Question 24066: A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet

Original question and givens

Use shifted_coefficients=[16, 32, 48, 64, 80, 96] of the explicit polynomial sum c_k*(x−a)^k, center a=3, evaluation point=1. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).

• G001 — Use shifted_coefficients=[16, 32, 48, 64, 80, 96] of the explicit polynomial sum c_k*(x−a)^k, center a=3, evaluation point=1. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, 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-polynomial

Step-by-step derivation

source roles = [taylor-exact-polynomial-reconstruction]

derived = [[shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate]]

source roles = [3, [16, 32, 48, 64, 80, 96]]

derived = [P(x)=sum c_k*(x-center)^k, 5]

source roles = [0]

derived = [1]

Taylor derivative0 from shifted coefficient := mul(16, 1)

Taylor derivative0 from shifted coefficient = 16

Taylor recover coefficient0 := div(16, 1)

Taylor recover coefficient0 = 16

Taylor k1 factorial step1 := mul(1, 1)

Taylor k1 factorial step1 = 1

Taylor derivative1 from shifted coefficient := mul(32, 1)

Taylor derivative1 from shifted coefficient = 32

Taylor recover coefficient1 := div(32, 1)

Taylor recover coefficient1 = 32

Taylor k2 factorial step1 := mul(1, 1)

Taylor k2 factorial step1 = 1

Taylor k2 factorial step2 := mul(1, 2)

Taylor k2 factorial step2 = 2

Taylor derivative2 from shifted coefficient := mul(48, 2)

Taylor derivative2 from shifted coefficient = 96

Taylor recover coefficient2 := div(96, 2)

Taylor recover coefficient2 = 48

Taylor k3 factorial step1 := mul(1, 1)

Taylor k3 factorial step1 = 1

Taylor k3 factorial step2 := mul(1, 2)

Taylor k3 factorial step2 = 2

Taylor k3 factorial step3 := mul(2, 3)

Taylor k3 factorial step3 = 6

Taylor derivative3 from shifted coefficient := mul(64, 6)

Taylor derivative3 from shifted coefficient = 384

Taylor recover coefficient3 := div(384, 6)

Taylor recover coefficient3 = 64

Taylor k4 factorial step1 := mul(1, 1)

Taylor k4 factorial step1 = 1

Taylor k4 factorial step2 := mul(1, 2)

Taylor k4 factorial step2 = 2

Taylor k4 factorial step3 := mul(2, 3)

Taylor k4 factorial step3 = 6

Taylor k4 factorial step4 := mul(6, 4)

Taylor k4 factorial step4 = 24

Taylor derivative4 from shifted coefficient := mul(80, 24)

Taylor derivative4 from shifted coefficient = 1920

Taylor recover coefficient4 := div(1920, 24)

Taylor recover coefficient4 = 80

Taylor k5 factorial step1 := mul(1, 1)

Taylor k5 factorial step1 = 1

Taylor k5 factorial step2 := mul(1, 2)

Taylor k5 factorial step2 = 2

Taylor k5 factorial step3 := mul(2, 3)

Taylor k5 factorial step3 = 6

Taylor k5 factorial step4 := mul(6, 4)

Taylor k5 factorial step4 = 24

Taylor k5 factorial step5 := mul(24, 5)

Taylor k5 factorial step5 = 120

Taylor derivative5 from shifted coefficient := mul(96, 120)

Taylor derivative5 from shifted coefficient = 11520

Taylor recover coefficient5 := div(11520, 120)

Taylor recover coefficient5 = 96

Taylor SAME point minus declared center := sub(1, 3)

Taylor SAME point minus declared center = -2

Taylor original Horner product5 := mul(0, -2)

Taylor original Horner product5 = 0

Taylor original Horner sum5 := sum(0, 96)

Taylor original Horner sum5 = 96

Taylor original Horner product4 := mul(96, -2)

Taylor original Horner product4 = -192

Taylor original Horner sum4 := sum(-192, 80)

Taylor original Horner sum4 = -112

Taylor original Horner product3 := mul(-112, -2)

Taylor original Horner product3 = 224

Taylor original Horner sum3 := sum(224, 64)

Taylor original Horner sum3 = 288

Taylor original Horner product2 := mul(288, -2)

Taylor original Horner product2 = -576

Taylor original Horner sum2 := sum(-576, 48)

Taylor original Horner sum2 = -528

Taylor original Horner product1 := mul(-528, -2)

Taylor original Horner product1 = 1056

Taylor original Horner sum1 := sum(1056, 32)

Taylor original Horner sum1 = 1088

Taylor original Horner product0 := mul(1088, -2)

Taylor original Horner product0 = -2176

Taylor original Horner sum0 := sum(-2176, 16)

Taylor original Horner sum0 = -2160

Taylor reconstructed Horner product5 := mul(0, -2)

Taylor reconstructed Horner product5 = 0

Taylor reconstructed Horner sum5 := sum(0, 96)

Taylor reconstructed Horner sum5 = 96

Taylor reconstructed Horner product4 := mul(96, -2)

Taylor reconstructed Horner product4 = -192

Taylor reconstructed Horner sum4 := sum(-192, 80)

Taylor reconstructed Horner sum4 = -112

Taylor reconstructed Horner product3 := mul(-112, -2)

Taylor reconstructed Horner product3 = 224

Taylor reconstructed Horner sum3 := sum(224, 64)

Taylor reconstructed Horner sum3 = 288

Taylor reconstructed Horner product2 := mul(288, -2)

