Key Equations

TRVIT Theorem 3.1:

$$\Phi(t_0,\tau) = \frac{|\sum_\alpha w_\alpha I_C(\alpha, t_0, \tau)|}{I_E(t_0, \tau) \cdot \sum_\alpha w_\alpha}$$

Energy Decay:

$$I_E(t_0,\tau) = \frac{e^{-\gamma t_0} - e^{-\gamma(t_0+\tau)}}{\gamma\tau}$$

Hamiltonian:

$$H = \sum \frac{p^2}{2m} + \frac{1}{2}\sum k_{ij}(q_i-q_j)^2$$

Virial (Clausius 1870):

$$\langle T \rangle = -\frac{1}{2}\langle \sum r_i \cdot F_i \rangle$$

GOMT Scalar:

$$\Omega = \frac{\sum_\alpha \omega_\alpha \cdot w_\alpha}{\sum w_\alpha}$$

About

Muhammad Umar Jabbar is an independent theoretical physicist working from Khanewal, Punjab, Pakistan. Without institutional affiliation or external funding, he has produced closed-form analytical results in classical mechanics that had not appeared anywhere in the literature since Clausius established the Virial Theorem in 1870.

His flagship result — Theorem 3.1 of TRVIT v2.2 — is a closed-form expression for the normalised kinetic-potential imbalance Φ(t₀, τ) in a deterministic N-body coupled damped harmonic system over an arbitrary finite observation window, extending the classical virial theorem to this driven-damped setting. The result is numerically verified to absolute error ≤ 3.44 × 10⁻⁵ against an O(ε²) analytical error bound.

This work was achieved through pure mathematical reasoning, from home, in one of Pakistan's smaller cities — demonstrating that the highest level of theoretical physics is determined by mathematical truth, not institutional prestige or national geography.

"A result is useful when it is exact, or when its error is bounded. The Zero-Overlap Index is not self-promotion — it is a falsifiable claim that shifts the burden of proof appropriately."

— Muhammad Umar Jabbar