The Arithmetic Vacuum
The Prime Lattice Substrate
From just two postulates — the Riemann Hypothesis and one length scale λ — the primes and zeta sieve quantum discreteness, gravity, electromagnetism, three dimensions, three generations, the weak interaction, and the exact observed speed of light. All constants emerge analytically; seven sharp, falsifiable predictions remain.
Bell/EPR: Resolving Quantum Paradoxes
No Ai assistance. 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.
What makes it unique: Unlike standard interpretations that debate collapse or many worlds, this derives resolution from pre-existing correlations in an infinite lattice, making quantum mechanics fully consistent without extra assumptions.
Entropy from the Arithmetic Vacuum
Hassall comfort zone. This paper derives Boltzmann entropy S = k ln W from gap multiplicity W ~ e^w / w, yielding partition Z ~ ζ(1/2 + i β) and laws like the 2nd law dS ≥0 from thinning dilution, with heat capacities dropping 20% at low T and the Boltzmann constant k ~1.38 × 10^{-23} J/K recovered analytically to exact precision from L_vac damping (no empirical input, a first in any framework).
What makes it unique: It unifies thermodynamics with quantum foundations by sieving S from prime rarity, predicting testable deviations like 20% C_v drops in IR spectra, while deriving k itself from the zeta tail.
Quantum Gravity from Jittered Gaps
Grok expansion on foundational work. The lattice’s jitter δK ~ ∑ e^{ρ w}/ρ sieves Wheeler-DeWitt H_ψ ψ=0, with gaps quantizing area √j(j+1) ~ √ln p (50% LQG match), diffeomorphism from thinning RG, and BH entropy S_BH ~ ln π(r) ~ r / ln r (10% Hawking match, 5% RH jitter testable in LIGO analogs).
What makes it unique: It supports QG as discrete jittered foam from zeta zeros, deriving G in ħ c units (0.1% CODATA) without tuning, linking to LQG without Immirzi parameter.
Quantum Mechanics
An Einsteinian Interpretation of QM
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.
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
Collatz Conjecture
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.
Riemann Hypothesis
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.
Goldbach Conjecture
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.
Fermat’s Last Theorem
This elementary argument reduces Fermat’s Last Theorem to a single, almost trivial observation about rational points on the curve using nothing more than high-school algebra and the concavity of the function: t1/nt^{1/n}t^{1/n}. It’s short, intuitive, and feels “too simple to be wrong,” yet it captures the essence of why no integer solutions exist without elliptic curves or modular forms.
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