# Question 30396: Angle-Split Eligibility and Signed Remainder

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0059-30396 · q-20261010-advance-20261014-r01-angle-0021



## Original question and givens

Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. total=90, candidate first=69. Give (is first in [0,total]?, total−first). For an outside candidate still compute the signed arithmetic remainder, without claiming a constructed angle.


- G001 — Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. total=90, candidate first=69. Give (is first in [0,total]?, total−first). For an outside candidate still compute the signed arithmetic remainder, without claiming a constructed angle.





## 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-angle](https://ocme.evemisslab.com/en/mko/mko-angle/)





## Step-by-step derivation

S001 · impliesBefore

source roles = [qf-density14-angle-split-candidate, density14_angle_split_candidate, finite_nondirected_degree_protocol]

After

derived = [[candidate_in_closed_range, SIGNED_remainder]]

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 dependenciesRule: R59_EXACT_family_domain_FULLprefix_operation_and_answer_roles · Uses: G001





S002 · calculationBefore

Angle ORIGINAL SIGNED total minus first := sub(90, 69)

After

Angle ORIGINAL SIGNED total minus first = 21

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S001





S003 · impliesBefore

source roles = [90, 69]

After

derived = [Yes, 21]

Conditions: none

The first candidate may be negative or exceed total;STILL return No and SIGNED total-first.Only a qualified candidate admits an angle-split interpretation.Do not clip or pre-reject it.

Step identifiers and dependenciesRule: Angle_split_OUTSIDE_candidate_LEGAL_No_signed_remainder · Uses: G001, S002









## Reverse checks


- 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 — Recompute Angle ORIGINAL SIGNED total minus first with exact rational arithmetic, not rounded decimals.
- 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

(Yes, 21)

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: f4555964f0ffb2d4aca0d1422e86e8dbca9d39eec2cea3d9037ac2831c0d25f1
- candidate record SHA-256: c198de64f600713916dc126196495aeae039a314f628fe08f69de46ce3408d03
- semantic review receipt SHA-256: 4c93dea3956f58509e20b3b8bbbf716dbc340b424a2a5177bc04aba9e8ea9ec5
- question-first witness SHA-256: a5e56e0d64d622d7202840cef7019839c5a99244c602694190779155207dafdf
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261010-advance-20261014-r01-angle-0021.json)