Question 26425: An Actual Error Inequality Is Different from a Qualified Derivative Bound

Original question and givens

Coefficients=[9, 17, 8, 16, 7, 15, 26] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=5; keep terms through that order, and define remainder=f(x)−Taylor_value. Mode=valid_bound, supplied M=18720≥0. Give (mode, remainder, derivative_supremum, M, proposed_error_bound, actual_error_inequality, qualified_bound_certificate, general_theorem_or_analyticity_claim, boundary).

• G001 — Coefficients=[9, 17, 8, 16, 7, 15, 26] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=5; keep terms through that order, and define remainder=f(x)−Taylor_value. Mode=valid_bound, supplied M=18720≥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(9, 1)

Taylor ORIGINAL derivative0 ORIGINAL coefficient term0 = 9

Taylor ORIGINAL derivative0 ascending derivative sum term0 := sum(0, 9)

Taylor ORIGINAL derivative0 ascending derivative sum term0 = 9

(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(17, 0)

Taylor ORIGINAL derivative0 ORIGINAL coefficient term1 = 0

Taylor ORIGINAL derivative0 ascending derivative sum term1 := sum(9, 0)

Taylor ORIGINAL derivative0 ascending derivative sum term1 = 9

(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(8, 0)

Taylor ORIGINAL derivative0 ORIGINAL coefficient term2 = 0

Taylor ORIGINAL derivative0 ascending derivative sum term2 := sum(9, 0)

Taylor ORIGINAL derivative0 ascending derivative sum term2 = 9

(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(16, 0)

Taylor ORIGINAL derivative0 ORIGINAL coefficient term3 = 0

Taylor ORIGINAL derivative0 ascending derivative sum term3 := sum(9, 0)

Taylor ORIGINAL derivative0 ascending derivative sum term3 = 9

(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(7, 0)

Taylor ORIGINAL derivative0 ORIGINAL coefficient term4 = 0

Taylor ORIGINAL derivative0 ascending derivative sum term4 := sum(9, 0)

Taylor ORIGINAL derivative0 ascending derivative sum term4 = 9

(0)^5

Taylor ORIGINAL derivative0 center power5 term5 = 0

Taylor ORIGINAL derivative0 falling factor times power term5 := mul(1, 0)

Taylor ORIGINAL derivative0 falling factor times power term5 = 0

Taylor ORIGINAL derivative0 ORIGINAL coefficient term5 := mul(15, 0)

Taylor ORIGINAL derivative0 ORIGINAL coefficient term5 = 0

Taylor ORIGINAL derivative0 ascending derivative sum term5 := sum(9, 0)

Taylor ORIGINAL derivative0 ascending derivative sum term5 = 9

(0)^6

Taylor ORIGINAL derivative0 center power6 term6 = 0

Taylor ORIGINAL derivative0 falling factor times power term6 := mul(1, 0)

Taylor ORIGINAL derivative0 falling factor times power term6 = 0

Taylor ORIGINAL derivative0 ORIGINAL coefficient term6 := mul(26, 0)

Taylor ORIGINAL derivative0 ORIGINAL coefficient term6 = 0

Taylor ORIGINAL derivative0 ascending derivative sum term6 := sum(9, 0)

Taylor ORIGINAL derivative0 ascending derivative sum term6 = 9

source roles = [0, 0, [9, 17, 8, 16, 7, 15, 26]]

derived = [[[0, 9, 1, 1, 9], [1, 17, 1, 0, 0], [2, 8, 1, 0, 0], [3, 16, 1, 0, 0], [4, 7, 1, 0, 0], [5, 15, 1, 0, 0], [6, 26, 1, 0, 0]], 9]

(1/2)^0

Taylor ORIGINAL displacement power0 = 1

Taylor ORIGINAL derivative times displacement0 := mul(9, 1)

Taylor ORIGINAL derivative times displacement0 = 9

Taylor ORIGINAL term0 divide factorial := div(9, 1)

Taylor ORIGINAL term0 divide factorial = 9

(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(17, 1)

