N12 · The CKM phase and its uncorrected control

Row

House note — R84 (S328a). Row 12a now prints the PDG global-fit comparator in the unit it is published in, 1.154 ± 0.025 rad, and computes the distance there: d = +1.40 (the degree cells quoted below give +1.41; R84-e extends R77-a — the comparator's published representation is primary). Rows 12a and 12b now carry Paper 3's own sentence in their status cells: “the 5φ⁻⁵ correction (+2.54°) is not supported by current data” (p3_ckm_pmns_mixing.tex:413-414; R84-d). Row 12c is unchanged. The cells quoted below are the Scorecard as it was supplied to the drafting lane; the sealed papers keep their figures until Rev32.12 (= Rev33.0, R156).

House correction — R77 (S327h). Row 12b: +2.60% → +2.61% (δ = 68.129749°, rounded once). Row 12c: −0.80% → −0.81% against the global fit, and the direct distance −0.30 → −0.29. R77 fixes the asymmetric-error convention: a distance uses the error on the side facing the theory value. So 12b keeps +0.64 (above the central value, upper error 2.7), and 12c uses the lower error 2.8. Of the two denominators this note audits below, the directional one is now the Scorecard's, and the "lower errors are used" sentence the note quotes is queued for Rev32.12 (= Rev33.0, R156) as wording. 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 12a

δ CKM (°) vs global fit arctan√(3G 7 ) theory 68.13 experiment 66.12 ± 1.43 deviation +3.04% · d = +1.41 Coincidence-class —

row 12b

δ CKM (°) vs direct γ arctan√(3G 7 ) theory 68.13 experiment 66.4 +2.7 −2.8 deviation +2.60% · d = +0.64 Coincidence-class —

row 12c

δ CKM (°) — bare-φ control (no G 7 correction) arctan√(3φ) theory 65.59 experiment 66.12 ± 1.43 (global) / 66.4 +2.7 −2.8 (direct) deviation −0.80% · d = −0.37 / −0.30 Control (printed beside 12a/b) not scored — the uncorrected argument sits closer than G 7 on both comparators; 28 of 64 sibling corrections land within 1σ, so the numeral’s agreement carries no significance (Paper 3, Rev32.2)

The asymmetric direct-angle convention is inconsistent across the printed summaries: the above-centre corrected value uses the upper error, while the below-centre control also matches the upper error. A nearby source note instead says lower errors are used. Both phase-distance conventions are exposed below. The direct corrected percentage also rounds to +2.61%, not the printed +2.60%; no comparator is reselected.

Theory value

row 12a

The suite prints

row 12b

The same \arctan\sqrt{3G_7} value, 68.13°, is compared with direct \gamma. This is a second comparator, not a second theory value.

row 12c

The uncorrected control is

It is printed beside the corrected argument and is not scored.

Derivation chain

row 12a

1. Take the retained corrected golden argument. Its first-principles motivation is explicitly incomplete. (appendix_x_zero_parameter_input_ledger.tex:75-82).
2. Evaluate the arctangent in degrees against the stated global-fit Dirac phase. (p3_ckm_pmns_mixing.tex:405-409).
3. NOT IN SUITE — a selection law for the correction. The printed sibling census and bare control are reasons not to treat agreement as evidence selecting that correction. (p3_ckm_pmns_mixing.tex:410-416).

row 12b

1. Reuse the corrected phase expression without changing its parameters. (appendix_x_zero_parameter_input_ledger.tex:75-82).
2. Keep the global Dirac phase and the direct unitarity-triangle angle as separately named comparison objects. (p3_ckm_pmns_mixing.tex:405-409).

row 12c

1. Remove the stated correction and evaluate the source’s bare-argument control. (p3_ckm_pmns_mixing.tex:410-416).
2. Keep both comparators and the supplied sibling census. The note does not rerun that search or select a replacement correction. (p3_ckm_pmns_mixing.tex:410-416).

Registrar sync

row 12a

LIB2-068 — Carries the corrected argument, degree-valued expression and coincidence classification.

Comparator LIB2-222 / SL-19 — value 66.12 (+1.43/−1.43) deg · scheme PDG global fit (=1.154±0.025 rad) · edition 2026.

row 12b

LIB2-068 — Carries the same corrected phase; this entry changes only the comparator.

Comparator LIB2-223 / SL-20 — value 66.4 (+2.7/−2.8) deg · scheme direct UT measurement · edition 2026.

row 12c

LIB2-075 — Carries the bare-argument control and its non-scored comparison, not a new physical selection law.

