Finite-Valued IID Strong-Law Scope and Fixed-n Audits
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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations · 2026-10-15 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.
Field-order and source-binding data
1. The Almost-Sure Limit Uses an Explicit Infinite IID Premise
Use the aligned finite law values=[-2, -1], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified).
A finite law may have zero variance. The almost-sure conclusion is conditional on the declared infinite IID premise and named theorem, not inferred from finite observations.
View answer and explanation
([[-2, 1/2], [-1, 1/2]], -3/2, 1/4, infinite_iid, P(lim_sample_mean=mu)=1, Yes)
Compute the common mean and variance of the finite law. Zero variance is allowed for this theorem. State P(lim sample_mean=mu)=1 as the conditional named-theorem conclusion, not an inference from finite observations or a proof of the infinite process. The exact result, retaining the requested order, is ([[-2, 1/2], [-1, 1/2]], -3/2, 1/4, infinite_iid, P(lim_sample_mean=mu)=1, Yes).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-process-law-qualification · candidate
2. The Almost-Sure Limit Uses an Explicit Infinite IID Premise
Use the aligned finite law values=[-1, 0], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified).
A finite law may have zero variance. The almost-sure conclusion is conditional on the declared infinite IID premise and named theorem, not inferred from finite observations.
View answer and explanation
([[-1, 1/2], [0, 1/2]], -1/2, 1/4, infinite_iid, P(lim_sample_mean=mu)=1, Yes)
Compute the common mean and variance of the finite law. Zero variance is allowed for this theorem. State P(lim sample_mean=mu)=1 as the conditional named-theorem conclusion, not an inference from finite observations or a proof of the infinite process. The exact result, retaining the requested order, is ([[-1, 1/2], [0, 1/2]], -1/2, 1/4, infinite_iid, P(lim_sample_mean=mu)=1, Yes).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-process-law-qualification · candidate
3. The Almost-Sure Limit Uses an Explicit Infinite IID Premise
Use the aligned finite law values=[0, 1], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified).
A finite law may have zero variance. The almost-sure conclusion is conditional on the declared infinite IID premise and named theorem, not inferred from finite observations.
View answer and explanation
([[0, 1/2], [1, 1/2]], 1/2, 1/4, infinite_iid, P(lim_sample_mean=mu)=1, Yes)
Compute the common mean and variance of the finite law. Zero variance is allowed for this theorem. State P(lim sample_mean=mu)=1 as the conditional named-theorem conclusion, not an inference from finite observations or a proof of the infinite process. The exact result, retaining the requested order, is ([[0, 1/2], [1, 1/2]], 1/2, 1/4, infinite_iid, P(lim_sample_mean=mu)=1, Yes).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-process-law-qualification · candidate
4. The Almost-Sure Limit Uses an Explicit Infinite IID Premise
Use the aligned finite law values=[1, 2], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified).
A finite law may have zero variance. The almost-sure conclusion is conditional on the declared infinite IID premise and named theorem, not inferred from finite observations.
View answer and explanation
([[1, 1/2], [2, 1/2]], 3/2, 1/4, infinite_iid, P(lim_sample_mean=mu)=1, Yes)
Compute the common mean and variance of the finite law. Zero variance is allowed for this theorem. State P(lim sample_mean=mu)=1 as the conditional named-theorem conclusion, not an inference from finite observations or a proof of the infinite process. The exact result, retaining the requested order, is ([[1, 1/2], [2, 1/2]], 3/2, 1/4, infinite_iid, P(lim_sample_mean=mu)=1, Yes).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-process-law-qualification · candidate
5. The Almost-Sure Limit Uses an Explicit Infinite IID Premise
Use the aligned finite law values=[2, 3], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified).
A finite law may have zero variance. The almost-sure conclusion is conditional on the declared infinite IID premise and named theorem, not inferred from finite observations.
View answer and explanation
([[2, 1/2], [3, 1/2]], 5/2, 1/4, infinite_iid, P(lim_sample_mean=mu)=1, Yes)
Compute the common mean and variance of the finite law. Zero variance is allowed for this theorem. State P(lim sample_mean=mu)=1 as the conditional named-theorem conclusion, not an inference from finite observations or a proof of the infinite process. The exact result, retaining the requested order, is ([[2, 1/2], [3, 1/2]], 5/2, 1/4, infinite_iid, P(lim_sample_mean=mu)=1, Yes).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-process-law-qualification · candidate
6. The Almost-Sure Limit Uses an Explicit Infinite IID Premise
Use the aligned finite law values=[-2, -1], probabilities=[1/3, 2/3], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified).
A finite law may have zero variance. The almost-sure conclusion is conditional on the declared infinite IID premise and named theorem, not inferred from finite observations.
View answer and explanation
([[-2, 1/3], [-1, 2/3]], -4/3, 2/9, infinite_iid, P(lim_sample_mean=mu)=1, Yes)
Compute the common mean and variance of the finite law. Zero variance is allowed for this theorem. State P(lim sample_mean=mu)=1 as the conditional named-theorem conclusion, not an inference from finite observations or a proof of the infinite process. The exact result, retaining the requested order, is ([[-2, 1/3], [-1, 2/3]], -4/3, 2/9, infinite_iid, P(lim_sample_mean=mu)=1, Yes).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-process-law-qualification · candidate
7. The Almost-Sure Limit Uses an Explicit Infinite IID Premise
Use the aligned finite law values=[-1, 0], probabilities=[1/3, 2/3], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified).
A finite law may have zero variance. The almost-sure conclusion is conditional on the declared infinite IID premise and named theorem, not inferred from finite observations.
View answer and explanation
([[-1, 1/3], [0, 2/3]], -1/3, 2/9, infinite_iid, P(lim_sample_mean=mu)=1, Yes)
Compute the common mean and variance of the finite law. Zero variance is allowed for this theorem. State P(lim sample_mean=mu)=1 as the conditional named-theorem conclusion, not an inference from finite observations or a proof of the infinite process. The exact result, retaining the requested order, is ([[-1, 1/3], [0, 2/3]], -1/3, 2/9, infinite_iid, P(lim_sample_mean=mu)=1, Yes).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-process-law-qualification · candidate
8. The Almost-Sure Limit Uses an Explicit Infinite IID Premise
Use the aligned finite law values=[0, 1], probabilities=[1/3, 2/3], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified).
A finite law may have zero variance. The almost-sure conclusion is conditional on the declared infinite IID premise and named theorem, not inferred from finite observations.
View answer and explanation
([[0, 1/3], [1, 2/3]], 2/3, 2/9, infinite_iid, P(lim_sample_mean=mu)=1, Yes)
Compute the common mean and variance of the finite law. Zero variance is allowed for this theorem. State P(lim sample_mean=mu)=1 as the conditional named-theorem conclusion, not an inference from finite observations or a proof of the infinite process. The exact result, retaining the requested order, is ([[0, 1/3], [1, 2/3]], 2/3, 2/9, infinite_iid, P(lim_sample_mean=mu)=1, Yes).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-process-law-qualification · candidate
9. The Almost-Sure Limit Uses an Explicit Infinite IID Premise
Use the aligned finite law values=[1, 2], probabilities=[1/3, 2/3], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified).
A finite law may have zero variance. The almost-sure conclusion is conditional on the declared infinite IID premise and named theorem, not inferred from finite observations.
View answer and explanation
([[1, 1/3], [2, 2/3]], 5/3, 2/9, infinite_iid, P(lim_sample_mean=mu)=1, Yes)
Compute the common mean and variance of the finite law. Zero variance is allowed for this theorem. State P(lim sample_mean=mu)=1 as the conditional named-theorem conclusion, not an inference from finite observations or a proof of the infinite process. The exact result, retaining the requested order, is ([[1, 1/3], [2, 2/3]], 5/3, 2/9, infinite_iid, P(lim_sample_mean=mu)=1, Yes).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-process-law-qualification · candidate
10. The Almost-Sure Limit Uses an Explicit Infinite IID Premise
Use the aligned finite law values=[2, 3], probabilities=[1/3, 2/3], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified).