Taylor ORIGINAL derivative1 ORIGINAL coefficient term1 = 17

Taylor ORIGINAL derivative1 ascending derivative sum term1 := sum(0, 17)

Taylor ORIGINAL derivative1 ascending derivative sum term1 = 17

(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(8, 0)

Taylor ORIGINAL derivative1 ORIGINAL coefficient term2 = 0

Taylor ORIGINAL derivative1 ascending derivative sum term2 := sum(17, 0)

Taylor ORIGINAL derivative1 ascending derivative sum term2 = 17

(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(16, 0)

Taylor ORIGINAL derivative1 ORIGINAL coefficient term3 = 0

Taylor ORIGINAL derivative1 ascending derivative sum term3 := sum(17, 0)

Taylor ORIGINAL derivative1 ascending derivative sum term3 = 17

(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(7, 0)

Taylor ORIGINAL derivative1 ORIGINAL coefficient term4 = 0

Taylor ORIGINAL derivative1 ascending derivative sum term4 := sum(17, 0)

Taylor ORIGINAL derivative1 ascending derivative sum term4 = 17

(0)^4

Taylor ORIGINAL derivative1 center power4 term5 = 0

Taylor ORIGINAL derivative1 falling factor times power term5 := mul(5, 0)

Taylor ORIGINAL derivative1 falling factor times power term5 = 0

Taylor ORIGINAL derivative1 ORIGINAL coefficient term5 := mul(15, 0)

Taylor ORIGINAL derivative1 ORIGINAL coefficient term5 = 0

Taylor ORIGINAL derivative1 ascending derivative sum term5 := sum(17, 0)

Taylor ORIGINAL derivative1 ascending derivative sum term5 = 17

(0)^5

Taylor ORIGINAL derivative1 center power5 term6 = 0

Taylor ORIGINAL derivative1 falling factor times power term6 := mul(6, 0)

Taylor ORIGINAL derivative1 falling factor times power term6 = 0

Taylor ORIGINAL derivative1 ORIGINAL coefficient term6 := mul(26, 0)

Taylor ORIGINAL derivative1 ORIGINAL coefficient term6 = 0

Taylor ORIGINAL derivative1 ascending derivative sum term6 := sum(17, 0)

Taylor ORIGINAL derivative1 ascending derivative sum term6 = 17

source roles = [1, 0, [9, 17, 8, 16, 7, 15, 26]]

derived = [[[1, 17, 1, 1, 17], [2, 8, 2, 0, 0], [3, 16, 3, 0, 0], [4, 7, 4, 0, 0], [5, 15, 5, 0, 0], [6, 26, 6, 0, 0]], 17]

(1/2)^1

Taylor ORIGINAL displacement power1 = 1/2

Taylor ORIGINAL derivative times displacement1 := mul(17, 1/2)

Taylor ORIGINAL derivative times displacement1 = 17/2

Taylor ORIGINAL term1 divide factorial := div(17/2, 1)

Taylor ORIGINAL term1 divide factorial = 17/2

(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(8, 2)

Taylor ORIGINAL derivative2 ORIGINAL coefficient term2 = 16

Taylor ORIGINAL derivative2 ascending derivative sum term2 := sum(0, 16)

Taylor ORIGINAL derivative2 ascending derivative sum term2 = 16

(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(16, 0)

Taylor ORIGINAL derivative2 ORIGINAL coefficient term3 = 0

Taylor ORIGINAL derivative2 ascending derivative sum term3 := sum(16, 0)

Taylor ORIGINAL derivative2 ascending derivative sum term3 = 16

(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(7, 0)

Taylor ORIGINAL derivative2 ORIGINAL coefficient term4 = 0

Taylor ORIGINAL derivative2 ascending derivative sum term4 := sum(16, 0)

Taylor ORIGINAL derivative2 ascending derivative sum term4 = 16

(0)^3

Taylor ORIGINAL derivative2 center power3 term5 = 0

Taylor ORIGINAL derivative2 falling factor times power term5 := mul(20, 0)

Taylor ORIGINAL derivative2 falling factor times power term5 = 0

Taylor ORIGINAL derivative2 ORIGINAL coefficient term5 := mul(15, 0)

