# Affine Residual Scaling, Forward Error and Tolerance Conventions This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed. 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-20261015-advance-20261119-r05-numerical-approximation-error-foundations · 2026-10-15 · 100 questions This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt. Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged. [Field-order and source-binding data](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json) ## 1. An Affine Residual Equals Absolute Slope Times Forward Error Use nonzero slope=-3, intercept=1, approximation=-4 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction). View answer and explanation(-3, 1, -4, 1/3, 13, 13/3, 13, Yes, residual_and_forward_error_are_distinct_quantities) Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (-3, 1, -4, 1/3, 13, 13/3, 13, Yes, residual_and_forward_error_are_distinct_quantities). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0001/) Numerical approximation and error · qf-mod50-numerr-affine-residual-forward · candidate ## 2. An Affine Residual Equals Absolute Slope Times Forward Error Use nonzero slope=-2, intercept=2, approximation=-3 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction). View answer and explanation(-2, 2, -3, 1, 8, 4, 8, Yes, residual_and_forward_error_are_distinct_quantities) Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (-2, 2, -3, 1, 8, 4, 8, Yes, residual_and_forward_error_are_distinct_quantities). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0002/) Numerical approximation and error · qf-mod50-numerr-affine-residual-forward · candidate ## 3. An Affine Residual Equals Absolute Slope Times Forward Error Use nonzero slope=-1, intercept=3, approximation=-2 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction). View answer and explanation(-1, 3, -2, 3, 5, 5, 5, Yes, residual_and_forward_error_are_distinct_quantities) Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (-1, 3, -2, 3, 5, 5, 5, Yes, residual_and_forward_error_are_distinct_quantities). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0003/) Numerical approximation and error · qf-mod50-numerr-affine-residual-forward · candidate ## 4. An Affine Residual Equals Absolute Slope Times Forward Error Use nonzero slope=2, intercept=4, approximation=-1 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction). View answer and explanation(2, 4, -1, -2, 2, 1, 2, Yes, residual_and_forward_error_are_distinct_quantities) Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (2, 4, -1, -2, 2, 1, 2, Yes, residual_and_forward_error_are_distinct_quantities). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0004/) Numerical approximation and error · qf-mod50-numerr-affine-residual-forward · candidate ## 5. An Affine Residual Equals Absolute Slope Times Forward Error Use nonzero slope=1, intercept=5, approximation=0 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction). View answer and explanation(1, 5, 0, -5, 5, 5, 5, Yes, residual_and_forward_error_are_distinct_quantities) Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (1, 5, 0, -5, 5, 5, 5, Yes, residual_and_forward_error_are_distinct_quantities). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0005/) Numerical approximation and error · qf-mod50-numerr-affine-residual-forward · candidate ## 6. An Affine Residual Equals Absolute Slope Times Forward Error Use nonzero slope=2, intercept=6, approximation=1 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction). View answer and explanation(2, 6, 1, -3, 8, 4, 8, Yes, residual_and_forward_error_are_distinct_quantities) Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (2, 6, 1, -3, 8, 4, 8, Yes, residual_and_forward_error_are_distinct_quantities). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0006/) Numerical approximation and error · qf-mod50-numerr-affine-residual-forward · candidate ## 7. An Affine Residual Equals Absolute Slope Times Forward Error Use nonzero slope=3, intercept=7, approximation=2 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction). View answer and explanation(3, 7, 2, -7/3, 13, 13/3, 13, Yes, residual_and_forward_error_are_distinct_quantities) Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (3, 7, 2, -7/3, 13, 13/3, 13, Yes, residual_and_forward_error_are_distinct_quantities). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0007/) Numerical approximation and error · qf-mod50-numerr-affine-residual-forward · candidate ## 8. An Affine Residual Equals Absolute Slope Times Forward Error Use nonzero slope=-3, intercept=8, approximation=3 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction). View answer and explanation(-3, 8, 3, 8/3, 1, 1/3, 1, Yes, residual_and_forward_error_are_distinct_quantities) Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (-3, 8, 3, 8/3, 1, 1/3, 1, Yes, residual_and_forward_error_are_distinct_quantities). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0008/) Numerical approximation and error · qf-mod50-numerr-affine-residual-forward · candidate ## 9. An Affine Residual Equals Absolute Slope Times Forward Error Use nonzero slope=-2, intercept=9, approximation=4 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction). View answer and explanation(-2, 9, 4, 9/2, 1, 1/2, 1, Yes, residual_and_forward_error_are_distinct_quantities) Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (-2, 9, 4, 9/2, 1, 1/2, 1, Yes, residual_and_forward_error_are_distinct_quantities). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0009/) Numerical approximation and error · qf-mod50-numerr-affine-residual-forward · candidate ## 10. An Affine Residual Equals Absolute Slope Times Forward Error Use nonzero slope=-1, intercept=10, approximation=5 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction). View answer and explanation(-1, 10, 5, 10, 5, 5, 5, Yes, residual_and_forward_error_are_distinct_quantities) Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (-1, 10, 5, 10, 5, 5, 5, Yes, residual_and_forward_error_are_distinct_quantities). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0010/) Numerical approximation and error · qf-mod50-numerr-affine-residual-forward · candidate ## 11. An Affine Residual Equals Absolute Slope Times Forward Error Use nonzero slope=2, intercept=11, approximation=6 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction). View answer and explanation(2, 11, 6, -11/2, 23, 23/2, 23, Yes, residual_and_forward_error_are_distinct_quantities) Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (2, 11, 6, -11/2, 23, 23/2, 23, Yes, residual_and_forward_error_are_distinct_quantities). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0011/) Numerical approximation and error · qf-mod50-numerr-affine-residual-forward · candidate ## 12. An Affine Residual Equals Absolute Slope Times Forward Error Use nonzero slope=1, intercept=12, approximation=7 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction). View answer and explanation(1, 12, 7, -12, 19, 19, 19, Yes, residual_and_forward_error_are_distinct_quantities) Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (1, 12, 7, -12, 19, 19, 19, Yes, residual_and_forward_error_are_distinct_quantities). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0012/) Numerical approximation and error · qf-mod50-numerr-affine-residual-forward · candidate ## 13. An Affine Residual Equals Absolute Slope Times Forward Error Use nonzero slope=2, intercept=13, approximation=8 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction). View answer and explanation(2, 13, 8, -13/2, 29, 29/2, 29, Yes, residual_and_forward_error_are_distinct_quantities) Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (2, 13, 8, -13/2, 29, 29/2, 29, Yes, residual_and_forward_error_are_distinct_quantities). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0013/) Numerical approximation and error · qf-mod50-numerr-affine-residual-forward · candidate ## 14. An Affine Residual Equals Absolute Slope Times Forward Error Use nonzero slope=3, intercept=14, approximation=9 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction). View answer and explanation(3, 14, 9, -14/3, 41, 41/3, 41, Yes, residual_and_forward_error_are_distinct_quantities) Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (3, 14, 9, -14/3, 41, 41/3, 41, Yes, residual_and_forward_error_are_distinct_quantities). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0014/) Numerical approximation and error · qf-mod50-numerr-affine-residual-forward · candidate ## 15. An Affine Residual Equals Absolute Slope Times Forward Error Use nonzero slope=-3, intercept=15, approximation=10 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction). View answer and explanation(-3, 15, 10, 5, 15, 5, 15, Yes, residual_and_forward_error_are_distinct_quantities) Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (-3, 15, 10, 5, 15, 5, 15, Yes, residual_and_forward_error_are_distinct_quantities). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0015/) Numerical approximation and error · qf-mod50-numerr-affine-residual-forward · candidate ## 16. An Affine Residual Equals Absolute Slope Times Forward Error Use nonzero slope=-2, intercept=16, approximation=11 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction). View answer and explanation(-2, 16, 11, 8, 6, 3, 6, Yes, residual_and_forward_error_are_distinct_quantities) Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (-2, 16, 11, 8, 6, 3, 6, Yes, residual_and_forward_error_are_distinct_quantities). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0016/) Numerical approximation and error · qf-mod50-numerr-affine-residual-forward · candidate ## 17. An Affine Residual Equals Absolute Slope Times Forward Error Use nonzero slope=-1, intercept=17, approximation=12 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction). View answer and explanation(-1, 17, 12, 17, 5, 5, 5, Yes, residual_and_forward_error_are_distinct_quantities) Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (-1, 17, 12, 17, 5, 5, 5, Yes, residual_and_forward_error_are_distinct_quantities). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0017/) Numerical approximation and error · qf-mod50-numerr-affine-residual-forward · candidate ## 18. An Affine Residual Equals Absolute Slope Times Forward Error Use nonzero slope=2, intercept=18, approximation=13 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction). View answer and explanation(2, 18, 13, -9, 44, 22, 44, Yes, residual_and_forward_error_are_distinct_quantities) Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (2, 18, 13, -9, 44, 22, 44, Yes, residual_and_forward_error_are_distinct_quantities). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0018/) Numerical approximation and error · qf-mod50-numerr-affine-residual-forward · candidate ## 19. An Affine Residual Equals Absolute Slope Times Forward Error Use nonzero slope=1, intercept=19, approximation=14 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction). View answer and explanation(1, 19, 14, -19, 33, 33, 33, Yes, residual_and_forward_error_are_distinct_quantities) Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (1, 19, 14, -19, 33, 33, 33, Yes, residual_and_forward_error_are_distinct_quantities). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0019/) Numerical approximation and error · qf-mod50-numerr-affine-residual-forward · candidate ## 20. An Affine Residual Equals Absolute Slope Times Forward Error Use nonzero slope=2, intercept=20, approximation=15 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction). View answer and explanation(2, 20, 15, -10, 50, 25, 50, Yes, residual_and_forward_error_are_distinct_quantities) Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (2, 20, 15, -10, 50, 25, 50, Yes, residual_and_forward_error_are_distinct_quantities). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0020/) Numerical approximation and error · qf-mod50-numerr-affine-residual-forward · candidate ## 21. An Affine Residual Equals Absolute Slope Times Forward Error Use nonzero slope=3, intercept=21, approximation=16 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction). View answer and explanation(3, 21, 16, -7, 69, 23, 69, Yes, residual_and_forward_error_are_distinct_quantities) Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (3, 21, 16, -7, 69, 23, 69, Yes, residual_and_forward_error_are_distinct_quantities). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0021/) Numerical approximation and error · qf-mod50-numerr-affine-residual-forward · candidate ## 22. An Affine Residual Equals Absolute Slope Times Forward Error Use nonzero slope=-3, intercept=22, approximation=17 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction). View answer and explanation(-3, 22, 17, 22/3, 29, 29/3, 29, Yes, residual_and_forward_error_are_distinct_quantities) Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (-3, 22, 17, 22/3, 29, 29/3, 29, Yes, residual_and_forward_error_are_distinct_quantities). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0022/) Numerical approximation and error · qf-mod50-numerr-affine-residual-forward · candidate ## 23. An Affine Residual Equals Absolute Slope Times Forward Error Use nonzero slope=-2, intercept=23, approximation=18 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction). View answer and explanation(-2, 23, 18, 23/2, 13, 13/2, 13, Yes, residual_and_forward_error_are_distinct_quantities) Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (-2, 23, 18, 23/2, 13, 13/2, 13, Yes, residual_and_forward_error_are_distinct_quantities). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0023/) Numerical approximation and error · qf-mod50-numerr-affine-residual-forward · candidate ## 24. An Affine Residual Equals Absolute Slope Times Forward Error Use nonzero slope=-1, intercept=24, approximation=19 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction). View answer and explanation(-1, 24, 19, 24, 5, 5, 5, Yes, residual_and_forward_error_are_distinct_quantities) Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (-1, 24, 19, 24, 5, 5, 5, Yes, residual_and_forward_error_are_distinct_quantities). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0024/) Numerical approximation and error · qf-mod50-numerr-affine-residual-forward · candidate ## 25. An Affine Residual Equals Absolute Slope Times Forward Error Use nonzero slope=2, intercept=25, approximation=20 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction). View answer and explanation(2, 25, 20, -25/2, 65, 65/2, 65, Yes, residual_and_forward_error_are_distinct_quantities) Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (2, 25, 20, -25/2, 65, 65/2, 65, Yes, residual_and_forward_error_are_distinct_quantities). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0025/) Numerical approximation and error · qf-mod50-numerr-affine-residual-forward · candidate ## 26. An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope Use slope=-3 and nonnegative absolute input error bound=1/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope). View answer and explanation(-3, 1/3, 1, affine_output_absolute_error_bound) Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (-3, 1/3, 1, affine_output_absolute_error_bound). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0026/) Numerical approximation and error · qf-mod50-numerr-affine-propagation · candidate ## 27. An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope Use slope=-2 and nonnegative absolute input error bound=2/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope). View answer and explanation(-2, 2/3, 4/3, affine_output_absolute_error_bound) Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (-2, 2/3, 4/3, affine_output_absolute_error_bound). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0027/) Numerical approximation and error · qf-mod50-numerr-affine-propagation · candidate ## 28. An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope Use slope=-1 and nonnegative absolute input error bound=3/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope). View answer and explanation(-1, 1, 1, affine_output_absolute_error_bound) Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (-1, 1, 1, affine_output_absolute_error_bound). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0028/) Numerical approximation and error · qf-mod50-numerr-affine-propagation · candidate ## 29. An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope Use slope=2 and nonnegative absolute input error bound=4/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope). View answer and explanation(2, 4/3, 8/3, affine_output_absolute_error_bound) Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (2, 4/3, 8/3, affine_output_absolute_error_bound). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0029/) Numerical approximation and error · qf-mod50-numerr-affine-propagation · candidate ## 30. An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope Use slope=1 and nonnegative absolute input error bound=5/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope). View answer and explanation(1, 5/3, 5/3, affine_output_absolute_error_bound) Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (1, 5/3, 5/3, affine_output_absolute_error_bound). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0030/) Numerical approximation and error · qf-mod50-numerr-affine-propagation · candidate ## 31. An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope Use slope=2 and nonnegative absolute input error bound=6/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope). View answer and explanation(2, 2, 4, affine_output_absolute_error_bound) Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (2, 2, 4, affine_output_absolute_error_bound). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0031/) Numerical approximation and error · qf-mod50-numerr-affine-propagation · candidate ## 32. An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope Use slope=3 and nonnegative absolute input error bound=7/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope). View answer and explanation(3, 7/3, 7, affine_output_absolute_error_bound) Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (3, 7/3, 7, affine_output_absolute_error_bound). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0032/) Numerical approximation and error · qf-mod50-numerr-affine-propagation · candidate ## 33. An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope Use slope=-3 and nonnegative absolute input error bound=8/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope). View answer and explanation(-3, 8/3, 8, affine_output_absolute_error_bound) Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (-3, 8/3, 8, affine_output_absolute_error_bound). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0033/) Numerical approximation and error · qf-mod50-numerr-affine-propagation · candidate ## 34. An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope Use slope=-2 and nonnegative absolute input error bound=9/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope). View answer and explanation(-2, 3, 6, affine_output_absolute_error_bound) Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (-2, 3, 6, affine_output_absolute_error_bound). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0034/) Numerical approximation and error · qf-mod50-numerr-affine-propagation · candidate ## 35. An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope Use slope=-1 and nonnegative absolute input error bound=10/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope). View answer and explanation(-1, 10/3, 10/3, affine_output_absolute_error_bound) Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (-1, 10/3, 10/3, affine_output_absolute_error_bound). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0035/) Numerical approximation and error · qf-mod50-numerr-affine-propagation · candidate ## 36. An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope Use slope=2 and nonnegative absolute input error bound=11/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope). View answer and explanation(2, 11/3, 22/3, affine_output_absolute_error_bound) Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (2, 11/3, 22/3, affine_output_absolute_error_bound). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0036/) Numerical approximation and error · qf-mod50-numerr-affine-propagation · candidate ## 37. An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope Use slope=1 and nonnegative absolute input error bound=12/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope). View answer and explanation(1, 4, 4, affine_output_absolute_error_bound) Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (1, 4, 4, affine_output_absolute_error_bound). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0037/) Numerical approximation and error · qf-mod50-numerr-affine-propagation · candidate ## 38. An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope Use slope=2 and nonnegative absolute input error bound=13/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope). View answer and explanation(2, 13/3, 26/3, affine_output_absolute_error_bound) Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (2, 13/3, 26/3, affine_output_absolute_error_bound). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0038/) Numerical approximation and error · qf-mod50-numerr-affine-propagation · candidate ## 39. An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope Use slope=3 and nonnegative absolute input error bound=14/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope). View answer and explanation(3, 14/3, 14, affine_output_absolute_error_bound) Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (3, 14/3, 14, affine_output_absolute_error_bound). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0039/) Numerical approximation and error · qf-mod50-numerr-affine-propagation · candidate ## 40. An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope Use slope=-3 and nonnegative absolute input error bound=15/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope). View answer and explanation(-3, 5, 15, affine_output_absolute_error_bound) Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (-3, 5, 15, affine_output_absolute_error_bound). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0040/) Numerical approximation and error · qf-mod50-numerr-affine-propagation · candidate ## 41. An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope Use slope=-2 and nonnegative absolute input error bound=16/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope). View answer and explanation(-2, 16/3, 32/3, affine_output_absolute_error_bound) Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (-2, 16/3, 32/3, affine_output_absolute_error_bound). