{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r07-lp-convergence-0001","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a2e9249d882ab0035f3e63fe5e194f909ae4dcdb0252d827f343c467952a935","family_id":"qf-mod35-lp-moment-at-n","seed":202611047001,"semantic_template":"qf-mod35-lp-moment-at-n/{\"coefficients\":[1,-2],\"masses\":[[1,2],[1,2]],\"n\":1,\"p\":1,\"shift\":2}"},"fields":{"title":"The p-Moment Is a Weighted Absolute-Power Error","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[1, -2], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. n=1. Give (p, n, p_moment, certificate).","answer":{"kind":"expression","value":"(1, 1, 1/2, exact_finite_atomic_moment)","explanation":"Compute each absolute amplitude raised to p, weight by that atom mass and add exactly. This output is E|error|^p, not its p-th-root norm or an unweighted atom average. The exact result, retaining the requested order, is (1, 1, 1/2, exact_finite_atomic_moment)."},"skills":["The p-Moment Is a Weighted Absolute-Power Error"],"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-20261014-advance-20261104-r07-lp-convergence-0002","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d86820aa91c74f49b700686e7611c38bfbfca71580d0773bf3ec9ee8faf76f9","family_id":"qf-mod35-lp-moment-at-n","seed":202611047002,"semantic_template":"qf-mod35-lp-moment-at-n/{\"coefficients\":[2,-3],\"masses\":[[1,2],[1,2]],\"n\":2,\"p\":2,\"shift\":2}"},"fields":{"title":"The p-Moment Is a Weighted Absolute-Power Error","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[2, -3], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. n=2. Give (p, n, p_moment, certificate).","answer":{"kind":"expression","value":"(2, 2, 13/32, exact_finite_atomic_moment)","explanation":"Compute each absolute amplitude raised to p, weight by that atom mass and add exactly. This output is E|error|^p, not its p-th-root norm or an unweighted atom average. The exact result, retaining the requested order, is (2, 2, 13/32, exact_finite_atomic_moment)."},"skills":["The p-Moment Is a Weighted Absolute-Power Error"],"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-20261014-advance-20261104-r07-lp-convergence-0003","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"711b4ac069c6abde56ecc03fb02ca5732c931a3f8fd819eb6d001bdc3434dd32","family_id":"qf-mod35-lp-moment-at-n","seed":202611047003,"semantic_template":"qf-mod35-lp-moment-at-n/{\"coefficients\":[3,-4],\"masses\":[[1,2],[1,2]],\"n\":3,\"p\":3,\"shift\":2}"},"fields":{"title":"The p-Moment Is a Weighted Absolute-Power Error","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[3, -4], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. n=3. Give (p, n, p_moment, certificate).","answer":{"kind":"expression","value":"(3, 3, 91/250, exact_finite_atomic_moment)","explanation":"Compute each absolute amplitude raised to p, weight by that atom mass and add exactly. This output is E|error|^p, not its p-th-root norm or an unweighted atom average. The exact result, retaining the requested order, is (3, 3, 91/250, exact_finite_atomic_moment)."},"skills":["The p-Moment Is a Weighted Absolute-Power Error"],"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-20261014-advance-20261104-r07-lp-convergence-0004","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"42909441adc3298240cc16c8875119d574e2e51a3101eb58a68a81862555a6f3","family_id":"qf-mod35-lp-moment-at-n","seed":202611047004,"semantic_template":"qf-mod35-lp-moment-at-n/{\"coefficients\":[4,-5],\"masses\":[[1,2],[1,2]],\"n\":4,\"p\":1,\"shift\":2}"},"fields":{"title":"The p-Moment Is a Weighted Absolute-Power Error","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[4, -5], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. n=4. Give (p, n, p_moment, certificate).","answer":{"kind":"expression","value":"(1, 4, 3/4, exact_finite_atomic_moment)","explanation":"Compute each absolute amplitude raised to p, weight by that atom mass and add exactly. This output is E|error|^p, not its p-th-root norm or an unweighted atom average. The exact result, retaining the requested order, is (1, 4, 3/4, exact_finite_atomic_moment)."},"skills":["The p-Moment Is a Weighted Absolute-Power Error"],"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-20261014-advance-20261104-r07-lp-convergence-0005","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e926bc462e568a438397cb4625837a5a37c4efe8082c43903ddbdae5fa424839","family_id":"qf-mod35-lp-moment-at-n","seed":202611047005,"semantic_template":"qf-mod35-lp-moment-at-n/{\"coefficients\":[5,-6],\"masses\":[[1,2],[1,2]],\"n\":5,\"p\":2,\"shift\":2}"},"fields":{"title":"The p-Moment Is a Weighted Absolute-Power Error","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[5, -6], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. n=5. Give (p, n, p_moment, certificate).","answer":{"kind":"expression","value":"(2, 5, 61/98, exact_finite_atomic_moment)","explanation":"Compute each absolute amplitude raised to p, weight by that atom mass and add exactly. This output is E|error|^p, not its p-th-root norm or an unweighted atom average. The exact result, retaining the requested order, is (2, 5, 61/98, exact_finite_atomic_moment)."},"skills":["The p-Moment Is a Weighted Absolute-Power Error"],"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-20261014-advance-20261104-r07-lp-convergence-0006","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2a5b2d8fdd56bdc5be3e9d2cd0602cc865911fdb59ca735c29a5ed2a0342288","family_id":"qf-mod35-lp-moment-at-n","seed":202611047006,"semantic_template":"qf-mod35-lp-moment-at-n/{\"coefficients\":[6,-7],\"masses\":[[1,2],[1,2]],\"n\":6,\"p\":3,\"shift\":2}"},"fields":{"title":"The p-Moment Is a Weighted Absolute-Power Error","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[6, -7], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. n=6. Give (p, n, p_moment, certificate).","answer":{"kind":"expression","value":"(3, 6, 559/1024, exact_finite_atomic_moment)","explanation":"Compute each absolute amplitude raised to p, weight by that atom mass and add exactly. This output is E|error|^p, not its p-th-root norm or an unweighted atom average. The exact result, retaining the requested order, is (3, 6, 559/1024, exact_finite_atomic_moment)."},"skills":["The p-Moment Is a Weighted Absolute-Power Error"],"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-20261014-advance-20261104-r07-lp-convergence-0007","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2db39bf50f4db712d6fdb5b4b4a41f81c75fc325fa1f978981c1af2bdbad5f2","family_id":"qf-mod35-lp-moment-at-n","seed":202611047007,"semantic_template":"qf-mod35-lp-moment-at-n/{\"coefficients\":[7,-8],\"masses\":[[1,2],[1,2]],\"n\":7,\"p\":1,\"shift\":2}"},"fields":{"title":"The p-Moment Is a Weighted Absolute-Power Error","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[7, -8], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. n=7. Give (p, n, p_moment, certificate).","answer":{"kind":"expression","value":"(1, 7, 5/6, exact_finite_atomic_moment)","explanation":"Compute each absolute amplitude raised to p, weight by that atom mass and add exactly. This output is E|error|^p, not its p-th-root norm or an unweighted atom average. The exact result, retaining the requested order, is (1, 7, 5/6, exact_finite_atomic_moment)."},"skills":["The p-Moment Is a Weighted Absolute-Power Error"],"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-20261014-advance-20261104-r07-lp-convergence-0008","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b713a76da70f00d0c97bd62cfe2c81945b39c8ea742a9fd32716d1d04809a14","family_id":"qf-mod35-lp-moment-at-n","seed":202611047008,"semantic_template":"qf-mod35-lp-moment-at-n/{\"coefficients\":[8,-9],\"masses\":[[1,2],[1,2]],\"n\":8,\"p\":2,\"shift\":2}"},"fields":{"title":"The p-Moment Is a Weighted Absolute-Power Error","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[8, -9], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. n=8. Give (p, n, p_moment, certificate).","answer":{"kind":"expression","value":"(2, 8, 29/40, exact_finite_atomic_moment)","explanation":"Compute each absolute amplitude raised to p, weight by that atom mass and add exactly. This output is E|error|^p, not its p-th-root norm or an unweighted atom average. The exact result, retaining the requested order, is (2, 8, 29/40, exact_finite_atomic_moment)."},"skills":["The p-Moment Is a Weighted Absolute-Power Error"],"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-20261014-advance-20261104-r07-lp-convergence-0009","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"53d6046ff53af9124b60f0c3cffa99d8f8ec08f430f567ca29f2d710c29f781c","family_id":"qf-mod35-lp-moment-at-n","seed":202611047009,"semantic_template":"qf-mod35-lp-moment-at-n/{\"coefficients\":[9,-10],\"masses\":[[1,2],[1,2]],\"n\":9,\"p\":3,\"shift\":2}"},"fields":{"title":"The p-Moment Is a Weighted Absolute-Power Error","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[9, -10], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. n=9. Give (p, n, p_moment, certificate).","answer":{"kind":"expression","value":"(3, 9, 1729/2662, exact_finite_atomic_moment)","explanation":"Compute each absolute amplitude raised to p, weight by that atom mass and add exactly. This output is E|error|^p, not its p-th-root norm or an unweighted atom average. The exact result, retaining the requested order, is (3, 9, 1729/2662, exact_finite_atomic_moment)."},"skills":["The p-Moment Is a Weighted Absolute-Power Error"],"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-20261014-advance-20261104-r07-lp-convergence-0010","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0485867751938a3c7770e8c89a8fd7573df073967c413e0a5b8ce1fb57e4f70c","family_id":"qf-mod35-lp-moment-at-n","seed":202611047010,"semantic_template":"qf-mod35-lp-moment-at-n/{\"coefficients\":[10,-11],\"masses\":[[1,2],[1,2]],\"n\":10,\"p\":1,\"shift\":2}"},"fields":{"title":"The p-Moment Is a Weighted Absolute-Power Error","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[10, -11], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. n=10. Give (p, n, p_moment, certificate).","answer":{"kind":"expression","value":"(1, 10, 7/8, exact_finite_atomic_moment)","explanation":"Compute each absolute amplitude raised to p, weight by that atom mass and add exactly. This output is E|error|^p, not its p-th-root norm or an unweighted atom average. The exact result, retaining the requested order, is (1, 10, 7/8, exact_finite_atomic_moment)."},"skills":["The p-Moment Is a Weighted Absolute-Power Error"],"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-20261014-advance-20261104-r07-lp-convergence-0011","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e53b7e9ea4def6225f8c6790ee3490b7917fb737afb78172a8c955560d06cff","family_id":"qf-mod35-lp-moment-at-n","seed":202611047011,"semantic_template":"qf-mod35-lp-moment-at-n/{\"coefficients\":[11,-12],\"masses\":[[1,2],[1,2]],\"n\":11,\"p\":2,\"shift\":2}"},"fields":{"title":"The p-Moment Is a Weighted Absolute-Power Error","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[11, -12], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. n=11. Give (p, n, p_moment, certificate).","answer":{"kind":"expression","value":"(2, 11, 265/338, exact_finite_atomic_moment)","explanation":"Compute each absolute amplitude raised to p, weight by that atom mass and add exactly. This output is E|error|^p, not its p-th-root norm or an unweighted atom average. The exact result, retaining the requested order, is (2, 11, 265/338, exact_finite_atomic_moment)."},"skills":["The p-Moment Is a Weighted Absolute-Power Error"],"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-20261014-advance-20261104-r07-lp-convergence-0012","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b21d1fae5c6a8a3be430f7ce3f927299aa8552686f7e02091069a6ea092e0330","family_id":"qf-mod35-lp-moment-at-n","seed":202611047012,"semantic_template":"qf-mod35-lp-moment-at-n/{\"coefficients\":[12,-13],\"masses\":[[1,2],[1,2]],\"n\":12,\"p\":3,\"shift\":2}"},"fields":{"title":"The p-Moment Is a Weighted Absolute-Power Error","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[12, -13], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. n=12. Give (p, n, p_moment, certificate).","answer":{"kind":"expression","value":"(3, 12, 3925/5488, exact_finite_atomic_moment)","explanation":"Compute each absolute amplitude raised to p, weight by that atom mass and add exactly. This output is E|error|^p, not its p-th-root norm or an unweighted atom average. The exact result, retaining the requested order, is (3, 12, 3925/5488, exact_finite_atomic_moment)."},"skills":["The p-Moment Is a Weighted Absolute-Power Error"],"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-20261014-advance-20261104-r07-lp-convergence-0013","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ed6b96b111107827350528de8995fe751536589f2c86652b4d91f5a95cd4c0bd","family_id":"qf-mod35-lp-moment-at-n","seed":202611047013,"semantic_template":"qf-mod35-lp-moment-at-n/{\"coefficients\":[13,-14],\"masses\":[[1,2],[1,2]],\"n\":13,\"p\":1,\"shift\":2}"},"fields":{"title":"The p-Moment Is a Weighted Absolute-Power Error","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[13, -14], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. n=13. Give (p, n, p_moment, certificate).","answer":{"kind":"expression","value":"(1, 13, 9/10, exact_finite_atomic_moment)","explanation":"Compute each absolute amplitude raised to p, weight by that atom mass and add exactly. This output is E|error|^p, not its p-th-root norm or an unweighted atom average. The exact result, retaining the requested order, is (1, 13, 9/10, exact_finite_atomic_moment)."