# Question 26369: An Exact Closed-Interval Root Witness Does Not Assert Uniqueness

[Return to catalogue page 132](https://ocme.evemisslab.com/en/questions/page/132/)

[← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations/#q-20261015-advance-20261116-r02-intermediate-value-theorem-0044)

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

R0051-26369 · q-20261015-advance-20261116-r02-intermediate-value-theorem-0044



## Original question and givens

Coefficients=[-6, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=5<b=7 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=6. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).


- G001 — Coefficients=[-6, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=5<b=7 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=6. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).





## Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.


- [mko-intermediate-value-theorem](https://ocme.evemisslab.com/en/mko/mko-intermediate-value-theorem/)





## Step-by-step derivation

S001 · impliesBefore

source roles = [ivt-exact-root]

After

derived = [[root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate]]

Conditions: none

List the full source field order without JavaScript.Original modes select contract6/4/6/6 and IVT8/7/8 roles,without padding or answer-shape inference.

Step identifiers and dependenciesRule: source_ordered_fields_without_JS · Uses: G001





S002 · impliesBefore

(5)^0

After

IVT ORIGINAL left endpoint ORIGINAL argument power0 = 1

Conditions: none

Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity.

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





S003 · calculationBefore

IVT ORIGINAL left endpoint coefficient term0 := mul(-6, 1)

After

IVT ORIGINAL left endpoint coefficient term0 = -6

Conditions: none

Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain.

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





S004 · calculationBefore

IVT ORIGINAL left endpoint ascending partial sum0 := sum(0, -6)

After

IVT ORIGINAL left endpoint ascending partial sum0 = -6

Conditions: none

Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain.

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





S005 · impliesBefore

(5)^1

After

IVT ORIGINAL left endpoint ORIGINAL argument power1 = 5

Conditions: none

Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity.

Step identifiers and dependenciesRule: integer_power · Uses: G001, S004





S006 · calculationBefore

IVT ORIGINAL left endpoint coefficient term1 := mul(1, 5)

After

IVT ORIGINAL left endpoint coefficient term1 = 5

Conditions: none

Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005





S007 · calculationBefore

IVT ORIGINAL left endpoint ascending partial sum1 := sum(-6, 5)

After

IVT ORIGINAL left endpoint ascending partial sum1 = -1

Conditions: none

Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006





S008 · impliesBefore

(7)^0

After

IVT ORIGINAL right endpoint ORIGINAL argument power0 = 1

Conditions: none

Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity.

Step identifiers and dependenciesRule: integer_power · Uses: G001, S007





S009 · calculationBefore

IVT ORIGINAL right endpoint coefficient term0 := mul(-6, 1)

After

IVT ORIGINAL right endpoint coefficient term0 = -6

Conditions: none

Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008





S010 · calculationBefore

IVT ORIGINAL right endpoint ascending partial sum0 := sum(0, -6)

After

IVT ORIGINAL right endpoint ascending partial sum0 = -6

Conditions: none

Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009





S011 · impliesBefore

(7)^1

After

IVT ORIGINAL right endpoint ORIGINAL argument power1 = 7

Conditions: none

Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity.

Step identifiers and dependenciesRule: integer_power · Uses: G001, S010





S012 · calculationBefore

IVT ORIGINAL right endpoint coefficient term1 := mul(1, 7)

After

IVT ORIGINAL right endpoint coefficient term1 = 7

Conditions: none

Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011





S013 · calculationBefore

IVT ORIGINAL right endpoint ascending partial sum1 := sum(-6, 7)

After

IVT ORIGINAL right endpoint ascending partial sum1 = 1

Conditions: none

Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012





S014 · impliesBefore

source roles = [[-6, 1], 5, 7, 0]

After

derived = [-1, 1, Yes, Yes]

Conditions: none

Every monomial and finite polynomial sum is continuous on the whole interval,not by sampled agreement.Target bracketing allows increasing or decreasing endpoint values and equality;qualification may report bracketFalse.

Step identifiers and dependenciesRule: IVT_WHOLE_polynomial_continuity_inclusive_both_endpoint_orientations · Uses: G001, S013





S015 · impliesBefore

(6)^0

After

IVT ORIGINAL supplied root ORIGINAL argument power0 = 1

Conditions: none

Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity.

