# Root-First Newton Updates, Exact Finite Orbits and Convergence Boundaries

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-r01-newton-method-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 Exact Root Stops before Any Newton Division

Coefficients in low-to-high degree order are=[-1, 0, 1]; all iterations use exact rational arithmetic. The current iterate is x=2. Give (coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention).

Test f(x)=0 first and stop, even when the derivative is also0; do not compute0/0. A nonroot with zero derivative has no Newton update. Coefficients run from low to high degree.

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

Evaluate f and f_prime exactly. Test f(x)=0 first; a root stops with next=null even when its derivative is0. Otherwise require a nonzero derivative and use the tangent-line zero x−f/f_prime. This is a qualified step, not a convergence or global-root certificate. The exact result, retaining the requested order, is ([-1, 0, 1], 2, 3, 4, updated, 5/4, root_test_precedes_division).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0001/)

Newton method for root finding · qf-mod50-newton-step · candidate





## 2. An Exact Root Stops before Any Newton Division

Coefficients in low-to-high degree order are=[-2, 0, 1]; all iterations use exact rational arithmetic. The current iterate is x=3. Give (coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention).

Test f(x)=0 first and stop, even when the derivative is also0; do not compute0/0. A nonroot with zero derivative has no Newton update. Coefficients run from low to high degree.

View answer and explanation([-2, 0, 1], 3, 7, 6, updated, 11/6, root_test_precedes_division)

Evaluate f and f_prime exactly. Test f(x)=0 first; a root stops with next=null even when its derivative is0. Otherwise require a nonzero derivative and use the tangent-line zero x−f/f_prime. This is a qualified step, not a convergence or global-root certificate. The exact result, retaining the requested order, is ([-2, 0, 1], 3, 7, 6, updated, 11/6, root_test_precedes_division).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0002/)

Newton method for root finding · qf-mod50-newton-step · candidate





## 3. An Exact Root Stops before Any Newton Division

Coefficients in low-to-high degree order are=[-3, 0, 1]; all iterations use exact rational arithmetic. The current iterate is x=4. Give (coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention).

Test f(x)=0 first and stop, even when the derivative is also0; do not compute0/0. A nonroot with zero derivative has no Newton update. Coefficients run from low to high degree.

View answer and explanation([-3, 0, 1], 4, 13, 8, updated, 19/8, root_test_precedes_division)

Evaluate f and f_prime exactly. Test f(x)=0 first; a root stops with next=null even when its derivative is0. Otherwise require a nonzero derivative and use the tangent-line zero x−f/f_prime. This is a qualified step, not a convergence or global-root certificate. The exact result, retaining the requested order, is ([-3, 0, 1], 4, 13, 8, updated, 19/8, root_test_precedes_division).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0003/)

Newton method for root finding · qf-mod50-newton-step · candidate





## 4. An Exact Root Stops before Any Newton Division

Coefficients in low-to-high degree order are=[-4, 0, 1]; all iterations use exact rational arithmetic. The current iterate is x=5. Give (coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention).

Test f(x)=0 first and stop, even when the derivative is also0; do not compute0/0. A nonroot with zero derivative has no Newton update. Coefficients run from low to high degree.

View answer and explanation([-4, 0, 1], 5, 21, 10, updated, 29/10, root_test_precedes_division)

Evaluate f and f_prime exactly. Test f(x)=0 first; a root stops with next=null even when its derivative is0. Otherwise require a nonzero derivative and use the tangent-line zero x−f/f_prime. This is a qualified step, not a convergence or global-root certificate. The exact result, retaining the requested order, is ([-4, 0, 1], 5, 21, 10, updated, 29/10, root_test_precedes_division).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0004/)

Newton method for root finding · qf-mod50-newton-step · candidate





## 5. An Exact Root Stops before Any Newton Division

Coefficients in low-to-high degree order are=[-5, 0, 1]; all iterations use exact rational arithmetic. The current iterate is x=1. Give (coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention).

Test f(x)=0 first and stop, even when the derivative is also0; do not compute0/0. A nonroot with zero derivative has no Newton update. Coefficients run from low to high degree.

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

Evaluate f and f_prime exactly. Test f(x)=0 first; a root stops with next=null even when its derivative is0. Otherwise require a nonzero derivative and use the tangent-line zero x−f/f_prime. This is a qualified step, not a convergence or global-root certificate. The exact result, retaining the requested order, is ([-5, 0, 1], 1, -4, 2, updated, 3, root_test_precedes_division).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0005/)

Newton method for root finding · qf-mod50-newton-step · candidate





## 6. An Exact Root Stops before Any Newton Division

Coefficients in low-to-high degree order are=[-6, 0, 1]; all iterations use exact rational arithmetic. The current iterate is x=2. Give (coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention).

Test f(x)=0 first and stop, even when the derivative is also0; do not compute0/0. A nonroot with zero derivative has no Newton update. Coefficients run from low to high degree.

View answer and explanation([-6, 0, 1], 2, -2, 4, updated, 5/2, root_test_precedes_division)

Evaluate f and f_prime exactly. Test f(x)=0 first; a root stops with next=null even when its derivative is0. Otherwise require a nonzero derivative and use the tangent-line zero x−f/f_prime. This is a qualified step, not a convergence or global-root certificate. The exact result, retaining the requested order, is ([-6, 0, 1], 2, -2, 4, updated, 5/2, root_test_precedes_division).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0006/)

Newton method for root finding · qf-mod50-newton-step · candidate





## 7. An Exact Root Stops before Any Newton Division

Coefficients in low-to-high degree order are=[-7, 0, 1]; all iterations use exact rational arithmetic. The current iterate is x=3. Give (coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention).

Test f(x)=0 first and stop, even when the derivative is also0; do not compute0/0. A nonroot with zero derivative has no Newton update. Coefficients run from low to high degree.

View answer and explanation([-7, 0, 1], 3, 2, 6, updated, 8/3, root_test_precedes_division)

Evaluate f and f_prime exactly. Test f(x)=0 first; a root stops with next=null even when its derivative is0. Otherwise require a nonzero derivative and use the tangent-line zero x−f/f_prime. This is a qualified step, not a convergence or global-root certificate. The exact result, retaining the requested order, is ([-7, 0, 1], 3, 2, 6, updated, 8/3, root_test_precedes_division).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0007/)

Newton method for root finding · qf-mod50-newton-step · candidate





## 8. An Exact Root Stops before Any Newton Division

Coefficients in low-to-high degree order are=[-8, 0, 1]; all iterations use exact rational arithmetic. The current iterate is x=4. Give (coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention).

Test f(x)=0 first and stop, even when the derivative is also0; do not compute0/0. A nonroot with zero derivative has no Newton update. Coefficients run from low to high degree.

View answer and explanation([-8, 0, 1], 4, 8, 8, updated, 3, root_test_precedes_division)

Evaluate f and f_prime exactly. Test f(x)=0 first; a root stops with next=null even when its derivative is0. Otherwise require a nonzero derivative and use the tangent-line zero x−f/f_prime. This is a qualified step, not a convergence or global-root certificate. The exact result, retaining the requested order, is ([-8, 0, 1], 4, 8, 8, updated, 3, root_test_precedes_division).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0008/)

Newton method for root finding · qf-mod50-newton-step · candidate





## 9. An Exact Root Stops before Any Newton Division

Coefficients in low-to-high degree order are=[-9, 0, 1]; all iterations use exact rational arithmetic. The current iterate is x=5. Give (coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention).

Test f(x)=0 first and stop, even when the derivative is also0; do not compute0/0. A nonroot with zero derivative has no Newton update. Coefficients run from low to high degree.

View answer and explanation([-9, 0, 1], 5, 16, 10, updated, 17/5, root_test_precedes_division)

Evaluate f and f_prime exactly. Test f(x)=0 first; a root stops with next=null even when its derivative is0. Otherwise require a nonzero derivative and use the tangent-line zero x−f/f_prime. This is a qualified step, not a convergence or global-root certificate. The exact result, retaining the requested order, is ([-9, 0, 1], 5, 16, 10, updated, 17/5, root_test_precedes_division).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0009/)

Newton method for root finding · qf-mod50-newton-step · candidate





## 10. An Exact Root Stops before Any Newton Division

Coefficients in low-to-high degree order are=[-10, 0, 1]; all iterations use exact rational arithmetic. The current iterate is x=1. Give (coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention).

Test f(x)=0 first and stop, even when the derivative is also0; do not compute0/0. A nonroot with zero derivative has no Newton update. Coefficients run from low to high degree.

View answer and explanation([-10, 0, 1], 1, -9, 2, updated, 11/2, root_test_precedes_division)

Evaluate f and f_prime exactly. Test f(x)=0 first; a root stops with next=null even when its derivative is0. Otherwise require a nonzero derivative and use the tangent-line zero x−f/f_prime. This is a qualified step, not a convergence or global-root certificate. The exact result, retaining the requested order, is ([-10, 0, 1], 1, -9, 2, updated, 11/2, root_test_precedes_division).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0010/)

Newton method for root finding · qf-mod50-newton-step · candidate





## 11. An Exact Root Stops before Any Newton Division

Coefficients in low-to-high degree order are=[-11, 0, 1]; all iterations use exact rational arithmetic. The current iterate is x=2. Give (coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention).

Test f(x)=0 first and stop, even when the derivative is also0; do not compute0/0. A nonroot with zero derivative has no Newton update. Coefficients run from low to high degree.

View answer and explanation([-11, 0, 1], 2, -7, 4, updated, 15/4, root_test_precedes_division)

Evaluate f and f_prime exactly. Test f(x)=0 first; a root stops with next=null even when its derivative is0. Otherwise require a nonzero derivative and use the tangent-line zero x−f/f_prime. This is a qualified step, not a convergence or global-root certificate. The exact result, retaining the requested order, is ([-11, 0, 1], 2, -7, 4, updated, 15/4, root_test_precedes_division).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0011/)

Newton method for root finding · qf-mod50-newton-step · candidate





## 12. An Exact Root Stops before Any Newton Division

Coefficients in low-to-high degree order are=[-12, 0, 1]; all iterations use exact rational arithmetic. The current iterate is x=3. Give (coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention).

Test f(x)=0 first and stop, even when the derivative is also0; do not compute0/0. A nonroot with zero derivative has no Newton update. Coefficients run from low to high degree.

View answer and explanation([-12, 0, 1], 3, -3, 6, updated, 7/2, root_test_precedes_division)

Evaluate f and f_prime exactly. Test f(x)=0 first; a root stops with next=null even when its derivative is0. Otherwise require a nonzero derivative and use the tangent-line zero x−f/f_prime. This is a qualified step, not a convergence or global-root certificate. The exact result, retaining the requested order, is ([-12, 0, 1], 3, -3, 6, updated, 7/2, root_test_precedes_division).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0012/)

Newton method for root finding · qf-mod50-newton-step · candidate





## 13. An Exact Root Stops before Any Newton Division

Coefficients in low-to-high degree order are=[-13, 0, 1]; all iterations use exact rational arithmetic. The current iterate is x=4. Give (coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention).

Test f(x)=0 first and stop, even when the derivative is also0; do not compute0/0. A nonroot with zero derivative has no Newton update. Coefficients run from low to high degree.

View answer and explanation([-13, 0, 1], 4, 3, 8, updated, 29/8, root_test_precedes_division)

Evaluate f and f_prime exactly. Test f(x)=0 first; a root stops with next=null even when its derivative is0. Otherwise require a nonzero derivative and use the tangent-line zero x−f/f_prime. This is a qualified step, not a convergence or global-root certificate. The exact result, retaining the requested order, is ([-13, 0, 1], 4, 3, 8, updated, 29/8, root_test_precedes_division).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0013/)

Newton method for root finding · qf-mod50-newton-step · candidate





## 14. An Exact Root Stops before Any Newton Division

Coefficients in low-to-high degree order are=[-14, 0, 1]; all iterations use exact rational arithmetic. The current iterate is x=5. Give (coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention).