},"skills":["The p-Moment Is a Weighted Absolute-Power Error"],"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-20261014-advance-20261104-r07-lp-convergence-0014","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba857ec5f3f25a00ae6da8a032c98b58c57829478ee073735b81798d48964b3f","family_id":"qf-mod35-lp-moment-at-n","seed":202611047014,"semantic_template":"qf-mod35-lp-moment-at-n/{\"coefficients\":[14,-15],\"masses\":[[1,2],[1,2]],\"n\":14,\"p\":2,\"shift\":2}"},"fields":{"title":"The p-Moment Is a Weighted Absolute-Power Error","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[14, -15], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. n=14. Give (p, n, p_moment, certificate).","answer":{"kind":"expression","value":"(2, 14, 421/512, exact_finite_atomic_moment)","explanation":"Compute each absolute amplitude raised to p, weight by that atom mass and add exactly. This output is E|error|^p, not its p-th-root norm or an unweighted atom average. The exact result, retaining the requested order, is (2, 14, 421/512, exact_finite_atomic_moment)."},"skills":["The p-Moment Is a Weighted Absolute-Power Error"],"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-20261014-advance-20261104-r07-lp-convergence-0015","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a8561dfc84f734def1e7fdb5c4c54f18fd96dde05ffc3db9283f31ff64a87dd","family_id":"qf-mod35-lp-moment-at-n","seed":202611047015,"semantic_template":"qf-mod35-lp-moment-at-n/{\"coefficients\":[15,-16],\"masses\":[[1,2],[1,2]],\"n\":15,\"p\":3,\"shift\":2}"},"fields":{"title":"The p-Moment Is a Weighted Absolute-Power Error","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[15, -16], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. n=15. Give (p, n, p_moment, certificate).","answer":{"kind":"expression","value":"(3, 15, 7471/9826, exact_finite_atomic_moment)","explanation":"Compute each absolute amplitude raised to p, weight by that atom mass and add exactly. This output is E|error|^p, not its p-th-root norm or an unweighted atom average. The exact result, retaining the requested order, is (3, 15, 7471/9826, exact_finite_atomic_moment)."},"skills":["The p-Moment Is a Weighted Absolute-Power Error"],"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-20261014-advance-20261104-r07-lp-convergence-0016","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ed9c3c301e7e70048af49567e98f5685977c2f3c4626c82e3f4efdd172447ce1","family_id":"qf-mod35-lp-moment-at-n","seed":202611047016,"semantic_template":"qf-mod35-lp-moment-at-n/{\"coefficients\":[16,-17],\"masses\":[[1,2],[1,2]],\"n\":16,\"p\":1,\"shift\":2}"},"fields":{"title":"The p-Moment Is a Weighted Absolute-Power Error","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[16, -17], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. n=16. Give (p, n, p_moment, certificate).","answer":{"kind":"expression","value":"(1, 16, 11/12, exact_finite_atomic_moment)","explanation":"Compute each absolute amplitude raised to p, weight by that atom mass and add exactly. This output is E|error|^p, not its p-th-root norm or an unweighted atom average. The exact result, retaining the requested order, is (1, 16, 11/12, exact_finite_atomic_moment)."},"skills":["The p-Moment Is a Weighted Absolute-Power Error"],"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-20261014-advance-20261104-r07-lp-convergence-0017","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9b7af7b66dc4fec31ed42cc534ddc0c067e2f29cedc9a8c9ca5a19a26f79a9f","family_id":"qf-mod35-lp-moment-at-n","seed":202611047017,"semantic_template":"qf-mod35-lp-moment-at-n/{\"coefficients\":[17,-18],\"masses\":[[1,2],[1,2]],\"n\":17,\"p\":2,\"shift\":2}"},"fields":{"title":"The p-Moment Is a Weighted Absolute-Power Error","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[17, -18], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. n=17. Give (p, n, p_moment, certificate).","answer":{"kind":"expression","value":"(2, 17, 613/722, exact_finite_atomic_moment)","explanation":"Compute each absolute amplitude raised to p, weight by that atom mass and add exactly. This output is E|error|^p, not its p-th-root norm or an unweighted atom average. The exact result, retaining the requested order, is (2, 17, 613/722, exact_finite_atomic_moment)."},"skills":["The p-Moment Is a Weighted Absolute-Power Error"],"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-20261014-advance-20261104-r07-lp-convergence-0018","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d61ddb0d6868d01650fa064d7d24fb1f586153e9507f31e6670b09053bbdae67","family_id":"qf-mod35-lp-moment-at-n","seed":202611047018,"semantic_template":"qf-mod35-lp-moment-at-n/{\"coefficients\":[18,-19],\"masses\":[[1,2],[1,2]],\"n\":18,\"p\":3,\"shift\":2}"},"fields":{"title":"The p-Moment Is a Weighted Absolute-Power Error","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[18, -19], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. n=18. Give (p, n, p_moment, certificate).","answer":{"kind":"expression","value":"(3, 18, 12691/16000, exact_finite_atomic_moment)","explanation":"Compute each absolute amplitude raised to p, weight by that atom mass and add exactly. This output is E|error|^p, not its p-th-root norm or an unweighted atom average. The exact result, retaining the requested order, is (3, 18, 12691/16000, exact_finite_atomic_moment)."},"skills":["The p-Moment Is a Weighted Absolute-Power Error"],"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-20261014-advance-20261104-r07-lp-convergence-0019","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fa0c4013682789e39771a38e54e43d6f9c33bd692309ff374febfd8638d36cf7","family_id":"qf-mod35-lp-moment-at-n","seed":202611047019,"semantic_template":"qf-mod35-lp-moment-at-n/{\"coefficients\":[19,-20],\"masses\":[[1,2],[1,2]],\"n\":19,\"p\":1,\"shift\":2}"},"fields":{"title":"The p-Moment Is a Weighted Absolute-Power Error","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[19, -20], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. n=19. Give (p, n, p_moment, certificate).","answer":{"kind":"expression","value":"(1, 19, 13/14, exact_finite_atomic_moment)","explanation":"Compute each absolute amplitude raised to p, weight by that atom mass and add exactly. This output is E|error|^p, not its p-th-root norm or an unweighted atom average. The exact result, retaining the requested order, is (1, 19, 13/14, exact_finite_atomic_moment)."},"skills":["The p-Moment Is a Weighted Absolute-Power Error"],"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-20261014-advance-20261104-r07-lp-convergence-0020","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"13a5b17f76345b49d8e49e05acac0a668f22a93bac053018337fe0f491fa6fd6","family_id":"qf-mod35-lp-moment-at-n","seed":202611047020,"semantic_template":"qf-mod35-lp-moment-at-n/{\"coefficients\":[20,-21],\"masses\":[[1,2],[1,2]],\"n\":20,\"p\":2,\"shift\":2}"},"fields":{"title":"The p-Moment Is a Weighted Absolute-Power Error","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[20, -21], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. n=20. Give (p, n, p_moment, certificate).","answer":{"kind":"expression","value":"(2, 20, 841/968, exact_finite_atomic_moment)","explanation":"Compute each absolute amplitude raised to p, weight by that atom mass and add exactly. This output is E|error|^p, not its p-th-root norm or an unweighted atom average. The exact result, retaining the requested order, is (2, 20, 841/968, exact_finite_atomic_moment)."},"skills":["The p-Moment Is a Weighted Absolute-Power Error"],"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-20261014-advance-20261104-r07-lp-convergence-0021","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c0b6d9d1ac8edeb6687fa4144768551e39f5db0457cafba008da7fc5d3742e7a","family_id":"qf-mod35-lp-moment-at-n","seed":202611047021,"semantic_template":"qf-mod35-lp-moment-at-n/{\"coefficients\":[21,-22],\"masses\":[[1,2],[1,2]],\"n\":21,\"p\":3,\"shift\":2}"},"fields":{"title":"The p-Moment Is a Weighted Absolute-Power Error","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[21, -22], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. n=21. Give (p, n, p_moment, certificate).","answer":{"kind":"expression","value":"(3, 21, 19909/24334, exact_finite_atomic_moment)","explanation":"Compute each absolute amplitude raised to p, weight by that atom mass and add exactly. This output is E|error|^p, not its p-th-root norm or an unweighted atom average. The exact result, retaining the requested order, is (3, 21, 19909/24334, exact_finite_atomic_moment)."},"skills":["The p-Moment Is a Weighted Absolute-Power Error"],"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-20261014-advance-20261104-r07-lp-convergence-0022","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"65f4f36112fecf918cc862655958edd7b31838ed4f4ae61d16ee78555a717019","family_id":"qf-mod35-lp-moment-at-n","seed":202611047022,"semantic_template":"qf-mod35-lp-moment-at-n/{\"coefficients\":[22,-23],\"masses\":[[1,2],[1,2]],\"n\":22,\"p\":1,\"shift\":2}"},"fields":{"title":"The p-Moment Is a Weighted Absolute-Power Error","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[22, -23], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. n=22. Give (p, n, p_moment, certificate).","answer":{"kind":"expression","value":"(1, 22, 15/16, exact_finite_atomic_moment)","explanation":"Compute each absolute amplitude raised to p, weight by that atom mass and add exactly. This output is E|error|^p, not its p-th-root norm or an unweighted atom average. The exact result, retaining the requested order, is (1, 22, 15/16, exact_finite_atomic_moment)."},"skills":["The p-Moment Is a Weighted Absolute-Power Error"],"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-20261014-advance-20261104-r07-lp-convergence-0023","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"23d53f20f0dca85f84d6cddc0049c6e201eef0e050957fc37c94262e9f11935b","family_id":"qf-mod35-lp-moment-at-n","seed":202611047023,"semantic_template":"qf-mod35-lp-moment-at-n/{\"coefficients\":[23,-24],\"masses\":[[1,2],[1,2]],\"n\":23,\"p\":2,\"shift\":2}"},"fields":{"title":"The p-Moment Is a Weighted Absolute-Power Error","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[23, -24], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. n=23. Give (p, n, p_moment, certificate).","answer":{"kind":"expression","value":"(2, 23, 221/250, exact_finite_atomic_moment)","explanation":"Compute each absolute amplitude raised to p, weight by that atom mass and add exactly. This output is E|error|^p, not its p-th-root norm or an unweighted atom average. The exact result, retaining the requested order, is (2, 23, 221/250, exact_finite_atomic_moment)."},"skills":["The p-Moment Is a Weighted Absolute-Power Error"],"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-20261014-advance-20261104-r07-lp-convergence-0024","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d48f58e4e8bfb2eb1ad66c0f5021b07ce147f0de83c54c1c56d8823fc125985","family_id":"qf-mod35-lp-moment-at-n","seed":202611047024,"semantic_template":"qf-mod35-lp-moment-at-n/{\"coefficients\":[24,-25],\"masses\":[[1,2],[1,2]],\"n\":24,\"p\":3,\"shift\":2}"},"fields":{"title":"The p-Moment Is a Weighted Absolute-Power Error","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[24, -25], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. n=24. Give (p, n, p_moment, certificate).","answer":{"kind":"expression","value":"(3, 24, 29449/35152, exact_finite_atomic_moment)","explanation":"Compute each absolute amplitude raised to p, weight by that atom mass and add exactly. This output is E|error|^p, not its p-th-root norm or an unweighted atom average. The exact result, retaining the requested order, is (3, 24, 29449/35152, exact_finite_atomic_moment)."},"skills":["The p-Moment Is a Weighted Absolute-Power Error"],"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-20261014-advance-20261104-r07-lp-convergence-0025","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd089f43972dcbb2cf205d6daf56f24adb57a5bf7e5e6a3ad8114cc6cdfe8a38","family_id":"qf-mod35-lp-moment-at-n","seed":202611047025,"semantic_template":"qf-mod35-lp-moment-at-n/{\"coefficients\":[25,-26],\"masses\":[[1,2],[1,2]],\"n\":25,\"p\":1,\"shift\":2}"},"fields":{"title":"The p-Moment Is a Weighted Absolute-Power Error","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[25, -26], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. n=25. Give (p, n, p_moment, certificate).","answer":{"kind":"expression","value":"(1, 25, 17/18, exact_finite_atomic_moment)","explanation":"Compute each absolute amplitude raised to p, weight by that atom mass and add exactly. This output is E|error|^p, not its p-th-root norm or an unweighted atom average. The exact result, retaining the requested order, is (1, 25, 17/18, exact_finite_atomic_moment)."