N06 · The reactor-angle round-trip formula

Row

row 6

sin²θ 13 sin⁴(π/8) = (3−2√2)/8 theory 0.0214466 experiment 0.02248 +0.00055 −0.00059 deviation −4.60% · d = −1.75 Structural —

Only LIB2-061 carries the retained formula. Historical proposals and the open question are not proofs of the adopted readout. LIB2-228 / SL-26 is a degree-valued angle, not this squared-sine comparator.

Theory value

row 6

The suite prints

at the Scorecard’s precision.

Derivation chain

row 6

1. Take the identified triality half-angle readout and round-trip squaring as the retained construction. (p0_framework_foundations.tex:702-706).
2. Evaluate the trigonometric expression and its printed radical form independently; the script checks that both routes give the same value. (p3_ckm_pmns_mixing.tex:583).
3. NOT IN SUITE — first-principles selection of the readout ray and squaring. The source explicitly withdraws the inference from the substrate half-turn to a canonical mixing half-angle. (appendix_b_independent_verification.tex:177-191; appendix_x_zero_parameter_input_ledger.tex:70).

Registrar sync

row 6

LIB2-061 — Carries the retained formula and the fact that its round-trip readout remains unforced.

Comparator LIB2-226 / SL-23 — value 0.02248 (+0.00055/−0.00059) · scheme NH global fit (IC24) · edition 6.1.

LIB2-252 is superseded wording; LIB2-280–LIB2-282 are archived proposals; LIB2-315 is an open question. None is mapped as the retained derivation.

Calculation

Run python3 N06_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
"""N06 — 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 6 ===')
    print('Scorecard (verbatim): sin²θ 13 sin⁴(π/8) = (3−2√2)/8 theory 0.0214466 experiment 0.02248 +0.00055 −0.00059 deviation −4.60% · d = −1.75 Structural —')
    trig = math.sin(math.pi/8.0)**4
    radical = (3.0-2.0*math.sqrt(2.0))/8.0
    assert abs(trig-radical) < 1e-14
    print("trigonometric_value = " + shown(trig))
    print("radical_value = " + shown(radical))
    print("theory_at_cell_precision = " + shown(trig,7))
    compare("full_formula", trig, "0.02248", "0.00059", "0.00055")
    compare("printed_cell", "0.0214466", "0.02248", "0.00059", "0.00055")
    print()


if __name__ == "__main__":
    main()

row 6

Actual stdout for this entry; the text blocks in entry order concatenate to N06_stdout.txt.

=== row 6 ===
Scorecard (verbatim): sin²θ 13 sin⁴(π/8) = (3−2√2)/8 theory 0.0214466 experiment 0.02248 +0.00055 −0.00059 deviation −4.60% · d = −1.75 Structural —
trigonometric_value = 0.021446609
radical_value = 0.021446609
theory_at_cell_precision = 0.0214466
full_formula.theory = 0.021446609
full_formula.reference = 0.022480000
full_formula.signed_percent = -4.596933244%; rounded = -4.60%
full_formula.uncertainty_used = 0.00059 (lower)
full_formula.d = -1.751509480; rounded = -1.75
printed_cell.theory = 0.021446600
printed_cell.reference = 0.022480000
printed_cell.signed_percent = -4.596975089%; rounded = -4.60%
printed_cell.uncertainty_used = 0.00059 (lower)
printed_cell.d = -1.751525424; rounded = -1.75

Comparison

row 6

Theory 0.0214466; comparator 0.02248 +0.00055 −0.00059, NuFIT 6.1 NH global fit, IC24, dimensionless (LIB2-226 / SL-23). Signed deviation −4.60%; naive d −1.75, lower error. It is a retained structural readout, not a derived selection result.

Tier and what this does not show

Literal tier: Structural; status: —. The half-angle normalization and squaring are chosen. Numerical agreement does not supply the missing physical readout selection or restore withdrawn canonicality.

Sources

LIB2-061; LIB2-226; SL-23; p0_framework_foundations.tex:702-706; p3_ckm_pmns_mixing.tex:583; appendix_b_independent_verification.tex:177-191; appendix_x_zero_parameter_input_ledger.tex:70; s502.