Question 15088: A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate

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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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R0029-15088 · q-20261014-advance-20261030-r07-convergence-in-probability-0063

Original question and givens

Use only the explicitly supplied finite list p_n=3/(n+7) for n=0 through prefix_length-1, with prefix_length=13. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).

G001 — Use only the explicitly supplied finite list p_n=3/(n+7) for n=0 through prefix_length-1, with prefix_length=13. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).

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-convergence-in-probability

Step-by-step derivation

S001 · implies

Before

positive profile parameters c=3;s=7;raw fraction=c/(n+s)

After

nonnegative valid probability indices start at=0;only there0<profile<=1

Conditions: none

A positive denominator and value≤1 require n+s≥c.Keep the original index;do not clip values above1 or silently shift them.This checks arithmetic domain only.The infinite-process/common-space interpretation remains declared,and the prefix-only family gains no infinite continuation premise.

Step identifiers and dependencies

Rule: deviation_profile_probability_domain_not_clipping · Uses: G001

S002 · calculation

Before

finite prefix index0 denominator := sum(0, 7)

After

finite prefix index0 denominator = 7

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

finite prefix index0 probability := div(3, 7)

After

finite prefix index0 probability = 3/7

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

finite prefix index1 denominator := sum(1, 7)

After

finite prefix index1 denominator = 8

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

finite prefix index1 probability := div(3, 8)

After

finite prefix index1 probability = 3/8

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

finite prefix index2 denominator := sum(2, 7)

After

finite prefix index2 denominator = 9

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

finite prefix index2 probability := div(3, 9)

After

finite prefix index2 probability = 1/3

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

Before

finite prefix index3 denominator := sum(3, 7)

After

finite prefix index3 denominator = 10

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

finite prefix index3 probability := div(3, 10)

After

finite prefix index3 probability = 3/10

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

finite prefix index4 denominator := sum(4, 7)

After

finite prefix index4 denominator = 11

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · calculation

Before

finite prefix index4 probability := div(3, 11)

After

finite prefix index4 probability = 3/11

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · calculation

Before

finite prefix index5 denominator := sum(5, 7)

After

finite prefix index5 denominator = 12

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · calculation

Before

finite prefix index5 probability := div(3, 12)

After

finite prefix index5 probability = 1/4

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · calculation

Before

finite prefix index6 denominator := sum(6, 7)

After

finite prefix index6 denominator = 13

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · calculation

Before

finite prefix index6 probability := div(3, 13)

After

finite prefix index6 probability = 3/13

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · calculation

Before

finite prefix index7 denominator := sum(7, 7)

After

finite prefix index7 denominator = 14

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · calculation

Before

finite prefix index7 probability := div(3, 14)

After

finite prefix index7 probability = 3/14

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · calculation

Before

finite prefix index8 denominator := sum(8, 7)

After

finite prefix index8 denominator = 15

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · calculation

Before

finite prefix index8 probability := div(3, 15)

After

finite prefix index8 probability = 1/5

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · calculation

Before

finite prefix index9 denominator := sum(9, 7)

After

finite prefix index9 denominator = 16

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · calculation

Before

finite prefix index9 probability := div(3, 16)

After

finite prefix index9 probability = 3/16

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · calculation

Before

finite prefix index10 denominator := sum(10, 7)

After

finite prefix index10 denominator = 17

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · calculation

Before

finite prefix index10 probability := div(3, 17)

After

finite prefix index10 probability = 3/17

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · calculation

Before

finite prefix index11 denominator := sum(11, 7)

After

finite prefix index11 denominator = 18

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S023

S025 · calculation

Before

finite prefix index11 probability := div(3, 18)

After

finite prefix index11 probability = 1/6

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · calculation

Before

finite prefix index12 denominator := sum(12, 7)

After

finite prefix index12 denominator = 19

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S025

S027 · calculation

Before

finite prefix index12 probability := div(3, 19)

After

finite prefix index12 probability = 3/19

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S026

S028 · implies

Before

only indices0 through 12 supplied;finite probability entries=[3/7, 3/8, 1/3, 3/10, 3/11, 1/4, 3/13, 3/14, 1/5, 3/16, 3/17, 1/6, 3/19]

After

general convergence established=No;no infinite continuation or every-epsilon tail supplied

Conditions: none

List the original zero-indexed prefix without dropping an invalid first entry.Even correct decreasing entries supply no every-epsilon/all-later obligation;an empty prefix is not a certificate either.