Taylor ORIGINAL derivative2 ORIGINAL coefficient term5 = 0

Taylor ORIGINAL derivative2 ascending derivative sum term5 := sum(16, 0)

Taylor ORIGINAL derivative2 ascending derivative sum term5 = 16

(0)^4

Taylor ORIGINAL derivative2 center power4 term6 = 0

Taylor ORIGINAL derivative2 falling factor times power term6 := mul(30, 0)

Taylor ORIGINAL derivative2 falling factor times power term6 = 0

Taylor ORIGINAL derivative2 ORIGINAL coefficient term6 := mul(26, 0)

Taylor ORIGINAL derivative2 ORIGINAL coefficient term6 = 0

Taylor ORIGINAL derivative2 ascending derivative sum term6 := sum(16, 0)

Taylor ORIGINAL derivative2 ascending derivative sum term6 = 16

source roles = [2, 0, [9, 17, 8, 16, 7, 15, 26]]

derived = [[[2, 8, 2, 1, 16], [3, 16, 6, 0, 0], [4, 7, 12, 0, 0], [5, 15, 20, 0, 0], [6, 26, 30, 0, 0]], 16]

(1/2)^2

Taylor ORIGINAL displacement power2 = 1/4

Taylor ORIGINAL derivative times displacement2 := mul(16, 1/4)

Taylor ORIGINAL derivative times displacement2 = 4

Taylor ORIGINAL term2 divide factorial := div(4, 2)

Taylor ORIGINAL term2 divide factorial = 2

(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(16, 6)

Taylor ORIGINAL derivative3 ORIGINAL coefficient term3 = 96

Taylor ORIGINAL derivative3 ascending derivative sum term3 := sum(0, 96)

Taylor ORIGINAL derivative3 ascending derivative sum term3 = 96

(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(7, 0)

Taylor ORIGINAL derivative3 ORIGINAL coefficient term4 = 0

Taylor ORIGINAL derivative3 ascending derivative sum term4 := sum(96, 0)

Taylor ORIGINAL derivative3 ascending derivative sum term4 = 96

(0)^2

Taylor ORIGINAL derivative3 center power2 term5 = 0

Taylor ORIGINAL derivative3 falling factor times power term5 := mul(60, 0)

Taylor ORIGINAL derivative3 falling factor times power term5 = 0

Taylor ORIGINAL derivative3 ORIGINAL coefficient term5 := mul(15, 0)

Taylor ORIGINAL derivative3 ORIGINAL coefficient term5 = 0

Taylor ORIGINAL derivative3 ascending derivative sum term5 := sum(96, 0)

Taylor ORIGINAL derivative3 ascending derivative sum term5 = 96

(0)^3

Taylor ORIGINAL derivative3 center power3 term6 = 0

Taylor ORIGINAL derivative3 falling factor times power term6 := mul(120, 0)

Taylor ORIGINAL derivative3 falling factor times power term6 = 0

Taylor ORIGINAL derivative3 ORIGINAL coefficient term6 := mul(26, 0)

Taylor ORIGINAL derivative3 ORIGINAL coefficient term6 = 0

Taylor ORIGINAL derivative3 ascending derivative sum term6 := sum(96, 0)

Taylor ORIGINAL derivative3 ascending derivative sum term6 = 96

source roles = [3, 0, [9, 17, 8, 16, 7, 15, 26]]

derived = [[[3, 16, 6, 1, 96], [4, 7, 24, 0, 0], [5, 15, 60, 0, 0], [6, 26, 120, 0, 0]], 96]

(1/2)^3

Taylor ORIGINAL displacement power3 = 1/8

Taylor ORIGINAL derivative times displacement3 := mul(96, 1/8)

Taylor ORIGINAL derivative times displacement3 = 12

Taylor ORIGINAL term3 divide factorial := div(12, 6)

Taylor ORIGINAL term3 divide factorial = 2

(0)^0

Taylor ORIGINAL derivative4 center power0 term4 = 1

Taylor ORIGINAL derivative4 falling factor times power term4 := mul(24, 1)

