{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r01-chromatic-number-0001","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b951ec1186d90f628aac83960ca1cb85cb346b6d0e8e8e4b59b548eafb47732","family_id":"qf-mod37-chromatic-proper-audit","seed":202611061001,"semantic_template":"qf-mod37-chromatic-proper-audit/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[0,5],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Proper Coloring Alone Does Not Prove the Chromatic Minimum","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [0, 1], 2, Yes, proper_coloring_only)","explanation":"Report the original color array, increasingly ordered used labels, their count and all-edge properness. A valid coloring supplies an upper bound, not automatically equality with the chromatic number. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [0, 1], 2, Yes, proper_coloring_only)."},"skills":["Proper Coloring Alone Does Not Prove the Chromatic Minimum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0002","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"50b0027d17dedd72804af1bf98e5313a3ca19527bf15cd4c4c0526c23763089b","family_id":"qf-mod37-chromatic-proper-audit","seed":202611061002,"semantic_template":"qf-mod37-chromatic-proper-audit/{\"colors\":[0,1,2,0,1,2],\"edges\":[[0,1],[0,2],[0,5],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Proper Coloring Alone Does Not Prove the Chromatic Minimum","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 0, 1, 2] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 0, 1, 2], [0, 1, 2], 3, Yes, proper_coloring_only)","explanation":"Report the original color array, increasingly ordered used labels, their count and all-edge properness. A valid coloring supplies an upper bound, not automatically equality with the chromatic number. The exact result, retaining the requested order, is ([0, 1, 2, 0, 1, 2], [0, 1, 2], 3, Yes, proper_coloring_only)."},"skills":["Proper Coloring Alone Does Not Prove the Chromatic Minimum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0003","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39a4c6db11841647496a473073fae4665ba32328939bf102a91f08af1b76ab47","family_id":"qf-mod37-chromatic-proper-audit","seed":202611061003,"semantic_template":"qf-mod37-chromatic-proper-audit/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[0,3],[0,5],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Proper Coloring Alone Does Not Prove the Chromatic Minimum","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [0, 5], [1, 2], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [0, 1], 2, Yes, proper_coloring_only)","explanation":"Report the original color array, increasingly ordered used labels, their count and all-edge properness. A valid coloring supplies an upper bound, not automatically equality with the chromatic number. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [0, 1], 2, Yes, proper_coloring_only)."},"skills":["Proper Coloring Alone Does Not Prove the Chromatic Minimum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0004","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a96edc4c7d6932f3305ab3f94e6468e3afa17f44b42cf23b737e4d525723b2ee","family_id":"qf-mod37-chromatic-proper-audit","seed":202611061004,"semantic_template":"qf-mod37-chromatic-proper-audit/{\"colors\":[0,1,2,1,0,1],\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Proper Coloring Alone Does Not Prove the Chromatic Minimum","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 1, 0, 1], [0, 1, 2], 3, Yes, proper_coloring_only)","explanation":"Report the original color array, increasingly ordered used labels, their count and all-edge properness. A valid coloring supplies an upper bound, not automatically equality with the chromatic number. The exact result, retaining the requested order, is ([0, 1, 2, 1, 0, 1], [0, 1, 2], 3, Yes, proper_coloring_only)."},"skills":["Proper Coloring Alone Does Not Prove the Chromatic Minimum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0005","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"53f6269b371da17e7e36adbb73fcf8002e88817b133577c5c2ce65927f645fa7","family_id":"qf-mod37-chromatic-proper-audit","seed":202611061005,"semantic_template":"qf-mod37-chromatic-proper-audit/{\"colors\":[0,1,0,2,0,1],\"edges\":[[0,1],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Proper Coloring Alone Does Not Prove the Chromatic Minimum","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [1, 3], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 2, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).","answer":{"kind":"expression","value":"([0, 1, 0, 2, 0, 1], [0, 1, 2], 3, Yes, proper_coloring_only)","explanation":"Report the original color array, increasingly ordered used labels, their count and all-edge properness. A valid coloring supplies an upper bound, not automatically equality with the chromatic number. The exact result, retaining the requested order, is ([0, 1, 0, 2, 0, 1], [0, 1, 2], 3, Yes, proper_coloring_only)."