},"skills":["The p-Moment Is a Weighted Absolute-Power Error"],"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-20261014-advance-20261104-r07-lp-convergence-0026","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8208fa991ca15974792a06e31a3091d798c3884987a6e8a793658368026c4ebf","family_id":"qf-mod35-lp-epsilon-witness","seed":202611047026,"semantic_template":"qf-mod35-lp-epsilon-witness/{\"coefficients\":[1,-2],\"epsilon\":[1,10],\"masses\":[[1,2],[1,2]],\"p\":1,\"shift\":2}"},"fields":{"title":"Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[1, -2], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/10 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).","answer":{"kind":"expression","value":"(1, 1/10, 14, 3/32, Yes, forall_n_ge_N_symbolic_monotone_tail)","explanation":"Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (1, 1/10, 14, 3/32, Yes, forall_n_ge_N_symbolic_monotone_tail)."},"skills":["Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number"],"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-20261014-advance-20261104-r07-lp-convergence-0027","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4634e86440d4a53a266cbbc5a580a49e08ec9c97bc94d63b882178e84006b4ba","family_id":"qf-mod35-lp-epsilon-witness","seed":202611047027,"semantic_template":"qf-mod35-lp-epsilon-witness/{\"coefficients\":[2,-3],\"epsilon\":[1,11],\"masses\":[[1,2],[1,2]],\"p\":2,\"shift\":2}"},"fields":{"title":"Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[2, -3], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/11 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).","answer":{"kind":"expression","value":"(2, 1/11, 7, 13/162, Yes, forall_n_ge_N_symbolic_monotone_tail)","explanation":"Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (2, 1/11, 7, 13/162, Yes, forall_n_ge_N_symbolic_monotone_tail)."},"skills":["Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number"],"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-20261014-advance-20261104-r07-lp-convergence-0028","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca9b95d65b84209d5919e733563176c82b05e1a062917c535954dc7dca8b9241","family_id":"qf-mod35-lp-epsilon-witness","seed":202611047028,"semantic_template":"qf-mod35-lp-epsilon-witness/{\"coefficients\":[3,-4],\"epsilon\":[1,12],\"masses\":[[1,2],[1,2]],\"p\":3,\"shift\":2}"},"fields":{"title":"Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[3, -4], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/12 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).","answer":{"kind":"expression","value":"(3, 1/12, 7, 91/1458, Yes, forall_n_ge_N_symbolic_monotone_tail)","explanation":"Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (3, 1/12, 7, 91/1458, Yes, forall_n_ge_N_symbolic_monotone_tail)."},"skills":["Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number"],"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-20261014-advance-20261104-r07-lp-convergence-0029","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e32f3ac6e3a775e5338b6bb61c6653c3a002dd9e2f3ef9112ab9ed3c7f54df5e","family_id":"qf-mod35-lp-epsilon-witness","seed":202611047029,"semantic_template":"qf-mod35-lp-epsilon-witness/{\"coefficients\":[4,-5],\"epsilon\":[1,13],\"masses\":[[1,2],[1,2]],\"p\":1,\"shift\":2}"},"fields":{"title":"Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[4, -5], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/13 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).","answer":{"kind":"expression","value":"(1, 1/13, 57, 9/118, Yes, forall_n_ge_N_symbolic_monotone_tail)","explanation":"Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (1, 1/13, 57, 9/118, Yes, forall_n_ge_N_symbolic_monotone_tail)."},"skills":["Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number"],"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-20261014-advance-20261104-r07-lp-convergence-0030","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5d1e0e2ca8477766cf7b0e4095b076d50b3d19d0c98fa7c20d2218d91f7fe81","family_id":"qf-mod35-lp-epsilon-witness","seed":202611047030,"semantic_template":"qf-mod35-lp-epsilon-witness/{\"coefficients\":[5,-6],\"epsilon\":[1,14],\"masses\":[[1,2],[1,2]],\"p\":2,\"shift\":2}"},"fields":{"title":"Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[5, -6], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/14 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).","answer":{"kind":"expression","value":"(2, 1/14, 19, 61/882, Yes, forall_n_ge_N_symbolic_monotone_tail)","explanation":"Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (2, 1/14, 19, 61/882, Yes, forall_n_ge_N_symbolic_monotone_tail)."},"skills":["Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number"],"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-20261014-advance-20261104-r07-lp-convergence-0031","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11b49d26b8268788277665c1f80cf9a88aaee057bdc1409aeab713826a1d1a64","family_id":"qf-mod35-lp-epsilon-witness","seed":202611047031,"semantic_template":"qf-mod35-lp-epsilon-witness/{\"coefficients\":[6,-7],\"epsilon\":[1,15],\"masses\":[[1,2],[1,2]],\"p\":3,\"shift\":2}"},"fields":{"title":"Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[6, -7], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/15 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).","answer":{"kind":"expression","value":"(3, 1/15, 15, 559/9826, Yes, forall_n_ge_N_symbolic_monotone_tail)","explanation":"Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (3, 1/15, 15, 559/9826, Yes, forall_n_ge_N_symbolic_monotone_tail)."},"skills":["Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number"],"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-20261014-advance-20261104-r07-lp-convergence-0032","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a6a0cacfc66926a49079732c5d992f3eaa13734c6a4abf18469d5bcbcaa5895","family_id":"qf-mod35-lp-epsilon-witness","seed":202611047032,"semantic_template":"qf-mod35-lp-epsilon-witness/{\"coefficients\":[7,-8],\"epsilon\":[1,16],\"masses\":[[1,2],[1,2]],\"p\":1,\"shift\":2}"},"fields":{"title":"Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[7, -8], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/16 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).","answer":{"kind":"expression","value":"(1, 1/16, 119, 15/242, Yes, forall_n_ge_N_symbolic_monotone_tail)","explanation":"Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (1, 1/16, 119, 15/242, Yes, forall_n_ge_N_symbolic_monotone_tail)."},"skills":["Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number"],"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-20261014-advance-20261104-r07-lp-convergence-0033","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d80061eb8beca8a3b3fc1988b9d1b0e92ca6e8e1c27a1512777514295d5f984","family_id":"qf-mod35-lp-epsilon-witness","seed":202611047033,"semantic_template":"qf-mod35-lp-epsilon-witness/{\"coefficients\":[8,-9],\"epsilon\":[1,17],\"masses\":[[1,2],[1,2]],\"p\":2,\"shift\":2}"},"fields":{"title":"Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[8, -9], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/17 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).","answer":{"kind":"expression","value":"(2, 1/17, 34, 145/2592, Yes, forall_n_ge_N_symbolic_monotone_tail)","explanation":"Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (2, 1/17, 34, 145/2592, Yes, forall_n_ge_N_symbolic_monotone_tail)."},"skills":["Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number"],"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-20261014-advance-20261104-r07-lp-convergence-0034","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71a564dd4442f7af02900f602e8600b1e2e12dd87b2663810f585f4c4a240868","family_id":"qf-mod35-lp-epsilon-witness","seed":202611047034,"semantic_template":"qf-mod35-lp-epsilon-witness/{\"coefficients\":[9,-10],\"epsilon\":[1,18],\"masses\":[[1,2],[1,2]],\"p\":3,\"shift\":2}"},"fields":{"title":"Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[9, -10], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/18 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).","answer":{"kind":"expression","value":"(3, 1/18, 23, 1729/31250, Yes, forall_n_ge_N_symbolic_monotone_tail)","explanation":"Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (3, 1/18, 23, 1729/31250, Yes, forall_n_ge_N_symbolic_monotone_tail)."},"skills":["Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number"],"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-20261014-advance-20261104-r07-lp-convergence-0035","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73517ae049993ca4a129791668bcfd3a2dea3678062fa1a6dba3ff98d0a41bd6","family_id":"qf-mod35-lp-epsilon-witness","seed":202611047035,"semantic_template":"qf-mod35-lp-epsilon-witness/{\"coefficients\":[10,-11],\"epsilon\":[1,19],\"masses\":[[1,2],[1,2]],\"p\":1,\"shift\":2}"},"fields":{"title":"Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[10, -11], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/19 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).","answer":{"kind":"expression","value":"(1, 1/19, 198, 21/400, Yes, forall_n_ge_N_symbolic_monotone_tail)","explanation":"Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (1, 1/19, 198, 21/400, Yes, forall_n_ge_N_symbolic_monotone_tail)."},"skills":["Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number"],"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-20261014-advance-20261104-r07-lp-convergence-0036","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b2631d638ebe66c88c8526f26bc31ac6c2fc41b9bf5f8afb797e893f47e4f98","family_id":"qf-mod35-lp-epsilon-witness","seed":202611047036,"semantic_template":"qf-mod35-lp-epsilon-witness/{\"coefficients\":[11,-12],\"epsilon\":[1,20],\"masses\":[[1,2],[1,2]],\"p\":2,\"shift\":2}"},"fields":{"title":"Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[11, -12], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/20 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).","answer":{"kind":"expression","value":"(2, 1/20, 50, 265/5408, Yes, forall_n_ge_N_symbolic_monotone_tail)","explanation":"Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (2, 1/20, 50, 265/5408, Yes, forall_n_ge_N_symbolic_monotone_tail)."},"skills":["Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number"],"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-20261014-advance-20261104-r07-lp-convergence-0037","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b13a5124a1dc966296ddf15bf292d19fd476bac3d4087f10869a0819f9672a08","family_id":"qf-mod35-lp-epsilon-witness","seed":202611047037,"semantic_template":"qf-mod35-lp-epsilon-witness/{\"coefficients\":[12,-13],\"epsilon\":[1,21],\"masses\":[[1,2],[1,2]],\"p\":3,\"shift\":2}"},"fields":{"title":"Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[12, -13], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/21 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).","answer":{"kind":"expression","value":"(3, 1/21, 33, 157/3430, Yes, forall_n_ge_N_symbolic_monotone_tail)","explanation":"Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (3, 1/21, 33, 157/3430, Yes, forall_n_ge_N_symbolic_monotone_tail)."},"skills":["Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number"],"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-20261014-advance-20261104-r07-lp-convergence-0038","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4f5aa7ecc4545049c41d8072fb350a06e15cfc1ba78e545671c9d89bfabb26bc","family_id":"qf-mod35-lp-epsilon-witness","seed":202611047038,"semantic_template":"qf-mod35-lp-epsilon-witness/{\"coefficients\":[13,-14],\"epsilon\":[1,22],\"masses\":[[1,2],[1,2]],\"p\":1,\"shift\":2}"},"fields":{"title":"Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[13, -14], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/22 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).","answer":{"kind":"expression","value":"(1, 1/22, 296, 27/596, Yes, forall_n_ge_N_symbolic_monotone_tail)","explanation":"Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (1, 1/22, 296, 27/596, Yes, forall_n_ge_N_symbolic_monotone_tail)."},"skills":["Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number"],"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-20261014-advance-20261104-r07-lp-convergence-0039","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e93ed06e712e99743b867a8630b8526f748af6019f2803432ec7a4dea1e27a99","family_id":"qf-mod35-lp-epsilon-witness","seed":202611047039,"semantic_template":"qf-mod35-lp-epsilon-witness/{\"coefficients\":[14,-15],\"epsilon\":[1,23],\"masses\":[[1,2],[1,2]],\"p\":2,\"shift\":2}"},"fields":{"title":"Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[14, -15], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/23 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).","answer":{"kind":"expression","value":"(2, 1/23, 68, 421/9800, Yes, forall_n_ge_N_symbolic_monotone_tail)","explanation":"Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (2, 1/23, 68, 421/9800, Yes, forall_n_ge_N_symbolic_monotone_tail)."},"skills":["Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number"],"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-20261014-advance-20261104-r07-lp-convergence-0040","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a9caf7611fb661aadb9337c395ca09fc3c29e9bbd51b13aefa638d118d37572","family_id":"qf-mod35-lp-epsilon-witness","seed":202611047040,"semantic_template":"qf-mod35-lp-epsilon-witness/{\"coefficients\":[15,-16],\"epsilon\":[1,24],\"masses\":[[1,2],[1,2]],\"p\":3,\"shift\":2}"},"fields":{"title":"Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[15, -16], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/24 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).","answer":{"kind":"expression","value":"(3, 1/24, 43, 7471/182250, Yes, forall_n_ge_N_symbolic_monotone_tail)","explanation":"Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (3, 1/24, 43, 7471/182250, Yes, forall_n_ge_N_symbolic_monotone_tail)."