Taylor ORIGINAL derivative4 falling factor times power term4 = 24

Taylor ORIGINAL derivative4 ORIGINAL coefficient term4 := mul(7, 24)

Taylor ORIGINAL derivative4 ORIGINAL coefficient term4 = 168

Taylor ORIGINAL derivative4 ascending derivative sum term4 := sum(0, 168)

Taylor ORIGINAL derivative4 ascending derivative sum term4 = 168

(0)^1

Taylor ORIGINAL derivative4 center power1 term5 = 0

Taylor ORIGINAL derivative4 falling factor times power term5 := mul(120, 0)

Taylor ORIGINAL derivative4 falling factor times power term5 = 0

Taylor ORIGINAL derivative4 ORIGINAL coefficient term5 := mul(15, 0)

Taylor ORIGINAL derivative4 ORIGINAL coefficient term5 = 0

Taylor ORIGINAL derivative4 ascending derivative sum term5 := sum(168, 0)

Taylor ORIGINAL derivative4 ascending derivative sum term5 = 168

(0)^2

Taylor ORIGINAL derivative4 center power2 term6 = 0

Taylor ORIGINAL derivative4 falling factor times power term6 := mul(360, 0)

Taylor ORIGINAL derivative4 falling factor times power term6 = 0

Taylor ORIGINAL derivative4 ORIGINAL coefficient term6 := mul(26, 0)

Taylor ORIGINAL derivative4 ORIGINAL coefficient term6 = 0

Taylor ORIGINAL derivative4 ascending derivative sum term6 := sum(168, 0)

Taylor ORIGINAL derivative4 ascending derivative sum term6 = 168

source roles = [4, 0, [9, 17, 8, 16, 7, 15, 26]]

derived = [[[4, 7, 24, 1, 168], [5, 15, 120, 0, 0], [6, 26, 360, 0, 0]], 168]

(1/2)^4

Taylor ORIGINAL displacement power4 = 1/16

Taylor ORIGINAL derivative times displacement4 := mul(168, 1/16)

Taylor ORIGINAL derivative times displacement4 = 21/2

Taylor ORIGINAL term4 divide factorial := div(21/2, 24)

Taylor ORIGINAL term4 divide factorial = 7/16

(0)^0

Taylor ORIGINAL derivative5 center power0 term5 = 1

Taylor ORIGINAL derivative5 falling factor times power term5 := mul(120, 1)

Taylor ORIGINAL derivative5 falling factor times power term5 = 120

Taylor ORIGINAL derivative5 ORIGINAL coefficient term5 := mul(15, 120)

Taylor ORIGINAL derivative5 ORIGINAL coefficient term5 = 1800

Taylor ORIGINAL derivative5 ascending derivative sum term5 := sum(0, 1800)

Taylor ORIGINAL derivative5 ascending derivative sum term5 = 1800

(0)^1

Taylor ORIGINAL derivative5 center power1 term6 = 0

Taylor ORIGINAL derivative5 falling factor times power term6 := mul(720, 0)

Taylor ORIGINAL derivative5 falling factor times power term6 = 0

Taylor ORIGINAL derivative5 ORIGINAL coefficient term6 := mul(26, 0)

Taylor ORIGINAL derivative5 ORIGINAL coefficient term6 = 0

Taylor ORIGINAL derivative5 ascending derivative sum term6 := sum(1800, 0)

Taylor ORIGINAL derivative5 ascending derivative sum term6 = 1800

source roles = [5, 0, [9, 17, 8, 16, 7, 15, 26]]

derived = [[[5, 15, 120, 1, 1800], [6, 26, 720, 0, 0]], 1800]

(1/2)^5

Taylor ORIGINAL displacement power5 = 1/32

Taylor ORIGINAL derivative times displacement5 := mul(1800, 1/32)

Taylor ORIGINAL derivative times displacement5 = 225/4

Taylor ORIGINAL term5 divide factorial := div(225/4, 120)

Taylor ORIGINAL term5 divide factorial = 15/32

Taylor ORIGINAL partial sum0 := sum(0, 9)

