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Question 33959: An Affine Residual Equals Absolute Slope Times Forward Error

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

R0066-33959 · q-20261015-pulse-20261119-r05-0009

Original question and givens

Use nonzero slope=-2, intercept=9, approximation=4 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).

  • G001 — Use nonzero slope=-2, intercept=9, approximation=4 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).

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-affine-residual-forward, [approximation, domain, intercept, operation, slope]]

After

derived = [numerr-affine-residual-forward, 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 dependencies

Rule: R66_EXACT_source_bank_keys_local_operation_domain · Uses: G001

S002 · implies

Before

source roles = [error ORIGINAL slope, 100000000, -2, 1]

After

derived = [[-2, 1], -2]

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 dependencies

Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S001

S003 · calculation

Before

error ORIGINAL slope ORIGINAL numerator over denominator := div(-2, 1)

After

error ORIGINAL slope ORIGINAL numerator over denominator = -2

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · implies

Before

source roles = [error ORIGINAL intercept, 100000000, 9, 1]

After

derived = [[9, 1], 9]

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 dependencies

Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S003

S005 · calculation

Before

error ORIGINAL intercept ORIGINAL numerator over denominator := div(9, 1)

After

error ORIGINAL intercept ORIGINAL numerator over denominator = 9

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · implies

Before

source roles = [error ORIGINAL approximation, 100000000, 4, 1]

After

derived = [[4, 1], 4]

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 dependencies

Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S005

S007 · calculation

Before

error ORIGINAL approximation ORIGINAL numerator over denominator := div(4, 1)

After

error ORIGINAL approximation ORIGINAL numerator over denominator = 4

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · implies

Before

source roles = [Affine exact root SOURCE naive div, 100000000, -9, -2]

After

derived = [[9, 2], 9/2]

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 dependencies

Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S007

S009 · calculation

Before

Affine exact root SOURCE naive div ORIGINAL numerator over denominator := div(-9, -2)

After

Affine exact root SOURCE naive div ORIGINAL numerator over denominator = 9/2

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

Affine exact root ORIGINAL rational div := div(-9, -2)

After

Affine exact root ORIGINAL rational div = 9/2

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · implies

Before

source roles = [Affine residual original slope times approximation SOURCE naive mul, 100000000, -8, 1]

After

derived = [[-8, 1], -8]

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 dependencies

Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S010

S012 · calculation

Before

Affine residual original slope times approximation SOURCE naive mul ORIGINAL numerator over denominator := div(-8, 1)

After

Affine residual original slope times approximation SOURCE naive mul ORIGINAL numerator over denominator = -8

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · calculation

Before

Affine residual original slope times approximation ORIGINAL rational mul := mul(-2, 4)

After

Affine residual original slope times approximation ORIGINAL rational mul = -8

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · implies

Before

source roles = [Affine residual add original intercept SOURCE naive sum, 100000000, 1, 1]

After

derived = [[1, 1], 1]

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 dependencies

Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S013

S015 · calculation

Before

Affine residual add original intercept SOURCE naive sum ORIGINAL numerator over denominator := div(1, 1)

After

Affine residual add original intercept SOURCE naive sum ORIGINAL numerator over denominator = 1

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · calculation

Before

Affine residual add original intercept ORIGINAL rational sum := sum(-8, 9)

After

Affine residual add original intercept ORIGINAL rational sum = 1

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · implies

Before

source roles = [Affine forward original approximation minus exact root SOURCE naive sub, 100000000, -1, 2]

After

derived = [[-1, 2], -1/2]

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 dependencies

Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S016

S018 · calculation

Before

Affine forward original approximation minus exact root SOURCE naive sub ORIGINAL numerator over denominator := div(-1, 2)

After

Affine forward original approximation minus exact root SOURCE naive sub ORIGINAL numerator over denominator = -1/2

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · calculation

Before

Affine forward original approximation minus exact root ORIGINAL rational sub := sub(4, 9/2)

After

Affine forward original approximation minus exact root ORIGINAL rational sub = -1/2

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · implies

Before

source roles = [Affine absolute slope times forward error SOURCE naive mul, 100000000, 2, 2]

After

derived = [[1, 1], 1]

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 dependencies

Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S019

S021 · calculation

Before

Affine absolute slope times forward error SOURCE naive mul ORIGINAL numerator over denominator := div(2, 2)

After

Affine absolute slope times forward error SOURCE naive mul ORIGINAL numerator over denominator = 1

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · calculation

Before

Affine absolute slope times forward error ORIGINAL rational mul := mul(2, 1/2)

After

Affine absolute slope times forward error ORIGINAL rational mul = 1

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · implies

Before

source roles = [-2, 9, 4]

After

derived = [9/2, 1, 1, -1/2, 1/2, 1, Yes]

Conditions: none

For m≠0 first solve the TRUE root−b/m.Calculate |m*xhat+b| and |xhat−root| SEPARATELY.Since m*root+b=0,residual=|m|*forward.Occasional numerical equality does not identify the definitions;the supplied model gives the true root.

Step identifiers and dependencies

Rule: Error_AFFINE_exact_root_residual_forward_scale_identity · Uses: G001, S022

Reverse checks

  • S023 · 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.
  • S022 · premise_and_result_audit — Recompute Affine absolute slope times forward error ORIGINAL rational mul with exact rational arithmetic, not rounded decimals.
  • S021 · premise_and_result_audit — Recompute Affine absolute slope times forward error SOURCE naive mul ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals.
  • S020 · 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.
  • S019 · premise_and_result_audit — Recompute Affine forward original approximation minus exact root ORIGINAL rational sub with exact rational arithmetic, not rounded decimals.
  • S018 · premise_and_result_audit — Recompute Affine forward original approximation minus exact root SOURCE naive sub ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals.
  • S017 · 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.
  • S016 · premise_and_result_audit — Recompute Affine residual add original intercept ORIGINAL rational sum with exact rational arithmetic, not rounded decimals.
  • S015 · premise_and_result_audit — Recompute Affine residual add original intercept SOURCE naive sum ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals.
  • S014 · 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.
  • S013 · premise_and_result_audit — Recompute Affine residual original slope times approximation ORIGINAL rational mul with exact rational arithmetic, not rounded decimals.
  • S012 · premise_and_result_audit — Recompute Affine residual original slope times approximation SOURCE naive mul ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals.
  • S011 · 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.
  • S010 · premise_and_result_audit — Recompute Affine exact root ORIGINAL rational div with exact rational arithmetic, not rounded decimals.
  • S009 · premise_and_result_audit — Recompute Affine exact root SOURCE naive div ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals.
  • S008 · 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.
  • S007 · premise_and_result_audit — Recompute error ORIGINAL approximation 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 error ORIGINAL intercept 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 error ORIGINAL slope 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

(-2, 9, 4, 9/2, 1, 1/2, 1, Yes, residual_and_forward_error_are_distinct_quantities)

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: 00c1ab9ea8bd7f75c398b974689203308a0f1cdabef471f3ad3855ead1bda567
  • candidate record SHA-256: 073d20f12f17416c779f51fdcd17103064d124ded374c6c9939d41ae5358c683
  • semantic review receipt SHA-256: 99b4f114b2ec93164b6aa5681e0107033a240e7a3f0a8988633ba65320ede267
  • question-first witness SHA-256: d9bd3a1940da8a73f790ee3fe460c9a8523791bdbe72c8e0304e3740b66710cc
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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