},"skills":["Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number"],"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-20261014-advance-20261104-r07-lp-convergence-0041","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0414be617fd21c8d32fbabdd3ccd898b741664f536373925bf3885f7b65adfa5","family_id":"qf-mod35-lp-epsilon-witness","seed":202611047041,"semantic_template":"qf-mod35-lp-epsilon-witness/{\"coefficients\":[16,-17],\"epsilon\":[1,25],\"masses\":[[1,2],[1,2]],\"p\":1,\"shift\":2}"},"fields":{"title":"Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[16, -17], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/25 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).","answer":{"kind":"expression","value":"(1, 1/25, 411, 33/826, Yes, forall_n_ge_N_symbolic_monotone_tail)","explanation":"Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (1, 1/25, 411, 33/826, Yes, forall_n_ge_N_symbolic_monotone_tail)."},"skills":["Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number"],"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-20261014-advance-20261104-r07-lp-convergence-0042","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"066e81d5ad94f439aa1c6c7c3e6c9ebe78e623e0a43858e4713feb0936cb93df","family_id":"qf-mod35-lp-epsilon-witness","seed":202611047042,"semantic_template":"qf-mod35-lp-epsilon-witness/{\"coefficients\":[17,-18],\"epsilon\":[1,26],\"masses\":[[1,2],[1,2]],\"p\":2,\"shift\":2}"},"fields":{"title":"Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[17, -18], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/26 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).","answer":{"kind":"expression","value":"(2, 1/26, 88, 613/16200, Yes, forall_n_ge_N_symbolic_monotone_tail)","explanation":"Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (2, 1/26, 88, 613/16200, Yes, forall_n_ge_N_symbolic_monotone_tail)."},"skills":["Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number"],"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-20261014-advance-20261104-r07-lp-convergence-0043","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8845e7b202d062031720422c83927bb386a894d855129103c054e53254a3babf","family_id":"qf-mod35-lp-epsilon-witness","seed":202611047043,"semantic_template":"qf-mod35-lp-epsilon-witness/{\"coefficients\":[18,-19],\"epsilon\":[1,27],\"masses\":[[1,2],[1,2]],\"p\":3,\"shift\":2}"},"fields":{"title":"Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[18, -19], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/27 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).","answer":{"kind":"expression","value":"(3, 1/27, 54, 37/1024, Yes, forall_n_ge_N_symbolic_monotone_tail)","explanation":"Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (3, 1/27, 54, 37/1024, Yes, forall_n_ge_N_symbolic_monotone_tail)."},"skills":["Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number"],"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-20261014-advance-20261104-r07-lp-convergence-0044","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6129f46fd81d5ed5cc8569b7ea59b282cb8e866b26cdbc04704830f42d99b083","family_id":"qf-mod35-lp-epsilon-witness","seed":202611047044,"semantic_template":"qf-mod35-lp-epsilon-witness/{\"coefficients\":[19,-20],\"epsilon\":[1,28],\"masses\":[[1,2],[1,2]],\"p\":1,\"shift\":2}"},"fields":{"title":"Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[19, -20], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/28 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).","answer":{"kind":"expression","value":"(1, 1/28, 545, 39/1094, Yes, forall_n_ge_N_symbolic_monotone_tail)","explanation":"Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (1, 1/28, 545, 39/1094, Yes, forall_n_ge_N_symbolic_monotone_tail)."},"skills":["Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number"],"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-20261014-advance-20261104-r07-lp-convergence-0045","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"69f5ae49b0756868c4fc05252973e8c9b8345fad77e59efe2171b5f7f8a82ed3","family_id":"qf-mod35-lp-epsilon-witness","seed":202611047045,"semantic_template":"qf-mod35-lp-epsilon-witness/{\"coefficients\":[20,-21],\"epsilon\":[1,29],\"masses\":[[1,2],[1,2]],\"p\":2,\"shift\":2}"},"fields":{"title":"Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[20, -21], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/29 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).","answer":{"kind":"expression","value":"(2, 1/29, 109, 841/24642, Yes, forall_n_ge_N_symbolic_monotone_tail)","explanation":"Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (2, 1/29, 109, 841/24642, Yes, forall_n_ge_N_symbolic_monotone_tail)."},"skills":["Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number"],"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-20261014-advance-20261104-r07-lp-convergence-0046","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"05f8d691eb2b74abfd31b0238f29b97df6199af1c9040ee6ebe7e4dbebbfae4e","family_id":"qf-mod35-lp-epsilon-witness","seed":202611047046,"semantic_template":"qf-mod35-lp-epsilon-witness/{\"coefficients\":[21,-22],\"epsilon\":[1,30],\"masses\":[[1,2],[1,2]],\"p\":3,\"shift\":2}"},"fields":{"title":"Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[21, -22], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/30 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).","answer":{"kind":"expression","value":"(3, 1/30, 65, 19909/601526, Yes, forall_n_ge_N_symbolic_monotone_tail)","explanation":"Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (3, 1/30, 65, 19909/601526, Yes, forall_n_ge_N_symbolic_monotone_tail)."},"skills":["Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number"],"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-20261014-advance-20261104-r07-lp-convergence-0047","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"77ebadf7306a1f0f489254d0e00ae16e49783fd9bbf7cbc3d12f04558e70d0d6","family_id":"qf-mod35-lp-epsilon-witness","seed":202611047047,"semantic_template":"qf-mod35-lp-epsilon-witness/{\"coefficients\":[22,-23],\"epsilon\":[1,31],\"masses\":[[1,2],[1,2]],\"p\":1,\"shift\":2}"},"fields":{"title":"Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[22, -23], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/31 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).","answer":{"kind":"expression","value":"(1, 1/31, 696, 45/1396, Yes, forall_n_ge_N_symbolic_monotone_tail)","explanation":"Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (1, 1/31, 696, 45/1396, Yes, forall_n_ge_N_symbolic_monotone_tail)."},"skills":["Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number"],"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-20261014-advance-20261104-r07-lp-convergence-0048","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ddd5c558be166675b60cac1b9c225ccfd7f098f20a58b4b0c68a9ca6e79e443","family_id":"qf-mod35-lp-epsilon-witness","seed":202611047048,"semantic_template":"qf-mod35-lp-epsilon-witness/{\"coefficients\":[23,-24],\"epsilon\":[1,32],\"masses\":[[1,2],[1,2]],\"p\":2,\"shift\":2}"},"fields":{"title":"Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[23, -24], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/32 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).","answer":{"kind":"expression","value":"(2, 1/32, 131, 1105/35378, Yes, forall_n_ge_N_symbolic_monotone_tail)","explanation":"Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (2, 1/32, 131, 1105/35378, Yes, forall_n_ge_N_symbolic_monotone_tail)."},"skills":["Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number"],"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-20261014-advance-20261104-r07-lp-convergence-0049","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"747414e4883d2514eb07458a270da1f6d45d3b15094f1262281c45d53f689054","family_id":"qf-mod35-lp-epsilon-witness","seed":202611047049,"semantic_template":"qf-mod35-lp-epsilon-witness/{\"coefficients\":[24,-25],\"epsilon\":[1,33],\"masses\":[[1,2],[1,2]],\"p\":3,\"shift\":2}"},"fields":{"title":"Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[24, -25], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/33 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).","answer":{"kind":"expression","value":"(3, 1/33, 77, 29449/986078, Yes, forall_n_ge_N_symbolic_monotone_tail)","explanation":"Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (3, 1/33, 77, 29449/986078, Yes, forall_n_ge_N_symbolic_monotone_tail)."},"skills":["Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number"],"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-20261014-advance-20261104-r07-lp-convergence-0050","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"964569d6911f327fbef27e661793e9915c3761bb76d94f1da5012f3292dba790","family_id":"qf-mod35-lp-epsilon-witness","seed":202611047050,"semantic_template":"qf-mod35-lp-epsilon-witness/{\"coefficients\":[25,-26],\"epsilon\":[1,34],\"masses\":[[1,2],[1,2]],\"p\":1,\"shift\":2}"},"fields":{"title":"Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[25, -26], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/34 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).","answer":{"kind":"expression","value":"(1, 1/34, 866, 51/1736, Yes, forall_n_ge_N_symbolic_monotone_tail)","explanation":"Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (1, 1/34, 866, 51/1736, Yes, forall_n_ge_N_symbolic_monotone_tail)."},"skills":["Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number"],"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-20261014-advance-20261104-r07-lp-convergence-0051","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b772bec409a985655d18d87da5567a728b418ba2bcbd37ecc5445f777721a91","family_id":"qf-mod35-lp-qualification","seed":202611047051,"semantic_template":"qf-mod35-lp-qualification/{\"coefficients\":[1,-2],\"masses\":[[1,2],[1,2]],\"p\":1,\"shift\":2}"},"fields":{"title":"Keep the Fixed p, Common Law and Whole Symbolic Tail Together","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[1, -2], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).","answer":{"kind":"expression","value":"(1, [1/2, 1/2], [1, -2], 2, Yes, common_probability_space_and_finite_integer_p_moments)","explanation":"The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (1, [1/2, 1/2], [1, -2], 2, Yes, common_probability_space_and_finite_integer_p_moments)."},"skills":["Keep the Fixed p, Common Law and Whole Symbolic Tail Together"],"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-20261014-advance-20261104-r07-lp-convergence-0052","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc31e11486ade0008b78f81ee6f4270cb72512247587d762c9dadffcd2a44e59","family_id":"qf-mod35-lp-qualification","seed":202611047052,"semantic_template":"qf-mod35-lp-qualification/{\"coefficients\":[2,-3],\"masses\":[[1,2],[1,2]],\"p\":2,\"shift\":2}"},"fields":{"title":"Keep the Fixed p, Common Law and Whole Symbolic Tail Together","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[2, -3], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).","answer":{"kind":"expression","value":"(2, [1/2, 1/2], [2, -3], 2, Yes, common_probability_space_and_finite_integer_p_moments)","explanation":"The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (2, [1/2, 1/2], [2, -3], 2, Yes, common_probability_space_and_finite_integer_p_moments)."},"skills":["Keep the Fixed p, Common Law and Whole Symbolic Tail Together"],"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-20261014-advance-20261104-r07-lp-convergence-0053","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d10dee86dfe77530e644328f11015e7960b585aba2a1ea382ba00e8ca9319ca5","family_id":"qf-mod35-lp-qualification","seed":202611047053,"semantic_template":"qf-mod35-lp-qualification/{\"coefficients\":[3,-4],\"masses\":[[1,2],[1,2]],\"p\":3,\"shift\":2}"},"fields":{"title":"Keep the Fixed p, Common Law and Whole Symbolic Tail Together","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[3, -4], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).","answer":{"kind":"expression","value":"(3, [1/2, 1/2], [3, -4], 2, Yes, common_probability_space_and_finite_integer_p_moments)","explanation":"The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (3, [1/2, 1/2], [3, -4], 2, Yes, common_probability_space_and_finite_integer_p_moments)."},"skills":["Keep the Fixed p, Common Law and Whole Symbolic Tail Together"],"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-20261014-advance-20261104-r07-lp-convergence-0054","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"974887a9efaf3423dfbf8895f7386990d91195e31d870d8305bf51eb0ffa972b","family_id":"qf-mod35-lp-qualification","seed":202611047054,"semantic_template":"qf-mod35-lp-qualification/{\"coefficients\":[4,-5],\"masses\":[[1,2],[1,2]],\"p\":1,\"shift\":2}"},"fields":{"title":"Keep the Fixed p, Common Law and Whole Symbolic Tail Together","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[4, -5], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).","answer":{"kind":"expression","value":"(1, [1/2, 1/2], [4, -5], 2, Yes, common_probability_space_and_finite_integer_p_moments)","explanation":"The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (1, [1/2, 1/2], [4, -5], 2, Yes, common_probability_space_and_finite_integer_p_moments)."