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R0004-02975 · q-20261005-advance-20261006-r05-circle-0025
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(12, -19) and radius r=23≥0. For P=(35, -19), is P a member of the circle boundary, not merely of the filled disk?
G001 — Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(12, -19) and radius r=23≥0. For P=(35, -19), is P a member of the circle boundary, not merely of the filled disk?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=23≥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(23)
After
radius squared = 529
Conditions: none
The square of the nonnegative radius is the distance-comparison threshold.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
coordinate difference0 := sub(35, 12)
After
coordinate difference0 = 23
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(23)
After
coordinate square0 = 529
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(-19, -19)
After
coordinate difference1 = 0
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(0)
After
coordinate square1 = 0
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(529, 0)
After
squared distance = 529
Conditions: none
In an orthonormal equal-unit frame,the Pythagorean relation adds the squared differences.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
d²=529;r²=529
After
requested classification=Yes
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.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: 330178faaf1253448f2bbd2b614dbc7e962d1968c46b7c1fa9053f180ff8031ccandidate record SHA-256: 551be1ec258efb239adb3393ecea2d56c6012e7b90df3437a7c63e4799b25cffsemantic review receipt SHA-256: 312ae86315346b8955842a547cb2d09246e1f826755f72c2a51f042e5c7d5c56question-first witness SHA-256: 831f449ed41ff9144a290e534aa32adf7252d15b81c16ed5886a7dfd65dee70cLean: NOT_RUN · human review: NOT_PERFORMED