Test f(x)=0 first and stop, even when the derivative is also0; do not compute0/0. A nonroot with zero derivative has no Newton update. Coefficients run from low to high degree.

View answer and explanation([-14, 0, 1], 5, 11, 10, updated, 39/10, root_test_precedes_division)

Evaluate f and f_prime exactly. Test f(x)=0 first; a root stops with next=null even when its derivative is0. Otherwise require a nonzero derivative and use the tangent-line zero x−f/f_prime. This is a qualified step, not a convergence or global-root certificate. The exact result, retaining the requested order, is ([-14, 0, 1], 5, 11, 10, updated, 39/10, root_test_precedes_division).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0014/)

Newton method for root finding · qf-mod50-newton-step · candidate





## 15. An Exact Root Stops before Any Newton Division

Coefficients in low-to-high degree order are=[-15, 0, 1]; all iterations use exact rational arithmetic. The current iterate is x=1. Give (coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention).

Test f(x)=0 first and stop, even when the derivative is also0; do not compute0/0. A nonroot with zero derivative has no Newton update. Coefficients run from low to high degree.

View answer and explanation([-15, 0, 1], 1, -14, 2, updated, 8, root_test_precedes_division)

Evaluate f and f_prime exactly. Test f(x)=0 first; a root stops with next=null even when its derivative is0. Otherwise require a nonzero derivative and use the tangent-line zero x−f/f_prime. This is a qualified step, not a convergence or global-root certificate. The exact result, retaining the requested order, is ([-15, 0, 1], 1, -14, 2, updated, 8, root_test_precedes_division).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0015/)

Newton method for root finding · qf-mod50-newton-step · candidate





## 16. An Exact Root Stops before Any Newton Division

Coefficients in low-to-high degree order are=[-16, 0, 1]; all iterations use exact rational arithmetic. The current iterate is x=2. Give (coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention).

Test f(x)=0 first and stop, even when the derivative is also0; do not compute0/0. A nonroot with zero derivative has no Newton update. Coefficients run from low to high degree.

View answer and explanation([-16, 0, 1], 2, -12, 4, updated, 5, root_test_precedes_division)

Evaluate f and f_prime exactly. Test f(x)=0 first; a root stops with next=null even when its derivative is0. Otherwise require a nonzero derivative and use the tangent-line zero x−f/f_prime. This is a qualified step, not a convergence or global-root certificate. The exact result, retaining the requested order, is ([-16, 0, 1], 2, -12, 4, updated, 5, root_test_precedes_division).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0016/)

Newton method for root finding · qf-mod50-newton-step · candidate





## 17. An Exact Root Stops before Any Newton Division

Coefficients in low-to-high degree order are=[-17, 0, 1]; all iterations use exact rational arithmetic. The current iterate is x=3. Give (coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention).

Test f(x)=0 first and stop, even when the derivative is also0; do not compute0/0. A nonroot with zero derivative has no Newton update. Coefficients run from low to high degree.

View answer and explanation([-17, 0, 1], 3, -8, 6, updated, 13/3, root_test_precedes_division)

Evaluate f and f_prime exactly. Test f(x)=0 first; a root stops with next=null even when its derivative is0. Otherwise require a nonzero derivative and use the tangent-line zero x−f/f_prime. This is a qualified step, not a convergence or global-root certificate. The exact result, retaining the requested order, is ([-17, 0, 1], 3, -8, 6, updated, 13/3, root_test_precedes_division).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0017/)

Newton method for root finding · qf-mod50-newton-step · candidate





## 18. An Exact Root Stops before Any Newton Division

Coefficients in low-to-high degree order are=[-18, 0, 1]; all iterations use exact rational arithmetic. The current iterate is x=4. Give (coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention).

Test f(x)=0 first and stop, even when the derivative is also0; do not compute0/0. A nonroot with zero derivative has no Newton update. Coefficients run from low to high degree.

View answer and explanation([-18, 0, 1], 4, -2, 8, updated, 17/4, root_test_precedes_division)

Evaluate f and f_prime exactly. Test f(x)=0 first; a root stops with next=null even when its derivative is0. Otherwise require a nonzero derivative and use the tangent-line zero x−f/f_prime. This is a qualified step, not a convergence or global-root certificate. The exact result, retaining the requested order, is ([-18, 0, 1], 4, -2, 8, updated, 17/4, root_test_precedes_division).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0018/)

Newton method for root finding · qf-mod50-newton-step · candidate





## 19. An Exact Root Stops before Any Newton Division

Coefficients in low-to-high degree order are=[-19, 0, 1]; all iterations use exact rational arithmetic. The current iterate is x=5. Give (coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention).

Test f(x)=0 first and stop, even when the derivative is also0; do not compute0/0. A nonroot with zero derivative has no Newton update. Coefficients run from low to high degree.

View answer and explanation([-19, 0, 1], 5, 6, 10, updated, 22/5, root_test_precedes_division)

Evaluate f and f_prime exactly. Test f(x)=0 first; a root stops with next=null even when its derivative is0. Otherwise require a nonzero derivative and use the tangent-line zero x−f/f_prime. This is a qualified step, not a convergence or global-root certificate. The exact result, retaining the requested order, is ([-19, 0, 1], 5, 6, 10, updated, 22/5, root_test_precedes_division).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0019/)

Newton method for root finding · qf-mod50-newton-step · candidate





## 20. An Exact Root Stops before Any Newton Division

Coefficients in low-to-high degree order are=[-20, 0, 1]; all iterations use exact rational arithmetic. The current iterate is x=1. Give (coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention).

Test f(x)=0 first and stop, even when the derivative is also0; do not compute0/0. A nonroot with zero derivative has no Newton update. Coefficients run from low to high degree.

View answer and explanation([-20, 0, 1], 1, -19, 2, updated, 21/2, root_test_precedes_division)

Evaluate f and f_prime exactly. Test f(x)=0 first; a root stops with next=null even when its derivative is0. Otherwise require a nonzero derivative and use the tangent-line zero x−f/f_prime. This is a qualified step, not a convergence or global-root certificate. The exact result, retaining the requested order, is ([-20, 0, 1], 1, -19, 2, updated, 21/2, root_test_precedes_division).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0020/)

Newton method for root finding · qf-mod50-newton-step · candidate





## 21. An Exact Root Stops before Any Newton Division

Coefficients in low-to-high degree order are=[-21, 0, 1]; all iterations use exact rational arithmetic. The current iterate is x=2. Give (coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention).

Test f(x)=0 first and stop, even when the derivative is also0; do not compute0/0. A nonroot with zero derivative has no Newton update. Coefficients run from low to high degree.

View answer and explanation([-21, 0, 1], 2, -17, 4, updated, 25/4, root_test_precedes_division)

Evaluate f and f_prime exactly. Test f(x)=0 first; a root stops with next=null even when its derivative is0. Otherwise require a nonzero derivative and use the tangent-line zero x−f/f_prime. This is a qualified step, not a convergence or global-root certificate. The exact result, retaining the requested order, is ([-21, 0, 1], 2, -17, 4, updated, 25/4, root_test_precedes_division).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0021/)

Newton method for root finding · qf-mod50-newton-step · candidate





## 22. An Exact Root Stops before Any Newton Division

Coefficients in low-to-high degree order are=[-22, 0, 1]; all iterations use exact rational arithmetic. The current iterate is x=3. Give (coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention).

Test f(x)=0 first and stop, even when the derivative is also0; do not compute0/0. A nonroot with zero derivative has no Newton update. Coefficients run from low to high degree.

View answer and explanation([-22, 0, 1], 3, -13, 6, updated, 31/6, root_test_precedes_division)

Evaluate f and f_prime exactly. Test f(x)=0 first; a root stops with next=null even when its derivative is0. Otherwise require a nonzero derivative and use the tangent-line zero x−f/f_prime. This is a qualified step, not a convergence or global-root certificate. The exact result, retaining the requested order, is ([-22, 0, 1], 3, -13, 6, updated, 31/6, root_test_precedes_division).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0022/)

Newton method for root finding · qf-mod50-newton-step · candidate





## 23. An Exact Root Stops before Any Newton Division

Coefficients in low-to-high degree order are=[-23, 0, 1]; all iterations use exact rational arithmetic. The current iterate is x=4. Give (coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention).

Test f(x)=0 first and stop, even when the derivative is also0; do not compute0/0. A nonroot with zero derivative has no Newton update. Coefficients run from low to high degree.

View answer and explanation([-23, 0, 1], 4, -7, 8, updated, 39/8, root_test_precedes_division)

Evaluate f and f_prime exactly. Test f(x)=0 first; a root stops with next=null even when its derivative is0. Otherwise require a nonzero derivative and use the tangent-line zero x−f/f_prime. This is a qualified step, not a convergence or global-root certificate. The exact result, retaining the requested order, is ([-23, 0, 1], 4, -7, 8, updated, 39/8, root_test_precedes_division).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0023/)

Newton method for root finding · qf-mod50-newton-step · candidate





## 24. An Exact Root Stops before Any Newton Division

Coefficients in low-to-high degree order are=[-24, 0, 1]; all iterations use exact rational arithmetic. The current iterate is x=5. Give (coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention).

Test f(x)=0 first and stop, even when the derivative is also0; do not compute0/0. A nonroot with zero derivative has no Newton update. Coefficients run from low to high degree.

View answer and explanation([-24, 0, 1], 5, 1, 10, updated, 49/10, root_test_precedes_division)

Evaluate f and f_prime exactly. Test f(x)=0 first; a root stops with next=null even when its derivative is0. Otherwise require a nonzero derivative and use the tangent-line zero x−f/f_prime. This is a qualified step, not a convergence or global-root certificate. The exact result, retaining the requested order, is ([-24, 0, 1], 5, 1, 10, updated, 49/10, root_test_precedes_division).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0024/)

Newton method for root finding · qf-mod50-newton-step · candidate





## 25. An Exact Root Stops before Any Newton Division

Coefficients in low-to-high degree order are=[-25, 0, 1]; all iterations use exact rational arithmetic. The current iterate is x=1. Give (coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention).

Test f(x)=0 first and stop, even when the derivative is also0; do not compute0/0. A nonroot with zero derivative has no Newton update. Coefficients run from low to high degree.

View answer and explanation([-25, 0, 1], 1, -24, 2, updated, 13, root_test_precedes_division)

Evaluate f and f_prime exactly. Test f(x)=0 first; a root stops with next=null even when its derivative is0. Otherwise require a nonzero derivative and use the tangent-line zero x−f/f_prime. This is a qualified step, not a convergence or global-root certificate. The exact result, retaining the requested order, is ([-25, 0, 1], 1, -24, 2, updated, 13, root_test_precedes_division).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0025/)

Newton method for root finding · qf-mod50-newton-step · candidate





## 26. A Finite Newton Orbit Records Steps without Certifying Convergence

Coefficients in low-to-high degree order are=[-1, 0, 1]; all iterations use exact rational arithmetic. Use initial iterate=2 and attempted-step budget=1. Give (coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope).

step_budget bounds attempted steps; an exact root may stop the records early. A finite trace or residual does not prove general convergence or a rate.

View answer and explanation([-1, 0, 1], 1, [[0, 2, 3, 4, updated, 5/4]], step_budget_exhausted, 5/4, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)

Recalculate f, f_prime and every qualified update. Each record gives index, current x, f, f_prime, status and next or null. Exact roots stop early; a nonroot zero derivative is rejected. A finite residual pattern proves neither a general limit nor quadratic convergence, a global basin or a forward-error bound. The exact result, retaining the requested order, is ([-1, 0, 1], 1, [[0, 2, 3, 4, updated, 5/4]], step_budget_exhausted, 5/4, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0026/)

Newton method for root finding · qf-mod50-newton-finite-orbit · candidate





## 27. A Finite Newton Orbit Records Steps without Certifying Convergence

Coefficients in low-to-high degree order are=[-2, 0, 1]; all iterations use exact rational arithmetic. Use initial iterate=3 and attempted-step budget=1. Give (coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope).

step_budget bounds attempted steps; an exact root may stop the records early. A finite trace or residual does not prove general convergence or a rate.

