# Question 24620: Coefficient Extraction Uses the Declared Complete Zero Tail

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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.

R0048-24620 · q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0020



## Original question and givens

Use finite_support_coefficients=[20, 40, 60, 80, 100, 120] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Extract index=1. Give (finite_support_coefficients, zero_tail_law, index, coefficient, certificate).


- G001 — Use finite_support_coefficients=[20, 40, 60, 80, 100, 120] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Extract index=1. Give (finite_support_coefficients, zero_tail_law, index, coefficient, 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-analytic-ordinary-generating-function](https://ocme.evemisslab.com/en/mko/mko-analytic-ordinary-generating-function/)





## Step-by-step derivation

S001 · impliesBefore

source roles = [ogf-coefficient-extraction]

After

derived = [[finite_support_coefficients, zero_tail_law, index, coefficient, certificate]]

Conditions: none

List the full source field order without JavaScript,keeping original data,Boolean flags,mathematical objects and nonclaims distinct.

Step identifiers and dependenciesRule: source_ordered_fields_without_JS · Uses: G001





S002 · impliesBefore

source roles = [[20, 40, 60, 80, 100, 120]]

After

derived = [[20, 40, 60, 80, 100, 120], a_n=0 for EVERY n beyond this finite list]

Conditions: none

Trim only canonical trailing zeros without changing inputs,retaining at least one coefficient.This defines the entire sequence with its zero tail,not an observed prefix or the EGF coefficient a_n/n!.

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





S003 · impliesBefore

source roles = [[20, 40, 60, 80, 100, 120], 1]

After

derived = [40]

Conditions: none

Read a_k inside support;outside it use the declared complete zero tail.

Step identifiers and dependenciesRule: OGF_exact_index_or_declared_zero_tail · 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

([20, 40, 60, 80, 100, 120], a_n=0_after_the_list, 1, 40, coefficient_extraction_uses_the_complete_finite_support_law)

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: 0783b56c749ba603e5c8f0cecdf1fae7bb0f9c0ae65820e9ae69eb9aee089ce5
- candidate record SHA-256: f66dc598bde5a427002c130c617c57d65b456b34dc9a31df0bb113bc3e5ff833
- semantic review receipt SHA-256: 049a9e07e69fead88d93b34f0937b9470a06d71c1a8d097058fc83554529bdd0
- question-first witness SHA-256: 0c71ebb418f3de347d232d159fc25e93036f4ab9166bcfb5ee5b4be86d5c55b9
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0020.json)