# Question 26448: A Constant Next Derivative Gives a Strictly Intermediate Lagrange Witness

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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.

R0051-26448 · q-20261015-advance-20261116-r03-taylor-remainder-theorem-0073



## Original question and givens

Coefficients=[7, 13, 2, 8, 24] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=3; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, constant_next_derivative, intermediate_point, strictly_between, Lagrange_remainder, actual_remainder, remainders_equal, certificate).


- G001 — Coefficients=[7, 13, 2, 8, 24] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=3; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, constant_next_derivative, intermediate_point, strictly_between, Lagrange_remainder, actual_remainder, remainders_equal, 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-taylor-remainder-theorem](https://ocme.evemisslab.com/en/mko/mko-taylor-remainder-theorem/)





## Step-by-step derivation

S001 · impliesBefore

source roles = [taylor-lagrange-witness]

After

derived = [[degree, order, constant_next_derivative, intermediate_point, strictly_between, Lagrange_remainder, actual_remainder, remainders_equal, 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 · calculationBefore

Taylor ORIGINAL x minus declared center := sub(1/2, 0)

After

Taylor ORIGINAL x minus declared center = 1/2

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, S001





S003 · impliesBefore

(0)^0

After

Taylor ORIGINAL derivative0 center power0 term0 = 1

Conditions: none

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

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





S004 · calculationBefore

Taylor ORIGINAL derivative0 falling factor times power term0 := mul(1, 1)

After

Taylor ORIGINAL derivative0 falling factor times power term0 = 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, S003





S005 · calculationBefore

Taylor ORIGINAL derivative0 ORIGINAL coefficient term0 := mul(7, 1)

After

Taylor ORIGINAL derivative0 ORIGINAL coefficient term0 = 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, S004





S006 · calculationBefore

Taylor ORIGINAL derivative0 ascending derivative sum term0 := sum(0, 7)

After

Taylor ORIGINAL derivative0 ascending derivative sum term0 = 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, S005





S007 · impliesBefore

(0)^1

After

Taylor ORIGINAL derivative0 center power1 term1 = 0

Conditions: none

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

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





S008 · calculationBefore

Taylor ORIGINAL derivative0 falling factor times power term1 := mul(1, 0)

After

Taylor ORIGINAL derivative0 falling factor times power term1 = 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, S007





S009 · calculationBefore

Taylor ORIGINAL derivative0 ORIGINAL coefficient term1 := mul(13, 0)

After

Taylor ORIGINAL derivative0 ORIGINAL coefficient term1 = 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, S008





S010 · calculationBefore

Taylor ORIGINAL derivative0 ascending derivative sum term1 := sum(7, 0)

After

Taylor ORIGINAL derivative0 ascending derivative sum 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, S009





S011 · impliesBefore

(0)^2

After

Taylor ORIGINAL derivative0 center power2 term2 = 0

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

Taylor ORIGINAL derivative0 falling factor times power term2 := mul(1, 0)

After

Taylor ORIGINAL derivative0 falling factor times power term2 = 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, S011





S013 · calculationBefore

Taylor ORIGINAL derivative0 ORIGINAL coefficient term2 := mul(2, 0)

After

Taylor ORIGINAL derivative0 ORIGINAL coefficient term2 = 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, S012





S014 · calculationBefore

Taylor ORIGINAL derivative0 ascending derivative sum term2 := sum(7, 0)

After

Taylor ORIGINAL derivative0 ascending derivative sum term2 = 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, S013





S015 · impliesBefore

(0)^3

After

Taylor ORIGINAL derivative0 center power3 term3 = 0

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

Taylor ORIGINAL derivative0 falling factor times power term3 := mul(1, 0)

After

Taylor ORIGINAL derivative0 falling factor times power term3 = 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, S015





S017 · calculationBefore

Taylor ORIGINAL derivative0 ORIGINAL coefficient term3 := mul(8, 0)

After

Taylor ORIGINAL derivative0 ORIGINAL coefficient term3 = 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, S016





S018 · calculationBefore

Taylor ORIGINAL derivative0 ascending derivative sum term3 := sum(7, 0)

After

Taylor ORIGINAL derivative0 ascending derivative sum term3 = 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, S017





