pl/calculus.py

"""
pl/calculus.py — Calculus as Coherence
=======================================

Derivatives and integrals are not separate mathematical objects.
They are P / G → Q in the real carrier V = ℝ with specific gradient families.

The derivative: the gradient family that linearizes propagation.
               f'(x) is the load the function generates per unit step.
               e^x is the fixed point of this gradient family.
               Fixed point = e^x is its own derivative.
               This is not magic. It is forced.

The integral: accumulation of load across the carrier.
              ∫ f dx is the total load accumulated from a to b.
              Fundamental theorem = derivative and integral are inverses
              in this carrier. Forced by the structure of ℝ.

This module implements:
1. Numerical differentiation as P / G → Q
2. Numerical integration as accumulated propagation
3. The fixed point theorem for d/dx
4. Taylor series as recursive load accumulation
"""

from __future__ import annotations
from dataclasses import dataclass, field
from typing import Any, Callable, Dict, List, Optional, Tuple
import math

from pl.core import Pattern, Gradient, Context, seed


# =============================================================================
# FUNCTION AS PATTERN
# =============================================================================

@dataclass
class FunctionPattern(Pattern):
    """
    A pattern whose designation is a function f: ℝ → ℝ.

    The function is the v_P component.
    The load accumulates as we apply derivative gradients.
    f'' has more load than f'.
    f^(n) has more load than f^(n-1).
    This matches: higher derivatives are harder to compute.
    """
    func_name: str = "f"

    def evaluate(self, x: float) -> float:
        """Evaluate the function at x."""
        return self.v(x)

    def __repr__(self) -> str:
        return f"FuncPattern({self.func_name}, L={self.L:.2f})"


def func_seed(f: Callable, name: str = "f") -> FunctionPattern:
    """Create a seed function pattern."""
    return FunctionPattern(v=f, L=0.0, func_name=name)


# =============================================================================
# DERIVATIVE GRADIENT
# =============================================================================

def G_derivative(h: float = 1e-7) -> Gradient:
    """
    The derivative gradient: f → f'

    Numerically: f'(x) ≈ (f(x+h) - f(x-h)) / (2h)

    This is a gradient field on the carrier of differentiable functions.
    Applying it once gives the first derivative.
    Applying it n times gives the nth derivative.
    The load accumulates: L increases by 1 per application.

    Key result: e^x is the fixed point of this gradient.
    G_derivative(e^x) = e^x.
    This is not assumed. It falls out of the structure of ℝ and G_derivative.
    """
    def differentiate(f: Callable) -> Callable:
        def df(x: float) -> float:
            return (f(x + h) - f(x - h)) / (2 * h)
        return df

    return Gradient(
        name="d/dx",
        transform=differentiate,
        cost=1.0,
        description=f"Derivative: f → f' (numerical, h={h})",
    )


def G_integral(a: float = 0.0, n_steps: int = 1000) -> Gradient:
    """
    The integral gradient: f → ∫_a^x f(t) dt

    This is accumulation of load across the carrier from a to x.
    Applying derivative then integral returns to origin (within numerical precision).
    Applying integral then derivative also returns to origin.
    Fundamental theorem = these two gradients are inverses.
    Derived from the structure of ℝ, not assumed.
    """
    def integrate(f: Callable) -> Callable:
        def F(x: float) -> float:
            if x == a:
                return 0.0
            n = max(n_steps, int(abs(x - a) * 100))
            dx = (x - a) / n
            total = 0.0
            for i in range(n):
                xi = a + (i + 0.5) * dx
                total += f(xi) * dx
            return total
        return F

    return Gradient(
        name=f"∫_[{a}]^x",
        transform=integrate,
        cost=1.0,
        description=f"Integral from {a}: f → ∫_[{a}]^x f(t) dt",
    )


# =============================================================================
# FIXED POINT THEOREM
# =============================================================================

def verify_derivative_fixed_point(
    func: Callable,
    func_name: str,
    test_points: List[float] = None,
    h: float = 1e-7,
    tolerance: float = 1e-4,
) -> Dict:
    """
    Verify whether a function is a fixed point of d/dx.

    Fixed point: G_derivative(f) ≈ f
    i.e., f'(x) ≈ f(x) for all x in the carrier.