A finite law may have zero variance. The almost-sure conclusion is conditional on the declared infinite IID premise and named theorem, not inferred from finite observations.
View answer and explanation
([[2, 1/3], [3, 2/3]], 8/3, 2/9, infinite_iid, P(lim_sample_mean=mu)=1, Yes)
Compute the common mean and variance of the finite law. Zero variance is allowed for this theorem. State P(lim sample_mean=mu)=1 as the conditional named-theorem conclusion, not an inference from finite observations or a proof of the infinite process. The exact result, retaining the requested order, is ([[2, 1/3], [3, 2/3]], 8/3, 2/9, infinite_iid, P(lim_sample_mean=mu)=1, Yes).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-process-law-qualification · candidate
11. The Almost-Sure Limit Uses an Explicit Infinite IID Premise
Use the aligned finite law values=[-2, -1], probabilities=[1/4, 3/4], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified).
A finite law may have zero variance. The almost-sure conclusion is conditional on the declared infinite IID premise and named theorem, not inferred from finite observations.
View answer and explanation
([[-2, 1/4], [-1, 3/4]], -5/4, 3/16, infinite_iid, P(lim_sample_mean=mu)=1, Yes)
Compute the common mean and variance of the finite law. Zero variance is allowed for this theorem. State P(lim sample_mean=mu)=1 as the conditional named-theorem conclusion, not an inference from finite observations or a proof of the infinite process. The exact result, retaining the requested order, is ([[-2, 1/4], [-1, 3/4]], -5/4, 3/16, infinite_iid, P(lim_sample_mean=mu)=1, Yes).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-process-law-qualification · candidate
12. The Almost-Sure Limit Uses an Explicit Infinite IID Premise
Use the aligned finite law values=[-1, 0], probabilities=[1/4, 3/4], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified).
A finite law may have zero variance. The almost-sure conclusion is conditional on the declared infinite IID premise and named theorem, not inferred from finite observations.
View answer and explanation
([[-1, 1/4], [0, 3/4]], -1/4, 3/16, infinite_iid, P(lim_sample_mean=mu)=1, Yes)
Compute the common mean and variance of the finite law. Zero variance is allowed for this theorem. State P(lim sample_mean=mu)=1 as the conditional named-theorem conclusion, not an inference from finite observations or a proof of the infinite process. The exact result, retaining the requested order, is ([[-1, 1/4], [0, 3/4]], -1/4, 3/16, infinite_iid, P(lim_sample_mean=mu)=1, Yes).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-process-law-qualification · candidate
13. The Almost-Sure Limit Uses an Explicit Infinite IID Premise
Use the aligned finite law values=[0, 1], probabilities=[1/4, 3/4], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified).
A finite law may have zero variance. The almost-sure conclusion is conditional on the declared infinite IID premise and named theorem, not inferred from finite observations.
View answer and explanation
([[0, 1/4], [1, 3/4]], 3/4, 3/16, infinite_iid, P(lim_sample_mean=mu)=1, Yes)
Compute the common mean and variance of the finite law. Zero variance is allowed for this theorem. State P(lim sample_mean=mu)=1 as the conditional named-theorem conclusion, not an inference from finite observations or a proof of the infinite process. The exact result, retaining the requested order, is ([[0, 1/4], [1, 3/4]], 3/4, 3/16, infinite_iid, P(lim_sample_mean=mu)=1, Yes).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-process-law-qualification · candidate
14. The Almost-Sure Limit Uses an Explicit Infinite IID Premise
Use the aligned finite law values=[1, 2], probabilities=[1/4, 3/4], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified).
A finite law may have zero variance. The almost-sure conclusion is conditional on the declared infinite IID premise and named theorem, not inferred from finite observations.
View answer and explanation
([[1, 1/4], [2, 3/4]], 7/4, 3/16, infinite_iid, P(lim_sample_mean=mu)=1, Yes)
Compute the common mean and variance of the finite law. Zero variance is allowed for this theorem. State P(lim sample_mean=mu)=1 as the conditional named-theorem conclusion, not an inference from finite observations or a proof of the infinite process. The exact result, retaining the requested order, is ([[1, 1/4], [2, 3/4]], 7/4, 3/16, infinite_iid, P(lim_sample_mean=mu)=1, Yes).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-process-law-qualification · candidate
15. The Almost-Sure Limit Uses an Explicit Infinite IID Premise
Use the aligned finite law values=[2, 3], probabilities=[1/4, 3/4], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified).
A finite law may have zero variance. The almost-sure conclusion is conditional on the declared infinite IID premise and named theorem, not inferred from finite observations.
View answer and explanation
([[2, 1/4], [3, 3/4]], 11/4, 3/16, infinite_iid, P(lim_sample_mean=mu)=1, Yes)
Compute the common mean and variance of the finite law. Zero variance is allowed for this theorem. State P(lim sample_mean=mu)=1 as the conditional named-theorem conclusion, not an inference from finite observations or a proof of the infinite process. The exact result, retaining the requested order, is ([[2, 1/4], [3, 3/4]], 11/4, 3/16, infinite_iid, P(lim_sample_mean=mu)=1, Yes).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-process-law-qualification · candidate
16. The Almost-Sure Limit Uses an Explicit Infinite IID Premise
Use the aligned finite law values=[-2, -1], 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. Give (law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified).
A finite law may have zero variance. The almost-sure conclusion is conditional on the declared infinite IID premise and named theorem, not inferred from finite observations.
View answer and explanation
([[-2, 2/5], [-1, 3/5]], -7/5, 6/25, infinite_iid, P(lim_sample_mean=mu)=1, Yes)
Compute the common mean and variance of the finite law. Zero variance is allowed for this theorem. State P(lim sample_mean=mu)=1 as the conditional named-theorem conclusion, not an inference from finite observations or a proof of the infinite process. The exact result, retaining the requested order, is ([[-2, 2/5], [-1, 3/5]], -7/5, 6/25, infinite_iid, P(lim_sample_mean=mu)=1, Yes).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-process-law-qualification · candidate
17. The Almost-Sure Limit Uses an Explicit Infinite IID Premise
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. Give (law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified).
A finite law may have zero variance. The almost-sure conclusion is conditional on the declared infinite IID premise and named theorem, not inferred from finite observations.
View answer and explanation
([[-1, 2/5], [0, 3/5]], -2/5, 6/25, infinite_iid, P(lim_sample_mean=mu)=1, Yes)
Compute the common mean and variance of the finite law. Zero variance is allowed for this theorem. State P(lim sample_mean=mu)=1 as the conditional named-theorem conclusion, not an inference from finite observations or a proof of the infinite process. The exact result, retaining the requested order, is ([[-1, 2/5], [0, 3/5]], -2/5, 6/25, infinite_iid, P(lim_sample_mean=mu)=1, Yes).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-process-law-qualification · candidate
18. The Almost-Sure Limit Uses an Explicit Infinite IID Premise
Use the aligned finite law values=[0, 1], 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. Give (law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified).
A finite law may have zero variance. The almost-sure conclusion is conditional on the declared infinite IID premise and named theorem, not inferred from finite observations.
View answer and explanation
([[0, 2/5], [1, 3/5]], 3/5, 6/25, infinite_iid, P(lim_sample_mean=mu)=1, Yes)
Compute the common mean and variance of the finite law. Zero variance is allowed for this theorem. State P(lim sample_mean=mu)=1 as the conditional named-theorem conclusion, not an inference from finite observations or a proof of the infinite process. The exact result, retaining the requested order, is ([[0, 2/5], [1, 3/5]], 3/5, 6/25, infinite_iid, P(lim_sample_mean=mu)=1, Yes).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-process-law-qualification · candidate
19. The Almost-Sure Limit Uses an Explicit Infinite IID Premise
Use the aligned finite law values=[1, 2], 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. Give (law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified).
A finite law may have zero variance. The almost-sure conclusion is conditional on the declared infinite IID premise and named theorem, not inferred from finite observations.
