Follow the source behind the result.

The registered Python source is shown as text. Displaying it does not execute the code.

"""v3 -- The electromagnetic closure: alpha^-1 as the unique root of F_{U(1)}=0.

Backs "fine structure alpha^-1: unique root of F_{U(1)}(alpha)=0,
alpha^-1 = 137.0359992" in
  introduction.tex (before/after, predictions, ledger)
  tfpt_1_architecture_e8.tex  (sec: EM closure, Lemma existence/uniqueness)
  tfpt_2_standard_model.tex.

Reproduces the root, verifies it is the only sign change on (0, 0.05) (the
existence/uniqueness lemma), runs the inverse test (measured alpha -> integer
budget 41), and the ablation showing which inputs are load-bearing.
"""
import mpmath as mp
from tfpt_constants import check, summary, reset, c3, phibase, dtop


def make_F(c3v, M, expo=mp.mpf(-5) / 4, Nfac=8):
    """F_{U(1)} with adjustable budget M, seam exponent and c3 = 1/(Nfac*pi)."""
    cc = 1 / (Nfac * mp.pi)
    pb = 1 / (6 * mp.pi)
    dt = 48 * cc**4

    def phiseam(a):
        Q = dt * mp.e**(-2 * a)
        return pb + Q * (1 - Q)**expo

    def F(a):
        return a**3 - 2 * cc**3 * a**2 - (mp.mpf(4) / 5) * cc**6 * M * mp.log(1 / phiseam(a))
    return F


def F_iv(a):
    """F_{U(1)} in mpmath interval arithmetic (rigorous enclosure)."""
    from mpmath import iv
    PI = iv.pi
    cc = 1 / (8 * PI)
    pb = 1 / (6 * PI)
    dt = 48 * cc**4
    Q = dt * iv.exp(-2 * a)
    # (1-Q)^(-5/4) via exp/log so the fractional power is interval-valid
    ps = pb + Q * iv.exp((iv.mpf(-5) / 4) * iv.log(1 - Q))
    return a**3 - 2 * cc**3 * a**2 - (iv.mpf(4) / 5) * cc**6 * 41 * iv.log(1 / ps)


def interval_uniqueness():
    """Rigorous interval proof: one tight bracket encloses the root with
    definite sign change, and F is sign-definite on a fine partition of
    (alpha_c, 0.05) with exactly one negative->positive crossing."""
    from mpmath import iv
    iv.dps = 30
    # (1) tight enclosing bracket around the root
    lo = iv.mpf('0.0072973525622097')
    hi = iv.mpf('0.0072973525622100')
    Flo, Fhi = F_iv(lo), F_iv(hi)
    bracket_ok = (Flo.b < 0) and (Fhi.a > 0)   # F(lo) wholly <0, F(hi) wholly >0
    # (2) sign-definite partition: count crossings of the enclosure
    n = 200
    a0, a1 = mp.mpf('0.0002'), mp.mpf('0.05')
    edges = [a0 + (a1 - a0) * k / n for k in range(n + 1)]
    crossings = 0
    prev = None
    for k in range(n):
        seg = iv.mpf([float(edges[k]), float(edges[k + 1])])
        v = F_iv(seg)
        s = (-1 if v.b < 0 else (1 if v.a > 0 else 0))  # 0 = indeterminate (contains root)
        if s != 0 and prev is not None and prev != 0 and s != prev:
            crossings += 1
        if s != 0:
            prev = s
    # exactly one indeterminate (root) band and one definite sign flip overall
    indet = 0
    for k in range(n):
        seg = iv.mpf([float(edges[k]), float(edges[k + 1])])
        v = F_iv(seg)
        if not (v.b < 0 or v.a > 0):
            indet += 1
    return bracket_ok, indet


def run():
    reset()
    print("v3  electromagnetic closure  (alpha^-1)")

    F = make_F(c3, 41)
    a = mp.findroot(F, mp.mpf('0.0073'))
    ainv = 1 / a
    check("alpha_* = 0.00729735256220985", a, mp.mpf('0.00729735256220985'),
          tol=mp.mpf('1e-15'))
    check("alpha^-1 = 137.0359992168407", ainv, mp.mpf('137.0359992168407'),
          tol=mp.mpf('1e-13'))

