#### Determinism and Uncertainty Reduction
# Resultant R = e^{±λZ}
"""

Key Features:
- λ: Control parameter that regulates the speed of transitions.
- Z: State variable that defines the position along the continuum.
- Vectors: Integral of primary directions (λD primary) and emerging possibilities
(λP possibilistic), reducing residual latency (λL latency).
"""

#### Formalization of Angular Momentum and Assonances
# Angular Momentum (θ_{NT})
def angular_momentum(R_t, omega):
   """
   Formalizes the cyclical equilibrium between observer and observed:
   θ_{NT} = lim_{t \to 0} \left( R(t) \cdot e^{i\omega t} \right)

   Parameters:
   R_t: Temporal resultant (function of time t).
   omega: Dominant frequency of oscillations.

   Returns:
   The formal angular momentum.
   """
   from sympy import limit, symbols, exp, I
   t = symbols('t')
   return limit(R_t * exp(I * omega * t), t, 0)

#### Optimization with the Principle of Least Action
# Unified Equation
def least_action_equation(delta, alpha, beta, gamma, f_dual_nond, f_movement, f_absorb_align, R_t, proto_axiom):
   """
   R(t+1) = δ(t) \left[ α ⋅ f_{Dual-NonDual}(A, B; λ) + β ⋅ f_{Movement}(R(t), P_{Proto-Axiom}) \right] + \
           (1 - δ(t)) \left[ γ ⋅ f_{Absorb-Align}(R(t), P_{Proto-Axiom}) \right]

   Parameters:
   delta: Transition factor.
   alpha, beta, gamma: Weights of the logical contributions.
   Other functions specified in the context.
   
   Returns:
   Updated resultant R.
   """
   return delta * (alpha * f_dual_nond + beta * f_movement) + (1 - delta) * gamma * f_absorb_align

#### Formalization of the Principle of Least Action
# Proposed Lagrangian
def lagrangian(dZ_dt, Z, theta_NT, lambda_param, f_theta, g_Z):
   """
   L = \frac{1}{2}\left(\frac{dZ}{dt}\right)^2 - V(Z, \theta_{NT}, \lambda)

   Potential V(Z, θ_{NT}, λ):
   V(Z, θ_{NT}, λ) = Z^2(1-Z)^2 + λ f(θ_{NT})g(Z)

   Returns:
   Lagrangian calculated for the given parameters.
   """
   return 0.5 * (dZ_dt ** 2) - (Z ** 2 * (1 - Z) ** 2 + lambda_param * f_theta(theta_NT) * g_Z(Z))

# NT Continuum
"""
The Nothing-Totality (NT) continuum represents the complete spectrum of dynamic possibilities.
Each resultant R updates the logical context and feeds the system by eliminating latency
and improving coherence. The D-ND model uses the NT to navigate between states of least
action, keeping the observer at the center of the system.
"""