View answer and explanation
([[1, 2/5], [2, 3/5]], 8/5, 6/25, infinite_iid, P(lim_sample_mean=mu)=1, Yes)
Compute the common mean and variance of the finite law. Zero variance is allowed for this theorem. State P(lim sample_mean=mu)=1 as the conditional named-theorem conclusion, not an inference from finite observations or a proof of the infinite process. The exact result, retaining the requested order, is ([[1, 2/5], [2, 3/5]], 8/5, 6/25, infinite_iid, P(lim_sample_mean=mu)=1, Yes).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-process-law-qualification · candidate
20. The Almost-Sure Limit Uses an Explicit Infinite IID Premise
Use the aligned finite law values=[2, 3], 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. Give (law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified).
A finite law may have zero variance. The almost-sure conclusion is conditional on the declared infinite IID premise and named theorem, not inferred from finite observations.
View answer and explanation
([[2, 2/5], [3, 3/5]], 13/5, 6/25, infinite_iid, P(lim_sample_mean=mu)=1, Yes)
Compute the common mean and variance of the finite law. Zero variance is allowed for this theorem. State P(lim sample_mean=mu)=1 as the conditional named-theorem conclusion, not an inference from finite observations or a proof of the infinite process. The exact result, retaining the requested order, is ([[2, 2/5], [3, 3/5]], 13/5, 6/25, infinite_iid, P(lim_sample_mean=mu)=1, Yes).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-process-law-qualification · candidate
21. The Almost-Sure Limit Uses an Explicit Infinite IID Premise
Use the aligned finite law values=[-2, -1], probabilities=[1/5, 4/5], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified).
A finite law may have zero variance. The almost-sure conclusion is conditional on the declared infinite IID premise and named theorem, not inferred from finite observations.
View answer and explanation
([[-2, 1/5], [-1, 4/5]], -6/5, 4/25, infinite_iid, P(lim_sample_mean=mu)=1, Yes)
Compute the common mean and variance of the finite law. Zero variance is allowed for this theorem. State P(lim sample_mean=mu)=1 as the conditional named-theorem conclusion, not an inference from finite observations or a proof of the infinite process. The exact result, retaining the requested order, is ([[-2, 1/5], [-1, 4/5]], -6/5, 4/25, infinite_iid, P(lim_sample_mean=mu)=1, Yes).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-process-law-qualification · candidate
22. The Almost-Sure Limit Uses an Explicit Infinite IID Premise
Use the aligned finite law values=[-1, 0], probabilities=[1/5, 4/5], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified).
A finite law may have zero variance. The almost-sure conclusion is conditional on the declared infinite IID premise and named theorem, not inferred from finite observations.
View answer and explanation
([[-1, 1/5], [0, 4/5]], -1/5, 4/25, infinite_iid, P(lim_sample_mean=mu)=1, Yes)
Compute the common mean and variance of the finite law. Zero variance is allowed for this theorem. State P(lim sample_mean=mu)=1 as the conditional named-theorem conclusion, not an inference from finite observations or a proof of the infinite process. The exact result, retaining the requested order, is ([[-1, 1/5], [0, 4/5]], -1/5, 4/25, infinite_iid, P(lim_sample_mean=mu)=1, Yes).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-process-law-qualification · candidate
23. The Almost-Sure Limit Uses an Explicit Infinite IID Premise
Use the aligned finite law values=[0, 1], probabilities=[1/5, 4/5], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified).
A finite law may have zero variance. The almost-sure conclusion is conditional on the declared infinite IID premise and named theorem, not inferred from finite observations.
View answer and explanation
([[0, 1/5], [1, 4/5]], 4/5, 4/25, infinite_iid, P(lim_sample_mean=mu)=1, Yes)
Compute the common mean and variance of the finite law. Zero variance is allowed for this theorem. State P(lim sample_mean=mu)=1 as the conditional named-theorem conclusion, not an inference from finite observations or a proof of the infinite process. The exact result, retaining the requested order, is ([[0, 1/5], [1, 4/5]], 4/5, 4/25, infinite_iid, P(lim_sample_mean=mu)=1, Yes).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-process-law-qualification · candidate
24. The Almost-Sure Limit Uses an Explicit Infinite IID Premise
Use the aligned finite law values=[1, 2], probabilities=[1/5, 4/5], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified).
A finite law may have zero variance. The almost-sure conclusion is conditional on the declared infinite IID premise and named theorem, not inferred from finite observations.
View answer and explanation
([[1, 1/5], [2, 4/5]], 9/5, 4/25, infinite_iid, P(lim_sample_mean=mu)=1, Yes)
Compute the common mean and variance of the finite law. Zero variance is allowed for this theorem. State P(lim sample_mean=mu)=1 as the conditional named-theorem conclusion, not an inference from finite observations or a proof of the infinite process. The exact result, retaining the requested order, is ([[1, 1/5], [2, 4/5]], 9/5, 4/25, infinite_iid, P(lim_sample_mean=mu)=1, Yes).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-process-law-qualification · candidate
25. The Almost-Sure Limit Uses an Explicit Infinite IID Premise
Use the aligned finite law values=[2, 3], probabilities=[1/5, 4/5], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified).
A finite law may have zero variance. The almost-sure conclusion is conditional on the declared infinite IID premise and named theorem, not inferred from finite observations.
View answer and explanation
([[2, 1/5], [3, 4/5]], 14/5, 4/25, infinite_iid, P(lim_sample_mean=mu)=1, Yes)
Compute the common mean and variance of the finite law. Zero variance is allowed for this theorem. State P(lim sample_mean=mu)=1 as the conditional named-theorem conclusion, not an inference from finite observations or a proof of the infinite process. The exact result, retaining the requested order, is ([[2, 1/5], [3, 4/5]], 14/5, 4/25, infinite_iid, P(lim_sample_mean=mu)=1, Yes).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-process-law-qualification · candidate
26. A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate
Use the aligned finite law values=[-2, -1], probabilities=[1/2, 1/2], 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=[-2]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).
View answer and explanation
([-2], -3/2, -2, 1/2, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)
Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([-2], -3/2, -2, 1/2, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-prefix-sample-mean · candidate
27. A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate
Use the aligned finite law values=[-1, 0], probabilities=[1/2, 1/2], 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=[0, -1]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).
View answer and explanation
([0, -1], -1/2, -1/2, 0, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)
Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([0, -1], -1/2, -1/2, 0, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-prefix-sample-mean · candidate
28. A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate
Use the aligned finite law values=[0, 1], probabilities=[1/2, 1/2], 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=[0, 1, 0]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).
View answer and explanation
([0, 1, 0], 1/2, 1/3, 1/6, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)
Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([0, 1, 0], 1/2, 1/3, 1/6, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-prefix-sample-mean · candidate
29. A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate
Use the aligned finite law values=[1, 2], probabilities=[1/2, 1/2], 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=[2, 1, 2, 1]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).
View answer and explanation
([2, 1, 2, 1], 3/2, 3/2, 0, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)
Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([2, 1, 2, 1], 3/2, 3/2, 0, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-prefix-sample-mean · candidate
30. A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate
Use the aligned finite law values=[2, 3], probabilities=[1/2, 1/2], 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=[2, 3, 2, 3, 2]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).
View answer and explanation
([2, 3, 2, 3, 2], 5/2, 12/5, 1/10, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)
Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([2, 3, 2, 3, 2], 5/2, 12/5, 1/10, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-prefix-sample-mean · candidate
31. A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate
Use the aligned finite law values=[-2, -1], probabilities=[1/3, 2/3], 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, -2, -1, -2, -1, -2]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).
View answer and explanation
([-1, -2, -1, -2, -1, -2], -4/3, -3/2, 1/6, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)
Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([-1, -2, -1, -2, -1, -2], -4/3, -3/2, 1/6, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-prefix-sample-mean · candidate
32. A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate
Use the aligned finite law values=[-1, 0], probabilities=[1/3, 2/3], 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]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).