},"skills":["Proper Coloring Alone Does Not Prove the Chromatic Minimum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0006","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"adc28406850c648a9e43fefe089280adaf0c169cf3c8cf8fa23d26fd08386ba4","family_id":"qf-mod37-chromatic-proper-audit","seed":202611061006,"semantic_template":"qf-mod37-chromatic-proper-audit/{\"colors\":[0,1,2,0,1,2],\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Proper Coloring Alone Does Not Prove the Chromatic Minimum","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 3], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 0, 1, 2] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 0, 1, 2], [0, 1, 2], 3, Yes, proper_coloring_only)","explanation":"Report the original color array, increasingly ordered used labels, their count and all-edge properness. A valid coloring supplies an upper bound, not automatically equality with the chromatic number. The exact result, retaining the requested order, is ([0, 1, 2, 0, 1, 2], [0, 1, 2], 3, Yes, proper_coloring_only)."},"skills":["Proper Coloring Alone Does Not Prove the Chromatic Minimum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0007","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf392b3c02b1e5cd4d21259e01474047ebe8de12f479c0d051dbf427e5ffb671","family_id":"qf-mod37-chromatic-proper-audit","seed":202611061007,"semantic_template":"qf-mod37-chromatic-proper-audit/{\"colors\":[0,1,0,2,0,1],\"edges\":[[0,1],[0,3],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Proper Coloring Alone Does Not Prove the Chromatic Minimum","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [0, 5], [1, 2], [1, 3], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 2, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).","answer":{"kind":"expression","value":"([0, 1, 0, 2, 0, 1], [0, 1, 2], 3, Yes, proper_coloring_only)","explanation":"Report the original color array, increasingly ordered used labels, their count and all-edge properness. A valid coloring supplies an upper bound, not automatically equality with the chromatic number. The exact result, retaining the requested order, is ([0, 1, 0, 2, 0, 1], [0, 1, 2], 3, Yes, proper_coloring_only)."},"skills":["Proper Coloring Alone Does Not Prove the Chromatic Minimum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0008","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b7ac2830bbc09c89add129f8e09e2e13bc421202c07276d27f5fac3f565ab804","family_id":"qf-mod37-chromatic-proper-audit","seed":202611061008,"semantic_template":"qf-mod37-chromatic-proper-audit/{\"colors\":[0,1,2,3,0,1],\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Proper Coloring Alone Does Not Prove the Chromatic Minimum","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 3], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 3, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 0, 1], [0, 1, 2, 3], 4, Yes, proper_coloring_only)","explanation":"Report the original color array, increasingly ordered used labels, their count and all-edge properness. A valid coloring supplies an upper bound, not automatically equality with the chromatic number. The exact result, retaining the requested order, is ([0, 1, 2, 3, 0, 1], [0, 1, 2, 3], 4, Yes, proper_coloring_only)."},"skills":["Proper Coloring Alone Does Not Prove the Chromatic Minimum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0009","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e4431da49bb25c52ad4a96fbde7dffc1801b08e6db486cb5bb5f7cc228b42ed","family_id":"qf-mod37-chromatic-proper-audit","seed":202611061009,"semantic_template":"qf-mod37-chromatic-proper-audit/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Proper Coloring Alone Does Not Prove the Chromatic Minimum","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [0, 1], 2, Yes, proper_coloring_only)","explanation":"Report the original color array, increasingly ordered used labels, their count and all-edge properness. A valid coloring supplies an upper bound, not automatically equality with the chromatic number. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [0, 1], 2, Yes, proper_coloring_only)."},"skills":["Proper Coloring Alone Does Not Prove the Chromatic Minimum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0010","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0be0a087957d3ed00943d475673c8224e37930150dc393eb1535fb7092ffba99","family_id":"qf-mod37-chromatic-proper-audit","seed":202611061010,"semantic_template":"qf-mod37-chromatic-proper-audit/{\"colors\":[0,1,2,0,2,1],\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Proper Coloring Alone Does Not Prove the Chromatic Minimum","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 0, 2, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 0, 2, 1], [0, 1, 2], 3, Yes, proper_coloring_only)","explanation":"Report the original color array, increasingly ordered used labels, their count and all-edge properness. A valid coloring supplies an upper bound, not automatically equality with the chromatic number. The exact result, retaining the requested order, is ([0, 1, 2, 0, 2, 1], [0, 1, 2], 3, Yes, proper_coloring_only)."