},"skills":["Keep the Fixed p, Common Law and Whole Symbolic Tail Together"],"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-20261014-advance-20261104-r07-lp-convergence-0055","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"736fdb2c302abcab533c2653d7b95f710f5fdd80460850eefecd36e00537e395","family_id":"qf-mod35-lp-qualification","seed":202611047055,"semantic_template":"qf-mod35-lp-qualification/{\"coefficients\":[5,-6],\"masses\":[[1,2],[1,2]],\"p\":2,\"shift\":2}"},"fields":{"title":"Keep the Fixed p, Common Law and Whole Symbolic Tail Together","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[5, -6], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).","answer":{"kind":"expression","value":"(2, [1/2, 1/2], [5, -6], 2, Yes, common_probability_space_and_finite_integer_p_moments)","explanation":"The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (2, [1/2, 1/2], [5, -6], 2, Yes, common_probability_space_and_finite_integer_p_moments)."},"skills":["Keep the Fixed p, Common Law and Whole Symbolic Tail Together"],"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-20261014-advance-20261104-r07-lp-convergence-0056","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b63300793848579b0d7c0c063ad484d39a003228bb9d6b68156ad1e84d7877a","family_id":"qf-mod35-lp-qualification","seed":202611047056,"semantic_template":"qf-mod35-lp-qualification/{\"coefficients\":[6,-7],\"masses\":[[1,2],[1,2]],\"p\":3,\"shift\":2}"},"fields":{"title":"Keep the Fixed p, Common Law and Whole Symbolic Tail Together","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[6, -7], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).","answer":{"kind":"expression","value":"(3, [1/2, 1/2], [6, -7], 2, Yes, common_probability_space_and_finite_integer_p_moments)","explanation":"The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (3, [1/2, 1/2], [6, -7], 2, Yes, common_probability_space_and_finite_integer_p_moments)."},"skills":["Keep the Fixed p, Common Law and Whole Symbolic Tail Together"],"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-20261014-advance-20261104-r07-lp-convergence-0057","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b50b5de95e84f65945fe52dff185398b660dafacb959d9f5bc63321db5b43d2f","family_id":"qf-mod35-lp-qualification","seed":202611047057,"semantic_template":"qf-mod35-lp-qualification/{\"coefficients\":[7,-8],\"masses\":[[1,2],[1,2]],\"p\":1,\"shift\":2}"},"fields":{"title":"Keep the Fixed p, Common Law and Whole Symbolic Tail Together","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[7, -8], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).","answer":{"kind":"expression","value":"(1, [1/2, 1/2], [7, -8], 2, Yes, common_probability_space_and_finite_integer_p_moments)","explanation":"The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (1, [1/2, 1/2], [7, -8], 2, Yes, common_probability_space_and_finite_integer_p_moments)."},"skills":["Keep the Fixed p, Common Law and Whole Symbolic Tail Together"],"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-20261014-advance-20261104-r07-lp-convergence-0058","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d211fd1c56b7edd3ec53f912e900d600f40a26e873552fac8e63d2b5503f687","family_id":"qf-mod35-lp-qualification","seed":202611047058,"semantic_template":"qf-mod35-lp-qualification/{\"coefficients\":[8,-9],\"masses\":[[1,2],[1,2]],\"p\":2,\"shift\":2}"},"fields":{"title":"Keep the Fixed p, Common Law and Whole Symbolic Tail Together","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[8, -9], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).","answer":{"kind":"expression","value":"(2, [1/2, 1/2], [8, -9], 2, Yes, common_probability_space_and_finite_integer_p_moments)","explanation":"The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (2, [1/2, 1/2], [8, -9], 2, Yes, common_probability_space_and_finite_integer_p_moments)."},"skills":["Keep the Fixed p, Common Law and Whole Symbolic Tail Together"],"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-20261014-advance-20261104-r07-lp-convergence-0059","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7f42a94bf8ea65c22132caafd89eb858a5d25e0e7c2f7e7e809d1d0ad426c409","family_id":"qf-mod35-lp-qualification","seed":202611047059,"semantic_template":"qf-mod35-lp-qualification/{\"coefficients\":[9,-10],\"masses\":[[1,2],[1,2]],\"p\":3,\"shift\":2}"},"fields":{"title":"Keep the Fixed p, Common Law and Whole Symbolic Tail Together","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[9, -10], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).","answer":{"kind":"expression","value":"(3, [1/2, 1/2], [9, -10], 2, Yes, common_probability_space_and_finite_integer_p_moments)","explanation":"The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (3, [1/2, 1/2], [9, -10], 2, Yes, common_probability_space_and_finite_integer_p_moments)."},"skills":["Keep the Fixed p, Common Law and Whole Symbolic Tail Together"],"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-20261014-advance-20261104-r07-lp-convergence-0060","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1522c05bb2288bb988fd4fa74cc07295d862ffdc32dfc1484e8bd68a094a09f6","family_id":"qf-mod35-lp-qualification","seed":202611047060,"semantic_template":"qf-mod35-lp-qualification/{\"coefficients\":[10,-11],\"masses\":[[1,2],[1,2]],\"p\":1,\"shift\":2}"},"fields":{"title":"Keep the Fixed p, Common Law and Whole Symbolic Tail Together","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[10, -11], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).","answer":{"kind":"expression","value":"(1, [1/2, 1/2], [10, -11], 2, Yes, common_probability_space_and_finite_integer_p_moments)","explanation":"The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (1, [1/2, 1/2], [10, -11], 2, Yes, common_probability_space_and_finite_integer_p_moments)."},"skills":["Keep the Fixed p, Common Law and Whole Symbolic Tail Together"],"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-20261014-advance-20261104-r07-lp-convergence-0061","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb6728838ffb2baa40fed9d7a647da41462de9acbff9254eaeabaca03cf6fc9c","family_id":"qf-mod35-lp-qualification","seed":202611047061,"semantic_template":"qf-mod35-lp-qualification/{\"coefficients\":[11,-12],\"masses\":[[1,2],[1,2]],\"p\":2,\"shift\":2}"},"fields":{"title":"Keep the Fixed p, Common Law and Whole Symbolic Tail Together","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[11, -12], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).","answer":{"kind":"expression","value":"(2, [1/2, 1/2], [11, -12], 2, Yes, common_probability_space_and_finite_integer_p_moments)","explanation":"The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (2, [1/2, 1/2], [11, -12], 2, Yes, common_probability_space_and_finite_integer_p_moments)."},"skills":["Keep the Fixed p, Common Law and Whole Symbolic Tail Together"],"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-20261014-advance-20261104-r07-lp-convergence-0062","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e7207441952da155c7ed0dd0c2e40451adb82ab3bce86cc9f7de05c370f1f67f","family_id":"qf-mod35-lp-qualification","seed":202611047062,"semantic_template":"qf-mod35-lp-qualification/{\"coefficients\":[12,-13],\"masses\":[[1,2],[1,2]],\"p\":3,\"shift\":2}"},"fields":{"title":"Keep the Fixed p, Common Law and Whole Symbolic Tail Together","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[12, -13], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).","answer":{"kind":"expression","value":"(3, [1/2, 1/2], [12, -13], 2, Yes, common_probability_space_and_finite_integer_p_moments)","explanation":"The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (3, [1/2, 1/2], [12, -13], 2, Yes, common_probability_space_and_finite_integer_p_moments)."},"skills":["Keep the Fixed p, Common Law and Whole Symbolic Tail Together"],"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-20261014-advance-20261104-r07-lp-convergence-0063","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"573cf02a2cd818d77caf5511db5f1a82921725afd4a815ede7e02b1ac6ca1162","family_id":"qf-mod35-lp-qualification","seed":202611047063,"semantic_template":"qf-mod35-lp-qualification/{\"coefficients\":[13,-14],\"masses\":[[1,2],[1,2]],\"p\":1,\"shift\":2}"},"fields":{"title":"Keep the Fixed p, Common Law and Whole Symbolic Tail Together","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[13, -14], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).","answer":{"kind":"expression","value":"(1, [1/2, 1/2], [13, -14], 2, Yes, common_probability_space_and_finite_integer_p_moments)","explanation":"The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (1, [1/2, 1/2], [13, -14], 2, Yes, common_probability_space_and_finite_integer_p_moments)."},"skills":["Keep the Fixed p, Common Law and Whole Symbolic Tail Together"],"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-20261014-advance-20261104-r07-lp-convergence-0064","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af46efed1dba35ca378cf1de702f878d62c1e4e5c5ca17a756f1178d009d56c3","family_id":"qf-mod35-lp-qualification","seed":202611047064,"semantic_template":"qf-mod35-lp-qualification/{\"coefficients\":[14,-15],\"masses\":[[1,2],[1,2]],\"p\":2,\"shift\":2}"},"fields":{"title":"Keep the Fixed p, Common Law and Whole Symbolic Tail Together","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[14, -15], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).","answer":{"kind":"expression","value":"(2, [1/2, 1/2], [14, -15], 2, Yes, common_probability_space_and_finite_integer_p_moments)","explanation":"The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (2, [1/2, 1/2], [14, -15], 2, Yes, common_probability_space_and_finite_integer_p_moments)."},"skills":["Keep the Fixed p, Common Law and Whole Symbolic Tail Together"],"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-20261014-advance-20261104-r07-lp-convergence-0065","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"222209a917cdcf08a7f0f707b50e0dd6533bc139c052f8d4571a6ee51b17ea86","family_id":"qf-mod35-lp-qualification","seed":202611047065,"semantic_template":"qf-mod35-lp-qualification/{\"coefficients\":[15,-16],\"masses\":[[1,2],[1,2]],\"p\":3,\"shift\":2}"},"fields":{"title":"Keep the Fixed p, Common Law and Whole Symbolic Tail Together","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[15, -16], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).","answer":{"kind":"expression","value":"(3, [1/2, 1/2], [15, -16], 2, Yes, common_probability_space_and_finite_integer_p_moments)","explanation":"The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (3, [1/2, 1/2], [15, -16], 2, Yes, common_probability_space_and_finite_integer_p_moments)."},"skills":["Keep the Fixed p, Common Law and Whole Symbolic Tail Together"],"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-20261014-advance-20261104-r07-lp-convergence-0066","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"08ac0ad0f92a4864fe10067e61d93bee6daa705102f4b0ddc6208eafac5ed44f","family_id":"qf-mod35-lp-qualification","seed":202611047066,"semantic_template":"qf-mod35-lp-qualification/{\"coefficients\":[16,-17],\"masses\":[[1,2],[1,2]],\"p\":1,\"shift\":2}"},"fields":{"title":"Keep the Fixed p, Common Law and Whole Symbolic Tail Together","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[16, -17], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).","answer":{"kind":"expression","value":"(1, [1/2, 1/2], [16, -17], 2, Yes, common_probability_space_and_finite_integer_p_moments)","explanation":"The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (1, [1/2, 1/2], [16, -17], 2, Yes, common_probability_space_and_finite_integer_p_moments)."},"skills":["Keep the Fixed p, Common Law and Whole Symbolic Tail Together"],"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-20261014-advance-20261104-r07-lp-convergence-0067","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"391ec406a772c19d575de06acf01713438960af2bfb199d2b7aae2d60a317021","family_id":"qf-mod35-lp-qualification","seed":202611047067,"semantic_template":"qf-mod35-lp-qualification/{\"coefficients\":[17,-18],\"masses\":[[1,2],[1,2]],\"p\":2,\"shift\":2}"},"fields":{"title":"Keep the Fixed p, Common Law and Whole Symbolic Tail Together","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[17, -18], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).","answer":{"kind":"expression","value":"(2, [1/2, 1/2], [17, -18], 2, Yes, common_probability_space_and_finite_integer_p_moments)","explanation":"The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (2, [1/2, 1/2], [17, -18], 2, Yes, common_probability_space_and_finite_integer_p_moments)."