View answer and explanation
([-1], -1/3, -1, 2/3, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)
Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([-1], -1/3, -1, 2/3, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-prefix-sample-mean · candidate
33. A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate
Use the aligned finite law values=[0, 1], probabilities=[1/3, 2/3], 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]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).
View answer and explanation
([1, 0], 2/3, 1/2, 1/6, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)
Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([1, 0], 2/3, 1/2, 1/6, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-prefix-sample-mean · candidate
34. A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate
Use the aligned finite law values=[1, 2], probabilities=[1/3, 2/3], 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, 2, 1]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).
View answer and explanation
([1, 2, 1], 5/3, 4/3, 1/3, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)
Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([1, 2, 1], 5/3, 4/3, 1/3, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-prefix-sample-mean · candidate
35. A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate
Use the aligned finite law values=[2, 3], probabilities=[1/3, 2/3], 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=[3, 2, 3, 2]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).
View answer and explanation
([3, 2, 3, 2], 8/3, 5/2, 1/6, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)
Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([3, 2, 3, 2], 8/3, 5/2, 1/6, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-prefix-sample-mean · candidate
36. A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate
Use the aligned finite law values=[-2, -1], probabilities=[1/4, 3/4], 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=[-2, -1, -2, -1, -2]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).
View answer and explanation
([-2, -1, -2, -1, -2], -5/4, -8/5, 7/20, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)
Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([-2, -1, -2, -1, -2], -5/4, -8/5, 7/20, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-prefix-sample-mean · candidate
37. A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate
Use the aligned finite law values=[-1, 0], probabilities=[1/4, 3/4], 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=[0, -1, 0, -1, 0, -1]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).
View answer and explanation
([0, -1, 0, -1, 0, -1], -1/4, -1/2, 1/4, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)
Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([0, -1, 0, -1, 0, -1], -1/4, -1/2, 1/4, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-prefix-sample-mean · candidate
38. A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate
Use the aligned finite law values=[0, 1], probabilities=[1/4, 3/4], 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=[0]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).
View answer and explanation
([0], 3/4, 0, 3/4, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)
Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([0], 3/4, 0, 3/4, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-prefix-sample-mean · candidate
39. A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate
Use the aligned finite law values=[1, 2], probabilities=[1/4, 3/4], 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=[2, 1]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).
View answer and explanation
([2, 1], 7/4, 3/2, 1/4, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)
Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([2, 1], 7/4, 3/2, 1/4, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-prefix-sample-mean · candidate
40. A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate
Use the aligned finite law values=[2, 3], probabilities=[1/4, 3/4], 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=[2, 3, 2]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).
View answer and explanation
([2, 3, 2], 11/4, 7/3, 5/12, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)
Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([2, 3, 2], 11/4, 7/3, 5/12, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-prefix-sample-mean · candidate
41. A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate
Use the aligned finite law values=[-2, -1], 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, -2, -1, -2]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).
View answer and explanation
([-1, -2, -1, -2], -7/5, -3/2, 1/10, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)
Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([-1, -2, -1, -2], -7/5, -3/2, 1/10, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-prefix-sample-mean · candidate
42. A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate
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).
View answer and explanation
([-1, 0, -1, 0, -1], -2/5, -3/5, 1/5, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)
Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([-1, 0, -1, 0, -1], -2/5, -3/5, 1/5, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-prefix-sample-mean · candidate
43. A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate
Use the aligned finite law values=[0, 1], 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, 0]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).
View answer and explanation
([1, 0, 1, 0, 1, 0], 3/5, 1/2, 1/10, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)
Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([1, 0, 1, 0, 1, 0], 3/5, 1/2, 1/10, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-prefix-sample-mean · candidate
44. A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate
Use the aligned finite law values=[1, 2], 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]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).
View answer and explanation
([1], 8/5, 1, 3/5, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)
Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([1], 8/5, 1, 3/5, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-prefix-sample-mean · candidate
45. A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate
Use the aligned finite law values=[2, 3], 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=[3, 2]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).
View answer and explanation
([3, 2], 13/5, 5/2, 1/10, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)
Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([3, 2], 13/5, 5/2, 1/10, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-prefix-sample-mean · candidate
46. A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate
Use the aligned finite law values=[-2, -1], probabilities=[1/5, 4/5], 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=[-2, -1, -2]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).
View answer and explanation
([-2, -1, -2], -6/5, -5/3, 7/15, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)
Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([-2, -1, -2], -6/5, -5/3, 7/15, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-prefix-sample-mean · candidate
47. A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate
Use the aligned finite law values=[-1, 0], probabilities=[1/5, 4/5], 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=[0, -1, 0, -1]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).
View answer and explanation
([0, -1, 0, -1], -1/5, -1/2, 3/10, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)
Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([0, -1, 0, -1], -1/5, -1/2, 3/10, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-prefix-sample-mean · candidate
48. A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate
Use the aligned finite law values=[0, 1], probabilities=[1/5, 4/5], 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=[0, 1, 0, 1, 0]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).
View answer and explanation
([0, 1, 0, 1, 0], 4/5, 2/5, 2/5, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)
Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0], 4/5, 2/5, 2/5, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-prefix-sample-mean · candidate
49. A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate
Use the aligned finite law values=[1, 2], probabilities=[1/5, 4/5], 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=[2, 1, 2, 1, 2, 1]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).
View answer and explanation
([2, 1, 2, 1, 2, 1], 9/5, 3/2, 3/10, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)
Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([2, 1, 2, 1, 2, 1], 9/5, 3/2, 3/10, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-prefix-sample-mean · candidate
50. A Fixed Prefix Mean Is a Finite Audit, Not an Almost-Sure Limit Certificate
Use the aligned finite law values=[2, 3], probabilities=[1/5, 4/5], 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=[2]. Give (observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary).
View answer and explanation
([2], 14/5, 2, 4/5, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence)
Compute its exact sample mean and absolute error from the common mean. All observations must belong to the declared carrier. A close prefix, or any one finite prefix, does not establish the infinite IID premise or a probability-one limit. The exact result, retaining the requested order, is ([2], 14/5, 2, 4/5, fixed_prefix_only, No, finite_prefix_does_not_prove_almost_sure_convergence).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-prefix-sample-mean · candidate
51. The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit
Use the aligned finite law values=[-2, -1], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=1 and epsilon=1/3>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).
The count includes all deviating joint words, including zero-mass words; probability is weighted separately. The event uses deviation≥epsilon, not>epsilon.
View answer and explanation
(1, -3/2, 1/3, 2, 1, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)
Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (1, -3/2, 1/3, 2, 1, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-fixed-n-deviation · candidate
52. The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit
Use the aligned finite law values=[-1, 0], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=1 and epsilon=2/4>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).
The count includes all deviating joint words, including zero-mass words; probability is weighted separately. The event uses deviation≥epsilon, not>epsilon.
View answer and explanation
(1, -1/2, 1/2, 2, 1, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)
Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (1, -1/2, 1/2, 2, 1, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-fixed-n-deviation · candidate
53. The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit
Use the aligned finite law values=[0, 1], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=1 and epsilon=3/5>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).
The count includes all deviating joint words, including zero-mass words; probability is weighted separately. The event uses deviation≥epsilon, not>epsilon.
View answer and explanation
(1, 1/2, 3/5, 0, 0, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)
Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (1, 1/2, 3/5, 0, 0, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-fixed-n-deviation · candidate
54. The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit
Use the aligned finite law values=[1, 2], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=1 and epsilon=1/3>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).
The count includes all deviating joint words, including zero-mass words; probability is weighted separately. The event uses deviation≥epsilon, not>epsilon.
View answer and explanation
(1, 3/2, 1/3, 2, 1, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)
Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (1, 3/2, 1/3, 2, 1, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-fixed-n-deviation · candidate
55. The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit
Use the aligned finite law values=[2, 3], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=1 and epsilon=2/4>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).
The count includes all deviating joint words, including zero-mass words; probability is weighted separately. The event uses deviation≥epsilon, not>epsilon.