Comparator LIB2-222 / SL-19 — value 66.12 (+1.43/−1.43) deg · scheme PDG global fit (=1.154±0.025 rad) · edition 2026.

Comparator LIB2-223 / SL-20 — value 66.4 (+2.7/−2.8) deg · scheme direct UT measurement · edition 2026.

LIB2-069 and LIB2-070 carry comparison summaries, not the correction’s derivation. Historical claims and open questions are not substituted for LIB2-068.

Calculation

Run python3 N12_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
"""N12 — 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 12a ===')
    print('Scorecard (verbatim): δ CKM (°) vs global fit arctan√(3G 7 ) theory 68.13 experiment 66.12 ± 1.43 deviation +3.04% · d = +1.41 Coincidence-class —')
    g7 = PHI+5.0*PHI**(-5)
    phase = math.degrees(math.atan(math.sqrt(3.0*g7)))
    print("G7 = " + shown(g7,12))
    print("theory_at_cell_precision_deg = " + shown(phase,2))
    compare("full_formula_global_deg", phase, "66.12", "1.43", "1.43")
    compare("printed_cell_global_deg", "68.13", "66.12", "1.43", "1.43")
    print()
    print('=== row 12b ===')
    print('Scorecard (verbatim): δ CKM (°) vs direct γ arctan√(3G 7 ) theory 68.13 experiment 66.4 +2.7 −2.8 deviation +2.60% · d = +0.64 Coincidence-class —')
    phase = math.degrees(math.atan(math.sqrt(3.0*(PHI+5.0*PHI**(-5)))))
    print("theory_at_cell_precision_deg = " + shown(phase,2))
    compare("full_formula_direct_deg", phase, "66.4", "2.8", "2.7")
    compare("printed_cell_direct_deg", "68.13", "66.4", "2.8", "2.7")
    print("alternate_lower_error_d = " + shown((phase-66.4)/2.8,9,True)
          + "; rounded = " + shown((phase-66.4)/2.8,2,True))
    print()
    print('=== row 12c ===')
    print('Scorecard (verbatim): δ CKM (°) — bare-φ control (no G 7 correction) arctan√(3φ) theory 65.59 experiment 66.12 ± 1.43 (global) / 66.4 +2.7 −2.8 (direct) deviation −0.80% · d = −0.37 / −0.30 Control (printed beside 12a/b) not scored — the uncorrected argument sits closer than G 7 on both comparators; 28 of 64 sibling corrections land within 1σ, so the numeral’s agreement carries no significance (Paper 3, Rev32.2)')
    phase = math.degrees(math.atan(math.sqrt(3.0*PHI)))
    print("theory_at_cell_precision_deg = " + shown(phase,2))
    compare("full_control_global_deg", phase, "66.12", "1.43", "1.43")
    compare("printed_control_global_deg", "65.59", "66.12", "1.43", "1.43")
    compare("full_control_direct_deg", phase, "66.4", "2.8", "2.7")
    compare("printed_control_direct_deg", "65.59", "66.4", "2.8", "2.7")
    print("alternate_upper_error_d = " + shown((phase-66.4)/2.7,9,True)
          + "; rounded = " + shown((phase-66.4)/2.7,2,True))
    print("source_census = 28 of 64 siblings within 1 sigma; reported, not rerun")
    print()


if __name__ == "__main__":
    main()

row 12a

Actual stdout for this entry; the text blocks in entry order concatenate to N12_stdout.txt.

=== row 12a ===
Scorecard (verbatim): δ CKM (°) vs global fit arctan√(3G 7 ) theory 68.13 experiment 66.12 ± 1.43 deviation +3.04% · d = +1.41 Coincidence-class —
G7 = 2.068883707497
theory_at_cell_precision_deg = 68.13
full_formula_global_deg.theory = 68.129749089
full_formula_global_deg.reference = 66.120000000
full_formula_global_deg.signed_percent = +3.039547926%; rounded = +3.04%
full_formula_global_deg.uncertainty_used = 1.43 (upper)
full_formula_global_deg.d = +1.405418943; rounded = +1.41
printed_cell_global_deg.theory = 68.130000000
printed_cell_global_deg.reference = 66.120000000
printed_cell_global_deg.signed_percent = +3.039927405%; rounded = +3.04%
printed_cell_global_deg.uncertainty_used = 1.43 (upper)
printed_cell_global_deg.d = +1.405594406; rounded = +1.41

row 12b

Actual stdout for this entry; the text blocks in entry order concatenate to N12_stdout.txt.