View answer and explanation([-2, 0, 1], 1, [[0, 3, 7, 6, updated, 11/6]], step_budget_exhausted, 11/6, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)

Recalculate f, f_prime and every qualified update. Each record gives index, current x, f, f_prime, status and next or null. Exact roots stop early; a nonroot zero derivative is rejected. A finite residual pattern proves neither a general limit nor quadratic convergence, a global basin or a forward-error bound. The exact result, retaining the requested order, is ([-2, 0, 1], 1, [[0, 3, 7, 6, updated, 11/6]], step_budget_exhausted, 11/6, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0027/)

Newton method for root finding · qf-mod50-newton-finite-orbit · candidate





## 28. A Finite Newton Orbit Records Steps without Certifying Convergence

Coefficients in low-to-high degree order are=[-3, 0, 1]; all iterations use exact rational arithmetic. Use initial iterate=4 and attempted-step budget=1. Give (coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope).

step_budget bounds attempted steps; an exact root may stop the records early. A finite trace or residual does not prove general convergence or a rate.

View answer and explanation([-3, 0, 1], 1, [[0, 4, 13, 8, updated, 19/8]], step_budget_exhausted, 19/8, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)

Recalculate f, f_prime and every qualified update. Each record gives index, current x, f, f_prime, status and next or null. Exact roots stop early; a nonroot zero derivative is rejected. A finite residual pattern proves neither a general limit nor quadratic convergence, a global basin or a forward-error bound. The exact result, retaining the requested order, is ([-3, 0, 1], 1, [[0, 4, 13, 8, updated, 19/8]], step_budget_exhausted, 19/8, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0028/)

Newton method for root finding · qf-mod50-newton-finite-orbit · candidate





## 29. A Finite Newton Orbit Records Steps without Certifying Convergence

Coefficients in low-to-high degree order are=[-4, 0, 1]; all iterations use exact rational arithmetic. Use initial iterate=5 and attempted-step budget=1. Give (coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope).

step_budget bounds attempted steps; an exact root may stop the records early. A finite trace or residual does not prove general convergence or a rate.

View answer and explanation([-4, 0, 1], 1, [[0, 5, 21, 10, updated, 29/10]], step_budget_exhausted, 29/10, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)

Recalculate f, f_prime and every qualified update. Each record gives index, current x, f, f_prime, status and next or null. Exact roots stop early; a nonroot zero derivative is rejected. A finite residual pattern proves neither a general limit nor quadratic convergence, a global basin or a forward-error bound. The exact result, retaining the requested order, is ([-4, 0, 1], 1, [[0, 5, 21, 10, updated, 29/10]], step_budget_exhausted, 29/10, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0029/)

Newton method for root finding · qf-mod50-newton-finite-orbit · candidate





## 30. A Finite Newton Orbit Records Steps without Certifying Convergence

Coefficients in low-to-high degree order are=[-5, 0, 1]; all iterations use exact rational arithmetic. Use initial iterate=1 and attempted-step budget=1. Give (coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope).

step_budget bounds attempted steps; an exact root may stop the records early. A finite trace or residual does not prove general convergence or a rate.

View answer and explanation([-5, 0, 1], 1, [[0, 1, -4, 2, updated, 3]], step_budget_exhausted, 3, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)

Recalculate f, f_prime and every qualified update. Each record gives index, current x, f, f_prime, status and next or null. Exact roots stop early; a nonroot zero derivative is rejected. A finite residual pattern proves neither a general limit nor quadratic convergence, a global basin or a forward-error bound. The exact result, retaining the requested order, is ([-5, 0, 1], 1, [[0, 1, -4, 2, updated, 3]], step_budget_exhausted, 3, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0030/)

Newton method for root finding · qf-mod50-newton-finite-orbit · candidate





## 31. A Finite Newton Orbit Records Steps without Certifying Convergence

Coefficients in low-to-high degree order are=[-6, 0, 1]; all iterations use exact rational arithmetic. Use initial iterate=2 and attempted-step budget=1. Give (coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope).

step_budget bounds attempted steps; an exact root may stop the records early. A finite trace or residual does not prove general convergence or a rate.

View answer and explanation([-6, 0, 1], 1, [[0, 2, -2, 4, updated, 5/2]], step_budget_exhausted, 5/2, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)

Recalculate f, f_prime and every qualified update. Each record gives index, current x, f, f_prime, status and next or null. Exact roots stop early; a nonroot zero derivative is rejected. A finite residual pattern proves neither a general limit nor quadratic convergence, a global basin or a forward-error bound. The exact result, retaining the requested order, is ([-6, 0, 1], 1, [[0, 2, -2, 4, updated, 5/2]], step_budget_exhausted, 5/2, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0031/)

Newton method for root finding · qf-mod50-newton-finite-orbit · candidate





## 32. A Finite Newton Orbit Records Steps without Certifying Convergence

Coefficients in low-to-high degree order are=[-7, 0, 1]; all iterations use exact rational arithmetic. Use initial iterate=3 and attempted-step budget=1. Give (coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope).

step_budget bounds attempted steps; an exact root may stop the records early. A finite trace or residual does not prove general convergence or a rate.

View answer and explanation([-7, 0, 1], 1, [[0, 3, 2, 6, updated, 8/3]], step_budget_exhausted, 8/3, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)

Recalculate f, f_prime and every qualified update. Each record gives index, current x, f, f_prime, status and next or null. Exact roots stop early; a nonroot zero derivative is rejected. A finite residual pattern proves neither a general limit nor quadratic convergence, a global basin or a forward-error bound. The exact result, retaining the requested order, is ([-7, 0, 1], 1, [[0, 3, 2, 6, updated, 8/3]], step_budget_exhausted, 8/3, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0032/)

Newton method for root finding · qf-mod50-newton-finite-orbit · candidate





## 33. A Finite Newton Orbit Records Steps without Certifying Convergence

Coefficients in low-to-high degree order are=[-8, 0, 1]; all iterations use exact rational arithmetic. Use initial iterate=4 and attempted-step budget=1. Give (coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope).

step_budget bounds attempted steps; an exact root may stop the records early. A finite trace or residual does not prove general convergence or a rate.

View answer and explanation([-8, 0, 1], 1, [[0, 4, 8, 8, updated, 3]], step_budget_exhausted, 3, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)

Recalculate f, f_prime and every qualified update. Each record gives index, current x, f, f_prime, status and next or null. Exact roots stop early; a nonroot zero derivative is rejected. A finite residual pattern proves neither a general limit nor quadratic convergence, a global basin or a forward-error bound. The exact result, retaining the requested order, is ([-8, 0, 1], 1, [[0, 4, 8, 8, updated, 3]], step_budget_exhausted, 3, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0033/)

Newton method for root finding · qf-mod50-newton-finite-orbit · candidate





## 34. A Finite Newton Orbit Records Steps without Certifying Convergence

Coefficients in low-to-high degree order are=[-9, 0, 1]; all iterations use exact rational arithmetic. Use initial iterate=5 and attempted-step budget=1. Give (coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope).

step_budget bounds attempted steps; an exact root may stop the records early. A finite trace or residual does not prove general convergence or a rate.

View answer and explanation([-9, 0, 1], 1, [[0, 5, 16, 10, updated, 17/5]], step_budget_exhausted, 17/5, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)

Recalculate f, f_prime and every qualified update. Each record gives index, current x, f, f_prime, status and next or null. Exact roots stop early; a nonroot zero derivative is rejected. A finite residual pattern proves neither a general limit nor quadratic convergence, a global basin or a forward-error bound. The exact result, retaining the requested order, is ([-9, 0, 1], 1, [[0, 5, 16, 10, updated, 17/5]], step_budget_exhausted, 17/5, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0034/)

Newton method for root finding · qf-mod50-newton-finite-orbit · candidate





## 35. A Finite Newton Orbit Records Steps without Certifying Convergence

Coefficients in low-to-high degree order are=[-10, 0, 1]; all iterations use exact rational arithmetic. Use initial iterate=1 and attempted-step budget=1. Give (coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope).

step_budget bounds attempted steps; an exact root may stop the records early. A finite trace or residual does not prove general convergence or a rate.

View answer and explanation([-10, 0, 1], 1, [[0, 1, -9, 2, updated, 11/2]], step_budget_exhausted, 11/2, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)

Recalculate f, f_prime and every qualified update. Each record gives index, current x, f, f_prime, status and next or null. Exact roots stop early; a nonroot zero derivative is rejected. A finite residual pattern proves neither a general limit nor quadratic convergence, a global basin or a forward-error bound. The exact result, retaining the requested order, is ([-10, 0, 1], 1, [[0, 1, -9, 2, updated, 11/2]], step_budget_exhausted, 11/2, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0035/)

Newton method for root finding · qf-mod50-newton-finite-orbit · candidate





## 36. A Finite Newton Orbit Records Steps without Certifying Convergence

Coefficients in low-to-high degree order are=[-11, 0, 1]; all iterations use exact rational arithmetic. Use initial iterate=2 and attempted-step budget=1. Give (coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope).

step_budget bounds attempted steps; an exact root may stop the records early. A finite trace or residual does not prove general convergence or a rate.

View answer and explanation([-11, 0, 1], 1, [[0, 2, -7, 4, updated, 15/4]], step_budget_exhausted, 15/4, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)

Recalculate f, f_prime and every qualified update. Each record gives index, current x, f, f_prime, status and next or null. Exact roots stop early; a nonroot zero derivative is rejected. A finite residual pattern proves neither a general limit nor quadratic convergence, a global basin or a forward-error bound. The exact result, retaining the requested order, is ([-11, 0, 1], 1, [[0, 2, -7, 4, updated, 15/4]], step_budget_exhausted, 15/4, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0036/)

Newton method for root finding · qf-mod50-newton-finite-orbit · candidate





## 37. A Finite Newton Orbit Records Steps without Certifying Convergence

Coefficients in low-to-high degree order are=[-12, 0, 1]; all iterations use exact rational arithmetic. Use initial iterate=3 and attempted-step budget=1. Give (coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope).

step_budget bounds attempted steps; an exact root may stop the records early. A finite trace or residual does not prove general convergence or a rate.

View answer and explanation([-12, 0, 1], 1, [[0, 3, -3, 6, updated, 7/2]], step_budget_exhausted, 7/2, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)

Recalculate f, f_prime and every qualified update. Each record gives index, current x, f, f_prime, status and next or null. Exact roots stop early; a nonroot zero derivative is rejected. A finite residual pattern proves neither a general limit nor quadratic convergence, a global basin or a forward-error bound. The exact result, retaining the requested order, is ([-12, 0, 1], 1, [[0, 3, -3, 6, updated, 7/2]], step_budget_exhausted, 7/2, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0037/)

Newton method for root finding · qf-mod50-newton-finite-orbit · candidate





## 38. A Finite Newton Orbit Records Steps without Certifying Convergence

Coefficients in low-to-high degree order are=[-13, 0, 1]; all iterations use exact rational arithmetic. Use initial iterate=4 and attempted-step budget=1. Give (coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope).

step_budget bounds attempted steps; an exact root may stop the records early. A finite trace or residual does not prove general convergence or a rate.

