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:

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

Education

Academic record, summarized. Full board-issued certificates are held privately and are available on request for verification purposes; scanned copies and roll/enrollment numbers are not published here to protect against identity misuse.

Intermediate Part-II (Pre-Medical) — 2024

Board of Intermediate & Secondary Education, Multan · Superior College for Boys, Khanewal · Group: Pre-Medical

Overall: 858 / 1200

Translation of the Holy Qur'an — 78/100 (Grade B)
Urdu — 144/200 (Grade B)
English — 138/200 (Grade C+)
Islamic Education — 46/50 (Grade A+)
Pakistan Studies — 41/50 (Grade B+)
Physics — 138/200 (Grade C)
Chemistry — 140/200 (Grade C+)
Biology — 133/200 (Grade C)

Secondary School Certificate (Matriculation, Science) — 2022

Board of Intermediate & Secondary Education, Multan · Govt. High School 2/AH, Khanewal · Group: Science

Overall: 1006 / 1100

Urdu — 130/150 (Grade B+)
English — 134/150 (Grade A)
Islamiyat — 97/100 (Grade A+)
Pakistan Studies — 87/100 (Grade A)
Mathematics — 147/150 (Grade A+)
Physics — 139/150 (Grade A+)
Chemistry — 147/150 (Grade A+)
Biology — 125/150 (Grade B+)

Self-Directed Study

Beyond formal schooling, ongoing self-directed study in classical mechanics, Hamiltonian dynamics, operator theory, and numerical methods (SciPy/Python), applied to the independent research presented on the Research page.

Verification

Board results are independently verifiable through BISE Multan's official result-checking service using the candidate's own roll number. For collaboration or credential verification inquiries, use the Contact page.

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