{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0001","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"abd48781924ae6559eb3e5ba7ba2689e54a98c93f581d65c09c0e6e46680dc0a","family_id":"qf-density13-pythagorean-leg-square-sum","seed":202610131001,"semantic_template":"qf-density13-pythagorean-leg-square-sum/[\"1\",\"1\"]"},"fields":{"title":"Ordered Leg Squares and Hypotenuse Square","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=1, b=1. Give (a², b², a²+b²), retaining leg order; the last field is the model’s hypotenuse square.","answer":{"kind":"expression","value":"(1, 1, 2)","explanation":"Square both positive legs separately and add; this finite side model is not a new universal proof or a completed affine geometry bridge. The exact result, retaining the requested order, is (1, 1, 2)."},"skills":["Ordered Leg Squares and Hypotenuse Square"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0002","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"abdc42e7798c484e4256f762547b4f70665522b093aee750921050acb2eae088","family_id":"qf-density13-pythagorean-leg-square-sum","seed":202610131002,"semantic_template":"qf-density13-pythagorean-leg-square-sum/[\"9\",\"16\"]"},"fields":{"title":"Ordered Leg Squares and Hypotenuse Square","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=3, b=4. Give (a², b², a²+b²), retaining leg order; the last field is the model’s hypotenuse square.","answer":{"kind":"expression","value":"(9, 16, 25)","explanation":"Square both positive legs separately and add; this finite side model is not a new universal proof or a completed affine geometry bridge. The exact result, retaining the requested order, is (9, 16, 25)."},"skills":["Ordered Leg Squares and Hypotenuse Square"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0003","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d395c22d61a17db0c0057e2c7f95ab065f22f4550f2f5ecda6cb8fb40d865f0","family_id":"qf-density13-pythagorean-leg-square-sum","seed":202610131003,"semantic_template":"qf-density13-pythagorean-leg-square-sum/[\"16\",\"16\"]"},"fields":{"title":"Ordered Leg Squares and Hypotenuse Square","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=4, b=4. Give (a², b², a²+b²), retaining leg order; the last field is the model’s hypotenuse square.","answer":{"kind":"expression","value":"(16, 16, 32)","explanation":"Square both positive legs separately and add; this finite side model is not a new universal proof or a completed affine geometry bridge. The exact result, retaining the requested order, is (16, 16, 32)."},"skills":["Ordered Leg Squares and Hypotenuse Square"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0004","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"623c845fc81d0a95cac4a81903eec49b9d13436ce3c4cc54e15e853eb8590034","family_id":"qf-density13-pythagorean-leg-square-sum","seed":202610131004,"semantic_template":"qf-density13-pythagorean-leg-square-sum/[\"25\",\"25\"]"},"fields":{"title":"Ordered Leg Squares and Hypotenuse Square","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=5, b=5. Give (a², b², a²+b²), retaining leg order; the last field is the model’s hypotenuse square.","answer":{"kind":"expression","value":"(25, 25, 50)","explanation":"Square both positive legs separately and add; this finite side model is not a new universal proof or a completed affine geometry bridge. The exact result, retaining the requested order, is (25, 25, 50)."},"skills":["Ordered Leg Squares and Hypotenuse Square"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0005","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8961789ecd8fafb5bae311d8b25a4d66745887d90f042acf996bbfcd0c43d1eb","family_id":"qf-density13-pythagorean-leg-square-sum","seed":202610131005,"semantic_template":"qf-density13-pythagorean-leg-square-sum/[\"36\",\"36\"]"},"fields":{"title":"Ordered Leg Squares and Hypotenuse Square","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=6, b=6. Give (a², b², a²+b²), retaining leg order; the last field is the model’s hypotenuse square.","answer":{"kind":"expression","value":"(36, 36, 72)","explanation":"Square both positive legs separately and add; this finite side model is not a new universal proof or a completed affine geometry bridge. The exact result, retaining the requested order, is (36, 36, 72)."},"skills":["Ordered Leg Squares and Hypotenuse Square"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0006","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b8222f5be4b6e6cf2912b2b09893fddd51b91729cf337ac8e27bf9c09f9b119b","family_id":"qf-density13-pythagorean-leg-square-sum","seed":202610131006,"semantic_template":"qf-density13-pythagorean-leg-square-sum/[\"49\",\"49\"]"},"fields":{"title":"Ordered Leg Squares and Hypotenuse Square","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=7, b=7. Give (a², b², a²+b²), retaining leg order; the last field is the model’s hypotenuse square.","answer":{"kind":"expression","value":"(49, 49, 98)","explanation":"Square both positive legs separately and add; this finite side model is not a new universal proof or a completed affine geometry bridge. The exact result, retaining the requested order, is (49, 49, 98)."},"skills":["Ordered Leg Squares and Hypotenuse Square"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0007","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"732a4b94e2d7488e1623bb362c14e365faf5a4e2392e300563f5d88f6c92581d","family_id":"qf-density13-pythagorean-leg-square-sum","seed":202610131007,"semantic_template":"qf-density13-pythagorean-leg-square-sum/[\"64\",\"64\"]"},"fields":{"title":"Ordered Leg Squares and Hypotenuse Square","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=8, b=8. Give (a², b², a²+b²), retaining leg order; the last field is the model’s hypotenuse square.","answer":{"kind":"expression","value":"(64, 64, 128)","explanation":"Square both positive legs separately and add; this finite side model is not a new universal proof or a completed affine geometry bridge. The exact result, retaining the requested order, is (64, 64, 128)."},"skills":["Ordered Leg Squares and Hypotenuse Square"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0008","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a1942fa03cf8897634eee5063da6f21dc60138ae54ba7bc8bdc19d349fc03fd","family_id":"qf-density13-pythagorean-leg-square-sum","seed":202610131008,"semantic_template":"qf-density13-pythagorean-leg-square-sum/[\"81\",\"4\"]"},"fields":{"title":"Ordered Leg Squares and Hypotenuse Square","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=9, b=2. Give (a², b², a²+b²), retaining leg order; the last field is the model’s hypotenuse square.","answer":{"kind":"expression","value":"(81, 4, 85)","explanation":"Square both positive legs separately and add; this finite side model is not a new universal proof or a completed affine geometry bridge. The exact result, retaining the requested order, is (81, 4, 85)."},"skills":["Ordered Leg Squares and Hypotenuse Square"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0009","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f24792a9720a7dc806f630cef34851892bd8e1816b9da2c305eeba9bfd0d637","family_id":"qf-density13-pythagorean-leg-square-sum","seed":202610131009,"semantic_template":"qf-density13-pythagorean-leg-square-sum/[\"100\",\"9\"]"},"fields":{"title":"Ordered Leg Squares and Hypotenuse Square","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=10, b=3. Give (a², b², a²+b²), retaining leg order; the last field is the model’s hypotenuse square.","answer":{"kind":"expression","value":"(100, 9, 109)","explanation":"Square both positive legs separately and add; this finite side model is not a new universal proof or a completed affine geometry bridge. The exact result, retaining the requested order, is (100, 9, 109)."},"skills":["Ordered Leg Squares and Hypotenuse Square"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0010","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"494d66e6c1ce28ae10cf0976fafbfc4116fb391783d07a75cc9c16e86d0f6056","family_id":"qf-density13-pythagorean-leg-square-sum","seed":202610131010,"semantic_template":"qf-density13-pythagorean-leg-square-sum/[\"121\",\"16\"]"},"fields":{"title":"Ordered Leg Squares and Hypotenuse Square","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=11, b=4. Give (a², b², a²+b²), retaining leg order; the last field is the model’s hypotenuse square.","answer":{"kind":"expression","value":"(121, 16, 137)","explanation":"Square both positive legs separately and add; this finite side model is not a new universal proof or a completed affine geometry bridge. The exact result, retaining the requested order, is (121, 16, 137)."},"skills":["Ordered Leg Squares and Hypotenuse Square"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0011","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d424475036149c1345d9c0ba7b5c05ea0fe813684aadd77394e65d0bcabed76","family_id":"qf-density13-pythagorean-leg-square-sum","seed":202610131011,"semantic_template":"qf-density13-pythagorean-leg-square-sum/[\"144\",\"25\"]"},"fields":{"title":"Ordered Leg Squares and Hypotenuse Square","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=12, b=5. Give (a², b², a²+b²), retaining leg order; the last field is the model’s hypotenuse square.","answer":{"kind":"expression","value":"(144, 25, 169)","explanation":"Square both positive legs separately and add; this finite side model is not a new universal proof or a completed affine geometry bridge. The exact result, retaining the requested order, is (144, 25, 169)."