# Exact Rational Order and Closed-Interval Boundaries

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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261012-advance-20261017-r04-order-relation-foundations · 2026-10-12 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Stable Rational Sort with Original Index Witnesses

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[0/1, 0/2, 0/3]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.

View answer and explanation([0/1, 0/1, 0/1], [0, 1, 2], [0/1, 0/1, 0/1], [=, =])

Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0/1, 0/1, 0/1], [0, 1, 2], [0/1, 0/1, 0/1], [=, =]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0001/)

Usual order on the real numbers · qf-density17-order-three-rational-sort · candidate





## 2. Stable Rational Sort with Original Index Witnesses

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[-1/2, 1/3, -2/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.

View answer and explanation([-1/2, 1/3, -1/2], [0, 2, 1], [-1/2, -1/2, 1/3], [=, <])

Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-1/2, 1/3, -1/2], [0, 2, 1], [-1/2, -1/2, 1/3], [=, <]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0002/)

Usual order on the real numbers · qf-density17-order-three-rational-sort · candidate





## 3. Stable Rational Sort with Original Index Witnesses

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[1/2, -1/2, 0/1]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.

View answer and explanation([1/2, -1/2, 0/1], [1, 2, 0], [-1/2, 0/1, 1/2], [<, <])

Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1/2, -1/2, 0/1], [1, 2, 0], [-1/2, 0/1, 1/2], [<, <]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0003/)

Usual order on the real numbers · qf-density17-order-three-rational-sort · candidate





## 4. Stable Rational Sort with Original Index Witnesses

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[-9/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.

View answer and explanation([-3/1, 1/2, -1/4], [0, 2, 1], [-3/1, -1/4, 1/2], [<, <])

Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-3/1, 1/2, -1/4], [0, 2, 1], [-3/1, -1/4, 1/2], [<, <]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0004/)

Usual order on the real numbers · qf-density17-order-three-rational-sort · candidate





## 5. Stable Rational Sort with Original Index Witnesses

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[-8/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.

View answer and explanation([-8/3, 1/2, -1/4], [0, 2, 1], [-8/3, -1/4, 1/2], [<, <])

Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-8/3, 1/2, -1/4], [0, 2, 1], [-8/3, -1/4, 1/2], [<, <]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0005/)

Usual order on the real numbers · qf-density17-order-three-rational-sort · candidate





## 6. Stable Rational Sort with Original Index Witnesses

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[-7/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.

View answer and explanation([-7/3, 1/2, -1/4], [0, 2, 1], [-7/3, -1/4, 1/2], [<, <])

Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-7/3, 1/2, -1/4], [0, 2, 1], [-7/3, -1/4, 1/2], [<, <]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0006/)

Usual order on the real numbers · qf-density17-order-three-rational-sort · candidate





## 7. Stable Rational Sort with Original Index Witnesses

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[-6/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.

View answer and explanation([-2/1, 1/2, -1/4], [0, 2, 1], [-2/1, -1/4, 1/2], [<, <])

Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-2/1, 1/2, -1/4], [0, 2, 1], [-2/1, -1/4, 1/2], [<, <]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0007/)

Usual order on the real numbers · qf-density17-order-three-rational-sort · candidate





## 8. Stable Rational Sort with Original Index Witnesses

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[-5/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.

View answer and explanation([-5/3, 1/2, -1/4], [0, 2, 1], [-5/3, -1/4, 1/2], [<, <])

Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-5/3, 1/2, -1/4], [0, 2, 1], [-5/3, -1/4, 1/2], [<, <]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0008/)

Usual order on the real numbers · qf-density17-order-three-rational-sort · candidate





## 9. Stable Rational Sort with Original Index Witnesses

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[-4/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.

View answer and explanation([-4/3, 1/2, -1/4], [0, 2, 1], [-4/3, -1/4, 1/2], [<, <])

Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-4/3, 1/2, -1/4], [0, 2, 1], [-4/3, -1/4, 1/2], [<, <]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0009/)

Usual order on the real numbers · qf-density17-order-three-rational-sort · candidate





## 10. Stable Rational Sort with Original Index Witnesses

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[-3/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.

View answer and explanation([-1/1, 1/2, -1/4], [0, 2, 1], [-1/1, -1/4, 1/2], [<, <])

Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-1/1, 1/2, -1/4], [0, 2, 1], [-1/1, -1/4, 1/2], [<, <]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0010/)

Usual order on the real numbers · qf-density17-order-three-rational-sort · candidate





## 11. Stable Rational Sort with Original Index Witnesses

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[-2/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.

View answer and explanation([-2/3, 1/2, -1/4], [0, 2, 1], [-2/3, -1/4, 1/2], [<, <])

Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-2/3, 1/2, -1/4], [0, 2, 1], [-2/3, -1/4, 1/2], [<, <]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0011/)

Usual order on the real numbers · qf-density17-order-three-rational-sort · candidate





## 12. Stable Rational Sort with Original Index Witnesses

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[-1/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.

View answer and explanation([-1/3, 1/2, -1/4], [0, 2, 1], [-1/3, -1/4, 1/2], [<, <])

Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-1/3, 1/2, -1/4], [0, 2, 1], [-1/3, -1/4, 1/2], [<, <]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0012/)

Usual order on the real numbers · qf-density17-order-three-rational-sort · candidate





## 13. Stable Rational Sort with Original Index Witnesses

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[0/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.

View answer and explanation([0/1, 1/2, -1/4], [2, 0, 1], [-1/4, 0/1, 1/2], [<, <])

Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0/1, 1/2, -1/4], [2, 0, 1], [-1/4, 0/1, 1/2], [<, <]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0013/)

Usual order on the real numbers · qf-density17-order-three-rational-sort · candidate





## 14. Stable Rational Sort with Original Index Witnesses

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[1/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.

