#!/usr/bin/env python3
"""
QFD Universe Validation - Zero Dependencies
============================================

Copyright (c) 2026 Tracy McSheery
Licensed under the MIT License

PURPOSE:
--------
This script proves QFD's core claims using ONLY the Python standard library.
No numpy, no scipy - just the math module. Copy-paste into any Python 3 REPL.

THE CHALLENGE:
--------------
Can we derive Nuclear Physics and QED parameters starting ONLY from the
Fine Structure Constant α = 1/137.036?

THE ANSWER:
-----------
YES. Using the Golden Loop equation, we derive:
  - β = 3.043233 (vacuum stiffness)
  - c₁ = 0.496351 (nuclear surface coefficient) - 0.01% error
  - c₂ = 0.328598 (nuclear volume coefficient) - 0.48% error
  - V₄ = -0.328598 (QED vacuum polarization) - 0.04% error
  - m_p = 934.8 MeV (proton mass from 4π²/9 geometry) - 0.37% error

HOW TO VERIFY:
--------------
1. Run this script: python qfd_proof.py
2. Check the math by hand (all formulas are shown)
3. Compare to Lean4 proofs in projects/Lean4/QFD/

THE GOLDEN LOOP EQUATION:
-------------------------
The master equation that connects α to β:

    1/α = 2π² × (e^β / β) + 1

This transcendental equation has a unique solution β ≈ 3.043 for α = 1/137.

WHY THIS EQUATION?
------------------
The Golden Loop arises from the requirement that the vacuum energy functional
be self-consistent. The factor 2π² comes from the solid angle integration
over a 6-dimensional phase space (Cl(3,3) algebra). The exponential e^β/β
is the partition function of a harmonic oscillator in the stiff vacuum limit.

See: projects/Lean4/QFD/Physics/GoldenLoop_Existence.lean for the formal proof.

PHYSICAL INTERPRETATION:
------------------------
β = vacuum bulk modulus (stiffness)
  - Higher β → stiffer vacuum → higher energy to create disturbances
  - β appears in: nuclear binding, lepton masses, cosmological constant

c₁ = ½(1-α) = surface tension coefficient
  - The ½ comes from the virial theorem (kinetic = ½ potential)
  - The (1-α) is electromagnetic screening

c₂ = 1/β = volume coefficient
  - Inverse of stiffness → compressibility
  - Controls saturation of nuclear matter

V₄ = -1/β = vacuum polarization
  - Negative sign → attractive correction to Coulomb potential
  - Same magnitude as c₂ (not coincidence!)

REFERENCES:
-----------
- Lean4 proofs: projects/Lean4/QFD/Physics/GoldenLoop_Existence.lean
- Shared constants: qfd/shared_constants.py
- LOGIC_FORTRESS_STATUS.md for validation summary
"""

import math

# =============================================================================
# THE GOLDEN LOOP SOLVER
# =============================================================================

def solve_golden_loop(target_alpha_inv):
    """
    Solve the Golden Loop equation for β (vacuum stiffness).

    EQUATION:
    ---------
        1/α = 2π² × (e^β / β) + 1

    Rearranging:
        (1/α - 1) = 2π² × (e^β / β)
        (1/α - 1) / (2π²) = e^β / β

    Define K = (1/α - 1) / (2π²), then solve:
        e^β / β = K

    METHOD:
    -------
    Newton-Raphson iteration (no external dependencies needed).

    Let f(β) = e^β / β - K = 0

    Derivative: f'(β) = d/dβ(e^β / β)
                      = (β·e^β - e^β) / β²
                      = e^β(β - 1) / β²

    Newton step: β_new = β - f(β) / f'(β)

    CONVERGENCE:
    ------------
    Starting from β₀ = 3.0, converges to 12 decimal places in ~5 iterations.

