{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r01-boundary-value-problem-0001","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"527aa7722e5d241a8927fc89a291b59d665a6d677eed060cb8fc70f6177d44b8","family_id":"qf-mod25-bvp-interior-pde-identity","seed":202610251001,"semantic_template":"qf-mod25-bvp-interior-pde-identity/{\"coefficients\":[-12,-2,-3,1,-1,-2],\"source\":-1}"},"fields":{"title":"An Interior Poisson Identity without a Boundary Claim","stem":"Use the quadratic candidate coefficients=[-12, -2, -3, 1, -1, -2] in fixed order [1,x,y,x²,x*y,y²]. Interior equation u_xx+u_yy=source has source=-1. Give (coefficients,u_xx+u_yy=source,constant Laplacian,source,signed residual,interior identity?,interior_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([-12, -2, -3, 1, -1, -2], u_xx+u_yy=source, -2, -1, -1, No, interior_only)","explanation":"The Laplacian is 2*c_xx+2*c_yy for this candidate, so a zero constant residual is an interior identity only. No boundary data have been checked. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-12, -2, -3, 1, -1, -2], u_xx+u_yy=source, -2, -1, -1, No, interior_only)."},"skills":["An Interior Poisson Identity without a Boundary Claim"],"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-20261013-advance-20261025-r01-boundary-value-problem-0002","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07618b544ad1bae7b6b549cd54ff95286ad88383c653ecb18012008b41b7abab","family_id":"qf-mod25-bvp-interior-pde-identity","seed":202610251002,"semantic_template":"qf-mod25-bvp-interior-pde-identity/{\"coefficients\":[-11,-1,-2,2,0,-1],\"source\":2}"},"fields":{"title":"An Interior Poisson Identity without a Boundary Claim","stem":"Use the quadratic candidate coefficients=[-11, -1, -2, 2, 0, -1] in fixed order [1,x,y,x²,x*y,y²]. Interior equation u_xx+u_yy=source has source=2. Give (coefficients,u_xx+u_yy=source,constant Laplacian,source,signed residual,interior identity?,interior_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([-11, -1, -2, 2, 0, -1], u_xx+u_yy=source, 2, 2, 0, Yes, interior_only)","explanation":"The Laplacian is 2*c_xx+2*c_yy for this candidate, so a zero constant residual is an interior identity only. No boundary data have been checked. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-11, -1, -2, 2, 0, -1], u_xx+u_yy=source, 2, 2, 0, Yes, interior_only)."},"skills":["An Interior Poisson Identity without a Boundary Claim"],"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-20261013-advance-20261025-r01-boundary-value-problem-0003","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"561fb6b5effe287364c5717172d221a59e595c9b8367ee2de6afe0021ddd34a0","family_id":"qf-mod25-bvp-interior-pde-identity","seed":202610251003,"semantic_template":"qf-mod25-bvp-interior-pde-identity/{\"coefficients\":[-10,0,-1,3,1,0],\"source\":6}"},"fields":{"title":"An Interior Poisson Identity without a Boundary Claim","stem":"Use the quadratic candidate coefficients=[-10, 0, -1, 3, 1, 0] in fixed order [1,x,y,x²,x*y,y²]. Interior equation u_xx+u_yy=source has source=6. Give (coefficients,u_xx+u_yy=source,constant Laplacian,source,signed residual,interior identity?,interior_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([-10, 0, -1, 3, 1, 0], u_xx+u_yy=source, 6, 6, 0, Yes, interior_only)","explanation":"The Laplacian is 2*c_xx+2*c_yy for this candidate, so a zero constant residual is an interior identity only. No boundary data have been checked. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-10, 0, -1, 3, 1, 0], u_xx+u_yy=source, 6, 6, 0, Yes, interior_only)."},"skills":["An Interior Poisson Identity without a Boundary Claim"],"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-20261013-advance-20261025-r01-boundary-value-problem-0004","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"42c4ca14be09cb08202230d26b2cc785622d71d2cb1f06d8145fdf82eac6232d","family_id":"qf-mod25-bvp-interior-pde-identity","seed":202610251004,"semantic_template":"qf-mod25-bvp-interior-pde-identity/{\"coefficients\":[-9,1,0,4,-1,1],\"source\":10}"},"fields":{"title":"An Interior Poisson Identity without a Boundary Claim","stem":"Use the quadratic candidate coefficients=[-9, 1, 0, 4, -1, 1] in fixed order [1,x,y,x²,x*y,y²]. Interior equation u_xx+u_yy=source has source=10. Give (coefficients,u_xx+u_yy=source,constant Laplacian,source,signed residual,interior identity?,interior_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([-9, 1, 0, 4, -1, 1], u_xx+u_yy=source, 10, 10, 0, Yes, interior_only)","explanation":"The Laplacian is 2*c_xx+2*c_yy for this candidate, so a zero constant residual is an interior identity only. No boundary data have been checked. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 1, 0, 4, -1, 1], u_xx+u_yy=source, 10, 10, 0, Yes, interior_only)."},"skills":["An Interior Poisson Identity without a Boundary Claim"],"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-20261013-advance-20261025-r01-boundary-value-problem-0005","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f6e4d25b9bd4ab279adf8ce2eac1d1fd22c0323862e532910f3511ff289c5d6","family_id":"qf-mod25-bvp-interior-pde-identity","seed":202610251005,"semantic_template":"qf-mod25-bvp-interior-pde-identity/{\"coefficients\":[-8,2,1,1,0,2],\"source\":7}"},"fields":{"title":"An Interior Poisson Identity without a Boundary Claim","stem":"Use the quadratic candidate coefficients=[-8, 2, 1, 1, 0, 2] in fixed order [1,x,y,x²,x*y,y²]. Interior equation u_xx+u_yy=source has source=7. Give (coefficients,u_xx+u_yy=source,constant Laplacian,source,signed residual,interior identity?,interior_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([-8, 2, 1, 1, 0, 2], u_xx+u_yy=source, 6, 7, -1, No, interior_only)","explanation":"The Laplacian is 2*c_xx+2*c_yy for this candidate, so a zero constant residual is an interior identity only. No boundary data have been checked. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 2, 1, 1, 0, 2], u_xx+u_yy=source, 6, 7, -1, No, interior_only)."},"skills":["An Interior Poisson Identity without a Boundary Claim"],"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-20261013-advance-20261025-r01-boundary-value-problem-0006","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"905a65c60bc1b1dfe4c9e181a813c51d36ed16610ac9b7b11e271c1c97c5a7be","family_id":"qf-mod25-bvp-interior-pde-identity","seed":202610251006,"semantic_template":"qf-mod25-bvp-interior-pde-identity/{\"coefficients\":[-7,-2,2,2,1,-2],\"source\":0}"},"fields":{"title":"An Interior Poisson Identity without a Boundary Claim","stem":"Use the quadratic candidate coefficients=[-7, -2, 2, 2, 1, -2] in fixed order [1,x,y,x²,x*y,y²]. Interior equation u_xx+u_yy=source has source=0. Give (coefficients,u_xx+u_yy=source,constant Laplacian,source,signed residual,interior identity?,interior_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([-7, -2, 2, 2, 1, -2], u_xx+u_yy=source, 0, 0, 0, Yes, interior_only)","explanation":"The Laplacian is 2*c_xx+2*c_yy for this candidate, so a zero constant residual is an interior identity only. No boundary data have been checked. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-7, -2, 2, 2, 1, -2], u_xx+u_yy=source, 0, 0, 0, Yes, interior_only)."},"skills":["An Interior Poisson Identity without a Boundary Claim"],"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-20261013-advance-20261025-r01-boundary-value-problem-0007","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8fcdd2c348626bcc5fb502f7c93f751cc46b38ca1fa9e2337fab4b0f48fdf801","family_id":"qf-mod25-bvp-interior-pde-identity","seed":202610251007,"semantic_template":"qf-mod25-bvp-interior-pde-identity/{\"coefficients\":[-6,-1,3,3,-1,-1],\"source\":4}"},"fields":{"title":"An Interior Poisson Identity without a Boundary Claim","stem":"Use the quadratic candidate coefficients=[-6, -1, 3, 3, -1, -1] in fixed order [1,x,y,x²,x*y,y²]. Interior equation u_xx+u_yy=source has source=4. Give (coefficients,u_xx+u_yy=source,constant Laplacian,source,signed residual,interior identity?,interior_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([-6, -1, 3, 3, -1, -1], u_xx+u_yy=source, 4, 4, 0, Yes, interior_only)","explanation":"The Laplacian is 2*c_xx+2*c_yy for this candidate, so a zero constant residual is an interior identity only. No boundary data have been checked. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-6, -1, 3, 3, -1, -1], u_xx+u_yy=source, 4, 4, 0, Yes, interior_only)."},"skills":["An Interior Poisson Identity without a Boundary Claim"],"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-20261013-advance-20261025-r01-boundary-value-problem-0008","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c8ebee76046c8e81864c7a3071373eb92d1f6a6ae8cbf68e891f2908f8ba934a","family_id":"qf-mod25-bvp-interior-pde-identity","seed":202610251008,"semantic_template":"qf-mod25-bvp-interior-pde-identity/{\"coefficients\":[-5,0,-3,4,0,0],\"source\":8}"},"fields":{"title":"An Interior Poisson Identity without a Boundary Claim","stem":"Use the quadratic candidate coefficients=[-5, 0, -3, 4, 0, 0] in fixed order [1,x,y,x²,x*y,y²]. Interior equation u_xx+u_yy=source has source=8. Give (coefficients,u_xx+u_yy=source,constant Laplacian,source,signed residual,interior identity?,interior_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([-5, 0, -3, 4, 0, 0], u_xx+u_yy=source, 8, 8, 0, Yes, interior_only)","explanation":"The Laplacian is 2*c_xx+2*c_yy for this candidate, so a zero constant residual is an interior identity only. No boundary data have been checked. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-5, 0, -3, 4, 0, 0], u_xx+u_yy=source, 8, 8, 0, Yes, interior_only)."},"skills":["An Interior Poisson Identity without a Boundary Claim"],"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-20261013-advance-20261025-r01-boundary-value-problem-0009","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ee97ad500a426b7dc0e9b024da14e42ef1bbbf292e1d01d261edf626913a5cf","family_id":"qf-mod25-bvp-interior-pde-identity","seed":202610251009,"semantic_template":"qf-mod25-bvp-interior-pde-identity/{\"coefficients\":[-4,1,-2,1,1,1],\"source\":5}"},"fields":{"title":"An Interior Poisson Identity without a Boundary Claim","stem":"Use the quadratic candidate coefficients=[-4, 1, -2, 1, 1, 1] in fixed order [1,x,y,x²,x*y,y²]. Interior equation u_xx+u_yy=source has source=5. Give (coefficients,u_xx+u_yy=source,constant Laplacian,source,signed residual,interior identity?,interior_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([-4, 1, -2, 1, 1, 1], u_xx+u_yy=source, 4, 5, -1, No, interior_only)","explanation":"The Laplacian is 2*c_xx+2*c_yy for this candidate, so a zero constant residual is an interior identity only. No boundary data have been checked. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-4, 1, -2, 1, 1, 1], u_xx+u_yy=source, 4, 5, -1, No, interior_only)."},"skills":["An Interior Poisson Identity without a Boundary Claim"],"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-20261013-advance-20261025-r01-boundary-value-problem-0010","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"054d9982248ae56f57a451af55f2c4078fa194a00907b6b97a4bb51834f50d7d","family_id":"qf-mod25-bvp-interior-pde-identity","seed":202610251010,"semantic_template":"qf-mod25-bvp-interior-pde-identity/{\"coefficients\":[-3,2,-1,2,-1,2],\"source\":8}"},"fields":{"title":"An Interior Poisson Identity without a Boundary Claim","stem":"Use the quadratic candidate coefficients=[-3, 2, -1, 2, -1, 2] in fixed order [1,x,y,x²,x*y,y²]. Interior equation u_xx+u_yy=source has source=8. Give (coefficients,u_xx+u_yy=source,constant Laplacian,source,signed residual,interior identity?,interior_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([-3, 2, -1, 2, -1, 2], u_xx+u_yy=source, 8, 8, 0, Yes, interior_only)","explanation":"The Laplacian is 2*c_xx+2*c_yy for this candidate, so a zero constant residual is an interior identity only. No boundary data have been checked. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-3, 2, -1, 2, -1, 2], u_xx+u_yy=source, 8, 8, 0, Yes, interior_only)."},"skills":["An Interior Poisson Identity without a Boundary Claim"],"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-20261013-advance-20261025-r01-boundary-value-problem-0011","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fff7addb2f81708c4e907f9cde756eb4869f93c87890b134c0eb1e53ab5d873c","family_id":"qf-mod25-bvp-interior-pde-identity","seed":202610251011,"semantic_template":"qf-mod25-bvp-interior-pde-identity/{\"coefficients\":[-2,-2,0,3,0,-2],\"source\":2}"},"fields":{"title":"An Interior Poisson Identity without a Boundary Claim","stem":"Use the quadratic candidate coefficients=[-2, -2, 0, 3, 0, -2] in fixed order [1,x,y,x²,x*y,y²]. Interior equation u_xx+u_yy=source has source=2. Give (coefficients,u_xx+u_yy=source,constant Laplacian,source,signed residual,interior identity?,interior_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([-2, -2, 0, 3, 0, -2], u_xx+u_yy=source, 2, 2, 0, Yes, interior_only)","explanation":"The Laplacian is 2*c_xx+2*c_yy for this candidate, so a zero constant residual is an interior identity only. No boundary data have been checked. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-2, -2, 0, 3, 0, -2], u_xx+u_yy=source, 2, 2, 0, Yes, interior_only)."},"skills":["An Interior Poisson Identity without a Boundary Claim"],"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-20261013-advance-20261025-r01-boundary-value-problem-0012","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dc32b0e3417b3c1447f87a44c815083dbe0ab90bc8acfaeb22e42d8016c4953f","family_id":"qf-mod25-bvp-interior-pde-identity","seed":202610251012,"semantic_template":"qf-mod25-bvp-interior-pde-identity/{\"coefficients\":[-1,-1,1,4,1,-1],\"source\":6}"},"fields":{"title":"An Interior Poisson Identity without a Boundary Claim","stem":"Use the quadratic candidate coefficients=[-1, -1, 1, 4, 1, -1] in fixed order [1,x,y,x²,x*y,y²]. Interior equation u_xx+u_yy=source has source=6. Give (coefficients,u_xx+u_yy=source,constant Laplacian,source,signed residual,interior identity?,interior_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([-1, -1, 1, 4, 1, -1], u_xx+u_yy=source, 6, 6, 0, Yes, interior_only)","explanation":"The Laplacian is 2*c_xx+2*c_yy for this candidate, so a zero constant residual is an interior identity only. No boundary data have been checked. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-1, -1, 1, 4, 1, -1], u_xx+u_yy=source, 6, 6, 0, Yes, interior_only)."},"skills":["An Interior Poisson Identity without a Boundary Claim"],"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-20261013-advance-20261025-r01-boundary-value-problem-0013","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc317875bdb5e4776fb128e5b4e02469996634f130dcb676621873b3953e0ef8","family_id":"qf-mod25-bvp-interior-pde-identity","seed":202610251013,"semantic_template":"qf-mod25-bvp-interior-pde-identity/{\"coefficients\":[0,0,2,1,-1,0],\"source\":3}"},"fields":{"title":"An Interior Poisson Identity without a Boundary Claim","stem":"Use the quadratic candidate coefficients=[0, 0, 2, 1, -1, 0] in fixed order [1,x,y,x²,x*y,y²]. Interior equation u_xx+u_yy=source has source=3. Give (coefficients,u_xx+u_yy=source,constant Laplacian,source,signed residual,interior identity?,interior_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([0, 0, 2, 1, -1, 0], u_xx+u_yy=source, 2, 3, -1, No, interior_only)","explanation":"The Laplacian is 2*c_xx+2*c_yy for this candidate, so a zero constant residual is an interior identity only. No boundary data have been checked. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0, 2, 1, -1, 0], u_xx+u_yy=source, 2, 3, -1, No, interior_only)."},"skills":["An Interior Poisson Identity without a Boundary Claim"],"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-20261013-advance-20261025-r01-boundary-value-problem-0014","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0073becba7ed54a224c7dd59828da8708f8a146a44385419e14f497b1249de36","family_id":"qf-mod25-bvp-interior-pde-identity","seed":202610251014,"semantic_template":"qf-mod25-bvp-interior-pde-identity/{\"coefficients\":[1,1,3,2,0,1],\"source\":6}"},"fields":{"title":"An Interior Poisson Identity without a Boundary Claim","stem":"Use the quadratic candidate coefficients=[1, 1, 3, 2, 0, 1] in fixed order [1,x,y,x²,x*y,y²]. Interior equation u_xx+u_yy=source has source=6. Give (coefficients,u_xx+u_yy=source,constant Laplacian,source,signed residual,interior identity?,interior_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([1, 1, 3, 2, 0, 1], u_xx+u_yy=source, 6, 6, 0, Yes, interior_only)","explanation":"The Laplacian is 2*c_xx+2*c_yy for this candidate, so a zero constant residual is an interior identity only. No boundary data have been checked. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1, 3, 2, 0, 1], u_xx+u_yy=source, 6, 6, 0, Yes, interior_only)."},"skills":["An Interior Poisson Identity without a Boundary Claim"],"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-20261013-advance-20261025-r01-boundary-value-problem-0015","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe72ee777b0a0b3b17fda394deab5ebfd102c4717b91d371ad0c943d71df2115","family_id":"qf-mod25-bvp-interior-pde-identity","seed":202610251015,"semantic_template":"qf-mod25-bvp-interior-pde-identity/{\"coefficients\":[2,2,-3,3,1,2],\"source\":10}"},"fields":{"title":"An Interior Poisson Identity without a Boundary Claim","stem":"Use the quadratic candidate coefficients=[2, 2, -3, 3, 1, 2] in fixed order [1,x,y,x²,x*y,y²]. Interior equation u_xx+u_yy=source has source=10. Give (coefficients,u_xx+u_yy=source,constant Laplacian,source,signed residual,interior identity?,interior_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([2, 2, -3, 3, 1, 2], u_xx+u_yy=source, 10, 10, 0, Yes, interior_only)","explanation":"The Laplacian is 2*c_xx+2*c_yy for this candidate, so a zero constant residual is an interior identity only. No boundary data have been checked. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 2, -3, 3, 1, 2], u_xx+u_yy=source, 10, 10, 0, Yes, interior_only)."},"skills":["An Interior Poisson Identity without a Boundary Claim"],"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-20261013-advance-20261025-r01-boundary-value-problem-0016","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"57908ee32aedc15fd9b2a30cc71b3f0e5e029aba6492243961cec154a3efde32","family_id":"qf-mod25-bvp-interior-pde-identity","seed":202610251016,"semantic_template":"qf-mod25-bvp-interior-pde-identity/{\"coefficients\":[3,-2,-2,4,-1,-2],\"source\":4}"},"fields":{"title":"An Interior Poisson Identity without a Boundary Claim","stem":"Use the quadratic candidate coefficients=[3, -2, -2, 4, -1, -2] in fixed order [1,x,y,x²,x*y,y²]. Interior equation u_xx+u_yy=source has source=4. Give (coefficients,u_xx+u_yy=source,constant Laplacian,source,signed residual,interior identity?,interior_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([3, -2, -2, 4, -1, -2], u_xx+u_yy=source, 4, 4, 0, Yes, interior_only)","explanation":"The Laplacian is 2*c_xx+2*c_yy for this candidate, so a zero constant residual is an interior identity only. No boundary data have been checked. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, -2, -2, 4, -1, -2], u_xx+u_yy=source, 4, 4, 0, Yes, interior_only)."},"skills":["An Interior Poisson Identity without a Boundary Claim"],"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-20261013-advance-20261025-r01-boundary-value-problem-0017","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f7b40eb11cfade82940ca786c06708f5f4d50d0cce09679dcefd7e997f2a658","family_id":"qf-mod25-bvp-interior-pde-identity","seed":202610251017,"semantic_template":"qf-mod25-bvp-interior-pde-identity/{\"coefficients\":[4,-1,-1,1,0,-1],\"source\":1}"},"fields":{"title":"An Interior Poisson Identity without a Boundary Claim","stem":"Use the quadratic candidate coefficients=[4, -1, -1, 1, 0, -1] in fixed order [1,x,y,x²,x*y,y²]. Interior equation u_xx+u_yy=source has source=1. Give (coefficients,u_xx+u_yy=source,constant Laplacian,source,signed residual,interior identity?,interior_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([4, -1, -1, 1, 0, -1], u_xx+u_yy=source, 0, 1, -1, No, interior_only)","explanation":"The Laplacian is 2*c_xx+2*c_yy for this candidate, so a zero constant residual is an interior identity only. No boundary data have been checked. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([4, -1, -1, 1, 0, -1], u_xx+u_yy=source, 0, 1, -1, No, interior_only)."