View answer and explanation([1/3, 1/2, -1/4], [2, 0, 1], [-1/4, 1/3, 1/2], [<, <])

Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1/3, 1/2, -1/4], [2, 0, 1], [-1/4, 1/3, 1/2], [<, <]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0014/)

Usual order on the real numbers · qf-density17-order-three-rational-sort · candidate





## 15. Stable Rational Sort with Original Index Witnesses

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[2/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.

View answer and explanation([2/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 2/3], [<, <])

Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 2/3], [<, <]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0015/)

Usual order on the real numbers · qf-density17-order-three-rational-sort · candidate





## 16. Stable Rational Sort with Original Index Witnesses

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[3/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.

View answer and explanation([1/1, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 1/1], [<, <])

Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1/1, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 1/1], [<, <]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0016/)

Usual order on the real numbers · qf-density17-order-three-rational-sort · candidate





## 17. Stable Rational Sort with Original Index Witnesses

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[4/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.

View answer and explanation([4/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 4/3], [<, <])

Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 4/3], [<, <]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0017/)

Usual order on the real numbers · qf-density17-order-three-rational-sort · candidate





## 18. Stable Rational Sort with Original Index Witnesses

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[5/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.

View answer and explanation([5/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 5/3], [<, <])

Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 5/3], [<, <]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0018/)

Usual order on the real numbers · qf-density17-order-three-rational-sort · candidate





## 19. Stable Rational Sort with Original Index Witnesses

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[6/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.

View answer and explanation([2/1, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 2/1], [<, <])

Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2/1, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 2/1], [<, <]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0019/)

Usual order on the real numbers · qf-density17-order-three-rational-sort · candidate





## 20. Stable Rational Sort with Original Index Witnesses

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[7/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.

View answer and explanation([7/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 7/3], [<, <])

Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([7/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 7/3], [<, <]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0020/)

Usual order on the real numbers · qf-density17-order-three-rational-sort · candidate





## 21. Stable Rational Sort with Original Index Witnesses

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[8/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.

View answer and explanation([8/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 8/3], [<, <])

Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([8/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 8/3], [<, <]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0021/)

Usual order on the real numbers · qf-density17-order-three-rational-sort · candidate





## 22. Stable Rational Sort with Original Index Witnesses

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[9/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.

View answer and explanation([3/1, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 3/1], [<, <])

Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3/1, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 3/1], [<, <]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0022/)

Usual order on the real numbers · qf-density17-order-three-rational-sort · candidate





## 23. Stable Rational Sort with Original Index Witnesses

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[10/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.

View answer and explanation([10/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 10/3], [<, <])

Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([10/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 10/3], [<, <]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0023/)

Usual order on the real numbers · qf-density17-order-three-rational-sort · candidate





## 24. Stable Rational Sort with Original Index Witnesses

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[11/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.

View answer and explanation([11/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 11/3], [<, <])

Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([11/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 11/3], [<, <]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0024/)

Usual order on the real numbers · qf-density17-order-three-rational-sort · candidate





## 25. Stable Rational Sort with Original Index Witnesses

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[12/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.

View answer and explanation([4/1, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 4/1], [<, <])

Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4/1, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 4/1], [<, <]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0025/)

Usual order on the real numbers · qf-density17-order-three-rational-sort · candidate





## 26. Closed-Interval Intersection Including Touching Endpoints

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [0/1, 1/1] and [1/1, 2/1]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).

View answer and explanation([0/1, 1/1], [1/1, 2/1], 1/1, 1/1, Yes, [1/1, 1/1])

Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0/1, 1/1], [1/1, 2/1], 1/1, 1/1, Yes, [1/1, 1/1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0026/)

Usual order on the real numbers · qf-density17-order-interval-intersection · candidate





## 27. Closed-Interval Intersection Including Touching Endpoints

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [-2/1, -1/1] and [0/1, 1/1]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).

View answer and explanation([-2/1, -1/1], [0/1, 1/1], 0/1, -1/1, No, empty-set)

Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-2/1, -1/1], [0/1, 1/1], 0/1, -1/1, No, empty-set).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0027/)

Usual order on the real numbers · qf-density17-order-interval-intersection · candidate





## 28. Closed-Interval Intersection Including Touching Endpoints

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [0/1, 0/1] and [0/1, 0/1]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).

View answer and explanation([0/1, 0/1], [0/1, 0/1], 0/1, 0/1, Yes, [0/1, 0/1])

Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0/1, 0/1], [0/1, 0/1], 0/1, 0/1, Yes, [0/1, 0/1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0028/)

Usual order on the real numbers · qf-density17-order-interval-intersection · candidate





## 29. Closed-Interval Intersection Including Touching Endpoints

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [-2/1, 2/1] and [-1/1, 3/1]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).

View answer and explanation([-2/1, 2/1], [-1/1, 3/1], -1/1, 2/1, Yes, [-1/1, 2/1])

Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-2/1, 2/1], [-1/1, 3/1], -1/1, 2/1, Yes, [-1/1, 2/1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0029/)

Usual order on the real numbers · qf-density17-order-interval-intersection · candidate





## 30. Closed-Interval Intersection Including Touching Endpoints

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [-8/2, -2/2] and [-5/2, 1/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).

View answer and explanation([-4/1, -1/1], [-5/2, 1/2], -5/2, -1/1, Yes, [-5/2, -1/1])

Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-4/1, -1/1], [-5/2, 1/2], -5/2, -1/1, Yes, [-5/2, -1/1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0030/)

Usual order on the real numbers · qf-density17-order-interval-intersection · candidate





## 31. Closed-Interval Intersection Including Touching Endpoints

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [-7/2, -1/2] and [-4/2, 2/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).

