simulation-theory/notebooks/page-01.md

Notebook Page 1 — Transcription

Source: → halting problem.pdf, page 1 of 24.
Transcribed by Alexa Louise Amundson.

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1. Computer Science & Logic: The Halting Problem

Complex & Imaginary Numbers

(a + ib)(a − ib) = a² − ibib
Imaginary: (y + x)² y
Real:      (y + x)²

Euler's formula expansion:

e^(ix) = 1 + ix − x²/2 − i(x³)/6 + x⁴/24 − ...

Paradoxes & Abstraction

• Golden Braid — a reference to levels of abstraction and paradoxes.
• "This sentence is false" → refers to its own truth value.
• Cantor diagonalization → linked to the Halting problem.

The Halting Problem

A thought experiment for a hypothetical program h that predicts whether another program will loop forever or halt.

Program 1 → [h]: Input I into program h.
h answers: will this problem halt, or will it not?

Examples:

x = 4
while x > 3: x += 1   → LOOPS FOREVER

x = 4
while x < 1000: x += 1   → Halts.

The Paradox (h+):

• Take the source code (e.g., 11001011) and use that code as both the program and the input.
• Feed x as data into itself: x = h+.
• If h halts → h+ begins an infinite loop.
• If h loops → h+ halts.

"Does it loop or halt? It's a paradox! But h does not exist!"

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2. Number Theory: The Möbius Function

Definitions & Rules

The Möbius function μ(n) is a multiplicative number-theoretic function.
For any positive integer n, define μ(n) as the sum of the primitive n-th roots of unity.

Factorization rules:

μ(n) = 0          if n has one or more repeated prime factors
μ(n) = 1          if n = 1
μ(n) = (−1)^k     if n is a product of k distinct primes

μ(n) ≠ 0 indicates that n is square-free.

First few values:

1, −1, −1, 0, −1, 1, −1, 0, 0, 1, −1, 0, ...

Formulas & Series

Mertens Function (summatory function of Möbius):

M(x) = Σ_{n ≤ x} μ(n)

Dirichlet Series (multiplicative inverse of the Riemann zeta function):

Σ_{n=1}^{∞} μ(n)/n^s = 1/ζ(s)   ;   Re(s) > 1

Lambert Series:

Σ_{n=1}^{∞} (μ(n) x^n) / (1 − x^n) = x   ;   |x| < 1

Kronecker Delta Relation:

Σ_{d|n} μ(d) = δ_{n,1}

Infinite Sums:

Σ_{n=1}^{∞} μ(n)/n        = 0
Σ_{n=1}^{∞} (μ(n) ln n)/n = −1
Σ_{n=1}^{∞} μ(n)/n²       = 6/π² = 1/ζ(2)

Historical note: Gauss considered the Möbius function over 30 years before Möbius, proving that for a prime number p, the sum of its primitive roots is congruent to μ(p − 1) (mod p).

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3. Probability & Math: Gaussian Functions & Fourier Transforms

Gaussian Basics

Used to represent the probability density function of a normally distributed random variable.

• Expected value: μ = b
• Variance: σ² = c²

Standard form:

f(x) = (1 / (σ √(2π))) · e^(−(1/2)((x−μ)/σ)²)

Arbitrary constants form (a = peak height, b = center, c = width):

f(x) = a · e^(−(x−b)² / 2c²)

Fourier Transform Proofs

Transform of a Gaussian:

F{ a · e^(−bx²) } = (a / √(2b)) · e^(−ω² / 4b)

The integration proof uses substitution t = x + iω/2b, showing that the Fourier transform of a Gaussian is also a Gaussian.

Derivative Properties:

Time domain:      F{ f′(x) }  = iω · F(ω)
Frequency domain: F{ x f(x) } = i · d/dω F(ω)

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4. Physics: Quantum Mechanics & Energy

Schrödinger Equation & Operators

Time-dependent equation:

iℏ (∂/∂t) Ψ = HΨ

Where:

• i = √(−1)
• ℏ = Planck's constant (reduced)
• Ψ = quantum wave function
• H = Hamiltonian operator

Harmonic Oscillator:

Classical energy:    (1/2)mv² + (1/2)kx² = E
Momentum operator:   p → (ℏ/i)(∂/∂x)
Quantum Hamiltonian: H → (−ℏ²/2m)(∂²/∂x²) + (1/2)kx²
Eigenvalue equation: HΨ = EΨ

Uncertainty & Photons

Heisenberg Uncertainty Principle:

Δp · Δx ≥ h / 4π   (= ℏ/2, where ℏ = h/2π)

Energy of a photon:

E = hν = hc/λ

Photoelectric effect:

(1/2) m v_max² = eV₀ = hf − φ

Fundamental Constants & Bohr Model

r = (n² h² ε₀) / (π m e²)   ∝ h²
v = e² / (2 ε₀ n h)          ∝ 1/n

Fine-Structure Constant (α):

α = (1 / 4πε₀) · (e² / ℏc) ≈ 1/137

Speed of light: c = 3 × 10⁸ m/s
Elementary charge: e = 1.602 × 10⁻¹⁹ C