},"skills":["An Interior Poisson Identity without a Boundary Claim"],"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-20261013-advance-20261025-r01-boundary-value-problem-0018","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7bdd052cc1dde29e49abee4d0ba0013b4ef2f18f3ca4824252f51c6aff40b58f","family_id":"qf-mod25-bvp-interior-pde-identity","seed":202610251018,"semantic_template":"qf-mod25-bvp-interior-pde-identity/{\"coefficients\":[5,0,0,2,1,0],\"source\":4}"},"fields":{"title":"An Interior Poisson Identity without a Boundary Claim","stem":"Use the quadratic candidate coefficients=[5, 0, 0, 2, 1, 0] in fixed order [1,x,y,x²,x*y,y²]. Interior equation u_xx+u_yy=source has source=4. Give (coefficients,u_xx+u_yy=source,constant Laplacian,source,signed residual,interior identity?,interior_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([5, 0, 0, 2, 1, 0], u_xx+u_yy=source, 4, 4, 0, Yes, interior_only)","explanation":"The Laplacian is 2*c_xx+2*c_yy for this candidate, so a zero constant residual is an interior identity only. No boundary data have been checked. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([5, 0, 0, 2, 1, 0], u_xx+u_yy=source, 4, 4, 0, Yes, interior_only)."},"skills":["An Interior Poisson Identity without a Boundary Claim"],"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-20261013-advance-20261025-r01-boundary-value-problem-0019","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"608e314dabf847c68324a330dd27f9468c28252b30c8414a7d507970f1969ed0","family_id":"qf-mod25-bvp-interior-pde-identity","seed":202610251019,"semantic_template":"qf-mod25-bvp-interior-pde-identity/{\"coefficients\":[6,1,1,3,-1,1],\"source\":8}"},"fields":{"title":"An Interior Poisson Identity without a Boundary Claim","stem":"Use the quadratic candidate coefficients=[6, 1, 1, 3, -1, 1] in fixed order [1,x,y,x²,x*y,y²]. Interior equation u_xx+u_yy=source has source=8. Give (coefficients,u_xx+u_yy=source,constant Laplacian,source,signed residual,interior identity?,interior_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([6, 1, 1, 3, -1, 1], u_xx+u_yy=source, 8, 8, 0, Yes, interior_only)","explanation":"The Laplacian is 2*c_xx+2*c_yy for this candidate, so a zero constant residual is an interior identity only. No boundary data have been checked. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([6, 1, 1, 3, -1, 1], u_xx+u_yy=source, 8, 8, 0, Yes, interior_only)."},"skills":["An Interior Poisson Identity without a Boundary Claim"],"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-20261013-advance-20261025-r01-boundary-value-problem-0020","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12dc34017cc372a3469de1393689c1a1c829997f98996420fca18b74f6bd9b23","family_id":"qf-mod25-bvp-interior-pde-identity","seed":202610251020,"semantic_template":"qf-mod25-bvp-interior-pde-identity/{\"coefficients\":[7,2,2,4,0,2],\"source\":12}"},"fields":{"title":"An Interior Poisson Identity without a Boundary Claim","stem":"Use the quadratic candidate coefficients=[7, 2, 2, 4, 0, 2] in fixed order [1,x,y,x²,x*y,y²]. Interior equation u_xx+u_yy=source has source=12. Give (coefficients,u_xx+u_yy=source,constant Laplacian,source,signed residual,interior identity?,interior_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([7, 2, 2, 4, 0, 2], u_xx+u_yy=source, 12, 12, 0, Yes, interior_only)","explanation":"The Laplacian is 2*c_xx+2*c_yy for this candidate, so a zero constant residual is an interior identity only. No boundary data have been checked. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([7, 2, 2, 4, 0, 2], u_xx+u_yy=source, 12, 12, 0, Yes, interior_only)."},"skills":["An Interior Poisson Identity without a Boundary Claim"],"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-20261013-advance-20261025-r01-boundary-value-problem-0021","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6da87fb77713e253f66ec8a4ba24af1eeeedec98ae1ab554d4ce5d7c26bd1072","family_id":"qf-mod25-bvp-interior-pde-identity","seed":202610251021,"semantic_template":"qf-mod25-bvp-interior-pde-identity/{\"coefficients\":[8,-2,3,1,1,-2],\"source\":-1}"},"fields":{"title":"An Interior Poisson Identity without a Boundary Claim","stem":"Use the quadratic candidate coefficients=[8, -2, 3, 1, 1, -2] in fixed order [1,x,y,x²,x*y,y²]. Interior equation u_xx+u_yy=source has source=-1. Give (coefficients,u_xx+u_yy=source,constant Laplacian,source,signed residual,interior identity?,interior_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([8, -2, 3, 1, 1, -2], u_xx+u_yy=source, -2, -1, -1, No, interior_only)","explanation":"The Laplacian is 2*c_xx+2*c_yy for this candidate, so a zero constant residual is an interior identity only. No boundary data have been checked. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([8, -2, 3, 1, 1, -2], u_xx+u_yy=source, -2, -1, -1, No, interior_only)."},"skills":["An Interior Poisson Identity without a Boundary Claim"],"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-20261013-advance-20261025-r01-boundary-value-problem-0022","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c737eb7c80c7d7698a69f899dfb5f15caa78b91c8670937724445067c9c5ca34","family_id":"qf-mod25-bvp-interior-pde-identity","seed":202610251022,"semantic_template":"qf-mod25-bvp-interior-pde-identity/{\"coefficients\":[9,-1,-3,2,-1,-1],\"source\":2}"},"fields":{"title":"An Interior Poisson Identity without a Boundary Claim","stem":"Use the quadratic candidate coefficients=[9, -1, -3, 2, -1, -1] in fixed order [1,x,y,x²,x*y,y²]. Interior equation u_xx+u_yy=source has source=2. Give (coefficients,u_xx+u_yy=source,constant Laplacian,source,signed residual,interior identity?,interior_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([9, -1, -3, 2, -1, -1], u_xx+u_yy=source, 2, 2, 0, Yes, interior_only)","explanation":"The Laplacian is 2*c_xx+2*c_yy for this candidate, so a zero constant residual is an interior identity only. No boundary data have been checked. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([9, -1, -3, 2, -1, -1], u_xx+u_yy=source, 2, 2, 0, Yes, interior_only)."},"skills":["An Interior Poisson Identity without a Boundary Claim"],"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-20261013-advance-20261025-r01-boundary-value-problem-0023","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1dcaeba8b2a9f75375c861e892b3836efc63d68626fe40d7c7c234fafb16762c","family_id":"qf-mod25-bvp-interior-pde-identity","seed":202610251023,"semantic_template":"qf-mod25-bvp-interior-pde-identity/{\"coefficients\":[10,0,-2,3,0,0],\"source\":6}"},"fields":{"title":"An Interior Poisson Identity without a Boundary Claim","stem":"Use the quadratic candidate coefficients=[10, 0, -2, 3, 0, 0] in fixed order [1,x,y,x²,x*y,y²]. Interior equation u_xx+u_yy=source has source=6. Give (coefficients,u_xx+u_yy=source,constant Laplacian,source,signed residual,interior identity?,interior_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([10, 0, -2, 3, 0, 0], u_xx+u_yy=source, 6, 6, 0, Yes, interior_only)","explanation":"The Laplacian is 2*c_xx+2*c_yy for this candidate, so a zero constant residual is an interior identity only. No boundary data have been checked. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([10, 0, -2, 3, 0, 0], u_xx+u_yy=source, 6, 6, 0, Yes, interior_only)."},"skills":["An Interior Poisson Identity without a Boundary Claim"],"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-20261013-advance-20261025-r01-boundary-value-problem-0024","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d5c8b0c1a1d061f0ca5c59d14f7d0dce467a8233b06f9a32a0d11ed929667330","family_id":"qf-mod25-bvp-interior-pde-identity","seed":202610251024,"semantic_template":"qf-mod25-bvp-interior-pde-identity/{\"coefficients\":[11,1,-1,4,1,1],\"source\":10}"},"fields":{"title":"An Interior Poisson Identity without a Boundary Claim","stem":"Use the quadratic candidate coefficients=[11, 1, -1, 4, 1, 1] in fixed order [1,x,y,x²,x*y,y²]. Interior equation u_xx+u_yy=source has source=10. Give (coefficients,u_xx+u_yy=source,constant Laplacian,source,signed residual,interior identity?,interior_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([11, 1, -1, 4, 1, 1], u_xx+u_yy=source, 10, 10, 0, Yes, interior_only)","explanation":"The Laplacian is 2*c_xx+2*c_yy for this candidate, so a zero constant residual is an interior identity only. No boundary data have been checked. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([11, 1, -1, 4, 1, 1], u_xx+u_yy=source, 10, 10, 0, Yes, interior_only)."},"skills":["An Interior Poisson Identity without a Boundary Claim"],"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-20261013-advance-20261025-r01-boundary-value-problem-0025","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"69d23fc9c4bdef6ee3bf46a8acec7d45b692d57add128c220fa1c852bd9cc0a4","family_id":"qf-mod25-bvp-interior-pde-identity","seed":202610251025,"semantic_template":"qf-mod25-bvp-interior-pde-identity/{\"coefficients\":[12,2,0,1,-1,2],\"source\":7}"},"fields":{"title":"An Interior Poisson Identity without a Boundary Claim","stem":"Use the quadratic candidate coefficients=[12, 2, 0, 1, -1, 2] in fixed order [1,x,y,x²,x*y,y²]. Interior equation u_xx+u_yy=source has source=7. Give (coefficients,u_xx+u_yy=source,constant Laplacian,source,signed residual,interior identity?,interior_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([12, 2, 0, 1, -1, 2], u_xx+u_yy=source, 6, 7, -1, No, interior_only)","explanation":"The Laplacian is 2*c_xx+2*c_yy for this candidate, so a zero constant residual is an interior identity only. No boundary data have been checked. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([12, 2, 0, 1, -1, 2], u_xx+u_yy=source, 6, 7, -1, No, interior_only)."},"skills":["An Interior Poisson Identity without a Boundary Claim"],"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-20261013-advance-20261025-r01-boundary-value-problem-0026","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e56d1f1d5b3b20d2f77182f2a184b8246f91c3d4244fccc593cea0add647821b","family_id":"qf-mod25-bvp-dirichlet-edge-identity","seed":202610251026,"semantic_template":"qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[-3,-1,-2],\"bounds\":[-2,2,-1,3],\"coefficients\":[-12,-2,-3,1,-1,-2],\"edge\":\"left\"}"},"fields":{"title":"A Whole-Edge Dirichlet Coefficient Identity","stem":"Use the quadratic candidate coefficients=[-12, -2, -3, 1, -1, -2] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-2, 2, -1, 3]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Selected edge=left; supplied coefficients=[-3, -1, -2]. Give (bounds,edge,remaining variable,restriction coefficients,supplied coefficients,coefficientwise identity?,declared_edge_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([-2, 2, -1, 3], left, y, [-4, -1, -2], [-3, -1, -2], No, declared_edge_only)","explanation":"Substitute the fixed edge coordinate and compare all three coefficients in the remaining variable. This proves only the declared edge identity, not all four boundaries or the interior equation. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-2, 2, -1, 3], left, y, [-4, -1, -2], [-3, -1, -2], No, declared_edge_only)."},"skills":["A Whole-Edge Dirichlet Coefficient Identity"],"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-20261013-advance-20261025-r01-boundary-value-problem-0027","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f9a0004e7d594001aaf87e5988f50f46cd227b256172593edfe2215b2065bf5a","family_id":"qf-mod25-bvp-dirichlet-edge-identity","seed":202610251027,"semantic_template":"qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[4,-2,-1],\"bounds\":[-3,3,-2,4],\"coefficients\":[-11,-1,-2,2,0,-1],\"edge\":\"right\"}"},"fields":{"title":"A Whole-Edge Dirichlet Coefficient Identity","stem":"Use the quadratic candidate coefficients=[-11, -1, -2, 2, 0, -1] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-3, 3, -2, 4]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Selected edge=right; supplied coefficients=[4, -2, -1]. Give (bounds,edge,remaining variable,restriction coefficients,supplied coefficients,coefficientwise identity?,declared_edge_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([-3, 3, -2, 4], right, y, [4, -2, -1], [4, -2, -1], Yes, declared_edge_only)","explanation":"Substitute the fixed edge coordinate and compare all three coefficients in the remaining variable. This proves only the declared edge identity, not all four boundaries or the interior equation. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-3, 3, -2, 4], right, y, [4, -2, -1], [4, -2, -1], Yes, declared_edge_only)."},"skills":["A Whole-Edge Dirichlet Coefficient Identity"],"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-20261013-advance-20261025-r01-boundary-value-problem-0028","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cdba3a6615ca0da0effe8300a861167d84fc35332ab16bd7de585d57f2fe930f","family_id":"qf-mod25-bvp-dirichlet-edge-identity","seed":202610251028,"semantic_template":"qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[-7,-3,3],\"bounds\":[-4,4,-3,5],\"coefficients\":[-10,0,-1,3,1,0],\"edge\":\"bottom\"}"},"fields":{"title":"A Whole-Edge Dirichlet Coefficient Identity","stem":"Use the quadratic candidate coefficients=[-10, 0, -1, 3, 1, 0] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-4, 4, -3, 5]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Selected edge=bottom; supplied coefficients=[-7, -3, 3]. Give (bounds,edge,remaining variable,restriction coefficients,supplied coefficients,coefficientwise identity?,declared_edge_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([-4, 4, -3, 5], bottom, x, [-7, -3, 3], [-7, -3, 3], Yes, declared_edge_only)","explanation":"Substitute the fixed edge coordinate and compare all three coefficients in the remaining variable. This proves only the declared edge identity, not all four boundaries or the interior equation. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-4, 4, -3, 5], bottom, x, [-7, -3, 3], [-7, -3, 3], Yes, declared_edge_only)."},"skills":["A Whole-Edge Dirichlet Coefficient Identity"],"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-20261013-advance-20261025-r01-boundary-value-problem-0029","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"881c50d4a51b870e70be9566baedc920af71c5444797780411eef1d7abc92ca9","family_id":"qf-mod25-bvp-dirichlet-edge-identity","seed":202610251029,"semantic_template":"qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[27,-5,4],\"bounds\":[-5,5,-4,6],\"coefficients\":[-9,1,0,4,-1,1],\"edge\":\"top\"}"},"fields":{"title":"A Whole-Edge Dirichlet Coefficient Identity","stem":"Use the quadratic candidate coefficients=[-9, 1, 0, 4, -1, 1] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-5, 5, -4, 6]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Selected edge=top; supplied coefficients=[27, -5, 4]. Give (bounds,edge,remaining variable,restriction coefficients,supplied coefficients,coefficientwise identity?,declared_edge_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([-5, 5, -4, 6], top, x, [27, -5, 4], [27, -5, 4], Yes, declared_edge_only)","explanation":"Substitute the fixed edge coordinate and compare all three coefficients in the remaining variable. This proves only the declared edge identity, not all four boundaries or the interior equation. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-5, 5, -4, 6], top, x, [27, -5, 4], [27, -5, 4], Yes, declared_edge_only)."},"skills":["A Whole-Edge Dirichlet Coefficient Identity"],"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-20261013-advance-20261025-r01-boundary-value-problem-0030","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f1a4221fb0e5fab5e1ba420be0eb2dcec689f59a1d1f7c3b36607b5d7da61eac","family_id":"qf-mod25-bvp-dirichlet-edge-identity","seed":202610251030,"semantic_template":"qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[16,1,2],\"bounds\":[-6,6,-5,7],\"coefficients\":[-8,2,1,1,0,2],\"edge\":\"left\"}"},"fields":{"title":"A Whole-Edge Dirichlet Coefficient Identity","stem":"Use the quadratic candidate coefficients=[-8, 2, 1, 1, 0, 2] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-6, 6, -5, 7]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Selected edge=left; supplied coefficients=[16, 1, 2]. Give (bounds,edge,remaining variable,restriction coefficients,supplied coefficients,coefficientwise identity?,declared_edge_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([-6, 6, -5, 7], left, y, [16, 1, 2], [16, 1, 2], Yes, declared_edge_only)","explanation":"Substitute the fixed edge coordinate and compare all three coefficients in the remaining variable. This proves only the declared edge identity, not all four boundaries or the interior equation. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-6, 6, -5, 7], left, y, [16, 1, 2], [16, 1, 2], Yes, declared_edge_only)."},"skills":["A Whole-Edge Dirichlet Coefficient Identity"],"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-20261013-advance-20261025-r01-boundary-value-problem-0031","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a2da7b3cc419c9fb49b158623f6578a6282f47ccf1ba8ef69477f2f32fa5c9e","family_id":"qf-mod25-bvp-dirichlet-edge-identity","seed":202610251031,"semantic_template":"qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[78,9,-2],\"bounds\":[-7,7,-6,8],\"coefficients\":[-7,-2,2,2,1,-2],\"edge\":\"right\"}"},"fields":{"title":"A Whole-Edge Dirichlet Coefficient Identity","stem":"Use the quadratic candidate coefficients=[-7, -2, 2, 2, 1, -2] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-7, 7, -6, 8]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Selected edge=right; supplied coefficients=[78, 9, -2]. Give (bounds,edge,remaining variable,restriction coefficients,supplied coefficients,coefficientwise identity?,declared_edge_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([-7, 7, -6, 8], right, y, [77, 9, -2], [78, 9, -2], No, declared_edge_only)","explanation":"Substitute the fixed edge coordinate and compare all three coefficients in the remaining variable. This proves only the declared edge identity, not all four boundaries or the interior equation. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-7, 7, -6, 8], right, y, [77, 9, -2], [78, 9, -2], No, declared_edge_only)."},"skills":["A Whole-Edge Dirichlet Coefficient Identity"],"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-20261013-advance-20261025-r01-boundary-value-problem-0032","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"094eb18d6616df852677c868224655275b2a0d450bb81a11bf5e1178064720dd","family_id":"qf-mod25-bvp-dirichlet-edge-identity","seed":202610251032,"semantic_template":"qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[-76,6,3],\"bounds\":[-8,8,-7,9],\"coefficients\":[-6,-1,3,3,-1,-1],\"edge\":\"bottom\"}"},"fields":{"title":"A Whole-Edge Dirichlet Coefficient Identity","stem":"Use the quadratic candidate coefficients=[-6, -1, 3, 3, -1, -1] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-8, 8, -7, 9]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Selected edge=bottom; supplied coefficients=[-76, 6, 3]. Give (bounds,edge,remaining variable,restriction coefficients,supplied coefficients,coefficientwise identity?,declared_edge_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([-8, 8, -7, 9], bottom, x, [-76, 6, 3], [-76, 6, 3], Yes, declared_edge_only)","explanation":"Substitute the fixed edge coordinate and compare all three coefficients in the remaining variable. This proves only the declared edge identity, not all four boundaries or the interior equation. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 8, -7, 9], bottom, x, [-76, 6, 3], [-76, 6, 3], Yes, declared_edge_only)."},"skills":["A Whole-Edge Dirichlet Coefficient Identity"],"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-20261013-advance-20261025-r01-boundary-value-problem-0033","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b9abb263739c3a6886f0184de74c4832eb4e69da958ade78a7ede605c26f8401","family_id":"qf-mod25-bvp-dirichlet-edge-identity","seed":202610251033,"semantic_template":"qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[-35,0,4],\"bounds\":[-9,9,-8,10],\"coefficients\":[-5,0,-3,4,0,0],\"edge\":\"top\"}"},"fields":{"title":"A Whole-Edge Dirichlet Coefficient Identity","stem":"Use the quadratic candidate coefficients=[-5, 0, -3, 4, 0, 0] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-9, 9, -8, 10]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Selected edge=top; supplied coefficients=[-35, 0, 4]. Give (bounds,edge,remaining variable,restriction coefficients,supplied coefficients,coefficientwise identity?,declared_edge_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([-9, 9, -8, 10], top, x, [-35, 0, 4], [-35, 0, 4], Yes, declared_edge_only)","explanation":"Substitute the fixed edge coordinate and compare all three coefficients in the remaining variable. This proves only the declared edge identity, not all four boundaries or the interior equation. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 9, -8, 10], top, x, [-35, 0, 4], [-35, 0, 4], Yes, declared_edge_only)."