# Question 22808: Strictly Ordered Query Points May Have Equal CDF Values [Return to catalogue page 115](https://ocme.evemisslab.com/en/questions/page/115/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations/#q-20261014-advance-20261111-r04-continuous-real-distribution-function-0033) 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. R0044-22808 · q-20261014-advance-20261111-r04-continuous-real-distribution-function-0033 ## Original question and givens Use strictly increasing knots=[-9, 0, 9], aligned exact values=[0, 8/9, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Exact query_points=[-10, 0, 10] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate). - G001 — Use strictly increasing knots=[-9, 0, 9], aligned exact values=[0, 8/9, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Exact query_points=[-10, 0, 10] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, 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-continuous-real-distribution-function](https://ocme.evemisslab.com/en/mko/mko-continuous-real-distribution-function/) ## Step-by-step derivation S001 · impliesBefore source operation=contcdf-monotone-endpoint-audit;original snapshot and mathematics retained After explicit ordered source output fields=[query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate] Conditions: none Keep the historical interface marker and list ordered source fields without JavaScript.Tags are not additional evidence;fixed No markers retain their scope.The ordering helper supplies protocol order,not mathematical correctness. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · calculationBefore CDF segment0 positive knot width := sub(0, -9) After CDF segment0 positive knot width = 9 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore CDF segment0 nonnegative value increment := sub(8/9, 0) After CDF segment0 nonnegative value increment = 8/9 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore CDF segment0 exact nonnegative slope := div(8/9, 9) After CDF segment0 exact nonnegative slope = 8/81 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore CDF segment1 positive knot width := sub(9, 0) After CDF segment1 positive knot width = 9 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore CDF segment1 nonnegative value increment := sub(1, 8/9) After CDF segment1 nonnegative value increment = 1/9 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore CDF segment1 exact nonnegative slope := div(1/9, 9) After CDF segment1 exact nonnegative slope = 1/81 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore ORIGINAL strictly increasing knots=[-9, 0, 9];aligned exact nondecreasing values=[0, 8/9, 1];ALL [width,height,slope] rows=[[9, 8/9, 8/81], [9, 1/9, 1/81]] After declare linear interpolation on EVERY segment,constant0 left and1 right;shared knot values make joins continuous;all slopes>=0 allow flats;continuous implies right-continuous and endpoints give limits0/1 Conditions: none These are properties of the declared whole interpolation law,not inferences from finite knot/query observations.Nonnegative slopes and matching joins give monotonicity and continuity;constant tails give the limits.This PL example may have a density without every continuous CDF having one. Step identifiers and dependenciesRule: contcdf_WHOLE_piecewise_law_continuous_flat_monotone_tails · Uses: G001, S007 S009 · impliesBefore CDF ORIGINAL ordered query0:ORIGINAL exact query x=-10;knots=[-9, 0, 9];values=[0, 8/9, 1] After case=left constant tail including first knot;source interval index=null;F(x)=0;this is CDF value,NOT singleton mass Conditions: none Evaluate the original query in its constant tail or first containing interval.The formulas agree at shared knots.F(x) is cumulative probability,not P(X=x). Step identifiers and dependenciesRule: contcdf_actual_query_exact_branch_not_empirical_mass · Uses: G001, S008 S010 · calculationBefore CDF ORIGINAL ordered query1 ORIGINAL query minus left knot := sub(0, -9) After CDF ORIGINAL ordered query1 ORIGINAL query minus left knot = 9 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore CDF ORIGINAL ordered query1 exact interval fraction := div(9, 9) After CDF ORIGINAL ordered query1 exact interval fraction = 1 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore CDF ORIGINAL ordered query1 fraction times ORIGINAL value increment := mul(1, 8/9) After CDF ORIGINAL ordered query1 fraction times ORIGINAL value increment = 8/9 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore CDF ORIGINAL ordered query1 plus ORIGINAL left-knot value := sum(0, 