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Φ(t₀, τ) = |Σ_α w_α I_C(α, t₀, τ)| / [I_E(t₀, τ) · Σ_α w_α] I_E(t₀,τ) = [e^{−γt₀} − e^{−γ(t₀+τ)}] / (γτ) H = Σ p²/2m + ½Σ k_ij(q_i−q_j)² ⟨T⟩ = −½⟨Σ r_i · F_i⟩ Ω = Σ_α ω_α · w_α / Σ w_α

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:

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

Research Timeline

1870

Clausius Establishes Virial Theorem

Finite-window imbalance left unsolved for 156 years.

Jan 2026

GOMT Framework Published

Global Oscillator Mode Trace framework on Zenodo. DOI: 10.5281/zenodo.19798586

Feb 2026

JAP Framework Published

Jabbar Adal Principle on Zenodo. DOI: 10.5281/zenodo.19815385

Apr 2026

TRVIT v2.2 Published & Submitted

Submitted to Journal of Physics A (IOP Publishing). Zenodo DOI: 10.5281/zenodo.19994556

May 2026

Manuscript Under Formal Consideration

Preprint deposited on Zenodo, open access, CC BY 4.0.