Taylor ORIGINAL partial sum0 = 9

Taylor ORIGINAL partial sum1 := sum(9, 17/2)

Taylor ORIGINAL partial sum1 = 35/2

Taylor ORIGINAL partial sum2 := sum(35/2, 2)

Taylor ORIGINAL partial sum2 = 39/2

Taylor ORIGINAL partial sum3 := sum(39/2, 2)

Taylor ORIGINAL partial sum3 = 43/2

Taylor ORIGINAL partial sum4 := sum(43/2, 7/16)

Taylor ORIGINAL partial sum4 = 351/16

Taylor ORIGINAL partial sum5 := sum(351/16, 15/32)

Taylor ORIGINAL partial sum5 = 717/32

(1/2)^0

Taylor ORIGINAL full function at x ORIGINAL argument power0 = 1

Taylor ORIGINAL full function at x coefficient term0 := mul(9, 1)

Taylor ORIGINAL full function at x coefficient term0 = 9

Taylor ORIGINAL full function at x ascending partial sum0 := sum(0, 9)

Taylor ORIGINAL full function at x ascending partial sum0 = 9

(1/2)^1

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

Taylor ORIGINAL full function at x coefficient term1 := mul(17, 1/2)

Taylor ORIGINAL full function at x coefficient term1 = 17/2

Taylor ORIGINAL full function at x ascending partial sum1 := sum(9, 17/2)

Taylor ORIGINAL full function at x ascending partial sum1 = 35/2

(1/2)^2

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

Taylor ORIGINAL full function at x coefficient term2 := mul(8, 1/4)

Taylor ORIGINAL full function at x coefficient term2 = 2

Taylor ORIGINAL full function at x ascending partial sum2 := sum(35/2, 2)

Taylor ORIGINAL full function at x ascending partial sum2 = 39/2

(1/2)^3

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

Taylor ORIGINAL full function at x coefficient term3 := mul(16, 1/8)

Taylor ORIGINAL full function at x coefficient term3 = 2

Taylor ORIGINAL full function at x ascending partial sum3 := sum(39/2, 2)

Taylor ORIGINAL full function at x ascending partial sum3 = 43/2

(1/2)^4

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

Taylor ORIGINAL full function at x coefficient term4 := mul(7, 1/16)

Taylor ORIGINAL full function at x coefficient term4 = 7/16

Taylor ORIGINAL full function at x ascending partial sum4 := sum(43/2, 7/16)

Taylor ORIGINAL full function at x ascending partial sum4 = 351/16

(1/2)^5

Taylor ORIGINAL full function at x ORIGINAL argument power5 = 1/32

Taylor ORIGINAL full function at x coefficient term5 := mul(15, 1/32)

Taylor ORIGINAL full function at x coefficient term5 = 15/32

Taylor ORIGINAL full function at x ascending partial sum5 := sum(351/16, 15/32)

Taylor ORIGINAL full function at x ascending partial sum5 = 717/32

(1/2)^6

Taylor ORIGINAL full function at x ORIGINAL argument power6 = 1/64

Taylor ORIGINAL full function at x coefficient term6 := mul(26, 1/64)

Taylor ORIGINAL full function at x coefficient term6 = 13/32

Taylor ORIGINAL full function at x ascending partial sum6 := sum(717/32, 13/32)

Taylor ORIGINAL full function at x ascending partial sum6 = 365/16

Taylor ORIGINAL f(x) minus Taylor value := sub(365/16, 717/32)

Taylor ORIGINAL f(x) minus Taylor value = 13/32

source roles = [[9, 17, 8, 16, 7, 15, 26], 0, 1/2, 5, [9, 17, 16, 96, 168, 1800], [9, 17/2, 2, 2, 7/16, 15/32]]

derived = [717/32, 365/16, 13/32, 6]

(0)^0

Taylor ORIGINAL next derivative center power0 term6 = 1

Taylor ORIGINAL next derivative falling factor times power term6 := mul(720, 1)

Taylor ORIGINAL next derivative falling factor times power term6 = 720

Taylor ORIGINAL next derivative ORIGINAL coefficient term6 := mul(26, 720)

