# Question 33981: Zero Reference and Equal Tolerances Require Explicit Error Conventions [Return to catalogue page 170](https://ocme.evemisslab.com/en/questions/page/170/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations/#q-20261015-pulse-20261119-r05-0081) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0066-33981 · q-20261015-pulse-20261119-r05-0081 ## Original question and givens Use mode=inclusive_tolerance, complete model={"error":[6,2],"tolerance":[6,2]}. Give (mode, absolute_error, tolerance, inclusive_test_result, equality_convention). - G001 — Use mode=inclusive_tolerance, complete model={"error":[6,2],"tolerance":[6,2]}. Give (mode, absolute_error, tolerance, inclusive_test_result, equality_convention). ## 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. - [mko-numerical-approximation-error](https://ocme.evemisslab.com/en/mko/mko-numerical-approximation-error/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-mod50-numerr-boundary, [domain, mode, model, operation]] After derived = [numerr-boundary, 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 dependenciesRule: R66_EXACT_source_bank_keys_local_operation_domain · Uses: G001 S002 · impliesBefore source roles = [tolerance ORIGINAL error, 100000000, 6, 2] 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. Step identifiers and dependenciesRule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S001 S003 · calculationBefore tolerance ORIGINAL error ORIGINAL numerator over denominator := div(6, 2) After tolerance ORIGINAL error 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · impliesBefore source roles = [tolerance ORIGINAL threshold, 100000000, 6, 2] 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. Step identifiers and dependenciesRule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S003 S005 · calculationBefore tolerance ORIGINAL threshold ORIGINAL numerator over denominator := div(6, 2) After tolerance ORIGINAL threshold 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore source roles = [inclusive_tolerance, 3, 3] After derived = [Yes] Conditions: none FIRST qualify nonnegative error/tolerance and THIS equality boundary.Inclusive uses≤ and is true;strict uses< and is false.Equal tuple arity does not allow dropping the mode or swapping results. Step identifiers and dependenciesRule: Error_TOLERANCE_equality_STRICT_vs_INCLUSIVE · Uses: G001, S005 ## Reverse checks - 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 tolerance ORIGINAL threshold 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 tolerance ORIGINAL error 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 (inclusive_tolerance, 3, 3, Yes, error<=tolerance_is_true_at_equality) 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 1699ecdf146eb8ce2bc0453969f00df85d003071d86687cf5d76d44f60aecb71 - candidate record SHA-256: 92dac2469b6114fbc2c587ef68596927d53876d67b271ac30b0e91790f4008d4 - semantic review receipt SHA-256: 99b4f114b2ec93164b6aa5681e0107033a240e7a3f0a8988633ba65320ede267 - question-first witness SHA-256: d9bd3a1940da8a73f790ee3fe460c9a8523791bdbe72c8e0304e3740b66710cc - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-pulse-20261119-r05-0081.json)