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0041/) Numerical approximation and error · qf-mod50-numerr-affine-propagation · candidate ## 42. An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope Use slope=-1 and nonnegative absolute input error bound=17/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope). View answer and explanation(-1, 17/3, 17/3, affine_output_absolute_error_bound) Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (-1, 17/3, 17/3, affine_output_absolute_error_bound). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0042/) Numerical approximation and error · qf-mod50-numerr-affine-propagation · candidate ## 43. An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope Use slope=2 and nonnegative absolute input error bound=18/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope). View answer and explanation(2, 6, 12, affine_output_absolute_error_bound) Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (2, 6, 12, affine_output_absolute_error_bound). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0043/) Numerical approximation and error · qf-mod50-numerr-affine-propagation · candidate ## 44. An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope Use slope=1 and nonnegative absolute input error bound=19/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope). View answer and explanation(1, 19/3, 19/3, affine_output_absolute_error_bound) Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (1, 19/3, 19/3, affine_output_absolute_error_bound). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0044/) Numerical approximation and error · qf-mod50-numerr-affine-propagation · candidate ## 45. An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope Use slope=2 and nonnegative absolute input error bound=20/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope). View answer and explanation(2, 20/3, 40/3, affine_output_absolute_error_bound) Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (2, 20/3, 40/3, affine_output_absolute_error_bound). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0045/) Numerical approximation and error · qf-mod50-numerr-affine-propagation · candidate ## 46. An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope Use slope=3 and nonnegative absolute input error bound=21/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope). View answer and explanation(3, 7, 21, affine_output_absolute_error_bound) Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (3, 7, 21, affine_output_absolute_error_bound). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0046/) Numerical approximation and error · qf-mod50-numerr-affine-propagation · candidate ## 47. An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope Use slope=-3 and nonnegative absolute input error bound=22/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope). View answer and explanation(-3, 22/3, 22, affine_output_absolute_error_bound) Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (-3, 22/3, 22, affine_output_absolute_error_bound). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0047/) Numerical approximation and error · qf-mod50-numerr-affine-propagation · candidate ## 48. An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope Use slope=-2 and nonnegative absolute input error bound=23/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope). View answer and explanation(-2, 23/3, 46/3, affine_output_absolute_error_bound) Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (-2, 23/3, 46/3, affine_output_absolute_error_bound). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0048/) Numerical approximation and error · qf-mod50-numerr-affine-propagation · candidate ## 49. An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope Use slope=-1 and nonnegative absolute input error bound=24/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope). View answer and explanation(-1, 8, 8, affine_output_absolute_error_bound) Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (-1, 8, 8, affine_output_absolute_error_bound). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0049/) Numerical approximation and error · qf-mod50-numerr-affine-propagation · candidate ## 50. An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope Use slope=2 and nonnegative absolute input error bound=25/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope). View answer and explanation(2, 25/3, 50/3, affine_output_absolute_error_bound) Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (2, 25/3, 50/3, affine_output_absolute_error_bound). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0050/) Numerical approximation and error · qf-mod50-numerr-affine-propagation · candidate ## 51. The Same Residual Can Correspond to Different Forward Errors Use positive absolute residual=1, nonzero first slope=1 and nonzero second slope=2, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope). View answer and explanation(1, 1, 1, 2, 1/2, Yes, same_residual_can_have_different_forward_error_under_different_conditioning) Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (1, 1, 1, 2, 1/2, Yes, same_residual_can_have_different_forward_error_under_different_conditioning). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0051/) Numerical approximation and error · qf-mod50-numerr-conditioning-witness · candidate ## 52. The Same Residual Can Correspond to Different Forward Errors Use positive absolute residual=2, nonzero first slope=1 and nonzero second slope=3, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope). View answer and explanation(2, 1, 2, 3, 2/3, Yes, same_residual_can_have_different_forward_error_under_different_conditioning) Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (2, 1, 2, 3, 2/3, Yes, same_residual_can_have_different_forward_error_under_different_conditioning). