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R0058-29921 · q-20260930-r07-coordinate-point-equality-0021
In the same explicit Cartesian coordinate system, are P=(2,-2) and Q=(2,-2) the same point? Compare corresponding coordinates.
G001 — In the same explicit Cartesian coordinate system, are P=(2,-2) and Q=(2,-2) the same point? Compare corresponding coordinates.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
source roles = [qf-coordinate-point-equality, point_equality]
After
derived = [[ordered_point_equality_yes_no]]
Conditions: none
Follow THIS bank exact keys,domain/frame,operation and answer role.STR22/App20 tuples differ from legacy scalars;retain or omit/1 by the ORIGINAL contract,not a generic visual convention.
Rule: R58_EXACT_source_interface_and_answer_ROLE · Uses: G001
Before
Coordinate ORIGINAL first component difference x minus u := sub(2, 2)
After
Coordinate ORIGINAL first component difference x minus u = 0
Conditions: none
Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
Coordinate ORIGINAL second component difference y minus v := sub(-2, -2)
After
Coordinate ORIGINAL second component difference y minus v = 0
Conditions: none
Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
source roles = [2, -2, 2, -2]
After
derived = [0, 0, [Yes, Yes], Yes, Yes]
Conditions: none
In the SAME coordinate system,(x,y)=(u,v) IFF x=u AND y=v,equivalently BOTH differences vanish.One matching component or swapped order is insufficient.Derived differences may exceed SAFE while raw inputs remain SAFE.
Rule: Coordinate_ORDERED_pair_equality_BOTH_components_SAME_system · Uses: G001, S003
S004 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.S003 · premise_and_result_audit — Recompute Coordinate ORIGINAL second component difference y minus v with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute Coordinate ORIGINAL first component difference x minus u with exact rational arithmetic, not rounded decimals.S001 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.Yes
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: 6bee90329de9fcf63a675a729c803d447a7d6356a4bf424142cbc57a2522d061candidate record SHA-256: 6b69a04a919a2ce525bdd30bb5ae278f7c5800dad1005d14eafe07d5789f29b5semantic review receipt SHA-256: 66bbb31c5fe644576dfd47ac71ca9e367fc23b948eac8e3eb648d96dadea05dfquestion-first witness SHA-256: b1e4fbdbca8d0ac3581e2f26466374d870f3c8498fab7cf6873feca5bbec44e7Lean: NOT_RUN · human review: NOT_PERFORMED