View answer and explanation([-7/2, -1/2], [-2/1, 1/1], -2/1, -1/2, Yes, [-2/1, -1/2])

Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-7/2, -1/2], [-2/1, 1/1], -2/1, -1/2, Yes, [-2/1, -1/2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0031/)

Usual order on the real numbers · qf-density17-order-interval-intersection · candidate





## 32. Closed-Interval Intersection Including Touching Endpoints

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [-6/2, 0/2] and [-3/2, 3/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).

View answer and explanation([-3/1, 0/1], [-3/2, 3/2], -3/2, 0/1, Yes, [-3/2, 0/1])

Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-3/1, 0/1], [-3/2, 3/2], -3/2, 0/1, Yes, [-3/2, 0/1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0032/)

Usual order on the real numbers · qf-density17-order-interval-intersection · candidate





## 33. Closed-Interval Intersection Including Touching Endpoints

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [-5/2, 1/2] and [-2/2, 4/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).

View answer and explanation([-5/2, 1/2], [-1/1, 2/1], -1/1, 1/2, Yes, [-1/1, 1/2])

Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-5/2, 1/2], [-1/1, 2/1], -1/1, 1/2, Yes, [-1/1, 1/2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0033/)

Usual order on the real numbers · qf-density17-order-interval-intersection · candidate





## 34. Closed-Interval Intersection Including Touching Endpoints

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [-4/2, 2/2] and [-1/2, 5/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).

View answer and explanation([-2/1, 1/1], [-1/2, 5/2], -1/2, 1/1, Yes, [-1/2, 1/1])

Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-2/1, 1/1], [-1/2, 5/2], -1/2, 1/1, Yes, [-1/2, 1/1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0034/)

Usual order on the real numbers · qf-density17-order-interval-intersection · candidate





## 35. Closed-Interval Intersection Including Touching Endpoints

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [-3/2, 3/2] and [0/2, 6/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).

View answer and explanation([-3/2, 3/2], [0/1, 3/1], 0/1, 3/2, Yes, [0/1, 3/2])

Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-3/2, 3/2], [0/1, 3/1], 0/1, 3/2, Yes, [0/1, 3/2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0035/)

Usual order on the real numbers · qf-density17-order-interval-intersection · candidate





## 36. Closed-Interval Intersection Including Touching Endpoints

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [-2/2, 4/2] and [1/2, 7/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).

View answer and explanation([-1/1, 2/1], [1/2, 7/2], 1/2, 2/1, Yes, [1/2, 2/1])

Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-1/1, 2/1], [1/2, 7/2], 1/2, 2/1, Yes, [1/2, 2/1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0036/)

Usual order on the real numbers · qf-density17-order-interval-intersection · candidate





## 37. Closed-Interval Intersection Including Touching Endpoints

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [-1/2, 5/2] and [2/2, 8/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).

View answer and explanation([-1/2, 5/2], [1/1, 4/1], 1/1, 5/2, Yes, [1/1, 5/2])

Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-1/2, 5/2], [1/1, 4/1], 1/1, 5/2, Yes, [1/1, 5/2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0037/)

Usual order on the real numbers · qf-density17-order-interval-intersection · candidate





## 38. Closed-Interval Intersection Including Touching Endpoints

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [0/2, 6/2] and [3/2, 9/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).

View answer and explanation([0/1, 3/1], [3/2, 9/2], 3/2, 3/1, Yes, [3/2, 3/1])

Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0/1, 3/1], [3/2, 9/2], 3/2, 3/1, Yes, [3/2, 3/1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0038/)

Usual order on the real numbers · qf-density17-order-interval-intersection · candidate





## 39. Closed-Interval Intersection Including Touching Endpoints

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [1/2, 7/2] and [4/2, 10/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).

View answer and explanation([1/2, 7/2], [2/1, 5/1], 2/1, 7/2, Yes, [2/1, 7/2])

Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1/2, 7/2], [2/1, 5/1], 2/1, 7/2, Yes, [2/1, 7/2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0039/)

Usual order on the real numbers · qf-density17-order-interval-intersection · candidate





## 40. Closed-Interval Intersection Including Touching Endpoints

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [2/2, 8/2] and [5/2, 11/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).

View answer and explanation([1/1, 4/1], [5/2, 11/2], 5/2, 4/1, Yes, [5/2, 4/1])

Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1/1, 4/1], [5/2, 11/2], 5/2, 4/1, Yes, [5/2, 4/1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0040/)

Usual order on the real numbers · qf-density17-order-interval-intersection · candidate





## 41. Closed-Interval Intersection Including Touching Endpoints

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [3/2, 9/2] and [6/2, 12/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).

View answer and explanation([3/2, 9/2], [3/1, 6/1], 3/1, 9/2, Yes, [3/1, 9/2])

Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3/2, 9/2], [3/1, 6/1], 3/1, 9/2, Yes, [3/1, 9/2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0041/)

Usual order on the real numbers · qf-density17-order-interval-intersection · candidate





## 42. Closed-Interval Intersection Including Touching Endpoints

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [4/2, 10/2] and [7/2, 13/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).

View answer and explanation([2/1, 5/1], [7/2, 13/2], 7/2, 5/1, Yes, [7/2, 5/1])

Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2/1, 5/1], [7/2, 13/2], 7/2, 5/1, Yes, [7/2, 5/1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0042/)

Usual order on the real numbers · qf-density17-order-interval-intersection · candidate





## 43. Closed-Interval Intersection Including Touching Endpoints

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [5/2, 11/2] and [8/2, 14/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).

