# Question 14996: A Finite-Law CDF Stays Constant Just to the Right [Return to catalogue page 75](https://ocme.evemisslab.com/en/questions/page/75/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations/#q-20261014-advance-20261030-r06-cumulative-distribution-function-0096) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0028-14996 · q-20261014-advance-20261030-r06-cumulative-distribution-function-0096 ## Original question and givens Use the complete finite discrete law with distinct integer support=[8, 15, 21] 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=21. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists. - G001 — Use the complete finite discrete law with distinct integer support=[8, 15, 21] 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=21. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists. ## 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-cumulative-distribution-function](https://ocme.evemisslab.com/en/mko/mko-cumulative-distribution-function/) ## Step-by-step derivation S001 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001 S002 · impliesBefore original ordered support=[8, 15, 21];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. Step identifiers and dependenciesRule: CDF_complete_probability_law_original_pairing · Uses: G001, S001 S003 · impliesBefore threshold CDF:original support index=0;value=8;threshold=21 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. Step identifiers and dependenciesRule: CDF_every_original_atom_threshold_guard · Uses: G001, S002 S004 · impliesBefore threshold CDF:original support index=1;value=15;threshold=21 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. Step identifiers and dependenciesRule: CDF_every_original_atom_threshold_guard · Uses: G001, S003 S005 · impliesBefore threshold CDF:original support index=2;value=21;threshold=21 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. Step identifiers and dependenciesRule: CDF_every_original_atom_threshold_guard · Uses: G001, S004 S006 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore threshold=21;no declared support strictly greater;F_t=1 After all right increments retain total mass1;choose any positive delta Conditions: none All support is at or below t,so every right increment retains total mass1 and any positive neighborhood works.This is not guessing a limit from finite right-side samples. Step identifiers and dependenciesRule: CDF_right_tail_no_next_support · Uses: G001, S006 ## Reverse checks - 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. ## Result (21, 1, null, constant_on_right_neighborhood_before_next_support, Yes) 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: c83db9e01656e2cf3416e287f4a8b0f7555070e353edcc2d20315dfc56d88371 - candidate record SHA-256: bb8022dadedb4994725902e333ab4be7999a95941b00ee7d67ff073e35dbaae3 - semantic review receipt SHA-256: bfea39236e49aa7e44aa18bfa0d7857ff34a6acb762d0599e20b0f5af0f209ae - question-first witness SHA-256: 61c433f5c1e55ed4d84d08557fb8c117d26bbd404fdf05519ad42c09324c36e0 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0096.json)