Question 27693: Outside-Scope Inputs Are Not General Strong-Law Counterexamples

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0054-27693 · q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0093

Original question and givens

Use mode=finite_prefix, complete model={"values":[0,1],"probabilities":[[2,5],[3,5]],"observations":[0,1,1]}. Give (mode, observations, common_mean, infinite_process_or_limit_proof_claim, boundary).

G001 — Use mode=finite_prefix, complete model={"values":[0,1],"probabilities":[[2,5],[3,5]],"observations":[0,1,1]}. 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

S001 · implies

Before

source roles = [slln-scope-boundary]

After

derived = [[mode, observations, common_mean, infinite_process_or_limit_proof_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 prefix law ORIGINAL mass sum0 := sum(0, 2/5)

After

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

After

SLLN boundary prefix 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 prefix law ORIGINAL weighted value0 := mul(0, 2/5)

After

SLLN boundary prefix law ORIGINAL weighted value0 = 0

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 prefix law ORIGINAL mean sum0 := sum(0, 0)

After

SLLN boundary prefix law ORIGINAL mean sum0 = 0

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 prefix law exact unbounded value square0 := mul(0, 0)

After

SLLN boundary prefix law exact unbounded value square0 = 0

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 prefix law ORIGINAL weighted square0 := mul(0, 2/5)

After

SLLN boundary prefix law ORIGINAL weighted square0 = 0

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 prefix law ORIGINAL second moment sum0 := sum(0, 0)

After

SLLN boundary prefix law ORIGINAL second moment sum0 = 0

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 prefix law ORIGINAL weighted value1 := mul(1, 3/5)

After

SLLN boundary prefix law ORIGINAL weighted value1 = 3/5

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 prefix law ORIGINAL mean sum1 := sum(0, 3/5)

After

SLLN boundary prefix law ORIGINAL mean sum1 = 3/5

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

After

SLLN boundary prefix law 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 prefix law ORIGINAL weighted square1 := mul(1, 3/5)

After

SLLN boundary prefix law ORIGINAL weighted square1 = 3/5

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 prefix law ORIGINAL second moment sum1 := sum(0, 3/5)

After

SLLN boundary prefix law ORIGINAL second moment sum1 = 3/5

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 prefix law ORIGINAL mean squared := mul(3/5, 3/5)

After

SLLN boundary prefix law ORIGINAL mean squared = 9/25

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 prefix law ORIGINAL second minus mean squared := sub(3/5, 9/25)

After

SLLN boundary prefix 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 prefix law, [0, 1], [[2, 5], [3, 5]]]

After

derived = [[[0, 2/5], [1, 3/5]], 1, 3/5, 3/5, 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 = [[0, 1, 1], [[0, 2/5], [1, 3/5]]]

After

derived = [3/5, No]

Conditions: none

This boundary permits an empty prefix.No finite prefix establishes an infinite IID process or almost-sure limit;No does not deny extendibility.

Step identifiers and dependencies

Rule: SLLN_boundary_finite_prefix_MAYempty_cannot_construct_process · 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 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, [0, 1, 1], 3/5, 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.

Source hashes and review receipts

input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
original question SHA-256: def272b63aec531e836f13c0bd8a0a07c218f332021d93aafcac702a2178b461
candidate record SHA-256: edc15f5336cdf6bc9ddb568346325a683ac785d5010d3b8a96990e71a623e178
semantic review receipt SHA-256: 0a76d8e93c08bb16998a0483f3dfc0504f269fc3e1982e6cd868cab0a2856d54
question-first witness SHA-256: 09127b736c225714dfa0b7ceb0e6a6e66df27438b480cad8217d3c2f99ce9437
Lean: NOT_RUN · human review: NOT_PERFORMED

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