View answer and explanation([5/2, 11/2], [4/1, 7/1], 4/1, 11/2, Yes, [4/1, 11/2])

Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5/2, 11/2], [4/1, 7/1], 4/1, 11/2, Yes, [4/1, 11/2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0043/)

Usual order on the real numbers · qf-density17-order-interval-intersection · candidate





## 44. Closed-Interval Intersection Including Touching Endpoints

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [6/2, 12/2] and [9/2, 15/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).

View answer and explanation([3/1, 6/1], [9/2, 15/2], 9/2, 6/1, Yes, [9/2, 6/1])

Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3/1, 6/1], [9/2, 15/2], 9/2, 6/1, Yes, [9/2, 6/1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0044/)

Usual order on the real numbers · qf-density17-order-interval-intersection · candidate





## 45. Closed-Interval Intersection Including Touching Endpoints

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [7/2, 13/2] and [10/2, 16/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).

View answer and explanation([7/2, 13/2], [5/1, 8/1], 5/1, 13/2, Yes, [5/1, 13/2])

Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([7/2, 13/2], [5/1, 8/1], 5/1, 13/2, Yes, [5/1, 13/2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0045/)

Usual order on the real numbers · qf-density17-order-interval-intersection · candidate





## 46. Closed-Interval Intersection Including Touching Endpoints

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [8/2, 14/2] and [11/2, 17/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).

View answer and explanation([4/1, 7/1], [11/2, 17/2], 11/2, 7/1, Yes, [11/2, 7/1])

Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4/1, 7/1], [11/2, 17/2], 11/2, 7/1, Yes, [11/2, 7/1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0046/)

Usual order on the real numbers · qf-density17-order-interval-intersection · candidate





## 47. Closed-Interval Intersection Including Touching Endpoints

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [9/2, 15/2] and [12/2, 18/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).

View answer and explanation([9/2, 15/2], [6/1, 9/1], 6/1, 15/2, Yes, [6/1, 15/2])

Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([9/2, 15/2], [6/1, 9/1], 6/1, 15/2, Yes, [6/1, 15/2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0047/)

Usual order on the real numbers · qf-density17-order-interval-intersection · candidate





## 48. Closed-Interval Intersection Including Touching Endpoints

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [10/2, 16/2] and [13/2, 19/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).

View answer and explanation([5/1, 8/1], [13/2, 19/2], 13/2, 8/1, Yes, [13/2, 8/1])

Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5/1, 8/1], [13/2, 19/2], 13/2, 8/1, Yes, [13/2, 8/1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0048/)

Usual order on the real numbers · qf-density17-order-interval-intersection · candidate





## 49. Closed-Interval Intersection Including Touching Endpoints

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [11/2, 17/2] and [14/2, 20/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).

View answer and explanation([11/2, 17/2], [7/1, 10/1], 7/1, 17/2, Yes, [7/1, 17/2])

Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([11/2, 17/2], [7/1, 10/1], 7/1, 17/2, Yes, [7/1, 17/2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0049/)

Usual order on the real numbers · qf-density17-order-interval-intersection · candidate





## 50. Closed-Interval Intersection Including Touching Endpoints

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [12/2, 18/2] and [15/2, 21/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).

View answer and explanation([6/1, 9/1], [15/2, 21/2], 15/2, 9/1, Yes, [15/2, 9/1])

Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([6/1, 9/1], [15/2, 21/2], 15/2, 9/1, Yes, [15/2, 9/1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0050/)

Usual order on the real numbers · qf-density17-order-interval-intersection · candidate





## 51. Three Closed-Interval Membership Tests and Signed Slacks

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[0/1, 0/1], queries=[-1/1, 0/1, 1/1]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).

View answer and explanation([0/1, 0/1], [-1/1, 0/1, 1/1], [No, Yes, No], [[-1/1, 1/1], [0/1, 0/1], [1/1, -1/1]])

Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0/1, 0/1], [-1/1, 0/1, 1/1], [No, Yes, No], [[-1/1, 1/1], [0/1, 0/1], [1/1, -1/1]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0051/)

Usual order on the real numbers · qf-density17-order-three-membership-slacks · candidate





## 52. Three Closed-Interval Membership Tests and Signed Slacks

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-1/2, 1/2], queries=[-1/2, 1/2, 1/1]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).

View answer and explanation([-1/2, 1/2], [-1/2, 1/2, 1/1], [Yes, Yes, No], [[0/1, 1/1], [1/1, 0/1], [3/2, -1/2]])

Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-1/2, 1/2], [-1/2, 1/2, 1/1], [Yes, Yes, No], [[0/1, 1/1], [1/1, 0/1], [3/2, -1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0052/)

Usual order on the real numbers · qf-density17-order-three-membership-slacks · candidate





## 53. Three Closed-Interval Membership Tests and Signed Slacks

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-2/3, 2/2], queries=[0/1, -2/3, 3/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).

View answer and explanation([-2/3, 1/1], [0/1, -2/3, 3/2], [Yes, Yes, No], [[2/3, 1/1], [0/1, 5/3], [13/6, -1/2]])

Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-2/3, 1/1], [0/1, -2/3, 3/2], [Yes, Yes, No], [[2/3, 1/1], [0/1, 5/3], [13/6, -1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0053/)

Usual order on the real numbers · qf-density17-order-three-membership-slacks · candidate





## 54. Three Closed-Interval Membership Tests and Signed Slacks

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-3/3, 3/2], queries=[0/1, -3/3, 4/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).

