Abstract

We develop a quantum–geometric cosmology in which two time-reversed semiclassical branches form a globally neutral (net-zero) energy bookkeeping, while each branch supports a locally time-directed history. The onset of classical description is identified with an emergence/partition event at $t=0$, where a cube–circle construction supplies branch-normalized sector weights $\bigl(\Omega_b,\Omega_{\rm dm}(0),\Omega_{\rm de}(0)\bigr)$ from simple $\pi$- and $\sqrt{2}$-based geometric ratios. An irreducible circle–square deficit

\[ \alpha \equiv 1-\frac{\pi}{4}, \]

acts as the intrinsic asymmetry seed that fixes boundary data at emergence. Post-emergence evolution is modeled as a bounded redistribution within the dark sector at fixed baryon weight. Writing $S\equiv\Omega_{\rm dm}+\Omega_{\rm de}=1-\Omega_b$ and defining the dark-matter share

\[ f(t)\equiv\frac{\Omega_{\rm dm}(t)}{S} =\frac{\rho_{\rm dm}}{\rho_{\rm dm}+\rho_{\rm de}}\in[0,1], \]

the branch evolution is postulated to follow a bounded logistic law

\[ f(t)=\frac{1}{1+\exp\!\big(-\kappa[t-\gamma]\big)}, \qquad \dot f=\kappa f(1-f). \]

The inflection time is fixed by definition as dark-sector equality, \[ f(\gamma)=\tfrac12 \quad\Longleftrightarrow\quad \rho_{\rm dm}(\gamma)=\rho_{\rm de}(\gamma), \] so the midpoint of the S–curve is the physically distinguished equality epoch. Anchoring the curve at emergence $f(0)$ and one present-day point $f(t_0)$ uniquely fixes $(\kappa,\gamma)$, yielding a finite mirror-closed branch duration

\[ L \equiv 2\gamma \simeq 111.18~\mathrm{Gyr}. \]

The logistic postulate is embedded covariantly as an interacting dark sector,

\[ \dot\rho_{\rm dm}+3H\rho_{\rm dm}=+Q,\qquad \dot\rho_{\rm de}=-Q, \]

with $Q(t)$ fixed (not freely chosen) by covariant conservation together with the imposed $f(t)$, yielding in the vacuum-like limit $w_{\rm de}=-1$

\[ Q(t)=\rho_D\,f(1-f)\,(\kappa+3H). \]

The framework is therefore $\Lambda$-free at the level of fundamental inputs: $\Lambda$CDM behavior appears only as a late-time approximation when exchange is suppressed away from the conversion window. A minimal variational relaxation model shows that the bounded kernel $f(1-f)$ arises naturally in an overdamped geometric order-parameter description.

A secondary bidirectional Viète residual $\beta_{\mathrm V}=9\sqrt{2}/(4\pi)$ introduces a percent-level curvature bias $\varepsilon_{\mathrm V}\sim10^{-2}$, which after microphysical processing and entropy dilution can plausibly project to the observed baryon asymmetry $\eta_B\sim10^{-10}$. Observationally, the model predicts localized interacting–dark–energy signatures: correlated departures from $\Lambda$CDM-like expansion and growth (in $H(z)$, $f\sigma_8$, and ISW cross-correlations) peaking near the equality window, with additional tensor or correlation imprints possible in cyclic renewal extensions.