Question 16013: A Finite Prefix Alone Does Not Establish Every Atomwise Tail

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0031-16013 · q-20261014-advance-20261101-r04-almost-sure-convergence-0088

Original question and givens

Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[13, 14, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=13. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).

G001 — Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[13, 14, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=13. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

mko-almost-sure-convergence

Step-by-step derivation

S001 · calculation

Before

almost-sure full probability mass := sum(1/2, 1/2, 0)

After

almost-sure full probability mass = 1

Conditions: none

Compute exactly from original field roles and established intermediates,retaining denominators,signs,indices and the complete carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001

S002 · implies

Before

same complete finite atomic space;raw masses=[[1, 2], [1, 2], [0, 1]];raw settling declarations=[13, 14, null]

After

normalized nonnegative masses=[1/2, 1/2, 0];total=1;integer N means equality to common limit at EVERY later index;null means model-declared nonconvergence

Conditions: none

Fix the same probability space and aligned atoms.An integer N declares equality on the entire later tail;null is model-declared nonconvergence,not an unchecked or missing receipt.These universal declarations cannot be manufactured from a finite observation.

Step identifiers and dependencies

Rule: asconv_declared_full_trajectory_model_not_missing_receipt · Uses: G001, S001

S003 · implies

Before

original atom=0;exact mass=1/2;full-tail declaration=13

After

atom row=[0, 1/2, 13, Yes, Yes]

Conditions: none

Every positive-mass atom must have its convergent-tail declaration.A zero-mass failure may be ignored only for an almost-sure claim.Keep every atom's original order,mass and declaration;do not delete null atoms and pretend the statement holds everywhere.

Step identifiers and dependencies

Rule: asconv_each_positive_atom_obligation_and_null_exception · Uses: G001, S002

S004 · implies

Before

original atom=1;exact mass=1/2;full-tail declaration=14

After

atom row=[1, 1/2, 14, Yes, Yes]

Conditions: none

Step identifiers and dependencies

Rule: asconv_each_positive_atom_obligation_and_null_exception · Uses: G001, S003

S005 · implies

Before

original atom=2;exact mass=0;full-tail declaration=null

After

atom row=[2, 0, null, No, Yes]

Conditions: none

Step identifiers and dependencies

Rule: asconv_each_positive_atom_obligation_and_null_exception · Uses: G001, S004

S006 · calculation

Before

almost-sure declared nonconvergent exceptional mass := sum(0)

After

almost-sure declared nonconvergent exceptional mass = 0

Conditions: none

Compute exactly from original field roles and established intermediates,retaining denominators,signs,indices and the complete carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · implies

Before

all model-declared nonconvergent atom indices=[2];their total mass=0

After

positive-mass failures=[];almost-sure qualification=Yes

Conditions: none

On this complete finite atomic space,the bad set has mass0 exactly when it contains no positive-mass atom.That gives probability1 without changing actual nonconvergence on a null atom.

Step identifiers and dependencies

Rule: asconv_probability_one_not_every_outcome · Uses: G001, S006

S008 · implies

Before

finite observed prefix length=13;separately declared full-tail rows=[[0, 1/2, 13, Yes, Yes], [1, 1/2, 14, Yes, Yes], [2, 0, null, No, Yes]]

After

finite-prefix sufficiency=No;finite_prefix_not_almost_sure_convergence_certificate

Conditions: none

This No denies sufficiency of the finite prefix alone,not the separately supplied universal-tail model.Nor does an unobserved tail mean a null nonconvergence declaration.Preserve the explicit common finite-atom hypotheses rather than erasing them with a general convergence-mode warning.

Step identifiers and dependencies

Rule: asconv_finite_observation_not_universal_tail_or_model_refutation · Uses: G001, S007

Reverse checks

S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S006 · premise_and_result_audit — Recompute almost-sure declared nonconvergent exceptional mass with exact rational arithmetic, not rounded decimals.
S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S001 · premise_and_result_audit — Recompute almost-sure full probability mass with exact rational arithmetic, not rounded decimals.

Result

(13, [[0, 1/2, 13, Yes, Yes], [1, 1/2, 14, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts

input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
original question SHA-256: cd8bcebc7beb17eefd1afae42e26287239117f2e459fe318450dfbaa97882625
candidate record SHA-256: 94e3e51a7a4d23cce31340b8c43d051b28b42139e83ac9807a755dc84aaf00c3
semantic review receipt SHA-256: 55bb21783e2125ecf1e67e42f83fd25bec0ccd493f911f4bb937a5e9d6f9369a
question-first witness SHA-256: 754ddf2a6039c5ef4aae78a538c6e9a1e9a58d4825aee194e869a27b8a4c95bb
Lean: NOT_RUN · human review: NOT_PERFORMED

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