View answer and explanation([-1/1, 3/2], [0/1, -1/1, 2/1], [Yes, Yes, No], [[1/1, 3/2], [0/1, 5/2], [3/1, -1/2]])

Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-1/1, 3/2], [0/1, -1/1, 2/1], [Yes, Yes, No], [[1/1, 3/2], [0/1, 5/2], [3/1, -1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0054/)

Usual order on the real numbers · qf-density17-order-three-membership-slacks · candidate





## 55. Three Closed-Interval Membership Tests and Signed Slacks

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-4/3, 4/2], queries=[0/1, -4/3, 5/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).

View answer and explanation([-4/3, 2/1], [0/1, -4/3, 5/2], [Yes, Yes, No], [[4/3, 2/1], [0/1, 10/3], [23/6, -1/2]])

Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-4/3, 2/1], [0/1, -4/3, 5/2], [Yes, Yes, No], [[4/3, 2/1], [0/1, 10/3], [23/6, -1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0055/)

Usual order on the real numbers · qf-density17-order-three-membership-slacks · candidate





## 56. Three Closed-Interval Membership Tests and Signed Slacks

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-5/3, 5/2], queries=[0/1, -5/3, 6/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).

View answer and explanation([-5/3, 5/2], [0/1, -5/3, 3/1], [Yes, Yes, No], [[5/3, 5/2], [0/1, 25/6], [14/3, -1/2]])

Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-5/3, 5/2], [0/1, -5/3, 3/1], [Yes, Yes, No], [[5/3, 5/2], [0/1, 25/6], [14/3, -1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0056/)

Usual order on the real numbers · qf-density17-order-three-membership-slacks · candidate





## 57. Three Closed-Interval Membership Tests and Signed Slacks

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-6/3, 6/2], queries=[0/1, -6/3, 7/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).

View answer and explanation([-2/1, 3/1], [0/1, -2/1, 7/2], [Yes, Yes, No], [[2/1, 3/1], [0/1, 5/1], [11/2, -1/2]])

Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-2/1, 3/1], [0/1, -2/1, 7/2], [Yes, Yes, No], [[2/1, 3/1], [0/1, 5/1], [11/2, -1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0057/)

Usual order on the real numbers · qf-density17-order-three-membership-slacks · candidate





## 58. Three Closed-Interval Membership Tests and Signed Slacks

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-7/3, 7/2], queries=[0/1, -7/3, 8/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).

View answer and explanation([-7/3, 7/2], [0/1, -7/3, 4/1], [Yes, Yes, No], [[7/3, 7/2], [0/1, 35/6], [19/3, -1/2]])

Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-7/3, 7/2], [0/1, -7/3, 4/1], [Yes, Yes, No], [[7/3, 7/2], [0/1, 35/6], [19/3, -1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0058/)

Usual order on the real numbers · qf-density17-order-three-membership-slacks · candidate





## 59. Three Closed-Interval Membership Tests and Signed Slacks

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-8/3, 8/2], queries=[0/1, -8/3, 9/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).

View answer and explanation([-8/3, 4/1], [0/1, -8/3, 9/2], [Yes, Yes, No], [[8/3, 4/1], [0/1, 20/3], [43/6, -1/2]])

Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-8/3, 4/1], [0/1, -8/3, 9/2], [Yes, Yes, No], [[8/3, 4/1], [0/1, 20/3], [43/6, -1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0059/)

Usual order on the real numbers · qf-density17-order-three-membership-slacks · candidate





## 60. Three Closed-Interval Membership Tests and Signed Slacks

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-9/3, 9/2], queries=[0/1, -9/3, 10/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).

View answer and explanation([-3/1, 9/2], [0/1, -3/1, 5/1], [Yes, Yes, No], [[3/1, 9/2], [0/1, 15/2], [8/1, -1/2]])

Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-3/1, 9/2], [0/1, -3/1, 5/1], [Yes, Yes, No], [[3/1, 9/2], [0/1, 15/2], [8/1, -1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0060/)

Usual order on the real numbers · qf-density17-order-three-membership-slacks · candidate





## 61. Three Closed-Interval Membership Tests and Signed Slacks

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-10/3, 10/2], queries=[0/1, -10/3, 11/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).

View answer and explanation([-10/3, 5/1], [0/1, -10/3, 11/2], [Yes, Yes, No], [[10/3, 5/1], [0/1, 25/3], [53/6, -1/2]])

Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-10/3, 5/1], [0/1, -10/3, 11/2], [Yes, Yes, No], [[10/3, 5/1], [0/1, 25/3], [53/6, -1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0061/)

Usual order on the real numbers · qf-density17-order-three-membership-slacks · candidate





## 62. Three Closed-Interval Membership Tests and Signed Slacks

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-11/3, 11/2], queries=[0/1, -11/3, 12/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).

View answer and explanation([-11/3, 11/2], [0/1, -11/3, 6/1], [Yes, Yes, No], [[11/3, 11/2], [0/1, 55/6], [29/3, -1/2]])

Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-11/3, 11/2], [0/1, -11/3, 6/1], [Yes, Yes, No], [[11/3, 11/2], [0/1, 55/6], [29/3, -1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0062/)

Usual order on the real numbers · qf-density17-order-three-membership-slacks · candidate





## 63. Three Closed-Interval Membership Tests and Signed Slacks

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-12/3, 12/2], queries=[0/1, -12/3, 13/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).

View answer and explanation([-4/1, 6/1], [0/1, -4/1, 13/2], [Yes, Yes, No], [[4/1, 6/1], [0/1, 10/1], [21/2, -1/2]])

Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-4/1, 6/1], [0/1, -4/1, 13/2], [Yes, Yes, No], [[4/1, 6/1], [0/1, 10/1], [21/2, -1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0063/)

Usual order on the real numbers · qf-density17-order-three-membership-slacks · candidate





## 64. Three Closed-Interval Membership Tests and Signed Slacks

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-13/3, 13/2], queries=[0/1, -13/3, 14/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).

