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Question 27686: 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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R0054-27686 · q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0086

Original question and givens

Use mode=non_iid, complete model={"values_a":[-2,-1],"probabilities_a":[[1,4],[3,4]],"values_b":[-2,-1],"probabilities_b":[[1,7],[6,7]]}. Give (mode, identical_marginals, this_finite_valued_IID_theorem_applicable, scope).

  • G001 — Use mode=non_iid, complete model={"values_a":[-2,-1],"probabilities_a":[[1,4],[3,4]],"values_b":[-2,-1],"probabilities_b":[[1,7],[6,7]]}. Give (mode, identical_marginals, this_finite_valued_IID_theorem_applicable, scope).

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, identical_marginals, this_finite_valued_IID_theorem_applicable, scope]]

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 first marginal ORIGINAL mass sum0 := sum(0, 1/4)

After

SLLN boundary first marginal ORIGINAL mass sum0 = 1/4

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, S001

S003 · calculation

Before

SLLN boundary first marginal ORIGINAL mass sum1 := sum(1/4, 3/4)

After

SLLN boundary first marginal 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 first marginal ORIGINAL weighted value0 := mul(-2, 1/4)

After

SLLN boundary first marginal ORIGINAL weighted value0 = -1/2

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 first marginal ORIGINAL mean sum0 := sum(0, -1/2)

After

SLLN boundary first marginal ORIGINAL mean sum0 = -1/2

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 first marginal exact unbounded value square0 := mul(-2, -2)

After

SLLN boundary first marginal exact unbounded value square0 = 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, S005

S007 · calculation

Before

SLLN boundary first marginal ORIGINAL weighted square0 := mul(4, 1/4)

After

SLLN boundary first marginal ORIGINAL weighted 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, S006

S008 · calculation

Before

SLLN boundary first marginal ORIGINAL second moment sum0 := sum(0, 1)

After

SLLN boundary first marginal ORIGINAL second moment sum0 = 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, S007

S009 · calculation

Before

SLLN boundary first marginal ORIGINAL weighted value1 := mul(-1, 3/4)

After

SLLN boundary first marginal ORIGINAL weighted value1 = -3/4

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, S008

S010 · calculation

Before

SLLN boundary first marginal ORIGINAL mean sum1 := sum(-1/2, -3/4)

After

SLLN boundary first marginal ORIGINAL mean sum1 = -5/4

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, S009

S011 · calculation

Before

SLLN boundary first marginal exact unbounded value square1 := mul(-1, -1)

After

SLLN boundary first marginal exact unbounded value square1 = 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, S010

S012 · calculation

Before

SLLN boundary first marginal ORIGINAL weighted square1 := mul(1, 3/4)

After

SLLN boundary first marginal ORIGINAL weighted square1 = 3/4

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, S011

S013 · calculation

Before

SLLN boundary first marginal ORIGINAL second moment sum1 := sum(1, 3/4)

After

SLLN boundary first marginal ORIGINAL second moment sum1 = 7/4

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, S012

S014 · calculation

Before

SLLN boundary first marginal ORIGINAL mean squared := mul(-5/4, -5/4)

After

SLLN boundary first marginal ORIGINAL mean squared = 25/16

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 first marginal ORIGINAL second minus mean squared := sub(7/4, 25/16)

After

SLLN boundary first marginal ORIGINAL second minus mean squared = 3/16

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 first marginal, [-2, -1], [[1, 4], [3, 4]]]

After

derived = [[[-2, 1/4], [-1, 3/4]], 1, -5/4, 7/4, 3/16]

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 · calculation

Before

SLLN boundary second marginal ORIGINAL mass sum0 := sum(0, 1/7)

After

SLLN boundary second marginal ORIGINAL mass sum0 = 1/7

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, S016

S018 · calculation

Before

SLLN boundary second marginal ORIGINAL mass sum1 := sum(1/7, 6/7)

After

SLLN boundary second marginal 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, S017

S019 · calculation

Before

SLLN boundary second marginal ORIGINAL weighted value0 := mul(-2, 1/7)

