# Question 24064: A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet [Return to catalogue page 121](https://ocme.evemisslab.com/en/questions/page/121/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261112-r02-taylor-polynomial-foundations/#q-20261014-advance-20261112-r02-taylor-polynomial-0064) 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. R0047-24064 · q-20261014-advance-20261112-r02-taylor-polynomial-0064 ## Original question and givens Use shifted_coefficients=[14, 28, 42, 56] of the explicit polynomial sum c_k*(x−a)^k, center a=1, evaluation point=2. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate). - G001 — Use shifted_coefficients=[14, 28, 42, 56] of the explicit polynomial sum c_k*(x−a)^k, center a=1, evaluation point=2. 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](https://ocme.evemisslab.com/en/mko/mko-taylor-polynomial/) ## Step-by-step derivation S001 · impliesBefore source roles = [taylor-exact-polynomial-reconstruction] After derived = [[shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate]] Conditions: none List the source field order without JavaScript.Gradient boundary uses its original mode to choose6,8 or7 fields;no padding or answer-derived mode. Step identifiers and dependenciesRule: source_ordered_fields_without_JS · Uses: G001 S002 · impliesBefore source roles = [1, [14, 28, 42, 56]] After derived = [P(x)=sum c_k*(x-center)^k, 3] Conditions: none The whole finite polynomial is declared.At a,only its degree-k shifted monomial contributes k!*c_k to the kth derivative.This is not an infinite expansion of an arbitrary smooth function. Step identifiers and dependenciesRule: Taylor_finite_shifted_polynomial_whole_law · Uses: G001, S001 S003 · impliesBefore source roles = [0] After derived = [1] Conditions: none The empty product0! is1,not0. Step identifiers and dependenciesRule: empty_factorial_is_multiplicative_unit · Uses: G001, S002 S004 · calculationBefore Taylor derivative0 from shifted coefficient := mul(14, 1) After Taylor derivative0 from shifted coefficient = 14 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Taylor recover coefficient0 := div(14, 1) After Taylor recover coefficient0 = 14 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Taylor k1 factorial step1 := mul(1, 1) After Taylor k1 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore Taylor derivative1 from shifted coefficient := mul(28, 1) After Taylor derivative1 from shifted coefficient = 28 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore Taylor recover coefficient1 := div(28, 1) After Taylor recover coefficient1 = 28 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore Taylor k2 factorial step1 := mul(1, 1) After Taylor k2 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore Taylor k2 factorial step2 := mul(1, 2) After Taylor k2 factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore Taylor derivative2 from shifted coefficient := mul(42, 2) After Taylor derivative2 from shifted coefficient = 84 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore Taylor recover coefficient2 := div(84, 2) After Taylor recover coefficient2 = 42 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore Taylor k3 factorial step1 := mul(1, 1) After Taylor k3 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore Taylor k3 factorial step2 := mul(1, 2) After Taylor k3 factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore Taylor k3 factorial step3 := mul(2, 3) After Taylor k3 factorial step3 = 6 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore Taylor derivative3 from shifted coefficient := mul(56, 6) After Taylor derivative3 from shifted coefficient = 336 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore Taylor recover coefficient3 := div(336, 6) After Taylor recover coefficient3 = 56 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore Taylor SAME point minus declared center := sub(2, 1) After Taylor SAME point minus declared center = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore Taylor original Horner product3 := mul(0, 1) After Taylor original Horner product3 = 0 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore Taylor original Horner sum3 := sum(0, 56) After Taylor original Horner sum3 = 56 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore Taylor original Horner product2 := mul(56, 1) After Taylor original Horner product2 = 56 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore Taylor original Horner sum2 := sum(56, 42) After Taylor original Horner sum2 = 98 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore Taylor original Horner product1 := mul(98, 1) After Taylor original Horner product1 = 98 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore Taylor original Horner sum1 := sum(98, 28) After Taylor original Horner sum1 = 126 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · calculationBefore Taylor original Horner product0 := mul(126, 1) After Taylor original Horner product0 = 126 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · calculationBefore Taylor original Horner sum0 := sum(126, 14) After Taylor original Horner sum0 = 140 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · calculationBefore Taylor reconstructed Horner product3 := mul(0, 1) After Taylor reconstructed Horner product3 = 0 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · calculationBefore Taylor reconstructed Horner sum3 := sum(0, 56) After Taylor reconstructed Horner sum3 = 56 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · calculationBefore Taylor reconstructed Horner product2 := mul(56, 1) After Taylor reconstructed Horner product2 = 56 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028 S030 · calculationBefore Taylor reconstructed Horner sum2 := sum(56, 42) After Taylor reconstructed Horner sum2 = 98 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S029 S031 · calculationBefore Taylor reconstructed Horner product1 := mul(98, 1) After Taylor reconstructed Horner product1 = 98 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030 S032 · calculationBefore Taylor reconstructed Horner sum1 := sum(98, 28) After Taylor reconstructed Horner sum1 = 126 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S031 S033 · calculationBefore Taylor reconstructed Horner product0 := mul(126, 1) After Taylor reconstructed Horner product0 = 126 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S032 S034 · calculationBefore Taylor reconstructed Horner sum0 := sum(126, 14) After Taylor reconstructed Horner sum0 = 140 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S033 S035 · impliesBefore source roles = [[14, 28, 42, 56], [14, 28, 42, 56], 1, 2] After derived = [Yes, 140, 140, Yes] Conditions: none Recover every coefficient first to identify the whole polynomial,then evaluate at the same point and center.One coincident point value alone would not establish polynomial identity. Step identifiers and dependenciesRule: Taylor_ALL_coefficients_recovered_before_point_equality · Uses: G001, S034 ## Reverse checks - 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 reconstructed Horner sum0 with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Recompute Taylor reconstructed Horner product0 with exact rational arithmetic, not rounded decimals. - S032 · premise_and_result_audit — Recompute Taylor reconstructed Horner sum1 with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Recompute Taylor reconstructed Horner product1 with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Recompute Taylor reconstructed Horner sum2 with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Recompute Taylor reconstructed Horner product2 with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Recompute Taylor reconstructed Horner sum3 with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Recompute Taylor reconstructed Horner product3 with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Recompute Taylor original Horner sum0 with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute Taylor original Horner product0 with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute Taylor original Horner sum1 with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute Taylor original Horner product1 with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute Taylor original Horner sum2 with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute Taylor original Horner product2 with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute Taylor original Horner sum3 with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute Taylor original Horner product3 with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute Taylor SAME point minus declared center 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 ([14, 28, 42, 56], [14, 28, 84, 336], [14, 28, 42, 56], 2, 140, 140, 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 94df77c9f7aebbb48f7453881db998aa643a3fd453147188983c731553813889 - candidate record SHA-256: bdf3431dc1b9a6a195b84d032a6a49f11995b3c601fe85c5452378b323ee4c80 - semantic review receipt SHA-256: 9b43b9823e9bd2478d6caa5b57ffed161c2c8956af86974e901d5662ad994386 - question-first witness SHA-256: eb97596068e2d5648033bab2b8f6ca9b653bbe8cb77c0f8e1d16d0c40e46f50b - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0064.json)