View answer and explanation([-13, 0, 1], 1, [[0, 4, 3, 8, updated, 29/8]], step_budget_exhausted, 29/8, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)

Recalculate f, f_prime and every qualified update. Each record gives index, current x, f, f_prime, status and next or null. Exact roots stop early; a nonroot zero derivative is rejected. A finite residual pattern proves neither a general limit nor quadratic convergence, a global basin or a forward-error bound. The exact result, retaining the requested order, is ([-13, 0, 1], 1, [[0, 4, 3, 8, updated, 29/8]], step_budget_exhausted, 29/8, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0038/)

Newton method for root finding · qf-mod50-newton-finite-orbit · candidate





## 39. A Finite Newton Orbit Records Steps without Certifying Convergence

Coefficients in low-to-high degree order are=[-14, 0, 1]; all iterations use exact rational arithmetic. Use initial iterate=5 and attempted-step budget=1. Give (coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope).

step_budget bounds attempted steps; an exact root may stop the records early. A finite trace or residual does not prove general convergence or a rate.

View answer and explanation([-14, 0, 1], 1, [[0, 5, 11, 10, updated, 39/10]], step_budget_exhausted, 39/10, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)

Recalculate f, f_prime and every qualified update. Each record gives index, current x, f, f_prime, status and next or null. Exact roots stop early; a nonroot zero derivative is rejected. A finite residual pattern proves neither a general limit nor quadratic convergence, a global basin or a forward-error bound. The exact result, retaining the requested order, is ([-14, 0, 1], 1, [[0, 5, 11, 10, updated, 39/10]], step_budget_exhausted, 39/10, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0039/)

Newton method for root finding · qf-mod50-newton-finite-orbit · candidate





## 40. A Finite Newton Orbit Records Steps without Certifying Convergence

Coefficients in low-to-high degree order are=[-15, 0, 1]; all iterations use exact rational arithmetic. Use initial iterate=1 and attempted-step budget=1. Give (coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope).

step_budget bounds attempted steps; an exact root may stop the records early. A finite trace or residual does not prove general convergence or a rate.

View answer and explanation([-15, 0, 1], 1, [[0, 1, -14, 2, updated, 8]], step_budget_exhausted, 8, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)

Recalculate f, f_prime and every qualified update. Each record gives index, current x, f, f_prime, status and next or null. Exact roots stop early; a nonroot zero derivative is rejected. A finite residual pattern proves neither a general limit nor quadratic convergence, a global basin or a forward-error bound. The exact result, retaining the requested order, is ([-15, 0, 1], 1, [[0, 1, -14, 2, updated, 8]], step_budget_exhausted, 8, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0040/)

Newton method for root finding · qf-mod50-newton-finite-orbit · candidate





## 41. A Finite Newton Orbit Records Steps without Certifying Convergence

Coefficients in low-to-high degree order are=[-16, 0, 1]; all iterations use exact rational arithmetic. Use initial iterate=2 and attempted-step budget=1. Give (coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope).

step_budget bounds attempted steps; an exact root may stop the records early. A finite trace or residual does not prove general convergence or a rate.

View answer and explanation([-16, 0, 1], 1, [[0, 2, -12, 4, updated, 5]], step_budget_exhausted, 5, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)

Recalculate f, f_prime and every qualified update. Each record gives index, current x, f, f_prime, status and next or null. Exact roots stop early; a nonroot zero derivative is rejected. A finite residual pattern proves neither a general limit nor quadratic convergence, a global basin or a forward-error bound. The exact result, retaining the requested order, is ([-16, 0, 1], 1, [[0, 2, -12, 4, updated, 5]], step_budget_exhausted, 5, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0041/)

Newton method for root finding · qf-mod50-newton-finite-orbit · candidate





## 42. A Finite Newton Orbit Records Steps without Certifying Convergence

Coefficients in low-to-high degree order are=[-17, 0, 1]; all iterations use exact rational arithmetic. Use initial iterate=3 and attempted-step budget=1. Give (coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope).

step_budget bounds attempted steps; an exact root may stop the records early. A finite trace or residual does not prove general convergence or a rate.

View answer and explanation([-17, 0, 1], 1, [[0, 3, -8, 6, updated, 13/3]], step_budget_exhausted, 13/3, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)

Recalculate f, f_prime and every qualified update. Each record gives index, current x, f, f_prime, status and next or null. Exact roots stop early; a nonroot zero derivative is rejected. A finite residual pattern proves neither a general limit nor quadratic convergence, a global basin or a forward-error bound. The exact result, retaining the requested order, is ([-17, 0, 1], 1, [[0, 3, -8, 6, updated, 13/3]], step_budget_exhausted, 13/3, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0042/)

Newton method for root finding · qf-mod50-newton-finite-orbit · candidate





## 43. A Finite Newton Orbit Records Steps without Certifying Convergence

Coefficients in low-to-high degree order are=[-18, 0, 1]; all iterations use exact rational arithmetic. Use initial iterate=4 and attempted-step budget=1. Give (coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope).

step_budget bounds attempted steps; an exact root may stop the records early. A finite trace or residual does not prove general convergence or a rate.

View answer and explanation([-18, 0, 1], 1, [[0, 4, -2, 8, updated, 17/4]], step_budget_exhausted, 17/4, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)

Recalculate f, f_prime and every qualified update. Each record gives index, current x, f, f_prime, status and next or null. Exact roots stop early; a nonroot zero derivative is rejected. A finite residual pattern proves neither a general limit nor quadratic convergence, a global basin or a forward-error bound. The exact result, retaining the requested order, is ([-18, 0, 1], 1, [[0, 4, -2, 8, updated, 17/4]], step_budget_exhausted, 17/4, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0043/)

Newton method for root finding · qf-mod50-newton-finite-orbit · candidate





## 44. A Finite Newton Orbit Records Steps without Certifying Convergence

Coefficients in low-to-high degree order are=[-19, 0, 1]; all iterations use exact rational arithmetic. Use initial iterate=5 and attempted-step budget=1. Give (coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope).

step_budget bounds attempted steps; an exact root may stop the records early. A finite trace or residual does not prove general convergence or a rate.

View answer and explanation([-19, 0, 1], 1, [[0, 5, 6, 10, updated, 22/5]], step_budget_exhausted, 22/5, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)

Recalculate f, f_prime and every qualified update. Each record gives index, current x, f, f_prime, status and next or null. Exact roots stop early; a nonroot zero derivative is rejected. A finite residual pattern proves neither a general limit nor quadratic convergence, a global basin or a forward-error bound. The exact result, retaining the requested order, is ([-19, 0, 1], 1, [[0, 5, 6, 10, updated, 22/5]], step_budget_exhausted, 22/5, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0044/)

Newton method for root finding · qf-mod50-newton-finite-orbit · candidate





## 45. A Finite Newton Orbit Records Steps without Certifying Convergence

Coefficients in low-to-high degree order are=[-20, 0, 1]; all iterations use exact rational arithmetic. Use initial iterate=1 and attempted-step budget=1. Give (coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope).

step_budget bounds attempted steps; an exact root may stop the records early. A finite trace or residual does not prove general convergence or a rate.

View answer and explanation([-20, 0, 1], 1, [[0, 1, -19, 2, updated, 21/2]], step_budget_exhausted, 21/2, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)

Recalculate f, f_prime and every qualified update. Each record gives index, current x, f, f_prime, status and next or null. Exact roots stop early; a nonroot zero derivative is rejected. A finite residual pattern proves neither a general limit nor quadratic convergence, a global basin or a forward-error bound. The exact result, retaining the requested order, is ([-20, 0, 1], 1, [[0, 1, -19, 2, updated, 21/2]], step_budget_exhausted, 21/2, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0045/)

Newton method for root finding · qf-mod50-newton-finite-orbit · candidate





## 46. A Finite Newton Orbit Records Steps without Certifying Convergence

Coefficients in low-to-high degree order are=[-21, 0, 1]; all iterations use exact rational arithmetic. Use initial iterate=2 and attempted-step budget=1. Give (coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope).

step_budget bounds attempted steps; an exact root may stop the records early. A finite trace or residual does not prove general convergence or a rate.

View answer and explanation([-21, 0, 1], 1, [[0, 2, -17, 4, updated, 25/4]], step_budget_exhausted, 25/4, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)

Recalculate f, f_prime and every qualified update. Each record gives index, current x, f, f_prime, status and next or null. Exact roots stop early; a nonroot zero derivative is rejected. A finite residual pattern proves neither a general limit nor quadratic convergence, a global basin or a forward-error bound. The exact result, retaining the requested order, is ([-21, 0, 1], 1, [[0, 2, -17, 4, updated, 25/4]], step_budget_exhausted, 25/4, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0046/)

Newton method for root finding · qf-mod50-newton-finite-orbit · candidate





## 47. A Finite Newton Orbit Records Steps without Certifying Convergence

Coefficients in low-to-high degree order are=[-22, 0, 1]; all iterations use exact rational arithmetic. Use initial iterate=3 and attempted-step budget=1. Give (coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope).

step_budget bounds attempted steps; an exact root may stop the records early. A finite trace or residual does not prove general convergence or a rate.

View answer and explanation([-22, 0, 1], 1, [[0, 3, -13, 6, updated, 31/6]], step_budget_exhausted, 31/6, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)

Recalculate f, f_prime and every qualified update. Each record gives index, current x, f, f_prime, status and next or null. Exact roots stop early; a nonroot zero derivative is rejected. A finite residual pattern proves neither a general limit nor quadratic convergence, a global basin or a forward-error bound. The exact result, retaining the requested order, is ([-22, 0, 1], 1, [[0, 3, -13, 6, updated, 31/6]], step_budget_exhausted, 31/6, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0047/)

Newton method for root finding · qf-mod50-newton-finite-orbit · candidate





## 48. A Finite Newton Orbit Records Steps without Certifying Convergence

Coefficients in low-to-high degree order are=[-23, 0, 1]; all iterations use exact rational arithmetic. Use initial iterate=4 and attempted-step budget=1. Give (coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope).

step_budget bounds attempted steps; an exact root may stop the records early. A finite trace or residual does not prove general convergence or a rate.

View answer and explanation([-23, 0, 1], 1, [[0, 4, -7, 8, updated, 39/8]], step_budget_exhausted, 39/8, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)

Recalculate f, f_prime and every qualified update. Each record gives index, current x, f, f_prime, status and next or null. Exact roots stop early; a nonroot zero derivative is rejected. A finite residual pattern proves neither a general limit nor quadratic convergence, a global basin or a forward-error bound. The exact result, retaining the requested order, is ([-23, 0, 1], 1, [[0, 4, -7, 8, updated, 39/8]], step_budget_exhausted, 39/8, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0048/)

Newton method for root finding · qf-mod50-newton-finite-orbit · candidate





## 49. A Finite Newton Orbit Records Steps without Certifying Convergence

Coefficients in low-to-high degree order are=[-24, 0, 1]; all iterations use exact rational arithmetic. Use initial iterate=5 and attempted-step budget=1. Give (coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope).

step_budget bounds attempted steps; an exact root may stop the records early. A finite trace or residual does not prove general convergence or a rate.

View answer and explanation([-24, 0, 1], 1, [[0, 5, 1, 10, updated, 49/10]], step_budget_exhausted, 49/10, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)

Recalculate f, f_prime and every qualified update. Each record gives index, current x, f, f_prime, status and next or null. Exact roots stop early; a nonroot zero derivative is rejected. A finite residual pattern proves neither a general limit nor quadratic convergence, a global basin or a forward-error bound. The exact result, retaining the requested order, is ([-24, 0, 1], 1, [[0, 5, 1, 10, updated, 49/10]], step_budget_exhausted, 49/10, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0049/)

Newton method for root finding · qf-mod50-newton-finite-orbit · candidate





## 50. A Finite Newton Orbit Records Steps without Certifying Convergence

Coefficients in low-to-high degree order are=[-25, 0, 1]; all iterations use exact rational arithmetic. Use initial iterate=1 and attempted-step budget=1. Give (coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope).

step_budget bounds attempted steps; an exact root may stop the records early. A finite trace or residual does not prove general convergence or a rate.

