N04 · The solar-angle readout in two coordinates

Row

row 4

sin²θ 12 3/(φ⁴+3) theory 0.304442 experiment 0.3088 +0.0067 −0.0066 deviation −1.41% · d = −0.66 Derived-conditional —

row 4′

θ 12 (°) arctan(√3/φ²) theory 33.488 experiment 33.76 +0.42 −0.41 deviation −0.81% · d = −0.66 Derived-conditional —

The candidates LIB2-063, LIB2-064 and LIB2-238 are Cabibbo readouts, not the solar derivation. The supplied ledger record LIB2-227 is an angle in degrees; no supplied comparator record carries the squared-sine cell itself.

Theory value

row 4

The suite prints \tan\theta_{12}=\sqrt3/\phi^2 and \tan^2\theta_{12}=3/\phi^4. Evaluating the Scorecard’s equivalent coordinate gives

at the cell’s precision.

row 4′

The suite’s angle formula is

at Scorecard precision. The coarser source display is 33.49^{\circ}; it is not a different formula.

Derivation chain

row 4

1. Use the golden-unit input and its reciprocal-square identity. (appendix_x_zero_parameter_input_ledger.tex:62; appendix_e_vacuum_selector.tex:149-153).
2. Apply the stated Peirce readout \tan\theta_{12}=\sqrt3/\phi^2, with the normal-ordering, resolvent-ladder and propagation-interface premises. (appendix_e_vacuum_selector.tex:154-156; p3_ckm_pmns_mixing.tex:581).
3. Evaluate the squared-sine coordinate of that angle; this is arithmetic re-expression of the supplied tangent relation, not an additional physical theorem. (p0_framework_foundations.tex:641-647).
4. NOT IN SUITE — a derived identification of the ladder with physical propagation eigenstates; the supplied source calls that interface undeclared. (p3_ckm_pmns_mixing.tex:581).

row 4′

1. Evaluate the same tangent readout in degrees, without using the measured solar angle as input. (appendix_e_vacuum_selector.tex:154-158).
2. The kernel-line passage prints the finer framework angle separately from its coarser display; the physical premises remain those of the solar row. (p3_ckm_pmns_mixing.tex:157; p3_ckm_pmns_mixing.tex:581).

Registrar sync

row 4

LIB2-046 — Carries the reciprocal-square input identity.

LIB2-060 — Carries the solar readout and its unclosed ordering/interface premises.

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

row 4′

LIB2-060 — Carries the tangent readout whose degree evaluation is shown.

Comparator LIB2-227 / SL-24 — value 33.76 (+0.42/−0.41) deg · scheme NH global fit · edition 6.1.

Calculation

Run python3 N04_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
"""N04 — 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 4 ===')
    print('Scorecard (verbatim): sin²θ 12 3/(φ⁴+3) theory 0.304442 experiment 0.3088 +0.0067 −0.0066 deviation −1.41% · d = −0.66 Derived-conditional —')
    value = 3.0/(PHI**4+3.0)
    print("formula = 3/(phi**4+3)")
    print("theory_at_cell_precision = " + shown(value,6))
    compare("full_formula", value, "0.3088", "0.0066", "0.0067")
    compare("printed_cell", "0.304442", "0.3088", "0.0066", "0.0067")
    print()
    print('=== row 4′ ===')
    print('Scorecard (verbatim): θ 12 (°) arctan(√3/φ²) theory 33.488 experiment 33.76 +0.42 −0.41 deviation −0.81% · d = −0.66 Derived-conditional —')
    value = math.degrees(math.atan(math.sqrt(3.0)/PHI**2))
    print("formula = degrees(atan(sqrt(3)/phi**2))")
    print("theory_at_cell_precision = " + shown(value,3))
    print("coarse_suite_display = " + shown(value,2))
    compare("full_formula_deg", value, "33.76", "0.41", "0.42")
    compare("printed_cell_deg", "33.488", "33.76", "0.41", "0.42")
    print()


if __name__ == "__main__":
    main()

row 4

Actual stdout for this entry; the text blocks in entry order concatenate to N04_stdout.txt.

=== row 4 ===
Scorecard (verbatim): sin²θ 12 3/(φ⁴+3) theory 0.304442 experiment 0.3088 +0.0067 −0.0066 deviation −1.41% · d = −0.66 Derived-conditional —
formula = 3/(phi**4+3)
theory_at_cell_precision = 0.304442
full_formula.theory = 0.304441745
full_formula.reference = 0.308800000
full_formula.signed_percent = -1.411351942%; rounded = -1.41%
full_formula.uncertainty_used = 0.0066 (lower)
full_formula.d = -0.660341636; rounded = -0.66
printed_cell.theory = 0.304442000
printed_cell.reference = 0.308800000
printed_cell.signed_percent = -1.411269430%; rounded = -1.41%
printed_cell.uncertainty_used = 0.0066 (lower)
printed_cell.d = -0.660303030; rounded = -0.66

row 4′

Actual stdout for this entry; the text blocks in entry order concatenate to N04_stdout.txt.

=== row 4′ ===
Scorecard (verbatim): θ 12 (°) arctan(√3/φ²) theory 33.488 experiment 33.76 +0.42 −0.41 deviation −0.81% · d = −0.66 Derived-conditional —
formula = degrees(atan(sqrt(3)/phi**2))
theory_at_cell_precision = 33.488
coarse_suite_display = 33.49
full_formula_deg.theory = 33.488005129
full_formula_deg.reference = 33.760000000
full_formula_deg.signed_percent = -0.805672011%; rounded = -0.81%
full_formula_deg.uncertainty_used = 0.41 (lower)
full_formula_deg.d = -0.663402124; rounded = -0.66
printed_cell_deg.theory = 33.488000000
printed_cell_deg.reference = 33.760000000
printed_cell_deg.signed_percent = -0.805687204%; rounded = -0.81%
printed_cell_deg.uncertainty_used = 0.41 (lower)
printed_cell_deg.d = -0.663414634; rounded = -0.66

Comparison

row 4

Theory 0.304442, comparator 0.3088 +0.0067 −0.0066, dimensionless; the supplied paired material identifies NuFIT 6.1, NH global fit. Signed offset rounds to −1.41% and naive d to −0.66, using the lower error. No scheme or running scale is assigned to this fit parameter. Its missing ledger ID is not replaced with the degree-valued record.

row 4′

Theory 33.488°; experiment 33.76 +0.42 −0.41°, NuFIT 6.1, NH global fit (LIB2-227 / SL-24). Signed offset −0.81%; naive d −0.66. These are two coordinate displays of one readout, not independent observations.

Tier and what this does not show

Both literal tiers: Derived-conditional; statuses: —. Normal ordering, the Structural ladder and the propagation attachment are consumed. The calculation does not derive those premises or double the evidence by printing an angle and its squared sine.

Sources

LIB2-046; LIB2-060; LIB2-227; SL-24; appendix_x_zero_parameter_input_ledger.tex:62; appendix_e_vacuum_selector.tex:149-158; p0_framework_foundations.tex:641-647; p3_ckm_pmns_mixing.tex:157; p3_ckm_pmns_mixing.tex:581; s1162.