R0044-22818 · q-20261014-advance-20261111-r04-continuous-real-distribution-function-0043
Step-by-step derivation
S001 · implies
Before
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 dependencies
Rule: source_ordered_interface_explicit_without_javascript · Uses: G001
S002 · calculation
Before
CDF segment0 positive knot width := sub(0, -19)
After
CDF segment0 positive knot width = 19
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S001
S003 · calculation
Before
CDF segment0 nonnegative value increment := sub(18/19, 0)
After
CDF segment0 nonnegative value increment = 18/19
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S002
S004 · calculation
Before
CDF segment0 exact nonnegative slope := div(18/19, 19)
After
CDF segment0 exact nonnegative slope = 18/361
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S003
S005 · calculation
Before
CDF segment1 positive knot width := sub(19, 0)
After
CDF segment1 positive knot width = 19
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S004
S006 · calculation
Before
CDF segment1 nonnegative value increment := sub(1, 18/19)
After
CDF segment1 nonnegative value increment = 1/19
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S005
S007 · calculation
Before
CDF segment1 exact nonnegative slope := div(1/19, 19)
After
CDF segment1 exact nonnegative slope = 1/361
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S006
S008 · implies
Before
ORIGINAL strictly increasing knots=[-19, 0, 19];aligned exact nondecreasing values=[0, 18/19, 1];ALL [width,height,slope] rows=[[19, 18/19, 18/361], [19, 1/19, 1/361]]
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 dependencies
Rule: contcdf_WHOLE_piecewise_law_continuous_flat_monotone_tails · Uses: G001, S007
S009 · implies
Before
CDF ORIGINAL ordered query0:ORIGINAL exact query x=-20;knots=[-19, 0, 19];values=[0, 18/19, 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 dependencies
Rule: contcdf_actual_query_exact_branch_not_empirical_mass · Uses: G001, S008
S010 · calculation
Before
CDF ORIGINAL ordered query1 ORIGINAL query minus left knot := sub(0, -19)
After
CDF ORIGINAL ordered query1 ORIGINAL query minus left knot = 19
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S009
S011 · calculation
Before
CDF ORIGINAL ordered query1 exact interval fraction := div(19, 19)
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S010
S012 · calculation
Before
CDF ORIGINAL ordered query1 fraction times ORIGINAL value increment := mul(1, 18/19)
After
CDF ORIGINAL ordered query1 fraction times ORIGINAL value increment = 18/19
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S011
S013 · calculation
Before
CDF ORIGINAL ordered query1 plus ORIGINAL left-knot value := sum(0, 18/19)
After
CDF ORIGINAL ordered query1 plus ORIGINAL left-knot value = 18/19
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S012
S014 · implies
Before
CDF ORIGINAL ordered query1:ORIGINAL exact query x=0;knots=[-19, 0, 19];values=[0, 18/19, 1]
After
case=FIRST source interval containing query,including shared knot;source interval index=0;F(x)=18/19;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 dependencies
Rule: contcdf_actual_query_exact_branch_not_empirical_mass · Uses: G001, S013
S015 · implies
Before
CDF ORIGINAL ordered query2:ORIGINAL exact query x=20;knots=[-19, 0, 19];values=[0, 18/19, 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 dependencies
Rule: contcdf_actual_query_exact_branch_not_empirical_mass · Uses: G001, S014
S016 · implies
Before
CDF ORIGINAL left-tail check:ORIGINAL exact query x=-20;knots=[-19, 0, 19];values=[0, 18/19, 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 dependencies
Rule: contcdf_actual_query_exact_branch_not_empirical_mass · Uses: G001, S015
S017 · implies
Before
CDF ORIGINAL right-tail check:ORIGINAL exact query x=20;knots=[-19, 0, 19];values=[0, 18/19, 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 dependencies
Rule: contcdf_actual_query_exact_branch_not_empirical_mass · Uses: G001, S016
S018 · implies
Before
ORIGINAL strictly increasing queries=[-20, 0, 20];exact values=[0, 18/19, 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 dependencies
Rule: contcdf_ordered_queries_allow_EQUAL_values_not_global_sample_proof · Uses: G001, S017