View answer and explanation([-25, 0, 1], 1, [[0, 1, -24, 2, updated, 13]], step_budget_exhausted, 13, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)

Recalculate f, f_prime and every qualified update. Each record gives index, current x, f, f_prime, status and next or null. Exact roots stop early; a nonroot zero derivative is rejected. A finite residual pattern proves neither a general limit nor quadratic convergence, a global basin or a forward-error bound. The exact result, retaining the requested order, is ([-25, 0, 1], 1, [[0, 1, -24, 2, updated, 13]], step_budget_exhausted, 13, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0050/)

Newton method for root finding · qf-mod50-newton-finite-orbit · candidate





## 51. Positive Square-Root Iterates Use Exact Rational Newton Steps

Use positive radicand A=2, positive initial iterate=2, and attempted-step budget=1; f(x)=x^2−A. Give (radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope).

View answer and explanation(2, 2, 1, [[0, 2, 2, updated, 3/2]], step_budget_exhausted, 3/2, 1/4, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)

At an exact root stop before dividing; otherwise update to (x+A/x)/2. Each record gives index, current x, signed residual, status and next or null. The final absolute residual is |x^2−A|, not an unprovided forward error to sqrt(A) or a general Newton convergence proof. The exact result, retaining the requested order, is (2, 2, 1, [[0, 2, 2, updated, 3/2]], step_budget_exhausted, 3/2, 1/4, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0051/)

Newton method for root finding · qf-mod50-newton-square-root-orbit · candidate





## 52. Positive Square-Root Iterates Use Exact Rational Newton Steps

Use positive radicand A=3, positive initial iterate=3, and attempted-step budget=1; f(x)=x^2−A. Give (radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope).

View answer and explanation(3, 3, 1, [[0, 3, 6, updated, 2]], step_budget_exhausted, 2, 1, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)

At an exact root stop before dividing; otherwise update to (x+A/x)/2. Each record gives index, current x, signed residual, status and next or null. The final absolute residual is |x^2−A|, not an unprovided forward error to sqrt(A) or a general Newton convergence proof. The exact result, retaining the requested order, is (3, 3, 1, [[0, 3, 6, updated, 2]], step_budget_exhausted, 2, 1, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0052/)

Newton method for root finding · qf-mod50-newton-square-root-orbit · candidate





## 53. Positive Square-Root Iterates Use Exact Rational Newton Steps

Use positive radicand A=4, positive initial iterate=4, and attempted-step budget=1; f(x)=x^2−A. Give (radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope).

View answer and explanation(4, 4, 1, [[0, 4, 12, updated, 5/2]], step_budget_exhausted, 5/2, 9/4, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)

At an exact root stop before dividing; otherwise update to (x+A/x)/2. Each record gives index, current x, signed residual, status and next or null. The final absolute residual is |x^2−A|, not an unprovided forward error to sqrt(A) or a general Newton convergence proof. The exact result, retaining the requested order, is (4, 4, 1, [[0, 4, 12, updated, 5/2]], step_budget_exhausted, 5/2, 9/4, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0053/)

Newton method for root finding · qf-mod50-newton-square-root-orbit · candidate





## 54. Positive Square-Root Iterates Use Exact Rational Newton Steps

Use positive radicand A=5, positive initial iterate=5, and attempted-step budget=1; f(x)=x^2−A. Give (radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope).

View answer and explanation(5, 5, 1, [[0, 5, 20, updated, 3]], step_budget_exhausted, 3, 4, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)

At an exact root stop before dividing; otherwise update to (x+A/x)/2. Each record gives index, current x, signed residual, status and next or null. The final absolute residual is |x^2−A|, not an unprovided forward error to sqrt(A) or a general Newton convergence proof. The exact result, retaining the requested order, is (5, 5, 1, [[0, 5, 20, updated, 3]], step_budget_exhausted, 3, 4, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0054/)

Newton method for root finding · qf-mod50-newton-square-root-orbit · candidate





## 55. Positive Square-Root Iterates Use Exact Rational Newton Steps

Use positive radicand A=6, positive initial iterate=1, and attempted-step budget=1; f(x)=x^2−A. Give (radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope).

View answer and explanation(6, 1, 1, [[0, 1, -5, updated, 7/2]], step_budget_exhausted, 7/2, 25/4, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)

At an exact root stop before dividing; otherwise update to (x+A/x)/2. Each record gives index, current x, signed residual, status and next or null. The final absolute residual is |x^2−A|, not an unprovided forward error to sqrt(A) or a general Newton convergence proof. The exact result, retaining the requested order, is (6, 1, 1, [[0, 1, -5, updated, 7/2]], step_budget_exhausted, 7/2, 25/4, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0055/)

Newton method for root finding · qf-mod50-newton-square-root-orbit · candidate





## 56. Positive Square-Root Iterates Use Exact Rational Newton Steps

Use positive radicand A=7, positive initial iterate=2, and attempted-step budget=1; f(x)=x^2−A. Give (radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope).

View answer and explanation(7, 2, 1, [[0, 2, -3, updated, 11/4]], step_budget_exhausted, 11/4, 9/16, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)

At an exact root stop before dividing; otherwise update to (x+A/x)/2. Each record gives index, current x, signed residual, status and next or null. The final absolute residual is |x^2−A|, not an unprovided forward error to sqrt(A) or a general Newton convergence proof. The exact result, retaining the requested order, is (7, 2, 1, [[0, 2, -3, updated, 11/4]], step_budget_exhausted, 11/4, 9/16, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0056/)

Newton method for root finding · qf-mod50-newton-square-root-orbit · candidate





## 57. Positive Square-Root Iterates Use Exact Rational Newton Steps

Use positive radicand A=8, positive initial iterate=3, and attempted-step budget=1; f(x)=x^2−A. Give (radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope).

View answer and explanation(8, 3, 1, [[0, 3, 1, updated, 17/6]], step_budget_exhausted, 17/6, 1/36, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)

At an exact root stop before dividing; otherwise update to (x+A/x)/2. Each record gives index, current x, signed residual, status and next or null. The final absolute residual is |x^2−A|, not an unprovided forward error to sqrt(A) or a general Newton convergence proof. The exact result, retaining the requested order, is (8, 3, 1, [[0, 3, 1, updated, 17/6]], step_budget_exhausted, 17/6, 1/36, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0057/)

Newton method for root finding · qf-mod50-newton-square-root-orbit · candidate





## 58. Positive Square-Root Iterates Use Exact Rational Newton Steps

Use positive radicand A=9, positive initial iterate=4, and attempted-step budget=1; f(x)=x^2−A. Give (radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope).

View answer and explanation(9, 4, 1, [[0, 4, 7, updated, 25/8]], step_budget_exhausted, 25/8, 49/64, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)

At an exact root stop before dividing; otherwise update to (x+A/x)/2. Each record gives index, current x, signed residual, status and next or null. The final absolute residual is |x^2−A|, not an unprovided forward error to sqrt(A) or a general Newton convergence proof. The exact result, retaining the requested order, is (9, 4, 1, [[0, 4, 7, updated, 25/8]], step_budget_exhausted, 25/8, 49/64, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0058/)

Newton method for root finding · qf-mod50-newton-square-root-orbit · candidate





## 59. Positive Square-Root Iterates Use Exact Rational Newton Steps

Use positive radicand A=10, positive initial iterate=5, and attempted-step budget=1; f(x)=x^2−A. Give (radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope).

View answer and explanation(10, 5, 1, [[0, 5, 15, updated, 7/2]], step_budget_exhausted, 7/2, 9/4, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)

At an exact root stop before dividing; otherwise update to (x+A/x)/2. Each record gives index, current x, signed residual, status and next or null. The final absolute residual is |x^2−A|, not an unprovided forward error to sqrt(A) or a general Newton convergence proof. The exact result, retaining the requested order, is (10, 5, 1, [[0, 5, 15, updated, 7/2]], step_budget_exhausted, 7/2, 9/4, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0059/)

Newton method for root finding · qf-mod50-newton-square-root-orbit · candidate





## 60. Positive Square-Root Iterates Use Exact Rational Newton Steps

Use positive radicand A=11, positive initial iterate=1, and attempted-step budget=1; f(x)=x^2−A. Give (radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope).

View answer and explanation(11, 1, 1, [[0, 1, -10, updated, 6]], step_budget_exhausted, 6, 25, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)

At an exact root stop before dividing; otherwise update to (x+A/x)/2. Each record gives index, current x, signed residual, status and next or null. The final absolute residual is |x^2−A|, not an unprovided forward error to sqrt(A) or a general Newton convergence proof. The exact result, retaining the requested order, is (11, 1, 1, [[0, 1, -10, updated, 6]], step_budget_exhausted, 6, 25, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0060/)

Newton method for root finding · qf-mod50-newton-square-root-orbit · candidate





## 61. Positive Square-Root Iterates Use Exact Rational Newton Steps

Use positive radicand A=12, positive initial iterate=2, and attempted-step budget=1; f(x)=x^2−A. Give (radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope).

View answer and explanation(12, 2, 1, [[0, 2, -8, updated, 4]], step_budget_exhausted, 4, 4, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)

At an exact root stop before dividing; otherwise update to (x+A/x)/2. Each record gives index, current x, signed residual, status and next or null. The final absolute residual is |x^2−A|, not an unprovided forward error to sqrt(A) or a general Newton convergence proof. The exact result, retaining the requested order, is (12, 2, 1, [[0, 2, -8, updated, 4]], step_budget_exhausted, 4, 4, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0061/)

Newton method for root finding · qf-mod50-newton-square-root-orbit · candidate





## 62. Positive Square-Root Iterates Use Exact Rational Newton Steps

Use positive radicand A=13, positive initial iterate=3, and attempted-step budget=1; f(x)=x^2−A. Give (radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope).

View answer and explanation(13, 3, 1, [[0, 3, -4, updated, 11/3]], step_budget_exhausted, 11/3, 4/9, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)

At an exact root stop before dividing; otherwise update to (x+A/x)/2. Each record gives index, current x, signed residual, status and next or null. The final absolute residual is |x^2−A|, not an unprovided forward error to sqrt(A) or a general Newton convergence proof. The exact result, retaining the requested order, is (13, 3, 1, [[0, 3, -4, updated, 11/3]], step_budget_exhausted, 11/3, 4/9, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0062/)

Newton method for root finding · qf-mod50-newton-square-root-orbit · candidate





## 63. Positive Square-Root Iterates Use Exact Rational Newton Steps

Use positive radicand A=14, positive initial iterate=4, and attempted-step budget=1; f(x)=x^2−A. Give (radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope).

View answer and explanation(14, 4, 1, [[0, 4, 2, updated, 15/4]], step_budget_exhausted, 15/4, 1/16, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)

At an exact root stop before dividing; otherwise update to (x+A/x)/2. Each record gives index, current x, signed residual, status and next or null. The final absolute residual is |x^2−A|, not an unprovided forward error to sqrt(A) or a general Newton convergence proof. The exact result, retaining the requested order, is (14, 4, 1, [[0, 4, 2, updated, 15/4]], step_budget_exhausted, 15/4, 1/16, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0063/)

Newton method for root finding · qf-mod50-newton-square-root-orbit · candidate





## 64. Positive Square-Root Iterates Use Exact Rational Newton Steps

Use positive radicand A=15, positive initial iterate=5, and attempted-step budget=1; f(x)=x^2−A. Give (radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope).

View answer and explanation(15, 5, 1, [[0, 5, 10, updated, 4]], step_budget_exhausted, 4, 1, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)

At an exact root stop before dividing; otherwise update to (x+A/x)/2. Each record gives index, current x, signed residual, status and next or null. The final absolute residual is |x^2−A|, not an unprovided forward error to sqrt(A) or a general Newton convergence proof. The exact result, retaining the requested order, is (15, 5, 1, [[0, 5, 10, updated, 4]], step_budget_exhausted, 4, 1, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0064/)

Newton method for root finding · qf-mod50-newton-square-root-orbit · candidate





## 65. Positive Square-Root Iterates Use Exact Rational Newton Steps

Use positive radicand A=16, positive initial iterate=1, and attempted-step budget=1; f(x)=x^2−A. Give (radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope).