},"skills":["A Whole-Edge Dirichlet Coefficient Identity"],"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-20261013-advance-20261025-r01-boundary-value-problem-0034","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"768a1261f8f0c01b31b86214df91de19cac71268b24bc733cdb537deb4430ef4","family_id":"qf-mod25-bvp-dirichlet-edge-identity","seed":202610251034,"semantic_template":"qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[86,-12,1],\"bounds\":[-10,10,-9,11],\"coefficients\":[-4,1,-2,1,1,1],\"edge\":\"left\"}"},"fields":{"title":"A Whole-Edge Dirichlet Coefficient Identity","stem":"Use the quadratic candidate coefficients=[-4, 1, -2, 1, 1, 1] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-10, 10, -9, 11]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Selected edge=left; supplied coefficients=[86, -12, 1]. Give (bounds,edge,remaining variable,restriction coefficients,supplied coefficients,coefficientwise identity?,declared_edge_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([-10, 10, -9, 11], left, y, [86, -12, 1], [86, -12, 1], Yes, declared_edge_only)","explanation":"Substitute the fixed edge coordinate and compare all three coefficients in the remaining variable. This proves only the declared edge identity, not all four boundaries or the interior equation. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-10, 10, -9, 11], left, y, [86, -12, 1], [86, -12, 1], Yes, declared_edge_only)."},"skills":["A Whole-Edge Dirichlet Coefficient Identity"],"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-20261013-advance-20261025-r01-boundary-value-problem-0035","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a0d7c44be1042b4ccf1ab16fd509e5df597644adb7a81e5721a6bb778a6d399","family_id":"qf-mod25-bvp-dirichlet-edge-identity","seed":202610251035,"semantic_template":"qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[261,-12,2],\"bounds\":[-11,11,-10,12],\"coefficients\":[-3,2,-1,2,-1,2],\"edge\":\"right\"}"},"fields":{"title":"A Whole-Edge Dirichlet Coefficient Identity","stem":"Use the quadratic candidate coefficients=[-3, 2, -1, 2, -1, 2] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-11, 11, -10, 12]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Selected edge=right; supplied coefficients=[261, -12, 2]. Give (bounds,edge,remaining variable,restriction coefficients,supplied coefficients,coefficientwise identity?,declared_edge_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([-11, 11, -10, 12], right, y, [261, -12, 2], [261, -12, 2], Yes, declared_edge_only)","explanation":"Substitute the fixed edge coordinate and compare all three coefficients in the remaining variable. This proves only the declared edge identity, not all four boundaries or the interior equation. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-11, 11, -10, 12], right, y, [261, -12, 2], [261, -12, 2], Yes, declared_edge_only)."},"skills":["A Whole-Edge Dirichlet Coefficient Identity"],"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-20261013-advance-20261025-r01-boundary-value-problem-0036","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ee720a8c4882934efda487379a65523417d0c7d4cc2b2a2c14c07714e07ebd6","family_id":"qf-mod25-bvp-dirichlet-edge-identity","seed":202610251036,"semantic_template":"qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[-243,-2,3],\"bounds\":[-12,12,-11,13],\"coefficients\":[-2,-2,0,3,0,-2],\"edge\":\"bottom\"}"},"fields":{"title":"A Whole-Edge Dirichlet Coefficient Identity","stem":"Use the quadratic candidate coefficients=[-2, -2, 0, 3, 0, -2] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-12, 12, -11, 13]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Selected edge=bottom; supplied coefficients=[-243, -2, 3]. Give (bounds,edge,remaining variable,restriction coefficients,supplied coefficients,coefficientwise identity?,declared_edge_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([-12, 12, -11, 13], bottom, x, [-244, -2, 3], [-243, -2, 3], No, declared_edge_only)","explanation":"Substitute the fixed edge coordinate and compare all three coefficients in the remaining variable. This proves only the declared edge identity, not all four boundaries or the interior equation. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-12, 12, -11, 13], bottom, x, [-244, -2, 3], [-243, -2, 3], No, declared_edge_only)."},"skills":["A Whole-Edge Dirichlet Coefficient Identity"],"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-20261013-advance-20261025-r01-boundary-value-problem-0037","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4905b57bf5164c9fc04dadff08fc1cf2ffb860f6a289135662fb8021bb412fa","family_id":"qf-mod25-bvp-dirichlet-edge-identity","seed":202610251037,"semantic_template":"qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[-183,13,4],\"bounds\":[-13,13,-12,14],\"coefficients\":[-1,-1,1,4,1,-1],\"edge\":\"top\"}"},"fields":{"title":"A Whole-Edge Dirichlet Coefficient Identity","stem":"Use the quadratic candidate coefficients=[-1, -1, 1, 4, 1, -1] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-13, 13, -12, 14]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Selected edge=top; supplied coefficients=[-183, 13, 4]. Give (bounds,edge,remaining variable,restriction coefficients,supplied coefficients,coefficientwise identity?,declared_edge_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([-13, 13, -12, 14], top, x, [-183, 13, 4], [-183, 13, 4], Yes, declared_edge_only)","explanation":"Substitute the fixed edge coordinate and compare all three coefficients in the remaining variable. This proves only the declared edge identity, not all four boundaries or the interior equation. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-13, 13, -12, 14], top, x, [-183, 13, 4], [-183, 13, 4], Yes, declared_edge_only)."},"skills":["A Whole-Edge Dirichlet Coefficient Identity"],"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-20261013-advance-20261025-r01-boundary-value-problem-0038","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"835c94f4bcc4ef12a820fd600f6b69522a6a5d44c475377e3ab97fafa510b945","family_id":"qf-mod25-bvp-dirichlet-edge-identity","seed":202610251038,"semantic_template":"qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[196,16,0],\"bounds\":[-14,14,-13,15],\"coefficients\":[0,0,2,1,-1,0],\"edge\":\"left\"}"},"fields":{"title":"A Whole-Edge Dirichlet Coefficient Identity","stem":"Use the quadratic candidate coefficients=[0, 0, 2, 1, -1, 0] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-14, 14, -13, 15]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Selected edge=left; supplied coefficients=[196, 16, 0]. Give (bounds,edge,remaining variable,restriction coefficients,supplied coefficients,coefficientwise identity?,declared_edge_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([-14, 14, -13, 15], left, y, [196, 16, 0], [196, 16, 0], Yes, declared_edge_only)","explanation":"Substitute the fixed edge coordinate and compare all three coefficients in the remaining variable. This proves only the declared edge identity, not all four boundaries or the interior equation. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-14, 14, -13, 15], left, y, [196, 16, 0], [196, 16, 0], Yes, declared_edge_only)."},"skills":["A Whole-Edge Dirichlet Coefficient Identity"],"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-20261013-advance-20261025-r01-boundary-value-problem-0039","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e850dd329c1b7f85368969f38447f605b21e344d5318386ffcbd3e409cb98541","family_id":"qf-mod25-bvp-dirichlet-edge-identity","seed":202610251039,"semantic_template":"qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[466,3,1],\"bounds\":[-15,15,-14,16],\"coefficients\":[1,1,3,2,0,1],\"edge\":\"right\"}"},"fields":{"title":"A Whole-Edge Dirichlet Coefficient Identity","stem":"Use the quadratic candidate coefficients=[1, 1, 3, 2, 0, 1] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-15, 15, -14, 16]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Selected edge=right; supplied coefficients=[466, 3, 1]. Give (bounds,edge,remaining variable,restriction coefficients,supplied coefficients,coefficientwise identity?,declared_edge_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([-15, 15, -14, 16], right, y, [466, 3, 1], [466, 3, 1], Yes, declared_edge_only)","explanation":"Substitute the fixed edge coordinate and compare all three coefficients in the remaining variable. This proves only the declared edge identity, not all four boundaries or the interior equation. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-15, 15, -14, 16], right, y, [466, 3, 1], [466, 3, 1], Yes, declared_edge_only)."},"skills":["A Whole-Edge Dirichlet Coefficient Identity"],"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-20261013-advance-20261025-r01-boundary-value-problem-0040","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b194e47c811f2484f1ddd20994fbf9019877a3957838782ffbf622535b8ed51","family_id":"qf-mod25-bvp-dirichlet-edge-identity","seed":202610251040,"semantic_template":"qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[497,-13,3],\"bounds\":[-16,16,-15,17],\"coefficients\":[2,2,-3,3,1,2],\"edge\":\"bottom\"}"},"fields":{"title":"A Whole-Edge Dirichlet Coefficient Identity","stem":"Use the quadratic candidate coefficients=[2, 2, -3, 3, 1, 2] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-16, 16, -15, 17]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Selected edge=bottom; supplied coefficients=[497, -13, 3]. Give (bounds,edge,remaining variable,restriction coefficients,supplied coefficients,coefficientwise identity?,declared_edge_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([-16, 16, -15, 17], bottom, x, [497, -13, 3], [497, -13, 3], Yes, declared_edge_only)","explanation":"Substitute the fixed edge coordinate and compare all three coefficients in the remaining variable. This proves only the declared edge identity, not all four boundaries or the interior equation. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-16, 16, -15, 17], bottom, x, [497, -13, 3], [497, -13, 3], Yes, declared_edge_only)."},"skills":["A Whole-Edge Dirichlet Coefficient Identity"],"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-20261013-advance-20261025-r01-boundary-value-problem-0041","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fda583687d5bb50e1fef985e1358286919280ff4793bf909800f306c25f087ea","family_id":"qf-mod25-bvp-dirichlet-edge-identity","seed":202610251041,"semantic_template":"qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[-680,-20,4],\"bounds\":[-17,17,-16,18],\"coefficients\":[3,-2,-2,4,-1,-2],\"edge\":\"top\"}"},"fields":{"title":"A Whole-Edge Dirichlet Coefficient Identity","stem":"Use the quadratic candidate coefficients=[3, -2, -2, 4, -1, -2] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-17, 17, -16, 18]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Selected edge=top; supplied coefficients=[-680, -20, 4]. Give (bounds,edge,remaining variable,restriction coefficients,supplied coefficients,coefficientwise identity?,declared_edge_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([-17, 17, -16, 18], top, x, [-681, -20, 4], [-680, -20, 4], No, declared_edge_only)","explanation":"Substitute the fixed edge coordinate and compare all three coefficients in the remaining variable. This proves only the declared edge identity, not all four boundaries or the interior equation. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-17, 17, -16, 18], top, x, [-681, -20, 4], [-680, -20, 4], No, declared_edge_only)."},"skills":["A Whole-Edge Dirichlet Coefficient Identity"],"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-20261013-advance-20261025-r01-boundary-value-problem-0042","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aaba14157b23cc0a16fe8caa4088e9dc90fdcde2c90f175e345bc62bf4288182","family_id":"qf-mod25-bvp-dirichlet-edge-identity","seed":202610251042,"semantic_template":"qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[346,-1,-1],\"bounds\":[-18,18,-17,19],\"coefficients\":[4,-1,-1,1,0,-1],\"edge\":\"left\"}"},"fields":{"title":"A Whole-Edge Dirichlet Coefficient Identity","stem":"Use the quadratic candidate coefficients=[4, -1, -1, 1, 0, -1] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-18, 18, -17, 19]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Selected edge=left; supplied coefficients=[346, -1, -1]. Give (bounds,edge,remaining variable,restriction coefficients,supplied coefficients,coefficientwise identity?,declared_edge_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([-18, 18, -17, 19], left, y, [346, -1, -1], [346, -1, -1], Yes, declared_edge_only)","explanation":"Substitute the fixed edge coordinate and compare all three coefficients in the remaining variable. This proves only the declared edge identity, not all four boundaries or the interior equation. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-18, 18, -17, 19], left, y, [346, -1, -1], [346, -1, -1], Yes, declared_edge_only)."},"skills":["A Whole-Edge Dirichlet Coefficient Identity"],"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-20261013-advance-20261025-r01-boundary-value-problem-0043","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9270bbb892ad4028777e87f484d74d5d8469df3c7684592221332db6d84ee0ae","family_id":"qf-mod25-bvp-dirichlet-edge-identity","seed":202610251043,"semantic_template":"qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[727,19,0],\"bounds\":[-19,19,-18,20],\"coefficients\":[5,0,0,2,1,0],\"edge\":\"right\"}"},"fields":{"title":"A Whole-Edge Dirichlet Coefficient Identity","stem":"Use the quadratic candidate coefficients=[5, 0, 0, 2, 1, 0] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-19, 19, -18, 20]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Selected edge=right; supplied coefficients=[727, 19, 0]. Give (bounds,edge,remaining variable,restriction coefficients,supplied coefficients,coefficientwise identity?,declared_edge_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([-19, 19, -18, 20], right, y, [727, 19, 0], [727, 19, 0], Yes, declared_edge_only)","explanation":"Substitute the fixed edge coordinate and compare all three coefficients in the remaining variable. This proves only the declared edge identity, not all four boundaries or the interior equation. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-19, 19, -18, 20], right, y, [727, 19, 0], [727, 19, 0], Yes, declared_edge_only)."},"skills":["A Whole-Edge Dirichlet Coefficient Identity"],"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-20261013-advance-20261025-r01-boundary-value-problem-0044","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ee3f7631d0c1cfce12688985232c9beeb81920975b68d3e57c893a0b79e1f5a","family_id":"qf-mod25-bvp-dirichlet-edge-identity","seed":202610251044,"semantic_template":"qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[348,20,3],\"bounds\":[-20,20,-19,21],\"coefficients\":[6,1,1,3,-1,1],\"edge\":\"bottom\"}"},"fields":{"title":"A Whole-Edge Dirichlet Coefficient Identity","stem":"Use the quadratic candidate coefficients=[6, 1, 1, 3, -1, 1] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-20, 20, -19, 21]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Selected edge=bottom; supplied coefficients=[348, 20, 3]. Give (bounds,edge,remaining variable,restriction coefficients,supplied coefficients,coefficientwise identity?,declared_edge_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([-20, 20, -19, 21], bottom, x, [348, 20, 3], [348, 20, 3], Yes, declared_edge_only)","explanation":"Substitute the fixed edge coordinate and compare all three coefficients in the remaining variable. This proves only the declared edge identity, not all four boundaries or the interior equation. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-20, 20, -19, 21], bottom, x, [348, 20, 3], [348, 20, 3], Yes, declared_edge_only)."},"skills":["A Whole-Edge Dirichlet Coefficient Identity"],"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-20261013-advance-20261025-r01-boundary-value-problem-0045","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5371d37d8a2956bb93eaad5e3e3e89caa044927c03b3ecfa3b69f75ffedbf20b","family_id":"qf-mod25-bvp-dirichlet-edge-identity","seed":202610251045,"semantic_template":"qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[1019,2,4],\"bounds\":[-21,21,-20,22],\"coefficients\":[7,2,2,4,0,2],\"edge\":\"top\"}"},"fields":{"title":"A Whole-Edge Dirichlet Coefficient Identity","stem":"Use the quadratic candidate coefficients=[7, 2, 2, 4, 0, 2] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-21, 21, -20, 22]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Selected edge=top; supplied coefficients=[1019, 2, 4]. Give (bounds,edge,remaining variable,restriction coefficients,supplied coefficients,coefficientwise identity?,declared_edge_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([-21, 21, -20, 22], top, x, [1019, 2, 4], [1019, 2, 4], Yes, declared_edge_only)","explanation":"Substitute the fixed edge coordinate and compare all three coefficients in the remaining variable. This proves only the declared edge identity, not all four boundaries or the interior equation. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-21, 21, -20, 22], top, x, [1019, 2, 4], [1019, 2, 4], Yes, declared_edge_only)."},"skills":["A Whole-Edge Dirichlet Coefficient Identity"],"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-20261013-advance-20261025-r01-boundary-value-problem-0046","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f65d8117ec7ce7ac6697694192a097b6260f9c107cf4f49215eea6cb9e6f5bc","family_id":"qf-mod25-bvp-dirichlet-edge-identity","seed":202610251046,"semantic_template":"qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[537,-19,-2],\"bounds\":[-22,22,-21,23],\"coefficients\":[8,-2,3,1,1,-2],\"edge\":\"left\"}"},"fields":{"title":"A Whole-Edge Dirichlet Coefficient Identity","stem":"Use the quadratic candidate coefficients=[8, -2, 3, 1, 1, -2] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-22, 22, -21, 23]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Selected edge=left; supplied coefficients=[537, -19, -2]. Give (bounds,edge,remaining variable,restriction coefficients,supplied coefficients,coefficientwise identity?,declared_edge_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([-22, 22, -21, 23], left, y, [536, -19, -2], [537, -19, -2], No, declared_edge_only)","explanation":"Substitute the fixed edge coordinate and compare all three coefficients in the remaining variable. This proves only the declared edge identity, not all four boundaries or the interior equation. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-22, 22, -21, 23], left, y, [536, -19, -2], [537, -19, -2], No, declared_edge_only)."},"skills":["A Whole-Edge Dirichlet Coefficient Identity"],"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-20261013-advance-20261025-r01-boundary-value-problem-0047","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7870e7b038bb2031ed58f7caa501631452208bb20eddf316d800a016a242b902","family_id":"qf-mod25-bvp-dirichlet-edge-identity","seed":202610251047,"semantic_template":"qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[1044,-26,-1],\"bounds\":[-23,23,-22,24],\"coefficients\":[9,-1,-3,2,-1,-1],\"edge\":\"right\"}"},"fields":{"title":"A Whole-Edge Dirichlet Coefficient Identity","stem":"Use the quadratic candidate coefficients=[9, -1, -3, 2, -1, -1] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-23, 23, -22, 24]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Selected edge=right; supplied coefficients=[1044, -26, -1]. Give (bounds,edge,remaining variable,restriction coefficients,supplied coefficients,coefficientwise identity?,declared_edge_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([-23, 23, -22, 24], right, y, [1044, -26, -1], [1044, -26, -1], Yes, declared_edge_only)","explanation":"Substitute the fixed edge coordinate and compare all three coefficients in the remaining variable. This proves only the declared edge identity, not all four boundaries or the interior equation. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-23, 23, -22, 24], right, y, [1044, -26, -1], [1044, -26, -1], Yes, declared_edge_only)."},"skills":["A Whole-Edge Dirichlet Coefficient Identity"],"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-20261013-advance-20261025-r01-boundary-value-problem-0048","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c471d11eec090eaf2e340cc0130329908a0e22a14773d1a8fb5402b1fd5e59d","family_id":"qf-mod25-bvp-dirichlet-edge-identity","seed":202610251048,"semantic_template":"qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[56,0,3],\"bounds\":[-24,24,-23,25],\"coefficients\":[10,0,-2,3,0,0],\"edge\":\"bottom\"}"},"fields":{"title":"A Whole-Edge Dirichlet Coefficient Identity","stem":"Use the quadratic candidate coefficients=[10, 0, -2, 3, 0, 0] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-24, 24, -23, 25]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Selected edge=bottom; supplied coefficients=[56, 0, 3]. Give (bounds,edge,remaining variable,restriction coefficients,supplied coefficients,coefficientwise identity?,declared_edge_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([-24, 24, -23, 25], bottom, x, [56, 0, 3], [56, 0, 3], Yes, declared_edge_only)","explanation":"Substitute the fixed edge coordinate and compare all three coefficients in the remaining variable. This proves only the declared edge identity, not all four boundaries or the interior equation. