N01 · The reciprocal-square identity

Row

row 1

φ² + φ⁻² = 3 (axiom) theory 3 experiment 3 deviation 0 · d = exact Axiom exact

This is an internal identity, not an experimental match. The Scorecard calls the row “Axiom”; LIB2-046 records the identity at the selected spectrum. Those are different descriptions, retained without changing the cell.

Theory value

row 1

The suite prints

The computed internal value is 3, exactly. The reference 3 is the same identity, not a measurement.

Derivation chain

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1. Use the named golden-unit input, AX1: the positive root of the stated quadratic defines the scalar used here. (p0_framework_foundations.tex:521; appendix_x_zero_parameter_input_ledger.tex:62).
2. Evaluate the reciprocal-square identity at that scalar. The supplied source explicitly gives \phi^2=\phi+1 and \phi^{-2}=2-\phi; adding them is the arithmetic check, not a physical selection step. (appendix_i_dynamics_bounce.tex:608-609; appendix_e_vacuum_selector.tex:149-153).
3. Read the result as the spectral corollary. The conditional vacuum-selection hypotheses and the ordered-frame loading remain separate. (appendix_e_vacuum_selector.tex:509).

Registrar sync

row 1

LIB2-046 — Carries the exact reciprocal-square corollary used in the addition step.

Comparator NONE — the reference is internal, structural or adopted, not a supplied experimental ledger entry.

LIB2-082, LIB2-164, LIB2-200 and LIB2-344 concern other predicates; their shared numerals do not carry this identity.

Calculation

Run python3 N01_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
"""N01 — 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

# Exact arithmetic in Q(phi), with phi^2=phi+1. A pair is a+b*phi.
def qmul(x, y):
    a, b = x
    c, d = y
    return (a*c + b*d, a*d + b*c + b*d)


def qadd(x, y):
    return (x[0]+y[0], x[1]+y[1])


def qstr(x):
    return str(x[0]) + " + (" + str(x[1]) + ")*phi"

def main():
    print('=== row 1 ===')
    print('Scorecard (verbatim): φ² + φ⁻² = 3 (axiom) theory 3 experiment 3 deviation 0 · d = exact Axiom exact')
    phi = (Fraction(0), Fraction(1))
    inv = (Fraction(-1), Fraction(1))
    square, inverse_square = qmul(phi, phi), qmul(inv, inv)
    result = qadd(square, inverse_square)
    print("phi_squared = " + qstr(square))
    print("phi_inverse_squared = " + qstr(inverse_square))
    print("identity = " + qstr(result))
    assert result == (Fraction(3), Fraction(0))
    print("theory = 3; internal_reference = 3; signed_percent = 0%")
    print("d = exact (identity label, not a statistical distance)")
    print()


if __name__ == "__main__":
    main()

row 1

Actual stdout for this entry; the text blocks in entry order concatenate to N01_stdout.txt.

=== row 1 ===
Scorecard (verbatim): φ² + φ⁻² = 3 (axiom) theory 3 experiment 3 deviation 0 · d = exact Axiom exact
phi_squared = 1 + (1)*phi
phi_inverse_squared = 2 + (-1)*phi
identity = 3 + (0)*phi
theory = 3; internal_reference = 3; signed_percent = 0%
d = exact (identity label, not a statistical distance)

Comparison

row 1

Theory 3; internal reference 3; signed deviation 0%. Unit, experimental uncertainty, renormalization scheme, scale and experimental edition are not applicable. “d = exact” means an exact algebraic check, not an error-normalized statistic.

Tier and what this does not show

Literal tier: Axiom; status: exact. AX1 is consumed. The calculation neither tests the axiom experimentally nor derives the physical vacuum, a mass scale, or a readout.

Sources

LIB2-046; p0_framework_foundations.tex:521; appendix_x_zero_parameter_input_ledger.tex:62; appendix_i_dynamics_bounce.tex:608-609; appendix_e_vacuum_selector.tex:149-153; appendix_e_vacuum_selector.tex:509.