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Question 30584: Exact Root Offsets and Original Substitution

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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R0060-30584 · q-20261010-advance-20261014-r04-linear-equation-0059

Original question and givens

Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=9≠0, b=-4, ordered offsets=[0/1, 9/2, -8/3]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.

  • G001 — Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=9≠0, b=-4, ordered offsets=[0/1, 9/2, -8/3]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.

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-density14-linear-root-substitution-grid, density14_linear_root_substitution_grid, finite_Q_affine_row_protocol]

After

derived = [[root, ALL_candidates, ALL_original_residuals, ALL_exact_solution_flags]]

Conditions: none

density14/15 use FULL-prefix underscores;density17 uses LOCAL hyphens.The coordinate and circle frames differ.Retain THIS family ordered answer roles/types and all original bilingual JSON/status.

Step identifiers and dependencies

Rule: R60_EXACT_perbank_operation_domain_FRAME_and_output_roles · Uses: G001

S002 · calculation

Before

Linear grid base ORIGINAL negated constant := sub(0, -4)

After

Linear grid base ORIGINAL negated constant = 4

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

Linear grid base ORIGINAL negative b over a := div(4, 9)

After

Linear grid base ORIGINAL negative b over a = 4/9

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

Linear grid base reverse ORIGINAL a times root := mul(9, 4/9)

After

Linear grid base reverse ORIGINAL a times root = 4

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

Linear grid base reverse ORIGINAL residual plus b := sum(4, -4)

After

Linear grid base reverse ORIGINAL residual plus b = 0

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · implies

Before

source roles = [Linear grid base, 9, -4]

After

derived = [4/9, 0/1]

Conditions: none

ONLY nonzero a permits division:ax+b=0 IFF x=-b/a.Substitute into the ORIGINAL coefficients to verify zero residual.Normalize a negative a to a positive denominator;derived roots are not SAFE-capped or approximated.

Step identifiers and dependencies

Rule: Linear_NONZERO_a_exact_Q_root_AND_original_substitution · Uses: G001, S005

S007 · implies

Before

source roles = [Linear grid original offset0, [0, 1]]

After

derived = [0/1]

Conditions: none

Both RAW pair components are SAFE and the denominator is nonzero.Negative denominators normalize sign/gcd;derived fractions are not SAFE/1e6-capped.Integer fractions retain/1.

Step identifiers and dependencies

Rule: R60_RAW_SAFE_rational_pair_NONZERO_den_SIGN_normalized · Uses: G001, S006

S008 · implies

Before

source roles = [Linear grid original offset1, [9, 2]]

After

derived = [9/2]

Conditions: none

Both RAW pair components are SAFE and the denominator is nonzero.Negative denominators normalize sign/gcd;derived fractions are not SAFE/1e6-capped.Integer fractions retain/1.

Step identifiers and dependencies

Rule: R60_RAW_SAFE_rational_pair_NONZERO_den_SIGN_normalized · Uses: G001, S007

S009 · implies

Before

source roles = [Linear grid original offset2, [-8, 3]]

After

derived = [-8/3]

Conditions: none

Both RAW pair components are SAFE and the denominator is nonzero.Negative denominators normalize sign/gcd;derived fractions are not SAFE/1e6-capped.Integer fractions retain/1.

Step identifiers and dependencies

Rule: R60_RAW_SAFE_rational_pair_NONZERO_den_SIGN_normalized · Uses: G001, S008

S010 · calculation

Before

Linear grid ORIGINAL root plus offset0 := sum(4/9, 0)

After

Linear grid ORIGINAL root plus offset0 = 4/9

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · calculation

Before

Linear grid ORIGINAL a times candidate0 := mul(9, 4/9)

After

Linear grid ORIGINAL a times candidate0 = 4

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · calculation

Before

Linear grid ORIGINAL candidate0 residual plus b := sum(4, -4)

After

Linear grid ORIGINAL candidate0 residual plus b = 0

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · calculation

Before

Linear grid reverse a times offset0 := mul(9, 0)

After

Linear grid reverse a times offset0 = 0

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · calculation

Before

Linear grid ORIGINAL root plus offset1 := sum(4/9, 9/2)

After

Linear grid ORIGINAL root plus offset1 = 89/18

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · calculation

Before

Linear grid ORIGINAL a times candidate1 := mul(9, 89/18)

After

Linear grid ORIGINAL a times candidate1 = 89/2

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · calculation

Before

Linear grid ORIGINAL candidate1 residual plus b := sum(89/2, -4)