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-24, 24, -23, 25], bottom, x, [56, 0, 3], [56, 0, 3], Yes, declared_edge_only)."},"skills":["A Whole-Edge Dirichlet Coefficient Identity"],"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-20261013-advance-20261025-r01-boundary-value-problem-0049","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7fd0b044450bf8f394dfe864d446a117659871151d9c3715c8d4843bc52854a5","family_id":"qf-mod25-bvp-dirichlet-edge-identity","seed":202610251049,"semantic_template":"qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[661,27,4],\"bounds\":[-25,25,-24,26],\"coefficients\":[11,1,-1,4,1,1],\"edge\":\"top\"}"},"fields":{"title":"A Whole-Edge Dirichlet Coefficient Identity","stem":"Use the quadratic candidate coefficients=[11, 1, -1, 4, 1, 1] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-25, 25, -24, 26]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Selected edge=top; supplied coefficients=[661, 27, 4]. Give (bounds,edge,remaining variable,restriction coefficients,supplied coefficients,coefficientwise identity?,declared_edge_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([-25, 25, -24, 26], top, x, [661, 27, 4], [661, 27, 4], Yes, declared_edge_only)","explanation":"Substitute the fixed edge coordinate and compare all three coefficients in the remaining variable. This proves only the declared edge identity, not all four boundaries or the interior equation. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-25, 25, -24, 26], top, x, [661, 27, 4], [661, 27, 4], Yes, declared_edge_only)."},"skills":["A Whole-Edge Dirichlet Coefficient Identity"],"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-20261013-advance-20261025-r01-boundary-value-problem-0050","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fdff92e5ee1f3867669977e68704b1ebd79a43479d66ee4a821b573bbb71bc57","family_id":"qf-mod25-bvp-dirichlet-edge-identity","seed":202610251050,"semantic_template":"qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[636,26,2],\"bounds\":[-26,26,-25,27],\"coefficients\":[12,2,0,1,-1,2],\"edge\":\"left\"}"},"fields":{"title":"A Whole-Edge Dirichlet Coefficient Identity","stem":"Use the quadratic candidate coefficients=[12, 2, 0, 1, -1, 2] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-26, 26, -25, 27]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Selected edge=left; supplied coefficients=[636, 26, 2]. Give (bounds,edge,remaining variable,restriction coefficients,supplied coefficients,coefficientwise identity?,declared_edge_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([-26, 26, -25, 27], left, y, [636, 26, 2], [636, 26, 2], Yes, declared_edge_only)","explanation":"Substitute the fixed edge coordinate and compare all three coefficients in the remaining variable. This proves only the declared edge identity, not all four boundaries or the interior equation. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-26, 26, -25, 27], left, y, [636, 26, 2], [636, 26, 2], Yes, declared_edge_only)."},"skills":["A Whole-Edge Dirichlet Coefficient Identity"],"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-20261013-advance-20261025-r01-boundary-value-problem-0051","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c977c5a76d1fd96613337a8fcce4fd0b666b7f516ec138e188e82ca6def1732b","family_id":"qf-mod25-bvp-joint-candidate-audit","seed":202610251051,"semantic_template":"qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[-3,-1,-2]],[\"right\",[-12,-5,-2]],[\"bottom\",[-11,-1,1]],[\"top\",[-39,-5,1]]],\"bounds\":[-2,2,-1,3],\"coefficients\":[-12,-2,-3,1,-1,-2],\"source\":-1}"},"fields":{"title":"Close Interior and All Four Boundary Obligations Separately","stem":"Use the quadratic candidate coefficients=[-12, -2, -3, 1, -1, -2] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-2, 2, -1, 3]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=-1; supplied [edge,coefficients] records=[[left, [-3, -1, -2]], [right, [-12, -5, -2]], [bottom, [-11, -1, 1]], [top, [-39, -5, 1]]] must include all four distinct edges. Preserve their supplied order. Give (coefficients,Laplacian,source,interior identity?,every [edge,remaining variable,restriction,supplied,identity?] row,all boundary identities?,both obligation groups?,candidate_satisfies_declared_BVP_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([-12, -2, -3, 1, -1, -2], -2, -1, No, [[left, y, [-4, -1, -2], [-3, -1, -2], No], [right, y, [-12, -5, -2], [-12, -5, -2], Yes], [bottom, x, [-11, -1, 1], [-11, -1, 1], Yes], [top, x, [-39, -5, 1], [-39, -5, 1], Yes]], No, No, candidate_satisfies_declared_BVP_only)","explanation":"Recompute the interior residual and every boundary restriction independently before taking their conjunction. This verifies a candidate for the declared problem without claiming general well-posedness. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-12, -2, -3, 1, -1, -2], -2, -1, No, [[left, y, [-4, -1, -2], [-3, -1, -2], No], [right, y, [-12, -5, -2], [-12, -5, -2], Yes], [bottom, x, [-11, -1, 1], [-11, -1, 1], Yes], [top, x, [-39, -5, 1], [-39, -5, 1], Yes]], No, No, candidate_satisfies_declared_BVP_only)."},"skills":["Close Interior and All Four Boundary Obligations Separately"],"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-20261013-advance-20261025-r01-boundary-value-problem-0052","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2b15a817f3336cac56a3bbcc25b03673b35fa66b2d2bffc6fd406fbf2bc48db","family_id":"qf-mod25-bvp-joint-candidate-audit","seed":202610251052,"semantic_template":"qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[10,-2,-1]],[\"right\",[4,-2,-1]],[\"bottom\",[-11,-1,2]],[\"top\",[-35,-1,2]]],\"bounds\":[-3,3,-2,4],\"coefficients\":[-11,-1,-2,2,0,-1],\"source\":2}"},"fields":{"title":"Close Interior and All Four Boundary Obligations Separately","stem":"Use the quadratic candidate coefficients=[-11, -1, -2, 2, 0, -1] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-3, 3, -2, 4]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=2; supplied [edge,coefficients] records=[[left, [10, -2, -1]], [right, [4, -2, -1]], [bottom, [-11, -1, 2]], [top, [-35, -1, 2]]] must include all four distinct edges. Preserve their supplied order. Give (coefficients,Laplacian,source,interior identity?,every [edge,remaining variable,restriction,supplied,identity?] row,all boundary identities?,both obligation groups?,candidate_satisfies_declared_BVP_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([-11, -1, -2, 2, 0, -1], 2, 2, Yes, [[left, y, [10, -2, -1], [10, -2, -1], Yes], [right, y, [4, -2, -1], [4, -2, -1], Yes], [bottom, x, [-11, -1, 2], [-11, -1, 2], Yes], [top, x, [-35, -1, 2], [-35, -1, 2], Yes]], Yes, Yes, candidate_satisfies_declared_BVP_only)","explanation":"Recompute the interior residual and every boundary restriction independently before taking their conjunction. This verifies a candidate for the declared problem without claiming general well-posedness. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-11, -1, -2, 2, 0, -1], 2, 2, Yes, [[left, y, [10, -2, -1], [10, -2, -1], Yes], [right, y, [4, -2, -1], [4, -2, -1], Yes], [bottom, x, [-11, -1, 2], [-11, -1, 2], Yes], [top, x, [-35, -1, 2], [-35, -1, 2], Yes]], Yes, Yes, candidate_satisfies_declared_BVP_only)."},"skills":["Close Interior and All Four Boundary Obligations Separately"],"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-20261013-advance-20261025-r01-boundary-value-problem-0053","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c4f7da91c4dd8175fe411ee551912922c30ebecd5bd5f5d04a30ae9d5650e99","family_id":"qf-mod25-bvp-joint-candidate-audit","seed":202610251053,"semantic_template":"qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[38,-5,0]],[\"right\",[38,3,0]],[\"bottom\",[-7,-3,3]],[\"top\",[-15,5,3]]],\"bounds\":[-4,4,-3,5],\"coefficients\":[-10,0,-1,3,1,0],\"source\":6}"},"fields":{"title":"Close Interior and All Four Boundary Obligations Separately","stem":"Use the quadratic candidate coefficients=[-10, 0, -1, 3, 1, 0] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-4, 4, -3, 5]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=6; supplied [edge,coefficients] records=[[left, [38, -5, 0]], [right, [38, 3, 0]], [bottom, [-7, -3, 3]], [top, [-15, 5, 3]]] must include all four distinct edges. Preserve their supplied order. Give (coefficients,Laplacian,source,interior identity?,every [edge,remaining variable,restriction,supplied,identity?] row,all boundary identities?,both obligation groups?,candidate_satisfies_declared_BVP_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([-10, 0, -1, 3, 1, 0], 6, 6, Yes, [[left, y, [38, -5, 0], [38, -5, 0], Yes], [right, y, [38, 3, 0], [38, 3, 0], Yes], [bottom, x, [-7, -3, 3], [-7, -3, 3], Yes], [top, x, [-15, 5, 3], [-15, 5, 3], Yes]], Yes, Yes, candidate_satisfies_declared_BVP_only)","explanation":"Recompute the interior residual and every boundary restriction independently before taking their conjunction. This verifies a candidate for the declared problem without claiming general well-posedness. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-10, 0, -1, 3, 1, 0], 6, 6, Yes, [[left, y, [38, -5, 0], [38, -5, 0], Yes], [right, y, [38, 3, 0], [38, 3, 0], Yes], [bottom, x, [-7, -3, 3], [-7, -3, 3], Yes], [top, x, [-15, 5, 3], [-15, 5, 3], Yes]], Yes, Yes, candidate_satisfies_declared_BVP_only)."},"skills":["Close Interior and All Four Boundary Obligations Separately"],"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-20261013-advance-20261025-r01-boundary-value-problem-0054","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"917dbcb6d268ccb5cb64a177b65545940b120f1e95f489b053cab2077a25e3db","family_id":"qf-mod25-bvp-joint-candidate-audit","seed":202610251054,"semantic_template":"qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[86,5,1]],[\"right\",[96,-5,1]],[\"bottom\",[7,5,4]],[\"top\",[27,-5,4]]],\"bounds\":[-5,5,-4,6],\"coefficients\":[-9,1,0,4,-1,1],\"source\":10}"},"fields":{"title":"Close Interior and All Four Boundary Obligations Separately","stem":"Use the quadratic candidate coefficients=[-9, 1, 0, 4, -1, 1] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-5, 5, -4, 6]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=10; supplied [edge,coefficients] records=[[left, [86, 5, 1]], [right, [96, -5, 1]], [bottom, [7, 5, 4]], [top, [27, -5, 4]]] must include all four distinct edges. Preserve their supplied order. Give (coefficients,Laplacian,source,interior identity?,every [edge,remaining variable,restriction,supplied,identity?] row,all boundary identities?,both obligation groups?,candidate_satisfies_declared_BVP_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([-9, 1, 0, 4, -1, 1], 10, 10, Yes, [[left, y, [86, 5, 1], [86, 5, 1], Yes], [right, y, [96, -5, 1], [96, -5, 1], Yes], [bottom, x, [7, 5, 4], [7, 5, 4], Yes], [top, x, [27, -5, 4], [27, -5, 4], Yes]], Yes, Yes, candidate_satisfies_declared_BVP_only)","explanation":"Recompute the interior residual and every boundary restriction independently before taking their conjunction. This verifies a candidate for the declared problem without claiming general well-posedness. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 1, 0, 4, -1, 1], 10, 10, Yes, [[left, y, [86, 5, 1], [86, 5, 1], Yes], [right, y, [96, -5, 1], [96, -5, 1], Yes], [bottom, x, [7, 5, 4], [7, 5, 4], Yes], [top, x, [27, -5, 4], [27, -5, 4], Yes]], Yes, Yes, candidate_satisfies_declared_BVP_only)."},"skills":["Close Interior and All Four Boundary Obligations Separately"],"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-20261013-advance-20261025-r01-boundary-value-problem-0055","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e412a102760da0971fcdfb40cd335efa9ff08aac473726ba0c6876a6a026cb1","family_id":"qf-mod25-bvp-joint-candidate-audit","seed":202610251055,"semantic_template":"qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[16,1,2]],[\"right\",[40,1,2]],[\"bottom\",[37,2,1]],[\"top\",[97,2,1]]],\"bounds\":[-6,6,-5,7],\"coefficients\":[-8,2,1,1,0,2],\"source\":6}"},"fields":{"title":"Close Interior and All Four Boundary Obligations Separately","stem":"Use the quadratic candidate coefficients=[-8, 2, 1, 1, 0, 2] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-6, 6, -5, 7]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=6; supplied [edge,coefficients] records=[[left, [16, 1, 2]], [right, [40, 1, 2]], [bottom, [37, 2, 1]], [top, [97, 2, 1]]] must include all four distinct edges. Preserve their supplied order. Give (coefficients,Laplacian,source,interior identity?,every [edge,remaining variable,restriction,supplied,identity?] row,all boundary identities?,both obligation groups?,candidate_satisfies_declared_BVP_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([-8, 2, 1, 1, 0, 2], 6, 6, Yes, [[left, y, [16, 1, 2], [16, 1, 2], Yes], [right, y, [40, 1, 2], [40, 1, 2], Yes], [bottom, x, [37, 2, 1], [37, 2, 1], Yes], [top, x, [97, 2, 1], [97, 2, 1], Yes]], Yes, Yes, candidate_satisfies_declared_BVP_only)","explanation":"Recompute the interior residual and every boundary restriction independently before taking their conjunction. This verifies a candidate for the declared problem without claiming general well-posedness. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 2, 1, 1, 0, 2], 6, 6, Yes, [[left, y, [16, 1, 2], [16, 1, 2], Yes], [right, y, [40, 1, 2], [40, 1, 2], Yes], [bottom, x, [37, 2, 1], [37, 2, 1], Yes], [top, x, [97, 2, 1], [97, 2, 1], Yes]], Yes, Yes, candidate_satisfies_declared_BVP_only)."},"skills":["Close Interior and All Four Boundary Obligations Separately"],"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-20261013-advance-20261025-r01-boundary-value-problem-0056","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"276c14b177aee387547c0e8d770e65d4975abf3b5122c9364a710be350365eab","family_id":"qf-mod25-bvp-joint-candidate-audit","seed":202610251056,"semantic_template":"qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[105,-5,-2]],[\"right\",[77,9,-2]],[\"bottom\",[-91,-8,2]],[\"top\",[-119,6,2]]],\"bounds\":[-7,7,-6,8],\"coefficients\":[-7,-2,2,2,1,-2],\"source\":0}"},"fields":{"title":"Close Interior and All Four Boundary Obligations Separately","stem":"Use the quadratic candidate coefficients=[-7, -2, 2, 2, 1, -2] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-7, 7, -6, 8]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=0; supplied [edge,coefficients] records=[[left, [105, -5, -2]], [right, [77, 9, -2]], [bottom, [-91, -8, 2]], [top, [-119, 6, 2]]] must include all four distinct edges. Preserve their supplied order. Give (coefficients,Laplacian,source,interior identity?,every [edge,remaining variable,restriction,supplied,identity?] row,all boundary identities?,both obligation groups?,candidate_satisfies_declared_BVP_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([-7, -2, 2, 2, 1, -2], 0, 0, Yes, [[left, y, [105, -5, -2], [105, -5, -2], Yes], [right, y, [77, 9, -2], [77, 9, -2], Yes], [bottom, x, [-91, -8, 2], [-91, -8, 2], Yes], [top, x, [-119, 6, 2], [-119, 6, 2], Yes]], Yes, Yes, candidate_satisfies_declared_BVP_only)","explanation":"Recompute the interior residual and every boundary restriction independently before taking their conjunction. This verifies a candidate for the declared problem without claiming general well-posedness. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-7, -2, 2, 2, 1, -2], 0, 0, Yes, [[left, y, [105, -5, -2], [105, -5, -2], Yes], [right, y, [77, 9, -2], [77, 9, -2], Yes], [bottom, x, [-91, -8, 2], [-91, -8, 2], Yes], [top, x, [-119, 6, 2], [-119, 6, 2], Yes]], Yes, Yes, candidate_satisfies_declared_BVP_only)."},"skills":["Close Interior and All Four Boundary Obligations Separately"],"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-20261013-advance-20261025-r01-boundary-value-problem-0057","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d40970f91c3c9b89ae13c7656bcfc9996483b0da8383122f27509ee66c44f022","family_id":"qf-mod25-bvp-joint-candidate-audit","seed":202610251057,"semantic_template":"qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[195,11,-1]],[\"right\",[178,-5,-1]],[\"bottom\",[-76,6,3]],[\"top\",[-60,-10,3]]],\"bounds\":[-8,8,-7,9],\"coefficients\":[-6,-1,3,3,-1,-1],\"source\":4}"},"fields":{"title":"Close Interior and All Four Boundary Obligations Separately","stem":"Use the quadratic candidate coefficients=[-6, -1, 3, 3, -1, -1] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-8, 8, -7, 9]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=4; supplied [edge,coefficients] records=[[left, [195, 11, -1]], [right, [178, -5, -1]], [bottom, [-76, 6, 3]], [top, [-60, -10, 3]]] must include all four distinct edges. Preserve their supplied order. Give (coefficients,Laplacian,source,interior identity?,every [edge,remaining variable,restriction,supplied,identity?] row,all boundary identities?,both obligation groups?,candidate_satisfies_declared_BVP_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([-6, -1, 3, 3, -1, -1], 4, 4, Yes, [[left, y, [194, 11, -1], [195, 11, -1], No], [right, y, [178, -5, -1], [178, -5, -1], Yes], [bottom, x, [-76, 6, 3], [-76, 6, 3], Yes], [top, x, [-60, -10, 3], [-60, -10, 3], Yes]], No, No, candidate_satisfies_declared_BVP_only)","explanation":"Recompute the interior residual and every boundary restriction independently before taking their conjunction. This verifies a candidate for the declared problem without claiming general well-posedness. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-6, -1, 3, 3, -1, -1], 4, 4, Yes, [[left, y, [194, 11, -1], [195, 11, -1], No], [right, y, [178, -5, -1], [178, -5, -1], Yes], [bottom, x, [-76, 6, 3], [-76, 6, 3], Yes], [top, x, [-60, -10, 3], [-60, -10, 3], Yes]], No, No, candidate_satisfies_declared_BVP_only)."},"skills":["Close Interior and All Four Boundary Obligations Separately"],"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-20261013-advance-20261025-r01-boundary-value-problem-0058","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4fd0cb1542ddd2a2909d41ae454d913b7d65675c83ee8cfb5882f1e5a8efc899","family_id":"qf-mod25-bvp-joint-candidate-audit","seed":202610251058,"semantic_template":"qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[319,-3,0]],[\"right\",[319,-3,0]],[\"bottom\",[19,0,4]],[\"top\",[-35,0,4]]],\"bounds\":[-9,9,-8,10],\"coefficients\":[-5,0,-3,4,0,0],\"source\":9}"},"fields":{"title":"Close Interior and All Four Boundary Obligations Separately","stem":"Use the quadratic candidate coefficients=[-5, 0, -3, 4, 0, 0] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-9, 9, -8, 10]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=9; supplied [edge,coefficients] records=[[left, [319, -3, 0]], [right, [319, -3, 0]], [bottom, [19, 0, 4]], [top, [-35, 0, 4]]] must include all four distinct edges. Preserve their supplied order. Give (coefficients,Laplacian,source,interior identity?,every [edge,remaining variable,restriction,supplied,identity?] row,all boundary identities?,both obligation groups?,candidate_satisfies_declared_BVP_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([-5, 0, -3, 4, 0, 0], 8, 9, No, [[left, y, [319, -3, 0], [319, -3, 0], Yes], [right, y, [319, -3, 0], [319, -3, 0], Yes], [bottom, x, [19, 0, 4], [19, 0, 4], Yes], [top, x, [-35, 0, 4], [-35, 0, 4], Yes]], Yes, No, candidate_satisfies_declared_BVP_only)","explanation":"Recompute the interior residual and every boundary restriction independently before taking their conjunction. This verifies a candidate for the declared problem without claiming general well-posedness. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-5, 0, -3, 4, 0, 0], 8, 9, No, [[left, y, [319, -3, 0], [319, -3, 0], Yes], [right, y, [319, -3, 0], [319, -3, 0], Yes], [bottom, x, [19, 0, 4], [19, 0, 4], Yes], [top, x, [-35, 0, 4], [-35, 0, 4], Yes]], Yes, No, candidate_satisfies_declared_BVP_only)."