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Question 27694: Outside-Scope Inputs Are Not General Strong-Law Counterexamples

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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← Question derivation index · Original question batch

R0054-27694 · q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0094

Original question and givens

Use mode=wrong_target, complete model={"values":[1,2],"probabilities":[[2,5],[3,5]],"target":20}. Give (mode, common_mean, proposed_target, target_equals_common_mean, proposed_target_claim, boundary).

  • G001 — Use mode=wrong_target, complete model={"values":[1,2],"probabilities":[[2,5],[3,5]],"target":20}. Give (mode, common_mean, proposed_target, target_equals_common_mean, proposed_target_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.

Step-by-step derivation

S001 · implies

Before

source roles = [slln-scope-boundary]

After

derived = [[mode, common_mean, proposed_target, target_equals_common_mean, proposed_target_claim, boundary]]

Conditions: none

List the complete operation/mode field order without JavaScript or answer-shape inference;Boolean,string,null and an unclaimed result remain distinct.

Step identifiers and dependencies

Rule: source_ordered_fields_without_JS · Uses: G001

S002 · calculation

Before

SLLN boundary target law ORIGINAL mass sum0 := sum(0, 2/5)

After

SLLN boundary target law ORIGINAL mass sum0 = 2/5

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

SLLN boundary target law ORIGINAL mass sum1 := sum(2/5, 3/5)

After

SLLN boundary target law ORIGINAL mass sum1 = 1

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

SLLN boundary target law ORIGINAL weighted value0 := mul(1, 2/5)

After

SLLN boundary target law ORIGINAL weighted value0 = 2/5

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

SLLN boundary target law ORIGINAL mean sum0 := sum(0, 2/5)

After

SLLN boundary target law ORIGINAL mean sum0 = 2/5

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

SLLN boundary target law exact unbounded value square0 := mul(1, 1)

After

SLLN boundary target law exact unbounded value square0 = 1

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

SLLN boundary target law ORIGINAL weighted square0 := mul(1, 2/5)

After

SLLN boundary target law ORIGINAL weighted square0 = 2/5

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

Before

SLLN boundary target law ORIGINAL second moment sum0 := sum(0, 2/5)

After

SLLN boundary target law ORIGINAL second moment sum0 = 2/5

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

SLLN boundary target law ORIGINAL weighted value1 := mul(2, 3/5)

After

SLLN boundary target law ORIGINAL weighted value1 = 6/5

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

SLLN boundary target law ORIGINAL mean sum1 := sum(2/5, 6/5)

After

SLLN boundary target law ORIGINAL mean sum1 = 8/5

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · calculation

Before

SLLN boundary target law exact unbounded value square1 := mul(2, 2)

After

SLLN boundary target law exact unbounded value square1 = 4

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · calculation

Before

SLLN boundary target law ORIGINAL weighted square1 := mul(4, 3/5)

After

SLLN boundary target law ORIGINAL weighted square1 = 12/5

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · calculation

Before

SLLN boundary target law ORIGINAL second moment sum1 := sum(2/5, 12/5)

After

SLLN boundary target law ORIGINAL second moment sum1 = 14/5

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · calculation

Before

SLLN boundary target law ORIGINAL mean squared := mul(8/5, 8/5)

After

SLLN boundary target law ORIGINAL mean squared = 64/25

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · calculation

Before

SLLN boundary target law ORIGINAL second minus mean squared := sub(14/5, 64/25)

After

SLLN boundary target law ORIGINAL second minus mean squared = 6/25

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · implies

Before

source roles = [SLLN boundary target law, [1, 2], [[2, 5], [3, 5]]]

After

derived = [[[1, 2/5], [2, 3/5]], 1, 8/5, 14/5, 6/25]

Conditions: none

Pair ORIGINAL values/probabilities before numeric sorting and retain zero-mass labels.Qualify the complete nonnegative mass1 law and its source second-minus-mean-square variance.SLLN allows variance0;do not import the CLT positive-variance condition.

Step identifiers and dependencies

Rule: SLLN_complete_ORIGINAL_numeric_value_probability_alignment_ALL_moments · Uses: G001, S015

S017 · implies

Before

source roles = [[[1, 2/5], [2, 3/5]], 8/5, 20]

After

derived = [No, No]

Conditions: none

First verify the proposed target actually differs from the common mean.The SLLN target is that mean,not a substituted value.

Step identifiers and dependencies

Rule: SLLN_ACTUAL_wrong_target_not_common_expectation · Uses: G001, S016

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 target law ORIGINAL second minus mean squared with exact rational arithmetic, not rounded decimals.
  • S014 · premise_and_result_audit — Recompute SLLN boundary target law ORIGINAL mean squared with exact rational arithmetic, not rounded decimals.
  • S013 · premise_and_result_audit — Recompute SLLN boundary target law ORIGINAL second moment sum1 with exact rational arithmetic, not rounded decimals.
  • S012 · premise_and_result_audit — Recompute SLLN boundary target law ORIGINAL weighted square1 with exact rational arithmetic, not rounded decimals.
  • S011 · premise_and_result_audit — Recompute SLLN boundary target law exact unbounded value square1 with exact rational arithmetic, not rounded decimals.
  • S010 · premise_and_result_audit — Recompute SLLN boundary target law ORIGINAL mean sum1 with exact rational arithmetic, not rounded decimals.
  • S009 · premise_and_result_audit — Recompute SLLN boundary target law ORIGINAL weighted value1 with exact rational arithmetic, not rounded decimals.
  • S008 · premise_and_result_audit — Recompute SLLN boundary target law ORIGINAL second moment sum0 with exact rational arithmetic, not rounded decimals.
  • S007 · premise_and_result_audit — Recompute SLLN boundary target law ORIGINAL weighted square0 with exact rational arithmetic, not rounded decimals.
  • S006 · premise_and_result_audit — Recompute SLLN boundary target law exact unbounded value square0 with exact rational arithmetic, not rounded decimals.
  • S005 · premise_and_result_audit — Recompute SLLN boundary target law ORIGINAL mean sum0 with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Recompute SLLN boundary target law ORIGINAL weighted value0 with exact rational arithmetic, not rounded decimals.
  • S003 · premise_and_result_audit — Recompute SLLN boundary target law ORIGINAL mass sum1 with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Recompute SLLN boundary target 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

(wrong_target, 8/5, 20, No, No, SLLN_target_is_common_expectation)

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: d65eb61077ffd10ad0209112ec6dd376cce7873b77e44e906601a3e3378c41ba
  • candidate record SHA-256: 6caad4c68db7ac91a8609846d492557c52827046987715cc690c24b8a563d5bb
  • semantic review receipt SHA-256: 0a76d8e93c08bb16998a0483f3dfc0504f269fc3e1982e6cd868cab0a2856d54
  • question-first witness SHA-256: 09127b736c225714dfa0b7ceb0e6a6e66df27438b480cad8217d3c2f99ce9437
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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