Step-by-step derivation
S001 · implies
Before
source roles = [qf-expectation-constant, no-domain-field, constant_expectation, no-form-field]
After
derived = [[reduced_fraction_RETAIN1_STRING]]
Conditions: none
Check THIS bank interface:BASE equations use form/domain;BASE events/complement have no operation;factorials have no domain.ext18 uses local hyphens,other legacy operations underscores.Do not force every answer into a tuple or fraction.
Step identifiers and dependencies
Rule: R62_EXACT_source_keys_domain_operation_form_AND_output_types · Uses: G001
S002 · calculation
Before
Expectation ORIGINAL weight total through slot0 := sum(0, 1)
After
Expectation ORIGINAL weight total through slot0 = 1
Conditions: none
Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S001
S003 · calculation
Before
Expectation ORIGINAL weight total through slot1 := sum(1, 3)
After
Expectation ORIGINAL weight total through slot1 = 4
Conditions: none
Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S002
S004 · implies
Before
source roles = [[0, 1], powerset(S), [1, 3], 4, not-supplied, not-supplied]
After
derived = [[0, 1], 4, Yes]
Conditions: none
S is nonempty,distinct and≤100;weights,X/Y align to ORIGINAL S slots.F is exactly powerset(S),weights nonnegative,denominator positive,and total EXACTLYden.EVEN at a=0 or null weights validate ALL raw fields;never silently uniformize.
Step identifiers and dependencies
Rule: Expectation_FULL_model_RAW_slots_SAME_alignment_and_normalized_law · Uses: G001, S003
S005 · implies
Before
source roles = [0, 0, constant_expectation, not-supplied, not-supplied, not-supplied, not-supplied, -7]
After
derived = [-7]
Conditions: none
Transform at EACH ORIGINAL outcome BEFORE using its aligned weight.Never sort S,X,Y,weights independently.Null-weight values remain original data;do not delete them to repair inputs or infer a range.
Step identifiers and dependencies
Rule: Expectation_EACH_original_outcome_transformation_BEFORE_weight · Uses: G001, S004
S006 · calculation
Before
Expectation ORIGINAL slot0 weight times transformed value := mul(1, -7)
After
Expectation ORIGINAL slot0 weight times transformed value = -7
Conditions: none
Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S005
S007 · calculation
Before
Expectation ORIGINAL weighted numerator through slot0 := sum(0, -7)
After
Expectation ORIGINAL weighted numerator through slot0 = -7
Conditions: none
Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S006
S008 · implies
Before
source roles = [1, 1, constant_expectation, not-supplied, not-supplied, not-supplied, not-supplied, -7]
After
derived = [-7]
Conditions: none
Transform at EACH ORIGINAL outcome BEFORE using its aligned weight.Never sort S,X,Y,weights independently.Null-weight values remain original data;do not delete them to repair inputs or infer a range.
Step identifiers and dependencies
Rule: Expectation_EACH_original_outcome_transformation_BEFORE_weight · Uses: G001, S007
S009 · calculation
Before
Expectation ORIGINAL slot1 weight times transformed value := mul(3, -7)
After
Expectation ORIGINAL slot1 weight times transformed value = -21
Conditions: none
Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S008
S010 · calculation
Before
Expectation ORIGINAL weighted numerator through slot1 := sum(-7, -21)
After
Expectation ORIGINAL weighted numerator through slot1 = -28
Conditions: none
Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S009
S011 · calculation
Before
Expectation ORIGINAL weighted numerator over denominator := div(-28, 4)
After
Expectation ORIGINAL weighted numerator over denominator = -7
Conditions: none
Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S010
S012 · implies
Before
source roles = [[0, 1], [1, 3], 4, constant_expectation, [-7, -7]]
After
derived = [[-7, -21], -28, -7/1]
Conditions: none
In the QUALIFIED SAME probability law,E[Z]=Σw_iZ(s_i)/q.Affine gives aE[X]+b and a same-model sum gives E[X]+E[Y]WITHOUT independence;this does not imply a product-expectation formula.A constant has mean c because total mass is1.Negative/unattained means are legal;retain/1.
Step identifiers and dependencies
Rule: Expectation_LINEARITY_qualified_SAME_law_NOT_independence_or_sample_mean · Uses: G001, S011