# Question 21376: Separate Zero, Sign Units and the Positive Prime Multiset

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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.

R0041-21376 · q-20261014-advance-20261108-r06-unique-prime-factorization-0076



## Original question and givens

Use original bounded integer n=0. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).


- G001 — Use original bounded integer n=0. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).





## Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.


- [mko-unique-prime-factorization](https://ocme.evemisslab.com/en/mko/mko-unique-prime-factorization/)





## Step-by-step derivation

S001 · impliesBefore

source operation=upf-unit-sign-boundary;original snapshot and mathematics retained

After

explicit ordered source output fields=[n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary]

Conditions: none

Retain the historical interface marker while listing ordered output fields without JavaScript.Fields and tags are not extra mathematical evidence;fixed No markers retain their claim scope.

Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001





S002 · impliesBefore

ORIGINAL signed integer=0;actual sign=null;positive prime multiset=[]

After

ordinary_positive_form_claim fixed No;zero has no finite prime product;units use empty product1;nonzero sign is separated from |n| prime factors

Conditions: none

Zero is not a finite prime product;±1 are units with an empty prime multiset.Other nonzero integers separate sign from positive factors.The fixed No is not a positivity test or a denial of factorization for a positive integer.

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









## Reverse checks


- S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

(0, No, null, [], zero_has_no_prime_factorization)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.





## 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: 6ee4e3cd6cb7ccc26d2df09f4e439335800d96fa461b96fc02d0d00b013d74bc
- candidate record SHA-256: d8e0b2957bea192f8946f1aa22fbfd2d9ce86b66ce3f5f56128c7aec39536444
- semantic review receipt SHA-256: 8f134403487694af4c70d2e00ec84bab542043ef4b3c54517c04f3c409c0262a
- question-first witness SHA-256: ad772845088ed8e99f87f90d4264b05ea97be35775305a02011af5ee35f0f4ce
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0076.json)