},"skills":["Proper Coloring Alone Does Not Prove the Chromatic Minimum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0011","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ef5a712cbe49789b7832672e4de45b2ae6c8249dcb95984fa2c264b7ec51097","family_id":"qf-mod37-chromatic-proper-audit","seed":202611061011,"semantic_template":"qf-mod37-chromatic-proper-audit/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[0,3],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Proper Coloring Alone Does Not Prove the Chromatic Minimum","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [0, 5], [1, 2], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [0, 1], 2, Yes, proper_coloring_only)","explanation":"Report the original color array, increasingly ordered used labels, their count and all-edge properness. A valid coloring supplies an upper bound, not automatically equality with the chromatic number. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [0, 1], 2, Yes, proper_coloring_only)."},"skills":["Proper Coloring Alone Does Not Prove the Chromatic Minimum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0012","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b5df7e81230a891a4deb0ef2faac0def35dcb349d0638b2d092189aa04a85217","family_id":"qf-mod37-chromatic-proper-audit","seed":202611061012,"semantic_template":"qf-mod37-chromatic-proper-audit/{\"colors\":[0,1,2,1,0,1],\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Proper Coloring Alone Does Not Prove the Chromatic Minimum","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 1, 0, 1], [0, 1, 2], 3, Yes, proper_coloring_only)","explanation":"Report the original color array, increasingly ordered used labels, their count and all-edge properness. A valid coloring supplies an upper bound, not automatically equality with the chromatic number. The exact result, retaining the requested order, is ([0, 1, 2, 1, 0, 1], [0, 1, 2], 3, Yes, proper_coloring_only)."},"skills":["Proper Coloring Alone Does Not Prove the Chromatic Minimum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0013","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"04f884742b90f58620ff40ecf875373f68dcc20d4a25d9f421809349001d0673","family_id":"qf-mod37-chromatic-proper-audit","seed":202611061013,"semantic_template":"qf-mod37-chromatic-proper-audit/{\"colors\":[0,1,0,2,0,1],\"edges\":[[0,1],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Proper Coloring Alone Does Not Prove the Chromatic Minimum","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 2, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).","answer":{"kind":"expression","value":"([0, 1, 0, 2, 0, 1], [0, 1, 2], 3, Yes, proper_coloring_only)","explanation":"Report the original color array, increasingly ordered used labels, their count and all-edge properness. A valid coloring supplies an upper bound, not automatically equality with the chromatic number. The exact result, retaining the requested order, is ([0, 1, 0, 2, 0, 1], [0, 1, 2], 3, Yes, proper_coloring_only)."},"skills":["Proper Coloring Alone Does Not Prove the Chromatic Minimum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0014","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28809b8b49129e2d55c680c2882c76f1226316875791a5a134f12a23f9c05dfa","family_id":"qf-mod37-chromatic-proper-audit","seed":202611061014,"semantic_template":"qf-mod37-chromatic-proper-audit/{\"colors\":[0,1,2,0,2,1],\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Proper Coloring Alone Does Not Prove the Chromatic Minimum","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 0, 2, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 0, 2, 1], [0, 1, 2], 3, Yes, proper_coloring_only)","explanation":"Report the original color array, increasingly ordered used labels, their count and all-edge properness. A valid coloring supplies an upper bound, not automatically equality with the chromatic number. The exact result, retaining the requested order, is ([0, 1, 2, 0, 2, 1], [0, 1, 2], 3, Yes, proper_coloring_only)."