},"skills":["Ordered Leg Squares and Hypotenuse Square"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0012","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"99ca8b5be0d0c72e9be5e48d63ea3d163572a1839766da7f180af7b3dc3d83fb","family_id":"qf-density13-pythagorean-leg-square-sum","seed":202610131012,"semantic_template":"qf-density13-pythagorean-leg-square-sum/[\"169\",\"36\"]"},"fields":{"title":"Ordered Leg Squares and Hypotenuse Square","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=13, b=6. Give (a², b², a²+b²), retaining leg order; the last field is the model’s hypotenuse square.","answer":{"kind":"expression","value":"(169, 36, 205)","explanation":"Square both positive legs separately and add; this finite side model is not a new universal proof or a completed affine geometry bridge. The exact result, retaining the requested order, is (169, 36, 205)."},"skills":["Ordered Leg Squares and Hypotenuse Square"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0013","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d6bd606dc316895dabee37b5cf570dc5f98232a0fc7f440d4537dfb21e1c44f","family_id":"qf-density13-pythagorean-leg-square-sum","seed":202610131013,"semantic_template":"qf-density13-pythagorean-leg-square-sum/[\"196\",\"49\"]"},"fields":{"title":"Ordered Leg Squares and Hypotenuse Square","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=14, b=7. Give (a², b², a²+b²), retaining leg order; the last field is the model’s hypotenuse square.","answer":{"kind":"expression","value":"(196, 49, 245)","explanation":"Square both positive legs separately and add; this finite side model is not a new universal proof or a completed affine geometry bridge. The exact result, retaining the requested order, is (196, 49, 245)."},"skills":["Ordered Leg Squares and Hypotenuse Square"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0014","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb44ffb8bc7fa967c7b0580a24847776971dec708e4d5408e706ac94f2138844","family_id":"qf-density13-pythagorean-leg-square-sum","seed":202610131014,"semantic_template":"qf-density13-pythagorean-leg-square-sum/[\"225\",\"64\"]"},"fields":{"title":"Ordered Leg Squares and Hypotenuse Square","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=15, b=8. Give (a², b², a²+b²), retaining leg order; the last field is the model’s hypotenuse square.","answer":{"kind":"expression","value":"(225, 64, 289)","explanation":"Square both positive legs separately and add; this finite side model is not a new universal proof or a completed affine geometry bridge. The exact result, retaining the requested order, is (225, 64, 289)."},"skills":["Ordered Leg Squares and Hypotenuse Square"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0015","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d1ed50bcf946ff21d32b9c4869f7d7c4d8d5dcd039ff873a883d30ec2092ae9","family_id":"qf-density13-pythagorean-leg-square-sum","seed":202610131015,"semantic_template":"qf-density13-pythagorean-leg-square-sum/[\"256\",\"4\"]"},"fields":{"title":"Ordered Leg Squares and Hypotenuse Square","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=16, b=2. Give (a², b², a²+b²), retaining leg order; the last field is the model’s hypotenuse square.","answer":{"kind":"expression","value":"(256, 4, 260)","explanation":"Square both positive legs separately and add; this finite side model is not a new universal proof or a completed affine geometry bridge. The exact result, retaining the requested order, is (256, 4, 260)."},"skills":["Ordered Leg Squares and Hypotenuse Square"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0016","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21f7c5b0027bef270648d6ce9e1502ae778c75f146e62cec50f0db110749c24a","family_id":"qf-density13-pythagorean-leg-square-sum","seed":202610131016,"semantic_template":"qf-density13-pythagorean-leg-square-sum/[\"289\",\"9\"]"},"fields":{"title":"Ordered Leg Squares and Hypotenuse Square","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=17, b=3. Give (a², b², a²+b²), retaining leg order; the last field is the model’s hypotenuse square.","answer":{"kind":"expression","value":"(289, 9, 298)","explanation":"Square both positive legs separately and add; this finite side model is not a new universal proof or a completed affine geometry bridge. The exact result, retaining the requested order, is (289, 9, 298)."},"skills":["Ordered Leg Squares and Hypotenuse Square"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0017","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4bc86713b9cb75e9822fdb5e6c9d8ccaddeed14408dbab4a22f7d7dafb78b159","family_id":"qf-density13-pythagorean-leg-square-sum","seed":202610131017,"semantic_template":"qf-density13-pythagorean-leg-square-sum/[\"324\",\"16\"]"},"fields":{"title":"Ordered Leg Squares and Hypotenuse Square","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=18, b=4. Give (a², b², a²+b²), retaining leg order; the last field is the model’s hypotenuse square.","answer":{"kind":"expression","value":"(324, 16, 340)","explanation":"Square both positive legs separately and add; this finite side model is not a new universal proof or a completed affine geometry bridge. The exact result, retaining the requested order, is (324, 16, 340)."},"skills":["Ordered Leg Squares and Hypotenuse Square"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0018","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"27f3f002bc0e5b314d80c92d9324a3ccef006cb3862545a229a3af298ee96fc1","family_id":"qf-density13-pythagorean-leg-square-sum","seed":202610131018,"semantic_template":"qf-density13-pythagorean-leg-square-sum/[\"361\",\"25\"]"},"fields":{"title":"Ordered Leg Squares and Hypotenuse Square","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=19, b=5. Give (a², b², a²+b²), retaining leg order; the last field is the model’s hypotenuse square.","answer":{"kind":"expression","value":"(361, 25, 386)","explanation":"Square both positive legs separately and add; this finite side model is not a new universal proof or a completed affine geometry bridge. The exact result, retaining the requested order, is (361, 25, 386)."},"skills":["Ordered Leg Squares and Hypotenuse Square"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0019","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"22af5cc6ef341b56c24c73052db905f579bbf1692865afa64bc5e31ecff6f20a","family_id":"qf-density13-pythagorean-leg-square-sum","seed":202610131019,"semantic_template":"qf-density13-pythagorean-leg-square-sum/[\"400\",\"36\"]"},"fields":{"title":"Ordered Leg Squares and Hypotenuse Square","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=20, b=6. Give (a², b², a²+b²), retaining leg order; the last field is the model’s hypotenuse square.","answer":{"kind":"expression","value":"(400, 36, 436)","explanation":"Square both positive legs separately and add; this finite side model is not a new universal proof or a completed affine geometry bridge. The exact result, retaining the requested order, is (400, 36, 436)."},"skills":["Ordered Leg Squares and Hypotenuse Square"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0020","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba67417772b9366aa155b6df7d6d0938186b50a4f5cdbb5340cc9f05d6689ccb","family_id":"qf-density13-pythagorean-leg-square-sum","seed":202610131020,"semantic_template":"qf-density13-pythagorean-leg-square-sum/[\"441\",\"49\"]"},"fields":{"title":"Ordered Leg Squares and Hypotenuse Square","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=21, b=7. Give (a², b², a²+b²), retaining leg order; the last field is the model’s hypotenuse square.","answer":{"kind":"expression","value":"(441, 49, 490)","explanation":"Square both positive legs separately and add; this finite side model is not a new universal proof or a completed affine geometry bridge. The exact result, retaining the requested order, is (441, 49, 490)."},"skills":["Ordered Leg Squares and Hypotenuse Square"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0021","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c37f0e9b11b48aa60899a8989d7dd5ba0d5646182f9aeb6e3dc31ae83e370672","family_id":"qf-density13-pythagorean-leg-square-sum","seed":202610131021,"semantic_template":"qf-density13-pythagorean-leg-square-sum/[\"484\",\"64\"]"},"fields":{"title":"Ordered Leg Squares and Hypotenuse Square","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=22, b=8. Give (a², b², a²+b²), retaining leg order; the last field is the model’s hypotenuse square.","answer":{"kind":"expression","value":"(484, 64, 548)","explanation":"Square both positive legs separately and add; this finite side model is not a new universal proof or a completed affine geometry bridge. The exact result, retaining the requested order, is (484, 64, 548)."