# Question 34006: The Same Residual Can Correspond to Different Forward Errors [Return to catalogue page 171](https://ocme.evemisslab.com/en/questions/page/171/) [← 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-r05-0056) 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. R0067-34006 · q-20261015-pulse-20261119-r05-0056 ## Original question and givens Use positive absolute residual=6, nonzero first slope=1 and nonzero second slope=2, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope). - G001 — Use positive absolute residual=6, nonzero first slope=1 and nonzero second slope=2, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_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. - [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-conditioning-witness, [domain, operation, residual, slope_a, slope_b]] After derived = [numerr-conditioning-witness, Exact_numerical_error_density] Conditions: none Check THIS bank’s keys/operation/domain:legacy table underscores,MOD50 local hyphens;permissive BASE data acquires no unknown-field restriction.Keep Number.isInteger and SAFE source guards distinct,while preserving ORIGINAL JSON number representations. Step identifiers and dependenciesRule: R67_SOURCE_specific_keys_operation_domain_and_numeric_bank · Uses: G001 S002 · impliesBefore source roles = [conditioning ORIGINAL residual, 6, 1] After derived = [[6, 1], 6] Conditions: none MOD50 bounds the PRE-reduction numerator/denominator by±100million and requires nonzero denominator BEFORE reduction.Do not cross-cancel first to evade THIS source envelope.Out-of-envelope is a source-slice hold,not mathematical falsity. Step identifiers and dependenciesRule: Conditioning_SOURCE_naive_PREreduction_1e8_then_sign_gcd · Uses: G001, S001 S003 · calculationBefore conditioning ORIGINAL residual ORIGINAL numerator over nonzero denominator := div(6, 1) After conditioning ORIGINAL residual ORIGINAL numerator over nonzero denominator = 6 Conditions: none Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · impliesBefore source roles = [conditioning ORIGINAL slope A, 1, 1] After derived = [[1, 1], 1] Conditions: none MOD50 bounds the PRE-reduction numerator/denominator by±100million and requires nonzero denominator BEFORE reduction.Do not cross-cancel first to evade THIS source envelope.Out-of-envelope is a source-slice hold,not mathematical falsity. Step identifiers and dependenciesRule: Conditioning_SOURCE_naive_PREreduction_1e8_then_sign_gcd · Uses: G001, S003 S005 · calculationBefore conditioning ORIGINAL slope A ORIGINAL numerator over nonzero denominator := div(1, 1) After conditioning ORIGINAL slope A ORIGINAL numerator over nonzero denominator = 1 Conditions: none Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore source roles = [conditioning ORIGINAL slope B, 2, 1] After derived = [[2, 1], 2] Conditions: none MOD50 bounds the PRE-reduction numerator/denominator by±100million and requires nonzero denominator BEFORE reduction.Do not cross-cancel first to evade THIS source envelope.Out-of-envelope is a source-slice hold,not mathematical falsity. Step identifiers and dependenciesRule: Conditioning_SOURCE_naive_PREreduction_1e8_then_sign_gcd · Uses: G001, S005 S007 · calculationBefore conditioning ORIGINAL slope B ORIGINAL numerator over nonzero denominator := div(2, 1) After conditioning ORIGINAL slope B ORIGINAL numerator over nonzero denominator = 2 Conditions: none Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore source roles = [A, 1] After derived = [1, Yes] Conditions: none FIRST require a nonzero slope,then take its absolute value.A positive residual and DIFFERENT absolute slopes discriminate the two forward errors.Opposite slopes of equal magnitude do not qualify. Step identifiers and dependenciesRule: Conditioning_NONzero_absolute_slope_before_division · Uses: G001, S007 S009 · impliesBefore source roles = [conditioning error A SOURCE naive division, 6, 1] After derived = [[6, 1], 6] Conditions: none MOD50 bounds the PRE-reduction numerator/denominator by±100million and requires nonzero denominator BEFORE reduction.Do not cross-cancel first to evade THIS source envelope.Out-of-envelope is a source-slice hold,not mathematical falsity. Step identifiers and dependenciesRule: Conditioning_SOURCE_naive_PREreduction_1e8_then_sign_gcd · Uses: G001, S008 S010 · calculationBefore conditioning error A SOURCE naive division ORIGINAL numerator over