After

SLLN boundary second marginal ORIGINAL weighted value0 = -2/7

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, S018

S020 · calculation

Before

SLLN boundary second marginal ORIGINAL mean sum0 := sum(0, -2/7)

After

SLLN boundary second marginal ORIGINAL mean sum0 = -2/7

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, S019

S021 · calculation

Before

SLLN boundary second marginal exact unbounded value square0 := mul(-2, -2)

After

SLLN boundary second marginal exact unbounded value square0 = 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, S020

S022 · calculation

Before

SLLN boundary second marginal ORIGINAL weighted square0 := mul(4, 1/7)

After

SLLN boundary second marginal ORIGINAL weighted square0 = 4/7

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, S021

S023 · calculation

Before

SLLN boundary second marginal ORIGINAL second moment sum0 := sum(0, 4/7)

After

SLLN boundary second marginal ORIGINAL second moment sum0 = 4/7

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, S022

S024 · calculation

Before

SLLN boundary second marginal ORIGINAL weighted value1 := mul(-1, 6/7)

After

SLLN boundary second marginal ORIGINAL weighted value1 = -6/7

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, S023

S025 · calculation

Before

SLLN boundary second marginal ORIGINAL mean sum1 := sum(-2/7, -6/7)

After

SLLN boundary second marginal ORIGINAL mean sum1 = -8/7

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, S024

S026 · calculation

Before

SLLN boundary second marginal exact unbounded value square1 := mul(-1, -1)

After

SLLN boundary second marginal exact unbounded value square1 = 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, S025

S027 · calculation

Before

SLLN boundary second marginal ORIGINAL weighted square1 := mul(1, 6/7)

After

SLLN boundary second marginal ORIGINAL weighted square1 = 6/7

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, S026

S028 · calculation

Before

SLLN boundary second marginal ORIGINAL second moment sum1 := sum(4/7, 6/7)

After

SLLN boundary second marginal ORIGINAL second moment sum1 = 10/7

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, S027

S029 · calculation

Before

SLLN boundary second marginal ORIGINAL mean squared := mul(-8/7, -8/7)

After

SLLN boundary second marginal ORIGINAL mean squared = 64/49

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, S028

S030 · calculation

Before

SLLN boundary second marginal ORIGINAL second minus mean squared := sub(10/7, 64/49)

After

SLLN boundary second marginal ORIGINAL second minus mean squared = 6/49

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, S029

S031 · implies

Before

source roles = [SLLN boundary second marginal, [-2, -1], [[1, 7], [6, 7]]]

After

derived = [[[-2, 1/7], [-1, 6/7]], 1, -8/7, 10/7, 6/49]

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, S030

S032 · implies

Before

source roles = [[[-2, 1/4], [-1, 3/4]], [[-2, 1/7], [-1, 6/7]]]

After

derived = [No, No]

Conditions: none

The complete source marginal rows actually differ,withholding this IID version,not every non-IID strong law.

Step identifiers and dependencies

Rule: SLLN_ACTUAL_different_full_marginal_rows_NOT_all_strong_laws · Uses: G001, S031