},"skills":["Keep the Fixed p, Common Law and Whole Symbolic Tail Together"],"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-20261014-advance-20261104-r07-lp-convergence-0068","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7098ef160a97d50b5fa43f664a7db46672d191220a40d0b11bb55c104e4b5ac5","family_id":"qf-mod35-lp-qualification","seed":202611047068,"semantic_template":"qf-mod35-lp-qualification/{\"coefficients\":[18,-19],\"masses\":[[1,2],[1,2]],\"p\":3,\"shift\":2}"},"fields":{"title":"Keep the Fixed p, Common Law and Whole Symbolic Tail Together","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[18, -19], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).","answer":{"kind":"expression","value":"(3, [1/2, 1/2], [18, -19], 2, Yes, common_probability_space_and_finite_integer_p_moments)","explanation":"The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (3, [1/2, 1/2], [18, -19], 2, Yes, common_probability_space_and_finite_integer_p_moments)."},"skills":["Keep the Fixed p, Common Law and Whole Symbolic Tail Together"],"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-20261014-advance-20261104-r07-lp-convergence-0069","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4f7d138124ded490c98283762970ddc1348b53ad3069ff6d952e73832dcb6e87","family_id":"qf-mod35-lp-qualification","seed":202611047069,"semantic_template":"qf-mod35-lp-qualification/{\"coefficients\":[19,-20],\"masses\":[[1,2],[1,2]],\"p\":1,\"shift\":2}"},"fields":{"title":"Keep the Fixed p, Common Law and Whole Symbolic Tail Together","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[19, -20], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).","answer":{"kind":"expression","value":"(1, [1/2, 1/2], [19, -20], 2, Yes, common_probability_space_and_finite_integer_p_moments)","explanation":"The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (1, [1/2, 1/2], [19, -20], 2, Yes, common_probability_space_and_finite_integer_p_moments)."},"skills":["Keep the Fixed p, Common Law and Whole Symbolic Tail Together"],"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-20261014-advance-20261104-r07-lp-convergence-0070","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b3661a359ec5fca2b9afbf9b6327602c95053e8385e3d3cc03416d99bc8bb8a","family_id":"qf-mod35-lp-qualification","seed":202611047070,"semantic_template":"qf-mod35-lp-qualification/{\"coefficients\":[20,-21],\"masses\":[[1,2],[1,2]],\"p\":2,\"shift\":2}"},"fields":{"title":"Keep the Fixed p, Common Law and Whole Symbolic Tail Together","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[20, -21], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).","answer":{"kind":"expression","value":"(2, [1/2, 1/2], [20, -21], 2, Yes, common_probability_space_and_finite_integer_p_moments)","explanation":"The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (2, [1/2, 1/2], [20, -21], 2, Yes, common_probability_space_and_finite_integer_p_moments)."},"skills":["Keep the Fixed p, Common Law and Whole Symbolic Tail Together"],"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-20261014-advance-20261104-r07-lp-convergence-0071","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6d6e2b93a641673ed5e99f19bba57aa6c061b43776e59f3d6fb277e86bf2bd85","family_id":"qf-mod35-lp-qualification","seed":202611047071,"semantic_template":"qf-mod35-lp-qualification/{\"coefficients\":[21,-22],\"masses\":[[1,2],[1,2]],\"p\":3,\"shift\":2}"},"fields":{"title":"Keep the Fixed p, Common Law and Whole Symbolic Tail Together","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[21, -22], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).","answer":{"kind":"expression","value":"(3, [1/2, 1/2], [21, -22], 2, Yes, common_probability_space_and_finite_integer_p_moments)","explanation":"The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (3, [1/2, 1/2], [21, -22], 2, Yes, common_probability_space_and_finite_integer_p_moments)."},"skills":["Keep the Fixed p, Common Law and Whole Symbolic Tail Together"],"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-20261014-advance-20261104-r07-lp-convergence-0072","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe2012fa53ddb51a47d2127a0fdea61c09b99e32b66b3f6e88e749ec915dfdca","family_id":"qf-mod35-lp-qualification","seed":202611047072,"semantic_template":"qf-mod35-lp-qualification/{\"coefficients\":[22,-23],\"masses\":[[1,2],[1,2]],\"p\":1,\"shift\":2}"},"fields":{"title":"Keep the Fixed p, Common Law and Whole Symbolic Tail Together","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[22, -23], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).","answer":{"kind":"expression","value":"(1, [1/2, 1/2], [22, -23], 2, Yes, common_probability_space_and_finite_integer_p_moments)","explanation":"The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (1, [1/2, 1/2], [22, -23], 2, Yes, common_probability_space_and_finite_integer_p_moments)."},"skills":["Keep the Fixed p, Common Law and Whole Symbolic Tail Together"],"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-20261014-advance-20261104-r07-lp-convergence-0073","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b7eb931d852690412e885b43ff16f0112f68d7002bf0f3ea0de09f8f8988cd9","family_id":"qf-mod35-lp-qualification","seed":202611047073,"semantic_template":"qf-mod35-lp-qualification/{\"coefficients\":[23,-24],\"masses\":[[1,2],[1,2]],\"p\":2,\"shift\":2}"},"fields":{"title":"Keep the Fixed p, Common Law and Whole Symbolic Tail Together","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[23, -24], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).","answer":{"kind":"expression","value":"(2, [1/2, 1/2], [23, -24], 2, Yes, common_probability_space_and_finite_integer_p_moments)","explanation":"The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (2, [1/2, 1/2], [23, -24], 2, Yes, common_probability_space_and_finite_integer_p_moments)."},"skills":["Keep the Fixed p, Common Law and Whole Symbolic Tail Together"],"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-20261014-advance-20261104-r07-lp-convergence-0074","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0515c9813b804cb5b52a72068934b98b00e6ff02243e86c40d776ee2843a5e5","family_id":"qf-mod35-lp-qualification","seed":202611047074,"semantic_template":"qf-mod35-lp-qualification/{\"coefficients\":[24,-25],\"masses\":[[1,2],[1,2]],\"p\":3,\"shift\":2}"},"fields":{"title":"Keep the Fixed p, Common Law and Whole Symbolic Tail Together","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[24, -25], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).","answer":{"kind":"expression","value":"(3, [1/2, 1/2], [24, -25], 2, Yes, common_probability_space_and_finite_integer_p_moments)","explanation":"The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (3, [1/2, 1/2], [24, -25], 2, Yes, common_probability_space_and_finite_integer_p_moments)."},"skills":["Keep the Fixed p, Common Law and Whole Symbolic Tail Together"],"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-20261014-advance-20261104-r07-lp-convergence-0075","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f78f3445f9989b70676b9beb9718244ebba46d6becd5bb8a48537e766882d990","family_id":"qf-mod35-lp-qualification","seed":202611047075,"semantic_template":"qf-mod35-lp-qualification/{\"coefficients\":[25,-26],\"masses\":[[1,2],[1,2]],\"p\":1,\"shift\":2}"},"fields":{"title":"Keep the Fixed p, Common Law and Whole Symbolic Tail Together","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[25, -26], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).","answer":{"kind":"expression","value":"(1, [1/2, 1/2], [25, -26], 2, Yes, common_probability_space_and_finite_integer_p_moments)","explanation":"The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (1, [1/2, 1/2], [25, -26], 2, Yes, common_probability_space_and_finite_integer_p_moments)."},"skills":["Keep the Fixed p, Common Law and Whole Symbolic Tail Together"],"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-20261014-advance-20261104-r07-lp-convergence-0076","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eeb3607137bfbeed5193214313cd1f2bac180154b48b77cefb93693b483a2ade","family_id":"qf-mod35-lp-finite-prefix-boundary","seed":202611047076,"semantic_template":"qf-mod35-lp-finite-prefix-boundary/{\"coefficients\":[1,-2],\"masses\":[[1,2],[1,2]],\"p\":1,\"prefix_length\":1,\"shift\":2}"},"fields":{"title":"A Finite Prefix Does Not Establish Lp Convergence or Mode Implications","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[1, -2], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=1. Give (prefix_length, sufficient, boundary).","answer":{"kind":"expression","value":"(1, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)","explanation":"The No concerns inference from a fixed finite prefix alone. It does not deny the separately declared rational-tail certificate or additional consequences available under explicit finite-space assumptions. General almost-sure, probability and other-Lq relations require their own stated hypotheses. The exact result, retaining the requested order, is (1, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)."},"skills":["A Finite Prefix Does Not Establish Lp Convergence or Mode Implications"],"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-20261014-advance-20261104-r07-lp-convergence-0077","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cdb7286ac23966f0cd6c956d380a2eab073456f56f06042ed2069fc0da607a0d","family_id":"qf-mod35-lp-finite-prefix-boundary","seed":202611047077,"semantic_template":"qf-mod35-lp-finite-prefix-boundary/{\"coefficients\":[2,-3],\"masses\":[[1,2],[1,2]],\"p\":2,\"prefix_length\":2,\"shift\":2}"},"fields":{"title":"A Finite Prefix Does Not Establish Lp Convergence or Mode Implications","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[2, -3], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=2. Give (prefix_length, sufficient, boundary).","answer":{"kind":"expression","value":"(2, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)","explanation":"The No concerns inference from a fixed finite prefix alone. It does not deny the separately declared rational-tail certificate or additional consequences available under explicit finite-space assumptions. General almost-sure, probability and other-Lq relations require their own stated hypotheses. The exact result, retaining the requested order, is (2, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)."},"skills":["A Finite Prefix Does Not Establish Lp Convergence or Mode Implications"],"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-20261014-advance-20261104-r07-lp-convergence-0078","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9f87aa53778c71add59882dc60385b4f19b6fea571caf48d8c6c04c03d73445e","family_id":"qf-mod35-lp-finite-prefix-boundary","seed":202611047078,"semantic_template":"qf-mod35-lp-finite-prefix-boundary/{\"coefficients\":[3,-4],\"masses\":[[1,2],[1,2]],\"p\":3,\"prefix_length\":3,\"shift\":2}"},"fields":{"title":"A Finite Prefix Does Not Establish Lp Convergence or Mode Implications","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[3, -4], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=3. Give (prefix_length, sufficient, boundary).","answer":{"kind":"expression","value":"(3, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)","explanation":"The No concerns inference from a fixed finite prefix alone. It does not deny the separately declared rational-tail certificate or additional consequences available under explicit finite-space assumptions. General almost-sure, probability and other-Lq relations require their own stated hypotheses. The exact result, retaining the requested order, is (3, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)."},"skills":["A Finite Prefix Does Not Establish Lp Convergence or Mode Implications"],"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-20261014-advance-20261104-r07-lp-convergence-0079","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8184fe67f98f26595b2195ce50841c06cf27f9892346198cd171c957048b4620","family_id":"qf-mod35-lp-finite-prefix-boundary","seed":202611047079,"semantic_template":"qf-mod35-lp-finite-prefix-boundary/{\"coefficients\":[4,-5],\"masses\":[[1,2],[1,2]],\"p\":1,\"prefix_length\":4,\"shift\":2}"},"fields":{"title":"A Finite Prefix Does Not Establish Lp Convergence or Mode Implications","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[4, -5], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=4. Give (prefix_length, sufficient, boundary).","answer":{"kind":"expression","value":"(4, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)","explanation":"The No concerns inference from a fixed finite prefix alone. It does not deny the separately declared rational-tail certificate or additional consequences available under explicit finite-space assumptions. General almost-sure, probability and other-Lq relations require their own stated hypotheses. The exact result, retaining the requested order, is (4, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)."},"skills":["A Finite Prefix Does Not Establish Lp Convergence or Mode Implications"],"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-20261014-advance-20261104-r07-lp-convergence-0080","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"80963c7c42cb2ef48991d5e1a3bef7b0d1819fc619fa9908d60bd5887145d409","family_id":"qf-mod35-lp-finite-prefix-boundary","seed":202611047080,"semantic_template":"qf-mod35-lp-finite-prefix-boundary/{\"coefficients\":[5,-6],\"masses\":[[1,2],[1,2]],\"p\":2,\"prefix_length\":5,\"shift\":2}"},"fields":{"title":"A Finite Prefix Does Not Establish Lp Convergence or Mode Implications","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[5, -6], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=5. Give (prefix_length, sufficient, boundary).","answer":{"kind":"expression","value":"(5, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)","explanation":"The No concerns inference from a fixed finite prefix alone. It does not deny the separately declared rational-tail certificate or additional consequences available under explicit finite-space assumptions. General almost-sure, probability and other-Lq relations require their own stated hypotheses. The exact result, retaining the requested order, is (5, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)."