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R0012-06851 · q-20261010-advance-20261014-r02-real-number-0026
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[0/1, 0/-2, 0/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.
G001 — Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[0/1, 0/-2, 0/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.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
explicit rational points embedded in the real line
After
exact finite order and distance data,not real completeness or irrational classification
Conditions: none
Use exact rational values for this finite slice of the real line,without rounded decimals or claims about real completeness or classification of irrationals.
Rule: rational_real_slice_scope · Uses: G001
Before
[[0, 1], [0, -2], [0, 3]]
After
[0/1]
Conditions: none
Sort and deduplicate by rational value,not fraction strings;0/−2 and0/3 represent the same point0.
Rule: rational_point_numeric_normalization · Uses: G001, S001
Before
smallest containing interval width := sub(0, 0)
After
smallest containing interval width = 0
Conditions: none
Rightmost minus leftmost point gives the nonnegative interval width.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
all points between 0 and 0
After
smallest containing closed interval=[0,0]
Conditions: none
Every containing closed interval must include the smallest and largest points,hence this interval is minimal.This is not a claim that topological closure fills the gaps of a finite point set.
Rule: containing_interval_not_topological_closure · 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 — Recompute smallest containing interval width with exact rational arithmetic, not rounded decimals.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.([0/1], 0/1, 0/1, 0/1)
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: 04bda6dc04bdc363469489cc075f4bd629e70e01dd62f73a094903773b122d83candidate record SHA-256: b86eb6dac5d2cd3cd54b6d6c084bfce7cc917c8b817190cacd4578c5550996e8semantic review receipt SHA-256: 90dbc494b2d34ef51f412dcff63cfcc1a2215d038bff71f38f5dded4c64faf00question-first witness SHA-256: b72a3b8bcf10f584396e0dd5198c657b0934a71864e5a3fa6321d7a613aef9f9Lean: NOT_RUN · human review: NOT_PERFORMED