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R0017-09101 · q-20261013-advance-20261018-r04-prime-number-0001
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. p=0. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).
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. p=0. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).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
p=0;d is an arbitrary positive natural
After
positive divisors=all positive natural numbers;p>=2=No;prime=No
Conditions: none
Every positive d divides0 via0=d*0,but0 fails the prime lower bound.An infinite positive-divisor collection does not make0 prime.
Rule: zero_positive_divisors_not_prime · Uses: G001, S001
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.(0, all-positive-naturals, No, No)
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: 32f566ad680b9b5246ca65b8e7d1a09592443b532a6bfde563d46397b88321b3candidate record SHA-256: 055452ba687e5f1acd2426de22c9b152d48cef9cba1f27789cebe0e3fc0de07esemantic review receipt SHA-256: abad28564830bb9b6ef069ca5a1aa128ce7b1a67a5cb8fe6d868a06db73bcfb8question-first witness SHA-256: 43126988479968837abe086263dac1b24c37463ad80e00d2e280a7b8e9036a04Lean: NOT_RUN · human review: NOT_PERFORMED