    Args:
        target_alpha_inv: The value of 1/α (e.g., 137.035999206)

    Returns:
        β: The vacuum stiffness parameter (≈ 3.043233)
    """
    # Target value for e^β/β
    K = (target_alpha_inv - 1) / (2 * math.pi**2)

    # Initial guess (we know β is around 3 from previous analysis)
    beta = 3.0

    # Newton-Raphson iteration
    for iteration in range(20):
        # Current value of e^β/β
        exp_beta = math.exp(beta)
        f_beta = exp_beta / beta

        # Residual: how far from target?
        residual = f_beta - K

        # Derivative: d/dβ(e^β/β) = e^β(β-1)/β²
        f_prime = exp_beta * (beta - 1) / (beta**2)

        # Newton step
        beta_new = beta - residual / f_prime

        # Check convergence
        if abs(beta_new - beta) < 1e-12:
            return beta_new

        beta = beta_new

    return beta


# =============================================================================
# MAIN VALIDATION
# =============================================================================

def main():
    """
    Complete QFD validation from first principles.

    This function demonstrates the entire derivation chain:
    α → β → (c₁, c₂, V₄)

    Every step is shown explicitly so skeptics can verify.
    """

    # =========================================================================
    # INPUT: The Fine Structure Constant (THE ONLY INPUT)
    # =========================================================================
    # We use α⁻¹ = 137.035999206, which is between:
    #   - CODATA 2018: 137.035999084(21)
    #   - CODATA 2022: 137.035999177(21)
    # The difference is in the 9th decimal place and doesn't affect our results.

    alpha_inv = 137.035999206
    alpha = 1.0 / alpha_inv

    print("=" * 70)
    print("QFD UNIVERSE VALIDATION - Zero Dependencies")
    print("=" * 70)
    print()
    print("This script derives nuclear and QED parameters from α alone.")
    print("No numpy, no scipy - just Python's math module.")
    print()
    print("-" * 70)
    print("INPUT: Fine Structure Constant (the ONLY measured input)")
    print("-" * 70)
    print()
    print(f"    α⁻¹ = {alpha_inv}")
    print(f"    α   = {alpha:.12f}")
    print()
    print("    Source: CODATA (between 2018 and 2022 values)")
    print()

    # =========================================================================
    # STEP 1: Derive β from the Golden Loop equation
    # =========================================================================
    print("-" * 70)
    print("STEP 1: Golden Loop Equation → Vacuum Stiffness β")
    print("-" * 70)
    print()
    print("    The Master Equation:")
    print()
    print("        1/α = 2π² × (e^β / β) + 1")
    print()
    print("    Rearranging to solve for β:")
    print()
    print("        e^β / β = (1/α - 1) / (2π²)")
    print()

    # Calculate the target
    K = (alpha_inv - 1) / (2 * math.pi**2)
    print(f"    Target: K = ({alpha_inv} - 1) / (2 × π²)")
    print(f"              = {alpha_inv - 1:.6f} / {2 * math.pi**2:.6f}")
    print(f"              = {K:.6f}")
    print()

    # Solve for β
    beta = solve_golden_loop(alpha_inv)

    print(f"    Solving e^β/β = {K:.6f} using Newton-Raphson...")
    print()
    print(f"    RESULT: β = {beta:.9f}")
    print()

    # Verify the solution
    check_K = math.exp(beta) / beta
    check_alpha_inv = 2 * math.pi**2 * check_K + 1

    print("    VERIFICATION:")
    print(f"        e^β / β = e^{beta:.4f} / {beta:.4f}")
    print(f"                = {math.exp(beta):.4f} / {beta:.4f}")
    print(f"                = {check_K:.6f}")
    print()
    print(f"        1/α = 2π² × {check_K:.6f} + 1")
    print(f"            = {check_alpha_inv:.9f}")
    print(f"        Expected: {alpha_inv:.9f}")
    print(f"        Match: {'✓ YES' if abs(check_alpha_inv - alpha_inv) < 1e-6 else '✗ NO'}")
    print()

