# Question 992: Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

P1000-0992 · q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0092



## Original question and givens

Use exact modulus n=40 and integer a=2. 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=noncoprime. 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=40 and integer a=2. 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=noncoprime. 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](https://ocme.evemisslab.com/en/mko/mko-multiplicative-order-mod-n/)





## Step-by-step derivation

S001 · impliesBefore

n=40; a=2

After

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

Conditions: none

Fix this node's domain first; do not confuse additive and multiplicative order.

Step identifiers and dependenciesRule: order_domain · Uses: G001





S002 · impliesBefore

a=2

After

normalized residue=2; gcd(2,40)=2

Conditions: none

Normalize even a negative representative into[0,n), then test coprimality.

Step identifiers and dependenciesRule: normalize_and_qualify · Uses: G001, S001





S003 · impliesBefore

g=2>1 divides both residue and n

After

no positive power can be congruent to1

Conditions: none

The common divisor divides each positive power and the modulus, so a congruence to1 is impossible.

Step identifiers and dependenciesRule: nonunit_power_obstruction · Uses: G001, S002





S004 · impliesBefore

not a unit

After

power table=null; order=undefined

Conditions: none

This rejects the domain qualification; it does not set an undefined order to1.

Step identifiers and dependenciesRule: nonunit_order_boundary · Uses: G001, S003









## 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

(noncoprime, 40, 2, 2, 2, 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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: a43046ff55a7f25dad0f8c63e86521eedb7ea6151cdfbe765e362a525eff0343
- candidate record SHA-256: 1e3714fe7d932071f995ccf8eb7600b1ea27f23823acf1ec834a5818beb123d1
- semantic review receipt SHA-256: 6c2c6d46aded7582a11ac41a8c0b3bdde2669a53bdb6e796eea2039994a02772
- question-first witness SHA-256: 77e43898df9efdbf8094ad81fe4ab91cf4b4bd82f481eb9b4699b73cb4da612b
- Lean: NOT_RUN · human review: NOT_PERFORMED

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