},"skills":["Close Interior and All Four Boundary Obligations Separately"],"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-20261013-advance-20261025-r01-boundary-value-problem-0059","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b0fec548cf85a74b0b24337846ec14fe2658b85f57d368f0ac62f97ef30a3f6","family_id":"qf-mod25-bvp-joint-candidate-audit","seed":202610251059,"semantic_template":"qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[86,-12,1]],[\"right\",[106,8,1]],[\"bottom\",[95,-8,1]],[\"top\",[95,12,1]]],\"bounds\":[-10,10,-9,11],\"coefficients\":[-4,1,-2,1,1,1],\"source\":4}"},"fields":{"title":"Close Interior and All Four Boundary Obligations Separately","stem":"Use the quadratic candidate coefficients=[-4, 1, -2, 1, 1, 1] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-10, 10, -9, 11]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=4; supplied [edge,coefficients] records=[[left, [86, -12, 1]], [right, [106, 8, 1]], [bottom, [95, -8, 1]], [top, [95, 12, 1]]] must include all four distinct edges. Preserve their supplied order. Give (coefficients,Laplacian,source,interior identity?,every [edge,remaining variable,restriction,supplied,identity?] row,all boundary identities?,both obligation groups?,candidate_satisfies_declared_BVP_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([-4, 1, -2, 1, 1, 1], 4, 4, Yes, [[left, y, [86, -12, 1], [86, -12, 1], Yes], [right, y, [106, 8, 1], [106, 8, 1], Yes], [bottom, x, [95, -8, 1], [95, -8, 1], Yes], [top, x, [95, 12, 1], [95, 12, 1], Yes]], Yes, Yes, candidate_satisfies_declared_BVP_only)","explanation":"Recompute the interior residual and every boundary restriction independently before taking their conjunction. This verifies a candidate for the declared problem without claiming general well-posedness. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-4, 1, -2, 1, 1, 1], 4, 4, Yes, [[left, y, [86, -12, 1], [86, -12, 1], Yes], [right, y, [106, 8, 1], [106, 8, 1], Yes], [bottom, x, [95, -8, 1], [95, -8, 1], Yes], [top, x, [95, 12, 1], [95, 12, 1], Yes]], Yes, Yes, candidate_satisfies_declared_BVP_only)."},"skills":["Close Interior and All Four Boundary Obligations Separately"],"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-20261013-advance-20261025-r01-boundary-value-problem-0060","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30c0457a5165d133ccf0409f41aebd7a16fc1a2e4b32c181acd91e7853ff2fb7","family_id":"qf-mod25-bvp-joint-candidate-audit","seed":202610251060,"semantic_template":"qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[217,10,2]],[\"right\",[261,-12,2]],[\"bottom\",[207,12,2]],[\"top\",[273,-10,2]]],\"bounds\":[-11,11,-10,12],\"coefficients\":[-3,2,-1,2,-1,2],\"source\":8}"},"fields":{"title":"Close Interior and All Four Boundary Obligations Separately","stem":"Use the quadratic candidate coefficients=[-3, 2, -1, 2, -1, 2] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-11, 11, -10, 12]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=8; supplied [edge,coefficients] records=[[left, [217, 10, 2]], [right, [261, -12, 2]], [bottom, [207, 12, 2]], [top, [273, -10, 2]]] must include all four distinct edges. Preserve their supplied order. Give (coefficients,Laplacian,source,interior identity?,every [edge,remaining variable,restriction,supplied,identity?] row,all boundary identities?,both obligation groups?,candidate_satisfies_declared_BVP_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([-3, 2, -1, 2, -1, 2], 8, 8, Yes, [[left, y, [217, 10, 2], [217, 10, 2], Yes], [right, y, [261, -12, 2], [261, -12, 2], Yes], [bottom, x, [207, 12, 2], [207, 12, 2], Yes], [top, x, [273, -10, 2], [273, -10, 2], Yes]], Yes, Yes, candidate_satisfies_declared_BVP_only)","explanation":"Recompute the interior residual and every boundary restriction independently before taking their conjunction. This verifies a candidate for the declared problem without claiming general well-posedness. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-3, 2, -1, 2, -1, 2], 8, 8, Yes, [[left, y, [217, 10, 2], [217, 10, 2], Yes], [right, y, [261, -12, 2], [261, -12, 2], Yes], [bottom, x, [207, 12, 2], [207, 12, 2], Yes], [top, x, [273, -10, 2], [273, -10, 2], Yes]], Yes, Yes, candidate_satisfies_declared_BVP_only)."},"skills":["Close Interior and All Four Boundary Obligations Separately"],"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-20261013-advance-20261025-r01-boundary-value-problem-0061","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e84a1f0fbcd5c7b1ec02ec586754a9906cb8b48744f7f8401d7d5cee087769fe","family_id":"qf-mod25-bvp-joint-candidate-audit","seed":202610251061,"semantic_template":"qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[454,0,-2]],[\"right\",[406,0,-2]],[\"bottom\",[-244,-2,3]],[\"top\",[-340,-2,3]]],\"bounds\":[-12,12,-11,13],\"coefficients\":[-2,-2,0,3,0,-2],\"source\":2}"},"fields":{"title":"Close Interior and All Four Boundary Obligations Separately","stem":"Use the quadratic candidate coefficients=[-2, -2, 0, 3, 0, -2] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-12, 12, -11, 13]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=2; supplied [edge,coefficients] records=[[left, [454, 0, -2]], [right, [406, 0, -2]], [bottom, [-244, -2, 3]], [top, [-340, -2, 3]]] must include all four distinct edges. Preserve their supplied order. Give (coefficients,Laplacian,source,interior identity?,every [edge,remaining variable,restriction,supplied,identity?] row,all boundary identities?,both obligation groups?,candidate_satisfies_declared_BVP_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([-2, -2, 0, 3, 0, -2], 2, 2, Yes, [[left, y, [454, 0, -2], [454, 0, -2], Yes], [right, y, [406, 0, -2], [406, 0, -2], Yes], [bottom, x, [-244, -2, 3], [-244, -2, 3], Yes], [top, x, [-340, -2, 3], [-340, -2, 3], Yes]], Yes, Yes, candidate_satisfies_declared_BVP_only)","explanation":"Recompute the interior residual and every boundary restriction independently before taking their conjunction. This verifies a candidate for the declared problem without claiming general well-posedness. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-2, -2, 0, 3, 0, -2], 2, 2, Yes, [[left, y, [454, 0, -2], [454, 0, -2], Yes], [right, y, [406, 0, -2], [406, 0, -2], Yes], [bottom, x, [-244, -2, 3], [-244, -2, 3], Yes], [top, x, [-340, -2, 3], [-340, -2, 3], Yes]], Yes, Yes, candidate_satisfies_declared_BVP_only)."},"skills":["Close Interior and All Four Boundary Obligations Separately"],"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-20261013-advance-20261025-r01-boundary-value-problem-0062","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2bb256935f287450621781516c7febd71d5398a0aec910de8a5289982811658f","family_id":"qf-mod25-bvp-joint-candidate-audit","seed":202610251062,"semantic_template":"qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[688,-12,-1]],[\"right\",[662,14,-1]],[\"bottom\",[-157,-13,4]],[\"top\",[-183,13,4]]],\"bounds\":[-13,13,-12,14],\"coefficients\":[-1,-1,1,4,1,-1],\"source\":6}"},"fields":{"title":"Close Interior and All Four Boundary Obligations Separately","stem":"Use the quadratic candidate coefficients=[-1, -1, 1, 4, 1, -1] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-13, 13, -12, 14]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=6; supplied [edge,coefficients] records=[[left, [688, -12, -1]], [right, [662, 14, -1]], [bottom, [-157, -13, 4]], [top, [-183, 13, 4]]] must include all four distinct edges. Preserve their supplied order. Give (coefficients,Laplacian,source,interior identity?,every [edge,remaining variable,restriction,supplied,identity?] row,all boundary identities?,both obligation groups?,candidate_satisfies_declared_BVP_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([-1, -1, 1, 4, 1, -1], 6, 6, Yes, [[left, y, [688, -12, -1], [688, -12, -1], Yes], [right, y, [662, 14, -1], [662, 14, -1], Yes], [bottom, x, [-157, -13, 4], [-157, -13, 4], Yes], [top, x, [-183, 13, 4], [-183, 13, 4], Yes]], Yes, Yes, candidate_satisfies_declared_BVP_only)","explanation":"Recompute the interior residual and every boundary restriction independently before taking their conjunction. This verifies a candidate for the declared problem without claiming general well-posedness. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-1, -1, 1, 4, 1, -1], 6, 6, Yes, [[left, y, [688, -12, -1], [688, -12, -1], Yes], [right, y, [662, 14, -1], [662, 14, -1], Yes], [bottom, x, [-157, -13, 4], [-157, -13, 4], Yes], [top, x, [-183, 13, 4], [-183, 13, 4], Yes]], Yes, Yes, candidate_satisfies_declared_BVP_only)."},"skills":["Close Interior and All Four Boundary Obligations Separately"],"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-20261013-advance-20261025-r01-boundary-value-problem-0063","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e59e79b203c6f882da792d9d032b08338ac78b107092ade08c7f4cb8d54d1a6b","family_id":"qf-mod25-bvp-joint-candidate-audit","seed":202610251063,"semantic_template":"qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[197,16,0]],[\"right\",[196,-12,0]],[\"bottom\",[-26,13,1]],[\"top\",[30,-15,1]]],\"bounds\":[-14,14,-13,15],\"coefficients\":[0,0,2,1,-1,0],\"source\":2}"},"fields":{"title":"Close Interior and All Four Boundary Obligations Separately","stem":"Use the quadratic candidate coefficients=[0, 0, 2, 1, -1, 0] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-14, 14, -13, 15]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=2; supplied [edge,coefficients] records=[[left, [197, 16, 0]], [right, [196, -12, 0]], [bottom, [-26, 13, 1]], [top, [30, -15, 1]]] must include all four distinct edges. Preserve their supplied order. Give (coefficients,Laplacian,source,interior identity?,every [edge,remaining variable,restriction,supplied,identity?] row,all boundary identities?,both obligation groups?,candidate_satisfies_declared_BVP_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([0, 0, 2, 1, -1, 0], 2, 2, Yes, [[left, y, [196, 16, 0], [197, 16, 0], No], [right, y, [196, -12, 0], [196, -12, 0], Yes], [bottom, x, [-26, 13, 1], [-26, 13, 1], Yes], [top, x, [30, -15, 1], [30, -15, 1], Yes]], No, No, candidate_satisfies_declared_BVP_only)","explanation":"Recompute the interior residual and every boundary restriction independently before taking their conjunction. This verifies a candidate for the declared problem without claiming general well-posedness. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0, 2, 1, -1, 0], 2, 2, Yes, [[left, y, [196, 16, 0], [197, 16, 0], No], [right, y, [196, -12, 0], [196, -12, 0], Yes], [bottom, x, [-26, 13, 1], [-26, 13, 1], Yes], [top, x, [30, -15, 1], [30, -15, 1], Yes]], No, No, candidate_satisfies_declared_BVP_only)."},"skills":["Close Interior and All Four Boundary Obligations Separately"],"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-20261013-advance-20261025-r01-boundary-value-problem-0064","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7dcb5c5edc8862dbd8ad7c1575e7d0591165d64d8757e7904fc7648cbac83a6c","family_id":"qf-mod25-bvp-joint-candidate-audit","seed":202610251064,"semantic_template":"qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[436,3,1]],[\"right\",[466,3,1]],[\"bottom\",[155,1,2]],[\"top\",[305,1,2]]],\"bounds\":[-15,15,-14,16],\"coefficients\":[1,1,3,2,0,1],\"source\":6}"},"fields":{"title":"Close Interior and All Four Boundary Obligations Separately","stem":"Use the quadratic candidate coefficients=[1, 1, 3, 2, 0, 1] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-15, 15, -14, 16]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=6; supplied [edge,coefficients] records=[[left, [436, 3, 1]], [right, [466, 3, 1]], [bottom, [155, 1, 2]], [top, [305, 1, 2]]] must include all four distinct edges. Preserve their supplied order. Give (coefficients,Laplacian,source,interior identity?,every [edge,remaining variable,restriction,supplied,identity?] row,all boundary identities?,both obligation groups?,candidate_satisfies_declared_BVP_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([1, 1, 3, 2, 0, 1], 6, 6, Yes, [[left, y, [436, 3, 1], [436, 3, 1], Yes], [right, y, [466, 3, 1], [466, 3, 1], Yes], [bottom, x, [155, 1, 2], [155, 1, 2], Yes], [top, x, [305, 1, 2], [305, 1, 2], Yes]], Yes, Yes, candidate_satisfies_declared_BVP_only)","explanation":"Recompute the interior residual and every boundary restriction independently before taking their conjunction. This verifies a candidate for the declared problem without claiming general well-posedness. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1, 3, 2, 0, 1], 6, 6, Yes, [[left, y, [436, 3, 1], [436, 3, 1], Yes], [right, y, [466, 3, 1], [466, 3, 1], Yes], [bottom, x, [155, 1, 2], [155, 1, 2], Yes], [top, x, [305, 1, 2], [305, 1, 2], Yes]], Yes, Yes, candidate_satisfies_declared_BVP_only)."},"skills":["Close Interior and All Four Boundary Obligations Separately"],"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-20261013-advance-20261025-r01-boundary-value-problem-0065","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"97d502f201dd81e1701c39c2b4e6e357a4e5bc066c07c6e0ff50fddf67c47dc9","family_id":"qf-mod25-bvp-joint-candidate-audit","seed":202610251065,"semantic_template":"qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[738,-19,2]],[\"right\",[802,13,2]],[\"bottom\",[497,-13,3]],[\"top\",[529,19,3]]],\"bounds\":[-16,16,-15,17],\"coefficients\":[2,2,-3,3,1,2],\"source\":11}"},"fields":{"title":"Close Interior and All Four Boundary Obligations Separately","stem":"Use the quadratic candidate coefficients=[2, 2, -3, 3, 1, 2] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-16, 16, -15, 17]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=11; supplied [edge,coefficients] records=[[left, [738, -19, 2]], [right, [802, 13, 2]], [bottom, [497, -13, 3]], [top, [529, 19, 3]]] must include all four distinct edges. Preserve their supplied order. Give (coefficients,Laplacian,source,interior identity?,every [edge,remaining variable,restriction,supplied,identity?] row,all boundary identities?,both obligation groups?,candidate_satisfies_declared_BVP_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([2, 2, -3, 3, 1, 2], 10, 11, No, [[left, y, [738, -19, 2], [738, -19, 2], Yes], [right, y, [802, 13, 2], [802, 13, 2], Yes], [bottom, x, [497, -13, 3], [497, -13, 3], Yes], [top, x, [529, 19, 3], [529, 19, 3], Yes]], Yes, No, candidate_satisfies_declared_BVP_only)","explanation":"Recompute the interior residual and every boundary restriction independently before taking their conjunction. This verifies a candidate for the declared problem without claiming general well-posedness. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 2, -3, 3, 1, 2], 10, 11, No, [[left, y, [738, -19, 2], [738, -19, 2], Yes], [right, y, [802, 13, 2], [802, 13, 2], Yes], [bottom, x, [497, -13, 3], [497, -13, 3], Yes], [top, x, [529, 19, 3], [529, 19, 3], Yes]], Yes, No, candidate_satisfies_declared_BVP_only)."},"skills":["Close Interior and All Four Boundary Obligations Separately"],"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-20261013-advance-20261025-r01-boundary-value-problem-0066","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b57df46451936f7a187714e3181e125318f13a886b36a3f0f8d1d3b52c91893c","family_id":"qf-mod25-bvp-joint-candidate-audit","seed":202610251066,"semantic_template":"qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[1193,15,-2]],[\"right\",[1125,-19,-2]],[\"bottom\",[-477,14,4]],[\"top\",[-681,-20,4]]],\"bounds\":[-17,17,-16,18],\"coefficients\":[3,-2,-2,4,-1,-2],\"source\":4}"},"fields":{"title":"Close Interior and All Four Boundary Obligations Separately","stem":"Use the quadratic candidate coefficients=[3, -2, -2, 4, -1, -2] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-17, 17, -16, 18]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=4; supplied [edge,coefficients] records=[[left, [1193, 15, -2]], [right, [1125, -19, -2]], [bottom, [-477, 14, 4]], [top, [-681, -20, 4]]] must include all four distinct edges. Preserve their supplied order. Give (coefficients,Laplacian,source,interior identity?,every [edge,remaining variable,restriction,supplied,identity?] row,all boundary identities?,both obligation groups?,candidate_satisfies_declared_BVP_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([3, -2, -2, 4, -1, -2], 4, 4, Yes, [[left, y, [1193, 15, -2], [1193, 15, -2], Yes], [right, y, [1125, -19, -2], [1125, -19, -2], Yes], [bottom, x, [-477, 14, 4], [-477, 14, 4], Yes], [top, x, [-681, -20, 4], [-681, -20, 4], Yes]], Yes, Yes, candidate_satisfies_declared_BVP_only)","explanation":"Recompute the interior residual and every boundary restriction independently before taking their conjunction. This verifies a candidate for the declared problem without claiming general well-posedness. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, -2, -2, 4, -1, -2], 4, 4, Yes, [[left, y, [1193, 15, -2], [1193, 15, -2], Yes], [right, y, [1125, -19, -2], [1125, -19, -2], Yes], [bottom, x, [-477, 14, 4], [-477, 14, 4], Yes], [top, x, [-681, -20, 4], [-681, -20, 4], Yes]], Yes, Yes, candidate_satisfies_declared_BVP_only)."},"skills":["Close Interior and All Four Boundary Obligations Separately"],"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-20261013-advance-20261025-r01-boundary-value-problem-0067","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5059d11892d6ec083a591c551ccb59c5113f9bf2cbb38cb357e940647a95a615","family_id":"qf-mod25-bvp-joint-candidate-audit","seed":202610251067,"semantic_template":"qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[346,-1,-1]],[\"right\",[310,-1,-1]],[\"bottom\",[-268,-1,1]],[\"top\",[-376,-1,1]]],\"bounds\":[-18,18,-17,19],\"coefficients\":[4,-1,-1,1,0,-1],\"source\":0}"},"fields":{"title":"Close Interior and All Four Boundary Obligations Separately","stem":"Use the quadratic candidate coefficients=[4, -1, -1, 1, 0, -1] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-18, 18, -17, 19]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=0; supplied [edge,coefficients] records=[[left, [346, -1, -1]], [right, [310, -1, -1]], [bottom, [-268, -1, 1]], [top, [-376, -1, 1]]] must include all four distinct edges. Preserve their supplied order. Give (coefficients,Laplacian,source,interior identity?,every [edge,remaining variable,restriction,supplied,identity?] row,all boundary identities?,both obligation groups?,candidate_satisfies_declared_BVP_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([4, -1, -1, 1, 0, -1], 0, 0, Yes, [[left, y, [346, -1, -1], [346, -1, -1], Yes], [right, y, [310, -1, -1], [310, -1, -1], Yes], [bottom, x, [-268, -1, 1], [-268, -1, 1], Yes], [top, x, [-376, -1, 1], [-376, -1, 1], Yes]], Yes, Yes, candidate_satisfies_declared_BVP_only)","explanation":"Recompute the interior residual and every boundary restriction independently before taking their conjunction. This verifies a candidate for the declared problem without claiming general well-posedness. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([4, -1, -1, 1, 0, -1], 0, 0, Yes, [[left, y, [346, -1, -1], [346, -1, -1], Yes], [right, y, [310, -1, -1], [310, -1, -1], Yes], [bottom, x, [-268, -1, 1], [-268, -1, 1], Yes], [top, x, [-376, -1, 1], [-376, -1, 1], Yes]], Yes, Yes, candidate_satisfies_declared_BVP_only)."},"skills":["Close Interior and All Four Boundary Obligations Separately"],"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-20261013-advance-20261025-r01-boundary-value-problem-0068","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2422355bb7c8ba86bd596bbfe4b2b2de6f04e4b648f19e4a6c48738ddce852e8","family_id":"qf-mod25-bvp-joint-candidate-audit","seed":202610251068,"semantic_template":"qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[727,-19,0]],[\"right\",[727,19,0]],[\"bottom\",[5,-18,2]],[\"top\",[5,20,2]]],\"bounds\":[-19,19,-18,20],\"coefficients\":[5,0,0,2,1,0],\"source\":4}"},"fields":{"title":"Close Interior and All Four Boundary Obligations Separately","stem":"Use the quadratic candidate coefficients=[5, 0, 0, 2, 1, 0] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-19, 19, -18, 20]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=4; supplied [edge,coefficients] records=[[left, [727, -19, 0]], [right, [727, 19, 0]], [bottom, [5, -18, 2]], [top, [5, 20, 2]]] must include all four distinct edges. Preserve their supplied order. Give (coefficients,Laplacian,source,interior identity?,every [edge,remaining variable,restriction,supplied,identity?] row,all boundary identities?,both obligation groups?,candidate_satisfies_declared_BVP_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([5, 0, 0, 2, 1, 0], 4, 4, Yes, [[left, y, [727, -19, 0], [727, -19, 0], Yes], [right, y, [727, 19, 0], [727, 19, 0], Yes], [bottom, x, [5, -18, 2], [5, -18, 2], Yes], [top, x, [5, 20, 2], [5, 20, 2], Yes]], Yes, Yes, candidate_satisfies_declared_BVP_only)","explanation":"Recompute the interior residual and every boundary restriction independently before taking their conjunction. This verifies a candidate for the declared problem without claiming general well-posedness. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([5, 0, 0, 2, 1, 0], 4, 4, Yes, [[left, y, [727, -19, 0], [727, -19, 0], Yes], [right, y, [727, 19, 0], [727, 19, 0], Yes], [bottom, x, [5, -18, 2], [5, -18, 2], Yes], [top, x, [5, 20, 2], [5, 20, 2], Yes]], Yes, Yes, candidate_satisfies_declared_BVP_only)."