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Question 30246: Inclusive Absolute-Value Threshold Mask

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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R0059-30246 · q-20261010-advance-20261014-r06-absolute-value-0071

Original question and givens

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. r=20/2≥0, ordered points=[-20/2, 20/2, 22/2, 0/1]. Give (r, four absolute values, four abs(point)≤r eligibility values), including endpoints.

  • G001 — Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. r=20/2≥0, ordered points=[-20/2, 20/2, 22/2, 0/1]. Give (r, four absolute values, four abs(point)≤r eligibility values), including endpoints.

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-closed-radius-mask, density14_absolute_closed_radius_mask, finite_rational_absolute_protocol]

After

derived = [[radius, ALL_absolute_values, ALL_inclusive_threshold_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, [20, 2]]

After

derived = [10/1, 10, 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 points slot0, [-20, 2]]

After

derived = [-10/1, -10, 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 points slot1, [20, 2]]

After

derived = [10/1, 10, 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 points slot2, [22, 2]]

After

derived = [11/1, 11, 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 points slot3, [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, S005

S007 · calculation

Before

Absolute ORIGINAL points slot0 ORIGINAL negative branch negation := sub(0, -10)

After

Absolute ORIGINAL points slot0 ORIGINAL negative branch negation = 10

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, S006

S008 · implies

Before

source roles = [Absolute ORIGINAL points slot0, -10]

After

derived = [No, 10]

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 · implies

Before

source roles = [Absolute ORIGINAL points slot1, 10]

After

derived = [Yes, 10]

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, S008

S010 · implies

Before

source roles = [Absolute ORIGINAL points slot2, 11]

After

derived = [Yes, 11]

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, S009

S011 · implies

Before

source roles = [Absolute ORIGINAL points slot3, 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, S010

S012 · calculation

Before

Absolute ORIGINAL magnitude minus target slot0 := sub(10, 10)

After

Absolute ORIGINAL magnitude minus target slot0 = 0

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, S011

S013 · calculation

Before

Absolute ORIGINAL magnitude minus target slot1 := sub(10, 10)

After

Absolute ORIGINAL magnitude minus target slot1 = 0

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, S012

S014 · calculation

Before

Absolute ORIGINAL magnitude minus target slot2 := sub(11, 10)

After

Absolute ORIGINAL magnitude minus target slot2 = 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, S013

S015 · calculation

Before

Absolute ORIGINAL magnitude minus target slot3 := sub(0, 10)

After

Absolute ORIGINAL magnitude minus target slot3 = -10

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, S014

S016 · implies

Before

source roles = [10/1, [-10/1, 10/1, 11/1, 0/1]]

After

derived = [[10/1, 10/1, 11/1, 0/1], [0/1, 0/1, 1/1, -10/1], [Yes, Yes, No, Yes]]

Conditions: none

The radius must be nonnegative,and abs(point)<=r INCLUDES equality.At r=0 only a zero value qualifies.Retain ALL four original positions;this finite mask is not a general inequality solution.

Step identifiers and dependencies

Rule: Absolute_CLOSED_radius_FOUR_original_positions_INCLUSIVE · Uses: G001, S015

Reverse checks

  • S016 · 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.
  • S015 · premise_and_result_audit — Recompute Absolute ORIGINAL magnitude minus target slot3 with exact rational arithmetic, not rounded decimals.
  • S014 · premise_and_result_audit — Recompute Absolute ORIGINAL magnitude minus target slot2 with exact rational arithmetic, not rounded decimals.
  • S013 · premise_and_result_audit — Recompute Absolute ORIGINAL magnitude minus target slot1 with exact rational arithmetic, not rounded decimals.
  • S012 · premise_and_result_audit — Recompute Absolute ORIGINAL magnitude minus target slot0 with exact rational arithmetic, not rounded decimals.
  • S011 · 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.
  • S010 · 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.
  • S009 · 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.
  • 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 — Recompute Absolute ORIGINAL points slot0 ORIGINAL negative branch negation with exact rational arithmetic, not rounded decimals.
  • 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

(10/1, [10/1, 10/1, 11/1, 0/1], [Yes, Yes, No, Yes])

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: a3718d8efa0a737507f7ed75cd908e41b01992e9dd690aee2262e8727662c9e6
  • candidate record SHA-256: bb8711efdc5d34ea134cd791cba0ba72a366f51d7540ae05253a6405210d0565
  • semantic review receipt SHA-256: 4c93dea3956f58509e20b3b8bbbf716dbc340b424a2a5177bc04aba9e8ea9ec5
  • question-first witness SHA-256: a5e56e0d64d622d7202840cef7019839c5a99244c602694190779155207dafdf
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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