The Probability-Cycle (Ps-Cycle) framework models the universe as an open thermodynamic, probabilistic engine. Instead of static localized values, mass, gravity, and cosmic acceleration are treated as cyclic energy outcomes dictated by fundamental state choices—such as staying, annihilating, or accelerating. This architecture establishes scale-invariance, cleanly bridging quantum mechanics and macro-astrophysics through unified constraints.
Core Mathematical Architecture
The framework operates under four foundational equations designed to resolve uncertainty propagation and flux dynamics across variable physical dimensions.
1. Core Mass-Probability Formulation
Defines the fundamental relationship linking physical mass scales directly to the underlying probability exponents of the cycling state system:
$$M = \left(\frac{a \cdot \hbar^2}{10^5}\right)^P$$
Variable Definitions:
- M : Total systemic mass output.
- a : Scale-invariant coupling coefficient.
- \hbar : Reduced Planck constant.
- P : Probabilistic cycling exponent state.
2. Scale-Inverse Gradient ($\delta$)
Tracks spatial changes across scaling barriers, assessing vacuum energy distribution relative to expansion velocities over structural node constraints:
$$\delta = \frac{1}{7\pi} \cdot \left(\frac{Q_{vi}}{H^z}\right)$$
Variable Definitions:
- \delta : Scale-Inverse structural gradient.
- Q_{vi} : Internal vacuum energy density index.
- H^z : Multi-dimensional Hubble expansion velocity parameter.
3. Multiverse Leakage Thermodynamics
Governs mass-asymmetric tunneling behaviors and thermodynamic losses escaping the open systemic bounds:
$$E_m = \int \Delta H_t + \left(\frac{-K}{E_{as9}}\right)$$
Variable Definitions:
- E_m : Total thermal leakage across the bounding boundary layers.
- \Delta H_t : Variable thermodynamic head differential over temporal steps.
- K : Systemic kinetic distribution boundary constant.
- E_{as9} : Spatial energy threshold coefficient tracking asymmetric tunneling states.
4. Chiral PP-Chain Trigger & Containment
Models the localized triggering thresholds and orbital containment required to maintain flux stability without collapsing:
$$\Delta T = \frac{T - \sigma_v}{t + \sigma^g} \quad \text{and} \quad A = \oint \gamma_l^\phi \, dl^r$$
Variable Definitions:
- \Delta T : Absolute temperature trigger delta required for systemic shift.
- \sigma_v, \sigma^g : Variance constraints tracking vacuum flux noise profiles.
- A : Orbital Angular Momentum containment area limit.
- \gamma_l^\phi : Phase-coherence function tracking localized angular mechanics.