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R0003-02016 · q-20261013-advance-20261021-r05-bisection-method-0016
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=15/1, right=19/1, left sign=1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).
G001 — Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=15/1, right=19/1, left sign=1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).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
left=15<right=19
After
all endpoints and midpoints are exact rationals
Conditions: none
Check strict endpoint order first;these are supplied data,not observations from an evaluated unknown function.
Rule: ordered_bracket_data · Uses: G001
Before
left sign=1; right sign=1
After
declared sign-bracket condition=No
Conditions: none
The criterion is a zero endpoint sign or opposite nonzero signs,not interval length.
Rule: recorded_sign_qualification · Uses: G001, S001
Before
a sign-bracket record is not a verified continuous function
After
root existence needs the additional continuity/evidence premises
Conditions: none
An opposite-sign record alone does not establish root existence;a discontinuous jump may change sign without attaining0.
Rule: sign_data_not_IVT_proof · Uses: G001, S002
Before
failure of this sign criterion
After
does not prove absence of an interior root
Conditions: none
Same-sign endpoints may still have interior roots,such as x²−1 on[−2,2];this task classifies only the declared signs.
Rule: qualification_not_nonexistence · 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.(15, 19, 1, 1, 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: 473e899381b6f9be1af53cca39d51bc7e00f88449acaa554cb7c5d8f9c3152bbcandidate record SHA-256: 7d3e39edea51e68b3de3769a550b071b5e270ae4e046e383225f3bbc5bed8b9csemantic review receipt SHA-256: 5dbc70ebd5f74b23384c93b63a2b1d6519e206c79f22ae16a6c2d7483cea783fquestion-first witness SHA-256: cf121c8dd2fec13aa118b04b049ee39494f1d6d9ecdc13514cafb78c0fc021faLean: NOT_RUN · human review: NOT_PERFORMED