},"skills":["Ordered Leg Squares and Hypotenuse Square"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0022","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"173c93c899cb1241a10a0c5e96f515938421f75668a460380acb44cf580eaedc","family_id":"qf-density13-pythagorean-leg-square-sum","seed":202610131022,"semantic_template":"qf-density13-pythagorean-leg-square-sum/[\"529\",\"4\"]"},"fields":{"title":"Ordered Leg Squares and Hypotenuse Square","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=23, b=2. Give (a², b², a²+b²), retaining leg order; the last field is the model’s hypotenuse square.","answer":{"kind":"expression","value":"(529, 4, 533)","explanation":"Square both positive legs separately and add; this finite side model is not a new universal proof or a completed affine geometry bridge. The exact result, retaining the requested order, is (529, 4, 533)."},"skills":["Ordered Leg Squares and Hypotenuse Square"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0023","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb82d40c8c65f5395f3228041bc68aff822a5d9a5bc2d3018c00a245516584a2","family_id":"qf-density13-pythagorean-leg-square-sum","seed":202610131023,"semantic_template":"qf-density13-pythagorean-leg-square-sum/[\"576\",\"9\"]"},"fields":{"title":"Ordered Leg Squares and Hypotenuse Square","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=24, b=3. Give (a², b², a²+b²), retaining leg order; the last field is the model’s hypotenuse square.","answer":{"kind":"expression","value":"(576, 9, 585)","explanation":"Square both positive legs separately and add; this finite side model is not a new universal proof or a completed affine geometry bridge. The exact result, retaining the requested order, is (576, 9, 585)."},"skills":["Ordered Leg Squares and Hypotenuse Square"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0024","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"84bd46daa50ce4525b85bd1dc75b5fb5f93709f330d61c74248ad0ac91dc9045","family_id":"qf-density13-pythagorean-leg-square-sum","seed":202610131024,"semantic_template":"qf-density13-pythagorean-leg-square-sum/[\"625\",\"16\"]"},"fields":{"title":"Ordered Leg Squares and Hypotenuse Square","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=25, b=4. Give (a², b², a²+b²), retaining leg order; the last field is the model’s hypotenuse square.","answer":{"kind":"expression","value":"(625, 16, 641)","explanation":"Square both positive legs separately and add; this finite side model is not a new universal proof or a completed affine geometry bridge. The exact result, retaining the requested order, is (625, 16, 641)."},"skills":["Ordered Leg Squares and Hypotenuse Square"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0025","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39f86dc097016d447f3e7db0888cc6ca4d76f6d33c9766a64a5e4fe515b9057a","family_id":"qf-density13-pythagorean-leg-square-sum","seed":202610131025,"semantic_template":"qf-density13-pythagorean-leg-square-sum/[\"676\",\"25\"]"},"fields":{"title":"Ordered Leg Squares and Hypotenuse Square","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=26, b=5. Give (a², b², a²+b²), retaining leg order; the last field is the model’s hypotenuse square.","answer":{"kind":"expression","value":"(676, 25, 701)","explanation":"Square both positive legs separately and add; this finite side model is not a new universal proof or a completed affine geometry bridge. The exact result, retaining the requested order, is (676, 25, 701)."},"skills":["Ordered Leg Squares and Hypotenuse Square"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0026","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ed8c15b44affd6e577cb688ed672b2b3278a76784c83d8a904ea95a3c16333c9","family_id":"qf-density13-pythagorean-hypotenuse-int","seed":202610131026,"semantic_template":"qf-density13-pythagorean-hypotenuse-int/\"25\""},"fields":{"title":"Positive Integer Hypotenuse","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=3, b=4, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.","answer":{"kind":"expression","value":"5","explanation":"Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 5."},"skills":["Positive Integer Hypotenuse"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0027","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2edb6ceb715c9bb5e9bde9aea64d0a53c29596587d8485304091694c72ab1476","family_id":"qf-density13-pythagorean-hypotenuse-int","seed":202610131027,"semantic_template":"qf-density13-pythagorean-hypotenuse-int/\"100\""},"fields":{"title":"Positive Integer Hypotenuse","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=6, b=8, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.","answer":{"kind":"expression","value":"10","explanation":"Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 10."},"skills":["Positive Integer Hypotenuse"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0028","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"100e95a0d42c2875d5e8a738401c4791e8ca471fee8ae45b6d60e2f033c6ba6f","family_id":"qf-density13-pythagorean-hypotenuse-int","seed":202610131028,"semantic_template":"qf-density13-pythagorean-hypotenuse-int/\"225\""},"fields":{"title":"Positive Integer Hypotenuse","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=9, b=12, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.","answer":{"kind":"expression","value":"15","explanation":"Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 15."},"skills":["Positive Integer Hypotenuse"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0029","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f52f3b07b653b8ca3c448535aeddb803bfec57d050dbfcf5c7b2e82b702b0897","family_id":"qf-density13-pythagorean-hypotenuse-int","seed":202610131029,"semantic_template":"qf-density13-pythagorean-hypotenuse-int/\"400\""},"fields":{"title":"Positive Integer Hypotenuse","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=12, b=16, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.","answer":{"kind":"expression","value":"20","explanation":"Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 20."},"skills":["Positive Integer Hypotenuse"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0030","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ab4490bfa7a99700ad9344cc5c9359945a54d9f53f92e4b67346bb7367b12aa","family_id":"qf-density13-pythagorean-hypotenuse-int","seed":202610131030,"semantic_template":"qf-density13-pythagorean-hypotenuse-int/\"625\""},"fields":{"title":"Positive Integer Hypotenuse","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=15, b=20, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.","answer":{"kind":"expression","value":"25","explanation":"Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 25."},"skills":["Positive Integer Hypotenuse"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0031","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1be6e9b1aed24b2b15ed51f7c8dce5fdac85b115c3a575747c8f49408dcafa39","family_id":"qf-density13-pythagorean-hypotenuse-int","seed":202610131031,"semantic_template":"qf-density13-pythagorean-hypotenuse-int/\"900\""},"fields":{"title":"Positive Integer Hypotenuse","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=18, b=24, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.","answer":{"kind":"expression","value":"30","explanation":"Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 30."},"skills":["Positive Integer Hypotenuse"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0032","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a23dba0c88bbf8421270ad149decd6d4f8eff2f94c95b406ea036410679b503","family_id":"qf-density13-pythagorean-hypotenuse-int","seed":202610131032,"semantic_template":"qf-density13-pythagorean-hypotenuse-int/\"1225\""},"fields":{"title":"Positive Integer Hypotenuse","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=21, b=28, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.","answer":{"kind":"expression","value":"35","explanation":"Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 35."},"skills":["Positive Integer Hypotenuse"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0033","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7870e1f60d0b414d16a321657973f8db9fc9340a6d6ef93d8d21f3c239c9ad35","family_id":"qf-density13-pythagorean-hypotenuse-int","seed":202610131033,"semantic_template":"qf-density13-pythagorean-hypotenuse-int/\"1600\""},"fields":{"title":"Positive Integer Hypotenuse","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=24, b=32, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.","answer":{"kind":"expression","value":"40","explanation":"Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 40."},"skills":["Positive Integer Hypotenuse"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0034","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"700e83311014ff757c166cf013147bef369d4494878b434000ea3c3043bf603a","family_id":"qf-density13-pythagorean-hypotenuse-int","seed":202610131034,"semantic_template":"qf-density13-pythagorean-hypotenuse-int/\"2025\""},"fields":{"title":"Positive Integer Hypotenuse","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=27, b=36, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.","answer":{"kind":"expression","value":"45","explanation":"Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 45."