# Question 5853: Exact π-Multiple Candidate without Periodic Wrapping

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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.

R0010-05853 · q-20261008-advance-20261012-r05-right-triangle-0078



## Original question and givens

Candidate θ=(0/1)π radians. Give (exact degree value, does it equal exactly 90°?), without taking a modulus or wrapping a period.


- G001 — Candidate θ=(0/1)π radians. Give (exact degree value, does it equal exactly 90°?), without taking a modulus or wrapping a period.





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





## Step-by-step derivation

S001 · impliesBefore

angle unit=pi_radians;exact declared protocol

After

check qualifications before geometric conclusions;no tolerance

Conditions: none

Use the declared exact angle protocol,not a sketch or floating tolerance.Candidate angle data do not by themselves construct a triangle.

Step identifiers and dependenciesRule: exact_right_angle_protocol · Uses: G001





S002 · calculationBefore

exact pi coefficient := div(0, 1)

After

exact pi coefficient = 0

Conditions: none

For θ=(n/d)π,first compute its exact rational coefficient with a positive nonzero denominator.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001





S003 · calculationBefore

unwrapped degree value := mul(180, 0)

After

unwrapped degree value = 0

Conditions: none

Convert using π radians=180 degrees,without a floating value for π.

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





S004 · impliesBefore

θ=0/1π;degrees=0

After

exactly90°=No

Conditions: none

Test whether the original angle is exactly90 degrees;do not reduce450 degrees or a negative angle modulo a full turn.

Step identifiers and dependenciesRule: unwrapped_pi_right_angle_test · Uses: G001, S003









## Reverse checks


- S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S003 · premise_and_result_audit — Recompute unwrapped degree value with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Recompute exact pi coefficient with exact rational arithmetic, not rounded decimals.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

(0/1, 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: f27ddd9dbe186bf58ed9c83ce30ce903f893242f1c3c47b4a62901f8f8834aaf
- candidate record SHA-256: 7c5160fbfcfb494824c07ef04cdaad42cb91878bc7aad8e4f42e58af2ba5d9e0
- semantic review receipt SHA-256: 0328166a2e335aac7a5fb34556e68c80ba7d16612e822b3999c8ac348a9ca06f
- question-first witness SHA-256: 0da18e4e6cf71ece5ec5f00dd0cd161217b6d251672bc0b46d79fe7e6253f5fc
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261008-advance-20261012-r05-right-triangle-0078.json)