View answer and explanation([-13/3, 13/2], [0/1, -13/3, 7/1], [Yes, Yes, No], [[13/3, 13/2], [0/1, 65/6], [34/3, -1/2]])

Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-13/3, 13/2], [0/1, -13/3, 7/1], [Yes, Yes, No], [[13/3, 13/2], [0/1, 65/6], [34/3, -1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0064/)

Usual order on the real numbers · qf-density17-order-three-membership-slacks · candidate





## 65. Three Closed-Interval Membership Tests and Signed Slacks

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-14/3, 14/2], queries=[0/1, -14/3, 15/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).

View answer and explanation([-14/3, 7/1], [0/1, -14/3, 15/2], [Yes, Yes, No], [[14/3, 7/1], [0/1, 35/3], [73/6, -1/2]])

Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-14/3, 7/1], [0/1, -14/3, 15/2], [Yes, Yes, No], [[14/3, 7/1], [0/1, 35/3], [73/6, -1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0065/)

Usual order on the real numbers · qf-density17-order-three-membership-slacks · candidate





## 66. Three Closed-Interval Membership Tests and Signed Slacks

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-15/3, 15/2], queries=[0/1, -15/3, 16/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).

View answer and explanation([-5/1, 15/2], [0/1, -5/1, 8/1], [Yes, Yes, No], [[5/1, 15/2], [0/1, 25/2], [13/1, -1/2]])

Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-5/1, 15/2], [0/1, -5/1, 8/1], [Yes, Yes, No], [[5/1, 15/2], [0/1, 25/2], [13/1, -1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0066/)

Usual order on the real numbers · qf-density17-order-three-membership-slacks · candidate





## 67. Three Closed-Interval Membership Tests and Signed Slacks

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-16/3, 16/2], queries=[0/1, -16/3, 17/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).

View answer and explanation([-16/3, 8/1], [0/1, -16/3, 17/2], [Yes, Yes, No], [[16/3, 8/1], [0/1, 40/3], [83/6, -1/2]])

Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-16/3, 8/1], [0/1, -16/3, 17/2], [Yes, Yes, No], [[16/3, 8/1], [0/1, 40/3], [83/6, -1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0067/)

Usual order on the real numbers · qf-density17-order-three-membership-slacks · candidate





## 68. Three Closed-Interval Membership Tests and Signed Slacks

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-17/3, 17/2], queries=[0/1, -17/3, 18/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).

View answer and explanation([-17/3, 17/2], [0/1, -17/3, 9/1], [Yes, Yes, No], [[17/3, 17/2], [0/1, 85/6], [44/3, -1/2]])

Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-17/3, 17/2], [0/1, -17/3, 9/1], [Yes, Yes, No], [[17/3, 17/2], [0/1, 85/6], [44/3, -1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0068/)

Usual order on the real numbers · qf-density17-order-three-membership-slacks · candidate





## 69. Three Closed-Interval Membership Tests and Signed Slacks

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-18/3, 18/2], queries=[0/1, -18/3, 19/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).

View answer and explanation([-6/1, 9/1], [0/1, -6/1, 19/2], [Yes, Yes, No], [[6/1, 9/1], [0/1, 15/1], [31/2, -1/2]])

Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-6/1, 9/1], [0/1, -6/1, 19/2], [Yes, Yes, No], [[6/1, 9/1], [0/1, 15/1], [31/2, -1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0069/)

Usual order on the real numbers · qf-density17-order-three-membership-slacks · candidate





## 70. Three Closed-Interval Membership Tests and Signed Slacks

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-19/3, 19/2], queries=[0/1, -19/3, 20/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).

View answer and explanation([-19/3, 19/2], [0/1, -19/3, 10/1], [Yes, Yes, No], [[19/3, 19/2], [0/1, 95/6], [49/3, -1/2]])

Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-19/3, 19/2], [0/1, -19/3, 10/1], [Yes, Yes, No], [[19/3, 19/2], [0/1, 95/6], [49/3, -1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0070/)

Usual order on the real numbers · qf-density17-order-three-membership-slacks · candidate





## 71. Three Closed-Interval Membership Tests and Signed Slacks

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-20/3, 20/2], queries=[0/1, -20/3, 21/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).

View answer and explanation([-20/3, 10/1], [0/1, -20/3, 21/2], [Yes, Yes, No], [[20/3, 10/1], [0/1, 50/3], [103/6, -1/2]])

Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-20/3, 10/1], [0/1, -20/3, 21/2], [Yes, Yes, No], [[20/3, 10/1], [0/1, 50/3], [103/6, -1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0071/)

Usual order on the real numbers · qf-density17-order-three-membership-slacks · candidate





## 72. Three Closed-Interval Membership Tests and Signed Slacks

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-21/3, 21/2], queries=[0/1, -21/3, 22/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).

View answer and explanation([-7/1, 21/2], [0/1, -7/1, 11/1], [Yes, Yes, No], [[7/1, 21/2], [0/1, 35/2], [18/1, -1/2]])

Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-7/1, 21/2], [0/1, -7/1, 11/1], [Yes, Yes, No], [[7/1, 21/2], [0/1, 35/2], [18/1, -1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0072/)

Usual order on the real numbers · qf-density17-order-three-membership-slacks · candidate





## 73. Three Closed-Interval Membership Tests and Signed Slacks

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-22/3, 22/2], queries=[0/1, -22/3, 23/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).

