Question 997: Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues

Original question and givens

Use exact modulus n=1 and integer a=23. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=invalid_modulus. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).

• G001 — Use exact modulus n=1 and integer a=23. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=invalid_modulus. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).

Definition and scope

For n<2 or a noncoprime residue the order is undefined; residue 1 has order 1; normalize a negative representative into [0,n) first.

• mko-multiplicative-order-mod-n

Step-by-step derivation

n=1; a=23

order is defined here only for n≥2 and units modulo n

n=1<2

modulus qualification fails

outside the declared unit-group domain

residue=null; gcd=null; power table=null

qualification=false

order=undefined

Reverse checks

• S004 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi.
• S003 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi.
• S002 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi.
• S001 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi.

Result

(invalid_modulus, 1, 23, null, null, No, null, undefined, mode_specific_multiplicative_order_boundary)

The full substitutions and premise audits recover the original answer, retaining its field order and verdict meanings.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.