# Question 30418: Three-Part Prefix Budget and Eligibility [Return to catalogue page 153](https://ocme.evemisslab.com/en/questions/page/153/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261010-advance-20261014-r01-angle-foundations/#q-20261010-advance-20261014-r01-angle-0093) 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. R0059-30418 · q-20261010-advance-20261014-r01-angle-0093 ## Original question and givens Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. Ordered parts=[17, 90, 18], target=90. Give (three prefix sums, is the total≤180?, target−total). - G001 — Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. Ordered parts=[17, 90, 18], target=90. Give (three prefix sums, is the total≤180?, target−total). ## 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-angle](https://ocme.evemisslab.com/en/mko/mko-angle/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-density14-angle-three-part-budget, density14_angle_three_part_budget, finite_nondirected_degree_protocol] After derived = [[ALL_prefix_sums, total_le180, SIGNED_target_minus_total]] Conditions: none Retain the ORIGINAL bank full operation prefix,domain and ordered answer roles.Use orthogonal_equal_unit only where a frame is declared,and degrees where an angle unit is declared.Integer fractions retain/1.Do not invent parameters or alter original bilingual sources. Step identifiers and dependenciesRule: R59_EXACT_family_domain_FULLprefix_operation_and_answer_roles · Uses: G001 S002 · calculationBefore Angle ORIGINAL ordered prefix0 := sum(0, 17) After Angle ORIGINAL ordered prefix0 = 17 Conditions: none Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore Angle ORIGINAL ordered prefix1 := sum(17, 90) After Angle ORIGINAL ordered prefix1 = 107 Conditions: none Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Angle ORIGINAL ordered prefix2 := sum(107, 18) After Angle ORIGINAL ordered prefix2 = 125 Conditions: none Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Angle ORIGINAL target minus total := sub(90, 125) After Angle ORIGINAL target minus total = -35 Conditions: none Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore source roles = [[17, 90, 18], 90] After derived = [[17, 107, 125], Yes, -35] Conditions: none The flag tests total<=180,NOT total<=target.With target90,total100 may have flagYes and remainder-10.Each raw part is within0..180;an accumulated540 is legal arithmetic but not an angle in this entry.Keep original prefix order without sorting/wrapping. Step identifiers and dependenciesRule: Angle_THREE_original_prefixes_budget180_NOT_target_threshold · Uses: G001, S005 ## Reverse checks - S006 · premise_and_result_audit — Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate. - S005 · premise_and_result_audit — Recompute Angle ORIGINAL target minus total with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Angle ORIGINAL ordered prefix2 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Angle ORIGINAL ordered prefix1 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute Angle ORIGINAL ordered prefix0 with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate. ## Result ([17, 107, 125], Yes, -35) 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: 46ceddde7a7db45388178edfc8b9a6ba32c4e3681978ef9dd832cc843748c800 - candidate record SHA-256: b17b8d838ff88527c18a87cf6aee8e3c33592f42c3e876f28d0dc2ad3b77cf60 - semantic review receipt SHA-256: 4c93dea3956f58509e20b3b8bbbf716dbc340b424a2a5177bc04aba9e8ea9ec5 - question-first witness SHA-256: a5e56e0d64d622d7202840cef7019839c5a99244c602694190779155207dafdf - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261010-advance-20261014-r01-angle-0093.json)