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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']}")
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