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R0013-07053 · q-20261011-advance-20261015-r02-ratio-proportion-0078
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=2, b=3, d=0, x=1/3. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).
G001 — Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=2, b=3, d=0, x=1/3. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).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
cross product ad := mul(2, 0)
After
cross product ad = 0
Conditions: none
Compute the cross term ad;this polynomial term exists even if the original ratio is illegal.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
cross product b times candidate := mul(3, 1/3)
After
cross product b times candidate = 1
Conditions: none
Use the original candidate x in the second term,not a substituted solved x*.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
ad minus b times candidate := sub(0, 1)
After
ad minus b times candidate = -1
Conditions: none
Keep the signed cross residual separate from eligibility of both original denominators.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
b=3;d=0;candidate=1/3;cross residual=-1
After
original ratios legal=No;x*=undefined;candidate valid=No
Conditions: none
If b or d is0,the original proportion is illegal even when the cross residual is0.Solving for x* and auditing the supplied candidate are distinct tasks.
Rule: missing_ratio_component_original_guard · Uses: G001, S004
S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S004 · premise_and_result_audit — Recompute ad minus b times candidate with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute cross product b times candidate with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute cross product ad 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.(No, undefined, -1/1, No)
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: a687ff4466968322ec5b97ebc2b2b063ee161a1063c35654d3562a0c236da748candidate record SHA-256: 03dbe906d5b5ae4d0d56b750b8eea014584b1e118688d498d839b04d3f1845c5semantic review receipt SHA-256: e89fc6d4a50a4a7fc942c6c8939a92f835e30105a9a5850e37dd198f31c1ed59question-first witness SHA-256: 0e1039f4cf022bf35af4e5531651c9a2b7ca7a5fdacac34fb1e4bcd9a81f6299Lean: NOT_RUN · human review: NOT_PERFORMED