# Question 30734: Three Ordered Coordinate Comparisons and First Mismatch [Return to catalogue page 154](https://ocme.evemisslab.com/en/questions/page/154/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261011-advance-20261015-r07-coordinate-plane-foundations/#q-20261011-advance-20261015-r07-coordinate-plane-0084) 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. R0060-30734 · q-20261011-advance-20261015-r07-coordinate-plane-0084 ## Original question and givens Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Three ordered [P,Q] pairs=[[[-4, 1], [0, 0]], [[0, 8], [0, 0]], [[1, -1], [0, 0]]]. Give (each P−Q component residual pair, pair-equality mask, first unequal zero-based index or none). - G001 — Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Three ordered [P,Q] pairs=[[[-4, 1], [0, 0]], [[0, 8], [0, 0]], [[1, -1], [0, 0]]]. Give (each P−Q component residual pair, pair-equality mask, first unequal zero-based index or none). ## 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-coordinate-plane](https://ocme.evemisslab.com/en/mko/mko-coordinate-plane/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-density15-coordinate-first-mismatch-index, density15_coordinate_first_mismatch_index, finite_ordered_coordinate_protocol] After derived = [[ALL_P_minus_Q_pairs, ALL_equal_flags, FIRST_unequal_PAIR_zero_index_or_none]] Conditions: none density14/15 use FULL-prefix underscores;density17 uses LOCAL hyphens.The coordinate and circle frames differ.Retain THIS family ordered answer roles/types and all original bilingual JSON/status. Step identifiers and dependenciesRule: R60_EXACT_perbank_operation_domain_FRAME_and_output_roles · Uses: G001 S002 · calculationBefore Coordinate original pair0 ORIGINAL P minus Q coordinate0 := sub(-4, 0) After Coordinate original pair0 ORIGINAL P minus Q coordinate0 = -4 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore Coordinate original pair0 ORIGINAL P minus Q coordinate1 := sub(1, 0) After Coordinate original pair0 ORIGINAL P minus Q coordinate1 = 1 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · impliesBefore source roles = [Coordinate original pair0, [-4, 1], [0, 0]] After derived = [[-4, 1]] Conditions: none THIS coordinate protocol retains both signs of ORIGINAL P-Q,OPPOSITE the length bank Q-P.A component difference is not distance or rendered size;derived values may exceed SAFE. Step identifiers and dependenciesRule: Coordinate_SOURCE_direction_PminusQ_not_length_QminusP · Uses: G001, S003 S005 · calculationBefore Coordinate original pair1 ORIGINAL P minus Q coordinate0 := sub(0, 0) After Coordinate original pair1 ORIGINAL P minus Q coordinate0 = 0 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Coordinate original pair1 ORIGINAL P minus Q coordinate1 := sub(8, 0) After Coordinate original pair1 ORIGINAL P minus Q coordinate1 = 8 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore source roles = [Coordinate original pair1, [0, 8], [0, 0]] After derived = [[0, 8]] Conditions: none THIS coordinate protocol retains both signs of ORIGINAL P-Q,OPPOSITE the length bank Q-P.A component difference is not distance or rendered size;derived values may exceed SAFE. Step identifiers and dependenciesRule: Coordinate_SOURCE_direction_PminusQ_not_length_QminusP · Uses: G001, S006 S008 · calculationBefore Coordinate original pair2 ORIGINAL P minus Q coordinate0 := sub(1, 0) After Coordinate original pair2 ORIGINAL P minus Q coordinate0 = 1 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore Coordinate original pair2 ORIGINAL P minus Q coordinate1 := sub(-1, 0) After Coordinate original pair2 ORIGINAL P minus Q coordinate1 = -1 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · impliesBefore source roles = [Coordinate original pair2, [1, -1], [0, 0]] After derived = [[1, -1]] Conditions: none THIS coordinate protocol retains both signs of ORIGINAL P-Q,OPPOSITE the length bank Q-P.A component difference is not distance or rendered size;derived values may exceed SAFE. Step identifiers and dependenciesRule: Coordinate_SOURCE_direction_PminusQ_not_length_QminusP · Uses: G001, S009 S011 · impliesBefore source roles = [[[[-4, 1], [0, 0]], [[0, 8], [0, 0]], [[1, -1], [0, 0]]]] After derived = [[[-4, 1], [0, 8], [1, -1]], [No, No, No], 0] Conditions: none First compare BOTH components within EACH pair,then find the FIRST unequal PAIR in original order using a ZERO-based index.Index0 is not none.All equal returns none;this is not a coordinate-component1/2 index. Step identifiers and dependenciesRule: Coordinate_FIRST_unequal_PAIR_zero_based_not_coordinate_index · Uses: G001, S010 ## Reverse checks - S011 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate. - S010 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate. - S009 · premise_and_result_audit — Recompute Coordinate original pair2 ORIGINAL P minus Q coordinate1 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute Coordinate original pair2 ORIGINAL P minus Q coordinate0 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate. - S006 · premise_and_result_audit — Recompute Coordinate original pair1 ORIGINAL P minus Q coordinate1 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Coordinate original pair1 ORIGINAL P minus Q coordinate0 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate. - S003 · premise_and_result_audit — Recompute Coordinate original pair0 ORIGINAL P minus Q coordinate1 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute Coordinate original pair0 ORIGINAL P minus Q coordinate0 with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate. ## Result ([[-4, 1], [0, 8], [1, -1]], [No, No, No], 0) 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: cf58ab9d9dc9a9e464142e613d6b84265559c31e5197e4fc8639394795991899 - candidate record SHA-256: 2ddbf9a686443d8b99b3f24da0258468d29a9b6098ca55ee9a338e58705e6c96 - semantic review receipt SHA-256: d6b5a409bf933281a46617cd715d8016fd5ca3d6fb76f77d179ac2ecd9d608bf - question-first witness SHA-256: f80ecf598c6893a580af018e1a82849b33ee6887efb48a6a1d69569aa17f12e4 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261011-advance-20261015-r07-coordinate-plane-0084.json)