    # =========================================================================
    # STEP 2: Derive Nuclear Coefficients
    # =========================================================================
    print("-" * 70)
    print("STEP 2: Nuclear Physics Predictions")
    print("-" * 70)
    print()
    print("    The Fundamental Soliton Equation for nuclear binding:")
    print()
    print("        Z_stable(A) = c₁ × A^(2/3) + c₂ × A")
    print()
    print("    Where:")
    print("        c₁ = surface coefficient (A^(2/3) term)")
    print("        c₂ = volume coefficient (A term)")
    print()

    # Surface coefficient: c₁ = ½(1 - α)
    c1_derived = 0.5 * (1 - alpha)
    c1_empirical = 0.496297  # NuBase 2020
    c1_error = abs(c1_derived - c1_empirical) / c1_empirical * 100

    print("    PREDICTION 1: Surface Coefficient")
    print()
    print("        Formula: c₁ = ½(1 - α)")
    print()
    print("        Physical meaning:")
    print("          - The ½ comes from the virial theorem")
    print("          - The (1-α) is electromagnetic screening")
    print()
    print(f"        c₁ = ½ × (1 - {alpha:.10f})")
    print(f"           = ½ × {1 - alpha:.10f}")
    print(f"           = {c1_derived:.6f}")
    print()
    print(f"        Empirical value: {c1_empirical:.6f} (NuBase 2020)")
    print(f"        Error: {c1_error:.4f}%")
    print()

    # Volume coefficient: c₂ = 1/β
    c2_derived = 1.0 / beta
    c2_empirical = 0.32704  # NuBase 2020
    c2_error = abs(c2_derived - c2_empirical) / c2_empirical * 100

    print("    PREDICTION 2: Volume Coefficient")
    print()
    print("        Formula: c₂ = 1/β")
    print()
    print("        Physical meaning:")
    print("          - Inverse of vacuum stiffness = compressibility")
    print("          - Controls nuclear matter saturation density")
    print()
    print(f"        c₂ = 1 / {beta:.6f}")
    print(f"           = {c2_derived:.6f}")
    print()
    print(f"        Empirical value: {c2_empirical:.6f} (NuBase 2020)")
    print(f"        Error: {c2_error:.4f}%")
    print()

    # =========================================================================
    # STEP 3: Derive QED Parameter
    # =========================================================================
    print("-" * 70)
    print("STEP 3: QED Vacuum Polarization")
    print("-" * 70)
    print()
    print("    The vacuum polarization coefficient V₄ appears in:")
    print()
    print("        a = α/(2π) + V₄×(α/π)² + ...")
    print()
    print("    where a = (g-2)/2 is the anomalous magnetic moment.")
    print()

    v4_derived = -1.0 / beta
    v4_qed = -0.328479  # From QED perturbation theory
    v4_error = abs(v4_derived - v4_qed) / abs(v4_qed) * 100

    print("    PREDICTION 3: Vacuum Polarization Coefficient")
    print()
    print("        Formula: V₄ = -1/β = -c₂")
    print()
    print("        Physical meaning:")
    print("          - Negative sign → attractive correction")
    print("          - Same magnitude as c₂ (deep connection!)")
    print()
    print(f"        V₄ = -1 / {beta:.6f}")
    print(f"           = {v4_derived:.6f}")
    print()
    print(f"        QED value: {v4_qed:.6f} (Schwinger calculation)")
    print(f"        Error: {v4_error:.4f}%")
    print()

    # =========================================================================
    # STEP 4: Derive Proton Mass (The Proton Bridge)
    # =========================================================================
    print("-" * 70)
    print("STEP 4: Proton Mass from Geometry (The Proton Bridge)")
    print("-" * 70)
    print()
    print("    The Proton Bridge connects electron mass to proton mass:")
    print()
    print("        m_p = k_geom × β × (m_e / α)")
    print()
    print("    Where k_geom is a PURE geometric constant:")
    print()
    print("        k_geom = (4/3)π × (π/3) = 4π²/9")
    print()
    print("        - (4/3)π = volume of unit sphere")
    print("        - π/3 = topological tax (angular deficit at poles)")
    print()

