Question 27681: Outside-Scope Inputs Are Not General Strong-Law Counterexamples

Original question and givens

Use mode=finite_prefix, complete model={"values":[-2,-1],"probabilities":[[1,3],[2,3]],"observations":[-2,-1,-1]}. Give (mode, observations, common_mean, infinite_process_or_limit_proof_claim, boundary).

• G001 — Use mode=finite_prefix, complete model={"values":[-2,-1],"probabilities":[[1,3],[2,3]],"observations":[-2,-1,-1]}. Give (mode, observations, common_mean, infinite_process_or_limit_proof_claim, boundary).

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-strong-law-large-numbers-finite-valued-iid

Step-by-step derivation

source roles = [slln-scope-boundary]

derived = [[mode, observations, common_mean, infinite_process_or_limit_proof_claim, boundary]]

SLLN boundary prefix law ORIGINAL mass sum0 := sum(0, 1/3)

SLLN boundary prefix law ORIGINAL mass sum0 = 1/3

SLLN boundary prefix law ORIGINAL mass sum1 := sum(1/3, 2/3)

SLLN boundary prefix law ORIGINAL mass sum1 = 1

SLLN boundary prefix law ORIGINAL weighted value0 := mul(-2, 1/3)

SLLN boundary prefix law ORIGINAL weighted value0 = -2/3

SLLN boundary prefix law ORIGINAL mean sum0 := sum(0, -2/3)

SLLN boundary prefix law ORIGINAL mean sum0 = -2/3

SLLN boundary prefix law exact unbounded value square0 := mul(-2, -2)

SLLN boundary prefix law exact unbounded value square0 = 4

SLLN boundary prefix law ORIGINAL weighted square0 := mul(4, 1/3)

SLLN boundary prefix law ORIGINAL weighted square0 = 4/3

SLLN boundary prefix law ORIGINAL second moment sum0 := sum(0, 4/3)

SLLN boundary prefix law ORIGINAL second moment sum0 = 4/3

SLLN boundary prefix law ORIGINAL weighted value1 := mul(-1, 2/3)

SLLN boundary prefix law ORIGINAL weighted value1 = -2/3

SLLN boundary prefix law ORIGINAL mean sum1 := sum(-2/3, -2/3)

SLLN boundary prefix law ORIGINAL mean sum1 = -4/3

SLLN boundary prefix law exact unbounded value square1 := mul(-1, -1)

SLLN boundary prefix law exact unbounded value square1 = 1

SLLN boundary prefix law ORIGINAL weighted square1 := mul(1, 2/3)

SLLN boundary prefix law ORIGINAL weighted square1 = 2/3

SLLN boundary prefix law ORIGINAL second moment sum1 := sum(4/3, 2/3)

SLLN boundary prefix law ORIGINAL second moment sum1 = 2

SLLN boundary prefix law ORIGINAL mean squared := mul(-4/3, -4/3)

SLLN boundary prefix law ORIGINAL mean squared = 16/9

SLLN boundary prefix law ORIGINAL second minus mean squared := sub(2, 16/9)

SLLN boundary prefix law ORIGINAL second minus mean squared = 2/9

source roles = [SLLN boundary prefix law, [-2, -1], [[1, 3], [2, 3]]]

derived = [[[-2, 1/3], [-1, 2/3]], 1, -4/3, 2, 2/9]

source roles = [[-2, -1, -1], [[-2, 1/3], [-1, 2/3]]]

derived = [-4/3, No]

Reverse checks

• S017 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S016 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S015 · premise_and_result_audit — Recompute SLLN boundary prefix law ORIGINAL second minus mean squared with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute SLLN boundary prefix law ORIGINAL mean squared with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute SLLN boundary prefix law ORIGINAL second moment sum1 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute SLLN boundary prefix law ORIGINAL weighted square1 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute SLLN boundary prefix law exact unbounded value square1 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute SLLN boundary prefix law ORIGINAL mean sum1 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute SLLN boundary prefix law ORIGINAL weighted value1 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute SLLN boundary prefix law ORIGINAL second moment sum0 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute SLLN boundary prefix law ORIGINAL weighted square0 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute SLLN boundary prefix law exact unbounded value square0 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute SLLN boundary prefix law ORIGINAL mean sum0 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute SLLN boundary prefix law ORIGINAL weighted value0 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute SLLN boundary prefix law ORIGINAL mass sum1 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute SLLN boundary prefix law ORIGINAL mass sum0 with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.

Result

(finite_prefix, [-2, -1, -1], -4/3, No, finite_prefix_cannot_establish_infinite_process_or_a.s._limit)

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.