View answer and explanation(16, 1, 1, [[0, 1, -15, updated, 17/2]], step_budget_exhausted, 17/2, 225/4, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)

At an exact root stop before dividing; otherwise update to (x+A/x)/2. Each record gives index, current x, signed residual, status and next or null. The final absolute residual is |x^2−A|, not an unprovided forward error to sqrt(A) or a general Newton convergence proof. The exact result, retaining the requested order, is (16, 1, 1, [[0, 1, -15, updated, 17/2]], step_budget_exhausted, 17/2, 225/4, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0065/)

Newton method for root finding · qf-mod50-newton-square-root-orbit · candidate





## 66. Positive Square-Root Iterates Use Exact Rational Newton Steps

Use positive radicand A=17, positive initial iterate=2, and attempted-step budget=1; f(x)=x^2−A. Give (radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope).

View answer and explanation(17, 2, 1, [[0, 2, -13, updated, 21/4]], step_budget_exhausted, 21/4, 169/16, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)

At an exact root stop before dividing; otherwise update to (x+A/x)/2. Each record gives index, current x, signed residual, status and next or null. The final absolute residual is |x^2−A|, not an unprovided forward error to sqrt(A) or a general Newton convergence proof. The exact result, retaining the requested order, is (17, 2, 1, [[0, 2, -13, updated, 21/4]], step_budget_exhausted, 21/4, 169/16, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0066/)

Newton method for root finding · qf-mod50-newton-square-root-orbit · candidate





## 67. Positive Square-Root Iterates Use Exact Rational Newton Steps

Use positive radicand A=18, positive initial iterate=3, and attempted-step budget=1; f(x)=x^2−A. Give (radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope).

View answer and explanation(18, 3, 1, [[0, 3, -9, updated, 9/2]], step_budget_exhausted, 9/2, 9/4, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)

At an exact root stop before dividing; otherwise update to (x+A/x)/2. Each record gives index, current x, signed residual, status and next or null. The final absolute residual is |x^2−A|, not an unprovided forward error to sqrt(A) or a general Newton convergence proof. The exact result, retaining the requested order, is (18, 3, 1, [[0, 3, -9, updated, 9/2]], step_budget_exhausted, 9/2, 9/4, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0067/)

Newton method for root finding · qf-mod50-newton-square-root-orbit · candidate





## 68. Positive Square-Root Iterates Use Exact Rational Newton Steps

Use positive radicand A=19, positive initial iterate=4, and attempted-step budget=1; f(x)=x^2−A. Give (radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope).

View answer and explanation(19, 4, 1, [[0, 4, -3, updated, 35/8]], step_budget_exhausted, 35/8, 9/64, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)

At an exact root stop before dividing; otherwise update to (x+A/x)/2. Each record gives index, current x, signed residual, status and next or null. The final absolute residual is |x^2−A|, not an unprovided forward error to sqrt(A) or a general Newton convergence proof. The exact result, retaining the requested order, is (19, 4, 1, [[0, 4, -3, updated, 35/8]], step_budget_exhausted, 35/8, 9/64, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0068/)

Newton method for root finding · qf-mod50-newton-square-root-orbit · candidate





## 69. Positive Square-Root Iterates Use Exact Rational Newton Steps

Use positive radicand A=20, positive initial iterate=5, and attempted-step budget=1; f(x)=x^2−A. Give (radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope).

View answer and explanation(20, 5, 1, [[0, 5, 5, updated, 9/2]], step_budget_exhausted, 9/2, 1/4, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)

At an exact root stop before dividing; otherwise update to (x+A/x)/2. Each record gives index, current x, signed residual, status and next or null. The final absolute residual is |x^2−A|, not an unprovided forward error to sqrt(A) or a general Newton convergence proof. The exact result, retaining the requested order, is (20, 5, 1, [[0, 5, 5, updated, 9/2]], step_budget_exhausted, 9/2, 1/4, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0069/)

Newton method for root finding · qf-mod50-newton-square-root-orbit · candidate





## 70. Positive Square-Root Iterates Use Exact Rational Newton Steps

Use positive radicand A=21, positive initial iterate=1, and attempted-step budget=1; f(x)=x^2−A. Give (radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope).

View answer and explanation(21, 1, 1, [[0, 1, -20, updated, 11]], step_budget_exhausted, 11, 100, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)

At an exact root stop before dividing; otherwise update to (x+A/x)/2. Each record gives index, current x, signed residual, status and next or null. The final absolute residual is |x^2−A|, not an unprovided forward error to sqrt(A) or a general Newton convergence proof. The exact result, retaining the requested order, is (21, 1, 1, [[0, 1, -20, updated, 11]], step_budget_exhausted, 11, 100, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0070/)

Newton method for root finding · qf-mod50-newton-square-root-orbit · candidate





## 71. Positive Square-Root Iterates Use Exact Rational Newton Steps

Use positive radicand A=22, positive initial iterate=2, and attempted-step budget=1; f(x)=x^2−A. Give (radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope).

View answer and explanation(22, 2, 1, [[0, 2, -18, updated, 13/2]], step_budget_exhausted, 13/2, 81/4, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)

At an exact root stop before dividing; otherwise update to (x+A/x)/2. Each record gives index, current x, signed residual, status and next or null. The final absolute residual is |x^2−A|, not an unprovided forward error to sqrt(A) or a general Newton convergence proof. The exact result, retaining the requested order, is (22, 2, 1, [[0, 2, -18, updated, 13/2]], step_budget_exhausted, 13/2, 81/4, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0071/)

Newton method for root finding · qf-mod50-newton-square-root-orbit · candidate





## 72. Positive Square-Root Iterates Use Exact Rational Newton Steps

Use positive radicand A=23, positive initial iterate=3, and attempted-step budget=1; f(x)=x^2−A. Give (radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope).

View answer and explanation(23, 3, 1, [[0, 3, -14, updated, 16/3]], step_budget_exhausted, 16/3, 49/9, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)

At an exact root stop before dividing; otherwise update to (x+A/x)/2. Each record gives index, current x, signed residual, status and next or null. The final absolute residual is |x^2−A|, not an unprovided forward error to sqrt(A) or a general Newton convergence proof. The exact result, retaining the requested order, is (23, 3, 1, [[0, 3, -14, updated, 16/3]], step_budget_exhausted, 16/3, 49/9, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0072/)

Newton method for root finding · qf-mod50-newton-square-root-orbit · candidate





## 73. Positive Square-Root Iterates Use Exact Rational Newton Steps

Use positive radicand A=24, positive initial iterate=4, and attempted-step budget=1; f(x)=x^2−A. Give (radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope).

View answer and explanation(24, 4, 1, [[0, 4, -8, updated, 5]], step_budget_exhausted, 5, 1, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)

At an exact root stop before dividing; otherwise update to (x+A/x)/2. Each record gives index, current x, signed residual, status and next or null. The final absolute residual is |x^2−A|, not an unprovided forward error to sqrt(A) or a general Newton convergence proof. The exact result, retaining the requested order, is (24, 4, 1, [[0, 4, -8, updated, 5]], step_budget_exhausted, 5, 1, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0073/)

Newton method for root finding · qf-mod50-newton-square-root-orbit · candidate





## 74. Positive Square-Root Iterates Use Exact Rational Newton Steps

Use positive radicand A=25, positive initial iterate=5, and attempted-step budget=1; f(x)=x^2−A. Give (radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope).

View answer and explanation(25, 5, 1, [[0, 5, 0, root_found, null]], root_found, 5, 0, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)

At an exact root stop before dividing; otherwise update to (x+A/x)/2. Each record gives index, current x, signed residual, status and next or null. The final absolute residual is |x^2−A|, not an unprovided forward error to sqrt(A) or a general Newton convergence proof. The exact result, retaining the requested order, is (25, 5, 1, [[0, 5, 0, root_found, null]], root_found, 5, 0, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0074/)

Newton method for root finding · qf-mod50-newton-square-root-orbit · candidate





## 75. Positive Square-Root Iterates Use Exact Rational Newton Steps

Use positive radicand A=26, positive initial iterate=1, and attempted-step budget=1; f(x)=x^2−A. Give (radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope).

View answer and explanation(26, 1, 1, [[0, 1, -25, updated, 27/2]], step_budget_exhausted, 27/2, 625/4, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)

At an exact root stop before dividing; otherwise update to (x+A/x)/2. Each record gives index, current x, signed residual, status and next or null. The final absolute residual is |x^2−A|, not an unprovided forward error to sqrt(A) or a general Newton convergence proof. The exact result, retaining the requested order, is (26, 1, 1, [[0, 1, -25, updated, 27/2]], step_budget_exhausted, 27/2, 625/4, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0075/)

Newton method for root finding · qf-mod50-newton-square-root-orbit · candidate





## 76. Root-First Stopping, Zero Derivatives and Residual Scaling Stay Distinct

Use mode=zero_derivative, complete model={"coefficients":[1,0,1],"x":0}. Give (mode, current_iterate, function_value, derivative_value, status, update_qualification_claim, boundary).

The four modes have7/6/6/7 fields. residual_equals_forward_error is an instance equality flag, not a universal identification; the correct affine relation includes abs(slope).

View answer and explanation(zero_derivative, 0, 1, 0, undefined_zero_derivative, No, division_by_zero_is_rejected_not_repaired)

A nonroot with zero derivative has no Newton update and is not silently repaired. An exact root stops even if the derivative is0. A finite trace is not a convergence/rate proof. For a nonzero affine slope, residual=abs(slope)*forward_error; numerical equality in one row does not identify the two quantities universally. The exact result, retaining the requested order, is (zero_derivative, 0, 1, 0, undefined_zero_derivative, No, division_by_zero_is_rejected_not_repaired).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0076/)

Newton method for root finding · qf-mod50-newton-boundary · candidate





## 77. Root-First Stopping, Zero Derivatives and Residual Scaling Stay Distinct

Use mode=finite_trace_only, complete model={"coefficients":[-2,0,1],"x0":3,"steps":1}. Give (mode, step_budget, step_records, status, convergence_or_rate_proof_claim, boundary).

The four modes have7/6/6/7 fields. residual_equals_forward_error is an instance equality flag, not a universal identification; the correct affine relation includes abs(slope).

View answer and explanation(finite_trace_only, 1, [[0, 3, 7, 6, updated, 11/6]], step_budget_exhausted, No, finite_residual_behavior_does_not_prove_convergence_or_rate)

A nonroot with zero derivative has no Newton update and is not silently repaired. An exact root stops even if the derivative is0. A finite trace is not a convergence/rate proof. For a nonzero affine slope, residual=abs(slope)*forward_error; numerical equality in one row does not identify the two quantities universally. The exact result, retaining the requested order, is (finite_trace_only, 1, [[0, 3, 7, 6, updated, 11/6]], step_budget_exhausted, No, finite_residual_behavior_does_not_prove_convergence_or_rate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0077/)

Newton method for root finding · qf-mod50-newton-boundary · candidate





## 78. Root-First Stopping, Zero Derivatives and Residual Scaling Stay Distinct

Use mode=root_stop, complete model={"coefficients":[-3,1],"x":3}. Give (mode, current_iterate, function_value, derivative_value, root_first_status, root_stop_qualified).

The four modes have7/6/6/7 fields. residual_equals_forward_error is an instance equality flag, not a universal identification; the correct affine relation includes abs(slope).

View answer and explanation(root_stop, 3, 0, 1, root_found_before_derivative_division, Yes)

A nonroot with zero derivative has no Newton update and is not silently repaired. An exact root stops even if the derivative is0. A finite trace is not a convergence/rate proof. For a nonzero affine slope, residual=abs(slope)*forward_error; numerical equality in one row does not identify the two quantities universally. The exact result, retaining the requested order, is (root_stop, 3, 0, 1, root_found_before_derivative_division, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0078/)

Newton method for root finding · qf-mod50-newton-boundary · candidate





## 79. Root-First Stopping, Zero Derivatives and Residual Scaling Stay Distinct

Use mode=residual_forward_confusion, complete model={"slope":5,"intercept":-4,"approximation":6}. Give (mode, slope, exact_root, absolute_residual, forward_error, residual_equals_forward_error, scale_relation).

