N18 · The external Koide rule and its light electron root

Row

House correction — R77 (S327h). The Scorecard printed −0.67%. With the electron mass kept at full precision (0.507607 MeV) it is −0.66%, which is also what the suite prints (p2:239). The Scorecard now prints −0.66%; the theory cell still displays 0.5076. The "forces the Koide sum rule" wording this note flags (p2:237–238) is queued for Rev32.12 (= Rev33.0, R156). The cell quoted below, and the cell the script echoes, are the text as it was supplied to the drafting lane (the Scorecard through S327g). Scorecard and Registrar are unchanged in every other respect; the sealed papers keep the earlier figure until Rev32.12 (= Rev33.0, R156).

row 18

m e (MeV) QED-Koide, K = 2/3 theory 0.5076 experiment 0.51099895069 deviation −0.67% Koide-consistent ext. selector

The cell is reproduced from the unrounded corrected muon chain. Substituting its displayed rounded muon mass changes the last digits. The cell’s percentage uses the rounded electron value; the full-chain percentage is separately printed. The comparator has no supplied uncertainty or ledger ID.

Theory value

row 18

The supplied Koide equation is

With the unrounded QED-corrected muon chain and the tau anchor, its selected light root is 0.5076 MeV at cell precision.

Derivation chain

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1. Obtain the tau-anchored tree muon value from the retained structural formula, keeping intermediate precision. (appendix_a_lagrangian_derivation.tex:701-704).
2. Divide by the supplied QED factor and insert the result into the printed Koide equation. (p0_framework_foundations.tex:968-977).
3. Solve that equation as a quadratic in the electron square-root mass and take the light positive branch; the source explicitly says a second positive branch exists. (p2_mass_hierarchy_resolvent_quintics.tex:240-243).
4. NOT IN SUITE — a derivation selecting the empirical Koide value or physical light branch. The current attachment is external; older “forces” wording in the intermediate chain does not supply this missing derivation. (appendix_a_lagrangian_derivation.tex:704; appendix_o_restmass_program.tex:117-118; p2_mass_hierarchy_resolvent_quintics.tex:237-243).

Registrar sync

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LIB2-122 — Carries the inherited muon ratio.

LIB2-125 — Carries its tau-anchored tree value.

LIB2-126 — Carries the supplied QED correction.

LIB2-127 — Carries the external Koide constraint, light-root choice and electron value.

Comparator NONE — no matching comparator record is supplied for this exact entry; the stated cell is retained without inventing an ID.

LIB2-267 is historical wording, not a replacement for the current external-selector record.

Calculation

Run python3 N18_calc.py. The complete self-contained script is below. All source cells are echoed unchanged. Computed displays use decimal half-up rounding; intermediate formula values are not display-rounded. A naive asymmetric distance uses the error toward the theory unless an explicit exception or alternate audit is printed.

#!/usr/bin/env python3
"""N18 — arithmetic from D1287 supplied sources.
No network, external packages, fitting operations or shared runtime files.
Printed source cells and unrounded arithmetic are distinct outputs.
"""

import math
from decimal import Decimal, ROUND_HALF_UP, getcontext
from fractions import Fraction

getcontext().prec = 40
PHI = (1.0 + math.sqrt(5.0)) / 2.0


def shown(value, places=9, signed=False):
    """Decimal half-up display; never use display-rounded inputs implicitly."""
    value = Decimal(str(value))
    rounded = value.quantize(Decimal(1).scaleb(-places), rounding=ROUND_HALF_UP)
    return format(rounded, ("+" if signed else "") + "." + str(places) + "f")


def compare(label, theory, reference, lower=None, upper=None):
    """Naive central-value arithmetic, not a likelihood or theory-error model.

    The asymmetric denominator points from the comparator toward the theory:
    lower error below the central value; upper error above it. Any different
    printed convention is audited separately, rather than silently substituted.
    """
    theory, reference = Decimal(str(theory)), Decimal(str(reference))
    if reference == 0:
        raise ValueError("A relative deviation needs a nonzero reference.")
    offset = theory - reference
    percent = 100 * offset / reference
    print(label + ".theory = " + shown(theory))
    print(label + ".reference = " + shown(reference))
    print(label + ".signed_percent = " + shown(percent, signed=True)
          + "%; rounded = " + shown(percent, 2, True) + "%")
    if lower is None or upper is None:
        print(label + ".d = NOT AVAILABLE (no uncertainty supplied for this comparison variable)")
        return
    side = "lower" if offset < 0 else "upper"
    uncertainty = Decimal(str(lower if offset < 0 else upper))
    if uncertainty <= 0:
        raise ValueError("The selected comparator uncertainty must be positive.")
    distance = offset / uncertainty
    print(label + ".uncertainty_used = " + str(uncertainty) + " (" + side + ")")
    print(label + ".d = " + shown(distance, signed=True)
          + "; rounded = " + shown(distance, 2, True))


