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R0028-14909 · q-20261014-advance-20261030-r06-cumulative-distribution-function-0009
Use the complete finite discrete law with distinct integer support=[-4, 3, 9] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=9. Give (t,F_t,uses_X_less_than_or_equal_t).
G001 — Use the complete finite discrete law with distinct integer support=[-4, 3, 9] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=9. Give (t,F_t,uses_X_less_than_or_equal_t).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
CDF law total mass := sum(1/4, 1/4, 1/2)
After
CDF law total mass = 1
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001
Before
original ordered support=[-4, 3, 9];raw aligned masses=[[1, 4], [1, 4], [1, 2]]
After
normalized aligned masses=[1/4, 1/4, 1/2];nonnegative=Yes;total=1
Conditions: none
Qualify distinct support,nonnegative masses and unit total.Preserve original support-mass pairing even if unsorted,and retain zero-mass support points.This independently enforces the probability-law premise declared by the source English scope.
Rule: CDF_complete_probability_law_original_pairing · Uses: G001, S001
Before
threshold CDF:original support index=0;value=-4;threshold=9
After
test value<=threshold=Yes;aligned mass=1/4
Conditions: none
Test the threshold at each original support index and use its aligned mass.Distinguish strict< from inclusive≤;do not sort support while retaining old mass positions.
Rule: CDF_every_original_atom_threshold_guard · Uses: G001, S002
Before
threshold CDF:original support index=1;value=3;threshold=9
After
test value<=threshold=Yes;aligned mass=1/4
Conditions: none
Test the threshold at each original support index and use its aligned mass.Distinguish strict< from inclusive≤;do not sort support while retaining old mass positions.
Rule: CDF_every_original_atom_threshold_guard · Uses: G001, S003
Before
threshold CDF:original support index=2;value=9;threshold=9
After
test value<=threshold=Yes;aligned mass=1/2
Conditions: none
Test the threshold at each original support index and use its aligned mass.Distinguish strict< from inclusive≤;do not sort support while retaining old mass positions.
Rule: CDF_every_original_atom_threshold_guard · Uses: G001, S004
Before
threshold CDF selected mass sum := sum(1/4, 1/4, 1/2)
After
threshold CDF selected mass sum = 1
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
threshold=9;cumulative mass=1
After
F(t)=P(X<=t),not point mass or density;finite-law tails0 and1
Conditions: none
Below all support the empty sum is0;at/above the maximum the total is1.F(t) sums every atom≤t,not only the point mass P(X=t) and not a density.
Rule: CDF_inclusive_value_and_endpoint_scope · Uses: G001, S006
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 threshold CDF selected mass sum 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 CDF law total mass with exact rational arithmetic, not rounded decimals.(9, 1, uses_X_less_than_or_equal_t)
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: fa73c74d7b4b396eaf79315a167a7e553fd801d27de67ee896169b2fd37857a9candidate record SHA-256: bcd37528482922801e556a1f54e8cc3b104fc363bfcfa40388403c3477ffd3dcsemantic review receipt SHA-256: bfea39236e49aa7e44aa18bfa0d7857ff34a6acb762d0599e20b0f5af0f209aequestion-first witness SHA-256: 61c433f5c1e55ed4d84d08557fb8c117d26bbd404fdf05519ad42c09324c36e0Lean: NOT_RUN · human review: NOT_PERFORMED