N09 · The data-selected Cabibbo route

Row

row 9

|V us | sin(θ 12 fw )/√6 theory 0.225256 experiment 0.22431 ± 0.00085 deviation +0.42% · d = +1.11 Loaded (data-selected route) (digit from kernel s1162, Rev32.6; the Rev32.5 print 0.225247 was a hand calculation) —

The finer cell agrees with the current kernel-line formula. The historical hand-calculated digit and the separate measured-solar-angle diagnostic are not used to obtain this row.

Theory value

row 9

The framework-angle route is

The superscript “fw” is essential: the measured solar angle is not the input.

Derivation chain

row 9

1. Evaluate the conditional framework tangent and convert it to its sine without rounding the angle. (appendix_x_zero_parameter_input_ledger.tex:68).
2. Apply the stated normalization in Route B. The source gives the precise value and distinguishes a coarsely rounded angle from the framework computation. (p3_ckm_pmns_mixing.tex:155-158).
3. Record why this route is retained: it was selected against kaon data. NOT IN SUITE — a data-independent rule selecting this physical route; the alternative measured-angle diagnostic is a different calculation. (p3_ckm_pmns_mixing.tex:155-162).

Registrar sync

row 9

LIB2-060 — Carries the conditional framework solar-angle input.

LIB2-063 — Carries the data-selected Route-B formula and corrected numeric value.

Comparator LIB2-214 / SL-11 — value 0.22431 (+0.00085/−0.00085) · scheme PDG first-row average · edition 2026 (A1546 lined).

LIB2-064 is the measured-angle diagnostic. LIB2-237, LIB2-238 and LIB2-239 describe superseded digits or routes; they do not carry the retained computation.

Calculation

Run python3 N09_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
"""N09 — 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 main():
    print('=== row 9 ===')
    print('Scorecard (verbatim): |V us | sin(θ 12 fw )/√6 theory 0.225256 experiment 0.22431 ± 0.00085 deviation +0.42% · d = +1.11 Loaded (data-selected route) (digit from kernel s1162, Rev32.6; the Rev32.5 print 0.225247 was a hand calculation) —')
    solar = 3.0/(PHI**4+3.0)
    angle = math.atan(math.sqrt(3.0)/PHI**2)
    vus = math.sin(angle)/math.sqrt(6.0)
    assert abs(vus-math.sqrt(solar)/math.sqrt(6.0)) < 1e-14
    print("framework_angle_deg = " + shown(math.degrees(angle),9))
    print("framework_sine_squared = " + shown(solar,12))
    print("theory_at_cell_precision = " + shown(vus,6))
    compare("full_formula", vus, "0.22431", "0.00085", "0.00085")
    compare("printed_cell", "0.225256", "0.22431", "0.00085", "0.00085")
    print()


if __name__ == "__main__":
    main()

row 9

Actual stdout for this entry; the text blocks in entry order concatenate to N09_stdout.txt.

=== row 9 ===
Scorecard (verbatim): |V us | sin(θ 12 fw )/√6 theory 0.225256 experiment 0.22431 ± 0.00085 deviation +0.42% · d = +1.11 Loaded (data-selected route) (digit from kernel s1162, Rev32.6; the Rev32.5 print 0.225247 was a hand calculation) —
framework_angle_deg = 33.488005129
framework_sine_squared = 0.304441745202
theory_at_cell_precision = 0.225256
full_formula.theory = 0.225256056
full_formula.reference = 0.224310000
full_formula.signed_percent = +0.421762840%; rounded = +0.42%
full_formula.uncertainty_used = 0.00085 (upper)
full_formula.d = +1.113007326; rounded = +1.11
printed_cell.theory = 0.225256000
printed_cell.reference = 0.224310000
printed_cell.signed_percent = +0.421737774%; rounded = +0.42%
printed_cell.uncertainty_used = 0.00085 (upper)
printed_cell.d = +1.112941176; rounded = +1.11

Comparison

row 9

Theory 0.225256; comparator 0.22431 ±0.00085, PDG 2026 first-row average, dimensionless (LIB2-214 / SL-11). Signed deviation +0.42%; naive d +1.11. The agreement was involved in selecting the route, so it is not independent evidence for that selection.

Tier and what this does not show

Literal tier: Loaded (data-selected route) (digit from kernel s1162, Rev32.6; the Rev32.5 print 0.225247 was a hand calculation). Status: —. This retains the route selection and the conditional solar attachment. It does not turn the separately obtained normalization into a physical route-selection law.

Sources

LIB2-060; LIB2-063; LIB2-214; SL-11; appendix_x_zero_parameter_input_ledger.tex:68; p3_ckm_pmns_mixing.tex:155-162; s1162.