S019 · impliesBefore

(0)^4

After

Taylor ORIGINAL derivative0 center power4 term4 = 0

Conditions: none

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

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





S020 · calculationBefore

Taylor ORIGINAL derivative0 falling factor times power term4 := mul(1, 0)

After

Taylor ORIGINAL derivative0 falling factor times power term4 = 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 · calculationBefore

Taylor ORIGINAL derivative0 ORIGINAL coefficient term4 := mul(24, 0)

After

Taylor ORIGINAL derivative0 ORIGINAL coefficient term4 = 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, S020





S022 · calculationBefore

Taylor ORIGINAL derivative0 ascending derivative sum term4 := sum(7, 0)

After

Taylor ORIGINAL derivative0 ascending derivative sum term4 = 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, S021





S023 · impliesBefore

source roles = [0, 0, [7, 13, 2, 8, 24]]

After

derived = [[[0, 7, 1, 1, 7], [1, 13, 1, 0, 0], [2, 2, 1, 0, 0], [3, 8, 1, 0, 0], [4, 24, 1, 0, 0]], 7]

Conditions: none

The kth derivative of x^j is j!/(j−k)!*x^(j−k);terms j<k vanish.Sum the complete ascending-order terms,with empty sum0.The whole polynomial is known,not just a center jet.

Step identifiers and dependenciesRule: Taylor_EACH_derivative_complete_falling_factorial_source_order · Uses: G001, S022





S024 · impliesBefore

(1/2)^0

After

Taylor ORIGINAL displacement 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, S023





S025 · calculationBefore

Taylor ORIGINAL derivative times displacement0 := mul(7, 1)

After

Taylor ORIGINAL derivative times displacement0 = 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, S024





S026 · calculationBefore

Taylor ORIGINAL term0 divide factorial := div(7, 1)

After

Taylor ORIGINAL term0 divide factorial = 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, S025





S027 · impliesBefore

(0)^0

After

Taylor ORIGINAL derivative1 center power0 term1 = 1

Conditions: none

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

Step identifiers and dependenciesRule: integer_power · Uses: G001, S026





S028 · calculationBefore

Taylor ORIGINAL derivative1 falling factor times power term1 := mul(1, 1)

After

Taylor ORIGINAL derivative1 falling factor times power term1 = 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, S027





S029 · calculationBefore

Taylor ORIGINAL derivative1 ORIGINAL coefficient term1 := mul(13, 1)

After

Taylor ORIGINAL derivative1 ORIGINAL coefficient term1 = 13

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, S028





S030 · calculationBefore

Taylor ORIGINAL derivative1 ascending derivative sum term1 := sum(0, 13)

After

Taylor ORIGINAL derivative1 ascending derivative sum term1 = 13

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, S029





S031 · impliesBefore

(0)^1

After

Taylor ORIGINAL derivative1 center power1 term2 = 0

Conditions: none

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

Step identifiers and dependenciesRule: integer_power · Uses: G001, S030





S032 · calculationBefore

Taylor ORIGINAL derivative1 falling factor times power term2 := mul(2, 0)

After

Taylor ORIGINAL derivative1 falling factor times power term2 = 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, S031





S033 · calculationBefore

Taylor ORIGINAL derivative1 ORIGINAL coefficient term2 := mul(2, 0)

After

Taylor ORIGINAL derivative1 ORIGINAL coefficient term2 = 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, S032





S034 · calculationBefore

Taylor ORIGINAL derivative1 ascending derivative sum term2 := sum(13, 0)

After

Taylor ORIGINAL derivative1 ascending derivative sum term2 = 13

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, S033





S035 · impliesBefore

(0)^2

After

Taylor ORIGINAL derivative1 center power2 term3 = 0

Conditions: none

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

Step identifiers and dependenciesRule: integer_power · Uses: G001, S034





S036 · calculationBefore

Taylor ORIGINAL derivative1 falling factor times power term3 := mul(3, 0)

After

Taylor ORIGINAL derivative1 falling factor times power term3 = 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, S035





S037 · calculationBefore

Taylor ORIGINAL derivative1 ORIGINAL coefficient term3 := mul(8, 0)

After

Taylor ORIGINAL derivative1 ORIGINAL coefficient term3 = 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, S036





S038 · calculationBefore

Taylor ORIGINAL derivative1 ascending derivative sum term3 := sum(13, 0)

