Between a circle and a square lies the smallest gap geometry allows — and perhaps the deepest one physics has yet to understand.

Dedicated to François Viète (1540–1603),

whose nested radicals revealed the infinite geometry

between the circle and the square,

and in doing so, hinted at the balance of opposites that underlies all creation.

Time has no beginning or end, there is only an in between.

Author’s Note

I have spent a lot of time thinking about theoretical physics and philosophy—reading, watching, and reflecting on the nature of existence.

Over the past few years, I have developed this paper gradually as an independent researcher with an academic background in the medical sciences. Although I am far removed from the world of professional physics, I believe that curiosity, logical discipline, and reductionist reasoning can reveal insights that sometimes lie outside established institutional perspectives.

In this work I propose a quantum–geometric model of the universe that aims to unify quantum mechanics and gravitation, address the cosmological constant problem, and describe both the origin and cyclic renewal of the cosmos within a single, self-consistent framework.

My starting assumptions were simple:

  1. The foundational principles of physics should reflect simplicity, not superfluous complexity.
  2. The constants $\sqrt{2}$ and $\pi$ are not mere numerical values but relational invariants encoding the symmetry between wave-like and particle-like domains.
  3. Creation can be expressed in three words—equal, opposite, asymmetric: the first two echoing Newton’s third law, and the last resonating with the second law of thermodynamics.
  4. A zero-energy universe provides a rational foundation for emergence ex nihilo.

From these principles the present model emerged. Although some philosophical interpretations are offered, they are included only insofar as they clarify the physical and mathematical framework.

Godspeed.

Quantum-Geometric Cyclic Cosmology

A Time-Symmetric Model of Expansion and Renewal

JIO

Independent Researcher

November 2025

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; its rescalings (e.g.\ $2-\pi/2=2\alpha$) arise whenever the construct aggregates multiple quadrant deficits and do not introduce additional independent parameters.

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

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

we impose a logistic composition history $f(t)=\bigl(1+e^{-\kappa(t-\gamma)}\bigr)^{-1}$. By construction the inflection time is dark-sector equality, $f(\gamma)=\tfrac12\iff \rho_{\rm dm}(\gamma)=\rho_{\rm de}(\gamma)$, while the conversion window is anchored by the emergence share $f(0)$ and one present-day point $f(t_0)$, fixing $(\kappa,\gamma)$. A finite epoch may be defined either by mirror closure $L\equiv 2\gamma$ (exchanging the initial dark weights across the endpoint) or, more generally, by an operational exhaustion threshold $f(L_{\varepsilon})=1-\varepsilon$; the mirror-closed realization corresponds to the special choice $\varepsilon=f(0)$.

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 $w_{\rm de}=-1$ the induced exchange $Q=\rho_D f(1-f)(\kappa+3H)$. The model is $\Lambda$-free at the level of fundamental inputs: $\Lambda$CDM behavior is recovered only as a late-time approximation when exchange is suppressed away from the conversion window. We also show that the logistic kernel arises as the overdamped limit of a minimal variational relaxation model with a cubic potential, providing a Ginzburg–Landau-type origin for bounded geometric repartitioning.

Viète-inspired finite-residual structure supplies a secondary geometric remnant through the bidirectional factor $\beta_{\mathrm V}=9\sqrt{2}/(4\pi)$, whose deviation $\varepsilon_{\mathrm V}=\beta_{\mathrm V}-1\sim 10^{-2}$ can be interpreted as a small curvature bias available to seed baryonic selection; after microphysical processing and entropy dilution it can plausibly project to $\eta_B\sim 10^{-10}$. Observationally, the framework predicts correlated, falsifiable signatures characteristic of interacting dark energy: localized departures from $\Lambda$CDM-like expansion and growth (e.g. in $H(z)$, $f\sigma_8$, and ISW cross-correlations) peaking near the equality window, with additional tensor/correlation imprints possible in model-dependent cyclic renewal extensions.