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R0013-07011 · q-20261011-advance-20261015-r02-ratio-proportion-0036
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=-2/2, complete ordered pairs (x,y)=[[0, 0], [1, 10], [2, -9]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.
G001 — Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=-2/2, complete ordered pairs (x,y)=[[0, 0], [1, 10], [2, -9]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.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
exact rational numerical ratios;each denominator needs its own guard
After
cross-multiplied equality does not legalize a zero ratio denominator
Conditions: none
A numerical ratio is not an automatic causal or geometric-similarity proof.Cross residual0 does not legalize a zero denominator,and the reverse ratio needs its own denominator guard.
Rule: rational_ratio_definition_scope · Uses: G001
Before
point0 kx := mul(-1, 0)
After
point0 kx = 0
Conditions: none
First evaluate k·x at this point,without assuming y/x is defined.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
point0 law residual := sub(0, 0)
After
point0 law residual = 0
Conditions: none
The point's proportional-law residual is y−kx.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
point=(0,0);k=-1;residual=0
After
law matches=Yes;ratio=undefined
Conditions: none
At x=0 one can still test whether y=0,but y/x is undefined.Agreement at these three points does not establish the law on unlisted inputs.
Rule: finite_point_law_and_ratio_guard · Uses: G001, S003
Before
point1 kx := mul(-1, 1)
After
point1 kx = -1
Conditions: none
First evaluate k·x at this point,without assuming y/x is defined.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
point1 law residual := sub(10, -1)
After
point1 law residual = 11
Conditions: none
The point's proportional-law residual is y−kx.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
point1 y/x := div(10, 1)
After
point1 y/x = 10
Conditions: none
Evaluate y/x only when this point's x is nonzero.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
point=(1,10);k=-1;residual=11
After
law matches=No;ratio=10/1
Conditions: none
At x=0 one can still test whether y=0,but y/x is undefined.Agreement at these three points does not establish the law on unlisted inputs.
Rule: finite_point_law_and_ratio_guard · Uses: G001, S007
Before
point2 kx := mul(-1, 2)
After
point2 kx = -2
Conditions: none
First evaluate k·x at this point,without assuming y/x is defined.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
point2 law residual := sub(-9, -2)
After
point2 law residual = -7
Conditions: none
The point's proportional-law residual is y−kx.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
point2 y/x := div(-9, 2)
After
point2 y/x = -9/2
Conditions: none
Evaluate y/x only when this point's x is nonzero.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
point=(2,-9);k=-1;residual=-7
After
law matches=No;ratio=-9/2
Conditions: none
At x=0 one can still test whether y=0,but y/x is undefined.Agreement at these three points does not establish the law on unlisted inputs.
Rule: finite_point_law_and_ratio_guard · Uses: G001, S011
S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S011 · premise_and_result_audit — Recompute point2 y/x with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Recompute point2 law residual with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Recompute point2 kx with exact rational arithmetic, not rounded decimals.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 point1 y/x with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute point1 law residual with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute point1 kx with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S003 · premise_and_result_audit — Recompute point0 law residual with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute point0 kx 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.([0/1, 11/1, -7/1], [Yes, No, No], [undefined, 10/1, -9/2])
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: 8e43b97590e83833769bfa8da95d6f1e38b1edeb58c7a05afb60562ad1e53165candidate record SHA-256: acee07ab67abb0e2560c5442b5dc17d0c102ccddcfd3604b3b327b13e328635bsemantic review receipt SHA-256: e89fc6d4a50a4a7fc942c6c8939a92f835e30105a9a5850e37dd198f31c1ed59question-first witness SHA-256: 0e1039f4cf022bf35af4e5531651c9a2b7ca7a5fdacac34fb1e4bcd9a81f6299Lean: NOT_RUN · human review: NOT_PERFORMED