""" 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']}")