View answer and explanation([-22/3, 11/1], [0/1, -22/3, 23/2], [Yes, Yes, No], [[22/3, 11/1], [0/1, 55/3], [113/6, -1/2]])

Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-22/3, 11/1], [0/1, -22/3, 23/2], [Yes, Yes, No], [[22/3, 11/1], [0/1, 55/3], [113/6, -1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0073/)

Usual order on the real numbers · qf-density17-order-three-membership-slacks · candidate





## 74. Three Closed-Interval Membership Tests and Signed Slacks

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-23/3, 23/2], queries=[0/1, -23/3, 24/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).

View answer and explanation([-23/3, 23/2], [0/1, -23/3, 12/1], [Yes, Yes, No], [[23/3, 23/2], [0/1, 115/6], [59/3, -1/2]])

Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-23/3, 23/2], [0/1, -23/3, 12/1], [Yes, Yes, No], [[23/3, 23/2], [0/1, 115/6], [59/3, -1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0074/)

Usual order on the real numbers · qf-density17-order-three-membership-slacks · candidate





## 75. Three Closed-Interval Membership Tests and Signed Slacks

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-24/3, 24/2], queries=[0/1, -24/3, 25/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).

View answer and explanation([-8/1, 12/1], [0/1, -8/1, 25/2], [Yes, Yes, No], [[8/1, 12/1], [0/1, 20/1], [41/2, -1/2]])

Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-8/1, 12/1], [0/1, -8/1, 25/2], [Yes, Yes, No], [[8/1, 12/1], [0/1, 20/1], [41/2, -1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0075/)

Usual order on the real numbers · qf-density17-order-three-membership-slacks · candidate





## 76. Scaled Rational Values and Order Direction

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=-1/2, b=1/3, t=0/1. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).

View answer and explanation(-1/2, 1/3, 0/1, 0/1, 0/1, <, =, collapsed)

Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-1/2, 1/3, 0/1, 0/1, 0/1, <, =, collapsed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0076/)

Usual order on the real numbers · qf-density17-order-scaling-direction · candidate





## 77. Scaled Rational Values and Order Direction

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=0/1, b=0/1, t=-2/1. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).

View answer and explanation(0/1, 0/1, -2/1, 0/1, 0/1, =, =, order-reversed)

Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (0/1, 0/1, -2/1, 0/1, 0/1, =, =, order-reversed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0077/)

Usual order on the real numbers · qf-density17-order-scaling-direction · candidate





## 78. Scaled Rational Values and Order Direction

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=-10/3, b=-9/2, t=3/3. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).

View answer and explanation(-10/3, -9/2, 1/1, -10/3, -9/2, >, >, order-preserved)

Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-10/3, -9/2, 1/1, -10/3, -9/2, >, >, order-preserved).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0078/)

Usual order on the real numbers · qf-density17-order-scaling-direction · candidate





## 79. Scaled Rational Values and Order Direction

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=-9/3, b=-8/2, t=0/4. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).

View answer and explanation(-3/1, -4/1, 0/1, 0/1, 0/1, >, =, collapsed)

Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-3/1, -4/1, 0/1, 0/1, 0/1, >, =, collapsed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0079/)

Usual order on the real numbers · qf-density17-order-scaling-direction · candidate





## 80. Scaled Rational Values and Order Direction

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=-8/3, b=-7/2, t=3/1. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).

View answer and explanation(-8/3, -7/2, 3/1, -8/1, -21/2, >, >, order-preserved)

Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-8/3, -7/2, 3/1, -8/1, -21/2, >, >, order-preserved).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0080/)

Usual order on the real numbers · qf-density17-order-scaling-direction · candidate





## 81. Scaled Rational Values and Order Direction

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=-7/3, b=-6/2, t=-2/2. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).

View answer and explanation(-7/3, -3/1, -1/1, 7/3, 3/1, >, <, order-reversed)

Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-7/3, -3/1, -1/1, 7/3, 3/1, >, <, order-reversed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0081/)

Usual order on the real numbers · qf-density17-order-scaling-direction · candidate





## 82. Scaled Rational Values and Order Direction

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=-6/3, b=-5/2, t=0/3. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).

View answer and explanation(-2/1, -5/2, 0/1, 0/1, 0/1, >, =, collapsed)

Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-2/1, -5/2, 0/1, 0/1, 0/1, >, =, collapsed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0082/)

Usual order on the real numbers · qf-density17-order-scaling-direction · candidate





## 83. Scaled Rational Values and Order Direction

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=-5/3, b=-4/2, t=-2/4. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).

View answer and explanation(-5/3, -2/1, -1/2, 5/6, 1/1, >, <, order-reversed)

Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-5/3, -2/1, -1/2, 5/6, 1/1, >, <, order-reversed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0083/)

Usual order on the real numbers · qf-density17-order-scaling-direction · candidate





## 84. Scaled Rational Values and Order Direction

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=-4/3, b=-3/2, t=3/1. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).

View answer and explanation(-4/3, -3/2, 3/1, -4/1, -9/2, >, >, order-preserved)

Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-4/3, -3/2, 3/1, -4/1, -9/2, >, >, order-preserved).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0084/)

Usual order on the real numbers · qf-density17-order-scaling-direction · candidate





## 85. Scaled Rational Values and Order Direction

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=-3/3, b=-2/2, t=0/2. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).

View answer and explanation(-1/1, -1/1, 0/1, 0/1, 0/1, =, =, collapsed)

Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-1/1, -1/1, 0/1, 0/1, 0/1, =, =, collapsed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0085/)

Usual order on the real numbers · qf-density17-order-scaling-direction · candidate





## 86. Scaled Rational Values and Order Direction

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=-2/3, b=-1/2, t=3/3. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).