After

Taylor ORIGINAL derivative1 ascending derivative sum term3 = 13

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, S037





S039 · impliesBefore

(0)^3

After

Taylor ORIGINAL derivative1 center power3 term4 = 0

Conditions: none

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

Step identifiers and dependenciesRule: integer_power · Uses: G001, S038





S040 · calculationBefore

Taylor ORIGINAL derivative1 falling factor times power term4 := mul(4, 0)

After

Taylor ORIGINAL derivative1 falling factor times power term4 = 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, S039





S041 · calculationBefore

Taylor ORIGINAL derivative1 ORIGINAL coefficient term4 := mul(24, 0)

After

Taylor ORIGINAL derivative1 ORIGINAL coefficient term4 = 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, S040





S042 · calculationBefore

Taylor ORIGINAL derivative1 ascending derivative sum term4 := sum(13, 0)

After

Taylor ORIGINAL derivative1 ascending derivative sum term4 = 13

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, S041





S043 · impliesBefore

source roles = [1, 0, [7, 13, 2, 8, 24]]

After

derived = [[[1, 13, 1, 1, 13], [2, 2, 2, 0, 0], [3, 8, 3, 0, 0], [4, 24, 4, 0, 0]], 13]

Conditions: none

The kth derivative of x^j is j!/(j−k)!*x^(j−k);terms j<k vanish.Sum the complete ascending-order terms,with empty sum0.The whole polynomial is known,not just a center jet.

Step identifiers and dependenciesRule: Taylor_EACH_derivative_complete_falling_factorial_source_order · Uses: G001, S042





S044 · impliesBefore

(1/2)^1

After

Taylor ORIGINAL displacement power1 = 1/2

Conditions: none

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

Step identifiers and dependenciesRule: integer_power · Uses: G001, S043





S045 · calculationBefore

Taylor ORIGINAL derivative times displacement1 := mul(13, 1/2)

After

Taylor ORIGINAL derivative times displacement1 = 13/2

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, S044





S046 · calculationBefore

Taylor ORIGINAL term1 divide factorial := div(13/2, 1)

After

Taylor ORIGINAL term1 divide factorial = 13/2

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, S045





S047 · impliesBefore

(0)^0

After

Taylor ORIGINAL derivative2 center power0 term2 = 1

Conditions: none

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

Step identifiers and dependenciesRule: integer_power · Uses: G001, S046





S048 · calculationBefore

Taylor ORIGINAL derivative2 falling factor times power term2 := mul(2, 1)

After

Taylor ORIGINAL derivative2 falling factor times power term2 = 2

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, S047





S049 · calculationBefore

Taylor ORIGINAL derivative2 ORIGINAL coefficient term2 := mul(2, 2)

After

Taylor ORIGINAL derivative2 ORIGINAL coefficient term2 = 4

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, S048





S050 · calculationBefore

Taylor ORIGINAL derivative2 ascending derivative sum term2 := sum(0, 4)

After

Taylor ORIGINAL derivative2 ascending derivative sum term2 = 4

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, S049





S051 · impliesBefore

(0)^1

After

Taylor ORIGINAL derivative2 center power1 term3 = 0

Conditions: none

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

Step identifiers and dependenciesRule: integer_power · Uses: G001, S050





S052 · calculationBefore

Taylor ORIGINAL derivative2 falling factor times power term3 := mul(6, 0)

After

Taylor ORIGINAL derivative2 falling factor times power term3 = 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, S051





S053 · calculationBefore

Taylor ORIGINAL derivative2 ORIGINAL coefficient term3 := mul(8, 0)

After

Taylor ORIGINAL derivative2 ORIGINAL coefficient term3 = 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, S052





S054 · calculationBefore

Taylor ORIGINAL derivative2 ascending derivative sum term3 := sum(4, 0)

After

Taylor ORIGINAL derivative2 ascending derivative sum term3 = 4

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, S053





S055 · impliesBefore

(0)^2

After

Taylor ORIGINAL derivative2 center power2 term4 = 0

Conditions: none

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

Step identifiers and dependenciesRule: integer_power · Uses: G001, S054





S056 · calculationBefore

Taylor ORIGINAL derivative2 falling factor times power term4 := mul(12, 0)

After

Taylor ORIGINAL derivative2 falling factor times power term4 = 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, S055