},"skills":["Positive Integer Hypotenuse"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0035","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"973c6d32afff5554cb736b62313ae54f916ed8ae5fc0dd376549a559000900c8","family_id":"qf-density13-pythagorean-hypotenuse-int","seed":202610131035,"semantic_template":"qf-density13-pythagorean-hypotenuse-int/\"2500\""},"fields":{"title":"Positive Integer Hypotenuse","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=30, b=40, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.","answer":{"kind":"expression","value":"50","explanation":"Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 50."},"skills":["Positive Integer Hypotenuse"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0036","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b1ffdf2fdfb7169c18769876a72460445f2f25c8951ec0de20bacda86701015","family_id":"qf-density13-pythagorean-hypotenuse-int","seed":202610131036,"semantic_template":"qf-density13-pythagorean-hypotenuse-int/\"3025\""},"fields":{"title":"Positive Integer Hypotenuse","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=33, b=44, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.","answer":{"kind":"expression","value":"55","explanation":"Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 55."},"skills":["Positive Integer Hypotenuse"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0037","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52fa038bf5d654e71fa05b9921e3c8028524a70557508950cb5ffcece57fd098","family_id":"qf-density13-pythagorean-hypotenuse-int","seed":202610131037,"semantic_template":"qf-density13-pythagorean-hypotenuse-int/\"3600\""},"fields":{"title":"Positive Integer Hypotenuse","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=36, b=48, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.","answer":{"kind":"expression","value":"60","explanation":"Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 60."},"skills":["Positive Integer Hypotenuse"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0038","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"53bce4a8621629a8f0a71308644e3f283a51dd8f82b8986d082245556c1ecaa4","family_id":"qf-density13-pythagorean-hypotenuse-int","seed":202610131038,"semantic_template":"qf-density13-pythagorean-hypotenuse-int/\"4225\""},"fields":{"title":"Positive Integer Hypotenuse","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=39, b=52, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.","answer":{"kind":"expression","value":"65","explanation":"Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 65."},"skills":["Positive Integer Hypotenuse"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0039","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"53b3b757e257bcc8b1c6aa9081ed0d2d74cf46b0743264e3114f9f86d9a30b63","family_id":"qf-density13-pythagorean-hypotenuse-int","seed":202610131039,"semantic_template":"qf-density13-pythagorean-hypotenuse-int/\"4900\""},"fields":{"title":"Positive Integer Hypotenuse","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=42, b=56, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.","answer":{"kind":"expression","value":"70","explanation":"Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 70."},"skills":["Positive Integer Hypotenuse"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0040","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d34b725e0e3b2f5fd264e3f0bb2cd4144d2d201eac6bf8dcec2ec585b5a024d","family_id":"qf-density13-pythagorean-hypotenuse-int","seed":202610131040,"semantic_template":"qf-density13-pythagorean-hypotenuse-int/\"5625\""},"fields":{"title":"Positive Integer Hypotenuse","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=45, b=60, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.","answer":{"kind":"expression","value":"75","explanation":"Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 75."},"skills":["Positive Integer Hypotenuse"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0041","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"18884de7a7cbd7cd9b3c31b3a6de4fb6d7ddb30b3c0619e8cb701a536e870e31","family_id":"qf-density13-pythagorean-hypotenuse-int","seed":202610131041,"semantic_template":"qf-density13-pythagorean-hypotenuse-int/\"6400\""},"fields":{"title":"Positive Integer Hypotenuse","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=48, b=64, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.","answer":{"kind":"expression","value":"80","explanation":"Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 80."},"skills":["Positive Integer Hypotenuse"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0042","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"282bf8fa8cf0a9a95879bcc27f81aa676cac762bcbb918dde233917b9b5f103e","family_id":"qf-density13-pythagorean-hypotenuse-int","seed":202610131042,"semantic_template":"qf-density13-pythagorean-hypotenuse-int/\"7225\""},"fields":{"title":"Positive Integer Hypotenuse","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=51, b=68, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.","answer":{"kind":"expression","value":"85","explanation":"Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 85."},"skills":["Positive Integer Hypotenuse"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0043","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"05b2c57e1a946d4a38a7e25e13552acd534371e62461a7cbab75c68b32b81d94","family_id":"qf-density13-pythagorean-hypotenuse-int","seed":202610131043,"semantic_template":"qf-density13-pythagorean-hypotenuse-int/\"8100\""},"fields":{"title":"Positive Integer Hypotenuse","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=54, b=72, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.","answer":{"kind":"expression","value":"90","explanation":"Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 90."},"skills":["Positive Integer Hypotenuse"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0044","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3baf004cf6f99ee7126aaec3d532a0531db90f70de523315021caa323188861","family_id":"qf-density13-pythagorean-hypotenuse-int","seed":202610131044,"semantic_template":"qf-density13-pythagorean-hypotenuse-int/\"9025\""},"fields":{"title":"Positive Integer Hypotenuse","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=57, b=76, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.","answer":{"kind":"expression","value":"95","explanation":"Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 95."},"skills":["Positive Integer Hypotenuse"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0045","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dc493aa1d4f637b980eb736c08a95ec0609153070feb438fb8a245b9aae3e4cd","family_id":"qf-density13-pythagorean-hypotenuse-int","seed":202610131045,"semantic_template":"qf-density13-pythagorean-hypotenuse-int/\"10000\""},"fields":{"title":"Positive Integer Hypotenuse","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=60, b=80, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.","answer":{"kind":"expression","value":"100","explanation":"Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 100."},"skills":["Positive Integer Hypotenuse"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0046","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02f7542c83b5af3759adb6f506f8fe0d3618fa26524b7ad5772e5995beb15e9a","family_id":"qf-density13-pythagorean-hypotenuse-int","seed":202610131046,"semantic_template":"qf-density13-pythagorean-hypotenuse-int/\"11025\""},"fields":{"title":"Positive Integer Hypotenuse","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=63, b=84, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.","answer":{"kind":"expression","value":"105","explanation":"Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 105."},"skills":["Positive Integer Hypotenuse"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0047","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b9bc71709e63d54b4ef665261c324afeea71d0e98a87e5c9c5b85117cc0edbd4","family_id":"qf-density13-pythagorean-hypotenuse-int","seed":202610131047,"semantic_template":"qf-density13-pythagorean-hypotenuse-int/\"12100\""},"fields":{"title":"Positive Integer Hypotenuse","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=66, b=88, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.","answer":{"kind":"expression","value":"110","explanation":"Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 110."},"skills":["Positive Integer Hypotenuse"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0048","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c2fe77878d8921e6d7741d13cbd98aba471ae1a695a4feac71c707159220208","family_id":"qf-density13-pythagorean-hypotenuse-int","seed":202610131048,"semantic_template":"qf-density13-pythagorean-hypotenuse-int/\"13225\""},"fields":{"title":"Positive Integer Hypotenuse","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=69, b=92, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.","answer":{"kind":"expression","value":"115","explanation":"Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 115."