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Question 15096: A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence 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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← Question derivation index · Original question batch

R0029-15096 · q-20261014-advance-20261030-r07-convergence-in-probability-0071

Original question and givens

Use only the explicitly supplied finite list p_n=1/(n+8) for n=0 through prefix_length-1, with prefix_length=21. 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=1/(n+8) for n=0 through prefix_length-1, with prefix_length=21. 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.

Step-by-step derivation

S001 · implies

Before

positive profile parameters c=1;s=8;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, 8)

After

finite prefix index0 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, S001

S003 · calculation

Before

finite prefix index0 probability := div(1, 8)

After

finite prefix index0 probability = 1/8

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, S002

S004 · calculation

Before

finite prefix index1 denominator := sum(1, 8)

After

finite prefix index1 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, S003

S005 · calculation

Before

finite prefix index1 probability := div(1, 9)

After

finite prefix index1 probability = 1/9

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, S004

S006 · calculation

Before

finite prefix index2 denominator := sum(2, 8)

After

finite prefix index2 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, S005

S007 · calculation

Before

finite prefix index2 probability := div(1, 10)

After

finite prefix index2 probability = 1/10

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, S006

S008 · calculation

Before

finite prefix index3 denominator := sum(3, 8)

After

finite prefix index3 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, S007

S009 · calculation

Before

finite prefix index3 probability := div(1, 11)

After

finite prefix index3 probability = 1/11

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, S008

S010 · calculation

Before

finite prefix index4 denominator := sum(4, 8)

After

finite prefix index4 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, S009

S011 · calculation

Before

finite prefix index4 probability := div(1, 12)

After

finite prefix index4 probability = 1/12

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, S010

S012 · calculation

Before

finite prefix index5 denominator := sum(5, 8)

After

finite prefix index5 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, S011

S013 · calculation

Before

finite prefix index5 probability := div(1, 13)

After

finite prefix index5 probability = 1/13

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, S012

S014 · calculation

Before

finite prefix index6 denominator := sum(6, 8)

After

finite prefix index6 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, S013

S015 · calculation

Before

finite prefix index6 probability := div(1, 14)

After

finite prefix index6 probability = 1/14

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, S014

S016 · calculation

Before

finite prefix index7 denominator := sum(7, 8)

After

finite prefix index7 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, S015

S017 · calculation

Before

finite prefix index7 probability := div(1, 15)

After

finite prefix index7 probability = 1/15

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, S016

S018 · calculation

Before

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

After

finite prefix index8 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, S017

S019 · calculation

Before

finite prefix index8 probability := div(1, 16)

After

finite prefix index8 probability = 1/16

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, S018

S020 · calculation

Before

finite prefix index9 denominator := sum(9, 8)

After

finite prefix index9 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, S019

S021 · calculation

Before

finite prefix index9 probability := div(1, 17)

After

finite prefix index9 probability = 1/17

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, S020

S022 · calculation

Before

finite prefix index10 denominator := sum(10, 8)

After

finite prefix index10 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, S021

S023 · calculation

Before

finite prefix index10 probability := div(1, 18)

After

finite prefix index10 probability = 1/18

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, S022

S024 · calculation

Before

finite prefix index11 denominator := sum(11, 8)

After

finite prefix index11 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, S023

S025 · calculation

Before

finite prefix index11 probability := div(1, 19)

After

finite prefix index11 probability = 1/19

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, S024

S026 · calculation

Before

finite prefix index12 denominator := sum(12, 8)

After

finite prefix index12 denominator = 20

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(1, 20)

After

finite prefix index12 probability = 1/20

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 · calculation

Before

finite prefix index13 denominator := sum(13, 8)

After

finite prefix index13 denominator = 21

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, S027

S029 · calculation

Before

finite prefix index13 probability := div(1, 21)

After

finite prefix index13 probability = 1/21

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, S028

S030 · calculation

Before

finite prefix index14 denominator := sum(14, 8)

After

finite prefix index14 denominator = 22

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, S029

S031 · calculation

Before

finite prefix index14 probability := div(1, 22)

After

finite prefix index14 probability = 1/22

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, S030

S032 · calculation

Before

finite prefix index15 denominator := sum(15, 8)

After

finite prefix index15 denominator = 23

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, S031

S033 · calculation

Before

finite prefix index15 probability := div(1, 23)

After

finite prefix index15 probability = 1/23

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, S032

S034 · calculation

Before

finite prefix index16 denominator := sum(16, 8)