S057 · calculationBefore

Taylor ORIGINAL derivative2 ORIGINAL coefficient term4 := mul(24, 0)

After

Taylor ORIGINAL derivative2 ORIGINAL coefficient term4 = 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, S056





S058 · calculationBefore

Taylor ORIGINAL derivative2 ascending derivative sum term4 := sum(4, 0)

After

Taylor ORIGINAL derivative2 ascending derivative sum term4 = 4

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, S057





S059 · impliesBefore

source roles = [2, 0, [7, 13, 2, 8, 24]]

After

derived = [[[2, 2, 2, 1, 4], [3, 8, 6, 0, 0], [4, 24, 12, 0, 0]], 4]

Conditions: none

The kth derivative of x^j is j!/(j−k)!*x^(j−k);terms j<k vanish.Sum the complete ascending-order terms,with empty sum0.The whole polynomial is known,not just a center jet.

Step identifiers and dependenciesRule: Taylor_EACH_derivative_complete_falling_factorial_source_order · Uses: G001, S058





S060 · impliesBefore

(1/2)^2

After

Taylor ORIGINAL displacement power2 = 1/4

Conditions: none

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

Step identifiers and dependenciesRule: integer_power · Uses: G001, S059





S061 · calculationBefore

Taylor ORIGINAL derivative times displacement2 := mul(4, 1/4)

After

Taylor ORIGINAL derivative times displacement2 = 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, S060





S062 · calculationBefore

Taylor ORIGINAL term2 divide factorial := div(1, 2)

After

Taylor ORIGINAL term2 divide factorial = 1/2

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, S061





S063 · impliesBefore

(0)^0

After

Taylor ORIGINAL derivative3 center power0 term3 = 1

Conditions: none

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

Step identifiers and dependenciesRule: integer_power · Uses: G001, S062





S064 · calculationBefore

Taylor ORIGINAL derivative3 falling factor times power term3 := mul(6, 1)

After

Taylor ORIGINAL derivative3 falling factor times power term3 = 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, S063





S065 · calculationBefore

Taylor ORIGINAL derivative3 ORIGINAL coefficient term3 := mul(8, 6)

After

Taylor ORIGINAL derivative3 ORIGINAL coefficient term3 = 48

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, S064





S066 · calculationBefore

Taylor ORIGINAL derivative3 ascending derivative sum term3 := sum(0, 48)

After

Taylor ORIGINAL derivative3 ascending derivative sum term3 = 48

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, S065





S067 · impliesBefore

(0)^1

After

Taylor ORIGINAL derivative3 center power1 term4 = 0

Conditions: none

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

Step identifiers and dependenciesRule: integer_power · Uses: G001, S066





S068 · calculationBefore

Taylor ORIGINAL derivative3 falling factor times power term4 := mul(24, 0)

After

Taylor ORIGINAL derivative3 falling factor times power term4 = 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, S067





S069 · calculationBefore

Taylor ORIGINAL derivative3 ORIGINAL coefficient term4 := mul(24, 0)

After

Taylor ORIGINAL derivative3 ORIGINAL coefficient term4 = 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, S068





S070 · calculationBefore

Taylor ORIGINAL derivative3 ascending derivative sum term4 := sum(48, 0)

After

Taylor ORIGINAL derivative3 ascending derivative sum term4 = 48

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, S069





S071 · impliesBefore

source roles = [3, 0, [7, 13, 2, 8, 24]]

After

derived = [[[3, 8, 6, 1, 48], [4, 24, 24, 0, 0]], 48]

Conditions: none

The kth derivative of x^j is j!/(j−k)!*x^(j−k);terms j<k vanish.Sum the complete ascending-order terms,with empty sum0.The whole polynomial is known,not just a center jet.

