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R0017-09130 · q-20261013-advance-20261018-r04-prime-number-0055
Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=28, candidate d=3. Give (n, d, n≥2 and 2≤d<n and d divides n?, quotient or not-applicable, composite-witness-valid/not-a-nontrivial-factor-witness).
G001 — Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=28, candidate d=3. Give (n, d, n≥2 and 2≤d<n and d divides n?, quotient or not-applicable, composite-witness-valid/not-a-nontrivial-factor-witness).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.
Before
natural-number prime protocol;0 and1 are not prime
After
a failed candidate witness alone is not a primality proof
Conditions: none
Naturals include0.Primality retains both n≥2 and a complete factor bound.A failed single factor candidate only invalidates that witness;it does not certify primality.
Rule: prime_complete_evidence_scope · Uses: G001
Before
n=28;candidate d=3
After
nontrivial typed range=Yes
Conditions: none
Require n≥2 and2≤d<n first.Zero,one or n cannot be this nontrivial witness;do not divide by an ineligible d.
Rule: nontrivial_factor_guard_before_division · Uses: G001, S001
Before
candidate factor division:natural dividend=28;positive divisor=3
After
28=9·3+1
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S002
Before
range eligible=Yes;valid factor=No
After
quotient/status=not applicable;classification=not a nontrivial factor witness
Conditions: none
An eligible divisor gives n=d*q with both factors>1 and witnesses compositeness.Otherwise report only that the candidate fails,not a primality certificate.
Rule: factor_witness_not_whole_prime_classification · Uses: G001, S003
S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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.(28, 3, No, not-applicable, not-a-nontrivial-factor-witness)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: f1435c085fc9450d16bdcad94afd948df76ce75f4ac32f465469ed3ebf99a793candidate record SHA-256: 87791bd1a4ab6754b31aec9cd2a6e16e9fdc455eef7b6dc56e0afcba4444ffbfsemantic review receipt SHA-256: abad28564830bb9b6ef069ca5a1aa128ce7b1a67a5cb8fe6d868a06db73bcfb8question-first witness SHA-256: 43126988479968837abe086263dac1b24c37463ad80e00d2e280a7b8e9036a04Lean: NOT_RUN · human review: NOT_PERFORMED