Step identifiers and dependenciesRule: integer_power · Uses: G001, S014





S016 · calculationBefore

IVT ORIGINAL supplied root coefficient term0 := mul(-6, 1)

After

IVT ORIGINAL supplied root coefficient term0 = -6

Conditions: none

Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015





S017 · calculationBefore

IVT ORIGINAL supplied root ascending partial sum0 := sum(0, -6)

After

IVT ORIGINAL supplied root ascending partial sum0 = -6

Conditions: none

Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016





S018 · impliesBefore

(6)^1

After

IVT ORIGINAL supplied root ORIGINAL argument power1 = 6

Conditions: none

Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity.

Step identifiers and dependenciesRule: integer_power · Uses: G001, S017





S019 · calculationBefore

IVT ORIGINAL supplied root coefficient term1 := mul(1, 6)

After

IVT ORIGINAL supplied root coefficient term1 = 6

Conditions: none

Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018





S020 · calculationBefore

IVT ORIGINAL supplied root ascending partial sum1 := sum(-6, 6)

After

IVT ORIGINAL supplied root ascending partial sum1 = 0

Conditions: none

Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019





S021 · impliesBefore

source roles = [5, 7, 0, 6]

After

derived = [0, Yes, Yes, No, No]

Conditions: none

Check the supplied exact root value and closed-interval membership.An endpoint-family witness may be interior,with flagFalse.The fixed No on exact-root withholds general uniqueness/rational closed-form guarantees,not this witness.

Step identifiers and dependenciesRule: IVT_actual_closed_interval_root_hit_endpoint_flag_separate · Uses: G001, S020









## Reverse checks


- S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S020 · premise_and_result_audit — Recompute IVT ORIGINAL supplied root ascending partial sum1 with exact rational arithmetic, not rounded decimals.
- S019 · premise_and_result_audit — Recompute IVT ORIGINAL supplied root coefficient term1 with exact rational arithmetic, not rounded decimals.
- S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S017 · premise_and_result_audit — Recompute IVT ORIGINAL supplied root ascending partial sum0 with exact rational arithmetic, not rounded decimals.
- S016 · premise_and_result_audit — Recompute IVT ORIGINAL supplied root coefficient term0 with exact rational arithmetic, not rounded decimals.
- S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S013 · premise_and_result_audit — Recompute IVT ORIGINAL right endpoint ascending partial sum1 with exact rational arithmetic, not rounded decimals.
- S012 · premise_and_result_audit — Recompute IVT ORIGINAL right endpoint coefficient term1 with exact rational arithmetic, not rounded decimals.
- S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S010 · premise_and_result_audit — Recompute IVT ORIGINAL right endpoint ascending partial sum0 with exact rational arithmetic, not rounded decimals.
- S009 · premise_and_result_audit — Recompute IVT ORIGINAL right endpoint coefficient term0 with exact rational arithmetic, not rounded decimals.
- S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S007 · premise_and_result_audit — Recompute IVT ORIGINAL left endpoint ascending partial sum1 with exact rational arithmetic, not rounded decimals.
- S006 · premise_and_result_audit — Recompute IVT ORIGINAL left endpoint coefficient term1 with exact rational arithmetic, not rounded decimals.
- S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S004 · premise_and_result_audit — Recompute IVT ORIGINAL left endpoint ascending partial sum0 with exact rational arithmetic, not rounded decimals.
- S003 · premise_and_result_audit — Recompute IVT ORIGINAL left endpoint coefficient term0 with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

(6, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)

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





## This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: bfa26985761a100e4a1fea7c1223d4b8365b7f47f373ef3c5ac2d3b5c81e976a
- candidate record SHA-256: 63bbde1f5b6dd1680c6cd3c8042741a04472e524a1ef206f16b754a08be0421d
- semantic review receipt SHA-256: 90df1e889242e9afdf01e068788ee8500ef366ffcfc0496e30a6800de32631c3
- question-first witness SHA-256: ae1e849608590ca856c9f3b6911ba58e3bf53c608eb7369606192a770f48bd5c
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261116-r02-intermediate-value-theorem-0044.json)