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

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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

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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}

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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

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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

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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

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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

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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)

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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

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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

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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]...

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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
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

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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.