After

Linear grid ORIGINAL candidate1 residual plus b = 81/2

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · calculation

Before

Linear grid reverse a times offset1 := mul(9, 9/2)

After

Linear grid reverse a times offset1 = 81/2

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · calculation

Before

Linear grid ORIGINAL root plus offset2 := sum(4/9, -8/3)

After

Linear grid ORIGINAL root plus offset2 = -20/9

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · calculation

Before

Linear grid ORIGINAL a times candidate2 := mul(9, -20/9)

After

Linear grid ORIGINAL a times candidate2 = -20

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · calculation

Before

Linear grid ORIGINAL candidate2 residual plus b := sum(-20, -4)

After

Linear grid ORIGINAL candidate2 residual plus b = -24

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · calculation

Before

Linear grid reverse a times offset2 := mul(9, -8/3)

After

Linear grid reverse a times offset2 = -24

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · implies

Before

source roles = [9, -4, 4/9, [0/1, 9/2, -8/3]]

After

derived = [[4/9, 89/18, -20/9], [0/1, 81/2, -24/1], [Yes, No, No]]

Conditions: none

For EVERY candidate evaluate the ORIGINAL a*x+b,not nearness to a root or raw offset appearance.The reverse check is residual=a*offset;with a!=0,ONLY offset0 gives a solution.Normalize negative denominators first.

Step identifiers and dependencies

Rule: Linear_grid_EVERY_original_equation_residual_exact_zero_NOT_nearness · Uses: G001, S021

Reverse checks

  • S022 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate.
  • S021 · premise_and_result_audit — Recompute Linear grid reverse a times offset2 with exact rational arithmetic, not rounded decimals.
  • S020 · premise_and_result_audit — Recompute Linear grid ORIGINAL candidate2 residual plus b with exact rational arithmetic, not rounded decimals.
  • S019 · premise_and_result_audit — Recompute Linear grid ORIGINAL a times candidate2 with exact rational arithmetic, not rounded decimals.
  • S018 · premise_and_result_audit — Recompute Linear grid ORIGINAL root plus offset2 with exact rational arithmetic, not rounded decimals.
  • S017 · premise_and_result_audit — Recompute Linear grid reverse a times offset1 with exact rational arithmetic, not rounded decimals.
  • S016 · premise_and_result_audit — Recompute Linear grid ORIGINAL candidate1 residual plus b with exact rational arithmetic, not rounded decimals.
  • S015 · premise_and_result_audit — Recompute Linear grid ORIGINAL a times candidate1 with exact rational arithmetic, not rounded decimals.
  • S014 · premise_and_result_audit — Recompute Linear grid ORIGINAL root plus offset1 with exact rational arithmetic, not rounded decimals.
  • S013 · premise_and_result_audit — Recompute Linear grid reverse a times offset0 with exact rational arithmetic, not rounded decimals.
  • S012 · premise_and_result_audit — Recompute Linear grid ORIGINAL candidate0 residual plus b with exact rational arithmetic, not rounded decimals.
  • S011 · premise_and_result_audit — Recompute Linear grid ORIGINAL a times candidate0 with exact rational arithmetic, not rounded decimals.
  • S010 · premise_and_result_audit — Recompute Linear grid ORIGINAL root plus offset0 with exact rational arithmetic, not rounded decimals.
  • S009 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate.
  • S008 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate.
  • S007 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate.
  • S006 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate.
  • S005 · premise_and_result_audit — Recompute Linear grid base reverse ORIGINAL residual plus b with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Recompute Linear grid base reverse ORIGINAL a times root with exact rational arithmetic, not rounded decimals.
  • S003 · premise_and_result_audit — Recompute Linear grid base ORIGINAL negative b over a with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Recompute Linear grid base ORIGINAL negated constant with exact rational arithmetic, not rounded decimals.
  • S001 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate.

Result

(4/9, [4/9, 89/18, -20/9], [0/1, 81/2, -24/1], [Yes, No, No])

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: b38e287e87317ca22971db1a552da8ddaae1ee3d28b779efa4c2fc4651e55e89
  • candidate record SHA-256: c17a25c22a324f3a6db0cb102456915a40262827c4d5f683b785d8de80c2e488
  • semantic review receipt SHA-256: d6b5a409bf933281a46617cd715d8016fd5ca3d6fb76f77d179ac2ecd9d608bf
  • question-first witness SHA-256: f80ecf598c6893a580af018e1a82849b33ee6887efb48a6a1d69569aa17f12e4
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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