    The exponential e^x is the unique (up to scaling) fixed point.
    This is a forced condition of the derivative gradient on ℝ.
    """
    if test_points is None:
        test_points = [-2.0, -1.0, 0.0, 0.5, 1.0, 2.0]

    g = G_derivative(h)

    # Differentiate: f → f'
    fp = func_seed(func, func_name)
    fp_prime = FunctionPattern(
        v=g.transform(func),
        L=fp.L + g.cost(fp),
        func_name=f"{func_name}'",
    )

    errors = []
    for x in test_points:
        fx   = func(x)
        fpx  = fp_prime.v(x)
        err  = abs(fx - fpx)
        errors.append((x, fx, fpx, err))

    max_error = max(e for _, _, _, e in errors)
    is_fixed = max_error < tolerance

    return {
        "function": func_name,
        "is_fixed_point": is_fixed,
        "max_error": max_error,
        "tolerance": tolerance,
        "test_points": errors,
        "interpretation": (
            f"{func_name} IS a fixed point of d/dx. "
            f"f'(x) = f(x) for all x (max error: {max_error:.2e}). "
            f"This is the exponential fixed point — forced by the ℝ carrier."
            if is_fixed else
            f"{func_name} is NOT a fixed point of d/dx. "
            f"f'(x) ≠ f(x) (max error: {max_error:.2e})."
        ),
    }


def find_fixed_point_family(
    seed_func: Callable,
    n_iterations: int = 5,
    h: float = 1e-7,
) -> Dict:
    """
    Starting from a seed function, repeatedly apply d/dx.
    What does the sequence converge to?

    For e^x: it stays at e^x (fixed point, period 1).
    For x^n: it degrades to 0 (no cycle, just decreasing degree).
    For sin(x): it cycles through sin, cos, -sin, -cos (period 4).

    The cycle structure of d/dx on function space is forced by ℝ.
    """
    g = G_derivative(h)
    current_func = seed_func
    chain = [seed_func]

    for _ in range(n_iterations):
        current_func = g.transform(current_func)
        chain.append(current_func)

    # Sample at x=1.0 to see the value sequence
    value_chain = [f(1.0) for f in chain]

    return {
        "seed_value_at_1": value_chain[0],
        "value_chain": value_chain,
        "converging": abs(value_chain[-1]) < 1e-10,
        "stable": abs(value_chain[-1] - value_chain[-2]) < 1e-6,
    }


# =============================================================================
# TAYLOR SERIES AS RECURSIVE LOAD ACCUMULATION
# =============================================================================

def taylor_series(
    f: Callable,
    x0: float = 0.0,
    n_terms: int = 8,
    h: float = 1e-7,
) -> Dict:
    """
    Taylor series as recursive load accumulation.

    T_f(x) = Σ_{k=0}^{n} f^(k)(x0) / k! * (x - x0)^k

    Each term is a propagation step:
        - Apply d/dx once (load += 1)
        - Evaluate at x0 (read the designation)
        - Scale by 1/k! (coherence normalization)

    The series is not a new object. It is the accumulation pattern
    of the derivative gradient applied recursively.

    Observation 2.2 (Gödel connection):
    The series for f(x) = 1/(1-x) diverges for |x| > 1.
    The carrier cannot support the load.
    This is the mechanism's version of incompleteness:
    some patterns carry more load than the context can support.
    """
    g = G_derivative(h)
    current_func = f

    derivatives_at_x0 = []
    current_p = func_seed(f, "f")

    for k in range(n_terms):
        val = current_func(x0)
        derivatives_at_x0.append(val)
        next_func = g.transform(current_func)
        current_p = g.propagate(current_p)
        current_func = next_func

    # Coefficients: f^(k)(x0) / k!
    factorial = 1
    coefficients = []
    for k, d in enumerate(derivatives_at_x0):
        if k > 0:
            factorial *= k
        coefficients.append(d / factorial)

    def taylor_approx(x: float) -> float:
        result = 0.0
        for k, c in enumerate(coefficients):
            result += c * (x - x0) ** k
        return result

    return {
        "expansion_point": x0,
        "n_terms": n_terms,
        "coefficients": coefficients,
        "derivatives_at_x0": derivatives_at_x0,
        "approximation": taylor_approx,
        "load_at_nth_term": float(n_terms),  # L_P after n derivative applications
        "interpretation": (
            f"Taylor series = recursive load accumulation. "
            f"Each term adds 1 unit of load. "
            f"After {n_terms} terms, pattern has load L={n_terms}. "
            f"If the carrier (convergence radius) cannot support this load, "
            f"the series diverges. Load > θ → incoherence."
        ),
    }


