Research-Stack/3-Mathematical-Models/extracted_equations.md

# Extracted Equations from Today's Articles and Theory Documents

## 1. From CERN / Physics Articles

### Muon g-2 (Fermilab 2025 / Nature / PNAS)

**Anomalous magnetic moment:**
```
a_μ = (|g| - 2) / 2 = 0.001165920705(114)
```

**Precision:**
```
σ = 0.127 ppm    (0.000000114)
```

**Standard Model prediction (now matched):**
```
a_μ^theory = a_μ^experiment     within 0.5σ
```

**g-factor:**
```
g_μ = -2.00233184122(82)
```

**Magnetic moment relation:**
```
μ = g · (eℏ / 2m) · S
```

---

## 2. From Fractional Unified Field Theory

### Unified fractional field equation

```
(D_t^α + (-∇²)^β) Ψ = λ |Ψ|^γ Ψ
```

Where:
- `D^α` = fractional derivative of order α
- `λ` = foam-level coupling constant
- `β` = self-interaction nonlinearity exponent
- `γ` = interaction power

### Riesz fractional derivative

```
D^α f(x) = F^{-1}[ |k|^α · F[f](k) ]
```

Eigenfunctions: `exp(i k x)` with eigenvalues `|k|^α`

### Force emergence (resonant quantization)

| Force | α | 1/α |
|-------|---|-----|
| Electromagnetism | 1 | 1 |
| Weak | 1/2 | 2 |
| Strong | 1/3 | 3 |
| Gravity | 1/4 | 4 |

### Anthropic shear transformation

```
Ψ_observed(x, t) = ∫ K_θ(x - x') Ψ_unified(x', t) dx'
```

### Mode weight under shear

```
w(α, θ) = sin(θ)^α · cos(θ)^{1-α}
```

### Coupling hierarchy

```
g_n(θ_obs) = g_0 · sin(θ_obs)^{1/n} · cos(θ_obs)^{1 - 1/n}
```

### Charge quantization

```
Q_n = (1/2π) ∮_C ∇_n φ · dn = m/n   for m ∈ ℤ
```

---

## 3. From Torsional Cosmology

### Torsional spacetime metric

```
ds² = -dθ²/ω(θ)² + a(θ)² [dr²/(1-kr²) + r² dΩ²] + ℓ_P² dθ² Γ(θ)
```

### Variable torsional rotation cases

| Case | ω(θ) | Expansion a(t) |
|------|-------|---------------|
| Constant | ω_0 | Exponential |
| Accelerating | ω_0 · θ | Super-exponential |
| Decelerating | ω_0 / θ | Power-law |
| Oscillating | ω_0 · sin(θ/θ_0) | Cyclic / bounce |
| Damping | ω_0 · exp(-θ/θ_c) | Asymptotic halt |

### Hubble parameter

```
H_eff = ω(θ)
```

### Dark energy as residual torsion

```
ρ_DE(θ) = ρ_foam · (1 - θ/θ_max)²
```

### Bekenstein bound on fractal

```
S(R) ≤ C' · R^{D_H} · T^{(D_H - 1)}
```

For D_H = 1.44:
```
S ≤ C' · R^{1.44} · T^{0.44}
```

---

## 4. From Recursive Branch-Cut Self-Similarity

### Hyperbolic area growth

```
A(r) = 2π (cosh(r) - 1) ≈ π · exp(r)    for r >> 1
```

### Scaling factor

```
L_{n+1} / L_n ≈ exp(d_inj) ≈ Φ² ≈ 2.618
```

### Spectral dimension

```
D_s = 2 D_H / (1 + D_H)
```

For D_H = 2 (sheet-like foam):
```
D_s = 4/3 ≈ 1.333
```

### Energy eigenvalue density

```
ρ(E) ~ E^{D_s/2 - 1} = E^{-1/3}
```

### Mode quantization

```
E_n ~ n³
```

### CMB spectral index

```
n_s = 1 - 2/(1 + θ_max/θ_recombination) ≈ 0.965
```

---

## 5. From Thermodynamic Tests

### Landauer limit

```
E_dissipated ≥ k_B T · ln(2)     per bit erased
```

### Shannon entropy

```
H(X) = -Σ p(x) log p(x)
```

### Bekenstein bound (standard)

```
S ≤ 2π R E / (ℏ c ln 2) = A / (4 G ℏ)
```

### Debye specific heat (3D)

```
C_V = (12π⁴/5) N k_B (T/Θ_D)³ ∝ T³
```

### Fractal specific heat

```
C_V ∝ T^{D_s}
```

For D_s = 1.18:
```
C_V ∝ T^{1.18}
```

### Jarzynski equality

```
⟨exp(-β W)⟩ = exp(-β ΔF)
```

### Thermodynamic uncertainty relation

```
(ΔJ)² / ⟨J⟩² · σ ≥ 2 k_B
```

---

## 6. From Quantum Uncertainty / Double-Slit

### Heisenberg uncertainty (derived from torsional sampling)

```
Δx · Δp ≥ ℏ/2
```

where `ℏ ≡ Δθ_min` (minimum resolvable phase interval)