},"skills":["Positive Integer Hypotenuse"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0049","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a724ab9b887a9cd11d2e36b4046559039c8b4e1821ad0200269db65bc3fd8c62","family_id":"qf-density13-pythagorean-hypotenuse-int","seed":202610131049,"semantic_template":"qf-density13-pythagorean-hypotenuse-int/\"14400\""},"fields":{"title":"Positive Integer Hypotenuse","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=72, b=96, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.","answer":{"kind":"expression","value":"120","explanation":"Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 120."},"skills":["Positive Integer Hypotenuse"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0050","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"23ea2b9d4a1b8337bf4455efc2033955fa6e87454c28f6b0dfc0818f345b6915","family_id":"qf-density13-pythagorean-hypotenuse-int","seed":202610131050,"semantic_template":"qf-density13-pythagorean-hypotenuse-int/\"15625\""},"fields":{"title":"Positive Integer Hypotenuse","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=75, b=100, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.","answer":{"kind":"expression","value":"125","explanation":"Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 125."},"skills":["Positive Integer Hypotenuse"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0051","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c30ad81acf39e64626285976b33cd77484b10072c3cdca215aa83165c042bd31","family_id":"qf-density13-pythagorean-missing-leg-int","seed":202610131051,"semantic_template":"qf-density13-pythagorean-missing-leg-int/\"16\""},"fields":{"title":"Positive Integer Missing Leg","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=3, positive hypotenuse c=5>a; c²−a² is a positive perfect square. Find the other positive leg b.","answer":{"kind":"expression","value":"4","explanation":"Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 4."},"skills":["Positive Integer Missing Leg"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0052","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"374cb220e59388031841fe15576f6675b01bb54d08821e899ea1935c5fe1187c","family_id":"qf-density13-pythagorean-missing-leg-int","seed":202610131052,"semantic_template":"qf-density13-pythagorean-missing-leg-int/\"64\""},"fields":{"title":"Positive Integer Missing Leg","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=6, positive hypotenuse c=10>a; c²−a² is a positive perfect square. Find the other positive leg b.","answer":{"kind":"expression","value":"8","explanation":"Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 8."},"skills":["Positive Integer Missing Leg"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0053","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6af580dac3d4725cc8833237fb3308e4fab20ad7bfc71fe5a7022fcb51ffe429","family_id":"qf-density13-pythagorean-missing-leg-int","seed":202610131053,"semantic_template":"qf-density13-pythagorean-missing-leg-int/\"144\""},"fields":{"title":"Positive Integer Missing Leg","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=9, positive hypotenuse c=15>a; c²−a² is a positive perfect square. Find the other positive leg b.","answer":{"kind":"expression","value":"12","explanation":"Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 12."},"skills":["Positive Integer Missing Leg"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0054","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"271b23b2b4e9e69e75094b21349d6e4715ba6c982fbf47efea5adc2f8a7ad539","family_id":"qf-density13-pythagorean-missing-leg-int","seed":202610131054,"semantic_template":"qf-density13-pythagorean-missing-leg-int/\"256\""},"fields":{"title":"Positive Integer Missing Leg","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=12, positive hypotenuse c=20>a; c²−a² is a positive perfect square. Find the other positive leg b.","answer":{"kind":"expression","value":"16","explanation":"Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 16."},"skills":["Positive Integer Missing Leg"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0055","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a00e4fab2b7aa51a8add1e5b7ba02c1d2c6d8256c537f45540120f102094712","family_id":"qf-density13-pythagorean-missing-leg-int","seed":202610131055,"semantic_template":"qf-density13-pythagorean-missing-leg-int/\"400\""},"fields":{"title":"Positive Integer Missing Leg","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=15, positive hypotenuse c=25>a; c²−a² is a positive perfect square. Find the other positive leg b.","answer":{"kind":"expression","value":"20","explanation":"Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 20."},"skills":["Positive Integer Missing Leg"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0056","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"97cbfe2548e432da59a60f864b6952efc1e41d6f3e1cd882f000feb58a07420e","family_id":"qf-density13-pythagorean-missing-leg-int","seed":202610131056,"semantic_template":"qf-density13-pythagorean-missing-leg-int/\"576\""},"fields":{"title":"Positive Integer Missing Leg","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=18, positive hypotenuse c=30>a; c²−a² is a positive perfect square. Find the other positive leg b.","answer":{"kind":"expression","value":"24","explanation":"Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 24."},"skills":["Positive Integer Missing Leg"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0057","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2dd20a45ff6a99ccf05d964a88962ef577694482dcbcf21b336b5675ff745fd9","family_id":"qf-density13-pythagorean-missing-leg-int","seed":202610131057,"semantic_template":"qf-density13-pythagorean-missing-leg-int/\"784\""},"fields":{"title":"Positive Integer Missing Leg","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=21, positive hypotenuse c=35>a; c²−a² is a positive perfect square. Find the other positive leg b.","answer":{"kind":"expression","value":"28","explanation":"Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 28."},"skills":["Positive Integer Missing Leg"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0058","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f750799b9a2a82a692eaff271721cf57786a482939b12ce5f81dfad215c4b4d5","family_id":"qf-density13-pythagorean-missing-leg-int","seed":202610131058,"semantic_template":"qf-density13-pythagorean-missing-leg-int/\"1024\""},"fields":{"title":"Positive Integer Missing Leg","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=24, positive hypotenuse c=40>a; c²−a² is a positive perfect square. Find the other positive leg b.","answer":{"kind":"expression","value":"32","explanation":"Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 32."},"skills":["Positive Integer Missing Leg"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0059","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c9084faad437cb9d917336c66a2be19a63a5e6b8ca3d6e7e94b351537df8d04","family_id":"qf-density13-pythagorean-missing-leg-int","seed":202610131059,"semantic_template":"qf-density13-pythagorean-missing-leg-int/\"1296\""},"fields":{"title":"Positive Integer Missing Leg","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=27, positive hypotenuse c=45>a; c²−a² is a positive perfect square. Find the other positive leg b.","answer":{"kind":"expression","value":"36","explanation":"Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 36."},"skills":["Positive Integer Missing Leg"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0060","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7f231c82cdae5c7d61f925b04913a7ba08f51628ea7fd3dbaf6331bad6e85de0","family_id":"qf-density13-pythagorean-missing-leg-int","seed":202610131060,"semantic_template":"qf-density13-pythagorean-missing-leg-int/\"1600\""},"fields":{"title":"Positive Integer Missing Leg","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=30, positive hypotenuse c=50>a; c²−a² is a positive perfect square. Find the other positive leg b.","answer":{"kind":"expression","value":"40","explanation":"Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 40."},"skills":["Positive Integer Missing Leg"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0061","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b8977d987826df7bce7c8a6c7704e09e964bb3f8f00ae4533e0944f28fe86b01","family_id":"qf-density13-pythagorean-missing-leg-int","seed":202610131061,"semantic_template":"qf-density13-pythagorean-missing-leg-int/\"1936\""},"fields":{"title":"Positive Integer Missing Leg","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=33, positive hypotenuse c=55>a; c²−a² is a positive perfect square. Find the other positive leg b.","answer":{"kind":"expression","value":"44","explanation":"Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 44."