N19 · The bottom bridge and its tau anchor

Row

House correction — R77 (S327h). Row 19: the Scorecard printed −0.96% · d = −6.67, computed from the displayed 4146 MeV. At full precision (m_b = 4146.17 MeV) it is −0.95% · d = −6.64; the theory cell still displays 4146. It is a naive cross-scheme comparator distance, not a tension, as before. The τ row is unchanged. 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 19

m b (MeV) (7/3)·m τ theory 4146 experiment 4186 ± 6 (m b (m b ), MS-bar) deviation −0.96% · d = −6.67 Structural cross-scheme

row τ

— m τ (MeV) anchor theory 1776.93 experiment 1776.93 ± 0.09 deviation 0 Input anchor

The bottom cell and its quoted distance use the integer-MeV display, not the unrounded bridge value. Both are shown. The tau entry is an input anchor and has no derivation record mapped as though it were a prediction.

Theory value

row 19

The suite prints

at cell precision. The unrounded product is 4146.17 MeV.

row τ

The entry is the external anchor

Its equality with the central comparator is by adoption, not an independent theory result.

Derivation chain

row 19

1. Take DET-7 and the stated Jordan rank ratio as the retained bridge coefficient, while preserving the invariant’s imposed status. (appendix_x_zero_parameter_input_ledger.tex:64; appendix_x_zero_parameter_input_ledger.tex:96).
2. Multiply the external tau anchor by the bridge coefficient without intermediate rounding. (appendix_x_zero_parameter_input_ledger.tex:419; appendix_x_zero_parameter_input_ledger.tex:427).
3. NOT IN SUITE — the on-shell/renormalization matching that makes this algebraic ratio a scheme-matched physical mass relation. The supplied source explicitly retains the physical attachment gap. (p2_mass_hierarchy_resolvent_quintics.tex:437-441; appendix_i_dynamics_bounce.tex:643-651).

row τ

1. Load the tau mass from the named operational-anchor inventory and use it unchanged in the formula chains. (Appendix X, Table 4, Named operational anchors, PDF p.334; appendix_x_zero_parameter_input_ledger.tex:419).
2. No derivation step is claimed for this input. The source distinguishes named empirical anchors from a derived relation tying their values together. (Appendix X, Ledger consequence: physical one-input reading, PDF p.331).

Registrar sync

row 19

LIB2-018 — Carries the imposed status of the invariant in the bridge coefficient.

LIB2-124 — Carries the operational tau anchor.

LIB2-130 — Carries the bottom bridge and its cross-scheme boundary.

Comparator LIB2-219 / SL-16 — value 4186 (+6/−6) MeV · scheme MS-bar at m_b · edition 2026.

row τ

NONE — this entry is the externally adopted tau anchor, not a derivation; LIB2-124 documents its use but does not derive its value.

Comparator LIB2-221 / SL-18 — value 1776.93 (+0.09/−0.09) MeV · scheme pole · edition 2026.

LIB2-132 is a scoped negative running test, not a conversion or a derivation of this bridge.

Calculation

Run python3 N19_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
"""N19 — 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 19 ===')
    print('Scorecard (verbatim): m b (MeV) (7/3)·m τ theory 4146 experiment 4186 ± 6 (m b (m b ), MS-bar) deviation −0.96% · d = −6.67 Structural cross-scheme')
    tau = Decimal("1776.93")
    value = Decimal(7)*tau/Decimal(3)
    print("tau_anchor_MeV = " + shown(tau,2))
    print("bridge_coefficient = 7/3")
    print("unrounded_bottom_MeV = " + shown(value,2))
    print("theory_at_cell_precision_MeV = " + shown(value,0))
    compare("unrounded_bridge_MeV", value, "4186", "6", "6")
    compare("printed_cell_MeV", "4146", "4186", "6", "6")
    print("comparison_type = cross-scheme; tau/matching uncertainty not propagated")
    print()
    print('=== row τ ===')
    print('Scorecard (verbatim): — m τ (MeV) anchor theory 1776.93 experiment 1776.93 ± 0.09 deviation 0 Input anchor')
    compare("adopted_tau_anchor_MeV", "1776.93", "1776.93", "0.09", "0.09")
    print("interpretation = input equality, not independent agreement")
    print()


if __name__ == "__main__":
    main()

row 19

Actual stdout for this entry; the text blocks in entry order concatenate to N19_stdout.txt.

