# Question 24088: A Taylor Jet Includes Every Derivative Order from Zero through n

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0047-24088 · q-20261014-advance-20261112-r02-taylor-polynomial-0013



## Original question and givens

Use the derivative values=[13, 26, 39] at the fixed center=0, through declared order=2≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).


- G001 — Use the derivative values=[13, 26, 39] at the fixed center=0, through declared order=2≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, 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-jet-qualification]

After

derived = [[center, order, derivatives, jet_length, complete_jet, 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 = [0, 2, [13, 26, 39]]

After

derived = [[0, 1, 2], 3, Yes]

Conditions: none

The data are assumed derivatives of the same function at the same a.A numeric list does not verify differentiability of an unknown function.Order0 is f(a).

Step identifiers and dependenciesRule: Taylor_assumed_complete_derivative_jet_not_differentiability_test · 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 coefficient0 derivative over factorial := div(13, 1)

After

Taylor coefficient0 derivative over factorial = 13

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 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, S004





S006 · calculationBefore

Taylor coefficient1 derivative over factorial := div(26, 1)

After

Taylor coefficient1 derivative over factorial = 26

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 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, S006





S008 · 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, S007





S009 · calculationBefore

Taylor coefficient2 derivative over factorial := div(39, 2)

After

Taylor coefficient2 derivative over factorial = 39/2

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, S008





S010 · impliesBefore

source roles = [0, 2, [13, 26, 39/2]]

After

derived = [T_n(x)=sum c_k*(x-center)^k, T_n^(j)(center)=declared derivative[j] for EVERY0<=j<=n, no remainder or infinite-series certificate]

Conditions: none

For each j,terms k<j differentiate to0 and k>j still contain a positive power at a;only k=j yields j!*c_j.This matches the full jet.Order0 retains f(a),without automatically identifying the original function or bounding a remainder.

Step identifiers and dependenciesRule: Taylor_shifted_basis_matches_derivatives_ALL_orders · Uses: G001, S009









## Reverse checks


- S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S009 · premise_and_result_audit — Recompute Taylor coefficient2 derivative over factorial with exact rational arithmetic, not rounded decimals.
- S008 · premise_and_result_audit — Recompute Taylor k2 factorial step2 with exact rational arithmetic, not rounded decimals.
- S007 · premise_and_result_audit — Recompute Taylor k2 factorial step1 with exact rational arithmetic, not rounded decimals.
- S006 · premise_and_result_audit — Recompute Taylor coefficient1 derivative over factorial with exact rational arithmetic, not rounded decimals.
- S005 · premise_and_result_audit — Recompute Taylor k1 factorial step1 with exact rational arithmetic, not rounded decimals.
- S004 · premise_and_result_audit — Recompute Taylor coefficient0 derivative over factorial 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

(0, 2, [13, 26, 39], 3, Yes, complete_zero_through_n_derivative_jet)

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: d5f49c9446bffc66cea69cf35557f541402ce4c981f00d9c982cd5f2a9eeb59d
- candidate record SHA-256: 69ff20113926c175c41a5cd7c39616a363fcb243a7f7527f3f02c58463543541
- 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-0013.json)