nonzero denominator := div(6, 1) After conditioning error A SOURCE naive division ORIGINAL numerator over nonzero denominator = 6 Conditions: none Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore conditioning error A ORIGINAL residual divided by absolute slope := div(6, 1) After conditioning error A ORIGINAL residual divided by absolute slope = 6 Conditions: none Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · impliesBefore source roles = [B, 2] After derived = [2, Yes] Conditions: none FIRST require a nonzero slope,then take its absolute value.A positive residual and DIFFERENT absolute slopes discriminate the two forward errors.Opposite slopes of equal magnitude do not qualify. Step identifiers and dependenciesRule: Conditioning_NONzero_absolute_slope_before_division · Uses: G001, S011 S013 · impliesBefore source roles = [conditioning error B SOURCE naive division, 6, 2] After derived = [[3, 1], 3] Conditions: none MOD50 bounds the PRE-reduction numerator/denominator by±100million and requires nonzero denominator BEFORE reduction.Do not cross-cancel first to evade THIS source envelope.Out-of-envelope is a source-slice hold,not mathematical falsity. Step identifiers and dependenciesRule: Conditioning_SOURCE_naive_PREreduction_1e8_then_sign_gcd · Uses: G001, S012 S014 · calculationBefore conditioning error B SOURCE naive division ORIGINAL numerator over nonzero denominator := div(6, 2) After conditioning error B SOURCE naive division ORIGINAL numerator over nonzero denominator = 3 Conditions: none Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore conditioning error B ORIGINAL residual divided by absolute slope := div(6, 2) After conditioning error B ORIGINAL residual divided by absolute slope = 3 Conditions: none Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · impliesBefore source roles = [6, 1, 2] After derived = [6, 3, Yes] Conditions: none The affine identity residual=abs(slope)*forward gives forward=residual/abs(slope).The SAME positive residual has different forward errors under different model scales.This witness does not supply general condition-number theory or convergence. Step identifiers and dependenciesRule: Conditioning_SAME_positive_residual_DIFFERENT_affine_forward_errors · Uses: G001, S015 ## Reverse checks - S016 · premise_and_result_audit — Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof. - S015 · premise_and_result_audit — Recompute conditioning error B ORIGINAL residual divided by absolute slope with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute conditioning error B SOURCE naive division ORIGINAL numerator over nonzero denominator with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof. - S012 · premise_and_result_audit — Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof. - S011 · premise_and_result_audit — Recompute conditioning error A ORIGINAL residual divided by absolute slope with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute conditioning error A SOURCE naive division ORIGINAL numerator over nonzero denominator with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof. - S008 · premise_and_result_audit — Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof. - S007 · premise_and_result_audit — Recompute conditioning ORIGINAL slope B ORIGINAL numerator over nonzero denominator with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof. - S005 · premise_and_result_audit — Recompute conditioning ORIGINAL slope A ORIGINAL numerator over nonzero denominator with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof. - S003 · premise_and_result_audit — Recompute conditioning ORIGINAL residual ORIGINAL numerator over nonzero denominator with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof. - S001 · premise_and_result_audit — Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof. ## Result (6, 1, 6, 2, 3, Yes, same_residual_can_have_different_forward_error_under_different_conditioning) 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: fee3346b33b32a19ab5b6379dd3364de5e83c89f77b14199dec833bfa258bbac - candidate record SHA-256: ec46e15bf99ee2240fca485330c085dfce79eed379e78527fe4aa7613f4ff61f - semantic review receipt SHA-256: c89efa383b67bb2cf25bcf16160b9f5f614bbe5e5e791f4d7f58234d17a3371c - question-first witness SHA-256: 12766d451c5375842b4fc2843ae94807ec55f15763a916558026912a13873d57 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-pulse-20261119-r05-0056.json)