Question 16015: 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
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R0031-16015 · q-20261014-advance-20261101-r04-almost-sure-convergence-0090

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=[15, 16, 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=15. 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=[15, 16, 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=15. 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=[15, 16, 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=15

After

atom row=[0, 1/2, 15, 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=16

After

atom row=[1, 1/2, 16, 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=15;separately declared full-tail rows=[[0, 1/2, 15, Yes, Yes], [1, 1/2, 16, 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

(15, [[0, 1/2, 15, Yes, Yes], [1, 1/2, 16, 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.

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input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
original question SHA-256: c5e919686d367cdf9bcff0e598718e9ebd9aaa7cf5b5c52295bed3d2613989da
candidate record SHA-256: ff08c0cbb936d4d54611b50b8c4d5cb14d6b8a7cf5732a5e5291c56c5fdb9db5
semantic review receipt SHA-256: 55bb21783e2125ecf1e67e42f83fd25bec0ccd493f911f4bb937a5e9d6f9369a
question-first witness SHA-256: 754ddf2a6039c5ef4aae78a538c6e9a1e9a58d4825aee194e869a27b8a4c95bb
Lean: NOT_RUN · human review: NOT_PERFORMED

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