Question 24638: Support Coverage Is Separate from Accidental Equality at One Point

Original question and givens

Use finite_support_coefficients=[13, 26, 39] 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=1/2, cutoff=2. Give (finite_support_coefficients, x, cutoff, term_rows, finite_sum, complete_support_included, certificate).

• G001 — Use finite_support_coefficients=[13, 26, 39] 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=1/2, 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

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 = [[13, 26, 39]]

derived = [[13, 26, 39], a_n=0 for EVERY n beyond this finite list]

(1/2)^0

OGF ORIGINAL x power0 = 1

OGF term0 coefficient times power := mul(13, 1)

OGF term0 coefficient times power = 13

OGF partial sum through0 := sum(0, 13)

OGF partial sum through0 = 13

(1/2)^1

OGF ORIGINAL x power1 = 1/2

OGF term1 coefficient times power := mul(26, 1/2)

OGF term1 coefficient times power = 13

OGF partial sum through1 := sum(13, 13)

OGF partial sum through1 = 26

(1/2)^2

OGF ORIGINAL x power2 = 1/4

OGF term2 coefficient times power := mul(39, 1/4)

OGF term2 coefficient times power = 39/4

OGF partial sum through2 := sum(26, 39/4)

OGF partial sum through2 = 143/4

source roles = [[13, 26, 39], 1/2, 2]

derived = [143/4, Yes]

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

([13, 26, 39], 1/2, 2, [[0, 13, 13], [1, 26, 13], [2, 39, 39/4]], 143/4, 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.