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Question 5873: Exact π-Multiple Candidate without Periodic Wrapping

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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R0010-05873 · q-20261008-advance-20261012-r05-right-triangle-0098

Original question and givens

Candidate θ=(19/7)π radians. Give (exact degree value, does it equal exactly 90°?), without taking a modulus or wrapping a period.

  • G001 — Candidate θ=(19/7)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.

Step-by-step derivation

S001 · implies

Before

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 dependencies

Rule: exact_right_angle_protocol · Uses: G001

S002 · calculation

Before

exact pi coefficient := div(19, 7)

After

exact pi coefficient = 19/7

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

unwrapped degree value := mul(180, 19/7)

After

unwrapped degree value = 3420/7

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · implies

Before

θ=19/7π;degrees=3420/7

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 dependencies

Rule: 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

(3420/7, 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: 65a6554dc679c257128e5da79854f41773f5b370facb7b39226ded695842f292
  • candidate record SHA-256: 13a95eb107ca9ad805c5446153a5ebf008c4332c98cdd2858390ccfd71e65c46
  • semantic review receipt SHA-256: 0328166a2e335aac7a5fb34556e68c80ba7d16612e822b3999c8ac348a9ca06f
  • question-first witness SHA-256: 0da18e4e6cf71ece5ec5f00dd0cd161217b6d251672bc0b46d79fe7e6253f5fc
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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