Fracurve, a nonlinear Schrödinger equation fractal-curve.
Appleby, Ethan, Appleby, Ethan
Sorry for the schizoposting, update on me. I survided a self inflicted chainsaw injury to neck/sholders anyway here is the conclusion to my work. my fracurve. i · dΨ(λ)/dλ = H[Ψ(λ)] · Ψ(λ) with H[Ψ] = H₀ + g · Σᵥ Pᵥ[Ψ] · |v⟩⟨v| Pᵥ[Ψ] = Σₙ |Ψ(v, n)|² if identical things follow identical rules then that begs what is a thing, if a thing has many gauges then what if the thing is all gauges then must evolve over a rule that is for all these gauges not the thing as a gauge. linear sketch Setup Work with: v = velocity index, v = 0..N_v-1 n = pattern index on a circle, n = 0..N_p-1 Psi(v, n; m) = state at discrete rule-step m Psi(v, n; lambda) = state at continuous rule-time lambda Define the DFT over velocities: F_vel is an N_v x N_v matrix with entries F_vel(v, v') = (1 / sqrt(N_v)) * exp( -2 * pi * i * v * v' / N_v ) Introduce a constant phase offset h_bar and define the universal velocity evolution operator: U_vel = exp(-i * h_bar) * F_vel This operator encodes the single rule applied uniformly to all velocities. 1. Discrete Fracurve equation The discrete Fracurve evolution is: Psi_{m+1}(v, n) = sum over v' [ U_vel(v, v') * Psi_m(v', n) ] with U_vel(v, v') = (exp(-i * h_bar) / sqrt(N_v)) * exp( -2 * pi * i * v * v' / N_v ) For each step m -> m+1, U_vel acts on the velocity index v, while the pattern index n is carried along. The sequence m = 0, 1, 2, ... → Psi_m(v, n) is the discrete Fracurve: a path through the combined (velocity x pattern) state space. 2. Continuous Fracurve equation Promote the step operator U_vel to a continuous generator H_F by: U_vel = exp( -i * H_F ) (H_F is the effective Hamiltonian acting on the velocity space.) Then the continuous Fracurve equation is: i * d/dlambda Psi(v, n; lambda) = sum over v' [ H_F(v, v') * Psi(v', n; lambda) ] In compact vector form: i * d/dlambda |Psi(lambda)> = H_F |Psi(lambda)> with solution: |Psi(lambda)> = exp( -i * H_F * lambda ) |Psi(0)> The curve lambda -> |Psi(lambda)> in the combined velocity–pattern state space is the Fracurve. i · dΨ_s/dλ = H_bit[{Ψ}] · Ψ_s H_bit[{Ψ}] = i · B · (M + μ I)^(-1) M = Σ_s Ψ_s Ψ_s^† B = Σ_s (dΨ_s/dλ) Ψ_s^† S[H, {Ψ_s}] = ∫ dλ [ Σ_s || i dΨ_s/dλ − H Ψ_s ||^2 + μ · Tr(H^† H) ] Key: Ψ_s : state vector for slice / observation s λ : rule-time (Fracurve parameter) H : candidate rule (Hamiltonian operator) H_bit : bit-minimizing / least-action rule derived from all Ψ_s μ : small positive regularization constant I : identity operator Σ_s : sum over slices / observations s (·)^† : Hermitian conjugate (conjugate transpose) Tr : trace of a matrix/operator S[H, {Ψ_s}] : action functional over rule H and states Ψ_s 1. Universe / Fracurve dynamics Continuous rule-time evolution i * dΨ/dλ = H * Ψ Discrete step form Ψ_{m+1} = U * Ψ_mU = exp( -i * H * Δλ ) 2. State–dependent / learned rule Action (least–action over rules and states) S[H, {Ψ_s}] = ∫ dλ [ Σ_s || i * dΨ_s/dλ - H * Ψ_s ||^2 + μ * Tr( H† * H )] Data matrices M = Σ_s Σ_t Ψ_s(t) * Ψ_s(t)†B = Σ_s Σ_t (dΨ_s/dλ)(t) * Ψ_s(t)† Least–action rule (bit Hamiltonian) H_bit = i * B * ( M + μ * I )^(-1) 3. Effective dimension of the universe Let {λ_j} be the eigenvalues of M. Effective dimension d_eff(μ) = Σ_j λ_j / ( λ_j + μ ) This is the “how many directions are really active” function. 4. Split into real and imaginary parts Write the universe state as: Ψ = a + i * b where a and b are real vectors. You can define separate covariances: M_R = Σ a * a^T (real-part covariance)M_I = Σ b * b^T (imag-part covariance) with eigenvalues: M_R → eigenvalues λ_j^(R)M_I → eigenvalues λ_j^(I) Then you get two effective dimensions: d_R(μ) = Σ_j λ_j^(R) / ( λ_j^(R) + μ ) d_I(μ) = Σ_j λ_j^(I) / ( λ_j^(I) + μ ) Universe / Fracurve dynamics Continuous rule–time evolution: i * dΨ/dλ = H * Ψ Discrete step form: Ψ_{m+1} = U * Ψ_mU = exp( -i * H * Δλ ) Action (least–action over rules and states) S[H, {Ψ_s}] = ∫ dλ [sum_s || i * dΨ_s/dλ - H * Ψ_s ||^2+ μ * Tr( H† * H )] Here: μ is a small positive regularization constant H† is the Hermitian conjugate of H Data matrices M = sum_s sum_t Ψ_s(t) * Ψ_s(t)† B = sum_s sum_t (dΨ_s/dλ)(t) * Ψ_s(t)† Least–action rule (bit Hamiltonian) H_bit = i * B * ( M + μ * I )^(-1) I is the identity matrix. Effective dimension of the universe Let {λ_j} be the eigenvalues of M. d_eff(μ) = sum_j [ λ_j / ( λ_j + μ ) ] This is the “how many directions are really active at scale μ” function. Split into real and imaginary parts Write the universe state as: Ψ = a + i * b where a and b are real vectors. Define separate real covariances: M_R = sum over samples ( a * a^T ) (real–part covariance)M_I = sum over samples ( b * b^T ) (imag–part covariance) Let the eigenvalues be: M_R → eigenvalues λ_j^(R)M_I → eigenvalues λ_j^(I) Then define two effective dimensions: d_R(μ) = sum_j [ λ_j^(R) / ( λ_j^(R) + μ ) ] d_I(μ) = sum_j [ λ_j^(I) / ( λ_j^(I) + μ ) ] 4 + 4 → complex 4–vector Take 4 real degrees of freedom: a ∈ R^4 Take another 4 real degrees of freedom: b ∈ R^4 Combine them into one complex 4–vector: Ψ = a + i * b ∈ C^4 λ↦{Ψ_s(λ)}↦{x_s^μ(λ)} x_s^μ(λ; Φ₀) = F^μ( U_s(λ) · exp( -i · H_phys · λ ) · S · |Φ₀⟩ ) Start from X(λ,u) = Σ_k G_k(λ) F_k(u) Differentiate with respect to λ: i ∂X(λ,u)/∂λ= i ∂/∂λ [ Σ_k G_k(λ) F_k(u) ]= Σ_k [ i dG_k(λ)/dλ ] F_k(u)= Σ_k [ Σ_ℓ H_bit(k,ℓ) G_ℓ(λ) ] F_k(u)= Σ_{k,ℓ} H_bit(k,ℓ) G_ℓ(λ) F_k(u) So the complete fracurve equation (discrete-mode form) is: i ∂X(λ,u)/∂λ = Σ_{k,ℓ} H_bit(k,ℓ) G_ℓ(λ) F_k(u) and you always have the reconstruction: X(λ,u) = Σ_k G_k(λ) F_k(u)
•Nov 15, 2025•0 citations