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R0011-06323 · q-20261009-advance-20261013-r01-pythagorean-theorem-0098
Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=23, b=24, candidate c=25. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.
G001 — Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=23, b=24, candidate c=25. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.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.
Before
Euclidean right-side model;positive leg a=23
After
use the supplied right-angle premise:a²+b²=c²
Conditions: none
Use the declared Euclidean right-side model with positive legs.This finite numerical check is not a new universal proof of the Pythagorean theorem or an arbitrary affine-geometric bridge.
Rule: pythagorean_right_side_premise · Uses: G001
Before
known leg a squared := square(23)
After
known leg a squared = 529
Conditions: none
Square the complete known positive leg a.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
leg b squared := square(24)
After
leg b squared = 576
Conditions: none
Square the other positive leg b separately.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
sum of leg squares := sum(529, 576)
After
sum of leg squares = 1105
Conditions: none
The sum of leg squares is the model's squared hypotenuse,not the hypotenuse itself.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
candidate c squared := square(25)
After
candidate c squared = 625
Conditions: none
A submitted candidate may be nonpositive;check its square separately from positive-length eligibility.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
candidate signed square residual := sub(625, 1105)
After
candidate signed square residual = -480
Conditions: none
The signed residual is c²−a²−b²;retain its sign.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
candidate c=25;positive=Yes;residual=-480
After
positive correct hypotenuse=No
Conditions: none
Require both positivity and residual0;for legs3,4,the candidate−5 has residual0 but is not a positive hypotenuse.
Rule: positive_hypotenuse_candidate_conjunction · Uses: G001, S006
S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S006 · premise_and_result_audit — Recompute candidate signed square residual with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute candidate c squared with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Recompute sum of leg squares with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute leg b squared with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute known leg a squared 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.(Yes, -480, No)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 0cf7828332994041b453f6c3ab321a3e56b529ceb6770ca778a418ed0711a4cccandidate record SHA-256: 27e91efd3b681d3a954094f47c6610caf0b8431a5eb7b90241739c56599ebc8csemantic review receipt SHA-256: b9bbc1c947f477a9c4bb5a579368ac5265f2da4572ce612c30f3633ecaf4ae80question-first witness SHA-256: 93f720df3ab3af13a792a3e6b443728e3c2a56276df839911f36569333236c55Lean: NOT_RUN · human review: NOT_PERFORMED