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Question 27642: A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate

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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R0054-27642 · q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0042

Original question and givens

Use the aligned finite law values=[-1, 0], probabilities=[2/5, 3/5], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. The observed finite prefix is=[-1, 0, -1, 0, -1]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).

  • G001 — Use the aligned finite law values=[-1, 0], probabilities=[2/5, 3/5], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. The observed finite prefix is=[-1, 0, -1, 0, -1]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_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.

Step-by-step derivation

S001 · implies

Before

source roles = [slln-prefix-sample-mean]

After

derived = [[observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_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 · implies

Before

source roles = [infinite_iid]

After

derived = [ASSUME whole infinite sequence is IID with this common marginal]

Conditions: none

infinite_iid is a GIVEN whole-process premise,not independence constructed or proved by finite data/this program.

Step identifiers and dependencies

Rule: SLLN_infinite_IID_EXACT_GIVEN_premise_not_finite_observation_inference · Uses: G001, S001

S003 · calculation

Before

SLLN ORIGINAL common marginal ORIGINAL mass sum0 := sum(0, 2/5)

After

SLLN ORIGINAL common marginal 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, S002

S004 · calculation

Before

SLLN ORIGINAL common marginal ORIGINAL mass sum1 := sum(2/5, 3/5)

After

SLLN ORIGINAL common 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, S003

S005 · calculation

Before

SLLN ORIGINAL common marginal ORIGINAL weighted value0 := mul(-1, 2/5)

After

SLLN ORIGINAL common marginal 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, S004

S006 · calculation

Before

SLLN ORIGINAL common marginal ORIGINAL mean sum0 := sum(0, -2/5)

After

SLLN ORIGINAL common marginal 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, S005

S007 · calculation

Before

SLLN ORIGINAL common marginal exact unbounded value square0 := mul(-1, -1)

After

SLLN ORIGINAL common marginal 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, S006

S008 · calculation

Before

SLLN ORIGINAL common marginal ORIGINAL weighted square0 := mul(1, 2/5)

After

SLLN ORIGINAL common marginal 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, S007

S009 · calculation

Before

SLLN ORIGINAL common marginal ORIGINAL second moment sum0 := sum(0, 2/5)

After

SLLN ORIGINAL common marginal 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, S008

S010 · calculation

Before

SLLN ORIGINAL common marginal ORIGINAL weighted value1 := mul(0, 3/5)

After

SLLN ORIGINAL common marginal ORIGINAL weighted value1 = 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, S009

S011 · calculation

Before

SLLN ORIGINAL common marginal ORIGINAL mean sum1 := sum(-2/5, 0)

After

SLLN ORIGINAL common marginal ORIGINAL mean sum1 = -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, S010

S012 · calculation

Before

SLLN ORIGINAL common marginal exact unbounded value square1 := mul(0, 0)

After

SLLN ORIGINAL common marginal exact unbounded value square1 = 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, S011

S013 · calculation

Before

SLLN ORIGINAL common marginal ORIGINAL weighted square1 := mul(0, 3/5)

After

SLLN ORIGINAL common marginal ORIGINAL weighted square1 = 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, S012

S014 · calculation

Before

SLLN ORIGINAL common marginal ORIGINAL second moment sum1 := sum(2/5, 0)

After

SLLN ORIGINAL common marginal ORIGINAL second moment sum1 = 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, S013

S015 · calculation

Before

SLLN ORIGINAL common marginal ORIGINAL mean squared := mul(-2/5, -2/5)

After

SLLN ORIGINAL common marginal ORIGINAL mean squared = 4/25

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 · calculation

Before

SLLN ORIGINAL common marginal ORIGINAL second minus mean squared := sub(2/5, 4/25)

After

SLLN ORIGINAL common marginal 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, S015

S017 · implies

Before

source roles = [SLLN ORIGINAL common marginal, [-1, 0], [[2, 5], [3, 5]]]

After

derived = [[[-1, 2/5], [0, 3/5]], 1, -2/5, 2/5, 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, S016

S018 · calculation

Before

SLLN ORIGINAL prefix integer sum := sum(-1, 0, -1, 0, -1)

After

SLLN ORIGINAL prefix integer sum = -3

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 ORIGINAL prefix sample mean := div(-3, 5)