Step identifiers and dependenciesRule: Taylor_EACH_derivative_complete_falling_factorial_source_order · Uses: G001, S070





S072 · impliesBefore

(1/2)^3

After

Taylor ORIGINAL displacement power3 = 1/8

Conditions: none

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

Step identifiers and dependenciesRule: integer_power · Uses: G001, S071





S073 · calculationBefore

Taylor ORIGINAL derivative times displacement3 := mul(48, 1/8)

After

Taylor ORIGINAL derivative times displacement3 = 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, S072





S074 · calculationBefore

Taylor ORIGINAL term3 divide factorial := div(6, 6)

After

Taylor ORIGINAL term3 divide factorial = 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, S073





S075 · calculationBefore

Taylor ORIGINAL partial sum0 := sum(0, 7)

After

Taylor ORIGINAL partial sum0 = 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, S074





S076 · calculationBefore

Taylor ORIGINAL partial sum1 := sum(7, 13/2)

After

Taylor ORIGINAL partial sum1 = 27/2

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, S075





S077 · calculationBefore

Taylor ORIGINAL partial sum2 := sum(27/2, 1/2)

After

Taylor ORIGINAL partial sum2 = 14

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, S076





S078 · calculationBefore

Taylor ORIGINAL partial sum3 := sum(14, 1)

After

Taylor ORIGINAL partial sum3 = 15

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, S077





S079 · impliesBefore

(1/2)^0

After

Taylor ORIGINAL full function at x 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, S078





S080 · calculationBefore

Taylor ORIGINAL full function at x coefficient term0 := mul(7, 1)

After

Taylor ORIGINAL full function at x coefficient term0 = 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, S079





S081 · calculationBefore

Taylor ORIGINAL full function at x ascending partial sum0 := sum(0, 7)

After

Taylor ORIGINAL full function at x ascending partial sum0 = 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, S080





S082 · impliesBefore

(1/2)^1

After

Taylor ORIGINAL full function at x ORIGINAL argument power1 = 1/2

Conditions: none

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

Step identifiers and dependenciesRule: integer_power · Uses: G001, S081





S083 · calculationBefore

Taylor ORIGINAL full function at x coefficient term1 := mul(13, 1/2)

After

Taylor ORIGINAL full function at x coefficient term1 = 13/2

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, S082





S084 · calculationBefore

Taylor ORIGINAL full function at x ascending partial sum1 := sum(7, 13/2)

After

Taylor ORIGINAL full function at x ascending partial sum1 = 27/2

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, S083





S085 · impliesBefore

(1/2)^2

After

Taylor ORIGINAL full function at x ORIGINAL argument power2 = 1/4

Conditions: none

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

Step identifiers and dependenciesRule: integer_power · Uses: G001, S084





S086 · calculationBefore

Taylor ORIGINAL full function at x coefficient term2 := mul(2, 1/4)

After

Taylor ORIGINAL full function at x coefficient term2 = 1/2

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, S085





S087 · calculationBefore

Taylor ORIGINAL full function at x ascending partial sum2 := sum(27/2, 1/2)

After

Taylor ORIGINAL full function at x ascending partial sum2 = 14

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, S086





S088 · impliesBefore

(1/2)^3

After

Taylor ORIGINAL full function at x ORIGINAL argument power3 = 1/8

Conditions: none

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

Step identifiers and dependenciesRule: integer_power · Uses: G001, S087





S089 · calculationBefore

Taylor ORIGINAL full function at x coefficient term3 := mul(8, 1/8)

After

Taylor ORIGINAL full function at x coefficient term3 = 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, S088





S090 · calculationBefore

Taylor ORIGINAL full function at x ascending partial sum3 := sum(14, 1)

After

Taylor ORIGINAL full function at x ascending partial sum3 = 15

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, S089





S091 · impliesBefore

(1/2)^4

After

Taylor ORIGINAL full function at x ORIGINAL argument power4 = 1/16

Conditions: none

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

Step identifiers and dependenciesRule: integer_power · Uses: G001, S090





S092 · calculationBefore

Taylor ORIGINAL full function at x coefficient term4 := mul(24, 1/16)

After

Taylor ORIGINAL full function at x coefficient term4 = 3/2

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, S091





S093 · calculationBefore

Taylor ORIGINAL full function at x ascending partial sum4 := sum(15, 3/2)

After

Taylor ORIGINAL full function at x ascending partial sum4 = 33/2

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, S092





S094 · calculationBefore

Taylor ORIGINAL f(x) minus Taylor value := sub(33/2, 15)

After

Taylor ORIGINAL f(x) minus Taylor value = 3/2

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, S093





S095 · impliesBefore

source roles = [[7, 13, 2, 8, 24], 0, 1/2, 3, [7, 13, 4, 48], [7, 13/2, 1/2, 1]]

After

derived = [15, 33/2, 3/2, 4]

Conditions: none

Use T_n=sum f^(k)(a)(x−a)^k/k!,including degree0 and all requested zero terms;remainder=f(x)−T_n.A polynomial differentiates on the whole interval,but a finite center jet of an arbitrary function does not supply interval hypotheses,series equality or analyticity.Empty coefficients have degree−1,distinct from[0]degree0.

