Question 30551: Parameter-Dependent Linear and Constant Cases

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0060-30551 · q-20261010-advance-20261014-r04-linear-equation-0026

Original question and givens

Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=0, b=0, ordered ts=[0, 1, -1]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.

G001 — Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=0, b=0, ordered ts=[0, 1, -1]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.

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-linear-equation

Step-by-step derivation

S001 · implies

Before

source roles = [qf-density14-linear-parameter-slice, density14_linear_parameter_slice, finite_Q_affine_row_protocol]

After

derived = [[ALL_derived_coefficients, ALL_linear_flags, ALL_solution_classes]]

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 parameter ORIGINAL t0 minus k := sub(0, 0)

After

Linear parameter ORIGINAL t0 minus k = 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, S001

S003 · implies

Before

source roles = [Linear parameter original slot0, 0, 0]

After

derived = [No, all Q]

Conditions: none

At a=0 this is NOT linear:b=0 gives0=0 and all Q solutions;nonzero b is a contradictory constant equation with no solution.Do not divide by0.

Step identifiers and dependencies

Rule: Linear_a0_CONSTANT_equation_separate_ALLQ_vs_NOsolution · Uses: G001, S002

S004 · calculation

Before

Linear parameter ORIGINAL t1 minus k := sub(1, 0)

After

Linear parameter ORIGINAL t1 minus k = 1

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 parameter original slot1 ORIGINAL negated constant := sub(0, 0)

After

Linear parameter original slot1 ORIGINAL negated constant = 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 · calculation

Before

Linear parameter original slot1 ORIGINAL negative b over a := div(0, 1)

After

Linear parameter original slot1 ORIGINAL negative b over a = 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, S005

S007 · calculation

Before

Linear parameter original slot1 reverse ORIGINAL a times root := mul(1, 0)

After

Linear parameter original slot1 reverse ORIGINAL a times root = 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, S006

S008 · calculation

Before

Linear parameter original slot1 reverse ORIGINAL residual plus b := sum(0, 0)

After

Linear parameter original slot1 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, S007

S009 · implies

Before

source roles = [Linear parameter original slot1, 1, 0]

After

derived = [0/1, 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, S008

S010 · calculation

Before

Linear parameter ORIGINAL t2 minus k := sub(-1, 0)

After

Linear parameter ORIGINAL t2 minus k = -1

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 parameter original slot2 ORIGINAL negated constant := sub(0, 0)

After

Linear parameter original slot2 ORIGINAL negated constant = 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, S010

S012 · calculation

Before

Linear parameter original slot2 ORIGINAL negative b over a := div(0, -1)

After

Linear parameter original slot2 ORIGINAL negative b over a = 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 parameter original slot2 reverse ORIGINAL a times root := mul(-1, 0)

After

Linear parameter original slot2 reverse ORIGINAL a times root = 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 parameter original slot2 reverse ORIGINAL residual plus b := sum(0, 0)

After

Linear parameter original slot2 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, S013

S015 · implies

Before

source roles = [Linear parameter original slot2, -1, 0]

After

derived = [0/1, 0/1]

Conditions: none

Step identifiers and dependencies

Rule: Linear_NONZERO_a_exact_Q_root_AND_original_substitution · Uses: G001, S014

S016 · implies

Before

source roles = [0, 0, [0, 1, -1]]

After

derived = [[0, 1, -1], [No, Yes, Yes], [all Q, single solution 0/1, single solution 0/1]]

Conditions: none

First compute a_i=t_i-k at EACH original position;derived coefficients may exceed SAFE.Then distinguish nonzero linear and zero-coefficient constant equations.Retain all three flags/classes without sorting/deduplicating parameters.

Step identifiers and dependencies

Rule: Linear_PARAMETER_derived_a_not_revalidated_RAW_and_ALL_classes · Uses: G001, S015

Reverse checks

S016 · 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.
S015 · 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.
S014 · premise_and_result_audit — Recompute Linear parameter original slot2 reverse ORIGINAL residual plus b with exact rational arithmetic, not rounded decimals.
S013 · premise_and_result_audit — Recompute Linear parameter original slot2 reverse ORIGINAL a times root with exact rational arithmetic, not rounded decimals.
S012 · premise_and_result_audit — Recompute Linear parameter original slot2 ORIGINAL negative b over a with exact rational arithmetic, not rounded decimals.
S011 · premise_and_result_audit — Recompute Linear parameter original slot2 ORIGINAL negated constant with exact rational arithmetic, not rounded decimals.
S010 · premise_and_result_audit — Recompute Linear parameter ORIGINAL t2 minus k 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 — Recompute Linear parameter original slot1 reverse ORIGINAL residual plus b with exact rational arithmetic, not rounded decimals.
S007 · premise_and_result_audit — Recompute Linear parameter original slot1 reverse ORIGINAL a times root with exact rational arithmetic, not rounded decimals.
S006 · premise_and_result_audit — Recompute Linear parameter original slot1 ORIGINAL negative b over a with exact rational arithmetic, not rounded decimals.
S005 · premise_and_result_audit — Recompute Linear parameter original slot1 ORIGINAL negated constant with exact rational arithmetic, not rounded decimals.
S004 · premise_and_result_audit — Recompute Linear parameter ORIGINAL t1 minus k with exact rational arithmetic, not rounded decimals.
S003 · 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.
S002 · premise_and_result_audit — Recompute Linear parameter ORIGINAL t0 minus k 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

([0, 1, -1], [No, Yes, Yes], [all Q, single-solution 0/1, single-solution 0/1])

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: 607699b752bada33b7182feae6f41ceae3a618a470a32f76885d151e43d8a875
candidate record SHA-256: 3e2ca28139397ec69048af67120e0b90a70863e0301fbb97c20dcbca80a89d0d
semantic review receipt SHA-256: d6b5a409bf933281a46617cd715d8016fd5ca3d6fb76f77d179ac2ecd9d608bf
question-first witness SHA-256: f80ecf598c6893a580af018e1a82849b33ee6887efb48a6a1d69569aa17f12e4
Lean: NOT_RUN · human review: NOT_PERFORMED

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