Question 26418: An Actual Error Inequality Is Different from a Qualified Derivative Bound

Original question and givens

Coefficients=[2, 3, 4, 5, 19] 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. Mode=undersized_M, supplied M=455≥0. Give (mode, remainder, derivative_supremum, M, proposed_error_bound, actual_error_inequality, qualified_bound_certificate, general_theorem_or_analyticity_claim, boundary).

• G001 — Coefficients=[2, 3, 4, 5, 19] 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. Mode=undersized_M, supplied M=455≥0. Give (mode, remainder, derivative_supremum, M, proposed_error_bound, actual_error_inequality, qualified_bound_certificate, general_theorem_or_analyticity_claim, boundary).

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

Step-by-step derivation

source roles = [taylor-bound-boundary]

derived = [[mode, remainder, derivative_supremum, M, proposed_error_bound, actual_error_inequality, qualified_bound_certificate, general_theorem_or_analyticity_claim, boundary]]

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

Taylor ORIGINAL x minus declared center = 1/2

(0)^0

Taylor ORIGINAL derivative0 center power0 term0 = 1

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

Taylor ORIGINAL derivative0 falling factor times power term0 = 1

Taylor ORIGINAL derivative0 ORIGINAL coefficient term0 := mul(2, 1)

Taylor ORIGINAL derivative0 ORIGINAL coefficient term0 = 2

Taylor ORIGINAL derivative0 ascending derivative sum term0 := sum(0, 2)

Taylor ORIGINAL derivative0 ascending derivative sum term0 = 2

(0)^1

Taylor ORIGINAL derivative0 center power1 term1 = 0

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

Taylor ORIGINAL derivative0 falling factor times power term1 = 0

Taylor ORIGINAL derivative0 ORIGINAL coefficient term1 := mul(3, 0)

Taylor ORIGINAL derivative0 ORIGINAL coefficient term1 = 0

Taylor ORIGINAL derivative0 ascending derivative sum term1 := sum(2, 0)

Taylor ORIGINAL derivative0 ascending derivative sum term1 = 2

(0)^2

Taylor ORIGINAL derivative0 center power2 term2 = 0

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

Taylor ORIGINAL derivative0 falling factor times power term2 = 0

Taylor ORIGINAL derivative0 ORIGINAL coefficient term2 := mul(4, 0)

Taylor ORIGINAL derivative0 ORIGINAL coefficient term2 = 0

Taylor ORIGINAL derivative0 ascending derivative sum term2 := sum(2, 0)

Taylor ORIGINAL derivative0 ascending derivative sum term2 = 2

(0)^3

Taylor ORIGINAL derivative0 center power3 term3 = 0

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

Taylor ORIGINAL derivative0 falling factor times power term3 = 0

Taylor ORIGINAL derivative0 ORIGINAL coefficient term3 := mul(5, 0)

Taylor ORIGINAL derivative0 ORIGINAL coefficient term3 = 0

Taylor ORIGINAL derivative0 ascending derivative sum term3 := sum(2, 0)

Taylor ORIGINAL derivative0 ascending derivative sum term3 = 2

(0)^4

Taylor ORIGINAL derivative0 center power4 term4 = 0

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

Taylor ORIGINAL derivative0 falling factor times power term4 = 0

Taylor ORIGINAL derivative0 ORIGINAL coefficient term4 := mul(19, 0)

Taylor ORIGINAL derivative0 ORIGINAL coefficient term4 = 0

Taylor ORIGINAL derivative0 ascending derivative sum term4 := sum(2, 0)

Taylor ORIGINAL derivative0 ascending derivative sum term4 = 2

source roles = [0, 0, [2, 3, 4, 5, 19]]

derived = [[[0, 2, 1, 1, 2], [1, 3, 1, 0, 0], [2, 4, 1, 0, 0], [3, 5, 1, 0, 0], [4, 19, 1, 0, 0]], 2]

(1/2)^0

Taylor ORIGINAL displacement power0 = 1

Taylor ORIGINAL derivative times displacement0 := mul(2, 1)

Taylor ORIGINAL derivative times displacement0 = 2

Taylor ORIGINAL term0 divide factorial := div(2, 1)

Taylor ORIGINAL term0 divide factorial = 2

(0)^0

Taylor ORIGINAL derivative1 center power0 term1 = 1

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

Taylor ORIGINAL derivative1 falling factor times power term1 = 1

Taylor ORIGINAL derivative1 ORIGINAL coefficient term1 := mul(3, 1)

Taylor ORIGINAL derivative1 ORIGINAL coefficient term1 = 3

Taylor ORIGINAL derivative1 ascending derivative sum term1 := sum(0, 3)

Taylor ORIGINAL derivative1 ascending derivative sum term1 = 3

(0)^1

Taylor ORIGINAL derivative1 center power1 term2 = 0

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

Taylor ORIGINAL derivative1 falling factor times power term2 = 0

Taylor ORIGINAL derivative1 ORIGINAL coefficient term2 := mul(4, 0)