Taylor reconstructed Horner product2 = -576

Taylor reconstructed Horner sum2 := sum(-576, 48)

Taylor reconstructed Horner sum2 = -528

Taylor reconstructed Horner product1 := mul(-528, -2)

Taylor reconstructed Horner product1 = 1056

Taylor reconstructed Horner sum1 := sum(1056, 32)

Taylor reconstructed Horner sum1 = 1088

Taylor reconstructed Horner product0 := mul(1088, -2)

Taylor reconstructed Horner product0 = -2176

Taylor reconstructed Horner sum0 := sum(-2176, 16)

Taylor reconstructed Horner sum0 = -2160

source roles = [[16, 32, 48, 64, 80, 96], [16, 32, 48, 64, 80, 96], 3, 1]

derived = [Yes, -2160, -2160, Yes]

Reverse checks

• S056 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S055 · premise_and_result_audit — Recompute Taylor reconstructed Horner sum0 with exact rational arithmetic, not rounded decimals.
• S054 · premise_and_result_audit — Recompute Taylor reconstructed Horner product0 with exact rational arithmetic, not rounded decimals.
• S053 · premise_and_result_audit — Recompute Taylor reconstructed Horner sum1 with exact rational arithmetic, not rounded decimals.
• S052 · premise_and_result_audit — Recompute Taylor reconstructed Horner product1 with exact rational arithmetic, not rounded decimals.
• S051 · premise_and_result_audit — Recompute Taylor reconstructed Horner sum2 with exact rational arithmetic, not rounded decimals.
• S050 · premise_and_result_audit — Recompute Taylor reconstructed Horner product2 with exact rational arithmetic, not rounded decimals.
• S049 · premise_and_result_audit — Recompute Taylor reconstructed Horner sum3 with exact rational arithmetic, not rounded decimals.
• S048 · premise_and_result_audit — Recompute Taylor reconstructed Horner product3 with exact rational arithmetic, not rounded decimals.
• S047 · premise_and_result_audit — Recompute Taylor reconstructed Horner sum4 with exact rational arithmetic, not rounded decimals.
• S046 · premise_and_result_audit — Recompute Taylor reconstructed Horner product4 with exact rational arithmetic, not rounded decimals.
• S045 · premise_and_result_audit — Recompute Taylor reconstructed Horner sum5 with exact rational arithmetic, not rounded decimals.
• S044 · premise_and_result_audit — Recompute Taylor reconstructed Horner product5 with exact rational arithmetic, not rounded decimals.
• S043 · premise_and_result_audit — Recompute Taylor original Horner sum0 with exact rational arithmetic, not rounded decimals.
• S042 · premise_and_result_audit — Recompute Taylor original Horner product0 with exact rational arithmetic, not rounded decimals.
• S041 · premise_and_result_audit — Recompute Taylor original Horner sum1 with exact rational arithmetic, not rounded decimals.
• S040 · premise_and_result_audit — Recompute Taylor original Horner product1 with exact rational arithmetic, not rounded decimals.
• S039 · premise_and_result_audit — Recompute Taylor original Horner sum2 with exact rational arithmetic, not rounded decimals.
• S038 · premise_and_result_audit — Recompute Taylor original Horner product2 with exact rational arithmetic, not rounded decimals.
• S037 · premise_and_result_audit — Recompute Taylor original Horner sum3 with exact rational arithmetic, not rounded decimals.
• S036 · premise_and_result_audit — Recompute Taylor original Horner product3 with exact rational arithmetic, not rounded decimals.
• S035 · premise_and_result_audit — Recompute Taylor original Horner sum4 with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Recompute Taylor original Horner product4 with exact rational arithmetic, not rounded decimals.
• S033 · premise_and_result_audit — Recompute Taylor original Horner sum5 with exact rational arithmetic, not rounded decimals.
• S032 · premise_and_result_audit — Recompute Taylor original Horner product5 with exact rational arithmetic, not rounded decimals.
• S031 · premise_and_result_audit — Recompute Taylor SAME point minus declared center with exact rational arithmetic, not rounded decimals.
• S030 · premise_and_result_audit — Recompute Taylor recover coefficient5 with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute Taylor derivative5 from shifted coefficient with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Recompute Taylor k5 factorial step5 with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute Taylor k5 factorial step4 with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Recompute Taylor k5 factorial step3 with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute Taylor k5 factorial step2 with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute Taylor k5 factorial step1 with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute Taylor recover coefficient4 with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute Taylor derivative4 from shifted coefficient with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute Taylor k4 factorial step4 with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute Taylor k4 factorial step3 with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute Taylor k4 factorial step2 with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute Taylor k4 factorial step1 with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute Taylor recover coefficient3 with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute Taylor derivative3 from shifted coefficient with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute Taylor k3 factorial step3 with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute Taylor k3 factorial step2 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute Taylor k3 factorial step1 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Taylor recover coefficient2 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute Taylor derivative2 from shifted coefficient with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Taylor k2 factorial step2 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Taylor k2 factorial step1 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Taylor recover coefficient1 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Taylor derivative1 from shifted coefficient with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Taylor k1 factorial step1 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Taylor recover coefficient0 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Taylor derivative0 from shifted coefficient 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

([16, 32, 48, 64, 80, 96], [16, 32, 96, 384, 1920, 11520], [16, 32, 48, 64, 80, 96], 1, -2160, -2160, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)

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.