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R0068-34855 · q-20260923-real-rational-closed-interval-0005
The closed interval [-8, -3] and rational point -4/1 are explicit. Does -4/1 belong to this closed interval?
G001 — The closed interval [-8, -3] and rational point -4/1 are explicit. Does -4/1 belong to this closed interval?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
source roles = [qf-real-rational-closed-interval, [a, b, operation, point]]
After
derived = [closed_interval, no-invented-domain]
Conditions: none
Separate source banks:legacy RAW SAFE with unbounded derived BigInts versus BASE Number.isInteger without invented SAFE/million caps.Operations,domains and permissive/exact keys follow THIS bank.Keep ORIGINAL JSON intact.
Rule: R68_BANK_specific_keys_operations_and_numeric_guards · Uses: G001
Before
Interval ORIGINAL point ORIGINAL exact numerator over nonzero denominator := div(-4, 1)
After
Interval ORIGINAL point ORIGINAL exact numerator over nonzero denominator = -4
Conditions: none
Calculate exactly from ORIGINAL ordered operands;mul/sub/div have EXPLICITLY TWO operands.Legacy derived BigInts are not recapped at RAW SAFE;check BASE Number type/representation conditions separately.Do not use floating approximations to force an answer.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
source roles = [Interval ORIGINAL point, [-4, 1]]
After
derived = [[-4, 1], -4/1]
Conditions: none
BASE normalizeFraction requires INTEGER components and nonzero denominator,without invented SAFE/million/100million caps.Normalize denominator sign and gcd;zero is0/1.This derived representation does not alter the ORIGINAL raw pair.
Rule: Real_BASE_fraction_sign_gcd_normalization_NO_invented_SAFEcap · Uses: G001, S002
Before
Real interval ORIGINAL left endpoint times normalized positive denominator := mul(-8, 1)
After
Real interval ORIGINAL left endpoint times normalized positive denominator = -8
Conditions: none
Calculate exactly from ORIGINAL ordered operands;mul/sub/div have EXPLICITLY TWO operands.Legacy derived BigInts are not recapped at RAW SAFE;check BASE Number type/representation conditions separately.Do not use floating approximations to force an answer.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
Real interval ORIGINAL right endpoint times normalized positive denominator := mul(-3, 1)
After
Real interval ORIGINAL right endpoint times normalized positive denominator = -3
Conditions: none
Calculate exactly from ORIGINAL ordered operands;mul/sub/div have EXPLICITLY TWO operands.Legacy derived BigInts are not recapped at RAW SAFE;check BASE Number type/representation conditions separately.Do not use floating approximations to force an answer.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
source roles = [-8, -3, -4/1]
After
derived = [-8, -4, -3, Yes]
Conditions: none
FIRST normalize the denominator positive,THEN check a*q≤p≤b*q,INCLUDING BOTH endpoints.A singleton a=b is valid,and an outside point is a LEGAL No.
Rule: Real_INTERVAL_BOTH_inclusive_boundaries_normalized_denominator · Uses: G001, S005
S006 · premise_and_result_audit — Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof.S005 · premise_and_result_audit — Recompute Real interval ORIGINAL right endpoint times normalized positive denominator with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Recompute Real interval ORIGINAL left endpoint times normalized positive denominator with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof.S002 · premise_and_result_audit — Recompute Interval ORIGINAL point ORIGINAL exact numerator over nonzero denominator with exact rational arithmetic, not rounded decimals.S001 · premise_and_result_audit — Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof.Yes
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: 7339ad6d73f8ddd0b1eef2ef13ff13257ca66feba00cb44743393039e0f85ad7candidate record SHA-256: 52a3888e783381178557804bf1685aac59368cf7db1df2ac9296dd4ca2b1aeeasemantic review receipt SHA-256: e7a54104fa02c235f7591329bd6db8d626dd74bf0fae29c557b1f269c85066edquestion-first witness SHA-256: 8401a3c2064c1275a6a7ea8f00412fa12286cec7b2db6c305ff9dba3724ae229Lean: NOT_RUN · human review: NOT_PERFORMED