View answer and explanation
(1, 5/2, 1/2, 2, 1, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)
Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (1, 5/2, 1/2, 2, 1, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-fixed-n-deviation · candidate
56. The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit
Use the aligned finite law values=[-2, -1], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=2 and epsilon=3/5>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).
The count includes all deviating joint words, including zero-mass words; probability is weighted separately. The event uses deviation≥epsilon, not>epsilon.
View answer and explanation
(2, -3/2, 3/5, 0, 0, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)
Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (2, -3/2, 3/5, 0, 0, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-fixed-n-deviation · candidate
57. The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit
Use the aligned finite law values=[-1, 0], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=2 and epsilon=1/3>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).
The count includes all deviating joint words, including zero-mass words; probability is weighted separately. The event uses deviation≥epsilon, not>epsilon.
View answer and explanation
(2, -1/2, 1/3, 2, 1/2, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)
Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (2, -1/2, 1/3, 2, 1/2, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-fixed-n-deviation · candidate
58. The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit
Use the aligned finite law values=[0, 1], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=2 and epsilon=2/4>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).
The count includes all deviating joint words, including zero-mass words; probability is weighted separately. The event uses deviation≥epsilon, not>epsilon.
View answer and explanation
(2, 1/2, 1/2, 2, 1/2, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)
Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (2, 1/2, 1/2, 2, 1/2, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-fixed-n-deviation · candidate
59. The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit
Use the aligned finite law values=[1, 2], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=2 and epsilon=3/5>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).
The count includes all deviating joint words, including zero-mass words; probability is weighted separately. The event uses deviation≥epsilon, not>epsilon.
View answer and explanation
(2, 3/2, 3/5, 0, 0, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)
Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (2, 3/2, 3/5, 0, 0, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-fixed-n-deviation · candidate
60. The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit
Use the aligned finite law values=[2, 3], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=2 and epsilon=1/3>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).
The count includes all deviating joint words, including zero-mass words; probability is weighted separately. The event uses deviation≥epsilon, not>epsilon.
View answer and explanation
(2, 5/2, 1/3, 2, 1/2, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)
Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (2, 5/2, 1/3, 2, 1/2, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-fixed-n-deviation · candidate
61. The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit
Use the aligned finite law values=[-2, -1], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=3 and epsilon=2/4>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).
The count includes all deviating joint words, including zero-mass words; probability is weighted separately. The event uses deviation≥epsilon, not>epsilon.
View answer and explanation
(3, -3/2, 1/2, 2, 1/4, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)
Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (3, -3/2, 1/2, 2, 1/4, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-fixed-n-deviation · candidate
62. The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit
Use the aligned finite law values=[-1, 0], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=3 and epsilon=3/5>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).
The count includes all deviating joint words, including zero-mass words; probability is weighted separately. The event uses deviation≥epsilon, not>epsilon.
View answer and explanation
(3, -1/2, 3/5, 0, 0, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)
Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (3, -1/2, 3/5, 0, 0, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-fixed-n-deviation · candidate
63. The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit
Use the aligned finite law values=[0, 1], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=3 and epsilon=1/3>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).
The count includes all deviating joint words, including zero-mass words; probability is weighted separately. The event uses deviation≥epsilon, not>epsilon.
View answer and explanation
(3, 1/2, 1/3, 2, 1/4, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)
Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (3, 1/2, 1/3, 2, 1/4, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-fixed-n-deviation · candidate
64. The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit
Use the aligned finite law values=[1, 2], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=3 and epsilon=2/4>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).
The count includes all deviating joint words, including zero-mass words; probability is weighted separately. The event uses deviation≥epsilon, not>epsilon.
View answer and explanation
(3, 3/2, 1/2, 2, 1/4, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)
Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (3, 3/2, 1/2, 2, 1/4, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-fixed-n-deviation · candidate
65. The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit
Use the aligned finite law values=[2, 3], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=3 and epsilon=3/5>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).
The count includes all deviating joint words, including zero-mass words; probability is weighted separately. The event uses deviation≥epsilon, not>epsilon.
View answer and explanation
(3, 5/2, 3/5, 0, 0, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)
Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (3, 5/2, 3/5, 0, 0, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-fixed-n-deviation · candidate
66. The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit
Use the aligned finite law values=[-2, -1], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=4 and epsilon=1/3>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).
The count includes all deviating joint words, including zero-mass words; probability is weighted separately. The event uses deviation≥epsilon, not>epsilon.
View answer and explanation
(4, -3/2, 1/3, 2, 1/8, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)
Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (4, -3/2, 1/3, 2, 1/8, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-fixed-n-deviation · candidate
67. The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit
Use the aligned finite law values=[-1, 0], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=4 and epsilon=2/4>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).
The count includes all deviating joint words, including zero-mass words; probability is weighted separately. The event uses deviation≥epsilon, not>epsilon.
View answer and explanation
(4, -1/2, 1/2, 2, 1/8, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)
Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (4, -1/2, 1/2, 2, 1/8, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-fixed-n-deviation · candidate
68. The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit
Use the aligned finite law values=[0, 1], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=4 and epsilon=3/5>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).
The count includes all deviating joint words, including zero-mass words; probability is weighted separately. The event uses deviation≥epsilon, not>epsilon.
View answer and explanation
(4, 1/2, 3/5, 0, 0, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)
Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (4, 1/2, 3/5, 0, 0, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-fixed-n-deviation · candidate
69. The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit
Use the aligned finite law values=[1, 2], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=4 and epsilon=1/3>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).
The count includes all deviating joint words, including zero-mass words; probability is weighted separately. The event uses deviation≥epsilon, not>epsilon.
View answer and explanation
(4, 3/2, 1/3, 2, 1/8, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)
Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (4, 3/2, 1/3, 2, 1/8, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-fixed-n-deviation · candidate
70. The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit
Use the aligned finite law values=[2, 3], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=4 and epsilon=2/4>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).
The count includes all deviating joint words, including zero-mass words; probability is weighted separately. The event uses deviation≥epsilon, not>epsilon.
View answer and explanation
(4, 5/2, 1/2, 2, 1/8, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)
Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (4, 5/2, 1/2, 2, 1/8, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-fixed-n-deviation · candidate
71. The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit
Use the aligned finite law values=[-2, -1], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=5 and epsilon=3/5>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).
The count includes all deviating joint words, including zero-mass words; probability is weighted separately. The event uses deviation≥epsilon, not>epsilon.
View answer and explanation
(5, -3/2, 3/5, 0, 0, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)
Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (5, -3/2, 3/5, 0, 0, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-fixed-n-deviation · candidate
72. The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit
Use the aligned finite law values=[-1, 0], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=5 and epsilon=1/3>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).
The count includes all deviating joint words, including zero-mass words; probability is weighted separately. The event uses deviation≥epsilon, not>epsilon.
View answer and explanation
(5, -1/2, 1/3, 2, 1/16, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)
Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (5, -1/2, 1/3, 2, 1/16, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-fixed-n-deviation · candidate
73. The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit
Use the aligned finite law values=[0, 1], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=5 and epsilon=2/4>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).
The count includes all deviating joint words, including zero-mass words; probability is weighted separately. The event uses deviation≥epsilon, not>epsilon.
View answer and explanation
(5, 1/2, 1/2, 2, 1/16, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)
Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (5, 1/2, 1/2, 2, 1/16, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-fixed-n-deviation · candidate
74. The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit
Use the aligned finite law values=[1, 2], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=5 and epsilon=3/5>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).
The count includes all deviating joint words, including zero-mass words; probability is weighted separately. The event uses deviation≥epsilon, not>epsilon.
View answer and explanation
(5, 3/2, 3/5, 0, 0, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)
Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (5, 3/2, 3/5, 0, 0, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-fixed-n-deviation · candidate
75. The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit
Use the aligned finite law values=[2, 3], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=5 and epsilon=1/3>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).
The count includes all deviating joint words, including zero-mass words; probability is weighted separately. The event uses deviation≥epsilon, not>epsilon.
