From 5c26b4108780b68d23bce1321c95b0312bf0746c Mon Sep 17 00:00:00 2001 From: "copilot-swe-agent[bot]" <198982749+Copilot@users.noreply.github.com> Date: Fri, 27 Feb 2026 20:22:02 +0000 Subject: [PATCH 1/3] Initial plan From c07e622c6c8d580a71bc3545c0bc428875c7927a Mon Sep 17 00:00:00 2001 From: "copilot-swe-agent[bot]" <198982749+Copilot@users.noreply.github.com> Date: Fri, 27 Feb 2026 20:24:51 +0000 Subject: [PATCH 2/3] =?UTF-8?q?Add=20notebook=20page=201=20transcription?= =?UTF-8?q?=20(Halting=20Problem,=20M=C3=B6bius,=20Gaussian/Fourier,=20Qua?= =?UTF-8?q?ntum)?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Co-authored-by: blackboxprogramming <118287761+blackboxprogramming@users.noreply.github.com> --- notebooks/README.md | 3 +- notebooks/page-01.md | 202 +++++++++++++++++++++++++++++++++++++++++++ 2 files changed, 204 insertions(+), 1 deletion(-) create mode 100644 notebooks/page-01.md diff --git a/notebooks/README.md b/notebooks/README.md index 10c3cfa..234fc26 100644 --- a/notebooks/README.md +++ b/notebooks/README.md @@ -6,7 +6,8 @@ Page 14 is a duplicate scan of page 13 and was skipped. | Page | Title / Topic | Section | Key Finding | |------|--------------|---------|-------------| -| 1–9 | Earlier sections | §1–§164 | Foundations, Riemann, Gödel, Turing, QWERTY encoding, halting problem | +| [1](./page-01.md) | Halting Problem, Möbius Function, Gaussian/Fourier, Quantum Mechanics | §1–§9 | Complex numbers, Cantor diagonalization, Möbius μ(n), Fourier transform of Gaussian, Schrödinger equation | +| 2–9 | Earlier sections | §10–§164 | Foundations, Riemann, Gödel, Turing, QWERTY encoding | | 10 | Trinary — she derived it | §165 | ELSE=REAL=37, TRINARY=63, UNKNOWN=3×REAL=111, z=m·w+b (b stays, she is b) | | 11 | Trinary as Trinomial, Dürer's Magic Square | §166 | FOUR=34=magic constant, MELANCHOLIA=169=13², PHI=FOUR=34, replaced 1 with 2000 | | 12 | Divisibility by 9, Birthday Reversal, Rohonc Codex | §167 | METHOD=72=reverse(27), ROHONC CODEX=153, CIRCULAR=REMAINDER=97 prime | diff --git a/notebooks/page-01.md b/notebooks/page-01.md new file mode 100644 index 0000000..a239aac --- /dev/null +++ b/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 From 78dd0f08f260bd31f4beab5eb43427ef8b4c7a16 Mon Sep 17 00:00:00 2001 From: "copilot-swe-agent[bot]" <198982749+Copilot@users.noreply.github.com> Date: Fri, 27 Feb 2026 21:18:14 +0000 Subject: [PATCH 3/3] =?UTF-8?q?Address=20review=20comments:=20fix=20attrib?= =?UTF-8?q?ution,=20notebook-verbatim=20M=C3=B6bius=20sum,=20add=20Fourier?= =?UTF-8?q?=20convention?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Co-authored-by: blackboxprogramming <118287761+blackboxprogramming@users.noreply.github.com> --- notebooks/page-01.md | 9 +++++++-- 1 file changed, 7 insertions(+), 2 deletions(-) diff --git a/notebooks/page-01.md b/notebooks/page-01.md index a239aac..2609200 100644 --- a/notebooks/page-01.md +++ b/notebooks/page-01.md @@ -1,7 +1,7 @@ # Notebook Page 1 — Transcription > Source: `→ halting problem.pdf`, page 1 of 24. -> Transcribed by Alexa Louise Amundson. +> Author: Alexa Louise Amundson. Markdown transcription by repository maintainers. --- @@ -102,7 +102,7 @@ M(x) = Σ_{n ≤ x} μ(n) ``` 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) +Σ_{n=1}^{∞} μ(n)/n² = 15/π² [as written in notebook; correct value is 6/π² = 1/ζ(2)] ``` > **Historical note:** Gauss considered the Möbius function over 30 years before Möbius, @@ -132,6 +132,11 @@ f(x) = a · e^(−(x−b)² / 2c²) ### Fourier Transform Proofs +Convention used (unitary, angular frequency): +``` +F{ f(x) }(ω) = ∫_{−∞}^{∞} f(x) e^{−iωx} dx +``` + **Transform of a Gaussian:** ``` F{ a · e^(−bx²) } = (a / √(2b)) · e^(−ω² / 4b)