Drive-Field Information Theory (DFIT)—a holographic entropic model in which gravitational dynamics emerge from a covariant entropy field s_{holo} that sources modified Einstein equations via an information-stress tensor I_{\mu\nu}—this paper develops a geometric account of temporal emergence. The construction connects the covariant entropy-field dynamics to an irreversible coarse-graining mechanism, identifies a distinguished sector-balance surface associated with the present, and yields phenomenological benchmarks linking the model to cosmological observables. In the adiabatic regime \sfrac{H}{\omega}\ll1 on an expanding Friedmann–Lemaître–Robertson–Walker (FLRW) background, the field equation \box s_{holo}+\gamma_0\ V\prime\left(s_{holo}\right)\ =0 admits oscillatory solutions whose analytic-signal embedding traces a logarithmic spiral r(\theta)=ae^{b\theta}, b=\sfrac{H}{\omega}, in the field's intrinsic phase space, with temporal evolution represented by the spiral phase parameter \theta. The propagating sector-balance surface \Sigma_L, defined by \beta\left|\nabla s_{holo}\right|^2=\alpha V\left(s_{holo}\right), marks this temporal boundary. Its forward advance is driven by covariant relaxation of the entropy field, while contraction of the quantum Fisher metric under MERA coarse-graining provides a boundary representation that yields the half-cycle closure condition e^{i\pi}+1=0—termed here the Euler–Poe Condition. A Holographic Entropy Theorem establishes that the total resolved holographic entropy S_{holo}\left(\mathcal{B},\ t\right)=\sfrac{A_{holo}\left(\mathcal{B},\ t\right)}{4l_P^2} is non-decreasing along all expanding DFIT trajectories, with equality only in the uniform-entropy de Sitter limit \nabla s_{holo}\rightarrow0. Within the model, this links the thermodynamic arrow to cosmological expansion dynamics: unresolved distinguishability is progressively expressed as resolved holographic boundary geometry through irreversible coarse-grained evolution. Finally, the framework suggests phenomenological avenues for investigation, including an illustrative redshift-dependent dark-energy equation of state and potential gravitational-wave phase-shift signatures, offering qualitative points of contact with datasets from DESI, Euclid, and future gravitational-wave observatories.