View answer and explanation
(5, 5/2, 1/3, 2, 1/16, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)
Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (5, 5/2, 1/3, 2, 1/16, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-fixed-n-deviation · candidate
76. Outside-Scope Inputs Are Not General Strong-Law Counterexamples
Use mode=nonfinite_support, complete model={"support_kind":"countably_infinite","moment_status":"integrable"}. Give (mode, support_kind, moment_status, this_finite_valued_theorem_applicable, scope).
The four modes have4/5/6/5 fields. Nonfinite support and non-IID are outside this scoped theorem, not counterexamples to every SLLN.
View answer and explanation
(nonfinite_support, countably_infinite, integrable, No, outside_finite_valued_scope_not_a_counterexample_to_general_SLLN)
Different marginals or nonfinite support do not meet this finite-valued IID theorem scope; this is not a claim that every general SLLN fails. A finite prefix does not establish an infinite process or limit. A proposed target different from the common expectation is not the theorem target. The exact result, retaining the requested order, is (nonfinite_support, countably_infinite, integrable, No, outside_finite_valued_scope_not_a_counterexample_to_general_SLLN).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-scope-boundary · candidate
77. Outside-Scope Inputs Are Not General Strong-Law Counterexamples
Use mode=nonfinite_support, complete model={"support_kind":"countably_infinite","moment_status":"unknown"}. Give (mode, support_kind, moment_status, this_finite_valued_theorem_applicable, scope).
The four modes have4/5/6/5 fields. Nonfinite support and non-IID are outside this scoped theorem, not counterexamples to every SLLN.
View answer and explanation
(nonfinite_support, countably_infinite, unknown, No, outside_finite_valued_scope_not_a_counterexample_to_general_SLLN)
Different marginals or nonfinite support do not meet this finite-valued IID theorem scope; this is not a claim that every general SLLN fails. A finite prefix does not establish an infinite process or limit. A proposed target different from the common expectation is not the theorem target. The exact result, retaining the requested order, is (nonfinite_support, countably_infinite, unknown, No, outside_finite_valued_scope_not_a_counterexample_to_general_SLLN).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-scope-boundary · candidate
78. Outside-Scope Inputs Are Not General Strong-Law Counterexamples
Use mode=nonfinite_support, complete model={"support_kind":"continuous","moment_status":"integrable"}. Give (mode, support_kind, moment_status, this_finite_valued_theorem_applicable, scope).
The four modes have4/5/6/5 fields. Nonfinite support and non-IID are outside this scoped theorem, not counterexamples to every SLLN.
View answer and explanation
(nonfinite_support, continuous, integrable, No, outside_finite_valued_scope_not_a_counterexample_to_general_SLLN)
Different marginals or nonfinite support do not meet this finite-valued IID theorem scope; this is not a claim that every general SLLN fails. A finite prefix does not establish an infinite process or limit. A proposed target different from the common expectation is not the theorem target. The exact result, retaining the requested order, is (nonfinite_support, continuous, integrable, No, outside_finite_valued_scope_not_a_counterexample_to_general_SLLN).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-scope-boundary · candidate
79. Outside-Scope Inputs Are Not General Strong-Law Counterexamples
Use mode=nonfinite_support, complete model={"support_kind":"continuous","moment_status":"unknown"}. Give (mode, support_kind, moment_status, this_finite_valued_theorem_applicable, scope).
The four modes have4/5/6/5 fields. Nonfinite support and non-IID are outside this scoped theorem, not counterexamples to every SLLN.
View answer and explanation
(nonfinite_support, continuous, unknown, No, outside_finite_valued_scope_not_a_counterexample_to_general_SLLN)
Different marginals or nonfinite support do not meet this finite-valued IID theorem scope; this is not a claim that every general SLLN fails. A finite prefix does not establish an infinite process or limit. A proposed target different from the common expectation is not the theorem target. The exact result, retaining the requested order, is (nonfinite_support, continuous, unknown, No, outside_finite_valued_scope_not_a_counterexample_to_general_SLLN).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-scope-boundary · candidate
80. Outside-Scope Inputs Are Not General Strong-Law Counterexamples
Use mode=non_iid, complete model={"values_a":[2,3],"probabilities_a":[[1,2],[1,2]],"values_b":[2,3],"probabilities_b":[[1,7],[6,7]]}. Give (mode, identical_marginals, this_finite_valued_IID_theorem_applicable, scope).
The four modes have4/5/6/5 fields. Nonfinite support and non-IID are outside this scoped theorem, not counterexamples to every SLLN.
View answer and explanation
(non_iid, No, No, finite_valued_IID_SLLN_not_applicable_not_general_failure)
Different marginals or nonfinite support do not meet this finite-valued IID theorem scope; this is not a claim that every general SLLN fails. A finite prefix does not establish an infinite process or limit. A proposed target different from the common expectation is not the theorem target. The exact result, retaining the requested order, is (non_iid, No, No, finite_valued_IID_SLLN_not_applicable_not_general_failure).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-scope-boundary · candidate
81. Outside-Scope Inputs Are Not General Strong-Law Counterexamples
Use mode=finite_prefix, complete model={"values":[-2,-1],"probabilities":[[1,3],[2,3]],"observations":[-2,-1,-1]}. Give (mode, observations, common_mean, infinite_process_or_limit_proof_claim, boundary).
The four modes have4/5/6/5 fields. Nonfinite support and non-IID are outside this scoped theorem, not counterexamples to every SLLN.
View answer and explanation
(finite_prefix, [-2, -1, -1], -4/3, No, finite_prefix_cannot_establish_infinite_process_or_a.s._limit)
Different marginals or nonfinite support do not meet this finite-valued IID theorem scope; this is not a claim that every general SLLN fails. A finite prefix does not establish an infinite process or limit. A proposed target different from the common expectation is not the theorem target. The exact result, retaining the requested order, is (finite_prefix, [-2, -1, -1], -4/3, No, finite_prefix_cannot_establish_infinite_process_or_a.s._limit).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-scope-boundary · candidate
82. Outside-Scope Inputs Are Not General Strong-Law Counterexamples
Use mode=wrong_target, complete model={"values":[-1,0],"probabilities":[[1,3],[2,3]],"target":8}. Give (mode, common_mean, proposed_target, target_equals_common_mean, proposed_target_claim, boundary).
The four modes have4/5/6/5 fields. Nonfinite support and non-IID are outside this scoped theorem, not counterexamples to every SLLN.
View answer and explanation
(wrong_target, -1/3, 8, No, No, SLLN_target_is_common_expectation)
Different marginals or nonfinite support do not meet this finite-valued IID theorem scope; this is not a claim that every general SLLN fails. A finite prefix does not establish an infinite process or limit. A proposed target different from the common expectation is not the theorem target. The exact result, retaining the requested order, is (wrong_target, -1/3, 8, No, No, SLLN_target_is_common_expectation).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-scope-boundary · candidate
83. Outside-Scope Inputs Are Not General Strong-Law Counterexamples
Use mode=non_iid, complete model={"values_a":[0,1],"probabilities_a":[[1,3],[2,3]],"values_b":[0,1],"probabilities_b":[[1,7],[6,7]]}. Give (mode, identical_marginals, this_finite_valued_IID_theorem_applicable, scope).
The four modes have4/5/6/5 fields. Nonfinite support and non-IID are outside this scoped theorem, not counterexamples to every SLLN.
View answer and explanation
(non_iid, No, No, finite_valued_IID_SLLN_not_applicable_not_general_failure)
Different marginals or nonfinite support do not meet this finite-valued IID theorem scope; this is not a claim that every general SLLN fails. A finite prefix does not establish an infinite process or limit. A proposed target different from the common expectation is not the theorem target. The exact result, retaining the requested order, is (non_iid, No, No, finite_valued_IID_SLLN_not_applicable_not_general_failure).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-scope-boundary · candidate
84. Outside-Scope Inputs Are Not General Strong-Law Counterexamples
Use mode=finite_prefix, complete model={"values":[1,2],"probabilities":[[1,3],[2,3]],"observations":[1,2,1]}. Give (mode, observations, common_mean, infinite_process_or_limit_proof_claim, boundary).