Step identifiers and dependenciesRule: Taylor_WHOLE_polynomial_shifted_basis_remainder_vs_interval_hypotheses · Uses: G001, S094





S096 · impliesBefore

(0)^0

After

Taylor ORIGINAL next derivative center power0 term4 = 1

Conditions: none

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

Step identifiers and dependenciesRule: integer_power · Uses: G001, S095





S097 · calculationBefore

Taylor ORIGINAL next derivative falling factor times power term4 := mul(24, 1)

After

Taylor ORIGINAL next derivative falling factor times power term4 = 24

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, S096





S098 · calculationBefore

Taylor ORIGINAL next derivative ORIGINAL coefficient term4 := mul(24, 24)

After

Taylor ORIGINAL next derivative ORIGINAL coefficient term4 = 576

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, S097





S099 · calculationBefore

Taylor ORIGINAL next derivative ascending derivative sum term4 := sum(0, 576)

After

Taylor ORIGINAL next derivative ascending derivative sum term4 = 576

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, S098





S100 · impliesBefore

source roles = [4, 0, [7, 13, 2, 8, 24]]

After

derived = [[[4, 24, 24, 1, 576]], 576]

Conditions: none

The kth derivative of x^j is j!/(j−k)!*x^(j−k);terms j<k vanish.Sum the complete ascending-order terms,with empty sum0.The whole polynomial is known,not just a center jet.

Step identifiers and dependenciesRule: Taylor_EACH_derivative_complete_falling_factorial_source_order · Uses: G001, S099





S101 · calculationBefore

Taylor witness ORIGINAL center plus x := sum(0, 1/2)

After

Taylor witness ORIGINAL center plus x = 1/2

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, S100





S102 · calculationBefore

Taylor witness ORIGINAL midpoint := div(1/2, 2)

After

Taylor witness ORIGINAL midpoint = 1/4

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, S101





S103 · impliesBefore

(1/2)^4

After

Taylor witness ORIGINAL displacement power = 1/16

Conditions: none

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

Step identifiers and dependenciesRule: integer_power · Uses: G001, S102





S104 · calculationBefore

Taylor witness constant derivative times power := mul(576, 1/16)

After

Taylor witness constant derivative times power = 36

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, S103





S105 · calculationBefore

Taylor witness divide next factorial := div(36, 24)

After

Taylor witness divide next factorial = 3/2

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, S104





S106 · impliesBefore

source roles = [4, 3, 0, 1/2, 576, 1/4]

After

derived = [Yes, 3/2, 3/2, Yes]

Conditions: none

Degree exactly n+1 makes the next derivative constant throughout the interval.The midpoint is strictly between either endpoint orientation and gives the exact Lagrange remainder.Do not apply this witness construction to an arbitrary nonconstant next derivative.

Step identifiers and dependenciesRule: Taylor_constant_next_derivative_strict_midpoint_EITHER_orientation · Uses: G001, S105