Step identifiers and dependencies

Rule: finite_probability_prefix_not_infinite_process_certificate · Uses: G001, S027

Reverse checks

S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S027 · premise_and_result_audit — Recompute finite prefix index12 probability with exact rational arithmetic, not rounded decimals.
S026 · premise_and_result_audit — Recompute finite prefix index12 denominator with exact rational arithmetic, not rounded decimals.
S025 · premise_and_result_audit — Recompute finite prefix index11 probability with exact rational arithmetic, not rounded decimals.
S024 · premise_and_result_audit — Recompute finite prefix index11 denominator with exact rational arithmetic, not rounded decimals.
S023 · premise_and_result_audit — Recompute finite prefix index10 probability with exact rational arithmetic, not rounded decimals.
S022 · premise_and_result_audit — Recompute finite prefix index10 denominator with exact rational arithmetic, not rounded decimals.
S021 · premise_and_result_audit — Recompute finite prefix index9 probability with exact rational arithmetic, not rounded decimals.
S020 · premise_and_result_audit — Recompute finite prefix index9 denominator with exact rational arithmetic, not rounded decimals.
S019 · premise_and_result_audit — Recompute finite prefix index8 probability with exact rational arithmetic, not rounded decimals.
S018 · premise_and_result_audit — Recompute finite prefix index8 denominator with exact rational arithmetic, not rounded decimals.
S017 · premise_and_result_audit — Recompute finite prefix index7 probability with exact rational arithmetic, not rounded decimals.
S016 · premise_and_result_audit — Recompute finite prefix index7 denominator with exact rational arithmetic, not rounded decimals.
S015 · premise_and_result_audit — Recompute finite prefix index6 probability with exact rational arithmetic, not rounded decimals.
S014 · premise_and_result_audit — Recompute finite prefix index6 denominator with exact rational arithmetic, not rounded decimals.
S013 · premise_and_result_audit — Recompute finite prefix index5 probability with exact rational arithmetic, not rounded decimals.
S012 · premise_and_result_audit — Recompute finite prefix index5 denominator with exact rational arithmetic, not rounded decimals.
S011 · premise_and_result_audit — Recompute finite prefix index4 probability with exact rational arithmetic, not rounded decimals.
S010 · premise_and_result_audit — Recompute finite prefix index4 denominator with exact rational arithmetic, not rounded decimals.
S009 · premise_and_result_audit — Recompute finite prefix index3 probability with exact rational arithmetic, not rounded decimals.
S008 · premise_and_result_audit — Recompute finite prefix index3 denominator with exact rational arithmetic, not rounded decimals.
S007 · premise_and_result_audit — Recompute finite prefix index2 probability with exact rational arithmetic, not rounded decimals.
S006 · premise_and_result_audit — Recompute finite prefix index2 denominator with exact rational arithmetic, not rounded decimals.
S005 · premise_and_result_audit — Recompute finite prefix index1 probability with exact rational arithmetic, not rounded decimals.
S004 · premise_and_result_audit — Recompute finite prefix index1 denominator with exact rational arithmetic, not rounded decimals.
S003 · premise_and_result_audit — Recompute finite prefix index0 probability with exact rational arithmetic, not rounded decimals.
S002 · premise_and_result_audit — Recompute finite prefix index0 denominator with exact rational arithmetic, not rounded decimals.
S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(3, 7, 13, [3/7, 3/8, 1/3, 3/10, 3/11, 1/4, 3/13, 3/14, 1/5, 3/16, 3/17, 1/6, 3/19], No, finite_prefix_not_convergence_certificate)

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: de438882ce98d750e197b0ce0584f20e787f84542b87d7342a6e4e469041102b
candidate record SHA-256: f551d5faa778986911506258eb1a39e37e29e6c48c4bd5c60a74b1481c9dbbf6
semantic review receipt SHA-256: 8c0ef6ca8c5ba2754e41f30c2b32bcb757756152d97421f7396d39038c11f725
question-first witness SHA-256: c7920a0ff8e724b7e566ebbda30b31d8087c136974b62f8c760fbd26ab816117
Lean: NOT_RUN · human review: NOT_PERFORMED

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