},"skills":["Proper Coloring Alone Does Not Prove the Chromatic Minimum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0015","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7425f08325da48cfd7feba1e4f78766e7a2efa5560bbcecdc81931ccc0cc69e7","family_id":"qf-mod37-chromatic-proper-audit","seed":202611061015,"semantic_template":"qf-mod37-chromatic-proper-audit/{\"colors\":[0,1,0,2,0,1],\"edges\":[[0,1],[0,3],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Proper Coloring Alone Does Not Prove the Chromatic Minimum","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [0, 5], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 2, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).","answer":{"kind":"expression","value":"([0, 1, 0, 2, 0, 1], [0, 1, 2], 3, Yes, proper_coloring_only)","explanation":"Report the original color array, increasingly ordered used labels, their count and all-edge properness. A valid coloring supplies an upper bound, not automatically equality with the chromatic number. The exact result, retaining the requested order, is ([0, 1, 0, 2, 0, 1], [0, 1, 2], 3, Yes, proper_coloring_only)."},"skills":["Proper Coloring Alone Does Not Prove the Chromatic Minimum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0016","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12b2b3297f9d40adeda67b442bc2da33d16918526bdadc7dfde8e7f1270a9ed6","family_id":"qf-mod37-chromatic-proper-audit","seed":202611061016,"semantic_template":"qf-mod37-chromatic-proper-audit/{\"colors\":[0,1,2,3,0,1],\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Proper Coloring Alone Does Not Prove the Chromatic Minimum","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 3, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 0, 1], [0, 1, 2, 3], 4, Yes, proper_coloring_only)","explanation":"Report the original color array, increasingly ordered used labels, their count and all-edge properness. A valid coloring supplies an upper bound, not automatically equality with the chromatic number. The exact result, retaining the requested order, is ([0, 1, 2, 3, 0, 1], [0, 1, 2, 3], 4, Yes, proper_coloring_only)."},"skills":["Proper Coloring Alone Does Not Prove the Chromatic Minimum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0017","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8cf0c738755a5a52c5bf09f4d4c18d4fce8d710af0fc833ba0ebf52a6f5714d2","family_id":"qf-mod37-chromatic-proper-audit","seed":202611061017,"semantic_template":"qf-mod37-chromatic-proper-audit/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Proper Coloring Alone Does Not Prove the Chromatic Minimum","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [0, 1], 2, Yes, proper_coloring_only)","explanation":"Report the original color array, increasingly ordered used labels, their count and all-edge properness. A valid coloring supplies an upper bound, not automatically equality with the chromatic number. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [0, 1], 2, Yes, proper_coloring_only)."},"skills":["Proper Coloring Alone Does Not Prove the Chromatic Minimum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0018","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"96aed4c28d1e299f263d4a2bb98f8422abc251559e4fafecb94a705d8a9585b0","family_id":"qf-mod37-chromatic-proper-audit","seed":202611061018,"semantic_template":"qf-mod37-chromatic-proper-audit/{\"colors\":[0,1,2,0,2,1],\"edges\":[[0,1],[0,2],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Proper Coloring Alone Does Not Prove the Chromatic Minimum","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 0, 2, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 0, 2, 1], [0, 1, 2], 3, Yes, proper_coloring_only)","explanation":"Report the original color array, increasingly ordered used labels, their count and all-edge properness. A valid coloring supplies an upper bound, not automatically equality with the chromatic number. The exact result, retaining the requested order, is ([0, 1, 2, 0, 2, 1], [0, 1, 2], 3, Yes, proper_coloring_only)."},"skills":["Proper Coloring Alone Does Not Prove the Chromatic Minimum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0019","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02cefbccfd7742c31740ba88584552fc5202568796f226183b4432fccbe8fdc8","family_id":"qf-mod37-chromatic-proper-audit","seed":202611061019,"semantic_template":"qf-mod37-chromatic-proper-audit/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[0,3],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Proper Coloring Alone Does Not Prove the Chromatic Minimum","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [0, 5], [1, 2], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [0, 1], 2, Yes, proper_coloring_only)","explanation":"Report the original color array, increasingly ordered used labels, their count and all-edge properness. A valid coloring supplies an upper bound, not automatically equality with the chromatic number. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [0, 1], 2, Yes, proper_coloring_only)."