Taylor ORIGINAL next derivative ORIGINAL coefficient term6 = 18720

Taylor ORIGINAL next derivative ascending derivative sum term6 := sum(0, 18720)

Taylor ORIGINAL next derivative ascending derivative sum term6 = 18720

source roles = [6, 0, [9, 17, 8, 16, 7, 15, 26]]

derived = [[[6, 26, 720, 1, 18720]], 18720]

(1/2)^6

Taylor bound ORIGINAL absolute displacement power = 1/64

Taylor bound supplied M times power := mul(18720, 1/64)

Taylor bound supplied M times power = 585/2

Taylor bound divide next factorial := div(585/2, 720)

Taylor bound divide next factorial = 13/32

source roles = [valid_bound, 13/32, 18720, 18720, 1/2]

derived = [13/32, Yes, Yes, No]

Reverse checks

• S172 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S171 · premise_and_result_audit — Recompute Taylor bound divide next factorial with exact rational arithmetic, not rounded decimals.
• S170 · premise_and_result_audit — Recompute Taylor bound supplied M times power with exact rational arithmetic, not rounded decimals.
• S169 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S168 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S167 · premise_and_result_audit — Recompute Taylor ORIGINAL next derivative ascending derivative sum term6 with exact rational arithmetic, not rounded decimals.
• S166 · premise_and_result_audit — Recompute Taylor ORIGINAL next derivative ORIGINAL coefficient term6 with exact rational arithmetic, not rounded decimals.
• S165 · premise_and_result_audit — Recompute Taylor ORIGINAL next derivative falling factor times power term6 with exact rational arithmetic, not rounded decimals.
• S164 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S163 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S162 · premise_and_result_audit — Recompute Taylor ORIGINAL f(x) minus Taylor value with exact rational arithmetic, not rounded decimals.
• S161 · premise_and_result_audit — Recompute Taylor ORIGINAL full function at x ascending partial sum6 with exact rational arithmetic, not rounded decimals.
• S160 · premise_and_result_audit — Recompute Taylor ORIGINAL full function at x coefficient term6 with exact rational arithmetic, not rounded decimals.
• S159 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S158 · premise_and_result_audit — Recompute Taylor ORIGINAL full function at x ascending partial sum5 with exact rational arithmetic, not rounded decimals.
• S157 · premise_and_result_audit — Recompute Taylor ORIGINAL full function at x coefficient term5 with exact rational arithmetic, not rounded decimals.
• S156 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S155 · premise_and_result_audit — Recompute Taylor ORIGINAL full function at x ascending partial sum4 with exact rational arithmetic, not rounded decimals.
• S154 · premise_and_result_audit — Recompute Taylor ORIGINAL full function at x coefficient term4 with exact rational arithmetic, not rounded decimals.
• S153 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S152 · premise_and_result_audit — Recompute Taylor ORIGINAL full function at x ascending partial sum3 with exact rational arithmetic, not rounded decimals.
• S151 · premise_and_result_audit — Recompute Taylor ORIGINAL full function at x coefficient term3 with exact rational arithmetic, not rounded decimals.
• S150 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S149 · premise_and_result_audit — Recompute Taylor ORIGINAL full function at x ascending partial sum2 with exact rational arithmetic, not rounded decimals.
• S148 · premise_and_result_audit — Recompute Taylor ORIGINAL full function at x coefficient term2 with exact rational arithmetic, not rounded decimals.
• S147 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S146 · premise_and_result_audit — Recompute Taylor ORIGINAL full function at x ascending partial sum1 with exact rational arithmetic, not rounded decimals.
• S145 · premise_and_result_audit — Recompute Taylor ORIGINAL full function at x coefficient term1 with exact rational arithmetic, not rounded decimals.
• S144 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S143 · premise_and_result_audit — Recompute Taylor ORIGINAL full function at x ascending partial sum0 with exact rational arithmetic, not rounded decimals.
• S142 · premise_and_result_audit — Recompute Taylor ORIGINAL full function at x coefficient term0 with exact rational arithmetic, not rounded decimals.
• S141 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S140 · premise_and_result_audit — Recompute Taylor ORIGINAL partial sum5 with exact rational arithmetic, not rounded decimals.
• S139 · premise_and_result_audit — Recompute Taylor ORIGINAL partial sum4 with exact rational arithmetic, not rounded decimals.
• S138 · premise_and_result_audit — Recompute Taylor ORIGINAL partial sum3 with exact rational arithmetic, not rounded decimals.
• S137 · premise_and_result_audit — Recompute Taylor ORIGINAL partial sum2 with exact rational arithmetic, not rounded decimals.
• S136 · premise_and_result_audit — Recompute Taylor ORIGINAL partial sum1 with exact rational arithmetic, not rounded decimals.
• S135 · premise_and_result_audit — Recompute Taylor ORIGINAL partial sum0 with exact rational arithmetic, not rounded decimals.
• S134 · premise_and_result_audit — Recompute Taylor ORIGINAL term5 divide factorial with exact rational arithmetic, not rounded decimals.
• S133 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative times displacement5 with exact rational arithmetic, not rounded decimals.
• S132 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S131 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S130 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative5 ascending derivative sum term6 with exact rational arithmetic, not rounded decimals.
