Question 24633: Support Coverage Is Separate from Accidental Equality at One Point

Original question and givens

Use finite_support_coefficients=[8, 16, 24, 32, 40, 48] 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=1/3, cutoff=4. Give (finite_support_coefficients, x, cutoff, term_rows, finite_sum, complete_support_included, certificate).

• G001 — Use finite_support_coefficients=[8, 16, 24, 32, 40, 48] 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=1/3, cutoff=4. 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

Step-by-step derivation

source roles = [ogf-finite-partial-sum]

derived = [[finite_support_coefficients, x, cutoff, term_rows, finite_sum, complete_support_included, certificate]]

source roles = [[8, 16, 24, 32, 40, 48]]

derived = [[8, 16, 24, 32, 40, 48], a_n=0 for EVERY n beyond this finite list]

(1/3)^0

OGF ORIGINAL x power0 = 1

OGF term0 coefficient times power := mul(8, 1)

OGF term0 coefficient times power = 8

OGF partial sum through0 := sum(0, 8)

OGF partial sum through0 = 8

(1/3)^1

OGF ORIGINAL x power1 = 1/3

OGF term1 coefficient times power := mul(16, 1/3)

OGF term1 coefficient times power = 16/3

OGF partial sum through1 := sum(8, 16/3)

OGF partial sum through1 = 40/3

(1/3)^2

OGF ORIGINAL x power2 = 1/9

OGF term2 coefficient times power := mul(24, 1/9)

OGF term2 coefficient times power = 8/3

OGF partial sum through2 := sum(40/3, 8/3)

OGF partial sum through2 = 16

(1/3)^3

OGF ORIGINAL x power3 = 1/27

OGF term3 coefficient times power := mul(32, 1/27)

OGF term3 coefficient times power = 32/27

OGF partial sum through3 := sum(16, 32/27)

OGF partial sum through3 = 464/27

(1/3)^4

OGF ORIGINAL x power4 = 1/81

OGF term4 coefficient times power := mul(40, 1/81)

OGF term4 coefficient times power = 40/81

OGF partial sum through4 := sum(464/27, 40/81)

OGF partial sum through4 = 1432/81

source roles = [[8, 16, 24, 32, 40, 48], 1/3, 4]

derived = [1432/81, No]

Reverse checks

• S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S017 · premise_and_result_audit — Recompute OGF partial sum through4 with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute OGF term4 coefficient times power with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S014 · premise_and_result_audit — Recompute OGF partial sum through3 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute OGF term3 coefficient times power with exact rational arithmetic, not rounded decimals.
• 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

([8, 16, 24, 32, 40, 48], 1/3, 4, [[0, 8, 8], [1, 16, 16/3], [2, 24, 8/3], [3, 32, 32/27], [4, 40, 40/81]], 1432/81, No, 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.