Research-Stack/0-Core-Formalism/lean/Semantics/EQUATIONS_EXTRACTION.md

Equation Extraction from ChatGPT Batch Files

Summary

Extracted all equation variations from ChatGPT-Batch-2026-04-22.zip files.

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1. ChatGPT-Making_It_Rigorous.md

Gabriel's Horn

H = {(x,y,z) ∈ R³ : x ≥ 1, y² + z² ≤ x⁻²}

• Finite volume, infinite boundary area

PIST / Hyperbola Index

k = ⌊√n⌋
a(n) = n - k²
b(n) = (k+1)² - n
m(n) = a(n)b(n)

• Square-bracket coordinates
• Hyperbola index / mass

Menger Sponge Dimension

dim_H(M) = log(20) / log(3) ≈ 2.7268

Euler's Product

ζ(s) = ∏_p 1/(1 - p⁻ˢ), Re(s) > 1
∑_{n=1}^∞ 1/n² = π²/6
∏_p 1/(1 - p⁻²) = π²/6

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2. ChatGPT-Time_Motion_Friction_Derivation.md

Friction Force

F_fric = γv
F = γv
v = F/γ
t = x/v
t = γx/F

Damped Harmonic Oscillator

M z̈ + C ż + K z = f(t)

Attenuation

I(r,t) = I₀(r) exp(-∫_ℓ(r) μ(x;z(t)) ds)

Overdamped System

m q̈ + c q̇ + k q = u(t)
c q̇ + kq ≈ u(t)
c q̇ ≈ u(t)
Δq ≈ (u/c) Δt

Scaling Law

t ∝ γL² / (k_B T)

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3. ChatGPT-Refinement_of_Update_Rule.md

State Update System

S_k = (G_k, C_k, L_k, P_k)
S_{k+1} = Φ(S_k, u_k)
Φ = Prune ∘ Canonicalize ∘ CacheUpdate ∘ LocalUpdate

Geometry Update

G_{k+1} = Π(U(G_k, u_k), P_k)
C_{k+1}(x) = α C_k(x) + (1-α) score(x, G_{k+1})
L_{k+1}[h(canon(G_{k+1}))] ← best(L_k, canon(G_{k+1}))

Pruning

P_{k+1}(x) = 
  1 if C_{k+1}(x) < τ or x induces forbidden alignment
  0 otherwise

Loss Function

min_θ L(θ) = 
  λ₁ N_unresolved(θ) + 
  λ₂ N_revisited(θ) + 
  λ₃ N_degenerate(θ) + 
  λ₄ T_update(θ)

Nutrient Evolution

N^{tot}_j(t+1) = (1-λ_j) N^{tot}_j(t) + ΔN^{gain}_j(t) - ΔN^{export}_j(t) - ΔN^{duty}_j(t) - ΔN^{vol}_j(t) + ΔN^{lock}_j(t)

ΔN^{gain}_i = ∑_j S(Q_i → Q_j,t) W_{myco}(Q_i → Q_j,t) (α₁ χ^{crc}_{ij} + α₂ χ^{btree}_{ij} + α₃ χ^{ammr}_{ij})

Nutrient Decay

N^{local}_j(t+1) = (1-λ_L) N^{local}_j(t) + ΔN^{local}_j(t) - β_L C_j(t)
N^{indexed}_j(t+1) = (1-λ_I) N^{indexed}_j(t) + ΔN^{indexed}_j(t) - β_I C_j(t)
N^{committed}_j(t+1) = (1-λ_C) N^{committed}_j(t) + ΔN^{committed}_j(t)
λ_C < λ_I < λ_L

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4. ChatGPT-Hutter_Prize_Compression_#1.md

Anisotropic Torsional Gradient Flow

∂_t φ = -∇_i (M^{ij}(x) ∇_j μ) + λ T[φ,g,T]
μ = δF/δφ

Embedding Dynamics

∂_t X^A = -Γ^A_{BC}(X) ∂_i X^B ∂^i X^C - Λ^{AB}(x) (X_B - X_{0B}) - δI_lock/δX_A + τ T^A[T,X]

Energy Functional

F[φ,X;g,T] = ∫_M [V(φ) + (κ_{ij}/2) ∇_i φ ∇_j φ + (C^{ij}_{AB}/2) ∇_i X^A ∇_j X^B + (μ/2) A^{ij} (X-X₀)_A (X-X₀)_B Q^{AB}_{ij} + α T_{ijk} T^{ijk} φ² + β I_lock(φ,X,A)] dvol_g

Locking Potential

I_lock = ∑_m ∫_M W(P_m(X,φ) - P_{m-1}(X,φ); A^{ij}) dvol_g
W(z;A) = ∑_r w_r(A) (1 - cos(k_r · z))

