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R0013-07036 · q-20261011-advance-20261015-r02-ratio-proportion-0011
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=9, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).
G001 — Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=9, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).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
forward ratio a/b := div(9, 3)
After
forward ratio a/b = 3
Conditions: none
The forward denominator b is nonzero,so compute the forward ratio.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
reverse ratio b/a := div(3, 9)
After
reverse ratio b/a = 1/3
Conditions: none
The reverse denominator is a and must separately be nonzero.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
forward times reverse := mul(3, 1/3)
After
forward times reverse = 1
Conditions: none
Multiply to audit reciprocity only when both directions are defined.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
a=9;b=3;forward=3/1
After
reverse defined=Yes;reverse=1/3
Conditions: none
When a=0,the valid forward ratio may be0,but this does not permit b/0 in reverse.Nor may the undefined product be assigned0.
Rule: reverse_ratio_independent_denominator · 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 forward times reverse with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute reverse ratio b/a with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute forward ratio a/b 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.(3/1, Yes, 1/3, 1/1)
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: edcb6d47aad60d91949ff2647f5bf8b124d2c1cc12bf1e7cd4a64185eb766040candidate record SHA-256: 17a8f796ecff03cc19542dad0cd4119fef9f7d6cc9a9581dad0cc0f0a84f6949semantic review receipt SHA-256: e89fc6d4a50a4a7fc942c6c8939a92f835e30105a9a5850e37dd198f31c1ed59question-first witness SHA-256: 0e1039f4cf022bf35af4e5531651c9a2b7ca7a5fdacac34fb1e4bcd9a81f6299Lean: NOT_RUN · human review: NOT_PERFORMED