Reverse checks

  • S032 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S031 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S030 · premise_and_result_audit — Recompute SLLN boundary second marginal ORIGINAL second minus mean squared with exact rational arithmetic, not rounded decimals.
  • S029 · premise_and_result_audit — Recompute SLLN boundary second marginal ORIGINAL mean squared with exact rational arithmetic, not rounded decimals.
  • S028 · premise_and_result_audit — Recompute SLLN boundary second marginal ORIGINAL second moment sum1 with exact rational arithmetic, not rounded decimals.
  • S027 · premise_and_result_audit — Recompute SLLN boundary second marginal ORIGINAL weighted square1 with exact rational arithmetic, not rounded decimals.
  • S026 · premise_and_result_audit — Recompute SLLN boundary second marginal exact unbounded value square1 with exact rational arithmetic, not rounded decimals.
  • S025 · premise_and_result_audit — Recompute SLLN boundary second marginal ORIGINAL mean sum1 with exact rational arithmetic, not rounded decimals.
  • S024 · premise_and_result_audit — Recompute SLLN boundary second marginal ORIGINAL weighted value1 with exact rational arithmetic, not rounded decimals.
  • S023 · premise_and_result_audit — Recompute SLLN boundary second marginal ORIGINAL second moment sum0 with exact rational arithmetic, not rounded decimals.
  • S022 · premise_and_result_audit — Recompute SLLN boundary second marginal ORIGINAL weighted square0 with exact rational arithmetic, not rounded decimals.
  • S021 · premise_and_result_audit — Recompute SLLN boundary second marginal exact unbounded value square0 with exact rational arithmetic, not rounded decimals.
  • S020 · premise_and_result_audit — Recompute SLLN boundary second marginal ORIGINAL mean sum0 with exact rational arithmetic, not rounded decimals.
  • S019 · premise_and_result_audit — Recompute SLLN boundary second marginal ORIGINAL weighted value0 with exact rational arithmetic, not rounded decimals.
  • S018 · premise_and_result_audit — Recompute SLLN boundary second marginal ORIGINAL mass sum1 with exact rational arithmetic, not rounded decimals.
  • S017 · premise_and_result_audit — Recompute SLLN boundary second marginal ORIGINAL mass sum0 with exact rational arithmetic, not rounded decimals.
  • 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 first marginal ORIGINAL second minus mean squared with exact rational arithmetic, not rounded decimals.
  • S014 · premise_and_result_audit — Recompute SLLN boundary first marginal ORIGINAL mean squared with exact rational arithmetic, not rounded decimals.
  • S013 · premise_and_result_audit — Recompute SLLN boundary first marginal ORIGINAL second moment sum1 with exact rational arithmetic, not rounded decimals.
  • S012 · premise_and_result_audit — Recompute SLLN boundary first marginal ORIGINAL weighted square1 with exact rational arithmetic, not rounded decimals.
  • S011 · premise_and_result_audit — Recompute SLLN boundary first marginal exact unbounded value square1 with exact rational arithmetic, not rounded decimals.
  • S010 · premise_and_result_audit — Recompute SLLN boundary first marginal ORIGINAL mean sum1 with exact rational arithmetic, not rounded decimals.
  • S009 · premise_and_result_audit — Recompute SLLN boundary first marginal ORIGINAL weighted value1 with exact rational arithmetic, not rounded decimals.
  • S008 · premise_and_result_audit — Recompute SLLN boundary first marginal ORIGINAL second moment sum0 with exact rational arithmetic, not rounded decimals.
  • S007 · premise_and_result_audit — Recompute SLLN boundary first marginal ORIGINAL weighted square0 with exact rational arithmetic, not rounded decimals.
  • S006 · premise_and_result_audit — Recompute SLLN boundary first marginal exact unbounded value square0 with exact rational arithmetic, not rounded decimals.
  • S005 · premise_and_result_audit — Recompute SLLN boundary first marginal ORIGINAL mean sum0 with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Recompute SLLN boundary first marginal ORIGINAL weighted value0 with exact rational arithmetic, not rounded decimals.
  • S003 · premise_and_result_audit — Recompute SLLN boundary first marginal ORIGINAL mass sum1 with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Recompute SLLN boundary first marginal 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

(non_iid, No, No, finite_valued_IID_SLLN_not_applicable_not_general_failure)

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: f9da71a77b9781a3619dcf23f7e58e83d004fa6c092e1b45d4cca8480b18ccdf
  • candidate record SHA-256: b550b8bae04347e21f05ec4a6c76120d3ed838a592604f1c5c7b0a6effeecaa0
  • semantic review receipt SHA-256: 0a76d8e93c08bb16998a0483f3dfc0504f269fc3e1982e6cd868cab0a2856d54
  • question-first witness SHA-256: 09127b736c225714dfa0b7ceb0e6a6e66df27438b480cad8217d3c2f99ce9437
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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