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R0002-01621 · q-20261013-advance-20261020-r03-numerical-approximation-error-0021
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=43/1, reference=3/1. Give (normalized approximation, normalized reference, absolute error).
G001 — Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=43/1, reference=3/1. Give (normalized approximation, normalized reference, absolute error).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.
Before
reference=3
After
all values are exact rationals
Conditions: none
This is exact error arithmetic against a specified reference,without floating-point rounding that could alter an equality boundary.
Rule: exact_error_model · Uses: G001
Before
approximation−reference := sub(43, 3)
After
approximation−reference = 40
Conditions: none
Compute the signed error as approximation minus reference.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
absolute error := mul(1, 40)
After
absolute error = 40
Conditions: none
Use the sign to take the absolute value;the result is nonnegative.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
signed error=40
After
absolute error=40≥0
Conditions: none
Absolute error has the original quantity's units;a negative reference does not make it negative.
Rule: absolute_error_definition · Uses: G001, S003
S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S003 · premise_and_result_audit — Recompute absolute error with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute approximation−reference with exact rational arithmetic, not rounded decimals.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.(43, 3, 40)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 4bf0533a49a1effb276449cecaa43cfed10972ac6be50e0529272b327dfafea9candidate record SHA-256: f4f675f569227c9b1b3db13da10f20941b8b5a0dd08759a6e66fa282fc926df7semantic review receipt SHA-256: 340d74dcc6e37df6249594a0b79a9ee2af804337b7847478b3c1342af9722861question-first witness SHA-256: 9a06233d190d31a16f13fb03b6c3ceba4cbb748fff6c997941f66ab1888dcca8Lean: NOT_RUN · human review: NOT_PERFORMED