Taylor ORIGINAL derivative1 ORIGINAL coefficient term2 = 0

Taylor ORIGINAL derivative1 ascending derivative sum term2 := sum(3, 0)

Taylor ORIGINAL derivative1 ascending derivative sum term2 = 3

(0)^2

Taylor ORIGINAL derivative1 center power2 term3 = 0

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

Taylor ORIGINAL derivative1 falling factor times power term3 = 0

Taylor ORIGINAL derivative1 ORIGINAL coefficient term3 := mul(5, 0)

Taylor ORIGINAL derivative1 ORIGINAL coefficient term3 = 0

Taylor ORIGINAL derivative1 ascending derivative sum term3 := sum(3, 0)

Taylor ORIGINAL derivative1 ascending derivative sum term3 = 3

(0)^3

Taylor ORIGINAL derivative1 center power3 term4 = 0

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

Taylor ORIGINAL derivative1 falling factor times power term4 = 0

Taylor ORIGINAL derivative1 ORIGINAL coefficient term4 := mul(19, 0)

Taylor ORIGINAL derivative1 ORIGINAL coefficient term4 = 0

Taylor ORIGINAL derivative1 ascending derivative sum term4 := sum(3, 0)

Taylor ORIGINAL derivative1 ascending derivative sum term4 = 3

source roles = [1, 0, [2, 3, 4, 5, 19]]

derived = [[[1, 3, 1, 1, 3], [2, 4, 2, 0, 0], [3, 5, 3, 0, 0], [4, 19, 4, 0, 0]], 3]

(1/2)^1

Taylor ORIGINAL displacement power1 = 1/2

Taylor ORIGINAL derivative times displacement1 := mul(3, 1/2)

Taylor ORIGINAL derivative times displacement1 = 3/2

Taylor ORIGINAL term1 divide factorial := div(3/2, 1)

Taylor ORIGINAL term1 divide factorial = 3/2

(0)^0

Taylor ORIGINAL derivative2 center power0 term2 = 1

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

Taylor ORIGINAL derivative2 falling factor times power term2 = 2

Taylor ORIGINAL derivative2 ORIGINAL coefficient term2 := mul(4, 2)

Taylor ORIGINAL derivative2 ORIGINAL coefficient term2 = 8

Taylor ORIGINAL derivative2 ascending derivative sum term2 := sum(0, 8)

Taylor ORIGINAL derivative2 ascending derivative sum term2 = 8

(0)^1

Taylor ORIGINAL derivative2 center power1 term3 = 0

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

Taylor ORIGINAL derivative2 falling factor times power term3 = 0

Taylor ORIGINAL derivative2 ORIGINAL coefficient term3 := mul(5, 0)

Taylor ORIGINAL derivative2 ORIGINAL coefficient term3 = 0

Taylor ORIGINAL derivative2 ascending derivative sum term3 := sum(8, 0)

Taylor ORIGINAL derivative2 ascending derivative sum term3 = 8

(0)^2

Taylor ORIGINAL derivative2 center power2 term4 = 0

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

Taylor ORIGINAL derivative2 falling factor times power term4 = 0

Taylor ORIGINAL derivative2 ORIGINAL coefficient term4 := mul(19, 0)

Taylor ORIGINAL derivative2 ORIGINAL coefficient term4 = 0

Taylor ORIGINAL derivative2 ascending derivative sum term4 := sum(8, 0)

Taylor ORIGINAL derivative2 ascending derivative sum term4 = 8

source roles = [2, 0, [2, 3, 4, 5, 19]]

derived = [[[2, 4, 2, 1, 8], [3, 5, 6, 0, 0], [4, 19, 12, 0, 0]], 8]

(1/2)^2

Taylor ORIGINAL displacement power2 = 1/4

Taylor ORIGINAL derivative times displacement2 := mul(8, 1/4)

Taylor ORIGINAL derivative times displacement2 = 2

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

Taylor ORIGINAL term2 divide factorial = 1

(0)^0

Taylor ORIGINAL derivative3 center power0 term3 = 1

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

Taylor ORIGINAL derivative3 falling factor times power term3 = 6

Taylor ORIGINAL derivative3 ORIGINAL coefficient term3 := mul(5, 6)

Taylor ORIGINAL derivative3 ORIGINAL coefficient term3 = 30

Taylor ORIGINAL derivative3 ascending derivative sum term3 := sum(0, 30)

Taylor ORIGINAL derivative3 ascending derivative sum term3 = 30

(0)^1

Taylor ORIGINAL derivative3 center power1 term4 = 0

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

Taylor ORIGINAL derivative3 falling factor times power term4 = 0

Taylor ORIGINAL derivative3 ORIGINAL coefficient term4 := mul(19, 0)

