Step-by-step derivation
S001 · implies
Before
source operation=markov-negative-variable-boundary;original snapshot and mathematics retained
After
explicit ordered source output fields=[values, pointwise_Markov_applicable, tail_probability, raw_expectation_over_threshold, boundary]
Conditions: none
Retain the historical interface marker in the original snapshot while listing ordered source fields here,without JavaScript or external lookup.Field labels are not additional mathematical evidence,and No retains the source claim scope.
Step identifiers and dependencies
Rule: source_ordered_interface_explicit_without_javascript · Uses: G001
S002 · calculation
Before
Markov complete probability mass := sum(1/2, 1/2)
After
Markov complete probability mass = 1
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S001
S003 · calculation
Before
Markov original atom0 weighted X := mul(1/2, -6)
After
Markov original atom0 weighted X = -3
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S002
S004 · calculation
Before
Markov original atom1 weighted X := mul(1/2, 8)
After
Markov original atom1 weighted X = 4
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S003
S005 · calculation
Before
Markov complete SAME-law expectation := sum(-3, 4)
After
Markov complete SAME-law expectation = 1
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S004
S006 · implies
Before
ORIGINAL complete nonempty masses=[1/2, 1/2];ALL aligned values X=[-6, 8];full mass=1;ALL weighted X terms=[-3, 4]
After
finite exact expectation=1;prototype EVERY-ENTRY nonnegative=No;actual positive-mass nonnegativity=No;null-negative atoms=[];positive-mass negative atoms=[0]
Conditions: none
The prototype flag tests every table entry,whereas almost-sure nonnegativity concerns positive-mass atoms only.The English source-review guard excludes null-only negative ambiguities from non-tail families rather than calling them a theorem-premise failure.The tail itself can still be computed for signed X.
Step identifiers and dependencies
Rule: markov_pointwise_source_gate_not_almost_sure_confusion · Uses: G001, S005
S007 · calculation
Before
Markov original atom0 threshold-event mass := mul(1/2, 0)
After
Markov original atom0 threshold-event mass = 0
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S006
S008 · implies
Before
ORIGINAL atom=0;value X=-6;positive threshold=7;original mass=1/2
After
event X>=threshold=No;EQUALITY INCLUDED;event mass contribution=0
Conditions: none
The event is X≥threshold,not X>threshold.Multiply by the event indicator at every original atom,including null atoms;use the complete law rather than observed frequencies.
Step identifiers and dependencies
Rule: markov_full_tail_event_includes_equality_not_strict_tail · Uses: G001, S007
S009 · calculation
Before
Markov original atom1 threshold-event mass := mul(1/2, 1)
After
Markov original atom1 threshold-event mass = 1/2
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S008
S010 · implies
Before
ORIGINAL atom=1;value X=8;positive threshold=7;original mass=1/2
After
event X>=threshold=Yes;EQUALITY INCLUDED;event mass contribution=1/2
Conditions: none
The event is X≥threshold,not X>threshold.Multiply by the event indicator at every original atom,including null atoms;use the complete law rather than observed frequencies.
Step identifiers and dependencies
Rule: markov_full_tail_event_includes_equality_not_strict_tail · Uses: G001, S009
S011 · calculation
Before
Markov complete exact tail probability := sum(0, 1/2)
After
Markov complete exact tail probability = 1/2
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S010
S012 · implies
Before
ALL original threshold-event flags=[No, Yes];ALL law-weighted indicator terms=[0, 1/2]
After
exact complete-law tail probability=1/2;this event calculation ALONE does not assert Markov applicability
Conditions: none
The complete event-mass sum is the actual tail probability,even for signed X.That calculation neither proves almost-sure nonnegativity nor identifies the tail with the later expectation-over-threshold bound.
Step identifiers and dependencies
Rule: markov_exact_event_probability_separate_from_theorem_bound · Uses: G001, S011
S013 · calculation
Before
Markov RAW expectation over positive threshold := div(1, 7)
After
Markov RAW expectation over positive threshold = 1/7
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S012
S014 · implies
Before
ACTUAL negative X at positive mass exists;original values=[-6, 8];tail=1/2;finite expectation=1;positive threshold=7
After
SOURCE pointwise applicability=No;raw E[X]/threshold=1/7;actual numeric tail<=raw quotient happens=No;NO general Markov guarantee or reverse-inequality claim
Conditions: none
A negative value at positive mass really violates the a.s.nonnegative premise,so Markov cannot be applied directly.This particular numeric inequality may coincidentally hold or fail.The No supplies no reverse inequality,and the unqualified raw quotient is not a certified probability bound.
Step identifiers and dependencies
Rule: markov_signed_inapplicability_not_numeric_failure_or_reverse · Uses: G001, S013