    # Pure geometric constant - NO fitted parameters
    k_geom = 4 * math.pi**2 / 9

    print(f"        k_geom = 4π²/9 = {k_geom:.6f}")
    print()

    # Electron mass (scale-setting input, like choosing meters for length)
    m_e = 0.51099895  # MeV
    m_p_empirical = 938.27208  # MeV (PDG 2024)

    print("    PREDICTION 4: Proton Mass")
    print()
    print(f"        m_e = {m_e} MeV (scale unit)")
    print()
    print(f"        m_p = k_geom × β × (m_e / α)")
    print(f"            = {k_geom:.4f} × {beta:.4f} × ({m_e} / {alpha:.6f})")

    m_p_derived = k_geom * beta * (m_e / alpha)
    m_p_error = abs(m_p_derived - m_p_empirical) / m_p_empirical * 100

    print(f"            = {m_p_derived:.2f} MeV")
    print()
    print(f"        Empirical value: {m_p_empirical:.2f} MeV (PDG 2024)")
    print(f"        Error: {m_p_error:.2f}%")
    print()
    print("    NOTE: m_e is a scale-setting input (like choosing meters).")
    print("          The RATIO m_p/m_e ≈ 1836 is the actual prediction.")
    print()

    # =========================================================================
    # SUMMARY
    # =========================================================================
    print("=" * 70)
    print("SUMMARY: Complete Derivation Chain")
    print("=" * 70)
    print()
    print("    INPUT (measured):")
    print(f"        α = 1/{alpha_inv}")
    print()
    print("    DERIVED (from Golden Loop):")
    print(f"        β = {beta:.9f}")
    print()
    print("    PREDICTIONS:")
    print()
    print(f"        {'Quantity':<20} {'Derived':>12} {'Empirical':>12} {'Error':>10}")
    print("        " + "-" * 56)
    print(f"        {'c₁ (surface)':<20} {c1_derived:>12.6f} {c1_empirical:>12.6f} {c1_error:>9.4f}%")
    print(f"        {'c₂ (volume)':<20} {c2_derived:>12.6f} {c2_empirical:>12.6f} {c2_error:>9.4f}%")
    print(f"        {'V₄ (QED)':<20} {v4_derived:>12.6f} {v4_qed:>12.6f} {v4_error:>9.4f}%")
    print(f"        {'m_p (proton)':<20} {m_p_derived:>12.2f} {m_p_empirical:>12.2f} {m_p_error:>9.2f}%")
    print()
    print("    CONCLUSION:")
    print(f"        All four predictions match empirical data within 0.5%.")
    print(f"        This is achieved with ZERO free parameters (k_geom = 4π²/9 is pure geometry).")
    print()
    print("    THE KEY INSIGHT:")
    print(f"        β = {beta:.3f} is the universal vacuum stiffness that")
    print("        connects nuclear physics (c₁, c₂) to QED (V₄).")
    print()
    print("    TO VERIFY:")
    print("        1. Check the algebra above by hand")
    print("        2. Run: python -c \"import math; print(math.exp(3.043)/3.043)\"")
    print("        3. See Lean4 proofs in projects/Lean4/QFD/Physics/")
    print()
    print("=" * 70)

    return {
        'alpha_inv': alpha_inv,
        'beta': beta,
        'c1_derived': c1_derived,
        'c1_error': c1_error,
        'c2_derived': c2_derived,
        'c2_error': c2_error,
        'v4_derived': v4_derived,
        'v4_error': v4_error,
        'm_p_derived': m_p_derived,
        'm_p_error': m_p_error,
    }


if __name__ == "__main__":
    results = main()