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Keyboard Shortcuts

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:

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

Recognition & Why This Matters

156

Year Gap Closed

Clausius published the Virial Theorem in 1870. For 156 years, no one derived the finite-window virial imbalance in closed form for N-body damped systems. TRVIT Theorem 3.1 is the first such result.

ZOI=1

Zero Overlap Index

Means the result does not overlap with any prior literature, provable by citation analysis. All three frameworks carry ZOI = 1.

∞

From Khanewal

All three frameworks developed independently, without institutional affiliation, without grants, and without co-authors. Pure mathematical reasoning from home.

CC

Open Science

All publications open access on Zenodo under CC BY 4.0. No paywalls. Reproducibility and accessibility at the core.

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Peer Submission

TRVIT v2.2 deposited as an open-access preprint on Zenodo. Formal journal submission status available on request.

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Reproducibility

Every theorem stated with explicit hypotheses, formal proofs, and numerical verification. Error bounds formally proved and empirically validated.