The four modes have7/6/6/7 fields. residual_equals_forward_error is an instance equality flag, not a universal identification; the correct affine relation includes abs(slope).

View answer and explanation(residual_forward_confusion, 5, 4/5, 26, 26/5, No, residual_equals_abs_slope_times_forward_error_not_forward_error_itself)

A nonroot with zero derivative has no Newton update and is not silently repaired. An exact root stops even if the derivative is0. A finite trace is not a convergence/rate proof. For a nonzero affine slope, residual=abs(slope)*forward_error; numerical equality in one row does not identify the two quantities universally. The exact result, retaining the requested order, is (residual_forward_confusion, 5, 4/5, 26, 26/5, No, residual_equals_abs_slope_times_forward_error_not_forward_error_itself).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0079/)

Newton method for root finding · qf-mod50-newton-boundary · candidate





## 80. Root-First Stopping, Zero Derivatives and Residual Scaling Stay Distinct

Use mode=zero_derivative, complete model={"coefficients":[5,0,1],"x":0}. Give (mode, current_iterate, function_value, derivative_value, status, update_qualification_claim, boundary).

The four modes have7/6/6/7 fields. residual_equals_forward_error is an instance equality flag, not a universal identification; the correct affine relation includes abs(slope).

View answer and explanation(zero_derivative, 0, 5, 0, undefined_zero_derivative, No, division_by_zero_is_rejected_not_repaired)

A nonroot with zero derivative has no Newton update and is not silently repaired. An exact root stops even if the derivative is0. A finite trace is not a convergence/rate proof. For a nonzero affine slope, residual=abs(slope)*forward_error; numerical equality in one row does not identify the two quantities universally. The exact result, retaining the requested order, is (zero_derivative, 0, 5, 0, undefined_zero_derivative, No, division_by_zero_is_rejected_not_repaired).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0080/)

Newton method for root finding · qf-mod50-newton-boundary · candidate





## 81. Root-First Stopping, Zero Derivatives and Residual Scaling Stay Distinct

Use mode=finite_trace_only, complete model={"coefficients":[-6,0,1],"x0":2,"steps":1}. Give (mode, step_budget, step_records, status, convergence_or_rate_proof_claim, boundary).

The four modes have7/6/6/7 fields. residual_equals_forward_error is an instance equality flag, not a universal identification; the correct affine relation includes abs(slope).

View answer and explanation(finite_trace_only, 1, [[0, 2, -2, 4, updated, 5/2]], step_budget_exhausted, No, finite_residual_behavior_does_not_prove_convergence_or_rate)

A nonroot with zero derivative has no Newton update and is not silently repaired. An exact root stops even if the derivative is0. A finite trace is not a convergence/rate proof. For a nonzero affine slope, residual=abs(slope)*forward_error; numerical equality in one row does not identify the two quantities universally. The exact result, retaining the requested order, is (finite_trace_only, 1, [[0, 2, -2, 4, updated, 5/2]], step_budget_exhausted, No, finite_residual_behavior_does_not_prove_convergence_or_rate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0081/)

Newton method for root finding · qf-mod50-newton-boundary · candidate





## 82. Root-First Stopping, Zero Derivatives and Residual Scaling Stay Distinct

Use mode=root_stop, complete model={"coefficients":[-7,1],"x":7}. Give (mode, current_iterate, function_value, derivative_value, root_first_status, root_stop_qualified).

The four modes have7/6/6/7 fields. residual_equals_forward_error is an instance equality flag, not a universal identification; the correct affine relation includes abs(slope).

View answer and explanation(root_stop, 7, 0, 1, root_found_before_derivative_division, Yes)

A nonroot with zero derivative has no Newton update and is not silently repaired. An exact root stops even if the derivative is0. A finite trace is not a convergence/rate proof. For a nonzero affine slope, residual=abs(slope)*forward_error; numerical equality in one row does not identify the two quantities universally. The exact result, retaining the requested order, is (root_stop, 7, 0, 1, root_found_before_derivative_division, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0082/)

Newton method for root finding · qf-mod50-newton-boundary · candidate





## 83. Root-First Stopping, Zero Derivatives and Residual Scaling Stay Distinct

Use mode=residual_forward_confusion, complete model={"slope":9,"intercept":-8,"approximation":10}. Give (mode, slope, exact_root, absolute_residual, forward_error, residual_equals_forward_error, scale_relation).

The four modes have7/6/6/7 fields. residual_equals_forward_error is an instance equality flag, not a universal identification; the correct affine relation includes abs(slope).

View answer and explanation(residual_forward_confusion, 9, 8/9, 82, 82/9, No, residual_equals_abs_slope_times_forward_error_not_forward_error_itself)

A nonroot with zero derivative has no Newton update and is not silently repaired. An exact root stops even if the derivative is0. A finite trace is not a convergence/rate proof. For a nonzero affine slope, residual=abs(slope)*forward_error; numerical equality in one row does not identify the two quantities universally. The exact result, retaining the requested order, is (residual_forward_confusion, 9, 8/9, 82, 82/9, No, residual_equals_abs_slope_times_forward_error_not_forward_error_itself).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0083/)

Newton method for root finding · qf-mod50-newton-boundary · candidate





## 84. Root-First Stopping, Zero Derivatives and Residual Scaling Stay Distinct

Use mode=zero_derivative, complete model={"coefficients":[9,0,1],"x":0}. Give (mode, current_iterate, function_value, derivative_value, status, update_qualification_claim, boundary).

The four modes have7/6/6/7 fields. residual_equals_forward_error is an instance equality flag, not a universal identification; the correct affine relation includes abs(slope).

View answer and explanation(zero_derivative, 0, 9, 0, undefined_zero_derivative, No, division_by_zero_is_rejected_not_repaired)

A nonroot with zero derivative has no Newton update and is not silently repaired. An exact root stops even if the derivative is0. A finite trace is not a convergence/rate proof. For a nonzero affine slope, residual=abs(slope)*forward_error; numerical equality in one row does not identify the two quantities universally. The exact result, retaining the requested order, is (zero_derivative, 0, 9, 0, undefined_zero_derivative, No, division_by_zero_is_rejected_not_repaired).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0084/)

Newton method for root finding · qf-mod50-newton-boundary · candidate





## 85. Root-First Stopping, Zero Derivatives and Residual Scaling Stay Distinct

Use mode=finite_trace_only, complete model={"coefficients":[-10,0,1],"x0":1,"steps":1}. Give (mode, step_budget, step_records, status, convergence_or_rate_proof_claim, boundary).

The four modes have7/6/6/7 fields. residual_equals_forward_error is an instance equality flag, not a universal identification; the correct affine relation includes abs(slope).

View answer and explanation(finite_trace_only, 1, [[0, 1, -9, 2, updated, 11/2]], step_budget_exhausted, No, finite_residual_behavior_does_not_prove_convergence_or_rate)

A nonroot with zero derivative has no Newton update and is not silently repaired. An exact root stops even if the derivative is0. A finite trace is not a convergence/rate proof. For a nonzero affine slope, residual=abs(slope)*forward_error; numerical equality in one row does not identify the two quantities universally. The exact result, retaining the requested order, is (finite_trace_only, 1, [[0, 1, -9, 2, updated, 11/2]], step_budget_exhausted, No, finite_residual_behavior_does_not_prove_convergence_or_rate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0085/)

Newton method for root finding · qf-mod50-newton-boundary · candidate





## 86. Root-First Stopping, Zero Derivatives and Residual Scaling Stay Distinct

Use mode=root_stop, complete model={"coefficients":[-11,1],"x":11}. Give (mode, current_iterate, function_value, derivative_value, root_first_status, root_stop_qualified).

The four modes have7/6/6/7 fields. residual_equals_forward_error is an instance equality flag, not a universal identification; the correct affine relation includes abs(slope).

View answer and explanation(root_stop, 11, 0, 1, root_found_before_derivative_division, Yes)

A nonroot with zero derivative has no Newton update and is not silently repaired. An exact root stops even if the derivative is0. A finite trace is not a convergence/rate proof. For a nonzero affine slope, residual=abs(slope)*forward_error; numerical equality in one row does not identify the two quantities universally. The exact result, retaining the requested order, is (root_stop, 11, 0, 1, root_found_before_derivative_division, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0086/)

Newton method for root finding · qf-mod50-newton-boundary · candidate





## 87. Root-First Stopping, Zero Derivatives and Residual Scaling Stay Distinct

Use mode=residual_forward_confusion, complete model={"slope":13,"intercept":-12,"approximation":14}. Give (mode, slope, exact_root, absolute_residual, forward_error, residual_equals_forward_error, scale_relation).

The four modes have7/6/6/7 fields. residual_equals_forward_error is an instance equality flag, not a universal identification; the correct affine relation includes abs(slope).

View answer and explanation(residual_forward_confusion, 13, 12/13, 170, 170/13, No, residual_equals_abs_slope_times_forward_error_not_forward_error_itself)

A nonroot with zero derivative has no Newton update and is not silently repaired. An exact root stops even if the derivative is0. A finite trace is not a convergence/rate proof. For a nonzero affine slope, residual=abs(slope)*forward_error; numerical equality in one row does not identify the two quantities universally. The exact result, retaining the requested order, is (residual_forward_confusion, 13, 12/13, 170, 170/13, No, residual_equals_abs_slope_times_forward_error_not_forward_error_itself).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0087/)

Newton method for root finding · qf-mod50-newton-boundary · candidate





## 88. Root-First Stopping, Zero Derivatives and Residual Scaling Stay Distinct

Use mode=zero_derivative, complete model={"coefficients":[13,0,1],"x":0}. Give (mode, current_iterate, function_value, derivative_value, status, update_qualification_claim, boundary).

The four modes have7/6/6/7 fields. residual_equals_forward_error is an instance equality flag, not a universal identification; the correct affine relation includes abs(slope).

View answer and explanation(zero_derivative, 0, 13, 0, undefined_zero_derivative, No, division_by_zero_is_rejected_not_repaired)

A nonroot with zero derivative has no Newton update and is not silently repaired. An exact root stops even if the derivative is0. A finite trace is not a convergence/rate proof. For a nonzero affine slope, residual=abs(slope)*forward_error; numerical equality in one row does not identify the two quantities universally. The exact result, retaining the requested order, is (zero_derivative, 0, 13, 0, undefined_zero_derivative, No, division_by_zero_is_rejected_not_repaired).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0088/)

Newton method for root finding · qf-mod50-newton-boundary · candidate





## 89. Root-First Stopping, Zero Derivatives and Residual Scaling Stay Distinct

Use mode=finite_trace_only, complete model={"coefficients":[-14,0,1],"x0":5,"steps":1}. Give (mode, step_budget, step_records, status, convergence_or_rate_proof_claim, boundary).

The four modes have7/6/6/7 fields. residual_equals_forward_error is an instance equality flag, not a universal identification; the correct affine relation includes abs(slope).

View answer and explanation(finite_trace_only, 1, [[0, 5, 11, 10, updated, 39/10]], step_budget_exhausted, No, finite_residual_behavior_does_not_prove_convergence_or_rate)

A nonroot with zero derivative has no Newton update and is not silently repaired. An exact root stops even if the derivative is0. A finite trace is not a convergence/rate proof. For a nonzero affine slope, residual=abs(slope)*forward_error; numerical equality in one row does not identify the two quantities universally. The exact result, retaining the requested order, is (finite_trace_only, 1, [[0, 5, 11, 10, updated, 39/10]], step_budget_exhausted, No, finite_residual_behavior_does_not_prove_convergence_or_rate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0089/)

Newton method for root finding · qf-mod50-newton-boundary · candidate





## 90. Root-First Stopping, Zero Derivatives and Residual Scaling Stay Distinct

Use mode=root_stop, complete model={"coefficients":[-15,1],"x":15}. Give (mode, current_iterate, function_value, derivative_value, root_first_status, root_stop_qualified).