},"skills":["Proper Coloring Alone Does Not Prove the Chromatic Minimum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0020","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a9436acf5bb405c834c057c4f9ccfa66d9c8776c8eaae194d091539813c997a","family_id":"qf-mod37-chromatic-proper-audit","seed":202611061020,"semantic_template":"qf-mod37-chromatic-proper-audit/{\"colors\":[0,1,2,1,0,1],\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Proper Coloring Alone Does Not Prove the Chromatic Minimum","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 1, 0, 1], [0, 1, 2], 3, Yes, proper_coloring_only)","explanation":"Report the original color array, increasingly ordered used labels, their count and all-edge properness. A valid coloring supplies an upper bound, not automatically equality with the chromatic number. The exact result, retaining the requested order, is ([0, 1, 2, 1, 0, 1], [0, 1, 2], 3, Yes, proper_coloring_only)."},"skills":["Proper Coloring Alone Does Not Prove the Chromatic Minimum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0021","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"68399e96d43b6c2886326e992024d4633dd4cc575acbd93ddae88db263bee507","family_id":"qf-mod37-chromatic-proper-audit","seed":202611061021,"semantic_template":"qf-mod37-chromatic-proper-audit/{\"colors\":[0,1,0,2,0,1],\"edges\":[[0,1],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Proper Coloring Alone Does Not Prove the Chromatic Minimum","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [1, 3], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 2, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).","answer":{"kind":"expression","value":"([0, 1, 0, 2, 0, 1], [0, 1, 2], 3, Yes, proper_coloring_only)","explanation":"Report the original color array, increasingly ordered used labels, their count and all-edge properness. A valid coloring supplies an upper bound, not automatically equality with the chromatic number. The exact result, retaining the requested order, is ([0, 1, 0, 2, 0, 1], [0, 1, 2], 3, Yes, proper_coloring_only)."},"skills":["Proper Coloring Alone Does Not Prove the Chromatic Minimum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0022","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b8424a1b4d36424c1b1d85bfad29e995b9d7a1df2251c008cc2c2fe1e8b35446","family_id":"qf-mod37-chromatic-proper-audit","seed":202611061022,"semantic_template":"qf-mod37-chromatic-proper-audit/{\"colors\":[0,1,2,0,2,1],\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Proper Coloring Alone Does Not Prove the Chromatic Minimum","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 3], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 0, 2, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 0, 2, 1], [0, 1, 2], 3, Yes, proper_coloring_only)","explanation":"Report the original color array, increasingly ordered used labels, their count and all-edge properness. A valid coloring supplies an upper bound, not automatically equality with the chromatic number. The exact result, retaining the requested order, is ([0, 1, 2, 0, 2, 1], [0, 1, 2], 3, Yes, proper_coloring_only)."},"skills":["Proper Coloring Alone Does Not Prove the Chromatic Minimum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0023","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b867b2860ea938e136968216c45fcfbaa609d711eddf9d5030ec0e44fe4a293c","family_id":"qf-mod37-chromatic-proper-audit","seed":202611061023,"semantic_template":"qf-mod37-chromatic-proper-audit/{\"colors\":[0,1,0,2,0,1],\"edges\":[[0,1],[0,3],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Proper Coloring Alone Does Not Prove the Chromatic Minimum","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [0, 5], [1, 2], [1, 3], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 2, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).","answer":{"kind":"expression","value":"([0, 1, 0, 2, 0, 1], [0, 1, 2], 3, Yes, proper_coloring_only)","explanation":"Report the original color array, increasingly ordered used labels, their count and all-edge properness. A valid coloring supplies an upper bound, not automatically equality with the chromatic number. The exact result, retaining the requested order, is ([0, 1, 0, 2, 0, 1], [0, 1, 2], 3, Yes, proper_coloring_only)."},"skills":["Proper Coloring Alone Does Not Prove the Chromatic Minimum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0024","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"db19e7f6be55ecef1c5e308ab2f3ca9a3b1175237061d795c9f3f27f5c55109f","family_id":"qf-mod37-chromatic-proper-audit","seed":202611061024,"semantic_template":"qf-mod37-chromatic-proper-audit/{\"colors\":[0,1,2,3,0,1],\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Proper Coloring Alone Does Not Prove the Chromatic Minimum","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 3], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 3, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 0, 1], [0, 1, 2, 3], 4, Yes, proper_coloring_only)","explanation":"Report the original color array, increasingly ordered used labels, their count and all-edge properness. A valid coloring supplies an upper bound, not automatically equality with the chromatic number. The exact result, retaining the requested order, is ([0, 1, 2, 3, 0, 1], [0, 1, 2, 3], 4, Yes, proper_coloring_only)."