N08 · The loaded first-row unitarity value

Row

row 8

|V ud | first-row unitarity theory 0.97429 experiment 0.97367 ± 0.00032 deviation +0.06% · d = +1.94 Loaded —

The candidate LIB2-065 carries the final square root. The prerequisite magnitudes are carried by records supplied with other notes; those records are named below rather than treating a unitarity identity as an independent selection theorem.

Theory value

row 8

The suite prints

The square root is evaluated from the unrounded framework magnitudes, not from measured magnitudes.

Derivation chain

row 8

1. Use the data-selected Route-B input |V_{us}|=\sin(\theta_{12}^{\rm fw})/\sqrt6 and the framework solar angle. (p3_ckm_pmns_mixing.tex:157).
2. Use the retained |V_{cb}|=1/(9\sqrt7) formula and |V_{ub}|=|V_{us}||V_{cb}|/\sqrt6. (p0_framework_foundations.tex:712-719).
3. Substitute these magnitudes into first-row unitarity and take the positive magnitude. The printed source explicitly retains the inherited loading. (p3_ckm_pmns_mixing.tex:145-154).

Registrar sync

row 8

LIB2-063 — Carries the selected Cabibbo input.

LIB2-066 — Carries the adjacent magnitude used in the hierarchy.

LIB2-067 — Carries the hierarchy expression for the small first-row magnitude.

LIB2-065 — Carries the final unitarity square root and inherited loading.

Comparator LIB2-217 / SL-14 — value 0.97367 (+0.00032/−0.00032) · scheme PDG first-row · edition 2026.

Calculation

Run python3 N08_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
"""N08 — 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 ckms():
    solar = 3.0 / (PHI**4 + 3.0)
    vus = math.sqrt(solar) / math.sqrt(6.0)
    vcb = 1.0 / (9.0 * math.sqrt(7.0))
    vub = vus * vcb / math.sqrt(6.0)
    vud = math.sqrt(1.0 - vus*vus - vub*vub)
    return vus, vcb, vub, vud

def main():
    print('=== row 8 ===')
    print('Scorecard (verbatim): |V ud | first-row unitarity theory 0.97429 experiment 0.97367 ± 0.00032 deviation +0.06% · d = +1.94 Loaded —')
    vus, vcb, vub, vud = ckms()
    print("input_vus = " + shown(vus,12))
    print("input_vcb = " + shown(vcb,12))
    print("input_vub = " + shown(vub,12))
    print("unitarity_sum = " + shown(vud*vud+vus*vus+vub*vub,12))
    print("theory_at_cell_precision = " + shown(vud,5))
    compare("full_formula", vud, "0.97367", "0.00032", "0.00032")
    compare("printed_cell", "0.97429", "0.97367", "0.00032", "0.00032")
    print()


if __name__ == "__main__":
    main()

row 8

Actual stdout for this entry; the text blocks in entry order concatenate to N08_stdout.txt.

=== row 8 ===
Scorecard (verbatim): |V ud | first-row unitarity theory 0.97429 experiment 0.97367 ± 0.00032 deviation +0.06% · d = +1.94 Loaded —
input_vus = 0.225256056227
input_vcb = 0.041996052557
input_vub = 0.003861973786
unitarity_sum = 1.000000000000
theory_at_cell_precision = 0.97429
full_formula.theory = 0.974291945
full_formula.reference = 0.973670000
full_formula.signed_percent = +0.063876375%; rounded = +0.06%
full_formula.uncertainty_used = 0.00032 (upper)
full_formula.d = +1.943578445; rounded = +1.94
printed_cell.theory = 0.974290000
printed_cell.reference = 0.973670000
printed_cell.signed_percent = +0.063676605%; rounded = +0.06%
printed_cell.uncertainty_used = 0.00032 (upper)
printed_cell.d = +1.937500000; rounded = +1.94

Comparison

row 8

Theory 0.97429; comparator 0.97367 ±0.00032, PDG 2026 first-row determination, dimensionless (LIB2-217 / SL-14). Signed deviation rounds to +0.06%, naive d to +1.94. No separate running scale is supplied. Exact normalization of the constructed row is not independent evidence for its data-selected inputs.

Tier and what this does not show

Literal tier: Loaded; status: —. Route choice and the hierarchy inputs are consumed. The algebraic unitarity operation does not make those choices physical consequences of the bulk algebra.

Sources

LIB2-063; LIB2-066; LIB2-067; LIB2-065; LIB2-217; SL-14; p3_ckm_pmns_mixing.tex:145-157; p0_framework_foundations.tex:712-719; s1162.