View answer and explanation(-2/3, -1/2, 1/1, -2/3, -1/2, <, <, order-preserved)

Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-2/3, -1/2, 1/1, -2/3, -1/2, <, <, order-preserved).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0086/)

Usual order on the real numbers · qf-density17-order-scaling-direction · candidate





## 87. Scaled Rational Values and Order Direction

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=-1/3, b=0/2, t=-2/4. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).

View answer and explanation(-1/3, 0/1, -1/2, 1/6, 0/1, <, >, order-reversed)

Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-1/3, 0/1, -1/2, 1/6, 0/1, <, >, order-reversed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0087/)

Usual order on the real numbers · qf-density17-order-scaling-direction · candidate





## 88. Scaled Rational Values and Order Direction

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=0/3, b=1/2, t=0/1. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).

View answer and explanation(0/1, 1/2, 0/1, 0/1, 0/1, <, =, collapsed)

Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (0/1, 1/2, 0/1, 0/1, 0/1, <, =, collapsed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0088/)

Usual order on the real numbers · qf-density17-order-scaling-direction · candidate





## 89. Scaled Rational Values and Order Direction

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=1/3, b=2/2, t=-2/2. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).

View answer and explanation(1/3, 1/1, -1/1, -1/3, -1/1, <, >, order-reversed)

Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (1/3, 1/1, -1/1, -1/3, -1/1, <, >, order-reversed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0089/)

Usual order on the real numbers · qf-density17-order-scaling-direction · candidate





## 90. Scaled Rational Values and Order Direction

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=2/3, b=3/2, t=3/3. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).

View answer and explanation(2/3, 3/2, 1/1, 2/3, 3/2, <, <, order-preserved)

Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (2/3, 3/2, 1/1, 2/3, 3/2, <, <, order-preserved).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0090/)

Usual order on the real numbers · qf-density17-order-scaling-direction · candidate





## 91. Scaled Rational Values and Order Direction

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=3/3, b=4/2, t=0/4. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).

View answer and explanation(1/1, 2/1, 0/1, 0/1, 0/1, <, =, collapsed)

Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (1/1, 2/1, 0/1, 0/1, 0/1, <, =, collapsed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0091/)

Usual order on the real numbers · qf-density17-order-scaling-direction · candidate





## 92. Scaled Rational Values and Order Direction

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=4/3, b=5/2, t=3/1. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).

View answer and explanation(4/3, 5/2, 3/1, 4/1, 15/2, <, <, order-preserved)

Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (4/3, 5/2, 3/1, 4/1, 15/2, <, <, order-preserved).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0092/)

Usual order on the real numbers · qf-density17-order-scaling-direction · candidate





## 93. Scaled Rational Values and Order Direction

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=5/3, b=6/2, t=-2/2. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).

View answer and explanation(5/3, 3/1, -1/1, -5/3, -3/1, <, >, order-reversed)

Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5/3, 3/1, -1/1, -5/3, -3/1, <, >, order-reversed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0093/)

Usual order on the real numbers · qf-density17-order-scaling-direction · candidate





## 94. Scaled Rational Values and Order Direction

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=6/3, b=7/2, t=0/3. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).

View answer and explanation(2/1, 7/2, 0/1, 0/1, 0/1, <, =, collapsed)

Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (2/1, 7/2, 0/1, 0/1, 0/1, <, =, collapsed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0094/)

Usual order on the real numbers · qf-density17-order-scaling-direction · candidate





## 95. Scaled Rational Values and Order Direction

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=7/3, b=8/2, t=-2/4. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).

View answer and explanation(7/3, 4/1, -1/2, -7/6, -2/1, <, >, order-reversed)

Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (7/3, 4/1, -1/2, -7/6, -2/1, <, >, order-reversed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0095/)

Usual order on the real numbers · qf-density17-order-scaling-direction · candidate





## 96. Scaled Rational Values and Order Direction

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=8/3, b=9/2, t=3/1. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).

View answer and explanation(8/3, 9/2, 3/1, 8/1, 27/2, <, <, order-preserved)

Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (8/3, 9/2, 3/1, 8/1, 27/2, <, <, order-preserved).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0096/)

Usual order on the real numbers · qf-density17-order-scaling-direction · candidate





## 97. Scaled Rational Values and Order Direction

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=9/3, b=10/2, t=0/2. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).

View answer and explanation(3/1, 5/1, 0/1, 0/1, 0/1, <, =, collapsed)

Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (3/1, 5/1, 0/1, 0/1, 0/1, <, =, collapsed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0097/)

Usual order on the real numbers · qf-density17-order-scaling-direction · candidate





## 98. Scaled Rational Values and Order Direction

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=10/3, b=11/2, t=3/3. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).

View answer and explanation(10/3, 11/2, 1/1, 10/3, 11/2, <, <, order-preserved)

Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (10/3, 11/2, 1/1, 10/3, 11/2, <, <, order-preserved).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0098/)

Usual order on the real numbers · qf-density17-order-scaling-direction · candidate





## 99. Scaled Rational Values and Order Direction

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=11/3, b=12/2, t=-2/4. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).

View answer and explanation(11/3, 6/1, -1/2, -11/6, -3/1, <, >, order-reversed)

Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (11/3, 6/1, -1/2, -11/6, -3/1, <, >, order-reversed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0099/)

Usual order on the real numbers · qf-density17-order-scaling-direction · candidate





## 100. Scaled Rational Values and Order Direction

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=12/3, b=13/2, t=0/1. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).

View answer and explanation(4/1, 13/2, 0/1, 0/1, 0/1, <, =, collapsed)

Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (4/1, 13/2, 0/1, 0/1, 0/1, <, =, collapsed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r04-order-relation-0100/)

Usual order on the real numbers · qf-density17-order-scaling-direction · candidate



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