},"skills":["Positive Integer Missing Leg"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0062","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e324711504f4f2cb45c753bad53b43936b1457d4fede7a21e2844f38de99f63","family_id":"qf-density13-pythagorean-missing-leg-int","seed":202610131062,"semantic_template":"qf-density13-pythagorean-missing-leg-int/\"2304\""},"fields":{"title":"Positive Integer Missing Leg","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=36, positive hypotenuse c=60>a; c²−a² is a positive perfect square. Find the other positive leg b.","answer":{"kind":"expression","value":"48","explanation":"Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 48."},"skills":["Positive Integer Missing Leg"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0063","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d23ddd0d926e08cc0cd1ad60eff681be5691552e692478c080eca2a90b0c2a38","family_id":"qf-density13-pythagorean-missing-leg-int","seed":202610131063,"semantic_template":"qf-density13-pythagorean-missing-leg-int/\"2704\""},"fields":{"title":"Positive Integer Missing Leg","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=39, positive hypotenuse c=65>a; c²−a² is a positive perfect square. Find the other positive leg b.","answer":{"kind":"expression","value":"52","explanation":"Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 52."},"skills":["Positive Integer Missing Leg"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0064","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"709d5ed68ffc9d34ccb8fdefbb741a16ab5c45fdf28dcfc70d2cdb9bfacf3b1d","family_id":"qf-density13-pythagorean-missing-leg-int","seed":202610131064,"semantic_template":"qf-density13-pythagorean-missing-leg-int/\"3136\""},"fields":{"title":"Positive Integer Missing Leg","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=42, positive hypotenuse c=70>a; c²−a² is a positive perfect square. Find the other positive leg b.","answer":{"kind":"expression","value":"56","explanation":"Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 56."},"skills":["Positive Integer Missing Leg"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0065","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"042c70a38c02a6aaa50995d84ba09967c2b2f45f590ecf6438a60bb11ab31622","family_id":"qf-density13-pythagorean-missing-leg-int","seed":202610131065,"semantic_template":"qf-density13-pythagorean-missing-leg-int/\"3600\""},"fields":{"title":"Positive Integer Missing Leg","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=45, positive hypotenuse c=75>a; c²−a² is a positive perfect square. Find the other positive leg b.","answer":{"kind":"expression","value":"60","explanation":"Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 60."},"skills":["Positive Integer Missing Leg"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0066","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"040851b8c367b176c3480202c248d458a4aae5b5e05d1c9441d65b73033849f3","family_id":"qf-density13-pythagorean-missing-leg-int","seed":202610131066,"semantic_template":"qf-density13-pythagorean-missing-leg-int/\"4096\""},"fields":{"title":"Positive Integer Missing Leg","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=48, positive hypotenuse c=80>a; c²−a² is a positive perfect square. Find the other positive leg b.","answer":{"kind":"expression","value":"64","explanation":"Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 64."},"skills":["Positive Integer Missing Leg"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0067","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9f52d90de45454c79119cb2d84bfa4d09f6bcb095603eb0fa47ef4739a96c1d8","family_id":"qf-density13-pythagorean-missing-leg-int","seed":202610131067,"semantic_template":"qf-density13-pythagorean-missing-leg-int/\"4624\""},"fields":{"title":"Positive Integer Missing Leg","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=51, positive hypotenuse c=85>a; c²−a² is a positive perfect square. Find the other positive leg b.","answer":{"kind":"expression","value":"68","explanation":"Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 68."},"skills":["Positive Integer Missing Leg"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0068","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bedde6bcf6a30bcdd3d9ad6d494750ddeb45ea9a70744cc36b756d0b96705a3f","family_id":"qf-density13-pythagorean-missing-leg-int","seed":202610131068,"semantic_template":"qf-density13-pythagorean-missing-leg-int/\"5184\""},"fields":{"title":"Positive Integer Missing Leg","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=54, positive hypotenuse c=90>a; c²−a² is a positive perfect square. Find the other positive leg b.","answer":{"kind":"expression","value":"72","explanation":"Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 72."},"skills":["Positive Integer Missing Leg"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0069","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"553037c7c75273a76fe342b2f6828c27672553d57c60edc82f7c98ab81de8fc0","family_id":"qf-density13-pythagorean-missing-leg-int","seed":202610131069,"semantic_template":"qf-density13-pythagorean-missing-leg-int/\"5776\""},"fields":{"title":"Positive Integer Missing Leg","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=57, positive hypotenuse c=95>a; c²−a² is a positive perfect square. Find the other positive leg b.","answer":{"kind":"expression","value":"76","explanation":"Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 76."},"skills":["Positive Integer Missing Leg"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0070","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0f3f41dd02e136bd8fa86b664d8703989a9a6548831184bbe0e7e96d15cac10","family_id":"qf-density13-pythagorean-missing-leg-int","seed":202610131070,"semantic_template":"qf-density13-pythagorean-missing-leg-int/\"6400\""},"fields":{"title":"Positive Integer Missing Leg","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=60, positive hypotenuse c=100>a; c²−a² is a positive perfect square. Find the other positive leg b.","answer":{"kind":"expression","value":"80","explanation":"Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 80."},"skills":["Positive Integer Missing Leg"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0071","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"56f1559d7faccab8163adf46b8e1a565df8cece9a90295b082dcd47b4c6c7fcc","family_id":"qf-density13-pythagorean-missing-leg-int","seed":202610131071,"semantic_template":"qf-density13-pythagorean-missing-leg-int/\"7056\""},"fields":{"title":"Positive Integer Missing Leg","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=63, positive hypotenuse c=105>a; c²−a² is a positive perfect square. Find the other positive leg b.","answer":{"kind":"expression","value":"84","explanation":"Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 84."},"skills":["Positive Integer Missing Leg"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0072","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c76eed47328b180b7e1b6aaaa7cd89ceb336adec1f41619108c3881f57112ef","family_id":"qf-density13-pythagorean-missing-leg-int","seed":202610131072,"semantic_template":"qf-density13-pythagorean-missing-leg-int/\"7744\""},"fields":{"title":"Positive Integer Missing Leg","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=66, positive hypotenuse c=110>a; c²−a² is a positive perfect square. Find the other positive leg b.","answer":{"kind":"expression","value":"88","explanation":"Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 88."},"skills":["Positive Integer Missing Leg"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0073","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1da90ce3406042fc99d226427c6c95ec1aab30951754f9e29e1f048594df94dd","family_id":"qf-density13-pythagorean-missing-leg-int","seed":202610131073,"semantic_template":"qf-density13-pythagorean-missing-leg-int/\"8464\""},"fields":{"title":"Positive Integer Missing Leg","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=69, positive hypotenuse c=115>a; c²−a² is a positive perfect square. Find the other positive leg b.","answer":{"kind":"expression","value":"92","explanation":"Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 92."},"skills":["Positive Integer Missing Leg"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0074","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"feafe49005720a3c397102fae44eac66850d90103339d37cdba2d5745da1d6dd","family_id":"qf-density13-pythagorean-missing-leg-int","seed":202610131074,"semantic_template":"qf-density13-pythagorean-missing-leg-int/\"9216\""},"fields":{"title":"Positive Integer Missing Leg","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=72, positive hypotenuse c=120>a; c²−a² is a positive perfect square. Find the other positive leg b.","answer":{"kind":"expression","value":"96","explanation":"Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 96."