The four modes have7/6/6/7 fields. residual_equals_forward_error is an instance equality flag, not a universal identification; the correct affine relation includes abs(slope).

View answer and explanation(root_stop, 15, 0, 1, root_found_before_derivative_division, Yes)

A nonroot with zero derivative has no Newton update and is not silently repaired. An exact root stops even if the derivative is0. A finite trace is not a convergence/rate proof. For a nonzero affine slope, residual=abs(slope)*forward_error; numerical equality in one row does not identify the two quantities universally. The exact result, retaining the requested order, is (root_stop, 15, 0, 1, root_found_before_derivative_division, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0090/)

Newton method for root finding · qf-mod50-newton-boundary · candidate





## 91. Root-First Stopping, Zero Derivatives and Residual Scaling Stay Distinct

Use mode=residual_forward_confusion, complete model={"slope":17,"intercept":-16,"approximation":18}. Give (mode, slope, exact_root, absolute_residual, forward_error, residual_equals_forward_error, scale_relation).

The four modes have7/6/6/7 fields. residual_equals_forward_error is an instance equality flag, not a universal identification; the correct affine relation includes abs(slope).

View answer and explanation(residual_forward_confusion, 17, 16/17, 290, 290/17, No, residual_equals_abs_slope_times_forward_error_not_forward_error_itself)

A nonroot with zero derivative has no Newton update and is not silently repaired. An exact root stops even if the derivative is0. A finite trace is not a convergence/rate proof. For a nonzero affine slope, residual=abs(slope)*forward_error; numerical equality in one row does not identify the two quantities universally. The exact result, retaining the requested order, is (residual_forward_confusion, 17, 16/17, 290, 290/17, No, residual_equals_abs_slope_times_forward_error_not_forward_error_itself).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0091/)

Newton method for root finding · qf-mod50-newton-boundary · candidate





## 92. Root-First Stopping, Zero Derivatives and Residual Scaling Stay Distinct

Use mode=zero_derivative, complete model={"coefficients":[17,0,1],"x":0}. Give (mode, current_iterate, function_value, derivative_value, status, update_qualification_claim, boundary).

The four modes have7/6/6/7 fields. residual_equals_forward_error is an instance equality flag, not a universal identification; the correct affine relation includes abs(slope).

View answer and explanation(zero_derivative, 0, 17, 0, undefined_zero_derivative, No, division_by_zero_is_rejected_not_repaired)

A nonroot with zero derivative has no Newton update and is not silently repaired. An exact root stops even if the derivative is0. A finite trace is not a convergence/rate proof. For a nonzero affine slope, residual=abs(slope)*forward_error; numerical equality in one row does not identify the two quantities universally. The exact result, retaining the requested order, is (zero_derivative, 0, 17, 0, undefined_zero_derivative, No, division_by_zero_is_rejected_not_repaired).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0092/)

Newton method for root finding · qf-mod50-newton-boundary · candidate





## 93. Root-First Stopping, Zero Derivatives and Residual Scaling Stay Distinct

Use mode=finite_trace_only, complete model={"coefficients":[-18,0,1],"x0":4,"steps":1}. Give (mode, step_budget, step_records, status, convergence_or_rate_proof_claim, boundary).

The four modes have7/6/6/7 fields. residual_equals_forward_error is an instance equality flag, not a universal identification; the correct affine relation includes abs(slope).

View answer and explanation(finite_trace_only, 1, [[0, 4, -2, 8, updated, 17/4]], step_budget_exhausted, No, finite_residual_behavior_does_not_prove_convergence_or_rate)

A nonroot with zero derivative has no Newton update and is not silently repaired. An exact root stops even if the derivative is0. A finite trace is not a convergence/rate proof. For a nonzero affine slope, residual=abs(slope)*forward_error; numerical equality in one row does not identify the two quantities universally. The exact result, retaining the requested order, is (finite_trace_only, 1, [[0, 4, -2, 8, updated, 17/4]], step_budget_exhausted, No, finite_residual_behavior_does_not_prove_convergence_or_rate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0093/)

Newton method for root finding · qf-mod50-newton-boundary · candidate





## 94. Root-First Stopping, Zero Derivatives and Residual Scaling Stay Distinct

Use mode=root_stop, complete model={"coefficients":[-19,1],"x":19}. Give (mode, current_iterate, function_value, derivative_value, root_first_status, root_stop_qualified).

The four modes have7/6/6/7 fields. residual_equals_forward_error is an instance equality flag, not a universal identification; the correct affine relation includes abs(slope).

View answer and explanation(root_stop, 19, 0, 1, root_found_before_derivative_division, Yes)

A nonroot with zero derivative has no Newton update and is not silently repaired. An exact root stops even if the derivative is0. A finite trace is not a convergence/rate proof. For a nonzero affine slope, residual=abs(slope)*forward_error; numerical equality in one row does not identify the two quantities universally. The exact result, retaining the requested order, is (root_stop, 19, 0, 1, root_found_before_derivative_division, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0094/)

Newton method for root finding · qf-mod50-newton-boundary · candidate





## 95. Root-First Stopping, Zero Derivatives and Residual Scaling Stay Distinct

Use mode=residual_forward_confusion, complete model={"slope":21,"intercept":-20,"approximation":22}. Give (mode, slope, exact_root, absolute_residual, forward_error, residual_equals_forward_error, scale_relation).

The four modes have7/6/6/7 fields. residual_equals_forward_error is an instance equality flag, not a universal identification; the correct affine relation includes abs(slope).

View answer and explanation(residual_forward_confusion, 21, 20/21, 442, 442/21, No, residual_equals_abs_slope_times_forward_error_not_forward_error_itself)

A nonroot with zero derivative has no Newton update and is not silently repaired. An exact root stops even if the derivative is0. A finite trace is not a convergence/rate proof. For a nonzero affine slope, residual=abs(slope)*forward_error; numerical equality in one row does not identify the two quantities universally. The exact result, retaining the requested order, is (residual_forward_confusion, 21, 20/21, 442, 442/21, No, residual_equals_abs_slope_times_forward_error_not_forward_error_itself).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0095/)

Newton method for root finding · qf-mod50-newton-boundary · candidate





## 96. Root-First Stopping, Zero Derivatives and Residual Scaling Stay Distinct

Use mode=zero_derivative, complete model={"coefficients":[21,0,1],"x":0}. Give (mode, current_iterate, function_value, derivative_value, status, update_qualification_claim, boundary).

The four modes have7/6/6/7 fields. residual_equals_forward_error is an instance equality flag, not a universal identification; the correct affine relation includes abs(slope).

View answer and explanation(zero_derivative, 0, 21, 0, undefined_zero_derivative, No, division_by_zero_is_rejected_not_repaired)

A nonroot with zero derivative has no Newton update and is not silently repaired. An exact root stops even if the derivative is0. A finite trace is not a convergence/rate proof. For a nonzero affine slope, residual=abs(slope)*forward_error; numerical equality in one row does not identify the two quantities universally. The exact result, retaining the requested order, is (zero_derivative, 0, 21, 0, undefined_zero_derivative, No, division_by_zero_is_rejected_not_repaired).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0096/)

Newton method for root finding · qf-mod50-newton-boundary · candidate





## 97. Root-First Stopping, Zero Derivatives and Residual Scaling Stay Distinct

Use mode=finite_trace_only, complete model={"coefficients":[-22,0,1],"x0":3,"steps":1}. Give (mode, step_budget, step_records, status, convergence_or_rate_proof_claim, boundary).

The four modes have7/6/6/7 fields. residual_equals_forward_error is an instance equality flag, not a universal identification; the correct affine relation includes abs(slope).

View answer and explanation(finite_trace_only, 1, [[0, 3, -13, 6, updated, 31/6]], step_budget_exhausted, No, finite_residual_behavior_does_not_prove_convergence_or_rate)

A nonroot with zero derivative has no Newton update and is not silently repaired. An exact root stops even if the derivative is0. A finite trace is not a convergence/rate proof. For a nonzero affine slope, residual=abs(slope)*forward_error; numerical equality in one row does not identify the two quantities universally. The exact result, retaining the requested order, is (finite_trace_only, 1, [[0, 3, -13, 6, updated, 31/6]], step_budget_exhausted, No, finite_residual_behavior_does_not_prove_convergence_or_rate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0097/)

Newton method for root finding · qf-mod50-newton-boundary · candidate





## 98. Root-First Stopping, Zero Derivatives and Residual Scaling Stay Distinct

Use mode=root_stop, complete model={"coefficients":[-23,1],"x":23}. Give (mode, current_iterate, function_value, derivative_value, root_first_status, root_stop_qualified).

The four modes have7/6/6/7 fields. residual_equals_forward_error is an instance equality flag, not a universal identification; the correct affine relation includes abs(slope).

View answer and explanation(root_stop, 23, 0, 1, root_found_before_derivative_division, Yes)

A nonroot with zero derivative has no Newton update and is not silently repaired. An exact root stops even if the derivative is0. A finite trace is not a convergence/rate proof. For a nonzero affine slope, residual=abs(slope)*forward_error; numerical equality in one row does not identify the two quantities universally. The exact result, retaining the requested order, is (root_stop, 23, 0, 1, root_found_before_derivative_division, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0098/)

Newton method for root finding · qf-mod50-newton-boundary · candidate





## 99. Root-First Stopping, Zero Derivatives and Residual Scaling Stay Distinct

Use mode=residual_forward_confusion, complete model={"slope":25,"intercept":-24,"approximation":26}. Give (mode, slope, exact_root, absolute_residual, forward_error, residual_equals_forward_error, scale_relation).

The four modes have7/6/6/7 fields. residual_equals_forward_error is an instance equality flag, not a universal identification; the correct affine relation includes abs(slope).

View answer and explanation(residual_forward_confusion, 25, 24/25, 626, 626/25, No, residual_equals_abs_slope_times_forward_error_not_forward_error_itself)

A nonroot with zero derivative has no Newton update and is not silently repaired. An exact root stops even if the derivative is0. A finite trace is not a convergence/rate proof. For a nonzero affine slope, residual=abs(slope)*forward_error; numerical equality in one row does not identify the two quantities universally. The exact result, retaining the requested order, is (residual_forward_confusion, 25, 24/25, 626, 626/25, No, residual_equals_abs_slope_times_forward_error_not_forward_error_itself).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0099/)

Newton method for root finding · qf-mod50-newton-boundary · candidate





## 100. Root-First Stopping, Zero Derivatives and Residual Scaling Stay Distinct

Use mode=zero_derivative, complete model={"coefficients":[25,0,1],"x":0}. Give (mode, current_iterate, function_value, derivative_value, status, update_qualification_claim, boundary).

The four modes have7/6/6/7 fields. residual_equals_forward_error is an instance equality flag, not a universal identification; the correct affine relation includes abs(slope).

View answer and explanation(zero_derivative, 0, 25, 0, undefined_zero_derivative, No, division_by_zero_is_rejected_not_repaired)

A nonroot with zero derivative has no Newton update and is not silently repaired. An exact root stops even if the derivative is0. A finite trace is not a convergence/rate proof. For a nonzero affine slope, residual=abs(slope)*forward_error; numerical equality in one row does not identify the two quantities universally. The exact result, retaining the requested order, is (zero_derivative, 0, 25, 0, undefined_zero_derivative, No, division_by_zero_is_rejected_not_repaired).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-pulse-20261119-r01-0100/)

Newton method for root finding · qf-mod50-newton-boundary · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261015-advance-20261119-r01-newton-method-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261015-advance-20261119-r01-newton-method-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261015-advance-20261119-r01-newton-method-foundations/validation-report.json)