=== row 19 ===
Scorecard (verbatim): m b (MeV) (7/3)·m τ theory 4146 experiment 4186 ± 6 (m b (m b ), MS-bar) deviation −0.96% · d = −6.67 Structural cross-scheme
tau_anchor_MeV = 1776.93
bridge_coefficient = 7/3
unrounded_bottom_MeV = 4146.17
theory_at_cell_precision_MeV = 4146
unrounded_bridge_MeV.theory = 4146.170000000
unrounded_bridge_MeV.reference = 4186.000000000
unrounded_bridge_MeV.signed_percent = -0.951505017%; rounded = -0.95%
unrounded_bridge_MeV.uncertainty_used = 6 (lower)
unrounded_bridge_MeV.d = -6.638333333; rounded = -6.64
printed_cell_MeV.theory = 4146.000000000
printed_cell_MeV.reference = 4186.000000000
printed_cell_MeV.signed_percent = -0.955566173%; rounded = -0.96%
printed_cell_MeV.uncertainty_used = 6 (lower)
printed_cell_MeV.d = -6.666666667; rounded = -6.67
comparison_type = cross-scheme; tau/matching uncertainty not propagated

row τ

Actual stdout for this entry; the text blocks in entry order concatenate to N19_stdout.txt.

=== row τ ===
Scorecard (verbatim): — m τ (MeV) anchor theory 1776.93 experiment 1776.93 ± 0.09 deviation 0 Input anchor
adopted_tau_anchor_MeV.theory = 1776.930000000
adopted_tau_anchor_MeV.reference = 1776.930000000
adopted_tau_anchor_MeV.signed_percent = +0.000000000%; rounded = +0.00%
adopted_tau_anchor_MeV.uncertainty_used = 0.09 (upper)
adopted_tau_anchor_MeV.d = +0.000000000; rounded = +0.00
interpretation = input equality, not independent agreement

Comparison

row 19

Theory cell 4146 MeV; comparator 4186 ±6 MeV, PDG 2026, m_b(m_b) in MS-bar (LIB2-219 / SL-16). The cell gives −0.96%, naive d −6.67. The unrounded product gives −0.95%, naive d −6.64 at the same display precision. These are cross-scheme comparator distances, not tensions; no conversion or theory uncertainty is supplied.

row τ

Adopted value 1776.93 MeV; comparator 1776.93 ±0.09 MeV, PDG 2026 tau pole mass (LIB2-221 / SL-18). Arithmetic deviation and naive d are 0 because the central value was used as input. A running scale is not applicable to this pole-mass anchor. Its uncertainty is not propagated into the bottom row’s naive experimental-error distance.

Tier and what this does not show

Row 19 literal tier: Structural; status cross-scheme. The tau anchor and structural bridge are consumed. The τ entry has literal tier Input, status anchor. Agreement at the anchor adds no evidence for the bridge and does not derive a shared dimensional calibration.

Sources

LIB2-018; LIB2-124; LIB2-130; LIB2-219; LIB2-221; SL-16; SL-18; appendix_x_zero_parameter_input_ledger.tex:64; appendix_x_zero_parameter_input_ledger.tex:96; appendix_x_zero_parameter_input_ledger.tex:419; appendix_x_zero_parameter_input_ledger.tex:427; p2_mass_hierarchy_resolvent_quintics.tex:437-441; appendix_i_dynamics_bounce.tex:643-651; Appendix X, Table 4, Named operational anchors, PDF p.334; Appendix X, Ledger consequence: physical one-input reading, PDF p.331.