},"skills":["Positive Integer Missing Leg"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0075","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"25e06a71cd9fb29d01505c1894b5ceef7f1419a036bc2cef9fa6f894d91063be","family_id":"qf-density13-pythagorean-missing-leg-int","seed":202610131075,"semantic_template":"qf-density13-pythagorean-missing-leg-int/\"10000\""},"fields":{"title":"Positive Integer Missing Leg","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=75, positive hypotenuse c=125>a; c²−a² is a positive perfect square. Find the other positive leg b.","answer":{"kind":"expression","value":"100","explanation":"Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 100."},"skills":["Positive Integer Missing Leg"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0076","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7dd641ca09133ddcee18b36d195a737555ea7ff65299c463ca36a38608d3039","family_id":"qf-density13-pythagorean-candidate-residual","seed":202610131076,"semantic_template":"qf-density13-pythagorean-candidate-residual/[\"25\",-5]"},"fields":{"title":"Hypotenuse Sign and Square Residual","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=3, b=4, candidate c=-5. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.","answer":{"kind":"expression","value":"(No, 0, No)","explanation":"Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (No, 0, No)."},"skills":["Hypotenuse Sign and Square Residual"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0077","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"415a2f500b3ca50b56b4392b88dd7378c254a38059c2e0017e7bd4684066c8d3","family_id":"qf-density13-pythagorean-candidate-residual","seed":202610131077,"semantic_template":"qf-density13-pythagorean-candidate-residual/[\"25\",5]"},"fields":{"title":"Hypotenuse Sign and Square Residual","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=3, b=4, candidate c=5. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.","answer":{"kind":"expression","value":"(Yes, 0, Yes)","explanation":"Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (Yes, 0, Yes)."},"skills":["Hypotenuse Sign and Square Residual"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0078","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e5be7f10b3f31714fe167da0c197f8d7bd4648b876802780ca31775327449021","family_id":"qf-density13-pythagorean-candidate-residual","seed":202610131078,"semantic_template":"qf-density13-pythagorean-candidate-residual/[\"25\",0]"},"fields":{"title":"Hypotenuse Sign and Square Residual","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=3, b=4, candidate c=0. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.","answer":{"kind":"expression","value":"(No, -25, No)","explanation":"Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (No, -25, No)."},"skills":["Hypotenuse Sign and Square Residual"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0079","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f907d5bd205077cbb3181472ae7559d4046b482297c0c7803a90c0d37f685775","family_id":"qf-density13-pythagorean-candidate-residual","seed":202610131079,"semantic_template":"qf-density13-pythagorean-candidate-residual/[\"25\",4]"},"fields":{"title":"Hypotenuse Sign and Square Residual","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=3, b=4, candidate c=4. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.","answer":{"kind":"expression","value":"(Yes, -9, No)","explanation":"Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (Yes, -9, No)."},"skills":["Hypotenuse Sign and Square Residual"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0080","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e4c4029b272728774cf8b479e5d5e03f500b2213bc1a751c0fabe08f680b3a6","family_id":"qf-density13-pythagorean-candidate-residual","seed":202610131080,"semantic_template":"qf-density13-pythagorean-candidate-residual/[\"25\",6]"},"fields":{"title":"Hypotenuse Sign and Square Residual","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=3, b=4, candidate c=6. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.","answer":{"kind":"expression","value":"(Yes, 11, No)","explanation":"Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (Yes, 11, No)."},"skills":["Hypotenuse Sign and Square Residual"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0081","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9baabbefc19ff856a9d9ac2900840ac51ed2bdf59603909c66a2f6cafae87c1","family_id":"qf-density13-pythagorean-candidate-residual","seed":202610131081,"semantic_template":"qf-density13-pythagorean-candidate-residual/[\"85\",-8]"},"fields":{"title":"Hypotenuse Sign and Square Residual","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=6, b=7, candidate c=-8. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.","answer":{"kind":"expression","value":"(No, -21, No)","explanation":"Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (No, -21, No)."},"skills":["Hypotenuse Sign and Square Residual"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0082","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5192718a0d8e237a5d7aeb49b1c3d4b71ebbaeda21c26768b4077ca14abb8f81","family_id":"qf-density13-pythagorean-candidate-residual","seed":202610131082,"semantic_template":"qf-density13-pythagorean-candidate-residual/[\"113\",9]"},"fields":{"title":"Hypotenuse Sign and Square Residual","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=7, b=8, candidate c=9. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.","answer":{"kind":"expression","value":"(Yes, -32, No)","explanation":"Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (Yes, -32, No)."},"skills":["Hypotenuse Sign and Square Residual"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0083","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f24e9625cf014e16bd2bb508c0116c109c0286601ade6ede5e3d0ba918668f02","family_id":"qf-density13-pythagorean-candidate-residual","seed":202610131083,"semantic_template":"qf-density13-pythagorean-candidate-residual/[\"145\",-10]"},"fields":{"title":"Hypotenuse Sign and Square Residual","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=8, b=9, candidate c=-10. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.","answer":{"kind":"expression","value":"(No, -45, No)","explanation":"Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (No, -45, No)."},"skills":["Hypotenuse Sign and Square Residual"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0084","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4c22c39796b006c31f51e2a54effdcf4152645c1eb27661cfab083095a1670bb","family_id":"qf-density13-pythagorean-candidate-residual","seed":202610131084,"semantic_template":"qf-density13-pythagorean-candidate-residual/[\"181\",11]"},"fields":{"title":"Hypotenuse Sign and Square Residual","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=9, b=10, candidate c=11. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.","answer":{"kind":"expression","value":"(Yes, -60, No)","explanation":"Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (Yes, -60, No)."},"skills":["Hypotenuse Sign and Square Residual"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0085","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e87ce6ffc3c6035ea91ed3be1f55f260f9da26d0ccb044ef315fae4c2114da44","family_id":"qf-density13-pythagorean-candidate-residual","seed":202610131085,"semantic_template":"qf-density13-pythagorean-candidate-residual/[\"221\",-12]"},"fields":{"title":"Hypotenuse Sign and Square Residual","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=10, b=11, candidate c=-12. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.","answer":{"kind":"expression","value":"(No, -77, No)","explanation":"Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (No, -77, No)."},"skills":["Hypotenuse Sign and Square Residual"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0086","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a84526268f1b98e51608d4d30c9b3dee95998df7ee1f2749df33018ef97a3f43","family_id":"qf-density13-pythagorean-candidate-residual","seed":202610131086,"semantic_template":"qf-density13-pythagorean-candidate-residual/[\"265\",13]"},"fields":{"title":"Hypotenuse Sign and Square Residual","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=11, b=12, candidate c=13. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.","answer":{"kind":"expression","value":"(Yes, -96, No)","explanation":"Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (Yes, -96, No)."},"skills":["Hypotenuse Sign and Square Residual"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0087","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07d005d4c4b54382a0c140657671d28fdcd9f735995c409dfdcb6e102ad1c5d3","family_id":"qf-density13-pythagorean-candidate-residual","seed":202610131087,"semantic_template":"qf-density13-pythagorean-candidate-residual/[\"313\",-14]"},"fields":{"title":"Hypotenuse Sign and Square Residual","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=12, b=13, candidate c=-14. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.","answer":{"kind":"expression","value":"(No, -117, No)","explanation":"Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (No, -117, No)."