# Question 6924: Ordered Fraction Differences and Endpoint Check [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-20261011-advance-20261015-r01-fraction-foundations/#q-20261011-advance-20261015-r01-fraction-0049) 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. R0012-06924 · q-20261011-advance-20261015-r01-fraction-0049 ## Original question and givens Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[-11/-3, 0/1, 1/2, 24/2]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort. - G001 — Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[-11/-3, 0/1, 1/2, 24/2]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort. ## 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-fraction](https://ocme.evemisslab.com/en/mko/mko-fraction/) ## Step-by-step derivation S001 · impliesBefore original integer numerator/denominator representations;original denominators nonzero After representation and rational value are distinct data Conditions: none Negative denominators are valid,but reduced values do not replace original representations.Candidate algorithms about raw numerators/denominators must use those raw data.This finite audit is not a general proof of quotient well-definedness. Step identifiers and dependenciesRule: fraction_representation_value_scope · Uses: G001 S002 · calculationBefore ordered adjacent difference0 := sub(0, 11/3) After ordered adjacent difference0 = -11/3 Conditions: none Subtract the preceding value from the next in original order,without sorting or deleting repeats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore ordered adjacent difference1 := sub(1/2, 0) After ordered adjacent difference1 = 1/2 Conditions: none Subtract the preceding value from the next in original order,without sorting or deleting repeats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore ordered adjacent difference2 := sub(12, 1/2) After ordered adjacent difference2 = 23/2 Conditions: none Subtract the preceding value from the next in original order,without sorting or deleting repeats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore adjacent difference chain sum := sum(-11/3, 1/2, 23/2) After adjacent difference chain sum = 25/3 Conditions: none Add all three signed adjacent differences. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore direct final minus initial := sub(12, 11/3) After direct final minus initial = 25/3 Conditions: none Separately subtract initial from final value to audit the endpoint difference. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore ordered values=[11/3, 0/1, 1/2, 12/1];differences=[-11/3, 1/2, 23/2] After chain sum=25/3;direct endpoint difference=25/3 Conditions: none The intermediate terms cancel in(v1−v0)+(v2−v1)+(v3−v2),retaining direction.This is not a difference chain of a sorted value set. Step identifiers and dependenciesRule: ordered_fraction_telescope · Uses: G001, S006 ## Reverse checks - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Recompute direct final minus initial with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute adjacent difference chain sum with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute ordered adjacent difference2 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute ordered adjacent difference1 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute ordered adjacent difference0 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 ([-11/3, 1/2, 23/2], 25/3, 25/3, Yes) 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: cf901153054ae8b94f8174bc7049ee0eefa47d7ae3bb7329f075a67b1d359889 - candidate record SHA-256: d30b394714066ddc79c5d6c22f914509243f0d3dec89fb69254a3ea08e51ed44 - 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-20261011-advance-20261015-r01-fraction-0049.json)