Quantum Mechanics
A synthesis of my notes on Quantum Mechanics since 2022. I derive key relations, e.g. Heisenberg Uncertainty Principle, the Rydberg Constant, from first principles. Then I show how the position-momentum commutation relation arises as a direct consequence, and how this (revived) framework can be used to justify experimental results.
Resolving Einstein’s famous EPR paradox and Bell inequalities by showing that quantum entanglement doesn’t require “spooky action at a distance” or hidden variables—it’s a single, unified wavefunction where measurements simply prune irrelevant parts, preserving completeness without non-locality.
Computer Science / Cryptography
P vs NP
An elementary proof of P /= NP within a decidable fragment of arithmetic. Collaboration by parts with Grok, building on an early, elementary proof by Hassall.
RSA analysis
1. The Divisor Staircase and Palindromic Symmetry of Semiprimes. The quiet discovery that multiplication hides a visible geometric fingerprint of RSA prime factors — and that fingerprint becomes impossible to conceal when the factors are too close or too unbalanced.
2. By multiplying a semiprime by a small known prime, we transform it into a three-prime integer. The floor function then exhibits a perfect algebraic and geometric symmetry with exactly eight terraces.
3. The study reveals a highly structured pattern consisting of universal low-x artefacts, a scalable quiet zone, and a prime-dependent spectral zone with regular troughs. The method provides profound insight into the divisor landscape of semiprimes.
Mathematics
This proof reduces the Collatz conjecture to a single question—“can two different odd-to-odd excursions ever land on the same odd number after wildly different numbers of halvings?”—and then shows that if the halving fluctuation is even modestly large (Δ ≥ 70 or so), the required earlier numbers explode in size, making such cycles impossible without violating basic arithmetic bounds.
This paper reframes the Collatz conjecture through an intuitive bitwise “sorting resolver” lens—revealing the mechanical elegance of how simple shifts, overlays, and carry cascades systematically untangle arbitrary binary patterns toward convergence, demystifying the problem.
We reframe the Collatz conjecture as a generative, bottom-up process starting from a single seed (1), using simple, mechanical operations to build an infinite tree that appears to cover all positive integers without gaps, duplicates, or crashes — we then offer a heuristic contradiction argument that makes the conjecture feel more tractable and inevitable in the limit.
This heuristic provides visually direct and completely elementary reasons why the critical line Re(s)=1/2 is mathematically privileged: it is the unique vertical line in the complex plane on which certain prime-power exponential roots p^{1/(2s)} remain exactly real for every height t, offering a fresh motivation for the location of all non-trivial zeros without requiring heavy analytic machinery.
This heuristic argument combines elementary explicit checks with a simple Prime Number Theorem estimate to show that the expected number of Goldbach representations grows like Z/(2(ln Z)²)—a quantity that rapidly becomes enormous—while bounded prime gaps ensure the probabilistic assumption holds uniformly, giving a concise “almost-proof” of the conjecture.
We reframe Goldbach’s conjecture through an inverse, generative “sum tree” perspective—starting from the seed 4 = 2 + 2 and using unbounded prime additions to build larger evens—offering a bottom-up heuristic that makes universal coverage feel inevitable in the limit, while providing a fresh, mechanical intuition that complements traditional forward-testing approaches and highlights the deep interplay between prime density and additive structure.
