Use slope=-3 and nonnegative absolute input error bound=15/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).
G001 — Use slope=-3 and nonnegative absolute input error bound=15/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).
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.
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.
source roles = [propagation ORIGINAL slope, 100000000, -3, 1]
After
derived = [[-3, 1], -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.
propagation ORIGINAL slope ORIGINAL numerator over denominator := div(-3, 1)
After
propagation ORIGINAL slope ORIGINAL numerator over denominator = -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.
source roles = [propagation ORIGINAL input bound, 100000000, 15, 3]
After
derived = [[5, 1], 5]
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.
propagation ORIGINAL input bound ORIGINAL numerator over denominator := div(15, 3)
After
propagation ORIGINAL input bound ORIGINAL numerator over denominator = 5
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.
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.
Affine output error bound SOURCE naive mul ORIGINAL numerator over denominator := div(15, 1)
After
Affine output error bound SOURCE naive mul ORIGINAL numerator over denominator = 15
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.
Affine output error bound ORIGINAL rational mul := mul(3, 5)
After
Affine output error bound ORIGINAL rational mul = 15
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.
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.
S009 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
S008 · premise_and_result_audit — Recompute Affine output error bound ORIGINAL rational mul with exact rational arithmetic, not rounded decimals.
S007 · premise_and_result_audit — Recompute Affine output error bound SOURCE naive mul ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals.
S006 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
S005 · premise_and_result_audit — Recompute propagation ORIGINAL input bound ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals.
S004 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
S003 · premise_and_result_audit — Recompute propagation ORIGINAL slope ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals.
S002 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
S001 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
Result
(-3, 5, 15, affine_output_absolute_error_bound)
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.