Stress Tensor

Σ_{ij} = Σ^{(φ)}_{ij} + Σ^{(X)}_{ij} + Σ^{(T)}_{ij} + Σ^{(lock)}_{ij}
Σ^{(T)}_{ij} = χ T_i^{ab} T_{jab} - (χ/2) g_{ij} T_{abc} T^{abc}

Recursive Menger Structure

P_{m+1} = R_{A_m}(P_m) ∩ C_m

Energy Dissipation

d/dt F(φ_t, X_t) ≤ 0

PDE System

∂_t φ = ∇ · (M(φ) ∇ μ) - σ ∂_φ W(φ, X)
∂_t X = -Λ(X - X₀) + ΔX + τ T(∇X)

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5. ChatGPT-Couch_as_Tetris_Manifold.md

Configuration Space

C = {x, y, θ, φ₁, φ₂, ...}

Potential Field

G(x) : Rⁿ → [0, ∞]
E_env(q) = ∫_{p ∈ M(q)} G(p) dμ(p)

Action

A[q] = ∫ (T(q, q̇) + E_env(q) + E_self(q)) dt

Gradient

ε(p) = ‖∇G(p)‖

Force Field

F(q) = ∑_i w_i δ(q - q_i)

Optimization

min_q (E_env(q) + E_self(q) + λ F(q))

Wall Potential

G(x) = 1 / d(x, ∂Ω)^k

Quaternion

q = (x, y, q_w, q_x, q_y, q_z)
‖q‖ = 1
Configuration space = R² × SO(3)

Normal Vector

n(x) = -∇G(x) / ‖∇G(x)‖

Alignment Energy

E_align = ∫ ρ(x,t) Φ(q(x,t), n(x)) dx

Total Energy

E = E_flow + E_stress + E_align + E_wall
E_wall = ∫ ρ(x,t) G(x) dx
E_stress = ∫ W(σ, ∇q, ∇ρ) dx
E_align = ∫ ρ Φ(q, ∇G) dx

Continuity Equation

∂_t ρ + ∇ · (ρ v) = 0

Momentum Balance

ρ(∂_t v + v · ∇v) = -∇p + ∇ · σ - ρ ∇G + f_align

Quaternion Transport

∂_t q + v · ∇q = ...

Update Rule

q_{t+1} = Π_A(q_t + Δq_t)
A = {q : E_wall(q) < ∞}

Manifold

M = {x | ρ(x,t) > 0 for a stable trajectory}

Wall Set

W := {x ∈ Ω : G(x) = ∞}
Ω_f := Ω \ W
G(x) = d(x, W)^{-k} for x ∈ Ω_f

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6. ChatGPT-Formal_Lean_Pipeline.md

Unified Field Potential

Φ(x) = (ρ² + v² + τ² + σ² + q²) / ((1 + κ²)(1 + ε))

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Equation Categories

1. Energy Functionals

• Gabriel's horn energy
• PIST energy
• HyperFabric energy functional
• Total energy (flow, stress, align, wall)
• Unified field potential

2. Differential Equations

• Gradient flow equations
• Heat equation variants
• Damped harmonic oscillator
• Continuity equation
• Momentum balance
• Quaternion transport

3. Optimization Problems

• Minimization with constraints
• Loss functions
• Action minimization

4. Geometric Equations

• Menger sponge dimension
• Configuration space
• Manifold definitions
• Normal vectors
• Curvature

5. Discrete Updates

• State update rules
• Nutrient evolution
• Pruning equations
• Cache updates

6. Stress/Force Equations

• Stress tensors
• Friction force
• Force fields
• Alignment forces

7. Probability/Statistics

• Euler's product
• Zeta function
• Probability distributions

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Key Equations for Hachimoji Pipeline Integration

1. Energy Dissipation Theorem

d/dt F ≤ 0

• Already integrated as isEnergyDissipating function

2. Stress Tensor

Σ = Σ_phase + Σ_elastic + Σ_torsional + Σ_locking

• Already integrated as NDStress structure

3. Nutrient Evolution

N(t+1) = (1-λ)N(t) + gain - cost

• Already integrated as updateNutrient function

4. Anisotropic Gradient Flow

∂_t φ = -∇ · (M ∇ μ) + λ T

• Could be integrated for adaptive encoding

5. Recursive Structure

P_{m+1} = R(P_m) ∩ C_m

• Could be integrated for hierarchical encoding

6. Unified Field Potential

Φ = (ρ² + v² + τ² + σ² + q²) / ((1 + κ²)(1 + ε))

• Could be integrated for compression efficiency scoring