### Double-slit interference

```
Ψ_total = Ψ_A + Ψ_B = A · exp(i ω_Ψ θ) · [exp(i k_Ψ x_A) + exp(i k_Ψ x_B)]
```

```
|Ψ_total|² = 2|A|² · [1 + cos(k_Ψ (x_A - x_B))]
```

### Planck relation

```
E = ℏ ω = ω_Ψ     (in natural units ℏ = 1)
```

### de Broglie relation

```
p = ℏ k = dθ_0/dx = k_Ψ
```

```
λ = 2π / k_Ψ = 2π / p
```

---

## 7. From Universal Evolutionary Equation

### Core equation

```
Phenotype(x, t) = Ψ_E [ Genotype(x) × Regulatory_State(t) ]
```

### Compression analog

```
Residual(n) = Ψ_decode [ Basis, Context(n) ] XOR Byte(n)
```

### Spectral entropy bound

```
H_Ψ(data) = -Σ_n p(n) log_2 p_Ψ(n) ≤ H_uniform(data) = 8 bits/byte
```

---

## 8. From van der Waals / Moiré Physics

### Moiré superlattice period

```
λ = a / (2 sin(θ/2))
```

For small θ:
```
λ ≈ a / θ
```

### Torsional force microscopy

Moiré period ~14.1 nm at twist angle ~0.99° (TBG)

---

## 9. From PIST Formalism

### Composite address

```
CompositeAddress = (Tree, Surface, Torus, Shell)
```

### Basis fusion operator

```
Ψ(A, B) = A ∩ B  ∪  (A \ B)  ∪  (B \ A)  ∪  Bridge(A, B)
```

### Mirror involution

```
t → 2k + 1 - t
```

### Gear ratio (AngrySphinx)

```
G_AS = 1 + α · L_FAMM + β · R + γ · U + δ · H
```

---

## 10. From Plant Acoustics / Biology

### Cavitation frequency (plant screams)

```
f_cav ≈ 30-50 clicks/hour   (stressed plants)
f_cav ≈ 0 clicks/hour     (healthy plants)
```

### Species classification

ML classifier:
```
P(species | sound_pattern) > threshold
```

Distinguishes:
- Dehydrated vs. cut
- Tomato vs. tobacco

---

## 11. From DNA / Genetics

### Genetic code mapping

```
f: ℤ₄ × ℤ₄ × ℤ₄ → ℤ₂₀ ∪ {stop}
```

64 codons → 20 amino acids + 1 start + 3 stop

### Supergene (inversion) structure

```
Normal:  A-B-C-D-E-F-G
Inverted: A-B-F-E-D-C-G
          └─inversion─┘
```

Recombination blocked: `P(recomb inside inversion) ≈ 0`

### Horizontal gene transfer

```
Beetle_Genome' = Beetle_Genome + Bacterial_Gene_HhMAN1
```

Flanked by transposable elements:
```
...[TE1]-HhMAN1-[TE2]...
```

---

## Summary Table: All Equations by Domain

| Domain | Key Equation | Physical Meaning |
|--------|-------------|------------------|
| Particle physics | `a_μ = 0.001165920705(114)` | Muon magnetic anomaly |
| Unified field | `D^α Ψ = λ|Ψ|^β Ψ` | Fractional field equation |
| Cosmology | `H_eff = ω(θ)` | Torsional Hubble parameter |
| Thermodynamics | `E_diss ≥ k_B T ln(2)` | Landauer limit |
| Quantum | `Δx·Δp ≥ ℏ/2` | Uncertainty from phase sampling |
| Evolution | `Phenotype = Ψ_E[Genotype × Context]` | Universal decode |
| Materials | `λ = a/(2 sin(θ/2))` | Moiré superlattice period |
| Compression | `Residual = Ψ[Basis, Context] XOR Data` | Moiré decoder |
| Biology | `f: ℤ₄³ → ℤ₂₀` | Genetic code as basis fusion |

---

*Compiled from: fractional_unified_field.md, torsional_cosmology_spin.md, thermodynamic_test_recursive_branch_cut.md, uncertainty_from_torsional_vibration.md, universal_evolutionary_equation.md, variable_omega_edge_anomalies.md, and SciTechDaily articles on muon g-2, plant acoustics, DNA inversions, evolution cheat sheet, horizontal gene transfer, and Sox9/Alzheimer's.*