},"skills":["A Finite Prefix Does Not Establish Lp Convergence or Mode Implications"],"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-20261014-advance-20261104-r07-lp-convergence-0081","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"958ce1467a7aae6138d507fabb2cda49cd84fdf32c95208fc7bd0eda6b57122a","family_id":"qf-mod35-lp-finite-prefix-boundary","seed":202611047081,"semantic_template":"qf-mod35-lp-finite-prefix-boundary/{\"coefficients\":[6,-7],\"masses\":[[1,2],[1,2]],\"p\":3,\"prefix_length\":6,\"shift\":2}"},"fields":{"title":"A Finite Prefix Does Not Establish Lp Convergence or Mode Implications","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[6, -7], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=6. Give (prefix_length, sufficient, boundary).","answer":{"kind":"expression","value":"(6, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)","explanation":"The No concerns inference from a fixed finite prefix alone. It does not deny the separately declared rational-tail certificate or additional consequences available under explicit finite-space assumptions. General almost-sure, probability and other-Lq relations require their own stated hypotheses. The exact result, retaining the requested order, is (6, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)."},"skills":["A Finite Prefix Does Not Establish Lp Convergence or Mode Implications"],"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-20261014-advance-20261104-r07-lp-convergence-0082","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78053063a27893daf4e6c79b4c4afaab9c40f08c9a82dfc2a354825257ac534e","family_id":"qf-mod35-lp-finite-prefix-boundary","seed":202611047082,"semantic_template":"qf-mod35-lp-finite-prefix-boundary/{\"coefficients\":[7,-8],\"masses\":[[1,2],[1,2]],\"p\":1,\"prefix_length\":7,\"shift\":2}"},"fields":{"title":"A Finite Prefix Does Not Establish Lp Convergence or Mode Implications","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[7, -8], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=7. Give (prefix_length, sufficient, boundary).","answer":{"kind":"expression","value":"(7, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)","explanation":"The No concerns inference from a fixed finite prefix alone. It does not deny the separately declared rational-tail certificate or additional consequences available under explicit finite-space assumptions. General almost-sure, probability and other-Lq relations require their own stated hypotheses. The exact result, retaining the requested order, is (7, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)."},"skills":["A Finite Prefix Does Not Establish Lp Convergence or Mode Implications"],"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-20261014-advance-20261104-r07-lp-convergence-0083","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"513ca7d6d92982e3553cfe4f821b2872220b13b359b8ee3ebd525c249092b8fc","family_id":"qf-mod35-lp-finite-prefix-boundary","seed":202611047083,"semantic_template":"qf-mod35-lp-finite-prefix-boundary/{\"coefficients\":[8,-9],\"masses\":[[1,2],[1,2]],\"p\":2,\"prefix_length\":8,\"shift\":2}"},"fields":{"title":"A Finite Prefix Does Not Establish Lp Convergence or Mode Implications","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[8, -9], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=8. Give (prefix_length, sufficient, boundary).","answer":{"kind":"expression","value":"(8, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)","explanation":"The No concerns inference from a fixed finite prefix alone. It does not deny the separately declared rational-tail certificate or additional consequences available under explicit finite-space assumptions. General almost-sure, probability and other-Lq relations require their own stated hypotheses. The exact result, retaining the requested order, is (8, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)."},"skills":["A Finite Prefix Does Not Establish Lp Convergence or Mode Implications"],"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-20261014-advance-20261104-r07-lp-convergence-0084","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a64120fbcb1fced3630b3dc107aa5c887b1c44a585b1395ea14806ecccd7def5","family_id":"qf-mod35-lp-finite-prefix-boundary","seed":202611047084,"semantic_template":"qf-mod35-lp-finite-prefix-boundary/{\"coefficients\":[9,-10],\"masses\":[[1,2],[1,2]],\"p\":3,\"prefix_length\":9,\"shift\":2}"},"fields":{"title":"A Finite Prefix Does Not Establish Lp Convergence or Mode Implications","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[9, -10], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=9. Give (prefix_length, sufficient, boundary).","answer":{"kind":"expression","value":"(9, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)","explanation":"The No concerns inference from a fixed finite prefix alone. It does not deny the separately declared rational-tail certificate or additional consequences available under explicit finite-space assumptions. General almost-sure, probability and other-Lq relations require their own stated hypotheses. The exact result, retaining the requested order, is (9, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)."},"skills":["A Finite Prefix Does Not Establish Lp Convergence or Mode Implications"],"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-20261014-advance-20261104-r07-lp-convergence-0085","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"96df8762510bd00fdeadf9fc26d9f37a337a5eeaaac5336bdeb8c55c715522d8","family_id":"qf-mod35-lp-finite-prefix-boundary","seed":202611047085,"semantic_template":"qf-mod35-lp-finite-prefix-boundary/{\"coefficients\":[10,-11],\"masses\":[[1,2],[1,2]],\"p\":1,\"prefix_length\":10,\"shift\":2}"},"fields":{"title":"A Finite Prefix Does Not Establish Lp Convergence or Mode Implications","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[10, -11], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=10. Give (prefix_length, sufficient, boundary).","answer":{"kind":"expression","value":"(10, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)","explanation":"The No concerns inference from a fixed finite prefix alone. It does not deny the separately declared rational-tail certificate or additional consequences available under explicit finite-space assumptions. General almost-sure, probability and other-Lq relations require their own stated hypotheses. The exact result, retaining the requested order, is (10, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)."},"skills":["A Finite Prefix Does Not Establish Lp Convergence or Mode Implications"],"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-20261014-advance-20261104-r07-lp-convergence-0086","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7655ace48416be1e88864e3ccc4c95016b1868439917d0d3a05fd539aa2fa696","family_id":"qf-mod35-lp-finite-prefix-boundary","seed":202611047086,"semantic_template":"qf-mod35-lp-finite-prefix-boundary/{\"coefficients\":[11,-12],\"masses\":[[1,2],[1,2]],\"p\":2,\"prefix_length\":11,\"shift\":2}"},"fields":{"title":"A Finite Prefix Does Not Establish Lp Convergence or Mode Implications","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[11, -12], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=11. Give (prefix_length, sufficient, boundary).","answer":{"kind":"expression","value":"(11, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)","explanation":"The No concerns inference from a fixed finite prefix alone. It does not deny the separately declared rational-tail certificate or additional consequences available under explicit finite-space assumptions. General almost-sure, probability and other-Lq relations require their own stated hypotheses. The exact result, retaining the requested order, is (11, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)."},"skills":["A Finite Prefix Does Not Establish Lp Convergence or Mode Implications"],"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-20261014-advance-20261104-r07-lp-convergence-0087","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"593dfe7db961f4df09fc2827f98b4ad2e1c01ed85564888d82416513c7751431","family_id":"qf-mod35-lp-finite-prefix-boundary","seed":202611047087,"semantic_template":"qf-mod35-lp-finite-prefix-boundary/{\"coefficients\":[12,-13],\"masses\":[[1,2],[1,2]],\"p\":3,\"prefix_length\":12,\"shift\":2}"},"fields":{"title":"A Finite Prefix Does Not Establish Lp Convergence or Mode Implications","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[12, -13], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=12. Give (prefix_length, sufficient, boundary).","answer":{"kind":"expression","value":"(12, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)","explanation":"The No concerns inference from a fixed finite prefix alone. It does not deny the separately declared rational-tail certificate or additional consequences available under explicit finite-space assumptions. General almost-sure, probability and other-Lq relations require their own stated hypotheses. The exact result, retaining the requested order, is (12, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)."},"skills":["A Finite Prefix Does Not Establish Lp Convergence or Mode Implications"],"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-20261014-advance-20261104-r07-lp-convergence-0088","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"89066457273f92315fcf5d0646fe548ac77845ff0a5d07a587785934ebd61e5c","family_id":"qf-mod35-lp-finite-prefix-boundary","seed":202611047088,"semantic_template":"qf-mod35-lp-finite-prefix-boundary/{\"coefficients\":[13,-14],\"masses\":[[1,2],[1,2]],\"p\":1,\"prefix_length\":13,\"shift\":2}"},"fields":{"title":"A Finite Prefix Does Not Establish Lp Convergence or Mode Implications","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[13, -14], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=13. Give (prefix_length, sufficient, boundary).","answer":{"kind":"expression","value":"(13, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)","explanation":"The No concerns inference from a fixed finite prefix alone. It does not deny the separately declared rational-tail certificate or additional consequences available under explicit finite-space assumptions. General almost-sure, probability and other-Lq relations require their own stated hypotheses. The exact result, retaining the requested order, is (13, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)."},"skills":["A Finite Prefix Does Not Establish Lp Convergence or Mode Implications"],"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-20261014-advance-20261104-r07-lp-convergence-0089","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7da5eaa3bef4e83445d85bd413b647978323b8fc0e25208f2804b134319bcaf9","family_id":"qf-mod35-lp-finite-prefix-boundary","seed":202611047089,"semantic_template":"qf-mod35-lp-finite-prefix-boundary/{\"coefficients\":[14,-15],\"masses\":[[1,2],[1,2]],\"p\":2,\"prefix_length\":14,\"shift\":2}"},"fields":{"title":"A Finite Prefix Does Not Establish Lp Convergence or Mode Implications","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[14, -15], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=14. Give (prefix_length, sufficient, boundary).","answer":{"kind":"expression","value":"(14, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)","explanation":"The No concerns inference from a fixed finite prefix alone. It does not deny the separately declared rational-tail certificate or additional consequences available under explicit finite-space assumptions. General almost-sure, probability and other-Lq relations require their own stated hypotheses. The exact result, retaining the requested order, is (14, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)."},"skills":["A Finite Prefix Does Not Establish Lp Convergence or Mode Implications"],"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-20261014-advance-20261104-r07-lp-convergence-0090","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"08e9eda2bfc9f6607e7b732f0e07b2ac8f0fe20f3fc1315c9681908eddec0a75","family_id":"qf-mod35-lp-finite-prefix-boundary","seed":202611047090,"semantic_template":"qf-mod35-lp-finite-prefix-boundary/{\"coefficients\":[15,-16],\"masses\":[[1,2],[1,2]],\"p\":3,\"prefix_length\":15,\"shift\":2}"},"fields":{"title":"A Finite Prefix Does Not Establish Lp Convergence or Mode Implications","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[15, -16], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=15. Give (prefix_length, sufficient, boundary).","answer":{"kind":"expression","value":"(15, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)","explanation":"The No concerns inference from a fixed finite prefix alone. It does not deny the separately declared rational-tail certificate or additional consequences available under explicit finite-space assumptions. General almost-sure, probability and other-Lq relations require their own stated hypotheses. The exact result, retaining the requested order, is (15, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)."