• S129 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative5 ORIGINAL coefficient term6 with exact rational arithmetic, not rounded decimals.
• S128 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative5 falling factor times power term6 with exact rational arithmetic, not rounded decimals.
• S127 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S126 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative5 ascending derivative sum term5 with exact rational arithmetic, not rounded decimals.
• S125 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative5 ORIGINAL coefficient term5 with exact rational arithmetic, not rounded decimals.
• S124 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative5 falling factor times power term5 with exact rational arithmetic, not rounded decimals.
• S123 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S122 · premise_and_result_audit — Recompute Taylor ORIGINAL term4 divide factorial with exact rational arithmetic, not rounded decimals.
• S121 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative times displacement4 with exact rational arithmetic, not rounded decimals.
• S120 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S119 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S118 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative4 ascending derivative sum term6 with exact rational arithmetic, not rounded decimals.
• S117 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative4 ORIGINAL coefficient term6 with exact rational arithmetic, not rounded decimals.
• S116 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative4 falling factor times power term6 with exact rational arithmetic, not rounded decimals.
• S115 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S114 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative4 ascending derivative sum term5 with exact rational arithmetic, not rounded decimals.
• S113 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative4 ORIGINAL coefficient term5 with exact rational arithmetic, not rounded decimals.
• S112 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative4 falling factor times power term5 with exact rational arithmetic, not rounded decimals.
• S111 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S110 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative4 ascending derivative sum term4 with exact rational arithmetic, not rounded decimals.
• S109 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative4 ORIGINAL coefficient term4 with exact rational arithmetic, not rounded decimals.
• S108 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative4 falling factor times power term4 with exact rational arithmetic, not rounded decimals.
• S107 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S106 · premise_and_result_audit — Recompute Taylor ORIGINAL term3 divide factorial with exact rational arithmetic, not rounded decimals.
• S105 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative times displacement3 with exact rational arithmetic, not rounded decimals.
• 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 — 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 ORIGINAL derivative3 ascending derivative sum term6 with exact rational arithmetic, not rounded decimals.
• S101 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative3 ORIGINAL coefficient term6 with exact rational arithmetic, not rounded decimals.
• S100 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative3 falling factor times power term6 with exact rational arithmetic, not rounded decimals.
• S099 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S098 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative3 ascending derivative sum term5 with exact rational arithmetic, not rounded decimals.
• S097 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative3 ORIGINAL coefficient term5 with exact rational arithmetic, not rounded decimals.
• S096 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative3 falling factor times power term5 with exact rational arithmetic, not rounded decimals.
• 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 derivative3 ascending derivative sum term4 with exact rational arithmetic, not rounded decimals.
• S093 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative3 ORIGINAL coefficient term4 with exact rational arithmetic, not rounded decimals.
• S092 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative3 falling factor times power 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 derivative3 ascending derivative sum term3 with exact rational arithmetic, not rounded decimals.
• S089 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative3 ORIGINAL coefficient term3 with exact rational arithmetic, not rounded decimals.
• S088 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative3 falling factor times power term3 with exact rational arithmetic, not rounded decimals.
• S087 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S086 · premise_and_result_audit — Recompute Taylor ORIGINAL term2 divide factorial with exact rational arithmetic, not rounded decimals.
• S085 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative times displacement2 with exact rational arithmetic, not rounded decimals.
• S084 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S083 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S082 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative2 ascending derivative sum term6 with exact rational arithmetic, not rounded decimals.
