# Question 6894: Ordered Rational Offsets and Absolute Distances [Return to catalogue page 35](https://ocme.evemisslab.com/en/questions/page/35/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261010-advance-20261014-r02-real-number-foundations/#q-20261010-advance-20261014-r02-real-number-0094) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0012-06894 · q-20261010-advance-20261014-r02-real-number-0094 ## Original question and givens Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=6/2, ordered points=[5/1, 5/2, 3/1, 11/3]. Give (each point−origin, each offset’s absolute value), retaining the four-field order. - G001 — Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=6/2, ordered points=[5/1, 5/2, 3/1, 11/3]. Give (each point−origin, each offset’s absolute value), retaining the four-field order. ## Definition and scope 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. - [mko-real-number](https://ocme.evemisslab.com/en/mko/mko-real-number/) ## Step-by-step derivation S001 · impliesBefore 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. Step identifiers and dependenciesRule: rational_real_slice_scope · Uses: G001 S002 · calculationBefore point0 signed offset := sub(5, 3) After point0 signed offset = 2 Conditions: none Retain the directed difference point−origin before taking an absolute value. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · impliesBefore point=5;origin=3;offset=2 After absolute distance=2/1 Conditions: none Keep a nonnegative offset and negate a negative one;distance is nonnegative while the first list retains direction. Step identifiers and dependenciesRule: signed_point_offset_absolute_distance · Uses: G001, S002 S004 · calculationBefore point1 signed offset := sub(5/2, 3) After point1 signed offset = -1/2 Conditions: none Retain the directed difference point−origin before taking an absolute value. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore point1 opposite negative offset := mul(-1, -1/2) After point1 opposite negative offset = 1/2 Conditions: none Multiply a negative offset by−1 to obtain nonnegative distance. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore point=5/2;origin=3;offset=-1/2 After absolute distance=1/2 Conditions: none Keep a nonnegative offset and negate a negative one;distance is nonnegative while the first list retains direction. Step identifiers and dependenciesRule: signed_point_offset_absolute_distance · Uses: G001, S005 S007 · calculationBefore point2 signed offset := sub(3, 3) After point2 signed offset = 0 Conditions: none Retain the directed difference point−origin before taking an absolute value. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore point=3;origin=3;offset=0 After absolute distance=0/1 Conditions: none Keep a nonnegative offset and negate a negative one;distance is nonnegative while the first list retains direction. Step identifiers and dependenciesRule: signed_point_offset_absolute_distance · Uses: G001, S007 S009 · calculationBefore point3 signed offset := sub(11/3, 3) After point3 signed offset = 2/3 Conditions: none Retain the directed difference point−origin before taking an absolute value. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · impliesBefore point=11/3;origin=3;offset=2/3 After absolute distance=2/3 Conditions: none Keep a nonnegative offset and negate a negative one;distance is nonnegative while the first list retains direction. Step identifiers and dependenciesRule: signed_point_offset_absolute_distance · Uses: G001, S009 ## Reverse checks - S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S009 · premise_and_result_audit — Recompute point3 signed offset with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Recompute point2 signed offset with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Recompute point1 opposite negative offset with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute point1 signed offset with exact rational arithmetic, not rounded decimals. - 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 — Recompute point0 signed offset with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result ([2/1, -1/2, 0/1, 2/3], [2/1, 1/2, 0/1, 2/3]) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 457a19dbba86896f0959d974ee4bffd0cda8912615be89fad3d07e4a479d5e57 - candidate record SHA-256: dc8f53cca1aa30a02dc6e2b4b4e840439160c01aaf9f76ad11908723fd3b8008 - semantic review receipt SHA-256: 90dbc494b2d34ef51f412dcff63cfcc1a2215d038bff71f38f5dded4c64faf00 - question-first witness SHA-256: b72a3b8bcf10f584396e0dd5198c657b0934a71864e5a3fa6321d7a613aef9f9 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261010-advance-20261014-r02-real-number-0094.json)