def lepton_chain():
    # The printed matching correction is an input, not re-derived here.
    tau = 1776.93
    ratio = (PHI / math.sqrt(5.0))**(8.0/3.0) * math.sqrt(2.0)/10.0
    tree = tau * ratio
    delta = 0.003125
    return tau, ratio, tree, delta, tree / (1.0 + delta)


def koide_roots(muon, tau):
    # Solve the supplied K=2/3 equation in x=sqrt(m_e), without a mass fit.
    s = math.sqrt(muon) + math.sqrt(tau)
    c = muon + tau - 4.0 * math.sqrt(muon*tau)
    disc = 4.0*s*s - c
    if disc < 0:
        raise ValueError("The supplied Koide inputs have no real roots.")
    lo, hi = 2.0*s - math.sqrt(disc), 2.0*s + math.sqrt(disc)
    if lo < 0:
        raise ValueError("The light square-root solution is not physical.")
    return lo*lo, hi*hi

def main():
    print('=== row 18 ===')
    print('Scorecard (verbatim): m e (MeV) QED-Koide, K = 2/3 theory 0.5076 experiment 0.51099895069 deviation −0.67% Koide-consistent ext. selector')
    tau, ratio, tree, delta, physical = lepton_chain()
    light, heavy = koide_roots(physical,tau)
    rounded_mu_light, unused = koide_roots(105.72,tau)
    rounded_tree_light, unused = koide_roots(106.05/(1+delta),tau)
    k = (light+physical+tau)/(math.sqrt(light)+math.sqrt(physical)+math.sqrt(tau))**2
    assert abs(k-2.0/3.0) < 1e-13
    print("tau_anchor_MeV = " + shown(tau,2))
    print("unrounded_mu_tree_MeV = " + shown(tree,12))
    print("supplied_delta_QED = " + shown(delta,6))
    print("unrounded_mu_physical_MeV = " + shown(physical,12))
    print("selected_light_root_MeV = " + shown(light,12))
    print("other_positive_root_MeV = " + shown(heavy,9))
    print("Koide_K_check = " + shown(k,12))
    print("theory_at_cell_precision_MeV = " + shown(light,4))
    print("light_root_from_rounded_mu_MeV = " + shown(rounded_mu_light,12)
          + "; rounded = " + shown(rounded_mu_light,4))
    print("light_root_from_rounded_tree_MeV = " + shown(rounded_tree_light,12)
          + "; rounded = " + shown(rounded_tree_light,4))
    compare("full_chain_MeV", light, "0.51099895069")
    compare("printed_cell_MeV", "0.5076", "0.51099895069")
    print()


if __name__ == "__main__":
    main()

row 18

Actual stdout for this entry; the text blocks in entry order concatenate to N18_stdout.txt.

=== row 18 ===
Scorecard (verbatim): m e (MeV) QED-Koide, K = 2/3 theory 0.5076 experiment 0.51099895069 deviation −0.67% Koide-consistent ext. selector
tau_anchor_MeV = 1776.93
unrounded_mu_tree_MeV = 106.053666035117
supplied_delta_QED = 0.003125
unrounded_mu_physical_MeV = 105.723280782672
selected_light_root_MeV = 0.507607012435
other_positive_root_MeV = 43693.898514467
Koide_K_check = 0.666666666667
theory_at_cell_precision_MeV = 0.5076
light_root_from_rounded_mu_MeV = 0.507771699106; rounded = 0.5078
light_root_from_rounded_tree_MeV = 0.507790466434; rounded = 0.5078
full_chain_MeV.theory = 0.507607012
full_chain_MeV.reference = 0.510998951
full_chain_MeV.signed_percent = -0.663785757%; rounded = -0.66%
full_chain_MeV.d = NOT AVAILABLE (no uncertainty supplied for this comparison variable)
printed_cell_MeV.theory = 0.507600000
printed_cell_MeV.reference = 0.510998951
printed_cell_MeV.signed_percent = -0.665158057%; rounded = -0.67%
printed_cell_MeV.d = NOT AVAILABLE (no uncertainty supplied for this comparison variable)

Comparison

row 18

Theory cell 0.5076 MeV, comparator cell 0.51099895069 MeV. Rounded-cell deviation is −0.67%; the unrounded chain rounds to −0.66%. The precise comparator’s uncertainty, edition and scheme/scale binding are not supplied, so d is unavailable. This is an externally constrained charged-lepton calculation, not three independently derived masses.

Tier and what this does not show

Literal tier: Koide-consistent; status: ext. selector. The tau anchor, structural muon ratio, supplied correction, empirical Koide value and light branch are consumed. Reproducing the equation and the cell does not derive those attachments. In particular, the older forcing sentence is not adopted as a tier upgrade.

Sources

LIB2-122; LIB2-125; LIB2-126; LIB2-127; appendix_a_lagrangian_derivation.tex:701-704; p0_framework_foundations.tex:968-977; p2_mass_hierarchy_resolvent_quintics.tex:237-243; appendix_o_restmass_program.tex:117-118.