},"skills":["A Finite Prefix Does Not Establish Lp Convergence or Mode Implications"],"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-20261014-advance-20261104-r07-lp-convergence-0091","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24cb50e053569715be53cc6d90f25855b805fbbc7c3a2d873a226f23a022594e","family_id":"qf-mod35-lp-finite-prefix-boundary","seed":202611047091,"semantic_template":"qf-mod35-lp-finite-prefix-boundary/{\"coefficients\":[16,-17],\"masses\":[[1,2],[1,2]],\"p\":1,\"prefix_length\":16,\"shift\":2}"},"fields":{"title":"A Finite Prefix Does Not Establish Lp Convergence or Mode Implications","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[16, -17], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=16. Give (prefix_length, sufficient, boundary).","answer":{"kind":"expression","value":"(16, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)","explanation":"The No concerns inference from a fixed finite prefix alone. It does not deny the separately declared rational-tail certificate or additional consequences available under explicit finite-space assumptions. General almost-sure, probability and other-Lq relations require their own stated hypotheses. The exact result, retaining the requested order, is (16, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)."},"skills":["A Finite Prefix Does Not Establish Lp Convergence or Mode Implications"],"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-20261014-advance-20261104-r07-lp-convergence-0092","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2f65c0477553d3fa9c72a1ca7f9c92140bf70857d20764f1515ac4e71201785","family_id":"qf-mod35-lp-finite-prefix-boundary","seed":202611047092,"semantic_template":"qf-mod35-lp-finite-prefix-boundary/{\"coefficients\":[17,-18],\"masses\":[[1,2],[1,2]],\"p\":2,\"prefix_length\":17,\"shift\":2}"},"fields":{"title":"A Finite Prefix Does Not Establish Lp Convergence or Mode Implications","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[17, -18], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=17. Give (prefix_length, sufficient, boundary).","answer":{"kind":"expression","value":"(17, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)","explanation":"The No concerns inference from a fixed finite prefix alone. It does not deny the separately declared rational-tail certificate or additional consequences available under explicit finite-space assumptions. General almost-sure, probability and other-Lq relations require their own stated hypotheses. The exact result, retaining the requested order, is (17, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)."},"skills":["A Finite Prefix Does Not Establish Lp Convergence or Mode Implications"],"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-20261014-advance-20261104-r07-lp-convergence-0093","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5293d5b1d4420f248e9502ab167e60a1d008165a5cdefd894217a8b541c53fc7","family_id":"qf-mod35-lp-finite-prefix-boundary","seed":202611047093,"semantic_template":"qf-mod35-lp-finite-prefix-boundary/{\"coefficients\":[18,-19],\"masses\":[[1,2],[1,2]],\"p\":3,\"prefix_length\":18,\"shift\":2}"},"fields":{"title":"A Finite Prefix Does Not Establish Lp Convergence or Mode Implications","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[18, -19], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=18. Give (prefix_length, sufficient, boundary).","answer":{"kind":"expression","value":"(18, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)","explanation":"The No concerns inference from a fixed finite prefix alone. It does not deny the separately declared rational-tail certificate or additional consequences available under explicit finite-space assumptions. General almost-sure, probability and other-Lq relations require their own stated hypotheses. The exact result, retaining the requested order, is (18, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)."},"skills":["A Finite Prefix Does Not Establish Lp Convergence or Mode Implications"],"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-20261014-advance-20261104-r07-lp-convergence-0094","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"443d9825291247c2cfdf8e97174787a40a1f9209ccb0b2f6db473b27958228ed","family_id":"qf-mod35-lp-finite-prefix-boundary","seed":202611047094,"semantic_template":"qf-mod35-lp-finite-prefix-boundary/{\"coefficients\":[19,-20],\"masses\":[[1,2],[1,2]],\"p\":1,\"prefix_length\":19,\"shift\":2}"},"fields":{"title":"A Finite Prefix Does Not Establish Lp Convergence or Mode Implications","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[19, -20], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=19. Give (prefix_length, sufficient, boundary).","answer":{"kind":"expression","value":"(19, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)","explanation":"The No concerns inference from a fixed finite prefix alone. It does not deny the separately declared rational-tail certificate or additional consequences available under explicit finite-space assumptions. General almost-sure, probability and other-Lq relations require their own stated hypotheses. The exact result, retaining the requested order, is (19, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)."},"skills":["A Finite Prefix Does Not Establish Lp Convergence or Mode Implications"],"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-20261014-advance-20261104-r07-lp-convergence-0095","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"03f2904a637f784587875cdee9ed5abbb7dab1cc2fd3d9bdeb09a22448a1b75d","family_id":"qf-mod35-lp-finite-prefix-boundary","seed":202611047095,"semantic_template":"qf-mod35-lp-finite-prefix-boundary/{\"coefficients\":[20,-21],\"masses\":[[1,2],[1,2]],\"p\":2,\"prefix_length\":20,\"shift\":2}"},"fields":{"title":"A Finite Prefix Does Not Establish Lp Convergence or Mode Implications","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[20, -21], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=20. Give (prefix_length, sufficient, boundary).","answer":{"kind":"expression","value":"(20, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)","explanation":"The No concerns inference from a fixed finite prefix alone. It does not deny the separately declared rational-tail certificate or additional consequences available under explicit finite-space assumptions. General almost-sure, probability and other-Lq relations require their own stated hypotheses. The exact result, retaining the requested order, is (20, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)."},"skills":["A Finite Prefix Does Not Establish Lp Convergence or Mode Implications"],"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-20261014-advance-20261104-r07-lp-convergence-0096","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"66359e1dafff6c0adfe4850e9d215a3809f772e1c53482d21ceaf2d341939290","family_id":"qf-mod35-lp-finite-prefix-boundary","seed":202611047096,"semantic_template":"qf-mod35-lp-finite-prefix-boundary/{\"coefficients\":[21,-22],\"masses\":[[1,2],[1,2]],\"p\":3,\"prefix_length\":21,\"shift\":2}"},"fields":{"title":"A Finite Prefix Does Not Establish Lp Convergence or Mode Implications","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[21, -22], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=21. Give (prefix_length, sufficient, boundary).","answer":{"kind":"expression","value":"(21, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)","explanation":"The No concerns inference from a fixed finite prefix alone. It does not deny the separately declared rational-tail certificate or additional consequences available under explicit finite-space assumptions. General almost-sure, probability and other-Lq relations require their own stated hypotheses. The exact result, retaining the requested order, is (21, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)."},"skills":["A Finite Prefix Does Not Establish Lp Convergence or Mode Implications"],"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-20261014-advance-20261104-r07-lp-convergence-0097","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be2acddff7a8113c30c904b22f13dd318d1843adbd552bfb0f51ac3a67cc741a","family_id":"qf-mod35-lp-finite-prefix-boundary","seed":202611047097,"semantic_template":"qf-mod35-lp-finite-prefix-boundary/{\"coefficients\":[22,-23],\"masses\":[[1,2],[1,2]],\"p\":1,\"prefix_length\":22,\"shift\":2}"},"fields":{"title":"A Finite Prefix Does Not Establish Lp Convergence or Mode Implications","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[22, -23], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=22. Give (prefix_length, sufficient, boundary).","answer":{"kind":"expression","value":"(22, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)","explanation":"The No concerns inference from a fixed finite prefix alone. It does not deny the separately declared rational-tail certificate or additional consequences available under explicit finite-space assumptions. General almost-sure, probability and other-Lq relations require their own stated hypotheses. The exact result, retaining the requested order, is (22, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)."},"skills":["A Finite Prefix Does Not Establish Lp Convergence or Mode Implications"],"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-20261014-advance-20261104-r07-lp-convergence-0098","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a206c8c6314dbafdcabed8f1c5f28a464e16c6ae8f28187d6b4871bb395dc63","family_id":"qf-mod35-lp-finite-prefix-boundary","seed":202611047098,"semantic_template":"qf-mod35-lp-finite-prefix-boundary/{\"coefficients\":[23,-24],\"masses\":[[1,2],[1,2]],\"p\":2,\"prefix_length\":23,\"shift\":2}"},"fields":{"title":"A Finite Prefix Does Not Establish Lp Convergence or Mode Implications","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[23, -24], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=23. Give (prefix_length, sufficient, boundary).","answer":{"kind":"expression","value":"(23, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)","explanation":"The No concerns inference from a fixed finite prefix alone. It does not deny the separately declared rational-tail certificate or additional consequences available under explicit finite-space assumptions. General almost-sure, probability and other-Lq relations require their own stated hypotheses. The exact result, retaining the requested order, is (23, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)."},"skills":["A Finite Prefix Does Not Establish Lp Convergence or Mode Implications"],"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-20261014-advance-20261104-r07-lp-convergence-0099","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee273fcfec6233f25f9cc654a3d5b8b2d9a3d9d1534fb0b3ac4216335fac7be8","family_id":"qf-mod35-lp-finite-prefix-boundary","seed":202611047099,"semantic_template":"qf-mod35-lp-finite-prefix-boundary/{\"coefficients\":[24,-25],\"masses\":[[1,2],[1,2]],\"p\":3,\"prefix_length\":24,\"shift\":2}"},"fields":{"title":"A Finite Prefix Does Not Establish Lp Convergence or Mode Implications","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[24, -25], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=24. Give (prefix_length, sufficient, boundary).","answer":{"kind":"expression","value":"(24, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)","explanation":"The No concerns inference from a fixed finite prefix alone. It does not deny the separately declared rational-tail certificate or additional consequences available under explicit finite-space assumptions. General almost-sure, probability and other-Lq relations require their own stated hypotheses. The exact result, retaining the requested order, is (24, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)."},"skills":["A Finite Prefix Does Not Establish Lp Convergence or Mode Implications"],"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-20261014-advance-20261104-r07-lp-convergence-0100","batch_id":"batch-20261014-advance-20261104-r07-lp-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a6bd34b7fc6ec8797ad9ad2597cefbdd55a2770d501237df8e7690062d3c09c","family_id":"qf-mod35-lp-finite-prefix-boundary","seed":202611047100,"semantic_template":"qf-mod35-lp-finite-prefix-boundary/{\"coefficients\":[25,-26],\"masses\":[[1,2],[1,2]],\"p\":1,\"prefix_length\":25,\"shift\":2}"},"fields":{"title":"A Finite Prefix Does Not Establish Lp Convergence or Mode Implications","stem":"Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[25, -26], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=25. Give (prefix_length, sufficient, boundary).","answer":{"kind":"expression","value":"(25, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)","explanation":"The No concerns inference from a fixed finite prefix alone. It does not deny the separately declared rational-tail certificate or additional consequences available under explicit finite-space assumptions. General almost-sure, probability and other-Lq relations require their own stated hypotheses. The exact result, retaining the requested order, is (25, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)."},"skills":["A Finite Prefix Does Not Establish Lp Convergence or Mode Implications"],"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."}}