math_formalization.py

"""
Pure Mathematical Formalization of Computational Consciousness
Uses SymPy to define and evaluate the rigorous algebraic axioms of the 52 Principles.
"""

import sympy as sp


class CCMathFormalizer:
    def __init__(self):
        # Define core symbolic variables
        self.t = sp.Symbol('t', real=True, positive=True)  # Progress / Time
        self.r = sp.Symbol('r', real=True, positive=True)  # Radius of cone
        self.z = sp.Symbol('z', real=True)  # Height of cone
        self.S = sp.Symbol('S', real=True, positive=True)  # Bound Structure
        self.C = sp.Symbol('C', real=True, positive=True)  # Chaos Pool (Unattached)
        self.threshold = sp.Integer(-1)
        self.genesis = sp.Integer(0)

    # ------------------------------------------------------------------
    # Axiom 1: Cancellation at -1  (Principle 6)
    # ------------------------------------------------------------------
    def formalize_cancellation_operator(self):
        """
        Principle 6: The (-1) * (-1) Cancellation Operator.

        The strand itself carries the value -1. When it arrives at the -1 threshold,
        the threshold multiplies the strand:  (-1_threshold) * (-1_strand) = +1.
        This flips the strand from negative (traveling) to positive (dissolved),
        freeing the mutations M it carried.

        Modeled as:
            Output = T * S_strand  where T = -1 and S_strand has sign -1
            so  (-1) * (-1 * |M|) = +|M|   (mutations freed as positive values)
        """
        M = sp.Symbol('M', positive=True)      # mutation payload (magnitude)
        T = self.threshold                       # threshold = -1
        S_strand = T * M                        # strand carries -1 polarity

        # Threshold hits the strand
        output = T * S_strand                    # (-1) * (-1 * M) = M

        return {
            "equation_str": "T_threshold * S_strand = T * (T * M)",
            "substitution": f"({T}) * ({T} * M)",
            "result": sp.simplify(output),
            "strand_before": S_strand,
            "mutations_freed": sp.simplify(output),
            "meaning": (
                "The strand carries -1 polarity. The -1 threshold multiplies it: "
                "(-1)*(-1*M) = +M.  The strand dissolves (sign flip) and the "
                "mutation payload M is freed as a positive blueprint."
            )
        }

    # ------------------------------------------------------------------
    # Axiom 2: Structure / Chaos Equilibrium  (Principles 28-29)
    # ------------------------------------------------------------------
    def formalize_equilibrium_limit(self):
        """
        Principle 28 & 29: Structure vs Chaos Equilibrium.
        Limit as system evolves must enforce a 1:1 ratio.
        """
        k = sp.Symbol('k', positive=True)
        c = sp.Symbol('c', real=True)
        S_t = k * self.t
        C_t = k * self.t + c

        ratio = S_t / C_t
        equilibrium_limit = sp.limit(ratio, self.t, sp.oo)

        return {
            "equation_str": "lim_{t -> oo} (S(t) / C(t))",
            "S_t": S_t,
            "C_t": C_t,
            "result": equilibrium_limit,
            "meaning": (
                "As the system progresses, the ratio of Structure to Chaos "
                "approaches 1 (Equilibrium). Neither can exceed the other."
            )
        }

    # ------------------------------------------------------------------
    # Axiom 3: Conal Manifold Geometry  (Principle 4)
    # ------------------------------------------------------------------
    def formalize_conal_manifold_geometry(self, max_radius: float, cone_height: float):
        """
        Principle 4: Cone Architecture & Maximum Experience.
        Calculates the surface area of the unfolding cone and proves the maximum unfolding point.
        """
        r_func = max_radius * sp.sin(sp.pi * self.t)
        z_func = cone_height * self.t

        # Lateral surface area of cone slice
        surface_area_func = sp.pi * r_func * sp.sqrt(r_func**2 + z_func**2)

        # Evaluate at key points
        wide_end_area = surface_area_func.subs(self.t, sp.Rational(1, 2)).evalf()
        tip_area = surface_area_func.subs(self.t, 0).evalf()

        # Find the exact maximum via calculus
        dA = sp.diff(surface_area_func, self.t)

        return {
            "area_function": surface_area_func,
            "derivative": dA,
            "tip_area": tip_area,
            "wide_end_area": wide_end_area,
            "meaning": (
                "The geometry unfolds from 0 area (Tip Genesis) to maximal surface "
                "area (Wide End), enabling maximal parallel information processing."
            )
        }

