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Question 30251: Absolute-Value Target Roots and Candidate Audit

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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R0059-30251 · q-20261010-advance-20261014-r06-absolute-value-0076

Original question and givens

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Target r=-1/1, ordered queries=[0/1, 1/1, 2/1]. Give (is r nonnegative?, increasing roots of |x|=r, three absolute query values, three equality-to-r values).

  • G001 — Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Target r=-1/1, ordered queries=[0/1, 1/1, 2/1]. Give (is r nonnegative?, increasing roots of |x|=r, three absolute query values, three equality-to-r values).

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-absolute-preimage-rational-audit, density14_absolute_preimage_rational_audit, finite_rational_absolute_protocol]

After

derived = [[target_nonnegative, ALL_increasing_roots, ALL_absolute_query_values, ALL_equal_target_flags]]

Conditions: none

Retain the ORIGINAL bank full operation prefix,domain and ordered answer roles.Use orthogonal_equal_unit only where a frame is declared,and degrees where an angle unit is declared.Integer fractions retain/1.Do not invent parameters or alter original bilingual sources.

Step identifiers and dependencies

Rule: R59_EXACT_family_domain_FULLprefix_operation_and_answer_roles · Uses: G001

S002 · implies

Before

source roles = [Absolute target r, [-1, 1]]

After

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

Conditions: none

Require a RAW SAFE pair with nonzero denominator.This bank ALLOWS negative denominators;normalize sign/gcd BEFORE deciding the value sign.Do not import a primitive positive-denominator rule or a STR22 component cap.

Step identifiers and dependencies

Rule: Absolute_RAW_SAFE_pair_NONZERO_den_normalize_SIGN_and_gcd · Uses: G001, S001

S003 · implies

Before

source roles = [Absolute ORIGINAL queries slot0, [0, 1]]

After

derived = [0/1, 0, 1]

Conditions: none

Require a RAW SAFE pair with nonzero denominator.This bank ALLOWS negative denominators;normalize sign/gcd BEFORE deciding the value sign.Do not import a primitive positive-denominator rule or a STR22 component cap.

Step identifiers and dependencies

Rule: Absolute_RAW_SAFE_pair_NONZERO_den_normalize_SIGN_and_gcd · Uses: G001, S002

S004 · implies

Before

source roles = [Absolute ORIGINAL queries slot1, [1, 1]]

After

derived = [1/1, 1, 1]

Conditions: none

Require a RAW SAFE pair with nonzero denominator.This bank ALLOWS negative denominators;normalize sign/gcd BEFORE deciding the value sign.Do not import a primitive positive-denominator rule or a STR22 component cap.

Step identifiers and dependencies

Rule: Absolute_RAW_SAFE_pair_NONZERO_den_normalize_SIGN_and_gcd · Uses: G001, S003

S005 · implies

Before

source roles = [Absolute ORIGINAL queries slot2, [2, 1]]

After

derived = [2/1, 2, 1]

Conditions: none

Require a RAW SAFE pair with nonzero denominator.This bank ALLOWS negative denominators;normalize sign/gcd BEFORE deciding the value sign.Do not import a primitive positive-denominator rule or a STR22 component cap.

Step identifiers and dependencies

Rule: Absolute_RAW_SAFE_pair_NONZERO_den_normalize_SIGN_and_gcd · Uses: G001, S004

S006 · implies

Before

source roles = [Absolute ORIGINAL queries slot0, 0]

After

derived = [Yes, 0]

Conditions: none

Choose the branch by the NORMALIZED value:x>=0 keeps x;for x<0 use0-x.Zero is in the nonnegative branch;do not misclassify a raw negative denominator.

Step identifiers and dependencies

Rule: Absolute_value_normalized_sign_branch_ZERO_is_nonnegative · Uses: G001, S005

S007 · implies

Before

source roles = [Absolute ORIGINAL queries slot1, 1]

After

derived = [Yes, 1]

Conditions: none

Choose the branch by the NORMALIZED value:x>=0 keeps x;for x<0 use0-x.Zero is in the nonnegative branch;do not misclassify a raw negative denominator.

Step identifiers and dependencies

Rule: Absolute_value_normalized_sign_branch_ZERO_is_nonnegative · Uses: G001, S006

S008 · implies

Before

source roles = [Absolute ORIGINAL queries slot2, 2]

After

derived = [Yes, 2]

Conditions: none

Choose the branch by the NORMALIZED value:x>=0 keeps x;for x<0 use0-x.Zero is in the nonnegative branch;do not misclassify a raw negative denominator.

Step identifiers and dependencies

Rule: Absolute_value_normalized_sign_branch_ZERO_is_nonnegative · Uses: G001, S007

S009 · calculation

Before

Absolute ORIGINAL magnitude minus target slot0 := sub(0, -1)

After

Absolute ORIGINAL magnitude minus target slot0 = 1

Conditions: none

Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

Absolute ORIGINAL magnitude minus target slot1 := sub(1, -1)

After

Absolute ORIGINAL magnitude minus target slot1 = 2

Conditions: none

Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · calculation

Before

Absolute ORIGINAL magnitude minus target slot2 := sub(2, -1)

After

Absolute ORIGINAL magnitude minus target slot2 = 3

Conditions: none

Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · implies

Before

source roles = [-1/1, [0/1, 1/1, 2/1]]

After

derived = [No, [], [0/1, 1/1, 2/1], [No, No, No]]

Conditions: none

Since |x|>=0,a negative target is a LEGAL No with an empty root list.At0 the ONLY root is0/1;a positive target gives increasing-r,r.STILL compute all three ORIGINAL absolute query values and masks;do not reject the negative target or deduplicate query positions.

Step identifiers and dependencies

Rule: Absolute_preimage_NEGATIVE_target_LEGAL_EMPTY_roots_ZERO_singleton · Uses: G001, S011

Reverse checks

  • S012 · premise_and_result_audit — Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate.
  • S011 · premise_and_result_audit — Recompute Absolute ORIGINAL magnitude minus target slot2 with exact rational arithmetic, not rounded decimals.
  • S010 · premise_and_result_audit — Recompute Absolute ORIGINAL magnitude minus target slot1 with exact rational arithmetic, not rounded decimals.
  • S009 · premise_and_result_audit — Recompute Absolute ORIGINAL magnitude minus target slot0 with exact rational arithmetic, not rounded decimals.
  • S008 · premise_and_result_audit — Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate.
  • S007 · premise_and_result_audit — Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate.
  • S006 · premise_and_result_audit — Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate.
  • S005 · premise_and_result_audit — Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate.
  • S004 · premise_and_result_audit — Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate.
  • S003 · premise_and_result_audit — Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate.
  • S002 · premise_and_result_audit — Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate.
  • S001 · premise_and_result_audit — Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate.

Result

(No, [], [0/1, 1/1, 2/1], [No, 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: d6f79f8299c25af316552b731f4eaa4b8b41363cfdee7107c6a6850f3b4626b6
  • candidate record SHA-256: ef76db710b854c06074107b9eb504c1ae25d12334c85a7110737cafa6902b646
  • semantic review receipt SHA-256: 4c93dea3956f58509e20b3b8bbbf716dbc340b424a2a5177bc04aba9e8ea9ec5
  • question-first witness SHA-256: a5e56e0d64d622d7202840cef7019839c5a99244c602694190779155207dafdf
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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