=== row 12b ===
Scorecard (verbatim): δ CKM (°) vs direct γ arctan√(3G 7 ) theory 68.13 experiment 66.4 +2.7 −2.8 deviation +2.60% · d = +0.64 Coincidence-class —
theory_at_cell_precision_deg = 68.13
full_formula_direct_deg.theory = 68.129749089
full_formula_direct_deg.reference = 66.400000000
full_formula_direct_deg.signed_percent = +2.605043809%; rounded = +2.61%
full_formula_direct_deg.uncertainty_used = 2.7 (upper)
full_formula_direct_deg.d = +0.640647811; rounded = +0.64
printed_cell_direct_deg.theory = 68.130000000
printed_cell_direct_deg.reference = 66.400000000
printed_cell_direct_deg.signed_percent = +2.605421687%; rounded = +2.61%
printed_cell_direct_deg.uncertainty_used = 2.7 (upper)
printed_cell_direct_deg.d = +0.640740741; rounded = +0.64
alternate_lower_error_d = +0.617767532; rounded = +0.62

row 12c

Actual stdout for this entry; the text blocks in entry order concatenate to N12_stdout.txt.

=== row 12c ===
Scorecard (verbatim): δ CKM (°) — bare-φ control (no G 7 correction) arctan√(3φ) theory 65.59 experiment 66.12 ± 1.43 (global) / 66.4 +2.7 −2.8 (direct) deviation −0.80% · d = −0.37 / −0.30 Control (printed beside 12a/b) not scored — the uncorrected argument sits closer than G 7 on both comparators; 28 of 64 sibling corrections land within 1σ, so the numeral’s agreement carries no significance (Paper 3, Rev32.2)
theory_at_cell_precision_deg = 65.59
full_control_global_deg.theory = 65.587428410
full_control_global_deg.reference = 66.120000000
full_control_global_deg.signed_percent = -0.805462176%; rounded = -0.81%
full_control_global_deg.uncertainty_used = 1.43 (lower)
full_control_global_deg.d = -0.372427686; rounded = -0.37
printed_control_global_deg.theory = 65.590000000
printed_control_global_deg.reference = 66.120000000
printed_control_global_deg.signed_percent = -0.801572898%; rounded = -0.80%
printed_control_global_deg.uncertainty_used = 1.43 (lower)
printed_control_global_deg.d = -0.370629371; rounded = -0.37
full_control_direct_deg.theory = 65.587428410
full_control_direct_deg.reference = 66.400000000
full_control_direct_deg.signed_percent = -1.223752395%; rounded = -1.22%
full_control_direct_deg.uncertainty_used = 2.8 (lower)
full_control_direct_deg.d = -0.290204139; rounded = -0.29
printed_control_direct_deg.theory = 65.590000000
printed_control_direct_deg.reference = 66.400000000
printed_control_direct_deg.signed_percent = -1.219879518%; rounded = -1.22%
printed_control_direct_deg.uncertainty_used = 2.8 (lower)
printed_control_direct_deg.d = -0.289285714; rounded = -0.29
alternate_upper_error_d = -0.300952441; rounded = -0.30
source_census = 28 of 64 siblings within 1 sigma; reported, not rerun

Comparison

row 12a

Global comparator: 66.12 ±1.43°, PDG 2026 Dirac-phase global fit (LIB2-222 / SL-19). Theory 68.13°, signed deviation +3.04%, naive d +1.41. No renormalization scale is supplied for this fit parameter.

row 12b

Direct comparator: 66.4 +2.7 −2.8°, PDG 2026 direct \gamma (LIB2-223 / SL-20). The cell prints +2.60%, but either full-formula or rounded-cell arithmetic gives +2.61% at two decimals; directional-upper d +0.64. Always using the lower error instead gives +0.62. The two named angles are not silently identified by their numerical proximity.

row 12c

The printed −0.80% refers to the rounded control against the global comparator; full-precision arithmetic is also shown. Global d rounds to −0.37. For the direct comparator the directional-lower d is −0.29, not the printed −0.30; the latter uses the upper error. The control is closer on both comparisons, but no significance or new selection is assigned.

Tier and what this does not show

Literal tiers: Coincidence-class for rows 12a and 12b; Control (printed beside 12a/b) for 12c. The corrected argument remains unselected from first principles; the control is not a hit. The supplied census is retained as a source report, not independently reproduced.

Sources

LIB2-068; LIB2-069; LIB2-070; LIB2-075; LIB2-222; LIB2-223; SL-19; SL-20; appendix_x_zero_parameter_input_ledger.tex:75-82; p3_ckm_pmns_mixing.tex:405-416.