• S081 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative2 ORIGINAL coefficient term6 with exact rational arithmetic, not rounded decimals.
• S080 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative2 falling factor times power term6 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 derivative2 ascending derivative sum term5 with exact rational arithmetic, not rounded decimals.
• S077 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative2 ORIGINAL coefficient term5 with exact rational arithmetic, not rounded decimals.
• S076 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative2 falling factor times power term5 with exact rational arithmetic, not rounded decimals.
• S075 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S074 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative2 ascending derivative sum term4 with exact rational arithmetic, not rounded decimals.
• S073 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative2 ORIGINAL coefficient term4 with exact rational arithmetic, not rounded decimals.
• S072 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative2 falling factor times power term4 with exact rational arithmetic, not rounded decimals.
• 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 derivative2 ascending derivative sum term3 with exact rational arithmetic, not rounded decimals.
• S069 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative2 ORIGINAL coefficient term3 with exact rational arithmetic, not rounded decimals.
• S068 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative2 falling factor times power term3 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 derivative2 ascending derivative sum term2 with exact rational arithmetic, not rounded decimals.
• S065 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative2 ORIGINAL coefficient term2 with exact rational arithmetic, not rounded decimals.
• S064 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative2 falling factor times power term2 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 term1 divide factorial with exact rational arithmetic, not rounded decimals.
• S061 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative times displacement1 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 derivative1 ascending derivative sum term6 with exact rational arithmetic, not rounded decimals.
• S057 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative1 ORIGINAL coefficient term6 with exact rational arithmetic, not rounded decimals.
• S056 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative1 falling factor times power term6 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 derivative1 ascending derivative sum term5 with exact rational arithmetic, not rounded decimals.
• S053 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative1 ORIGINAL coefficient term5 with exact rational arithmetic, not rounded decimals.
• S052 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative1 falling factor times power term5 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 derivative1 ascending derivative sum term4 with exact rational arithmetic, not rounded decimals.
• S049 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative1 ORIGINAL coefficient term4 with exact rational arithmetic, not rounded decimals.
• S048 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative1 falling factor times power term4 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 derivative1 ascending derivative sum term3 with exact rational arithmetic, not rounded decimals.
• S045 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative1 ORIGINAL coefficient term3 with exact rational arithmetic, not rounded decimals.
• S044 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative1 falling factor times power term3 with exact rational arithmetic, not rounded decimals.
• 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 term2 with exact rational arithmetic, not rounded decimals.
• S041 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative1 ORIGINAL coefficient term2 with exact rational arithmetic, not rounded decimals.
• S040 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative1 falling factor times power term2 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 term1 with exact rational arithmetic, not rounded decimals.
• S037 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative1 ORIGINAL coefficient term1 with exact rational arithmetic, not rounded decimals.
• S036 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative1 falling factor times power term1 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 term0 divide factorial with exact rational arithmetic, not rounded decimals.
• S033 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative times displacement0 with exact rational arithmetic, not rounded decimals.
• S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 derivative0 ascending derivative sum term6 with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative0 ORIGINAL coefficient term6 with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative0 falling factor times power term6 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 derivative0 ascending derivative sum term5 with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative0 ORIGINAL coefficient term5 with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative0 falling factor times power term5 with exact rational arithmetic, not rounded decimals.
• 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

(valid_bound, 13/32, 18720, 18720, 13/32, Yes, Yes, 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.