},"skills":["Proper Coloring Alone Does Not Prove the Chromatic Minimum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0025","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"68ccb2497e1091bef3a60418355a4affd19fb4cf0f9a3d743bec04e7b55d5f02","family_id":"qf-mod37-chromatic-proper-audit","seed":202611061025,"semantic_template":"qf-mod37-chromatic-proper-audit/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[0,5],[1,2],[1,4],[2,3],[2,5],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Proper Coloring Alone Does Not Prove the Chromatic Minimum","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [1, 4], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [0, 1], 2, Yes, proper_coloring_only)","explanation":"Report the original color array, increasingly ordered used labels, their count and all-edge properness. A valid coloring supplies an upper bound, not automatically equality with the chromatic number. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [0, 1], 2, Yes, proper_coloring_only)."},"skills":["Proper Coloring Alone Does Not Prove the Chromatic Minimum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0026","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f132819205d9f3eca6fcf56dd483fdf4635df7af3d1e5f6384124028717e2e3b","family_id":"qf-mod37-chromatic-upper-bound","seed":202611061026,"semantic_template":"qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[0,5],[1,2],[2,3],[3,4],[4,5]],\"k\":3,\"n\":6}"},"fields":{"title":"A Supplied k-Palette Is a Candidate Upper Certificate","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=3. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).","answer":{"kind":"expression","value":"(3, Yes, Yes, Yes, chi_at_most_k_not_equality)","explanation":"Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (3, Yes, Yes, Yes, chi_at_most_k_not_equality)."},"skills":["A Supplied k-Palette Is a Candidate Upper Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0027","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d828ecaa4e11f668847ce9e6e82c1b575f673fbfe7c0866a46e362ea5ad31bb","family_id":"qf-mod37-chromatic-upper-bound","seed":202611061027,"semantic_template":"qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,2,0,1,2],\"edges\":[[0,1],[0,2],[0,5],[1,2],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"},"fields":{"title":"A Supplied k-Palette Is a Candidate Upper Certificate","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 0, 1, 2] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=4. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).","answer":{"kind":"expression","value":"(4, Yes, Yes, Yes, chi_at_most_k_not_equality)","explanation":"Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (4, Yes, Yes, Yes, chi_at_most_k_not_equality)."},"skills":["A Supplied k-Palette Is a Candidate Upper Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0028","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"770361f06560c0dbbfaa4e037a72a8b2da7c9c3287baad6d1061094a64bf0734","family_id":"qf-mod37-chromatic-upper-bound","seed":202611061028,"semantic_template":"qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[0,3],[0,5],[1,2],[2,3],[3,4],[4,5]],\"k\":3,\"n\":6}"},"fields":{"title":"A Supplied k-Palette Is a Candidate Upper Certificate","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [0, 5], [1, 2], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=3. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).","answer":{"kind":"expression","value":"(3, Yes, Yes, Yes, chi_at_most_k_not_equality)","explanation":"Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (3, Yes, Yes, Yes, chi_at_most_k_not_equality)."},"skills":["A Supplied k-Palette Is a Candidate Upper Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0029","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"253a1fa124f85d1409d3defa23f549209bc57b04159d4c738f1257c230e5f6b3","family_id":"qf-mod37-chromatic-upper-bound","seed":202611061029,"semantic_template":"qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,2,1,0,1],\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"},"fields":{"title":"A Supplied k-Palette Is a Candidate Upper Certificate","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=4. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).","answer":{"kind":"expression","value":"(4, Yes, Yes, Yes, chi_at_most_k_not_equality)","explanation":"Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (4, Yes, Yes, Yes, chi_at_most_k_not_equality)."