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R0004-02987 · q-20261005-advance-20261006-r05-circle-0037
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-1, -6) and radius r=10≥0. Classify P=(10, -5) as inside, on boundary, or outside.
G001 — Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-1, -6) and radius r=10≥0. Classify P=(10, -5) as inside, on boundary, or outside.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
orthonormal equal-length units;r=10≥0
After
C={P:distance(P,O)=r} is a boundary,not a filled diskConditions: none
The circle is the boundary at distance exactly r from the center,not the filled disk.For r=0 use the degenerate singleton{O},with no inside points and diameter length0.
Rule: circle_boundary_convention · Uses: G001
Before
radius squared := square(10)
After
radius squared = 100
Conditions: none
The square of the nonnegative radius is the distance-comparison threshold.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
coordinate difference0 := sub(10, -1)
After
coordinate difference0 = 11
Conditions: none
Subtract the center coordinate from the point coordinate;do not square absolute coordinates instead.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
coordinate square0 := square(11)
After
coordinate square0 = 121
Conditions: none
Square this coordinate difference;a negative difference still contributes a nonnegative square.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
coordinate difference1 := sub(-5, -6)
After
coordinate difference1 = 1
Conditions: none
Subtract the center coordinate from the point coordinate;do not square absolute coordinates instead.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
coordinate square1 := square(1)
After
coordinate square1 = 1
Conditions: none
Square this coordinate difference;a negative difference still contributes a nonnegative square.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
squared distance := sum(121, 1)
After
squared distance = 122
Conditions: none
In an orthonormal equal-unit frame,the Pythagorean relation adds the squared differences.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
d²=122;r²=100
After
requested classification=outside
Conditions: none
Distance and radius are nonnegative,so squaring preserves their order.Equality means boundary,strictly smaller means inside,and larger means outside.No rounded square root is needed.
Rule: nonnegative_squared_distance_comparison · Uses: G001, S007
S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S007 · premise_and_result_audit — Recompute squared distance with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute coordinate square1 with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute coordinate difference1 with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Recompute coordinate square0 with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute coordinate difference0 with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute radius squared 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.outside
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: 3e98bbadc3fb98f930c8dc4931bb7e9b6be4cabb27d792c0ec68e03622294bf1candidate record SHA-256: a06b4975d3755d3b435f2244e312a61fcc5e1cd10fe36ee65de2dd068dd801cdsemantic review receipt SHA-256: 312ae86315346b8955842a547cb2d09246e1f826755f72c2a51f042e5c7d5c56question-first witness SHA-256: 831f449ed41ff9144a290e534aa32adf7252d15b81c16ed5886a7dfd65dee70cLean: NOT_RUN · human review: NOT_PERFORMED