# Question 34112: Exact Order of Positive-Denominator Rationals

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0067-34112 · q-20261005-advance-20261006-r01-order-relation-0012



## Original question and givens

Under the usual real order, compare (-12)/3 and (1)/4. Both denominators are positive. Give <, =, or > in that order.


- G001 — Under the usual real order, compare (-12)/3 and (1)/4. Both denominators are positive. Give <, =, or > in that order.





## Definition and scope

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.


- [mko-order-relation](https://ocme.evemisslab.com/en/mko/mko-order-relation/)





## Step-by-step derivation

S001 · impliesBefore

source roles = [qf-order-rational-comparison, [operation, p, q, r, s]]

After

derived = [rational_comparison, no-invented-domain]

Conditions: none

Check THIS bank’s keys/operation/domain:legacy table underscores,MOD50 local hyphens;permissive BASE data acquires no unknown-field restriction.Keep Number.isInteger and SAFE source guards distinct,while preserving ORIGINAL JSON number representations.

Step identifiers and dependenciesRule: R67_SOURCE_specific_keys_operation_domain_and_numeric_bank · Uses: G001





S002 · calculationBefore

Order ORIGINAL first numerator times second positive denominator := mul(-12, 4)

After

Order ORIGINAL first numerator times second positive denominator = -48

Conditions: none

Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001





S003 · calculationBefore

Order ORIGINAL second numerator times first positive denominator := mul(1, 3)

After

Order ORIGINAL second numerator times first positive denominator = 3

Conditions: none

Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · impliesBefore

source roles = [-12, 3, 1, 4]

After

derived = [-48, 3, <]

Conditions: none

Both denominators are POSITIVE,so multiplying by q*s preserves direction.Compare p*s with r*q.Equality includes unreduced equivalent fractions.Derived cross products are not recapped at RAW SAFE.

Step identifiers and dependenciesRule: Order_POSITIVE_denominators_cross_products_preserve_direction · Uses: G001, S003









## Reverse checks


- S004 · premise_and_result_audit — Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof.
- S003 · premise_and_result_audit — Recompute Order ORIGINAL second numerator times first positive denominator with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Recompute Order ORIGINAL first numerator times second positive denominator with exact rational arithmetic, not rounded decimals.
- S001 · premise_and_result_audit — Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof.





## Result

<

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.





## This question review and source binding

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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: a9cb009f05067dca59c1111ca2d3891e0f72d7537410fd36a4f43d9d69e5ba24
- candidate record SHA-256: b1b28c20ac2ef5f378cb9edd9bdf85aec367d720b38bee7172088ed59aa287fa
- semantic review receipt SHA-256: c89efa383b67bb2cf25bcf16160b9f5f614bbe5e5e791f4d7f58234d17a3371c
- question-first witness SHA-256: 12766d451c5375842b4fc2843ae94807ec55f15763a916558026912a13873d57
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261005-advance-20261006-r01-order-relation-0012.json)