## Reverse checks


- S106 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S105 · premise_and_result_audit — Recompute Taylor witness divide next factorial with exact rational arithmetic, not rounded decimals.
- S104 · premise_and_result_audit — Recompute Taylor witness constant derivative times power with exact rational arithmetic, not rounded decimals.
- S103 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S102 · premise_and_result_audit — Recompute Taylor witness ORIGINAL midpoint with exact rational arithmetic, not rounded decimals.
- S101 · premise_and_result_audit — Recompute Taylor witness ORIGINAL center plus x with exact rational arithmetic, not rounded decimals.
- S100 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S099 · premise_and_result_audit — Recompute Taylor ORIGINAL next derivative ascending derivative sum term4 with exact rational arithmetic, not rounded decimals.
- S098 · premise_and_result_audit — Recompute Taylor ORIGINAL next derivative ORIGINAL coefficient term4 with exact rational arithmetic, not rounded decimals.
- S097 · premise_and_result_audit — Recompute Taylor ORIGINAL next derivative falling factor times power term4 with exact rational arithmetic, not rounded decimals.
- S096 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S095 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S094 · premise_and_result_audit — Recompute Taylor ORIGINAL f(x) minus Taylor value with exact rational arithmetic, not rounded decimals.
- S093 · premise_and_result_audit — Recompute Taylor ORIGINAL full function at x ascending partial sum4 with exact rational arithmetic, not rounded decimals.
- S092 · premise_and_result_audit — Recompute Taylor ORIGINAL full function at x coefficient term4 with exact rational arithmetic, not rounded decimals.
- S091 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S090 · premise_and_result_audit — Recompute Taylor ORIGINAL full function at x ascending partial sum3 with exact rational arithmetic, not rounded decimals.
- S089 · premise_and_result_audit — Recompute Taylor ORIGINAL full function at x coefficient term3 with exact rational arithmetic, not rounded decimals.
- S088 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S087 · premise_and_result_audit — Recompute Taylor ORIGINAL full function at x ascending partial sum2 with exact rational arithmetic, not rounded decimals.
- S086 · premise_and_result_audit — Recompute Taylor ORIGINAL full function at x coefficient term2 with exact rational arithmetic, not rounded decimals.
- S085 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S084 · premise_and_result_audit — Recompute Taylor ORIGINAL full function at x ascending partial sum1 with exact rational arithmetic, not rounded decimals.
- S083 · premise_and_result_audit — Recompute Taylor ORIGINAL full function at x coefficient term1 with exact rational arithmetic, not rounded decimals.
- S082 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S081 · premise_and_result_audit — Recompute Taylor ORIGINAL full function at x ascending partial sum0 with exact rational arithmetic, not rounded decimals.
- S080 · premise_and_result_audit — Recompute Taylor ORIGINAL full function at x coefficient term0 with exact rational arithmetic, not rounded decimals.
- S079 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S078 · premise_and_result_audit — Recompute Taylor ORIGINAL partial sum3 with exact rational arithmetic, not rounded decimals.
- S077 · premise_and_result_audit — Recompute Taylor ORIGINAL partial sum2 with exact rational arithmetic, not rounded decimals.
- S076 · premise_and_result_audit — Recompute Taylor ORIGINAL partial sum1 with exact rational arithmetic, not rounded decimals.
- S075 · premise_and_result_audit — Recompute Taylor ORIGINAL partial sum0 with exact rational arithmetic, not rounded decimals.
- S074 · premise_and_result_audit — Recompute Taylor ORIGINAL term3 divide factorial with exact rational arithmetic, not rounded decimals.
- S073 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative times displacement3 with exact rational arithmetic, not rounded decimals.
- S072 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S071 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S070 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative3 ascending derivative sum term4 with exact rational arithmetic, not rounded decimals.
- S069 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative3 ORIGINAL coefficient term4 with exact rational arithmetic, not rounded decimals.
- S068 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative3 falling factor times power term4 with exact rational arithmetic, not rounded decimals.
- S067 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S066 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative3 ascending derivative sum term3 with exact rational arithmetic, not rounded decimals.
- S065 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative3 ORIGINAL coefficient term3 with exact rational arithmetic, not rounded decimals.
- S064 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative3 falling factor times power term3 with exact rational arithmetic, not rounded decimals.
- S063 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S062 · premise_and_result_audit — Recompute Taylor ORIGINAL term2 divide factorial with exact rational arithmetic, not rounded decimals.
- S061 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative times displacement2 with exact rational arithmetic, not rounded decimals.
- S060 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S059 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S058 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative2 ascending derivative sum term4 with exact rational arithmetic, not rounded decimals.
- S057 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative2 ORIGINAL coefficient term4 with exact rational arithmetic, not rounded decimals.
- S056 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative2 falling factor times power term4 with exact rational arithmetic, not rounded decimals.
- S055 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S054 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative2 ascending derivative sum term3 with exact rational arithmetic, not rounded decimals.
- S053 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative2 ORIGINAL coefficient term3 with exact rational arithmetic, not rounded decimals.