The four modes have4/5/6/5 fields. Nonfinite support and non-IID are outside this scoped theorem, not counterexamples to every SLLN.
View answer and explanation
(finite_prefix, [1, 2, 1], 5/3, No, finite_prefix_cannot_establish_infinite_process_or_a.s._limit)
Different marginals or nonfinite support do not meet this finite-valued IID theorem scope; this is not a claim that every general SLLN fails. A finite prefix does not establish an infinite process or limit. A proposed target different from the common expectation is not the theorem target. The exact result, retaining the requested order, is (finite_prefix, [1, 2, 1], 5/3, No, finite_prefix_cannot_establish_infinite_process_or_a.s._limit).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-scope-boundary · candidate
85. Outside-Scope Inputs Are Not General Strong-Law Counterexamples
Use mode=wrong_target, complete model={"values":[2,3],"probabilities":[[1,3],[2,3]],"target":11}. Give (mode, common_mean, proposed_target, target_equals_common_mean, proposed_target_claim, boundary).
The four modes have4/5/6/5 fields. Nonfinite support and non-IID are outside this scoped theorem, not counterexamples to every SLLN.
View answer and explanation
(wrong_target, 8/3, 11, No, No, SLLN_target_is_common_expectation)
Different marginals or nonfinite support do not meet this finite-valued IID theorem scope; this is not a claim that every general SLLN fails. A finite prefix does not establish an infinite process or limit. A proposed target different from the common expectation is not the theorem target. The exact result, retaining the requested order, is (wrong_target, 8/3, 11, No, No, SLLN_target_is_common_expectation).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-scope-boundary · candidate
86. Outside-Scope Inputs Are Not General Strong-Law Counterexamples
Use mode=non_iid, complete model={"values_a":[-2,-1],"probabilities_a":[[1,4],[3,4]],"values_b":[-2,-1],"probabilities_b":[[1,7],[6,7]]}. Give (mode, identical_marginals, this_finite_valued_IID_theorem_applicable, scope).
The four modes have4/5/6/5 fields. Nonfinite support and non-IID are outside this scoped theorem, not counterexamples to every SLLN.
View answer and explanation
(non_iid, No, No, finite_valued_IID_SLLN_not_applicable_not_general_failure)
Different marginals or nonfinite support do not meet this finite-valued IID theorem scope; this is not a claim that every general SLLN fails. A finite prefix does not establish an infinite process or limit. A proposed target different from the common expectation is not the theorem target. The exact result, retaining the requested order, is (non_iid, No, No, finite_valued_IID_SLLN_not_applicable_not_general_failure).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-scope-boundary · candidate
87. Outside-Scope Inputs Are Not General Strong-Law Counterexamples
Use mode=finite_prefix, complete model={"values":[-1,0],"probabilities":[[1,4],[3,4]],"observations":[-1,0,0]}. Give (mode, observations, common_mean, infinite_process_or_limit_proof_claim, boundary).
The four modes have4/5/6/5 fields. Nonfinite support and non-IID are outside this scoped theorem, not counterexamples to every SLLN.
View answer and explanation
(finite_prefix, [-1, 0, 0], -1/4, No, finite_prefix_cannot_establish_infinite_process_or_a.s._limit)
Different marginals or nonfinite support do not meet this finite-valued IID theorem scope; this is not a claim that every general SLLN fails. A finite prefix does not establish an infinite process or limit. A proposed target different from the common expectation is not the theorem target. The exact result, retaining the requested order, is (finite_prefix, [-1, 0, 0], -1/4, No, finite_prefix_cannot_establish_infinite_process_or_a.s._limit).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-scope-boundary · candidate
88. Outside-Scope Inputs Are Not General Strong-Law Counterexamples
Use mode=wrong_target, complete model={"values":[0,1],"probabilities":[[1,4],[3,4]],"target":14}. Give (mode, common_mean, proposed_target, target_equals_common_mean, proposed_target_claim, boundary).
The four modes have4/5/6/5 fields. Nonfinite support and non-IID are outside this scoped theorem, not counterexamples to every SLLN.
View answer and explanation
(wrong_target, 3/4, 14, No, No, SLLN_target_is_common_expectation)
Different marginals or nonfinite support do not meet this finite-valued IID theorem scope; this is not a claim that every general SLLN fails. A finite prefix does not establish an infinite process or limit. A proposed target different from the common expectation is not the theorem target. The exact result, retaining the requested order, is (wrong_target, 3/4, 14, No, No, SLLN_target_is_common_expectation).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-scope-boundary · candidate
89. Outside-Scope Inputs Are Not General Strong-Law Counterexamples
Use mode=non_iid, complete model={"values_a":[1,2],"probabilities_a":[[1,4],[3,4]],"values_b":[1,2],"probabilities_b":[[1,7],[6,7]]}. Give (mode, identical_marginals, this_finite_valued_IID_theorem_applicable, scope).
The four modes have4/5/6/5 fields. Nonfinite support and non-IID are outside this scoped theorem, not counterexamples to every SLLN.
View answer and explanation
(non_iid, No, No, finite_valued_IID_SLLN_not_applicable_not_general_failure)
Different marginals or nonfinite support do not meet this finite-valued IID theorem scope; this is not a claim that every general SLLN fails. A finite prefix does not establish an infinite process or limit. A proposed target different from the common expectation is not the theorem target. The exact result, retaining the requested order, is (non_iid, No, No, finite_valued_IID_SLLN_not_applicable_not_general_failure).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-scope-boundary · candidate
90. Outside-Scope Inputs Are Not General Strong-Law Counterexamples
Use mode=finite_prefix, complete model={"values":[2,3],"probabilities":[[1,4],[3,4]],"observations":[2,3,2]}. Give (mode, observations, common_mean, infinite_process_or_limit_proof_claim, boundary).
The four modes have4/5/6/5 fields. Nonfinite support and non-IID are outside this scoped theorem, not counterexamples to every SLLN.
View answer and explanation
(finite_prefix, [2, 3, 2], 11/4, No, finite_prefix_cannot_establish_infinite_process_or_a.s._limit)
Different marginals or nonfinite support do not meet this finite-valued IID theorem scope; this is not a claim that every general SLLN fails. A finite prefix does not establish an infinite process or limit. A proposed target different from the common expectation is not the theorem target. The exact result, retaining the requested order, is (finite_prefix, [2, 3, 2], 11/4, No, finite_prefix_cannot_establish_infinite_process_or_a.s._limit).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-scope-boundary · candidate
91. Outside-Scope Inputs Are Not General Strong-Law Counterexamples
Use mode=wrong_target, complete model={"values":[-2,-1],"probabilities":[[2,5],[3,5]],"target":17}. Give (mode, common_mean, proposed_target, target_equals_common_mean, proposed_target_claim, boundary).
The four modes have4/5/6/5 fields. Nonfinite support and non-IID are outside this scoped theorem, not counterexamples to every SLLN.
View answer and explanation
(wrong_target, -7/5, 17, No, No, SLLN_target_is_common_expectation)
Different marginals or nonfinite support do not meet this finite-valued IID theorem scope; this is not a claim that every general SLLN fails. A finite prefix does not establish an infinite process or limit. A proposed target different from the common expectation is not the theorem target. The exact result, retaining the requested order, is (wrong_target, -7/5, 17, No, No, SLLN_target_is_common_expectation).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-scope-boundary · candidate
92. Outside-Scope Inputs Are Not General Strong-Law Counterexamples
Use mode=non_iid, complete model={"values_a":[-1,0],"probabilities_a":[[2,5],[3,5]],"values_b":[-1,0],"probabilities_b":[[1,7],[6,7]]}. Give (mode, identical_marginals, this_finite_valued_IID_theorem_applicable, scope).
The four modes have4/5/6/5 fields. Nonfinite support and non-IID are outside this scoped theorem, not counterexamples to every SLLN.