Taylor ORIGINAL derivative3 ORIGINAL coefficient term4 = 0

Taylor ORIGINAL derivative3 ascending derivative sum term4 := sum(30, 0)

Taylor ORIGINAL derivative3 ascending derivative sum term4 = 30

source roles = [3, 0, [2, 3, 4, 5, 19]]

derived = [[[3, 5, 6, 1, 30], [4, 19, 24, 0, 0]], 30]

(1/2)^3

Taylor ORIGINAL displacement power3 = 1/8

Taylor ORIGINAL derivative times displacement3 := mul(30, 1/8)

Taylor ORIGINAL derivative times displacement3 = 15/4

Taylor ORIGINAL term3 divide factorial := div(15/4, 6)

Taylor ORIGINAL term3 divide factorial = 5/8

Taylor ORIGINAL partial sum0 := sum(0, 2)

Taylor ORIGINAL partial sum0 = 2

Taylor ORIGINAL partial sum1 := sum(2, 3/2)

Taylor ORIGINAL partial sum1 = 7/2

Taylor ORIGINAL partial sum2 := sum(7/2, 1)

Taylor ORIGINAL partial sum2 = 9/2

Taylor ORIGINAL partial sum3 := sum(9/2, 5/8)

Taylor ORIGINAL partial sum3 = 41/8

(1/2)^0

Taylor ORIGINAL full function at x ORIGINAL argument power0 = 1

Taylor ORIGINAL full function at x coefficient term0 := mul(2, 1)

Taylor ORIGINAL full function at x coefficient term0 = 2

Taylor ORIGINAL full function at x ascending partial sum0 := sum(0, 2)

Taylor ORIGINAL full function at x ascending partial sum0 = 2

(1/2)^1

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

Taylor ORIGINAL full function at x coefficient term1 := mul(3, 1/2)

Taylor ORIGINAL full function at x coefficient term1 = 3/2

Taylor ORIGINAL full function at x ascending partial sum1 := sum(2, 3/2)

Taylor ORIGINAL full function at x ascending partial sum1 = 7/2

(1/2)^2

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

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

Taylor ORIGINAL full function at x coefficient term2 = 1

Taylor ORIGINAL full function at x ascending partial sum2 := sum(7/2, 1)

Taylor ORIGINAL full function at x ascending partial sum2 = 9/2

(1/2)^3

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

Taylor ORIGINAL full function at x coefficient term3 := mul(5, 1/8)

Taylor ORIGINAL full function at x coefficient term3 = 5/8

Taylor ORIGINAL full function at x ascending partial sum3 := sum(9/2, 5/8)

Taylor ORIGINAL full function at x ascending partial sum3 = 41/8

(1/2)^4

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

Taylor ORIGINAL full function at x coefficient term4 := mul(19, 1/16)

Taylor ORIGINAL full function at x coefficient term4 = 19/16

Taylor ORIGINAL full function at x ascending partial sum4 := sum(41/8, 19/16)

Taylor ORIGINAL full function at x ascending partial sum4 = 101/16

Taylor ORIGINAL f(x) minus Taylor value := sub(101/16, 41/8)

Taylor ORIGINAL f(x) minus Taylor value = 19/16

source roles = [[2, 3, 4, 5, 19], 0, 1/2, 3, [2, 3, 8, 30], [2, 3/2, 1, 5/8]]

derived = [41/8, 101/16, 19/16, 4]

(0)^0

Taylor ORIGINAL next derivative center power0 term4 = 1

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

Taylor ORIGINAL next derivative falling factor times power term4 = 24

Taylor ORIGINAL next derivative ORIGINAL coefficient term4 := mul(19, 24)

Taylor ORIGINAL next derivative ORIGINAL coefficient term4 = 456

Taylor ORIGINAL next derivative ascending derivative sum term4 := sum(0, 456)

Taylor ORIGINAL next derivative ascending derivative sum term4 = 456

source roles = [4, 0, [2, 3, 4, 5, 19]]

derived = [[[4, 19, 24, 1, 456]], 456]

(1/2)^4

Taylor bound ORIGINAL absolute displacement power = 1/16

Taylor bound supplied M times power := mul(455, 1/16)

Taylor bound supplied M times power = 455/16

Taylor bound divide next factorial := div(455/16, 24)

Taylor bound divide next factorial = 455/384

source roles = [undersized_M, 19/16, 456, 455, 1/2]

derived = [455/384, No, No, No]

Reverse checks

• S104 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S103 · premise_and_result_audit — Recompute Taylor bound divide next factorial with exact rational arithmetic, not rounded decimals.
• S102 · premise_and_result_audit — Recompute Taylor bound supplied M times power with exact rational arithmetic, not rounded decimals.
• S101 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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

(undersized_M, 19/16, 456, 455, 455/384, No, No, No, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)

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.