},"skills":["Close Interior and All Four Boundary Obligations Separately"],"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-20261013-advance-20261025-r01-boundary-value-problem-0069","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34ce6b3d8df94d34e21e50f3cfe9d0a5459750fd6e3aa15c3ef13c04af5ee120","family_id":"qf-mod25-bvp-joint-candidate-audit","seed":202610251069,"semantic_template":"qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[1187,21,1]],[\"right\",[1226,-19,1]],[\"bottom\",[348,20,3]],[\"top\",[468,-20,3]]],\"bounds\":[-20,20,-19,21],\"coefficients\":[6,1,1,3,-1,1],\"source\":8}"},"fields":{"title":"Close Interior and All Four Boundary Obligations Separately","stem":"Use the quadratic candidate coefficients=[6, 1, 1, 3, -1, 1] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-20, 20, -19, 21]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=8; supplied [edge,coefficients] records=[[left, [1187, 21, 1]], [right, [1226, -19, 1]], [bottom, [348, 20, 3]], [top, [468, -20, 3]]] must include all four distinct edges. Preserve their supplied order. Give (coefficients,Laplacian,source,interior identity?,every [edge,remaining variable,restriction,supplied,identity?] row,all boundary identities?,both obligation groups?,candidate_satisfies_declared_BVP_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([6, 1, 1, 3, -1, 1], 8, 8, Yes, [[left, y, [1186, 21, 1], [1187, 21, 1], No], [right, y, [1226, -19, 1], [1226, -19, 1], Yes], [bottom, x, [348, 20, 3], [348, 20, 3], Yes], [top, x, [468, -20, 3], [468, -20, 3], Yes]], No, No, candidate_satisfies_declared_BVP_only)","explanation":"Recompute the interior residual and every boundary restriction independently before taking their conjunction. This verifies a candidate for the declared problem without claiming general well-posedness. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([6, 1, 1, 3, -1, 1], 8, 8, Yes, [[left, y, [1186, 21, 1], [1187, 21, 1], No], [right, y, [1226, -19, 1], [1226, -19, 1], Yes], [bottom, x, [348, 20, 3], [348, 20, 3], Yes], [top, x, [468, -20, 3], [468, -20, 3], Yes]], No, No, candidate_satisfies_declared_BVP_only)."},"skills":["Close Interior and All Four Boundary Obligations Separately"],"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-20261013-advance-20261025-r01-boundary-value-problem-0070","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fddf9f28473d5903f958ec85b33c34fe8a5ea43d1543263c78990dca6f15b09b","family_id":"qf-mod25-bvp-joint-candidate-audit","seed":202610251070,"semantic_template":"qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[1729,2,2]],[\"right\",[1813,2,2]],[\"bottom\",[767,2,4]],[\"top\",[1019,2,4]]],\"bounds\":[-21,21,-20,22],\"coefficients\":[7,2,2,4,0,2],\"source\":12}"},"fields":{"title":"Close Interior and All Four Boundary Obligations Separately","stem":"Use the quadratic candidate coefficients=[7, 2, 2, 4, 0, 2] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-21, 21, -20, 22]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=12; supplied [edge,coefficients] records=[[left, [1729, 2, 2]], [right, [1813, 2, 2]], [bottom, [767, 2, 4]], [top, [1019, 2, 4]]] must include all four distinct edges. Preserve their supplied order. Give (coefficients,Laplacian,source,interior identity?,every [edge,remaining variable,restriction,supplied,identity?] row,all boundary identities?,both obligation groups?,candidate_satisfies_declared_BVP_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([7, 2, 2, 4, 0, 2], 12, 12, Yes, [[left, y, [1729, 2, 2], [1729, 2, 2], Yes], [right, y, [1813, 2, 2], [1813, 2, 2], Yes], [bottom, x, [767, 2, 4], [767, 2, 4], Yes], [top, x, [1019, 2, 4], [1019, 2, 4], Yes]], Yes, Yes, candidate_satisfies_declared_BVP_only)","explanation":"Recompute the interior residual and every boundary restriction independently before taking their conjunction. This verifies a candidate for the declared problem without claiming general well-posedness. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([7, 2, 2, 4, 0, 2], 12, 12, Yes, [[left, y, [1729, 2, 2], [1729, 2, 2], Yes], [right, y, [1813, 2, 2], [1813, 2, 2], Yes], [bottom, x, [767, 2, 4], [767, 2, 4], Yes], [top, x, [1019, 2, 4], [1019, 2, 4], Yes]], Yes, Yes, candidate_satisfies_declared_BVP_only)."},"skills":["Close Interior and All Four Boundary Obligations Separately"],"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-20261013-advance-20261025-r01-boundary-value-problem-0071","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c7849076523a7ae1757432b109128b0375bae876cebc7a87eaaadc1636378121","family_id":"qf-mod25-bvp-joint-candidate-audit","seed":202610251071,"semantic_template":"qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[536,-19,-2]],[\"right\",[448,25,-2]],[\"bottom\",[-937,-23,1]],[\"top\",[-981,21,1]]],\"bounds\":[-22,22,-21,23],\"coefficients\":[8,-2,3,1,1,-2],\"source\":-2}"},"fields":{"title":"Close Interior and All Four Boundary Obligations Separately","stem":"Use the quadratic candidate coefficients=[8, -2, 3, 1, 1, -2] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-22, 22, -21, 23]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=-2; supplied [edge,coefficients] records=[[left, [536, -19, -2]], [right, [448, 25, -2]], [bottom, [-937, -23, 1]], [top, [-981, 21, 1]]] must include all four distinct edges. Preserve their supplied order. Give (coefficients,Laplacian,source,interior identity?,every [edge,remaining variable,restriction,supplied,identity?] row,all boundary identities?,both obligation groups?,candidate_satisfies_declared_BVP_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([8, -2, 3, 1, 1, -2], -2, -2, Yes, [[left, y, [536, -19, -2], [536, -19, -2], Yes], [right, y, [448, 25, -2], [448, 25, -2], Yes], [bottom, x, [-937, -23, 1], [-937, -23, 1], Yes], [top, x, [-981, 21, 1], [-981, 21, 1], Yes]], Yes, Yes, candidate_satisfies_declared_BVP_only)","explanation":"Recompute the interior residual and every boundary restriction independently before taking their conjunction. This verifies a candidate for the declared problem without claiming general well-posedness. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([8, -2, 3, 1, 1, -2], -2, -2, Yes, [[left, y, [536, -19, -2], [536, -19, -2], Yes], [right, y, [448, 25, -2], [448, 25, -2], Yes], [bottom, x, [-937, -23, 1], [-937, -23, 1], Yes], [top, x, [-981, 21, 1], [-981, 21, 1], Yes]], Yes, Yes, candidate_satisfies_declared_BVP_only)."},"skills":["Close Interior and All Four Boundary Obligations Separately"],"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-20261013-advance-20261025-r01-boundary-value-problem-0072","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0dd5f980fad37e147b992f9718d6a5f768c5626f58b24c009f30e2faeb48b11a","family_id":"qf-mod25-bvp-joint-candidate-audit","seed":202610251072,"semantic_template":"qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[1090,20,-1]],[\"right\",[1044,-26,-1]],[\"bottom\",[-409,21,2]],[\"top\",[-639,-25,2]]],\"bounds\":[-23,23,-22,24],\"coefficients\":[9,-1,-3,2,-1,-1],\"source\":3}"},"fields":{"title":"Close Interior and All Four Boundary Obligations Separately","stem":"Use the quadratic candidate coefficients=[9, -1, -3, 2, -1, -1] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-23, 23, -22, 24]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=3; supplied [edge,coefficients] records=[[left, [1090, 20, -1]], [right, [1044, -26, -1]], [bottom, [-409, 21, 2]], [top, [-639, -25, 2]]] must include all four distinct edges. Preserve their supplied order. Give (coefficients,Laplacian,source,interior identity?,every [edge,remaining variable,restriction,supplied,identity?] row,all boundary identities?,both obligation groups?,candidate_satisfies_declared_BVP_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([9, -1, -3, 2, -1, -1], 2, 3, No, [[left, y, [1090, 20, -1], [1090, 20, -1], Yes], [right, y, [1044, -26, -1], [1044, -26, -1], Yes], [bottom, x, [-409, 21, 2], [-409, 21, 2], Yes], [top, x, [-639, -25, 2], [-639, -25, 2], Yes]], Yes, No, candidate_satisfies_declared_BVP_only)","explanation":"Recompute the interior residual and every boundary restriction independently before taking their conjunction. This verifies a candidate for the declared problem without claiming general well-posedness. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([9, -1, -3, 2, -1, -1], 2, 3, No, [[left, y, [1090, 20, -1], [1090, 20, -1], Yes], [right, y, [1044, -26, -1], [1044, -26, -1], Yes], [bottom, x, [-409, 21, 2], [-409, 21, 2], Yes], [top, x, [-639, -25, 2], [-639, -25, 2], Yes]], Yes, No, candidate_satisfies_declared_BVP_only)."},"skills":["Close Interior and All Four Boundary Obligations Separately"],"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-20261013-advance-20261025-r01-boundary-value-problem-0073","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d6353eb3094f44205c438a1f4e4666698e532cbc0e3395edde8b4fe5616795fb","family_id":"qf-mod25-bvp-joint-candidate-audit","seed":202610251073,"semantic_template":"qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[1738,-2,0]],[\"right\",[1738,-2,0]],[\"bottom\",[56,0,3]],[\"top\",[-40,0,3]]],\"bounds\":[-24,24,-23,25],\"coefficients\":[10,0,-2,3,0,0],\"source\":6}"},"fields":{"title":"Close Interior and All Four Boundary Obligations Separately","stem":"Use the quadratic candidate coefficients=[10, 0, -2, 3, 0, 0] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-24, 24, -23, 25]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=6; supplied [edge,coefficients] records=[[left, [1738, -2, 0]], [right, [1738, -2, 0]], [bottom, [56, 0, 3]], [top, [-40, 0, 3]]] must include all four distinct edges. Preserve their supplied order. Give (coefficients,Laplacian,source,interior identity?,every [edge,remaining variable,restriction,supplied,identity?] row,all boundary identities?,both obligation groups?,candidate_satisfies_declared_BVP_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([10, 0, -2, 3, 0, 0], 6, 6, Yes, [[left, y, [1738, -2, 0], [1738, -2, 0], Yes], [right, y, [1738, -2, 0], [1738, -2, 0], Yes], [bottom, x, [56, 0, 3], [56, 0, 3], Yes], [top, x, [-40, 0, 3], [-40, 0, 3], Yes]], Yes, Yes, candidate_satisfies_declared_BVP_only)","explanation":"Recompute the interior residual and every boundary restriction independently before taking their conjunction. This verifies a candidate for the declared problem without claiming general well-posedness. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([10, 0, -2, 3, 0, 0], 6, 6, Yes, [[left, y, [1738, -2, 0], [1738, -2, 0], Yes], [right, y, [1738, -2, 0], [1738, -2, 0], Yes], [bottom, x, [56, 0, 3], [56, 0, 3], Yes], [top, x, [-40, 0, 3], [-40, 0, 3], Yes]], Yes, Yes, candidate_satisfies_declared_BVP_only)."},"skills":["Close Interior and All Four Boundary Obligations Separately"],"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-20261013-advance-20261025-r01-boundary-value-problem-0074","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"42b13908a9b584e189eaae9ab40ae6b595b583fa41888c8a831e22bb38153a18","family_id":"qf-mod25-bvp-joint-candidate-audit","seed":202610251074,"semantic_template":"qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[2486,-26,1]],[\"right\",[2536,24,1]],[\"bottom\",[611,-23,4]],[\"top\",[661,27,4]]],\"bounds\":[-25,25,-24,26],\"coefficients\":[11,1,-1,4,1,1],\"source\":10}"},"fields":{"title":"Close Interior and All Four Boundary Obligations Separately","stem":"Use the quadratic candidate coefficients=[11, 1, -1, 4, 1, 1] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-25, 25, -24, 26]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=10; supplied [edge,coefficients] records=[[left, [2486, -26, 1]], [right, [2536, 24, 1]], [bottom, [611, -23, 4]], [top, [661, 27, 4]]] must include all four distinct edges. Preserve their supplied order. Give (coefficients,Laplacian,source,interior identity?,every [edge,remaining variable,restriction,supplied,identity?] row,all boundary identities?,both obligation groups?,candidate_satisfies_declared_BVP_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([11, 1, -1, 4, 1, 1], 10, 10, Yes, [[left, y, [2486, -26, 1], [2486, -26, 1], Yes], [right, y, [2536, 24, 1], [2536, 24, 1], Yes], [bottom, x, [611, -23, 4], [611, -23, 4], Yes], [top, x, [661, 27, 4], [661, 27, 4], Yes]], Yes, Yes, candidate_satisfies_declared_BVP_only)","explanation":"Recompute the interior residual and every boundary restriction independently before taking their conjunction. This verifies a candidate for the declared problem without claiming general well-posedness. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([11, 1, -1, 4, 1, 1], 10, 10, Yes, [[left, y, [2486, -26, 1], [2486, -26, 1], Yes], [right, y, [2536, 24, 1], [2536, 24, 1], Yes], [bottom, x, [611, -23, 4], [611, -23, 4], Yes], [top, x, [661, 27, 4], [661, 27, 4], Yes]], Yes, Yes, candidate_satisfies_declared_BVP_only)."},"skills":["Close Interior and All Four Boundary Obligations Separately"],"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-20261013-advance-20261025-r01-boundary-value-problem-0075","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"42b3d337e56129850e61e882d4b5aaca2ba5ca16ac5f8e55ac716ed28316d722","family_id":"qf-mod25-bvp-joint-candidate-audit","seed":202610251075,"semantic_template":"qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[637,26,2]],[\"right\",[740,-26,2]],[\"bottom\",[1262,27,1]],[\"top\",[1470,-25,1]]],\"bounds\":[-26,26,-25,27],\"coefficients\":[12,2,0,1,-1,2],\"source\":6}"},"fields":{"title":"Close Interior and All Four Boundary Obligations Separately","stem":"Use the quadratic candidate coefficients=[12, 2, 0, 1, -1, 2] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-26, 26, -25, 27]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=6; supplied [edge,coefficients] records=[[left, [637, 26, 2]], [right, [740, -26, 2]], [bottom, [1262, 27, 1]], [top, [1470, -25, 1]]] must include all four distinct edges. Preserve their supplied order. Give (coefficients,Laplacian,source,interior identity?,every [edge,remaining variable,restriction,supplied,identity?] row,all boundary identities?,both obligation groups?,candidate_satisfies_declared_BVP_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"([12, 2, 0, 1, -1, 2], 6, 6, Yes, [[left, y, [636, 26, 2], [637, 26, 2], No], [right, y, [740, -26, 2], [740, -26, 2], Yes], [bottom, x, [1262, 27, 1], [1262, 27, 1], Yes], [top, x, [1470, -25, 1], [1470, -25, 1], Yes]], No, No, candidate_satisfies_declared_BVP_only)","explanation":"Recompute the interior residual and every boundary restriction independently before taking their conjunction. This verifies a candidate for the declared problem without claiming general well-posedness. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([12, 2, 0, 1, -1, 2], 6, 6, Yes, [[left, y, [636, 26, 2], [637, 26, 2], No], [right, y, [740, -26, 2], [740, -26, 2], Yes], [bottom, x, [1262, 27, 1], [1262, 27, 1], Yes], [top, x, [1470, -25, 1], [1470, -25, 1], Yes]], No, No, candidate_satisfies_declared_BVP_only)."},"skills":["Close Interior and All Four Boundary Obligations Separately"],"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-20261013-advance-20261025-r01-boundary-value-problem-0076","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f0c9f79ae6af149b2e501397da944322a5653d168eeae3f61f6d0ba4fe55232","family_id":"qf-mod25-bvp-obligation-separation","seed":202610251076,"semantic_template":"qf-mod25-bvp-obligation-separation/{\"boundary_data\":[[\"left\",[-4,-1,-2]],[\"right\",[-12,-5,-2]],[\"bottom\",[-11,-1,1]],[\"top\",[-39,-5,1]]],\"bounds\":[-2,2,-1,3],\"coefficients\":[-12,-2,-3,1,-1,-2],\"source\":-1}"},"fields":{"title":"Interior Pass and Boundary Pass Must Not Be Conflated","stem":"Use the quadratic candidate coefficients=[-12, -2, -3, 1, -1, -2] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-2, 2, -1, 3]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=-1, boundary data=[[left, [-4, -1, -2]], [right, [-12, -5, -2]], [bottom, [-11, -1, 1]], [top, [-39, -5, 1]]] with all four distinct edges. Give (Laplacian,source,interior identity?,boundary rows in supplied order,boundary identities?,one of both_pass / interior_pass_boundary_fail / boundary_pass_interior_fail / both_fail). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"(-2, -1, No, [[left, y, [-4, -1, -2], [-4, -1, -2], Yes], [right, y, [-12, -5, -2], [-12, -5, -2], Yes], [bottom, x, [-11, -1, 1], [-11, -1, 1], Yes], [top, x, [-39, -5, 1], [-39, -5, 1], Yes]], Yes, boundary_pass_interior_fail)","explanation":"Report both groups even when only one succeeds. A boundary pass does not repair a failed interior equation, or conversely. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (-2, -1, No, [[left, y, [-4, -1, -2], [-4, -1, -2], Yes], [right, y, [-12, -5, -2], [-12, -5, -2], Yes], [bottom, x, [-11, -1, 1], [-11, -1, 1], Yes], [top, x, [-39, -5, 1], [-39, -5, 1], Yes]], Yes, boundary_pass_interior_fail)."},"skills":["Interior Pass and Boundary Pass Must Not Be Conflated"],"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-20261013-advance-20261025-r01-boundary-value-problem-0077","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"907a8285dfa25fc1f97334318ef46e75febd1d8b9c52098ea5730e8ab1b02f8a","family_id":"qf-mod25-bvp-obligation-separation","seed":202610251077,"semantic_template":"qf-mod25-bvp-obligation-separation/{\"boundary_data\":[[\"left\",[11,-2,-1]],[\"right\",[4,-2,-1]],[\"bottom\",[-11,-1,2]],[\"top\",[-35,-1,2]]],\"bounds\":[-3,3,-2,4],\"coefficients\":[-11,-1,-2,2,0,-1],\"source\":2}"},"fields":{"title":"Interior Pass and Boundary Pass Must Not Be Conflated","stem":"Use the quadratic candidate coefficients=[-11, -1, -2, 2, 0, -1] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-3, 3, -2, 4]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=2, boundary data=[[left, [11, -2, -1]], [right, [4, -2, -1]], [bottom, [-11, -1, 2]], [top, [-35, -1, 2]]] with all four distinct edges. Give (Laplacian,source,interior identity?,boundary rows in supplied order,boundary identities?,one of both_pass / interior_pass_boundary_fail / boundary_pass_interior_fail / both_fail). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"(2, 2, Yes, [[left, y, [10, -2, -1], [11, -2, -1], No], [right, y, [4, -2, -1], [4, -2, -1], Yes], [bottom, x, [-11, -1, 2], [-11, -1, 2], Yes], [top, x, [-35, -1, 2], [-35, -1, 2], Yes]], No, interior_pass_boundary_fail)","explanation":"Report both groups even when only one succeeds. A boundary pass does not repair a failed interior equation, or conversely. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (2, 2, Yes, [[left, y, [10, -2, -1], [11, -2, -1], No], [right, y, [4, -2, -1], [4, -2, -1], Yes], [bottom, x, [-11, -1, 2], [-11, -1, 2], Yes], [top, x, [-35, -1, 2], [-35, -1, 2], Yes]], No, interior_pass_boundary_fail)."},"skills":["Interior Pass and Boundary Pass Must Not Be Conflated"],"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-20261013-advance-20261025-r01-boundary-value-problem-0078","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0918ec418afacb40a002c9407c804b1ef4126d34eb7d39ae5c2b78b9c28e2e8b","family_id":"qf-mod25-bvp-obligation-separation","seed":202610251078,"semantic_template":"qf-mod25-bvp-obligation-separation/{\"boundary_data\":[[\"left\",[38,-5,0]],[\"right\",[38,3,0]],[\"bottom\",[-7,-3,3]],[\"top\",[-15,5,3]]],\"bounds\":[-4,4,-3,5],\"coefficients\":[-10,0,-1,3,1,0],\"source\":7}"},"fields":{"title":"Interior Pass and Boundary Pass Must Not Be Conflated","stem":"Use the quadratic candidate coefficients=[-10, 0, -1, 3, 1, 0] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-4, 4, -3, 5]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=7, boundary data=[[left, [38, -5, 0]], [right, [38, 3, 0]], [bottom, [-7, -3, 3]], [top, [-15, 5, 3]]] with all four distinct edges. Give (Laplacian,source,interior identity?,boundary rows in supplied order,boundary identities?,one of both_pass / interior_pass_boundary_fail / boundary_pass_interior_fail / both_fail). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"(6, 7, No, [[left, y, [38, -5, 0], [38, -5, 0], Yes], [right, y, [38, 3, 0], [38, 3, 0], Yes], [bottom, x, [-7, -3, 3], [-7, -3, 3], Yes], [top, x, [-15, 5, 3], [-15, 5, 3], Yes]], Yes, boundary_pass_interior_fail)","explanation":"Report both groups even when only one succeeds. A boundary pass does not repair a failed interior equation, or conversely. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (6, 7, No, [[left, y, [38, -5, 0], [38, -5, 0], Yes], [right, y, [38, 3, 0], [38, 3, 0], Yes], [bottom, x, [-7, -3, 3], [-7, -3, 3], Yes], [top, x, [-15, 5, 3], [-15, 5, 3], Yes]], Yes, boundary_pass_interior_fail)."