Question 30227: Inclusive Absolute-Value Threshold Mask

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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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R0059-30227 · q-20261010-advance-20261014-r06-absolute-value-0052

Original question and givens

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. r=1/2≥0, ordered points=[-1/2, 1/2, 3/2, 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=1/2≥0, ordered points=[-1/2, 1/2, 3/2, 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.

mko-absolute-value

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, [1, 2]]

After

derived = [1/2, 1, 2]

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, [-1, 2]]

After

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

Conditions: none

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, [1, 2]]

After

derived = [1/2, 1, 2]

Conditions: none

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, [3, 2]]

After

derived = [3/2, 3, 2]

Conditions: none

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

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, -1/2)

After

Absolute ORIGINAL points slot0 ORIGINAL negative branch negation = 1/2

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, S006

S008 · implies

Before

source roles = [Absolute ORIGINAL points slot0, -1/2]

After

derived = [No, 1/2]

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 · implies

Before

source roles = [Absolute ORIGINAL points slot1, 1/2]

After

derived = [Yes, 1/2]

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, S008

S010 · implies

Before

source roles = [Absolute ORIGINAL points slot2, 3/2]

After

derived = [Yes, 3/2]

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, 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(1/2, 1/2)

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(1/2, 1/2)

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(3/2, 1/2)

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, 1/2)

After

Absolute ORIGINAL magnitude minus target slot3 = -1/2

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, S014

S016 · implies

Before

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

After

derived = [[1/2, 1/2, 3/2, 0/1], [0/1, 0/1, 1/1, -1/2], [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

(1/2, [1/2, 1/2, 3/2, 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: 728cfd1a7636706a251d5540fbb76f23eb7ac149baab1d015cf0e723c7420d42
candidate record SHA-256: 398370330dd4241a950f67657ea52f840db72eb23dea37b1eb5438fdcef64462
semantic review receipt SHA-256: 4c93dea3956f58509e20b3b8bbbf716dbc340b424a2a5177bc04aba9e8ea9ec5
question-first witness SHA-256: a5e56e0d64d622d7202840cef7019839c5a99244c602694190779155207dafdf
Lean: NOT_RUN · human review: NOT_PERFORMED

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