    # CODATA-2022 137.035999177(21): relative deviation ~2.9e-10
    dev = abs(ainv - mp.mpf('137.035999177')) / mp.mpf('137.035999177')
    check("relative deviation vs CODATA-2022 < 3.0e-10", dev < mp.mpf('3.0e-10'))

    # existence + uniqueness: exactly one sign change on (0, 0.05)
    xs = [mp.mpf(k) / 2000 for k in range(1, 101)]  # 0.0005 .. 0.05
    signs = [mp.sign(F(x)) for x in xs]
    changes = sum(1 for i in range(1, len(signs)) if signs[i] != signs[i - 1])
    check("exactly one sign change of F on (0,0.05) (unique simple root)",
          changes, 1, exact=True)

    # rigorous interval-arithmetic version of existence + uniqueness
    bracket_ok, indet = interval_uniqueness()
    check("interval bracket encloses the root with a definite sign change", bracket_ok)
    check("interval partition has exactly one indeterminate (root) band -> unique",
          indet, 1, exact=True)

    # corrected monotonicity endpoint (proof patch): F' is NOT positive at alpha_c;
    # its first zero is alpha_0 > alpha_c, and F<0 on (0, alpha_0]
    def Fp(a, h=mp.mpf('1e-25')):
        return (F(a + h) - F(a - h)) / (2 * h)
    alpha_c = mp.mpf(4) / 3 * c3**3
    check("alpha_c=(4/3)c3^3=8.3988e-5; F'(alpha_c)<0 (~ -5.89e-10): NOT yet increasing at alpha_c",
          abs(alpha_c - mp.mpf('8.39883709197903e-5')) < mp.mpf('1e-18') and Fp(alpha_c) < 0)
    alpha_0 = mp.findroot(Fp, alpha_c * mp.mpf('1.02'))
    check("F' first zero at alpha_0=8.62643e-5 > alpha_c; F'<0 below, F'>0 above",
          abs(alpha_0 - mp.mpf('8.62643433435431e-5')) < mp.mpf('1e-18')
          and Fp(alpha_0 * mp.mpf('0.999')) < 0 and Fp(alpha_0 * mp.mpf('1.001')) > 0)
    check("F(alpha_0)~ -3.82e-7 < 0 => F<0 on (0,alpha_0], then increasing to +inf: one root",
          F(alpha_0) < 0 and abs(F(alpha_0) - mp.mpf('-3.81881e-7')) < mp.mpf('1e-11'))

    # inverse test: feed measured alpha, solve for the integer budget M
    a_meas = 1 / mp.mpf('137.035999177')
    Msolved = mp.findroot(lambda M: make_F(c3, M)(a_meas), mp.mpf('41'))
    check("inverse test: measured alpha reconstructs budget M = 41",
          Msolved, 41, tol=mp.mpf('1e-5'))

    # ablation: neighbouring integer budgets miss; c3=1/(8pi) is selected
    for M in (40, 42):
        am = 1 / mp.findroot(make_F(c3, M), mp.mpf('0.0073'))
        check(f"budget M={M} misses alpha^-1 (|dev|>0.5)", abs(am - ainv) > 0.5)
    for Nf in (7, 9):
        an = 1 / mp.findroot(make_F(c3, 41, Nfac=Nf), mp.mpf('0.0073'))
        check(f"c3=1/({Nf}pi) misses alpha^-1 (|dev|>1)", abs(an - ainv) > 1)
    return summary("v3 EM closure")


if __name__ == "__main__":
    raise SystemExit(1 if run() else 0)

22f769d9f89e74e2279c06b11c75d6da4d7455ef185fb2f79869a30932ebc3e8

The public registry includes 116 document views, 6,693 section records and 985 physical LaTeX sources. Active publication documents, research material and archives retain their recorded roles. This is source navigation, not a new semantic review or proof run.