- S052 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative2 falling factor times power term3 with exact rational arithmetic, not rounded decimals.
- S051 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S050 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative2 ascending derivative sum term2 with exact rational arithmetic, not rounded decimals.
- S049 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative2 ORIGINAL coefficient term2 with exact rational arithmetic, not rounded decimals.
- S048 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative2 falling factor times power term2 with exact rational arithmetic, not rounded decimals.
- S047 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S046 · premise_and_result_audit — Recompute Taylor ORIGINAL term1 divide factorial with exact rational arithmetic, not rounded decimals.
- S045 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative times displacement1 with exact rational arithmetic, not rounded decimals.
- S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S042 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative1 ascending derivative sum term4 with exact rational arithmetic, not rounded decimals.
- S041 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative1 ORIGINAL coefficient term4 with exact rational arithmetic, not rounded decimals.
- S040 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative1 falling factor times power term4 with exact rational arithmetic, not rounded decimals.
- S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S038 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative1 ascending derivative sum term3 with exact rational arithmetic, not rounded decimals.
- S037 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative1 ORIGINAL coefficient term3 with exact rational arithmetic, not rounded decimals.
- S036 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative1 falling factor times power term3 with exact rational arithmetic, not rounded decimals.
- S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S034 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative1 ascending derivative sum term2 with exact rational arithmetic, not rounded decimals.
- S033 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative1 ORIGINAL coefficient term2 with exact rational arithmetic, not rounded decimals.
- S032 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative1 falling factor times power term2 with exact rational arithmetic, not rounded decimals.
- S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S030 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative1 ascending derivative sum term1 with exact rational arithmetic, not rounded decimals.
- S029 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative1 ORIGINAL coefficient term1 with exact rational arithmetic, not rounded decimals.
- S028 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative1 falling factor times power term1 with exact rational arithmetic, not rounded decimals.
- S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S026 · premise_and_result_audit — Recompute Taylor ORIGINAL term0 divide factorial with exact rational arithmetic, not rounded decimals.
- S025 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative times displacement0 with exact rational arithmetic, not rounded decimals.
- S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S022 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative0 ascending derivative sum term4 with exact rational arithmetic, not rounded decimals.
- S021 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative0 ORIGINAL coefficient term4 with exact rational arithmetic, not rounded decimals.
- S020 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative0 falling factor times power term4 with exact rational arithmetic, not rounded decimals.
- S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S018 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative0 ascending derivative sum term3 with exact rational arithmetic, not rounded decimals.
- S017 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative0 ORIGINAL coefficient term3 with exact rational arithmetic, not rounded decimals.
- S016 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative0 falling factor times power term3 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 — Recompute Taylor ORIGINAL derivative0 ascending derivative sum term2 with exact rational arithmetic, not rounded decimals.
- S013 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative0 ORIGINAL coefficient term2 with exact rational arithmetic, not rounded decimals.
- S012 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative0 falling factor times power term2 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 Taylor ORIGINAL derivative0 ascending derivative sum term1 with exact rational arithmetic, not rounded decimals.
- S009 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative0 ORIGINAL coefficient term1 with exact rational arithmetic, not rounded decimals.
- S008 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative0 falling factor times power term1 with exact rational arithmetic, not rounded decimals.
- S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S006 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative0 ascending derivative sum term0 with exact rational arithmetic, not rounded decimals.
- S005 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative0 ORIGINAL coefficient term0 with exact rational arithmetic, not rounded decimals.
- S004 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative0 falling factor times power term0 with exact rational arithmetic, not rounded decimals.
- S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S002 · premise_and_result_audit — Recompute Taylor ORIGINAL x minus declared center with exact rational arithmetic, not rounded decimals.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

(4, 3, 576, 1/4, Yes, 3/2, 3/2, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)

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: 1029bf2967c62609661b2fed0471c4a1452c6ed843f98c90ba12dae6c3f99100
- candidate record SHA-256: a98142033274abfa915dd0936f11b3e095612a26babd74ec001653f99ec1ac93
- semantic review receipt SHA-256: 90df1e889242e9afdf01e068788ee8500ef366ffcfc0496e30a6800de32631c3
- question-first witness SHA-256: ae1e849608590ca856c9f3b6911ba58e3bf53c608eb7369606192a770f48bd5c
- Lean: NOT_RUN · human review: NOT_PERFORMED

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