# Question 24634: Support Coverage Is Separate from Accidental Equality at One Point [Return to catalogue page 124](https://ocme.evemisslab.com/en/questions/page/124/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations/#q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0034) 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. R0048-24634 · q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0034 ## Original question and givens Use finite_support_coefficients=[9, 18, 27] 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. Use x=1/4, cutoff=2. Give (finite_support_coefficients, x, cutoff, term_rows, finite_sum, complete_support_included, certificate). - G001 — Use finite_support_coefficients=[9, 18, 27] 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. Use x=1/4, cutoff=2. Give (finite_support_coefficients, x, cutoff, term_rows, finite_sum, complete_support_included, 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-finite-partial-sum] After derived = [[finite_support_coefficients, x, cutoff, term_rows, finite_sum, complete_support_included, 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 = [[9, 18, 27]] After derived = [[9, 18, 27], 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 (1/4)^0 After OGF ORIGINAL x power0 = 1 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S002 S004 · calculationBefore OGF term0 coefficient times power := mul(9, 1) After OGF term0 coefficient times power = 9 Conditions: none Substitute original roles and established intermediates exactly,checking nonzero denominators without floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore OGF partial sum through0 := sum(0, 9) After OGF partial sum through0 = 9 Conditions: none Substitute original roles and established intermediates exactly,checking nonzero denominators without floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore (1/4)^1 After OGF ORIGINAL x power1 = 1/4 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S005 S007 · calculationBefore OGF term1 coefficient times power := mul(18, 1/4) After OGF term1 coefficient times power = 9/2 Conditions: none Substitute original roles and established intermediates exactly,checking nonzero denominators without floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore OGF partial sum through1 := sum(9, 9/2) After OGF partial sum through1 = 27/2 Conditions: none Substitute original roles and established intermediates exactly,checking nonzero denominators without floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · impliesBefore (1/4)^2 After OGF ORIGINAL x power2 = 1/16 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S008 S010 · calculationBefore OGF term2 coefficient times power := mul(27, 1/16) After OGF term2 coefficient times power = 27/16 Conditions: none Substitute original roles and established intermediates exactly,checking nonzero denominators without floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore OGF partial sum through2 := sum(27/2, 27/16) After OGF partial sum through2 = 243/16 Conditions: none Substitute original roles and established intermediates exactly,checking nonzero denominators without floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · impliesBefore source roles = [[9, 18, 27], 1/4, 2] After derived = [243/16, Yes] Conditions: none The coverage flag compares cutoff with the final nonzero support,not accidental point equality.At x=0 or by cancellation,a shorter sum may have the same value. Step identifiers and dependenciesRule: OGF_support_coverage_NOT_accidental_pointwise_equality · Uses: G001, S011 ## Reverse checks - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S011 · premise_and_result_audit — Recompute OGF partial sum through2 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute OGF term2 coefficient times power with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S008 · premise_and_result_audit — Recompute OGF partial sum through1 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute OGF term1 coefficient times power with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Recompute OGF partial sum through0 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute OGF term0 coefficient times power 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 ([9, 18, 27], 1/4, 2, [[0, 9, 9], [1, 18, 9/2], [2, 27, 27/16]], 243/16, Yes, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included) 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: eb5634a368af543bedfb7c6ef8da82b8958805924676893f275e98e37b36667a - candidate record SHA-256: 82b468122468c8230177d77177d767e01598cd01e0a907b6e1ca6295d53f4d1a - 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-0034.json)