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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:

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

Frequently Asked Questions

Independent theoretical physicist from Khanewal, Punjab, Pakistan, working outside any university affiliation. Author of the TRVIT, JAP, and GOMT frameworks in classical mechanics, published open access on Zenodo in 2026. No institutional affiliation, no external funding, no co-authors.

A closed-form expression for the normalised kinetic-potential imbalance Φ(t₀, τ) in an N-body coupled damped harmonic oscillator system over any finite observation window, extending virial-theorem analysis (originating with Clausius, 1870) to this driven-damped regime. Error bound ≤ 3.44 × 10⁻⁵ against a formally derived O(ε²) bound.

A falsifiability metric for scientific novelty. ZOI of 1 means no prior literature contains the same result. All three frameworks carry ZOI = 1, provable through systematic citation analysis and literature review.

All freely available on Zenodo under CC BY 4.0. TRVIT: doi.org/10.5281/zenodo.19994556 · JAP: doi.org/10.5281/zenodo.19815385 · GOMT: doi.org/10.5281/zenodo.19798586

Two formally proved theorems in non-relativistic Hamiltonian mechanics establishing structural symmetry constraints on phase-space trajectories. Provides exact necessary conditions for canonical transformations that preserve the action-angle variable structure, proved from first principles without approximation.

Global Oscillator Mode Trace — a single-scalar spectral fingerprint for N coupled harmonic oscillators. Collapses the full normal-mode spectrum into one invariant scalar, enabling mode identification, degeneracy detection, and structural comparison with O(1) storage. Analytically derived with exact closed-form expressions.

Every publication includes explicit hypotheses, formal proofs, and numerical verification code. Error bounds are formally proved and empirically validated. All code and data are reproducible from the Zenodo repositories. The Zero-Overlap Index claim is falsifiable through systematic literature review.

Matriculation (Science Group) from Govt. High School 2/AH, Khanewal, BISE Multan, 2022 — 1006/1100. Intermediate Part-II (Pre-Medical) from Superior College for Boys, Khanewal, BISE Multan, 2024 — 858/1200. Full subject-wise breakdown is on the Education page.

No. All research here is produced independently, without university affiliation, grant funding, or co-authors. This is stated explicitly in every manuscript's acknowledgments and conflict-of-interest sections.

Status varies by piece and is stated on each page. The virial identity manuscript is currently submitted to Physica Scripta and under double-blind review — it has not yet been accepted or formally published. Zenodo deposits are open-access preprints with a DOI but are not peer-reviewed by Zenodo itself. Peer-review status is never implied where it hasn't happened.

Python with SciPy's solve_ivp, typically the adaptive eighth-order Runge–Kutta method DOP853, at tight absolute/relative tolerances (10⁻¹² / 10⁻¹⁰). Verification runs are independently re-implemented from scratch as a second check before results are reported.

Yes, in places — for drafting assistance, code scaffolding, and website development, similar to how many independent researchers use modern tooling. Any manuscript that involved substantial AI drafting assistance discloses that explicitly in its authorship or acknowledgments rather than presenting it as unaided work. The mathematics and numerical results themselves are independently checkable regardless of what tools helped produce the prose around them.

Yes. Zenodo-deposited work has a citable DOI (see the Publications page). The Physica Scripta submission should be cited as "manuscript submitted for review" until a formal citation is available, in line with normal academic convention for unpublished work under review.

Independent, open-access publication (Zenodo, arXiv-style preprints) lowers the barrier for researchers without institutional access while still allowing formal journal submission in parallel. It does not replace peer review — it makes work available for scrutiny while that process runs.

English for technical writing and research; Urdu for translation work and some creative writing. Bilingual content is noted where it appears on the site.

Email umarjaumofficial@gmail.com, or use the form on the Contact page. Review requests, error reports, and collaboration proposals are all welcome — corrections to published work are taken seriously and will be acknowledged.

Primarily classical and Hamiltonian mechanics — virial-theorem identities, N-body oscillator systems, and related closed-form derivations — with numerical verification via adaptive Runge–Kutta integration. See the Research page for the full portfolio.

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Submitted to Physica Scripta and currently under double-blind review, as of 2026. This page will be updated when a decision is issued — no earlier claim of acceptance is made or implied.

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