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Question 34012: The Same Residual Can Correspond to Different Forward Errors

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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← Question derivation index · Original question batch

R0067-34012 · q-20261015-pulse-20261119-r05-0062

Original question and givens

Use positive absolute residual=12, nonzero first slope=1 and nonzero second slope=3, 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=12, nonzero first slope=1 and nonzero second slope=3, 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.

Step-by-step derivation

S001 · implies

Before

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 dependencies

Rule: R67_SOURCE_specific_keys_operation_domain_and_numeric_bank · Uses: G001

S002 · implies

Before

source roles = [conditioning ORIGINAL residual, 12, 1]

After

derived = [[12, 1], 12]

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 dependencies

Rule: Conditioning_SOURCE_naive_PREreduction_1e8_then_sign_gcd · Uses: G001, S001

S003 · calculation

Before

conditioning ORIGINAL residual ORIGINAL numerator over nonzero denominator := div(12, 1)

After

conditioning ORIGINAL residual ORIGINAL numerator over nonzero denominator = 12

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · implies

Before

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 dependencies

Rule: Conditioning_SOURCE_naive_PREreduction_1e8_then_sign_gcd · Uses: G001, S003

S005 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · implies

Before

source roles = [conditioning ORIGINAL slope B, 3, 1]

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 dependencies

Rule: Conditioning_SOURCE_naive_PREreduction_1e8_then_sign_gcd · Uses: G001, S005

S007 · calculation

Before

conditioning ORIGINAL slope B ORIGINAL numerator over nonzero denominator := div(3, 1)

After

conditioning ORIGINAL slope B 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · implies

Before

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 dependencies

Rule: Conditioning_NONzero_absolute_slope_before_division · Uses: G001, S007

S009 · implies

Before

source roles = [conditioning error A SOURCE naive division, 12, 1]

After

derived = [[12, 1], 12]

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 dependencies

Rule: Conditioning_SOURCE_naive_PREreduction_1e8_then_sign_gcd · Uses: G001, S008

S010 · calculation

Before

conditioning error A SOURCE naive division ORIGINAL numerator over nonzero denominator := div(12, 1)

After

conditioning error A SOURCE naive division ORIGINAL numerator over nonzero denominator = 12

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · calculation

Before

conditioning error A ORIGINAL residual divided by absolute slope := div(12, 1)

After

conditioning error A ORIGINAL residual divided by absolute slope = 12

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · implies

Before

source roles = [B, 3]

After

derived = [3, 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 dependencies

Rule: Conditioning_NONzero_absolute_slope_before_division · Uses: G001, S011

S013 · implies

Before

source roles = [conditioning error B SOURCE naive division, 12, 3]

After

derived = [[4, 1], 4]

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 dependencies

Rule: Conditioning_SOURCE_naive_PREreduction_1e8_then_sign_gcd · Uses: G001, S012

S014 · calculation

Before

conditioning error B SOURCE naive division ORIGINAL numerator over nonzero denominator := div(12, 3)

After

conditioning error B SOURCE naive division ORIGINAL numerator over nonzero denominator = 4

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · calculation

Before

conditioning error B ORIGINAL residual divided by absolute slope := div(12, 3)

After

conditioning error B ORIGINAL residual divided by absolute slope = 4

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · implies

Before

source roles = [12, 1, 3]

After

derived = [12, 4, 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 dependencies

Rule: 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

(12, 1, 12, 3, 4, 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: 3b6614e87af1d84e6a81d38fb9c1d707e9c94afc3a4a7e3f4506ac67ebc3e4fd
  • candidate record SHA-256: 3d4ac5cf3b0cf653cf307352d455c724439177e0b3defcbad5413647207ee3e6
  • semantic review receipt SHA-256: c89efa383b67bb2cf25bcf16160b9f5f614bbe5e5e791f4d7f58234d17a3371c
  • question-first witness SHA-256: 12766d451c5375842b4fc2843ae94807ec55f15763a916558026912a13873d57
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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