Use mode=signed_vs_absolute, complete model={"approximation":16,"reference":-16}. Give (mode, approximation, reference, signed_error, absolute_error, quantity_identification_claim, boundary).
G001 — Use mode=signed_vs_absolute, complete model={"approximation":16,"reference":-16}. Give (mode, approximation, reference, signed_error, absolute_error, quantity_identification_claim, boundary).
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 = [signed ORIGINAL approximation, 100000000, 16, 1]
After
derived = [[16, 1], 16]
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.
signed ORIGINAL approximation ORIGINAL numerator over denominator := div(16, 1)
After
signed ORIGINAL approximation ORIGINAL numerator over denominator = 16
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 = [signed ORIGINAL reference, 100000000, -16, 1]
After
derived = [[-16, 1], -16]
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.
signed ORIGINAL reference ORIGINAL numerator over denominator := div(-16, 1)
After
signed ORIGINAL reference ORIGINAL numerator over denominator = -16
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 = [Signed original approximation minus reference SOURCE naive sub, 100000000, 32, 1]
After
derived = [[32, 1], 32]
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.
Signed original approximation minus reference SOURCE naive sub ORIGINAL numerator over denominator := div(32, 1)
After
Signed original approximation minus reference SOURCE naive sub ORIGINAL numerator over denominator = 32
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.
Signed original approximation minus reference ORIGINAL rational sub := sub(16, -16)
After
Signed original approximation minus reference ORIGINAL rational sub = 32
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.
Signed error=xhat−reference retains direction;absolute error=|signed| does not.For a positive signed error,the NUMBERS may coincide.The No rejects identifying the quantities/information,not a claim that THIS pair must be numerically unequal.
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 Signed original approximation minus reference ORIGINAL rational sub with exact rational arithmetic, not rounded decimals.
S007 · premise_and_result_audit — Recompute Signed original approximation minus reference SOURCE naive sub 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 signed ORIGINAL reference 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 signed ORIGINAL approximation 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
(signed_vs_absolute, 16, -16, 32, 32, No, signed_error_and_absolute_error_preserve_different_information)
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.