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R0058-29768 · q-20261006-advance-20261007-r07-geometric-transformations-0018
Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(5, -16) and translation vector t=[1, 2]. Find the coordinates of P+t.
G001 — Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(5, -16) and translation vector t=[1, 2]. Find the coordinates of P+t.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-transform-translation-point, translation_point]
After
derived = [[image_POINT_string]]
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
Transform ORIGINAL translation coordinate0 := sum(5, 1)
After
Transform ORIGINAL translation coordinate0 = 6
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
Transform ORIGINAL translation coordinate1 := sum(-16, 2)
After
Transform ORIGINAL translation coordinate1 = -14
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 = [[5, -16], [1, 2]]
After
derived = [[6, -14]]
Conditions: none
In the SAME origin/basis,add each original translation component to obtain the ACTUAL image.Returning only v is not P+t.Derived sums may exceed SAFE while raw inputs remain SAFE.
Rule: Transform_translation_ACTUAL_image_not_only_displacement · 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 Transform ORIGINAL translation coordinate1 with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute Transform ORIGINAL translation coordinate0 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.(6, -14)
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: dba0255a16525739d99237c55118b2d210c26e6b7c8bbc5c61ff8f1fc3f2d2e9candidate record SHA-256: 129758d8ab7c734856c39103bf1f0d28104eacdd4f0aa234809da0875f3daac6semantic review receipt SHA-256: 66bbb31c5fe644576dfd47ac71ca9e367fc23b948eac8e3eb648d96dadea05dfquestion-first witness SHA-256: b1e4fbdbca8d0ac3581e2f26466374d870f3c8498fab7cf6873feca5bbec44e7Lean: NOT_RUN · human review: NOT_PERFORMED