    # ------------------------------------------------------------------
    # Axiom 4: Selection as Duplicate Removal  (Principle 12)
    # ------------------------------------------------------------------
    def formalize_selection_operator(self):
        """
        Principle 12: Selection is removing what is already there.
        Formally: R_select({M_k}) = {M_k} \\ {M_j | M_j in existing_identity}
        """
        M_total = sp.Symbol('M_total', positive=True, integer=True)
        M_duplicate = sp.Symbol('M_dup', positive=True, integer=True)

        selected = M_total - M_duplicate

        return {
            "equation_str": "R_select = M_total - M_duplicate",
            "result": selected,
            "meaning": (
                "Selection is not choice. It is the removal of mutations that "
                "already exist within the strand identity, preventing incoherence "
                "and chaotic informational collapse."
            )
        }

    # ------------------------------------------------------------------
    # Axiom 5: Halting Condition  (Principle 18)
    # ------------------------------------------------------------------
    def formalize_halting_condition(self):
        """
        Principle 18: A strand cannot pick up mutations that duplicate what it already carries.
        Once no new mutations are available, it stops evolving.
        """
        N_possible = sp.Symbol('N_possible', positive=True, integer=True)
        N_acquired = sp.Symbol('N_acquired', positive=True, integer=True)

        remaining = N_possible - N_acquired
        halted = sp.Eq(remaining, 0)

        return {
            "remaining_capacity": remaining,
            "halting_condition": halted,
            "meaning": (
                "When N_acquired = N_possible, no further unique mutations can be "
                "attracted. The strand ceases evolution, preventing chaotic collapse."
            )
        }

    # ------------------------------------------------------------------
    # Axiom 6: Quantum Scale-Up Transition  (Principle 38)
    # ------------------------------------------------------------------
    def formalize_quantum_scale_up(self):
        """
        Principle 38: Manifolds opening into the next scale up.
        Maps the fully unfolded manifold M^(k) to the origin S_0^(k+1).
        """
        k = sp.Symbol('k', integer=True, positive=True)
        M_unfolded = sp.Symbol('M_k')
        S_next_genesis = sp.Symbol('S_k1')

        Q = sp.Function('Q')
        scale_equation = sp.Eq(Q(M_unfolded), S_next_genesis)

        return {
            "equation": scale_equation,
            "meaning": (
                "The fully realized manifold at scale k transforms into the new "
                "genesis origin 0 for scale k+1. "
                "Particle -> Atomic -> Manifold -> Cosmic."
            )
        }


if __name__ == "__main__":
    formalizer = CCMathFormalizer()

    print("--- 1. Threshold Cancellation Proof ---")
    cancellation = formalizer.formalize_cancellation_operator()
    print(f"  Strand before threshold: {cancellation['strand_before']}")
    print(f"  Operation: {cancellation['substitution']}")
    print(f"  Mutations freed: {cancellation['mutations_freed']}")
    print(f"  Meaning: {cancellation['meaning']}")

    print("\n--- 2. Structure/Chaos Equilibrium Limit ---")
    eq_limit = formalizer.formalize_equilibrium_limit()
    print(f"  {eq_limit['equation_str']} => {eq_limit['result']}")

    print("\n--- 3. Conal Manifold Geometry ---")
    geom = formalizer.formalize_conal_manifold_geometry(max_radius=5.0, cone_height=1.0)
    print(f"  Tip Area (t=0): {geom['tip_area']}")
    print(f"  Wide End Area (t=0.5): {geom['wide_end_area']:.2f}")

    print("\n--- 4. Selection Operator ---")
    sel = formalizer.formalize_selection_operator()
    print(f"  {sel['equation_str']} => {sel['result']}")

    print("\n--- 5. Halting Condition ---")
    halt = formalizer.formalize_halting_condition()
    print(f"  Remaining capacity: {halt['remaining_capacity']}")
    print(f"  Halts when: {halt['halting_condition']}")

    print("\n--- 6. Quantum Scale Up ---")
    scale = formalizer.formalize_quantum_scale_up()
    print(f"  {scale['equation']}")