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R0031-16054 · q-20261014-advance-20261101-r04-almost-sure-convergence-0029
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=[4, 5, 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. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.
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=[4, 5, 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. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.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.
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.
Rule: exact_rational_arithmetic · Uses: G001
Before
same complete finite atomic space;raw masses=[[1, 2], [1, 2], [0, 1]];raw settling declarations=[4, 5, 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.
Rule: asconv_declared_full_trajectory_model_not_missing_receipt · Uses: G001, S001
Before
original atom=0;exact mass=1/2;full-tail declaration=4
After
atom row=[0, 1/2, 4, 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.
Rule: asconv_each_positive_atom_obligation_and_null_exception · Uses: G001, S002
Before
original atom=1;exact mass=1/2;full-tail declaration=5
After
atom row=[1, 1/2, 5, 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.
Rule: asconv_each_positive_atom_obligation_and_null_exception · Uses: G001, S003
Before
original atom=2;exact mass=0;full-tail declaration=null
After
atom row=[2, 0, null, No, 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.
Rule: asconv_each_positive_atom_obligation_and_null_exception · Uses: G001, S004
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.
Rule: exact_rational_arithmetic · Uses: G001, S005
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.
Rule: asconv_probability_one_not_every_outcome · Uses: G001, S006
Before
positive-mass rows only=[[0, 1/2, 4, Yes, Yes], [1, 1/2, 5, Yes, Yes]];positive failures=[]
After
maximum settling index=5;certificate=uniform_finite_atom_tail_after_maximum
Conditions: none
When every positive atom has an integer cutoff,take the maximum of finitely many cutoffs.All positive atoms then equal the common limit together.Null-atom cutoffs are irrelevant.If a positive atom is declared nonconvergent,return null rather than invent a finite bound.This finite maximum is not a theorem about arbitrary infinite atomic spaces.
Rule: asconv_finite_positive_atom_maximum_not_arbitrary_infinite_supremum · Uses: G001, S007
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.([[0, 1/2, 4, Yes, Yes], [1, 1/2, 5, Yes, Yes]], 5, Yes, uniform_finite_atom_tail_after_maximum)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 9fb18fea3a409f7ad69632e8e68ad77b19e21260b4b23cdb35953c3722ee0f5dcandidate record SHA-256: fdbec3182cad33d130e8379e6f10001bf6617bb87e2e4934bd016926b4dd69e9semantic review receipt SHA-256: 55bb21783e2125ecf1e67e42f83fd25bec0ccd493f911f4bb937a5e9d6f9369aquestion-first witness SHA-256: 754ddf2a6039c5ef4aae78a538c6e9a1e9a58d4825aee194e869a27b8a4c95bbLean: NOT_RUN · human review: NOT_PERFORMED