},"skills":["Interior Pass and Boundary Pass Must Not Be Conflated"],"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-20261013-advance-20261025-r01-boundary-value-problem-0079","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5de572879f2406d2089ec51eb23331dd8de61b16871c9e1a0ce85419532004ce","family_id":"qf-mod25-bvp-obligation-separation","seed":202610251079,"semantic_template":"qf-mod25-bvp-obligation-separation/{\"boundary_data\":[[\"left\",[87,5,1]],[\"right\",[96,-5,1]],[\"bottom\",[7,5,4]],[\"top\",[27,-5,4]]],\"bounds\":[-5,5,-4,6],\"coefficients\":[-9,1,0,4,-1,1],\"source\":10}"},"fields":{"title":"Interior Pass and Boundary Pass Must Not Be Conflated","stem":"Use the quadratic candidate coefficients=[-9, 1, 0, 4, -1, 1] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-5, 5, -4, 6]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=10, boundary data=[[left, [87, 5, 1]], [right, [96, -5, 1]], [bottom, [7, 5, 4]], [top, [27, -5, 4]]] with all four distinct edges. Give (Laplacian,source,interior identity?,boundary rows in supplied order,boundary identities?,one of both_pass / interior_pass_boundary_fail / boundary_pass_interior_fail / both_fail). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"(10, 10, Yes, [[left, y, [86, 5, 1], [87, 5, 1], No], [right, y, [96, -5, 1], [96, -5, 1], Yes], [bottom, x, [7, 5, 4], [7, 5, 4], Yes], [top, x, [27, -5, 4], [27, -5, 4], Yes]], No, interior_pass_boundary_fail)","explanation":"Report both groups even when only one succeeds. A boundary pass does not repair a failed interior equation, or conversely. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (10, 10, Yes, [[left, y, [86, 5, 1], [87, 5, 1], No], [right, y, [96, -5, 1], [96, -5, 1], Yes], [bottom, x, [7, 5, 4], [7, 5, 4], Yes], [top, x, [27, -5, 4], [27, -5, 4], Yes]], No, interior_pass_boundary_fail)."},"skills":["Interior Pass and Boundary Pass Must Not Be Conflated"],"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-20261013-advance-20261025-r01-boundary-value-problem-0080","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"35f5a6129bec09b3970cecc129113e809bb74f8d27a513bba23935de3462ee5e","family_id":"qf-mod25-bvp-obligation-separation","seed":202610251080,"semantic_template":"qf-mod25-bvp-obligation-separation/{\"boundary_data\":[[\"left\",[16,1,2]],[\"right\",[40,1,2]],[\"bottom\",[37,2,1]],[\"top\",[97,2,1]]],\"bounds\":[-6,6,-5,7],\"coefficients\":[-8,2,1,1,0,2],\"source\":7}"},"fields":{"title":"Interior Pass and Boundary Pass Must Not Be Conflated","stem":"Use the quadratic candidate coefficients=[-8, 2, 1, 1, 0, 2] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-6, 6, -5, 7]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=7, boundary data=[[left, [16, 1, 2]], [right, [40, 1, 2]], [bottom, [37, 2, 1]], [top, [97, 2, 1]]] with all four distinct edges. Give (Laplacian,source,interior identity?,boundary rows in supplied order,boundary identities?,one of both_pass / interior_pass_boundary_fail / boundary_pass_interior_fail / both_fail). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"(6, 7, No, [[left, y, [16, 1, 2], [16, 1, 2], Yes], [right, y, [40, 1, 2], [40, 1, 2], Yes], [bottom, x, [37, 2, 1], [37, 2, 1], Yes], [top, x, [97, 2, 1], [97, 2, 1], Yes]], Yes, boundary_pass_interior_fail)","explanation":"Report both groups even when only one succeeds. A boundary pass does not repair a failed interior equation, or conversely. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (6, 7, No, [[left, y, [16, 1, 2], [16, 1, 2], Yes], [right, y, [40, 1, 2], [40, 1, 2], Yes], [bottom, x, [37, 2, 1], [37, 2, 1], Yes], [top, x, [97, 2, 1], [97, 2, 1], Yes]], Yes, boundary_pass_interior_fail)."},"skills":["Interior Pass and Boundary Pass Must Not Be Conflated"],"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-20261013-advance-20261025-r01-boundary-value-problem-0081","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7927868e00517542a0dc52d0cdadf964d100ccea6950f361619f62cf3311783a","family_id":"qf-mod25-bvp-obligation-separation","seed":202610251081,"semantic_template":"qf-mod25-bvp-obligation-separation/{\"boundary_data\":[[\"left\",[106,-5,-2]],[\"right\",[77,9,-2]],[\"bottom\",[-91,-8,2]],[\"top\",[-119,6,2]]],\"bounds\":[-7,7,-6,8],\"coefficients\":[-7,-2,2,2,1,-2],\"source\":0}"},"fields":{"title":"Interior Pass and Boundary Pass Must Not Be Conflated","stem":"Use the quadratic candidate coefficients=[-7, -2, 2, 2, 1, -2] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-7, 7, -6, 8]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=0, boundary data=[[left, [106, -5, -2]], [right, [77, 9, -2]], [bottom, [-91, -8, 2]], [top, [-119, 6, 2]]] with all four distinct edges. Give (Laplacian,source,interior identity?,boundary rows in supplied order,boundary identities?,one of both_pass / interior_pass_boundary_fail / boundary_pass_interior_fail / both_fail). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"(0, 0, Yes, [[left, y, [105, -5, -2], [106, -5, -2], No], [right, y, [77, 9, -2], [77, 9, -2], Yes], [bottom, x, [-91, -8, 2], [-91, -8, 2], Yes], [top, x, [-119, 6, 2], [-119, 6, 2], Yes]], No, interior_pass_boundary_fail)","explanation":"Report both groups even when only one succeeds. A boundary pass does not repair a failed interior equation, or conversely. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (0, 0, Yes, [[left, y, [105, -5, -2], [106, -5, -2], No], [right, y, [77, 9, -2], [77, 9, -2], Yes], [bottom, x, [-91, -8, 2], [-91, -8, 2], Yes], [top, x, [-119, 6, 2], [-119, 6, 2], Yes]], No, interior_pass_boundary_fail)."},"skills":["Interior Pass and Boundary Pass Must Not Be Conflated"],"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-20261013-advance-20261025-r01-boundary-value-problem-0082","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba7975830ed78f6b75d222bfe920bff55187c1f5aa4b42fc93a41c9db49fcc13","family_id":"qf-mod25-bvp-obligation-separation","seed":202610251082,"semantic_template":"qf-mod25-bvp-obligation-separation/{\"boundary_data\":[[\"left\",[194,11,-1]],[\"right\",[178,-5,-1]],[\"bottom\",[-76,6,3]],[\"top\",[-60,-10,3]]],\"bounds\":[-8,8,-7,9],\"coefficients\":[-6,-1,3,3,-1,-1],\"source\":5}"},"fields":{"title":"Interior Pass and Boundary Pass Must Not Be Conflated","stem":"Use the quadratic candidate coefficients=[-6, -1, 3, 3, -1, -1] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-8, 8, -7, 9]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=5, boundary data=[[left, [194, 11, -1]], [right, [178, -5, -1]], [bottom, [-76, 6, 3]], [top, [-60, -10, 3]]] with all four distinct edges. Give (Laplacian,source,interior identity?,boundary rows in supplied order,boundary identities?,one of both_pass / interior_pass_boundary_fail / boundary_pass_interior_fail / both_fail). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"(4, 5, No, [[left, y, [194, 11, -1], [194, 11, -1], Yes], [right, y, [178, -5, -1], [178, -5, -1], Yes], [bottom, x, [-76, 6, 3], [-76, 6, 3], Yes], [top, x, [-60, -10, 3], [-60, -10, 3], Yes]], Yes, boundary_pass_interior_fail)","explanation":"Report both groups even when only one succeeds. A boundary pass does not repair a failed interior equation, or conversely. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 5, No, [[left, y, [194, 11, -1], [194, 11, -1], Yes], [right, y, [178, -5, -1], [178, -5, -1], Yes], [bottom, x, [-76, 6, 3], [-76, 6, 3], Yes], [top, x, [-60, -10, 3], [-60, -10, 3], Yes]], Yes, boundary_pass_interior_fail)."},"skills":["Interior Pass and Boundary Pass Must Not Be Conflated"],"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-20261013-advance-20261025-r01-boundary-value-problem-0083","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b1ed6f114a1fef6b2a530a13744ef1cb79deb9c8c7751098f70c3afaa882068","family_id":"qf-mod25-bvp-obligation-separation","seed":202610251083,"semantic_template":"qf-mod25-bvp-obligation-separation/{\"boundary_data\":[[\"left\",[320,-3,0]],[\"right\",[319,-3,0]],[\"bottom\",[19,0,4]],[\"top\",[-35,0,4]]],\"bounds\":[-9,9,-8,10],\"coefficients\":[-5,0,-3,4,0,0],\"source\":8}"},"fields":{"title":"Interior Pass and Boundary Pass Must Not Be Conflated","stem":"Use the quadratic candidate coefficients=[-5, 0, -3, 4, 0, 0] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-9, 9, -8, 10]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=8, boundary data=[[left, [320, -3, 0]], [right, [319, -3, 0]], [bottom, [19, 0, 4]], [top, [-35, 0, 4]]] with all four distinct edges. Give (Laplacian,source,interior identity?,boundary rows in supplied order,boundary identities?,one of both_pass / interior_pass_boundary_fail / boundary_pass_interior_fail / both_fail). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"(8, 8, Yes, [[left, y, [319, -3, 0], [320, -3, 0], No], [right, y, [319, -3, 0], [319, -3, 0], Yes], [bottom, x, [19, 0, 4], [19, 0, 4], Yes], [top, x, [-35, 0, 4], [-35, 0, 4], Yes]], No, interior_pass_boundary_fail)","explanation":"Report both groups even when only one succeeds. A boundary pass does not repair a failed interior equation, or conversely. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (8, 8, Yes, [[left, y, [319, -3, 0], [320, -3, 0], No], [right, y, [319, -3, 0], [319, -3, 0], Yes], [bottom, x, [19, 0, 4], [19, 0, 4], Yes], [top, x, [-35, 0, 4], [-35, 0, 4], Yes]], No, interior_pass_boundary_fail)."},"skills":["Interior Pass and Boundary Pass Must Not Be Conflated"],"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-20261013-advance-20261025-r01-boundary-value-problem-0084","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f19f862a165278c035553cbc7277b9685c649ebb79f51d61ffec3de5de4cb0f","family_id":"qf-mod25-bvp-obligation-separation","seed":202610251084,"semantic_template":"qf-mod25-bvp-obligation-separation/{\"boundary_data\":[[\"left\",[86,-12,1]],[\"right\",[106,8,1]],[\"bottom\",[95,-8,1]],[\"top\",[95,12,1]]],\"bounds\":[-10,10,-9,11],\"coefficients\":[-4,1,-2,1,1,1],\"source\":5}"},"fields":{"title":"Interior Pass and Boundary Pass Must Not Be Conflated","stem":"Use the quadratic candidate coefficients=[-4, 1, -2, 1, 1, 1] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-10, 10, -9, 11]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=5, boundary data=[[left, [86, -12, 1]], [right, [106, 8, 1]], [bottom, [95, -8, 1]], [top, [95, 12, 1]]] with all four distinct edges. Give (Laplacian,source,interior identity?,boundary rows in supplied order,boundary identities?,one of both_pass / interior_pass_boundary_fail / boundary_pass_interior_fail / both_fail). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"(4, 5, No, [[left, y, [86, -12, 1], [86, -12, 1], Yes], [right, y, [106, 8, 1], [106, 8, 1], Yes], [bottom, x, [95, -8, 1], [95, -8, 1], Yes], [top, x, [95, 12, 1], [95, 12, 1], Yes]], Yes, boundary_pass_interior_fail)","explanation":"Report both groups even when only one succeeds. A boundary pass does not repair a failed interior equation, or conversely. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 5, No, [[left, y, [86, -12, 1], [86, -12, 1], Yes], [right, y, [106, 8, 1], [106, 8, 1], Yes], [bottom, x, [95, -8, 1], [95, -8, 1], Yes], [top, x, [95, 12, 1], [95, 12, 1], Yes]], Yes, boundary_pass_interior_fail)."},"skills":["Interior Pass and Boundary Pass Must Not Be Conflated"],"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-20261013-advance-20261025-r01-boundary-value-problem-0085","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb07ca84bf0cb668753331bc7b2c65d4a35994c0d8a205abe93d5144877663b3","family_id":"qf-mod25-bvp-obligation-separation","seed":202610251085,"semantic_template":"qf-mod25-bvp-obligation-separation/{\"boundary_data\":[[\"left\",[218,10,2]],[\"right\",[261,-12,2]],[\"bottom\",[207,12,2]],[\"top\",[273,-10,2]]],\"bounds\":[-11,11,-10,12],\"coefficients\":[-3,2,-1,2,-1,2],\"source\":8}"},"fields":{"title":"Interior Pass and Boundary Pass Must Not Be Conflated","stem":"Use the quadratic candidate coefficients=[-3, 2, -1, 2, -1, 2] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-11, 11, -10, 12]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=8, boundary data=[[left, [218, 10, 2]], [right, [261, -12, 2]], [bottom, [207, 12, 2]], [top, [273, -10, 2]]] with all four distinct edges. Give (Laplacian,source,interior identity?,boundary rows in supplied order,boundary identities?,one of both_pass / interior_pass_boundary_fail / boundary_pass_interior_fail / both_fail). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"(8, 8, Yes, [[left, y, [217, 10, 2], [218, 10, 2], No], [right, y, [261, -12, 2], [261, -12, 2], Yes], [bottom, x, [207, 12, 2], [207, 12, 2], Yes], [top, x, [273, -10, 2], [273, -10, 2], Yes]], No, interior_pass_boundary_fail)","explanation":"Report both groups even when only one succeeds. A boundary pass does not repair a failed interior equation, or conversely. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (8, 8, Yes, [[left, y, [217, 10, 2], [218, 10, 2], No], [right, y, [261, -12, 2], [261, -12, 2], Yes], [bottom, x, [207, 12, 2], [207, 12, 2], Yes], [top, x, [273, -10, 2], [273, -10, 2], Yes]], No, interior_pass_boundary_fail)."},"skills":["Interior Pass and Boundary Pass Must Not Be Conflated"],"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-20261013-advance-20261025-r01-boundary-value-problem-0086","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0056c878a9d248f2810dd7b04a36439ff0254a7829971699415d01936523fc80","family_id":"qf-mod25-bvp-obligation-separation","seed":202610251086,"semantic_template":"qf-mod25-bvp-obligation-separation/{\"boundary_data\":[[\"left\",[454,0,-2]],[\"right\",[406,0,-2]],[\"bottom\",[-244,-2,3]],[\"top\",[-340,-2,3]]],\"bounds\":[-12,12,-11,13],\"coefficients\":[-2,-2,0,3,0,-2],\"source\":3}"},"fields":{"title":"Interior Pass and Boundary Pass Must Not Be Conflated","stem":"Use the quadratic candidate coefficients=[-2, -2, 0, 3, 0, -2] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-12, 12, -11, 13]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=3, boundary data=[[left, [454, 0, -2]], [right, [406, 0, -2]], [bottom, [-244, -2, 3]], [top, [-340, -2, 3]]] with all four distinct edges. Give (Laplacian,source,interior identity?,boundary rows in supplied order,boundary identities?,one of both_pass / interior_pass_boundary_fail / boundary_pass_interior_fail / both_fail). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"(2, 3, No, [[left, y, [454, 0, -2], [454, 0, -2], Yes], [right, y, [406, 0, -2], [406, 0, -2], Yes], [bottom, x, [-244, -2, 3], [-244, -2, 3], Yes], [top, x, [-340, -2, 3], [-340, -2, 3], Yes]], Yes, boundary_pass_interior_fail)","explanation":"Report both groups even when only one succeeds. A boundary pass does not repair a failed interior equation, or conversely. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (2, 3, No, [[left, y, [454, 0, -2], [454, 0, -2], Yes], [right, y, [406, 0, -2], [406, 0, -2], Yes], [bottom, x, [-244, -2, 3], [-244, -2, 3], Yes], [top, x, [-340, -2, 3], [-340, -2, 3], Yes]], Yes, boundary_pass_interior_fail)."},"skills":["Interior Pass and Boundary Pass Must Not Be Conflated"],"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-20261013-advance-20261025-r01-boundary-value-problem-0087","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9feed45d25d51c311b19d3d411cd2bb281c6c73d9b03c586ffb64222687514e1","family_id":"qf-mod25-bvp-obligation-separation","seed":202610251087,"semantic_template":"qf-mod25-bvp-obligation-separation/{\"boundary_data\":[[\"left\",[689,-12,-1]],[\"right\",[662,14,-1]],[\"bottom\",[-157,-13,4]],[\"top\",[-183,13,4]]],\"bounds\":[-13,13,-12,14],\"coefficients\":[-1,-1,1,4,1,-1],\"source\":6}"},"fields":{"title":"Interior Pass and Boundary Pass Must Not Be Conflated","stem":"Use the quadratic candidate coefficients=[-1, -1, 1, 4, 1, -1] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-13, 13, -12, 14]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=6, boundary data=[[left, [689, -12, -1]], [right, [662, 14, -1]], [bottom, [-157, -13, 4]], [top, [-183, 13, 4]]] with all four distinct edges. Give (Laplacian,source,interior identity?,boundary rows in supplied order,boundary identities?,one of both_pass / interior_pass_boundary_fail / boundary_pass_interior_fail / both_fail). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"(6, 6, Yes, [[left, y, [688, -12, -1], [689, -12, -1], No], [right, y, [662, 14, -1], [662, 14, -1], Yes], [bottom, x, [-157, -13, 4], [-157, -13, 4], Yes], [top, x, [-183, 13, 4], [-183, 13, 4], Yes]], No, interior_pass_boundary_fail)","explanation":"Report both groups even when only one succeeds. A boundary pass does not repair a failed interior equation, or conversely. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (6, 6, Yes, [[left, y, [688, -12, -1], [689, -12, -1], No], [right, y, [662, 14, -1], [662, 14, -1], Yes], [bottom, x, [-157, -13, 4], [-157, -13, 4], Yes], [top, x, [-183, 13, 4], [-183, 13, 4], Yes]], No, interior_pass_boundary_fail)."},"skills":["Interior Pass and Boundary Pass Must Not Be Conflated"],"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-20261013-advance-20261025-r01-boundary-value-problem-0088","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b6949c6400ccaa23e0441fa350a8a223d5efaee162e5c1e991fc4c950085276","family_id":"qf-mod25-bvp-obligation-separation","seed":202610251088,"semantic_template":"qf-mod25-bvp-obligation-separation/{\"boundary_data\":[[\"left\",[196,16,0]],[\"right\",[196,-12,0]],[\"bottom\",[-26,13,1]],[\"top\",[30,-15,1]]],\"bounds\":[-14,14,-13,15],\"coefficients\":[0,0,2,1,-1,0],\"source\":3}"},"fields":{"title":"Interior Pass and Boundary Pass Must Not Be Conflated","stem":"Use the quadratic candidate coefficients=[0, 0, 2, 1, -1, 0] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-14, 14, -13, 15]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=3, boundary data=[[left, [196, 16, 0]], [right, [196, -12, 0]], [bottom, [-26, 13, 1]], [top, [30, -15, 1]]] with all four distinct edges. Give (Laplacian,source,interior identity?,boundary rows in supplied order,boundary identities?,one of both_pass / interior_pass_boundary_fail / boundary_pass_interior_fail / both_fail). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"(2, 3, No, [[left, y, [196, 16, 0], [196, 16, 0], Yes], [right, y, [196, -12, 0], [196, -12, 0], Yes], [bottom, x, [-26, 13, 1], [-26, 13, 1], Yes], [top, x, [30, -15, 1], [30, -15, 1], Yes]], Yes, boundary_pass_interior_fail)","explanation":"Report both groups even when only one succeeds. A boundary pass does not repair a failed interior equation, or conversely. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (2, 3, No, [[left, y, [196, 16, 0], [196, 16, 0], Yes], [right, y, [196, -12, 0], [196, -12, 0], Yes], [bottom, x, [-26, 13, 1], [-26, 13, 1], Yes], [top, x, [30, -15, 1], [30, -15, 1], Yes]], Yes, boundary_pass_interior_fail)."},"skills":["Interior Pass and Boundary Pass Must Not Be Conflated"],"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-20261013-advance-20261025-r01-boundary-value-problem-0089","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c95d73d7c16c8a850c3f1eb437b471b4646f9ce0f15b93cbe318601e523d5b58","family_id":"qf-mod25-bvp-obligation-separation","seed":202610251089,"semantic_template":"qf-mod25-bvp-obligation-separation/{\"boundary_data\":[[\"left\",[437,3,1]],[\"right\",[466,3,1]],[\"bottom\",[155,1,2]],[\"top\",[305,1,2]]],\"bounds\":[-15,15,-14,16],\"coefficients\":[1,1,3,2,0,1],\"source\":6}"},"fields":{"title":"Interior Pass and Boundary Pass Must Not Be Conflated","stem":"Use the quadratic candidate coefficients=[1, 1, 3, 2, 0, 1] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-15, 15, -14, 16]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=6, boundary data=[[left, [437, 3, 1]], [right, [466, 3, 1]], [bottom, [155, 1, 2]], [top, [305, 1, 2]]] with all four distinct edges. Give (Laplacian,source,interior identity?,boundary rows in supplied order,boundary identities?,one of both_pass / interior_pass_boundary_fail / boundary_pass_interior_fail / both_fail). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"(6, 6, Yes, [[left, y, [436, 3, 1], [437, 3, 1], No], [right, y, [466, 3, 1], [466, 3, 1], Yes], [bottom, x, [155, 1, 2], [155, 1, 2], Yes], [top, x, [305, 1, 2], [305, 1, 2], Yes]], No, interior_pass_boundary_fail)","explanation":"Report both groups even when only one succeeds. A boundary pass does not repair a failed interior equation, or conversely. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (6, 6, Yes, [[left, y, [436, 3, 1], [437, 3, 1], No], [right, y, [466, 3, 1], [466, 3, 1], Yes], [bottom, x, [155, 1, 2], [155, 1, 2], Yes], [top, x, [305, 1, 2], [305, 1, 2], Yes]], No, interior_pass_boundary_fail)."