View answer and explanation
(non_iid, No, No, finite_valued_IID_SLLN_not_applicable_not_general_failure)
Different marginals or nonfinite support do not meet this finite-valued IID theorem scope; this is not a claim that every general SLLN fails. A finite prefix does not establish an infinite process or limit. A proposed target different from the common expectation is not the theorem target. The exact result, retaining the requested order, is (non_iid, No, No, finite_valued_IID_SLLN_not_applicable_not_general_failure).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-scope-boundary · candidate
93. Outside-Scope Inputs Are Not General Strong-Law Counterexamples
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).
The four modes have4/5/6/5 fields. Nonfinite support and non-IID are outside this scoped theorem, not counterexamples to every SLLN.
View answer and explanation
(finite_prefix, [0, 1, 1], 3/5, No, finite_prefix_cannot_establish_infinite_process_or_a.s._limit)
Different marginals or nonfinite support do not meet this finite-valued IID theorem scope; this is not a claim that every general SLLN fails. A finite prefix does not establish an infinite process or limit. A proposed target different from the common expectation is not the theorem target. The exact result, retaining the requested order, is (finite_prefix, [0, 1, 1], 3/5, No, finite_prefix_cannot_establish_infinite_process_or_a.s._limit).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-scope-boundary · candidate
94. Outside-Scope Inputs Are Not General Strong-Law Counterexamples
Use mode=wrong_target, complete model={"values":[1,2],"probabilities":[[2,5],[3,5]],"target":20}. Give (mode, common_mean, proposed_target, target_equals_common_mean, proposed_target_claim, boundary).
The four modes have4/5/6/5 fields. Nonfinite support and non-IID are outside this scoped theorem, not counterexamples to every SLLN.
View answer and explanation
(wrong_target, 8/5, 20, No, No, SLLN_target_is_common_expectation)
Different marginals or nonfinite support do not meet this finite-valued IID theorem scope; this is not a claim that every general SLLN fails. A finite prefix does not establish an infinite process or limit. A proposed target different from the common expectation is not the theorem target. The exact result, retaining the requested order, is (wrong_target, 8/5, 20, No, No, SLLN_target_is_common_expectation).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-scope-boundary · candidate
95. Outside-Scope Inputs Are Not General Strong-Law Counterexamples
Use mode=non_iid, complete model={"values_a":[2,3],"probabilities_a":[[2,5],[3,5]],"values_b":[2,3],"probabilities_b":[[1,7],[6,7]]}. Give (mode, identical_marginals, this_finite_valued_IID_theorem_applicable, scope).
The four modes have4/5/6/5 fields. Nonfinite support and non-IID are outside this scoped theorem, not counterexamples to every SLLN.
View answer and explanation
(non_iid, No, No, finite_valued_IID_SLLN_not_applicable_not_general_failure)
Different marginals or nonfinite support do not meet this finite-valued IID theorem scope; this is not a claim that every general SLLN fails. A finite prefix does not establish an infinite process or limit. A proposed target different from the common expectation is not the theorem target. The exact result, retaining the requested order, is (non_iid, No, No, finite_valued_IID_SLLN_not_applicable_not_general_failure).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-scope-boundary · candidate
96. Outside-Scope Inputs Are Not General Strong-Law Counterexamples
Use mode=finite_prefix, complete model={"values":[-2,-1],"probabilities":[[1,5],[4,5]],"observations":[-2,-1,-2]}. Give (mode, observations, common_mean, infinite_process_or_limit_proof_claim, boundary).
The four modes have4/5/6/5 fields. Nonfinite support and non-IID are outside this scoped theorem, not counterexamples to every SLLN.
View answer and explanation
(finite_prefix, [-2, -1, -2], -6/5, No, finite_prefix_cannot_establish_infinite_process_or_a.s._limit)
Different marginals or nonfinite support do not meet this finite-valued IID theorem scope; this is not a claim that every general SLLN fails. A finite prefix does not establish an infinite process or limit. A proposed target different from the common expectation is not the theorem target. The exact result, retaining the requested order, is (finite_prefix, [-2, -1, -2], -6/5, No, finite_prefix_cannot_establish_infinite_process_or_a.s._limit).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-scope-boundary · candidate
97. Outside-Scope Inputs Are Not General Strong-Law Counterexamples
Use mode=wrong_target, complete model={"values":[-1,0],"probabilities":[[1,5],[4,5]],"target":23}. Give (mode, common_mean, proposed_target, target_equals_common_mean, proposed_target_claim, boundary).
The four modes have4/5/6/5 fields. Nonfinite support and non-IID are outside this scoped theorem, not counterexamples to every SLLN.
View answer and explanation
(wrong_target, -1/5, 23, No, No, SLLN_target_is_common_expectation)
Different marginals or nonfinite support do not meet this finite-valued IID theorem scope; this is not a claim that every general SLLN fails. A finite prefix does not establish an infinite process or limit. A proposed target different from the common expectation is not the theorem target. The exact result, retaining the requested order, is (wrong_target, -1/5, 23, No, No, SLLN_target_is_common_expectation).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-scope-boundary · candidate
98. Outside-Scope Inputs Are Not General Strong-Law Counterexamples
Use mode=non_iid, complete model={"values_a":[0,1],"probabilities_a":[[1,5],[4,5]],"values_b":[0,1],"probabilities_b":[[1,7],[6,7]]}. Give (mode, identical_marginals, this_finite_valued_IID_theorem_applicable, scope).
The four modes have4/5/6/5 fields. Nonfinite support and non-IID are outside this scoped theorem, not counterexamples to every SLLN.
View answer and explanation
(non_iid, No, No, finite_valued_IID_SLLN_not_applicable_not_general_failure)
Different marginals or nonfinite support do not meet this finite-valued IID theorem scope; this is not a claim that every general SLLN fails. A finite prefix does not establish an infinite process or limit. A proposed target different from the common expectation is not the theorem target. The exact result, retaining the requested order, is (non_iid, No, No, finite_valued_IID_SLLN_not_applicable_not_general_failure).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-scope-boundary · candidate
99. Outside-Scope Inputs Are Not General Strong-Law Counterexamples
Use mode=finite_prefix, complete model={"values":[1,2],"probabilities":[[1,5],[4,5]],"observations":[1,2,2]}. Give (mode, observations, common_mean, infinite_process_or_limit_proof_claim, boundary).
The four modes have4/5/6/5 fields. Nonfinite support and non-IID are outside this scoped theorem, not counterexamples to every SLLN.
View answer and explanation
(finite_prefix, [1, 2, 2], 9/5, No, finite_prefix_cannot_establish_infinite_process_or_a.s._limit)
Different marginals or nonfinite support do not meet this finite-valued IID theorem scope; this is not a claim that every general SLLN fails. A finite prefix does not establish an infinite process or limit. A proposed target different from the common expectation is not the theorem target. The exact result, retaining the requested order, is (finite_prefix, [1, 2, 2], 9/5, No, finite_prefix_cannot_establish_infinite_process_or_a.s._limit).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-scope-boundary · candidate
100. Outside-Scope Inputs Are Not General Strong-Law Counterexamples
Use mode=wrong_target, complete model={"values":[2,3],"probabilities":[[1,5],[4,5]],"target":26}. Give (mode, common_mean, proposed_target, target_equals_common_mean, proposed_target_claim, boundary).
The four modes have4/5/6/5 fields. Nonfinite support and non-IID are outside this scoped theorem, not counterexamples to every SLLN.
View answer and explanation
(wrong_target, 14/5, 26, No, No, SLLN_target_is_common_expectation)
Different marginals or nonfinite support do not meet this finite-valued IID theorem scope; this is not a claim that every general SLLN fails. A finite prefix does not establish an infinite process or limit. A proposed target different from the common expectation is not the theorem target. The exact result, retaining the requested order, is (wrong_target, 14/5, 26, No, No, SLLN_target_is_common_expectation).
Reviewed step-by-step derivation
Strong law of large numbers for finite-valued IID sequences · qf-mod49-slln-scope-boundary · candidate
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