Step-by-step derivation
S001 · implies
Before
source roles = [qf-mod50-numerr-affine-propagation, [domain, input_error_bound, operation, slope]]
After
derived = [numerr-affine-propagation, Exact_numerical_error_density]
Conditions: none
THIS source uses LOCAL-hyphen operations with exact keys/domain.Keep source field order and numeric/Boolean/string roles;integer rational texts omit/1.Original questions,English,MKOs,ASTs and statuses remain unchanged.
Step identifiers and dependencies
Rule: R66_EXACT_source_bank_keys_local_operation_domain · Uses: G001
S002 · implies
Before
source roles = [propagation ORIGINAL slope, 100000000, -2, 1]
After
derived = [[-2, 1], -2]
Conditions: none
Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity.
Step identifiers and dependencies
Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S001
S003 · calculation
Before
propagation ORIGINAL slope ORIGINAL numerator over denominator := div(-2, 1)
After
propagation ORIGINAL slope ORIGINAL numerator over denominator = -2
Conditions: none
Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S002
S004 · implies
Before
source roles = [propagation ORIGINAL input bound, 100000000, 23, 3]
After
derived = [[23, 3], 23/3]
Conditions: none
Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity.
Step identifiers and dependencies
Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S003
S005 · calculation
Before
propagation ORIGINAL input bound ORIGINAL numerator over denominator := div(23, 3)
After
propagation ORIGINAL input bound ORIGINAL numerator over denominator = 23/3
Conditions: none
Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S004
S006 · implies
Before
source roles = [Affine output error bound SOURCE naive mul, 100000000, 46, 3]
After
derived = [[46, 3], 46/3]
Conditions: none
Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity.
Step identifiers and dependencies
Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S005
S007 · calculation
Before
Affine output error bound SOURCE naive mul ORIGINAL numerator over denominator := div(46, 3)
After
Affine output error bound SOURCE naive mul ORIGINAL numerator over denominator = 46/3
Conditions: none
Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S006
S008 · calculation
Before
Affine output error bound ORIGINAL rational mul := mul(2, 23/3)
After
Affine output error bound ORIGINAL rational mul = 46/3
Conditions: none
Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S007
S009 · implies
Before
source roles = [-2, 23/3]
After
derived = [46/3]
Conditions: none
Use |m(x+δ)+b−(mx+b)|=|m|*|δ|≤|m|*bound.The bound is nonnegative;zero slope or bound is allowed.This is THIS affine model’s absolute bound,not identification of rounding/truncation/measurement sources or a convergence proof.
Step identifiers and dependencies
Rule: Error_AFFINE_propagation_absolute_bound_INCLUDES_zero · Uses: G001, S008