After

finite prefix index16 denominator = 24

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, S033

S035 · calculation

Before

finite prefix index16 probability := div(1, 24)

After

finite prefix index16 probability = 1/24

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, S034

S036 · calculation

Before

finite prefix index17 denominator := sum(17, 8)

After

finite prefix index17 denominator = 25

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, S035

S037 · calculation

Before

finite prefix index17 probability := div(1, 25)

After

finite prefix index17 probability = 1/25

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, S036

S038 · calculation

Before

finite prefix index18 denominator := sum(18, 8)

After

finite prefix index18 denominator = 26

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, S037

S039 · calculation

Before

finite prefix index18 probability := div(1, 26)

After

finite prefix index18 probability = 1/26

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, S038

S040 · calculation

Before

finite prefix index19 denominator := sum(19, 8)

After

finite prefix index19 denominator = 27

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, S039

S041 · calculation

Before

finite prefix index19 probability := div(1, 27)

After

finite prefix index19 probability = 1/27

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, S040

S042 · calculation

Before

finite prefix index20 denominator := sum(20, 8)

After

finite prefix index20 denominator = 28

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, S041

S043 · calculation

Before

finite prefix index20 probability := div(1, 28)

After

finite prefix index20 probability = 1/28

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, S042

S044 · implies

Before

only indices0 through 20 supplied;finite probability entries=[1/8, 1/9, 1/10, 1/11, 1/12, 1/13, 1/14, 1/15, 1/16, 1/17, 1/18, 1/19, 1/20, 1/21, 1/22, 1/23, 1/24, 1/25, 1/26, 1/27, 1/28]

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, S043

Reverse checks

  • S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S043 · premise_and_result_audit — Recompute finite prefix index20 probability with exact rational arithmetic, not rounded decimals.
  • S042 · premise_and_result_audit — Recompute finite prefix index20 denominator with exact rational arithmetic, not rounded decimals.
  • S041 · premise_and_result_audit — Recompute finite prefix index19 probability with exact rational arithmetic, not rounded decimals.
  • S040 · premise_and_result_audit — Recompute finite prefix index19 denominator with exact rational arithmetic, not rounded decimals.
  • S039 · premise_and_result_audit — Recompute finite prefix index18 probability with exact rational arithmetic, not rounded decimals.
  • S038 · premise_and_result_audit — Recompute finite prefix index18 denominator with exact rational arithmetic, not rounded decimals.
  • S037 · premise_and_result_audit — Recompute finite prefix index17 probability with exact rational arithmetic, not rounded decimals.
  • S036 · premise_and_result_audit — Recompute finite prefix index17 denominator with exact rational arithmetic, not rounded decimals.
  • S035 · premise_and_result_audit — Recompute finite prefix index16 probability with exact rational arithmetic, not rounded decimals.
  • S034 · premise_and_result_audit — Recompute finite prefix index16 denominator with exact rational arithmetic, not rounded decimals.
  • S033 · premise_and_result_audit — Recompute finite prefix index15 probability with exact rational arithmetic, not rounded decimals.
  • S032 · premise_and_result_audit — Recompute finite prefix index15 denominator with exact rational arithmetic, not rounded decimals.
  • S031 · premise_and_result_audit — Recompute finite prefix index14 probability with exact rational arithmetic, not rounded decimals.
  • S030 · premise_and_result_audit — Recompute finite prefix index14 denominator with exact rational arithmetic, not rounded decimals.
  • S029 · premise_and_result_audit — Recompute finite prefix index13 probability with exact rational arithmetic, not rounded decimals.
  • S028 · premise_and_result_audit — Recompute finite prefix index13 denominator with exact rational arithmetic, not rounded decimals.
  • 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

(1, 8, 21, [1/8, 1/9, 1/10, 1/11, 1/12, 1/13, 1/14, 1/15, 1/16, 1/17, 1/18, 1/19, 1/20, 1/21, 1/22, 1/23, 1/24, 1/25, 1/26, 1/27, 1/28], 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: 8817de706d3ca95904acb8f2ab1c09e4f4e92b70ce961fa79367f03637cb31a9
  • candidate record SHA-256: 167c8854734634b6f0bedd049d9603e4e18d04bdf8f47fad5c65888a19663ede
  • semantic review receipt SHA-256: 8c0ef6ca8c5ba2754e41f30c2b32bcb757756152d97421f7396d39038c11f725
  • question-first witness SHA-256: c7920a0ff8e724b7e566ebbda30b31d8087c136974b62f8c760fbd26ab816117
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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