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R0051-26357 · q-20261015-advance-20261116-r02-intermediate-value-theorem-0032
Coefficients=[6, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-7<b=-5 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=-7<b=-5 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).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.
Before
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.
Rule: source_ordered_fields_without_JS · Uses: G001
Before
(-7)^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.
Rule: integer_power · Uses: G001, S001
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
(-7)^1
After
IVT ORIGINAL left endpoint ORIGINAL argument power1 = -7
Conditions: none
Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity.
Rule: integer_power · Uses: G001, S004
Before
IVT ORIGINAL left endpoint coefficient term1 := mul(1, -7)
After
IVT ORIGINAL left endpoint coefficient term1 = -7
Conditions: none
Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
IVT ORIGINAL left endpoint ascending partial sum1 := sum(6, -7)
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.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
(-5)^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.
Rule: integer_power · Uses: G001, S007
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
(-5)^1
After
IVT ORIGINAL right endpoint ORIGINAL argument power1 = -5
Conditions: none
Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity.
Rule: integer_power · Uses: G001, S010
Before
IVT ORIGINAL right endpoint coefficient term1 := mul(1, -5)
After
IVT ORIGINAL right endpoint coefficient term1 = -5
Conditions: none
Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
IVT ORIGINAL right endpoint ascending partial sum1 := sum(6, -5)
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.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
source roles = [[6, 1], -7, -5, 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.
Rule: IVT_WHOLE_polynomial_continuity_inclusive_both_endpoint_orientations · Uses: G001, S013
Before
(-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.
Rule: integer_power · Uses: G001, S014
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
(-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.
Rule: integer_power · Uses: G001, S017
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S019
Before
source roles = [-7, -5, 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.
Rule: IVT_actual_closed_interval_root_hit_endpoint_flag_separate · Uses: G001, S020
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.(-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.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 390d61f6146a1002dcca6ceaa05658ef6bff1aca14dc7e61584eec68776e22d7candidate record SHA-256: 546298c9050a47418e4011cc5db406d12517cdf4ae6f132c9ca8e0a695a318e3semantic review receipt SHA-256: 90df1e889242e9afdf01e068788ee8500ef366ffcfc0496e30a6800de32631c3question-first witness SHA-256: ae1e849608590ca856c9f3b6911ba58e3bf53c608eb7369606192a770f48bd5cLean: NOT_RUN · human review: NOT_PERFORMED