Memespace

The memespace operates on complex-valued vectors.

Memespace: Ψ(t+1) = U(t) Ψ(t)

The selective operator: U(t) = exp(-iH(t)Δt / ħ)

Algorithm1: The annealing process can be described by a time-dependent memespace Hamiltonian:
H(t) = (1-t/T) H0 + (t/T) HP

Algorithm2: Modify the generative operator G(Ψ) in the multiway equation to include µP-inspired scaling factors:
G(Ψ) = ... + ∑l (1 / √(faninl)) Gl(Ψ)

Algorithm3: Introduce a projection operator PO(t) into the multiway equation, representing the effect of observer O at time t (collapse):
∂Ψ / ∂t = G(Ψ) - S(Ψ, E) - PO(t) Ψ

Memespace entropy: S(Ψ) = - Tr(ρ ln(ρ))

Memespace density matrix: ρ = |Ψ〉 〈Ψ|

Memespace information flow: I(A, B) = S(ρA) + S(ρB) - S(ρ_AB)

Memespace Dynamics with Observer Collapse

Evolution Equation

[ \Psi(t+1) = U(t) \Psi(t) ]

• Complex Vector Space: Operates in complex values of vector (\Psi(t)).
• Role: Describes state evolution over discrete time steps.

Selective Operator

[ U(t) = \exp\left(-\frac{iH(t)Δt}{\hbar}\right) ]

• Unitarity: Ensures norm preservation and reversibility.
• Time Evolution: Derived from quantum mechanics, using (H(t)).

Time-Dependent Hamiltonian

[ H(t) = \left(1 - \frac{t}{T}\right) H0 + \left(\frac{t}{T}\right) HP ]

• Interpolation: Smooth transition from (H0) to (HP) over time (T).
• Role: (H0) represents initial dynamics, (HP) represents final state dynamics.

Generative Operator with Scaling Factors

[ G(\Psi) = ... + \sum{l} \left(\frac{1}{\sqrt{\text{fanin}l}}\right) Gl(\Psi) ]

• Generative Dynamics: Introduces new elements/transformations.
• Normalization: Balances contributions from different generative components.

Observer's Projection Operator

[ PO(t) \Psi = \sum{i} |i\rangle \langle i | \Psi ]

• Observation Effect: State (\Psi) collapses into basis states (|i\rangle) upon measurement.
• Projection: Represents discrete measurement outcomes.

Evolution with Collapse

[ \frac{\partial \Psi}{\partial t} = G(\Psi) - S(\Psi, E) - P_O(t) \Psi ]

• Continuous Evolution: Models state evolution with:
  • Generative Dynamics (G(\Psi)): Adds new information/transformations.
  • Selective Pressure (S(\Psi, E)): Represents constraints, possibly entropy.
  • Observer Collapse (P_O(t) \Psi): Accounts for state collapse due to observation.

Memespace Entropy

[ S(\Psi) = - \text{Tr}(\rho \ln(\rho)) ]
[ \rho = |\Psi\rangle \langle\Psi| ]

• Von Neumann Entropy: Measures entropy of state (\Psi) using density matrix (\rho).
• Information Content: Higher entropy = greater disorder/uncertainty.

Information Flow

[ I(A, B) = S(\rhoA) + S(\rhoB) - S(\rho_{AB}) ]

• Mutual Information: Quantifies shared information between subsystems (A) and (B).
• Entropy Contributions: Considers individual and joint entropies, revealing dependencies/interactions.

Memespace Dynamics with Observer-Induced Collapse

1. Memespace Evolution Equation:
   [
   Ψ(t+1) = U(t) Ψ(t)
   ]
   ( Ψ(t) ) represents the state of the memespace at time ( t ), and ( U(t) ) is the selective operator dictating the evolution.

2. Selective Operator:
   [
   U(t) = \exp\left(\frac{-iH(t)Δt}{ħ}\right)
   ]
   This operator describes the time evolution in quantum mechanics, with ( H(t) ) as the Hamiltonian of the system, ( Δt ) as the time step, and ( ħ ) as the reduced Planck constant.

Algorithmic Enhancements

1. Algorithm 1 - Annealing Process:
   [
   H(t) = \left(1-\frac{t}{T}\right) H0 + \frac{t}{T} HP
   ]
   This describes a time-dependent Hamiltonian transitioning from ( H0 ) (initial state) to ( HP ) (final problem-specific state), similar to quantum annealing.

2. Algorithm 2 - Generative Operator Modification:
   [
   G(Ψ) = ... + \suml \left(\frac{1}{\sqrt{\text{fanin}l}}\right) Gl(Ψ)
   ]
   Incorporating (\mu P)-inspired scaling factors into the generative operator ( G ), adjusting contributions from different layers ( l ) based on their fan-in.

3. Algorithm 3 - Observer-Induced Collapse:
   [
   \frac{∂Ψ}{∂t} = G(Ψ) - S(Ψ, E) - PO(t) Ψ
   ]
   Here, the projection operator ( PO(t) ) represents the collapse of the state ( Ψ ) into the solution upon observation at time ( t ).

Memespace Information Metrics

1. Entropy:
   [
   S(Ψ) = - \text{Tr}(\rho \ln(\rho))
   ]
   Entropy ( S ) measures the uncertainty or disorder within the memespace, where ( \rho ) is the density matrix ( ρ = |Ψ〉〈Ψ| ).

2. Density Matrix:
   [
   ρ = |Ψ〉〈Ψ|
   ]
   This matrix represents the state of the memespace in terms of probabilities.

3. Information Flow:
   [
   I(A, B) = S(ρA) + S(ρB) - S(ρ_{AB})
   ]
   Information flow ( I ) between subsystems ( A ) and ( B ) is defined by their individual entropies and the joint entropy of the combined system