# Question 23046: Order Zero Is the Original Function; Later Orders Need the Recursive Chain

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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.

R0045-23046 · q-20261014-advance-20261111-r02-higher-order-derivative-0021



## Original question and givens

Use original low-to-high coefficients=[21, 42, 63, 84] for f(t)=sum coefficients[k]*t^k and derivative order=0≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).


- G001 — Use original low-to-high coefficients=[21, 42, 63, 84] for f(t)=sum coefficients[k]*t^k and derivative order=0≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_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-higher-order-derivative](https://ocme.evemisslab.com/en/mko/mko-higher-order-derivative/)





## Step-by-step derivation

S001 · impliesBefore

source operation=higher-order-qualification;original snapshot and mathematics retained

After

explicit ordered source output fields=[declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate]

Conditions: none

Retain the historical interface marker while listing source fields without JavaScript.Tags and No retain their claim scope;source-protocol ordering is not a mathematical proof.

Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001





S002 · impliesBefore

ORIGINAL low-to-high coefficients=[21, 42, 63, 84];declared order=0

After

declared degree BOUND=3=ORIGINAL length-1;initialize derivative order0 with the UNTRIMMED original list

Conditions: none

Length minus1 is the declared bound;do not trim trailing zeros.Order0 is the original function.The monomial rule and finite sums justify each polynomial derivative,not an infinite extrapolation for arbitrary functions.

Step identifiers and dependenciesRule: higher_ORIGINAL_low_to_high_coefficients_degree_BOUND_order0 · Uses: G001, S001





S003 · impliesBefore

ORIGINAL coefficients=[21, 42, 63, 84];order=0;declared degree bound=3

After

final derivative coefficients=[21, 42, 63, 84];for each original power k>=order multiplier=k*(k-1)*...*(k-order+1);lower powers vanish

Conditions: none

Repeated differentiation gives the falling factorial in original coefficient order;orders above the declared bound vanish.This gives no analyticity claim for arbitrary differentiable functions and does not turn a trailing-zero bound into the minimal degree.

Step identifiers and dependenciesRule: higher_falling_factorial_rule_all_orders_polynomial_only · Uses: G001, S002









## Reverse checks


- 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

(3, 0, Yes, f_zero_equals_f, Yes)

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: 9d942ab97e384ea8a9b9a9597e31e77c9d10ed541204c2c93b139984832dadaa
- candidate record SHA-256: 1206fb3951fe370f573f96260cec952682cdf3b7f99ed2ac60d1772fbf978ee4
- semantic review receipt SHA-256: 357d7ad7a6726b2a757558d3b9586a408ade06526cc023da093719c292704438
- question-first witness SHA-256: 99cefe17e98c7a74321801cf72df6d02fc95b6a6b78aa335accb7e11ed467b2a
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0021.json)