},"skills":["Hypotenuse Sign and Square Residual"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0088","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2f4716533c137fa1318b2d5c4da04404ca97fd8e336ddb688631ca456cd7785","family_id":"qf-density13-pythagorean-candidate-residual","seed":202610131088,"semantic_template":"qf-density13-pythagorean-candidate-residual/[\"365\",15]"},"fields":{"title":"Hypotenuse Sign and Square Residual","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=13, b=14, candidate c=15. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.","answer":{"kind":"expression","value":"(Yes, -140, No)","explanation":"Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (Yes, -140, No)."},"skills":["Hypotenuse Sign and Square Residual"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0089","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e59395e4f9f737f4713065d24fb3e255d530e703e01da8e829c08ce2b4d1ae8","family_id":"qf-density13-pythagorean-candidate-residual","seed":202610131089,"semantic_template":"qf-density13-pythagorean-candidate-residual/[\"421\",-16]"},"fields":{"title":"Hypotenuse Sign and Square Residual","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=14, b=15, candidate c=-16. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.","answer":{"kind":"expression","value":"(No, -165, No)","explanation":"Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (No, -165, No)."},"skills":["Hypotenuse Sign and Square Residual"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0090","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8691b44d41be57239d5c6086087b232e31014bc4a8e9080e771124dfc7cfa555","family_id":"qf-density13-pythagorean-candidate-residual","seed":202610131090,"semantic_template":"qf-density13-pythagorean-candidate-residual/[\"481\",17]"},"fields":{"title":"Hypotenuse Sign and Square Residual","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=15, b=16, candidate c=17. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.","answer":{"kind":"expression","value":"(Yes, -192, No)","explanation":"Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (Yes, -192, No)."},"skills":["Hypotenuse Sign and Square Residual"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0091","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c98445a820d2d672927445df33f931fe4756290d74e1ac0209b42dfc50f0216","family_id":"qf-density13-pythagorean-candidate-residual","seed":202610131091,"semantic_template":"qf-density13-pythagorean-candidate-residual/[\"545\",-18]"},"fields":{"title":"Hypotenuse Sign and Square Residual","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=16, b=17, candidate c=-18. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.","answer":{"kind":"expression","value":"(No, -221, No)","explanation":"Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (No, -221, No)."},"skills":["Hypotenuse Sign and Square Residual"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0092","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45d2b23e21689ada7a72ccc57d7383dc5416c4365a11861cb24d277c5f741086","family_id":"qf-density13-pythagorean-candidate-residual","seed":202610131092,"semantic_template":"qf-density13-pythagorean-candidate-residual/[\"613\",19]"},"fields":{"title":"Hypotenuse Sign and Square Residual","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=17, b=18, candidate c=19. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.","answer":{"kind":"expression","value":"(Yes, -252, No)","explanation":"Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (Yes, -252, No)."},"skills":["Hypotenuse Sign and Square Residual"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0093","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c8e7508b25e6390a1c0ae9fce4634215cd10e3bd0c0ba080d82ad1a29bef39e","family_id":"qf-density13-pythagorean-candidate-residual","seed":202610131093,"semantic_template":"qf-density13-pythagorean-candidate-residual/[\"685\",-20]"},"fields":{"title":"Hypotenuse Sign and Square Residual","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=18, b=19, candidate c=-20. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.","answer":{"kind":"expression","value":"(No, -285, No)","explanation":"Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (No, -285, No)."},"skills":["Hypotenuse Sign and Square Residual"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0094","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1345837fb06eec9f5d4515e23c1e79cbca9d82a52aaa0c833c4c1763db921d41","family_id":"qf-density13-pythagorean-candidate-residual","seed":202610131094,"semantic_template":"qf-density13-pythagorean-candidate-residual/[\"761\",21]"},"fields":{"title":"Hypotenuse Sign and Square Residual","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=19, b=20, candidate c=21. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.","answer":{"kind":"expression","value":"(Yes, -320, No)","explanation":"Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (Yes, -320, No)."},"skills":["Hypotenuse Sign and Square Residual"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0095","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e5d8c70013e5caa1bc175ac7a8065663c7c4947c182a2d802e6d4155f0af71f8","family_id":"qf-density13-pythagorean-candidate-residual","seed":202610131095,"semantic_template":"qf-density13-pythagorean-candidate-residual/[\"841\",-22]"},"fields":{"title":"Hypotenuse Sign and Square Residual","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=20, b=21, candidate c=-22. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.","answer":{"kind":"expression","value":"(No, -357, No)","explanation":"Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (No, -357, No)."},"skills":["Hypotenuse Sign and Square Residual"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0096","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4582a26170d24a49a29857adeb9645dbe88d7a928d5e7d71037b962f1c38a1f4","family_id":"qf-density13-pythagorean-candidate-residual","seed":202610131096,"semantic_template":"qf-density13-pythagorean-candidate-residual/[\"925\",23]"},"fields":{"title":"Hypotenuse Sign and Square Residual","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=21, b=22, candidate c=23. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.","answer":{"kind":"expression","value":"(Yes, -396, No)","explanation":"Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (Yes, -396, No)."},"skills":["Hypotenuse Sign and Square Residual"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0097","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71ec92d73837bd990713e65868cfa38aebd64fa0f80215b7ba0da87af62af8c1","family_id":"qf-density13-pythagorean-candidate-residual","seed":202610131097,"semantic_template":"qf-density13-pythagorean-candidate-residual/[\"1013\",-24]"},"fields":{"title":"Hypotenuse Sign and Square Residual","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=22, b=23, candidate c=-24. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.","answer":{"kind":"expression","value":"(No, -437, No)","explanation":"Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (No, -437, No)."},"skills":["Hypotenuse Sign and Square Residual"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0098","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0cf7828332994041b453f6c3ab321a3e56b529ceb6770ca778a418ed0711a4cc","family_id":"qf-density13-pythagorean-candidate-residual","seed":202610131098,"semantic_template":"qf-density13-pythagorean-candidate-residual/[\"1105\",25]"},"fields":{"title":"Hypotenuse Sign and Square Residual","stem":"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.","answer":{"kind":"expression","value":"(Yes, -480, No)","explanation":"Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (Yes, -480, No)."},"skills":["Hypotenuse Sign and Square Residual"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0099","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"77ecea405617548fdb7286fa0b70648d770e4cdb28ad8f0641eccc846208cf03","family_id":"qf-density13-pythagorean-candidate-residual","seed":202610131099,"semantic_template":"qf-density13-pythagorean-candidate-residual/[\"1201\",-26]"},"fields":{"title":"Hypotenuse Sign and Square Residual","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=24, b=25, candidate c=-26. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.","answer":{"kind":"expression","value":"(No, -525, No)","explanation":"Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (No, -525, No)."},"skills":["Hypotenuse Sign and Square Residual"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r01-pythagorean-theorem-0100","batch_id":"batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6c21b15f7a191841ea9116913b7b3956d54e6eedda51d248e3f6ea6251f0ebd","family_id":"qf-density13-pythagorean-candidate-residual","seed":202610131100,"semantic_template":"qf-density13-pythagorean-candidate-residual/[\"1301\",27]"},"fields":{"title":"Hypotenuse Sign and Square Residual","stem":"Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=25, b=26, candidate c=27. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.","answer":{"kind":"expression","value":"(Yes, -572, No)","explanation":"Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (Yes, -572, No)."},"skills":["Hypotenuse Sign and Square Residual"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}