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0052/) Numerical approximation and error · qf-mod50-numerr-conditioning-witness · candidate ## 53. The Same Residual Can Correspond to Different Forward Errors Use positive absolute residual=3, nonzero first slope=1 and nonzero second slope=4, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope). View answer and explanation(3, 1, 3, 4, 3/4, Yes, same_residual_can_have_different_forward_error_under_different_conditioning) Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (3, 1, 3, 4, 3/4, Yes, same_residual_can_have_different_forward_error_under_different_conditioning). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0053/) Numerical approximation and error · qf-mod50-numerr-conditioning-witness · candidate ## 54. The Same Residual Can Correspond to Different Forward Errors Use positive absolute residual=4, nonzero first slope=1 and nonzero second slope=5, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope). View answer and explanation(4, 1, 4, 5, 4/5, Yes, same_residual_can_have_different_forward_error_under_different_conditioning) Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (4, 1, 4, 5, 4/5, Yes, same_residual_can_have_different_forward_error_under_different_conditioning). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0054/) Numerical approximation and error · qf-mod50-numerr-conditioning-witness · candidate ## 55. The Same Residual Can Correspond to Different Forward Errors Use positive absolute residual=5, nonzero first slope=1 and nonzero second slope=6, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope). View answer and explanation(5, 1, 5, 6, 5/6, Yes, same_residual_can_have_different_forward_error_under_different_conditioning) Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (5, 1, 5, 6, 5/6, Yes, same_residual_can_have_different_forward_error_under_different_conditioning). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0055/) Numerical approximation and error · qf-mod50-numerr-conditioning-witness · candidate ## 56. The Same Residual Can Correspond to Different Forward Errors Use positive absolute residual=6, nonzero first slope=1 and nonzero second slope=2, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope). View answer and explanation(6, 1, 6, 2, 3, Yes, same_residual_can_have_different_forward_error_under_different_conditioning) Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (6, 1, 6, 2, 3, Yes, same_residual_can_have_different_forward_error_under_different_conditioning). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0056/) Numerical approximation and error · qf-mod50-numerr-conditioning-witness · candidate ## 57. The Same Residual Can Correspond to Different Forward Errors Use positive absolute residual=7, nonzero first slope=1 and nonzero second slope=3, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope). View answer and explanation(7, 1, 7, 3, 7/3, Yes, same_residual_can_have_different_forward_error_under_different_conditioning) Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (7, 1, 7, 3, 7/3, Yes, same_residual_can_have_different_forward_error_under_different_conditioning). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0057/) Numerical approximation and error · qf-mod50-numerr-conditioning-witness · candidate ## 58. The Same Residual Can Correspond to Different Forward Errors Use positive absolute residual=8, nonzero first slope=1 and nonzero second slope=4, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope). View answer and explanation(8, 1, 8, 4, 2, Yes, same_residual_can_have_different_forward_error_under_different_conditioning) Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (8, 1, 8, 4, 2, Yes, same_residual_can_have_different_forward_error_under_different_conditioning). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0058/) Numerical approximation and error · qf-mod50-numerr-conditioning-witness · candidate ## 59. The Same Residual Can Correspond to Different Forward Errors Use positive absolute residual=9, nonzero first slope=1 and nonzero second slope=5, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope). View answer and explanation(9, 1, 9, 5, 9/5, Yes, same_residual_can_have_different_forward_error_under_different_conditioning) Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (9, 1, 9, 5, 9/5, Yes, same_residual_can_have_different_forward_error_under_different_conditioning). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0059/) Numerical approximation and error · qf-mod50-numerr-conditioning-witness · candidate ## 60. The Same Residual Can Correspond to Different Forward Errors Use positive absolute residual=10, nonzero first slope=1 and nonzero second slope=6, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope). View answer and explanation(10, 1, 10, 6, 5/3, Yes, same_residual_can_have_different_forward_error_under_different_conditioning) Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (10, 1, 10, 6, 5/3, Yes, same_residual_can_have_different_forward_error_under_different_conditioning). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0060/) Numerical approximation and error · qf-mod50-numerr-conditioning-witness · candidate ## 61. The Same Residual Can Correspond to Different Forward Errors Use positive absolute residual=11, nonzero first slope=1 and nonzero second slope=2, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope). View answer and explanation(11, 1, 11, 2, 11/2, Yes, same_residual_can_have_different_forward_error_under_different_conditioning) Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (11, 1, 11, 2, 11/2, Yes, same_residual_can_have_different_forward_error_under_different_conditioning). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0061/) Numerical approximation and error · qf-mod50-numerr-conditioning-witness · candidate ## 62. The Same Residual Can Correspond to Different Forward Errors Use positive absolute residual=12, nonzero first slope=1 and nonzero second slope=3, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope). View answer and explanation(12, 1, 12, 3, 4, Yes, same_residual_can_have_different_forward_error_under_different_conditioning) Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (12, 1, 12, 3, 4, Yes, same_residual_can_have_different_forward_error_under_different_conditioning). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0062/) Numerical approximation and error · qf-mod50-numerr-conditioning-witness · candidate ## 63. The Same Residual Can Correspond to Different Forward Errors Use positive absolute residual=13, nonzero first slope=1 and nonzero second slope=4, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope). View answer and explanation(13, 1, 13, 4, 13/4, Yes, same_residual_can_have_different_forward_error_under_different_conditioning) Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (13, 1, 13, 4, 13/4, Yes, same_residual_can_have_different_forward_error_under_different_conditioning). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0063/) Numerical approximation and error · qf-mod50-numerr-conditioning-witness · candidate ## 64. The Same Residual Can Correspond to Different Forward Errors Use positive absolute residual=14, nonzero first slope=1 and nonzero second slope=5, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope). View answer and explanation(14, 1, 14, 5, 14/5, Yes, same_residual_can_have_different_forward_error_under_different_conditioning) Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (14, 1, 14, 5, 14/5, Yes, same_residual_can_have_different_forward_error_under_different_conditioning). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0064/) Numerical approximation and error · qf-mod50-numerr-conditioning-witness · candidate ## 65. The Same Residual Can Correspond to Different Forward Errors Use positive absolute residual=15, nonzero first slope=1 and nonzero second slope=6, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope). View answer and explanation(15, 1, 15, 6, 5/2, Yes, same_residual_can_have_different_forward_error_under_different_conditioning) Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (15, 1, 15, 6, 5/2, Yes, same_residual_can_have_different_forward_error_under_different_conditioning). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0065/) Numerical approximation and error · qf-mod50-numerr-conditioning-witness · candidate ## 66. The Same Residual Can Correspond to Different Forward Errors Use positive absolute residual=16, nonzero first slope=1 and nonzero second slope=2, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope). View answer and explanation(16, 1, 16, 2, 8, Yes, same_residual_can_have_different_forward_error_under_different_conditioning) Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (16, 1, 16, 2, 8, Yes, same_residual_can_have_different_forward_error_under_different_conditioning). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0066/) Numerical approximation and error · qf-mod50-numerr-conditioning-witness · candidate ## 67. The Same Residual Can Correspond to Different Forward Errors Use positive absolute residual=17, nonzero first slope=1 and nonzero second slope=3, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope). View answer and explanation(17, 1, 17, 3, 17/3, Yes, same_residual_can_have_different_forward_error_under_different_conditioning) Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (17, 1, 17, 3, 17/3, Yes, same_residual_can_have_different_forward_error_under_different_conditioning). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0067/) Numerical approximation and error · qf-mod50-numerr-conditioning-witness · candidate ## 68. The Same Residual Can Correspond to Different Forward Errors Use positive absolute residual=18, nonzero first slope=1 and nonzero second slope=4, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope). View answer and explanation(18, 1, 18, 4, 9/2, Yes, same_residual_can_have_different_forward_error_under_different_conditioning) Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (18, 1, 18, 4, 9/2, Yes, same_residual_can_have_different_forward_error_under_different_conditioning). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0068/) Numerical approximation and error · qf-mod50-numerr-conditioning-witness · candidate ## 69. The Same Residual Can Correspond to Different Forward Errors Use positive absolute residual=19, nonzero first slope=1 and nonzero second slope=5, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope). View answer and explanation(19, 1, 19, 5, 19/5, Yes, same_residual_can_have_different_forward_error_under_different_conditioning) Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (19, 1, 19, 5, 19/5, Yes, same_residual_can_have_different_forward_error_under_different_conditioning). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0069/) Numerical approximation and error · qf-mod50-numerr-conditioning-witness · candidate ## 70. The Same Residual Can Correspond to Different Forward Errors Use positive absolute residual=20, nonzero first slope=1 and nonzero second slope=6, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope). View answer and explanation(20, 1, 20, 6, 10/3, Yes, same_residual_can_have_different_forward_error_under_different_conditioning) Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (20, 1, 20, 6, 10/3, Yes, same_residual_can_have_different_forward_error_under_different_conditioning). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0070/) Numerical approximation and error · qf-mod50-numerr-conditioning-witness · candidate ## 71. The Same Residual Can Correspond to Different Forward Errors Use positive absolute residual=21, nonzero first slope=1 and nonzero second slope=2, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope). View answer and explanation(21, 1, 21, 2, 21/2, Yes, same_residual_can_have_different_forward_error_under_different_conditioning) Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (21, 1, 21, 2, 21/2, Yes, same_residual_can_have_different_forward_error_under_different_conditioning). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0071/) Numerical approximation and error · qf-mod50-numerr-conditioning-witness · candidate ## 72. The Same Residual Can Correspond to Different Forward Errors Use positive absolute residual=22, nonzero first slope=1 and nonzero second slope=3, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope). View answer and explanation(22, 1, 22, 3, 22/3, Yes, same_residual_can_have_different_forward_error_under_different_conditioning) Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (22, 1, 22, 3, 22/3, Yes, same_residual_can_have_different_forward_error_under_different_conditioning). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0072/) Numerical approximation and error · qf-mod50-numerr-conditioning-witness · candidate ## 73. The Same Residual Can Correspond to Different Forward Errors Use positive absolute residual=23, nonzero first slope=1 and nonzero second slope=4, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope). View answer and explanation(23, 1, 23, 4, 23/4, Yes, same_residual_can_have_different_forward_error_under_different_conditioning) Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (23, 1, 23, 4, 23/4, Yes, same_residual_can_have_different_forward_error_under_different_conditioning). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0073/) Numerical approximation and error · qf-mod50-numerr-conditioning-witness · candidate ## 74. The Same Residual Can Correspond to Different Forward Errors Use positive absolute residual=24, nonzero first slope=1 and nonzero second slope=5, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope). View answer and explanation(24, 1, 24, 5, 24/5, Yes, same_residual_can_have_different_forward_error_under_different_conditioning) Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (24, 1, 24, 5, 24/5, Yes, same_residual_can_have_different_forward_error_under_different_conditioning). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0074/) Numerical approximation and error · qf-mod50-numerr-conditioning-witness · candidate ## 75. The Same Residual Can Correspond to Different Forward Errors Use positive absolute residual=25, nonzero first slope=1 and nonzero second slope=6, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope). View answer and explanation(25, 1, 25, 6, 25/6, Yes, same_residual_can_have_different_forward_error_under_different_conditioning) Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (25, 1, 25, 6, 25/6, Yes, same_residual_can_have_different_forward_error_under_different_conditioning). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0075/) Numerical approximation and error · qf-mod50-numerr-conditioning-witness · candidate ## 76. Zero Reference and Equal Tolerances Require Explicit Error Conventions Use mode=zero_reference, complete model={"approximation":1}. Give (mode, approximation, reference_description, relative_error_status, relative_error_defined_claim). The four modes have5/5/5/7 fields. Strict and inclusive modes have equal arity but opposite results at equality. Relative error is undefined at a zero reference. View answer and explanation(zero_reference, 1, reference=0, relative_error=undefined, No) Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (zero_reference, 1, reference=0, relative_error=undefined, No). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0076/) Numerical approximation and error · qf-mod50-numerr-boundary · candidate ## 77. Zero Reference and Equal Tolerances Require Explicit Error Conventions Use mode=inclusive_tolerance, complete model={"error":[2,2],"tolerance":[2,2]}. Give (mode, absolute_error, tolerance, inclusive_test_result, equality_convention). The four modes have5/5/5/7 fields. Strict and inclusive modes have equal arity but opposite results at equality. Relative error is undefined at a zero reference. View answer and explanation(inclusive_tolerance, 1, 1, Yes, error<=tolerance_is_true_at_equality) Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (inclusive_tolerance, 1, 1, Yes, error<=tolerance_is_true_at_equality). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r05-0077/) Numerical approximation and error · qf-mod50-numerr-boundary · candidate ## 78. Zero Reference and Equal Tolerances Require Explicit Error Conventions Use mode=strict_tolerance, complete model={"error":[3,2],"tolerance":[3,2]}. Give (mode, absolute_error, tolerance, strict_test_result, equality_convention). The four modes have5/5/5/7 fields. Strict and inclusive modes have equal arity but opposite results at equality. Relative error is undefined at a zero reference. View answer and explanation(strict_tolerance, 3/2, 3/2, No, error