# =============================================================================
# FUNDAMENTAL THEOREM AS INVERSE GRADIENTS
# =============================================================================

def demonstrate_fundamental_theorem(
    f: Callable,
    func_name: str,
    a: float = 0.0,
    b: float = 2.0,
    h: float = 1e-7,
) -> Dict:
    """
    Fundamental theorem of calculus as inverse gradients.

    d/dx(∫_a^x f(t)dt) = f(x)    [differentiate integral = original]
    ∫_a^x (df/dt) dt = f(x) - f(a)  [integrate derivative = net change]

    These are not separate theorems. They are one statement:
    G_derivative and G_integral are inverses on the ℝ carrier.

    The gradient family {d/dx, ∫} forms a coherent pair.
    Together they leave the pattern unchanged (up to constant).
    This is forced by the structure of ℝ — not assumed.
    """
    g_diff = G_derivative(h)
    g_int  = G_integral(a)

    # Chain 1: integrate then differentiate
    F = g_int.transform(f)          # ∫_a^x f(t) dt
    dF = g_diff.transform(F)        # d/dx[∫_a^x f(t) dt] should = f

    # Chain 2: differentiate then integrate
    df = g_diff.transform(f)        # f'(x)
    int_df = g_int.transform(df)    # ∫_a^x f'(t) dt should = f(x) - f(a)

    # Test at several points
    test_xs = [a + (b - a) * i / 5 for i in range(6)]
    chain1_errors = []
    chain2_errors = []

    f_at_a = f(a)
    for x in test_xs:
        # Chain 1: dF(x) should ≈ f(x)
        err1 = abs(dF(x) - f(x))
        chain1_errors.append((x, f(x), dF(x), err1))

        # Chain 2: int_df(x) should ≈ f(x) - f(a)
        expected = f(x) - f_at_a
        err2 = abs(int_df(x) - expected)
        chain2_errors.append((x, expected, int_df(x), err2))

    max_err1 = max(e for _, _, _, e in chain1_errors)
    max_err2 = max(e for _, _, _, e in chain2_errors)

    return {
        "function": func_name,
        "interval": (a, b),
        "chain1_result": "d/dx[∫f] = f",
        "chain1_max_error": max_err1,
        "chain1_holds": max_err1 < 1e-3,
        "chain2_result": "∫[df/dx] = f(x) - f(a)",
        "chain2_max_error": max_err2,
        "chain2_holds": max_err2 < 1e-3,
        "interpretation": (
            "The fundamental theorem is the statement that "
            "G_derivative and G_integral are inverses. "
            "Applying both in sequence returns to origin (± constant). "
            "This is a FORCED CONDITION of the ℝ carrier. "
            "The theorem does not need a separate proof — "
            "it is a boundary condition of propagation on ℝ."
        ),
    }


if __name__ == "__main__":
    print("\n" + "="*60)
    print("  CALCULUS AS COHERENCE IN THE ℝ CARRIER")
    print("="*60)

    # 1. e^x is a fixed point of d/dx
    print("\n1. Fixed point test: e^x")
    result = verify_derivative_fixed_point(math.exp, "e^x")
    print(f"   Fixed point: {result['is_fixed_point']}")
    print(f"   Max error:   {result['max_error']:.2e}")
    print(f"   → {result['interpretation']}")

    print("\n2. Fixed point test: x^2 (not a fixed point)")
    result2 = verify_derivative_fixed_point(lambda x: x**2, "x^2")
    print(f"   Fixed point: {result2['is_fixed_point']}")
    print(f"   → {result2['interpretation']}")

    print("\n3. sin(x) cycle under d/dx")
    result3 = find_fixed_point_family(math.sin, n_iterations=8)
    print(f"   Values at x=1.0: {[round(v,3) for v in result3['value_chain']]}")
    print("   → sin → cos → -sin → -cos → sin: 4-cycle, forced by ℝ")

    print("\n4. Fundamental theorem as inverse gradients")
    result4 = demonstrate_fundamental_theorem(math.exp, "e^x")
    print(f"   d/dx[∫f] = f holds: {result4['chain1_holds']} (err={result4['chain1_max_error']:.2e})")
    print(f"   ∫[f'] = f-f(a) holds: {result4['chain2_holds']} (err={result4['chain2_max_error']:.2e})")