# Question 15050: A Strict Rational Tail Cutoff Is Different from a Finite Prefix

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0029-15050 · q-20261014-advance-20261030-r07-convergence-in-probability-0050



## Original question and givens

Use the declared rational deviation-probability tail p_n=5/(n+5) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/34. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).


- G001 — Use the declared rational deviation-probability tail p_n=5/(n+5) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/34. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).





## 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](https://ocme.evemisslab.com/en/mko/mko-convergence-in-probability/)





## Step-by-step derivation

S001 · impliesBefore

positive profile parameters c=5;s=5;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 dependenciesRule: deviation_profile_probability_domain_not_clipping · Uses: G001





S002 · calculationBefore

strict profile cross-product numerator := mul(5, 34)

After

strict profile cross-product numerator = 170

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001





S003 · calculationBefore

strict profile reciprocal threshold := div(170, 1)

After

strict profile reciprocal threshold = 170

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · calculationBefore

strict profile unclamped cutoff := sum(170, -5, 1)

After

strict profile unclamped cutoff = 166

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003





S005 · impliesBefore

raw target delta=[1, 34];normalized delta=1/34;positive denominator target numerator=1

After

floor(c*delta_denominator/delta_numerator)=170;threshold=170;raw least nonnegative strict N=166

Conditions: none

The strict inequality c/(N+s)<a/b is equivalent to N+s>cb/a,hence floor(cb/a)−s+1 clamped only at0.Equality still requires the next index.

Step identifiers and dependenciesRule: strict_integer_cutoff_floor_and_nonnegative_boundary · Uses: G001, S004





S006 · calculationBefore

strict witness denominator := sum(166, 5)

After

strict witness denominator = 171

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005





S007 · calculationBefore

strict witness valid probability bound := div(5, 171)

After

strict witness valid probability bound = 5/171

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006





S008 · calculationBefore

previous raw reciprocal denominator := sum(165, 5)

After

previous raw reciprocal denominator = 170

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007





S009 · calculationBefore

previous raw reciprocal value := div(5, 170)

After

previous raw reciprocal value = 1/34

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008





S010 · impliesBefore

N=166;current valid probability=5/171;previous RAW reciprocal=1/34;delta=1/34

After

current strict bound holds,previous raw threshold fails;N is minimal

Conditions: none

The previous arithmetic reciprocal is still≥delta,so every smaller index fails the original strict threshold.If that earlier index is outside the probability domain,it is only a raw reciprocal,not a probability above1.The returned N must pass the original English domain guard;do not max-shift the source answer.

Step identifiers and dependenciesRule: strict_profile_minimality_without_invalid_probability_claim · Uses: G001, S009





S011 · impliesBefore

declared same-space deviation profile on valid tail;n>=N=166;positive denominators increase

After

every later profile<=bound=5/171<delta=1/34

Conditions: none

Conditionally on the declared deviation profile,increasing denominators bound every later index.This selected-delta witness neither constructs a common process from numbers nor proves all-epsilon convergence of an unknown sequence from finite data.

Step identifiers and dependenciesRule: probability_tail_all_later_indices_not_samples · Uses: G001, S010









## Reverse checks


- S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S009 · premise_and_result_audit — Recompute previous raw reciprocal value with exact rational arithmetic, not rounded decimals.
- S008 · premise_and_result_audit — Recompute previous raw reciprocal denominator with exact rational arithmetic, not rounded decimals.
- S007 · premise_and_result_audit — Recompute strict witness valid probability bound with exact rational arithmetic, not rounded decimals.
- S006 · premise_and_result_audit — Recompute strict witness denominator with exact rational arithmetic, not rounded decimals.
- S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S004 · premise_and_result_audit — Recompute strict profile unclamped cutoff with exact rational arithmetic, not rounded decimals.
- S003 · premise_and_result_audit — Recompute strict profile reciprocal threshold with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Recompute strict profile cross-product numerator 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

(5, 5, 1/34, 166, 5/171, Yes, forall_n_ge_N_tail_bound)

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: 37a2eb671b9c16a4ad16447ef3bd3fb161960a2b5d2a4a2b10e2f07aa70d0511
- candidate record SHA-256: ae9d6efbfb39d46903ebdaa53487d92861a028639e924ae7156265178b059315
- semantic review receipt SHA-256: 8c0ef6ca8c5ba2754e41f30c2b32bcb757756152d97421f7396d39038c11f725
- question-first witness SHA-256: c7920a0ff8e724b7e566ebbda30b31d8087c136974b62f8c760fbd26ab816117
- Lean: NOT_RUN · human review: NOT_PERFORMED

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