Add notebook page 1 transcription (Halting Problem, Möbius, Gaussian/Fourier, Quantum)

Co-authored-by: blackboxprogramming <118287761+blackboxprogramming@users.noreply.github.com>

notebooks/page-01.md

@@ -0,0 +1,202 @@
+# Notebook Page 1 — Transcription
+
+> Source: `→ halting problem.pdf`, page 1 of 24.  
+> Transcribed by Alexa Louise Amundson.
+
+---
+
+## 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!"*
+
+---
+
+## 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).
+
+---
+
+## 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)
+```
+
+---
+
+## 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