After

SLLN ORIGINAL prefix sample mean = -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, S018

S020 · calculation

Before

SLLN ORIGINAL sample mean minus common mean := sub(-3/5, -2/5)

After

SLLN ORIGINAL sample mean minus common mean = -1/5

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, S019

S021 · implies

Before

source roles = [[-1, 0, -1, 0, -1], -2/5, -3/5, -1/5]

After

derived = [1/5, No]

Conditions: none

Every observation is correctly typed and in the carrier;the prefix must be nonempty.The source uses the UNREDUCED[sum,length]1e6 profile,not an invented8-term cap.One close/equal prefix does not prove an almost-sure limit.

Step identifiers and dependencies

Rule: SLLN_fixed_nonempty_prefix_mean_error_NOT_as_limit_certificate · Uses: G001, S020

Reverse checks

  • S021 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S020 · premise_and_result_audit — Recompute SLLN ORIGINAL sample mean minus common mean with exact rational arithmetic, not rounded decimals.
  • S019 · premise_and_result_audit — Recompute SLLN ORIGINAL prefix sample mean with exact rational arithmetic, not rounded decimals.
  • S018 · premise_and_result_audit — Recompute SLLN ORIGINAL prefix integer sum with exact rational arithmetic, not rounded decimals.
  • 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 — Recompute SLLN ORIGINAL common marginal ORIGINAL second minus mean squared with exact rational arithmetic, not rounded decimals.
  • S015 · premise_and_result_audit — Recompute SLLN ORIGINAL common marginal ORIGINAL mean squared with exact rational arithmetic, not rounded decimals.
  • S014 · premise_and_result_audit — Recompute SLLN ORIGINAL common marginal ORIGINAL second moment sum1 with exact rational arithmetic, not rounded decimals.
  • S013 · premise_and_result_audit — Recompute SLLN ORIGINAL common marginal ORIGINAL weighted square1 with exact rational arithmetic, not rounded decimals.
  • S012 · premise_and_result_audit — Recompute SLLN ORIGINAL common marginal exact unbounded value square1 with exact rational arithmetic, not rounded decimals.
  • S011 · premise_and_result_audit — Recompute SLLN ORIGINAL common marginal ORIGINAL mean sum1 with exact rational arithmetic, not rounded decimals.
  • S010 · premise_and_result_audit — Recompute SLLN ORIGINAL common marginal ORIGINAL weighted value1 with exact rational arithmetic, not rounded decimals.
  • S009 · premise_and_result_audit — Recompute SLLN ORIGINAL common marginal ORIGINAL second moment sum0 with exact rational arithmetic, not rounded decimals.
  • S008 · premise_and_result_audit — Recompute SLLN ORIGINAL common marginal ORIGINAL weighted square0 with exact rational arithmetic, not rounded decimals.
  • S007 · premise_and_result_audit — Recompute SLLN ORIGINAL common marginal exact unbounded value square0 with exact rational arithmetic, not rounded decimals.
  • S006 · premise_and_result_audit — Recompute SLLN ORIGINAL common marginal ORIGINAL mean sum0 with exact rational arithmetic, not rounded decimals.
  • S005 · premise_and_result_audit — Recompute SLLN ORIGINAL common marginal ORIGINAL weighted value0 with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Recompute SLLN ORIGINAL common marginal ORIGINAL mass sum1 with exact rational arithmetic, not rounded decimals.
  • S003 · premise_and_result_audit — Recompute SLLN ORIGINAL common marginal ORIGINAL mass sum0 with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • 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

([-1, 0, -1, 0, -1], -2/5, -3/5, 1/5, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)

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: 3870a60b42b76ebf4e70d56099fdde869b2f9e405afb0fffb5e1d0cbb154262e
  • candidate record SHA-256: 2c3cc5f58380e1fd8d75780b591fb82e2b0cca26a18dc7a5b12c4084f53a6365
  • semantic review receipt SHA-256: 0a76d8e93c08bb16998a0483f3dfc0504f269fc3e1982e6cd868cab0a2856d54
  • question-first witness SHA-256: 09127b736c225714dfa0b7ceb0e6a6e66df27438b480cad8217d3c2f99ce9437
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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