},"skills":["Interior Pass and Boundary Pass Must Not Be Conflated"],"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-20261013-advance-20261025-r01-boundary-value-problem-0090","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f8e334daecb7275cd423097da1d40dafb7c622e4c5ef238727fa49db56d5ffd","family_id":"qf-mod25-bvp-obligation-separation","seed":202610251090,"semantic_template":"qf-mod25-bvp-obligation-separation/{\"boundary_data\":[[\"left\",[738,-19,2]],[\"right\",[802,13,2]],[\"bottom\",[497,-13,3]],[\"top\",[529,19,3]]],\"bounds\":[-16,16,-15,17],\"coefficients\":[2,2,-3,3,1,2],\"source\":11}"},"fields":{"title":"Interior Pass and Boundary Pass Must Not Be Conflated","stem":"Use the quadratic candidate coefficients=[2, 2, -3, 3, 1, 2] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-16, 16, -15, 17]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=11, boundary data=[[left, [738, -19, 2]], [right, [802, 13, 2]], [bottom, [497, -13, 3]], [top, [529, 19, 3]]] with all four distinct edges. Give (Laplacian,source,interior identity?,boundary rows in supplied order,boundary identities?,one of both_pass / interior_pass_boundary_fail / boundary_pass_interior_fail / both_fail). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"(10, 11, No, [[left, y, [738, -19, 2], [738, -19, 2], Yes], [right, y, [802, 13, 2], [802, 13, 2], Yes], [bottom, x, [497, -13, 3], [497, -13, 3], Yes], [top, x, [529, 19, 3], [529, 19, 3], Yes]], Yes, boundary_pass_interior_fail)","explanation":"Report both groups even when only one succeeds. A boundary pass does not repair a failed interior equation, or conversely. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (10, 11, No, [[left, y, [738, -19, 2], [738, -19, 2], Yes], [right, y, [802, 13, 2], [802, 13, 2], Yes], [bottom, x, [497, -13, 3], [497, -13, 3], Yes], [top, x, [529, 19, 3], [529, 19, 3], Yes]], Yes, boundary_pass_interior_fail)."},"skills":["Interior Pass and Boundary Pass Must Not Be Conflated"],"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-20261013-advance-20261025-r01-boundary-value-problem-0091","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4a2117cfbace2781e055d8bbc4ed97522a3f6cb27f71e3434c6ed759d304be9","family_id":"qf-mod25-bvp-obligation-separation","seed":202610251091,"semantic_template":"qf-mod25-bvp-obligation-separation/{\"boundary_data\":[[\"left\",[1194,15,-2]],[\"right\",[1125,-19,-2]],[\"bottom\",[-477,14,4]],[\"top\",[-681,-20,4]]],\"bounds\":[-17,17,-16,18],\"coefficients\":[3,-2,-2,4,-1,-2],\"source\":4}"},"fields":{"title":"Interior Pass and Boundary Pass Must Not Be Conflated","stem":"Use the quadratic candidate coefficients=[3, -2, -2, 4, -1, -2] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-17, 17, -16, 18]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=4, boundary data=[[left, [1194, 15, -2]], [right, [1125, -19, -2]], [bottom, [-477, 14, 4]], [top, [-681, -20, 4]]] with all four distinct edges. Give (Laplacian,source,interior identity?,boundary rows in supplied order,boundary identities?,one of both_pass / interior_pass_boundary_fail / boundary_pass_interior_fail / both_fail). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"(4, 4, Yes, [[left, y, [1193, 15, -2], [1194, 15, -2], No], [right, y, [1125, -19, -2], [1125, -19, -2], Yes], [bottom, x, [-477, 14, 4], [-477, 14, 4], Yes], [top, x, [-681, -20, 4], [-681, -20, 4], Yes]], No, interior_pass_boundary_fail)","explanation":"Report both groups even when only one succeeds. A boundary pass does not repair a failed interior equation, or conversely. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, Yes, [[left, y, [1193, 15, -2], [1194, 15, -2], No], [right, y, [1125, -19, -2], [1125, -19, -2], Yes], [bottom, x, [-477, 14, 4], [-477, 14, 4], Yes], [top, x, [-681, -20, 4], [-681, -20, 4], Yes]], No, interior_pass_boundary_fail)."},"skills":["Interior Pass and Boundary Pass Must Not Be Conflated"],"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-20261013-advance-20261025-r01-boundary-value-problem-0092","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f3ef787d14aaaec7fe60ab336d0ebaefbb29661d1a2d213dd711626bb1211cae","family_id":"qf-mod25-bvp-obligation-separation","seed":202610251092,"semantic_template":"qf-mod25-bvp-obligation-separation/{\"boundary_data\":[[\"left\",[346,-1,-1]],[\"right\",[310,-1,-1]],[\"bottom\",[-268,-1,1]],[\"top\",[-376,-1,1]]],\"bounds\":[-18,18,-17,19],\"coefficients\":[4,-1,-1,1,0,-1],\"source\":1}"},"fields":{"title":"Interior Pass and Boundary Pass Must Not Be Conflated","stem":"Use the quadratic candidate coefficients=[4, -1, -1, 1, 0, -1] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-18, 18, -17, 19]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=1, boundary data=[[left, [346, -1, -1]], [right, [310, -1, -1]], [bottom, [-268, -1, 1]], [top, [-376, -1, 1]]] with all four distinct edges. Give (Laplacian,source,interior identity?,boundary rows in supplied order,boundary identities?,one of both_pass / interior_pass_boundary_fail / boundary_pass_interior_fail / both_fail). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"(0, 1, No, [[left, y, [346, -1, -1], [346, -1, -1], Yes], [right, y, [310, -1, -1], [310, -1, -1], Yes], [bottom, x, [-268, -1, 1], [-268, -1, 1], Yes], [top, x, [-376, -1, 1], [-376, -1, 1], Yes]], Yes, boundary_pass_interior_fail)","explanation":"Report both groups even when only one succeeds. A boundary pass does not repair a failed interior equation, or conversely. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, No, [[left, y, [346, -1, -1], [346, -1, -1], Yes], [right, y, [310, -1, -1], [310, -1, -1], Yes], [bottom, x, [-268, -1, 1], [-268, -1, 1], Yes], [top, x, [-376, -1, 1], [-376, -1, 1], Yes]], Yes, boundary_pass_interior_fail)."},"skills":["Interior Pass and Boundary Pass Must Not Be Conflated"],"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-20261013-advance-20261025-r01-boundary-value-problem-0093","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"898b53c8120c35169ad50e4060b68c252460670c9e960f44add2df8c47e60897","family_id":"qf-mod25-bvp-obligation-separation","seed":202610251093,"semantic_template":"qf-mod25-bvp-obligation-separation/{\"boundary_data\":[[\"left\",[728,-19,0]],[\"right\",[727,19,0]],[\"bottom\",[5,-18,2]],[\"top\",[5,20,2]]],\"bounds\":[-19,19,-18,20],\"coefficients\":[5,0,0,2,1,0],\"source\":4}"},"fields":{"title":"Interior Pass and Boundary Pass Must Not Be Conflated","stem":"Use the quadratic candidate coefficients=[5, 0, 0, 2, 1, 0] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-19, 19, -18, 20]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=4, boundary data=[[left, [728, -19, 0]], [right, [727, 19, 0]], [bottom, [5, -18, 2]], [top, [5, 20, 2]]] with all four distinct edges. Give (Laplacian,source,interior identity?,boundary rows in supplied order,boundary identities?,one of both_pass / interior_pass_boundary_fail / boundary_pass_interior_fail / both_fail). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"(4, 4, Yes, [[left, y, [727, -19, 0], [728, -19, 0], No], [right, y, [727, 19, 0], [727, 19, 0], Yes], [bottom, x, [5, -18, 2], [5, -18, 2], Yes], [top, x, [5, 20, 2], [5, 20, 2], Yes]], No, interior_pass_boundary_fail)","explanation":"Report both groups even when only one succeeds. A boundary pass does not repair a failed interior equation, or conversely. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, Yes, [[left, y, [727, -19, 0], [728, -19, 0], No], [right, y, [727, 19, 0], [727, 19, 0], Yes], [bottom, x, [5, -18, 2], [5, -18, 2], Yes], [top, x, [5, 20, 2], [5, 20, 2], Yes]], No, interior_pass_boundary_fail)."},"skills":["Interior Pass and Boundary Pass Must Not Be Conflated"],"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-20261013-advance-20261025-r01-boundary-value-problem-0094","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dc7c2a08ecb0061330d514ff24e4b6d11978d57fd3b4338ce36db630377d6e86","family_id":"qf-mod25-bvp-obligation-separation","seed":202610251094,"semantic_template":"qf-mod25-bvp-obligation-separation/{\"boundary_data\":[[\"left\",[1186,21,1]],[\"right\",[1226,-19,1]],[\"bottom\",[348,20,3]],[\"top\",[468,-20,3]]],\"bounds\":[-20,20,-19,21],\"coefficients\":[6,1,1,3,-1,1],\"source\":9}"},"fields":{"title":"Interior Pass and Boundary Pass Must Not Be Conflated","stem":"Use the quadratic candidate coefficients=[6, 1, 1, 3, -1, 1] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-20, 20, -19, 21]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=9, boundary data=[[left, [1186, 21, 1]], [right, [1226, -19, 1]], [bottom, [348, 20, 3]], [top, [468, -20, 3]]] with all four distinct edges. Give (Laplacian,source,interior identity?,boundary rows in supplied order,boundary identities?,one of both_pass / interior_pass_boundary_fail / boundary_pass_interior_fail / both_fail). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"(8, 9, No, [[left, y, [1186, 21, 1], [1186, 21, 1], Yes], [right, y, [1226, -19, 1], [1226, -19, 1], Yes], [bottom, x, [348, 20, 3], [348, 20, 3], Yes], [top, x, [468, -20, 3], [468, -20, 3], Yes]], Yes, boundary_pass_interior_fail)","explanation":"Report both groups even when only one succeeds. A boundary pass does not repair a failed interior equation, or conversely. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (8, 9, No, [[left, y, [1186, 21, 1], [1186, 21, 1], Yes], [right, y, [1226, -19, 1], [1226, -19, 1], Yes], [bottom, x, [348, 20, 3], [348, 20, 3], Yes], [top, x, [468, -20, 3], [468, -20, 3], Yes]], Yes, boundary_pass_interior_fail)."},"skills":["Interior Pass and Boundary Pass Must Not Be Conflated"],"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-20261013-advance-20261025-r01-boundary-value-problem-0095","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9da01a49eecaf2e1ee6d9c3be037f21dceacc7efe383f1f666355e306da68e5f","family_id":"qf-mod25-bvp-obligation-separation","seed":202610251095,"semantic_template":"qf-mod25-bvp-obligation-separation/{\"boundary_data\":[[\"left\",[1730,2,2]],[\"right\",[1813,2,2]],[\"bottom\",[767,2,4]],[\"top\",[1019,2,4]]],\"bounds\":[-21,21,-20,22],\"coefficients\":[7,2,2,4,0,2],\"source\":12}"},"fields":{"title":"Interior Pass and Boundary Pass Must Not Be Conflated","stem":"Use the quadratic candidate coefficients=[7, 2, 2, 4, 0, 2] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-21, 21, -20, 22]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=12, boundary data=[[left, [1730, 2, 2]], [right, [1813, 2, 2]], [bottom, [767, 2, 4]], [top, [1019, 2, 4]]] with all four distinct edges. Give (Laplacian,source,interior identity?,boundary rows in supplied order,boundary identities?,one of both_pass / interior_pass_boundary_fail / boundary_pass_interior_fail / both_fail). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"(12, 12, Yes, [[left, y, [1729, 2, 2], [1730, 2, 2], No], [right, y, [1813, 2, 2], [1813, 2, 2], Yes], [bottom, x, [767, 2, 4], [767, 2, 4], Yes], [top, x, [1019, 2, 4], [1019, 2, 4], Yes]], No, interior_pass_boundary_fail)","explanation":"Report both groups even when only one succeeds. A boundary pass does not repair a failed interior equation, or conversely. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (12, 12, Yes, [[left, y, [1729, 2, 2], [1730, 2, 2], No], [right, y, [1813, 2, 2], [1813, 2, 2], Yes], [bottom, x, [767, 2, 4], [767, 2, 4], Yes], [top, x, [1019, 2, 4], [1019, 2, 4], Yes]], No, interior_pass_boundary_fail)."},"skills":["Interior Pass and Boundary Pass Must Not Be Conflated"],"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-20261013-advance-20261025-r01-boundary-value-problem-0096","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eab8cc79b71b0cdd1895d49c7333b10910d8f94396574062f21b0056add3a3da","family_id":"qf-mod25-bvp-obligation-separation","seed":202610251096,"semantic_template":"qf-mod25-bvp-obligation-separation/{\"boundary_data\":[[\"left\",[536,-19,-2]],[\"right\",[448,25,-2]],[\"bottom\",[-937,-23,1]],[\"top\",[-981,21,1]]],\"bounds\":[-22,22,-21,23],\"coefficients\":[8,-2,3,1,1,-2],\"source\":-1}"},"fields":{"title":"Interior Pass and Boundary Pass Must Not Be Conflated","stem":"Use the quadratic candidate coefficients=[8, -2, 3, 1, 1, -2] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-22, 22, -21, 23]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=-1, boundary data=[[left, [536, -19, -2]], [right, [448, 25, -2]], [bottom, [-937, -23, 1]], [top, [-981, 21, 1]]] with all four distinct edges. Give (Laplacian,source,interior identity?,boundary rows in supplied order,boundary identities?,one of both_pass / interior_pass_boundary_fail / boundary_pass_interior_fail / both_fail). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"(-2, -1, No, [[left, y, [536, -19, -2], [536, -19, -2], Yes], [right, y, [448, 25, -2], [448, 25, -2], Yes], [bottom, x, [-937, -23, 1], [-937, -23, 1], Yes], [top, x, [-981, 21, 1], [-981, 21, 1], Yes]], Yes, boundary_pass_interior_fail)","explanation":"Report both groups even when only one succeeds. A boundary pass does not repair a failed interior equation, or conversely. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (-2, -1, No, [[left, y, [536, -19, -2], [536, -19, -2], Yes], [right, y, [448, 25, -2], [448, 25, -2], Yes], [bottom, x, [-937, -23, 1], [-937, -23, 1], Yes], [top, x, [-981, 21, 1], [-981, 21, 1], Yes]], Yes, boundary_pass_interior_fail)."},"skills":["Interior Pass and Boundary Pass Must Not Be Conflated"],"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-20261013-advance-20261025-r01-boundary-value-problem-0097","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8927b9fd9f1d14bebb751ea831100d0b0e4e967c6a68ce854cb6c1e75d8f1a7c","family_id":"qf-mod25-bvp-obligation-separation","seed":202610251097,"semantic_template":"qf-mod25-bvp-obligation-separation/{\"boundary_data\":[[\"left\",[1091,20,-1]],[\"right\",[1044,-26,-1]],[\"bottom\",[-409,21,2]],[\"top\",[-639,-25,2]]],\"bounds\":[-23,23,-22,24],\"coefficients\":[9,-1,-3,2,-1,-1],\"source\":2}"},"fields":{"title":"Interior Pass and Boundary Pass Must Not Be Conflated","stem":"Use the quadratic candidate coefficients=[9, -1, -3, 2, -1, -1] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-23, 23, -22, 24]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=2, boundary data=[[left, [1091, 20, -1]], [right, [1044, -26, -1]], [bottom, [-409, 21, 2]], [top, [-639, -25, 2]]] with all four distinct edges. Give (Laplacian,source,interior identity?,boundary rows in supplied order,boundary identities?,one of both_pass / interior_pass_boundary_fail / boundary_pass_interior_fail / both_fail). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"(2, 2, Yes, [[left, y, [1090, 20, -1], [1091, 20, -1], No], [right, y, [1044, -26, -1], [1044, -26, -1], Yes], [bottom, x, [-409, 21, 2], [-409, 21, 2], Yes], [top, x, [-639, -25, 2], [-639, -25, 2], Yes]], No, interior_pass_boundary_fail)","explanation":"Report both groups even when only one succeeds. A boundary pass does not repair a failed interior equation, or conversely. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (2, 2, Yes, [[left, y, [1090, 20, -1], [1091, 20, -1], No], [right, y, [1044, -26, -1], [1044, -26, -1], Yes], [bottom, x, [-409, 21, 2], [-409, 21, 2], Yes], [top, x, [-639, -25, 2], [-639, -25, 2], Yes]], No, interior_pass_boundary_fail)."},"skills":["Interior Pass and Boundary Pass Must Not Be Conflated"],"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-20261013-advance-20261025-r01-boundary-value-problem-0098","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1e3eb89ad092e3ee8a571cf29549db5916730d6e733fbc835419f76ebfb6fb0","family_id":"qf-mod25-bvp-obligation-separation","seed":202610251098,"semantic_template":"qf-mod25-bvp-obligation-separation/{\"boundary_data\":[[\"left\",[1738,-2,0]],[\"right\",[1738,-2,0]],[\"bottom\",[56,0,3]],[\"top\",[-40,0,3]]],\"bounds\":[-24,24,-23,25],\"coefficients\":[10,0,-2,3,0,0],\"source\":7}"},"fields":{"title":"Interior Pass and Boundary Pass Must Not Be Conflated","stem":"Use the quadratic candidate coefficients=[10, 0, -2, 3, 0, 0] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-24, 24, -23, 25]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=7, boundary data=[[left, [1738, -2, 0]], [right, [1738, -2, 0]], [bottom, [56, 0, 3]], [top, [-40, 0, 3]]] with all four distinct edges. Give (Laplacian,source,interior identity?,boundary rows in supplied order,boundary identities?,one of both_pass / interior_pass_boundary_fail / boundary_pass_interior_fail / both_fail). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"(6, 7, No, [[left, y, [1738, -2, 0], [1738, -2, 0], Yes], [right, y, [1738, -2, 0], [1738, -2, 0], Yes], [bottom, x, [56, 0, 3], [56, 0, 3], Yes], [top, x, [-40, 0, 3], [-40, 0, 3], Yes]], Yes, boundary_pass_interior_fail)","explanation":"Report both groups even when only one succeeds. A boundary pass does not repair a failed interior equation, or conversely. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (6, 7, No, [[left, y, [1738, -2, 0], [1738, -2, 0], Yes], [right, y, [1738, -2, 0], [1738, -2, 0], Yes], [bottom, x, [56, 0, 3], [56, 0, 3], Yes], [top, x, [-40, 0, 3], [-40, 0, 3], Yes]], Yes, boundary_pass_interior_fail)."},"skills":["Interior Pass and Boundary Pass Must Not Be Conflated"],"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-20261013-advance-20261025-r01-boundary-value-problem-0099","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e5470a74ae99151c439cbf813245ddc232a0f5e70cf8fd473d09f06a3e52c48","family_id":"qf-mod25-bvp-obligation-separation","seed":202610251099,"semantic_template":"qf-mod25-bvp-obligation-separation/{\"boundary_data\":[[\"left\",[2487,-26,1]],[\"right\",[2536,24,1]],[\"bottom\",[611,-23,4]],[\"top\",[661,27,4]]],\"bounds\":[-25,25,-24,26],\"coefficients\":[11,1,-1,4,1,1],\"source\":10}"},"fields":{"title":"Interior Pass and Boundary Pass Must Not Be Conflated","stem":"Use the quadratic candidate coefficients=[11, 1, -1, 4, 1, 1] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-25, 25, -24, 26]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=10, boundary data=[[left, [2487, -26, 1]], [right, [2536, 24, 1]], [bottom, [611, -23, 4]], [top, [661, 27, 4]]] with all four distinct edges. Give (Laplacian,source,interior identity?,boundary rows in supplied order,boundary identities?,one of both_pass / interior_pass_boundary_fail / boundary_pass_interior_fail / both_fail). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"(10, 10, Yes, [[left, y, [2486, -26, 1], [2487, -26, 1], No], [right, y, [2536, 24, 1], [2536, 24, 1], Yes], [bottom, x, [611, -23, 4], [611, -23, 4], Yes], [top, x, [661, 27, 4], [661, 27, 4], Yes]], No, interior_pass_boundary_fail)","explanation":"Report both groups even when only one succeeds. A boundary pass does not repair a failed interior equation, or conversely. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (10, 10, Yes, [[left, y, [2486, -26, 1], [2487, -26, 1], No], [right, y, [2536, 24, 1], [2536, 24, 1], Yes], [bottom, x, [611, -23, 4], [611, -23, 4], Yes], [top, x, [661, 27, 4], [661, 27, 4], Yes]], No, interior_pass_boundary_fail)."},"skills":["Interior Pass and Boundary Pass Must Not Be Conflated"],"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-20261013-advance-20261025-r01-boundary-value-problem-0100","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81e002aa38e9e97d8afe821394f63e6a416c6cee39dc3d5259f53bdd381a16f2","family_id":"qf-mod25-bvp-obligation-separation","seed":202610251100,"semantic_template":"qf-mod25-bvp-obligation-separation/{\"boundary_data\":[[\"left\",[636,26,2]],[\"right\",[740,-26,2]],[\"bottom\",[1262,27,1]],[\"top\",[1470,-25,1]]],\"bounds\":[-26,26,-25,27],\"coefficients\":[12,2,0,1,-1,2],\"source\":7}"},"fields":{"title":"Interior Pass and Boundary Pass Must Not Be Conflated","stem":"Use the quadratic candidate coefficients=[12, 2, 0, 1, -1, 2] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-26, 26, -25, 27]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=7, boundary data=[[left, [636, 26, 2]], [right, [740, -26, 2]], [bottom, [1262, 27, 1]], [top, [1470, -25, 1]]] with all four distinct edges. Give (Laplacian,source,interior identity?,boundary rows in supplied order,boundary identities?,one of both_pass / interior_pass_boundary_fail / boundary_pass_interior_fail / both_fail). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.","answer":{"kind":"expression","value":"(6, 7, No, [[left, y, [636, 26, 2], [636, 26, 2], Yes], [right, y, [740, -26, 2], [740, -26, 2], Yes], [bottom, x, [1262, 27, 1], [1262, 27, 1], Yes], [top, x, [1470, -25, 1], [1470, -25, 1], Yes]], Yes, boundary_pass_interior_fail)","explanation":"Report both groups even when only one succeeds. A boundary pass does not repair a failed interior equation, or conversely. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (6, 7, No, [[left, y, [636, 26, 2], [636, 26, 2], Yes], [right, y, [740, -26, 2], [740, -26, 2], Yes], [bottom, x, [1262, 27, 1], [1262, 27, 1], Yes], [top, x, [1470, -25, 1], [1470, -25, 1], Yes]], Yes, boundary_pass_interior_fail)."},"skills":["Interior Pass and Boundary Pass Must Not Be Conflated"],"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."}}