8/9) After CDF ORIGINAL ordered query1 plus ORIGINAL left-knot value = 8/9 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · impliesBefore CDF ORIGINAL ordered query1:ORIGINAL exact query x=0;knots=[-9, 0, 9];values=[0, 8/9, 1] After case=FIRST source interval containing query,including shared knot;source interval index=0;F(x)=8/9;this is CDF value,NOT singleton mass Conditions: none Evaluate the original query in its constant tail or first containing interval.The formulas agree at shared knots.F(x) is cumulative probability,not P(X=x). Step identifiers and dependenciesRule: contcdf_actual_query_exact_branch_not_empirical_mass · Uses: G001, S013 S015 · impliesBefore CDF ORIGINAL ordered query2:ORIGINAL exact query x=10;knots=[-9, 0, 9];values=[0, 8/9, 1] After case=right constant tail including last knot;source interval index=null;F(x)=1;this is CDF value,NOT singleton mass Conditions: none Evaluate the original query in its constant tail or first containing interval.The formulas agree at shared knots.F(x) is cumulative probability,not P(X=x). Step identifiers and dependenciesRule: contcdf_actual_query_exact_branch_not_empirical_mass · Uses: G001, S014 S016 · impliesBefore CDF ORIGINAL left-tail check:ORIGINAL exact query x=-10;knots=[-9, 0, 9];values=[0, 8/9, 1] After case=left constant tail including first knot;source interval index=null;F(x)=0;this is CDF value,NOT singleton mass Conditions: none Evaluate the original query in its constant tail or first containing interval.The formulas agree at shared knots.F(x) is cumulative probability,not P(X=x). Step identifiers and dependenciesRule: contcdf_actual_query_exact_branch_not_empirical_mass · Uses: G001, S015 S017 · impliesBefore CDF ORIGINAL right-tail check:ORIGINAL exact query x=10;knots=[-9, 0, 9];values=[0, 8/9, 1] After case=right constant tail including last knot;source interval index=null;F(x)=1;this is CDF value,NOT singleton mass Conditions: none Evaluate the original query in its constant tail or first containing interval.The formulas agree at shared knots.F(x) is cumulative probability,not P(X=x). Step identifiers and dependenciesRule: contcdf_actual_query_exact_branch_not_empirical_mass · Uses: G001, S016 S018 · impliesBefore ORIGINAL strictly increasing queries=[-10, 0, 10];exact values=[0, 8/9, 1] After NONSTRICT sample value monotonicity=Yes;left tail=0,right tail=1;global monotonicity already proved from the WHOLE law Conditions: none Query coordinates must be strictly increasing,but flat segments may give equal CDF values.An empty query list has a vacuous conjunction.Global properties came from the whole law,not this finite list alone. Step identifiers and dependenciesRule: contcdf_ordered_queries_allow_EQUAL_values_not_global_sample_proof · Uses: G001, S017 ## Reverse checks - S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S013 · premise_and_result_audit — Recompute CDF ORIGINAL ordered query1 plus ORIGINAL left-knot value with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute CDF ORIGINAL ordered query1 fraction times ORIGINAL value increment with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute CDF ORIGINAL ordered query1 exact interval fraction with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute CDF ORIGINAL ordered query1 ORIGINAL query minus left knot with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 — Recompute CDF segment1 exact nonnegative slope with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute CDF segment1 nonnegative value increment with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute CDF segment1 positive knot width with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute CDF segment0 exact nonnegative slope with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute CDF segment0 nonnegative value increment with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute CDF segment0 positive knot width with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result ([[-10, 0], [0, 8/9], [10, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits) 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: 59d111ebcb767a318fdd3cbc8d7f930dfae610cefb8650c5a7d90a36e737e1d5 - candidate record SHA-256: be7932fc3427cb5af92b68387522e190f4c16aa0285e384878bc51a231cb3fd5 - semantic review receipt SHA-256: 89b8e810eced1a69dbfef28d4b112e829edf156c7905a86e39a3f4958cd75e81 - question-first witness SHA-256: b2e8b769c463242f05172564ae96e310403eae151e1ec30bcb50be3ffe12cfc7 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261111-r04-continuous-real-distribution-function-0033.json)