},"skills":["A Supplied k-Palette Is a Candidate Upper Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0030","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"acfc6f261b29e9339aa662eb9a482bfd1332e715da7a1a4f78ecdff482cb2e6a","family_id":"qf-mod37-chromatic-upper-bound","seed":202611061030,"semantic_template":"qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,0,2,0,1],\"edges\":[[0,1],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"},"fields":{"title":"A Supplied k-Palette Is a Candidate Upper Certificate","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [1, 3], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 2, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=4. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).","answer":{"kind":"expression","value":"(4, Yes, Yes, Yes, chi_at_most_k_not_equality)","explanation":"Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (4, Yes, Yes, Yes, chi_at_most_k_not_equality)."},"skills":["A Supplied k-Palette Is a Candidate Upper Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0031","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af9dc27f1485cbd8615d2f79117cb2e72edfe89ab7f6bfc2536142c1a4302826","family_id":"qf-mod37-chromatic-upper-bound","seed":202611061031,"semantic_template":"qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,2,0,1,2],\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"},"fields":{"title":"A Supplied k-Palette Is a Candidate Upper Certificate","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 3], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 0, 1, 2] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=4. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).","answer":{"kind":"expression","value":"(4, Yes, Yes, Yes, chi_at_most_k_not_equality)","explanation":"Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (4, Yes, Yes, Yes, chi_at_most_k_not_equality)."},"skills":["A Supplied k-Palette Is a Candidate Upper Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0032","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dfecbcbde1a7bf008fc88213240350638abb450a36c28acbed91fe9c098f8312","family_id":"qf-mod37-chromatic-upper-bound","seed":202611061032,"semantic_template":"qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,0,2,0,1],\"edges\":[[0,1],[0,3],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"},"fields":{"title":"A Supplied k-Palette Is a Candidate Upper Certificate","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [0, 5], [1, 2], [1, 3], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 2, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=4. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).","answer":{"kind":"expression","value":"(4, Yes, Yes, Yes, chi_at_most_k_not_equality)","explanation":"Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (4, Yes, Yes, Yes, chi_at_most_k_not_equality)."},"skills":["A Supplied k-Palette Is a Candidate Upper Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0033","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc56ae1d65a20d96f372ca1c3f525420fa3227a0394658277f2f18e80475489b","family_id":"qf-mod37-chromatic-upper-bound","seed":202611061033,"semantic_template":"qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,2,3,0,1],\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],\"k\":5,\"n\":6}"},"fields":{"title":"A Supplied k-Palette Is a Candidate Upper Certificate","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 3], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 3, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=5. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).","answer":{"kind":"expression","value":"(5, Yes, Yes, Yes, chi_at_most_k_not_equality)","explanation":"Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (5, Yes, Yes, Yes, chi_at_most_k_not_equality)."},"skills":["A Supplied k-Palette Is a Candidate Upper Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0034","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f83f5cba9e6efb3d7b9363463a04bad229f3b1e967e2355fd32f36e48882e94","family_id":"qf-mod37-chromatic-upper-bound","seed":202611061034,"semantic_template":"qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],\"k\":3,\"n\":6}"},"fields":{"title":"A Supplied k-Palette Is a Candidate Upper Certificate","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=3. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).","answer":{"kind":"expression","value":"(3, Yes, Yes, Yes, chi_at_most_k_not_equality)","explanation":"Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (3, Yes, Yes, Yes, chi_at_most_k_not_equality)."},"skills":["A Supplied k-Palette Is a Candidate Upper Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0035","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"343c849918c2fc651bb15e13a2c10ec3b93c6b59f3396ad445546d5431e625de","family_id":"qf-mod37-chromatic-upper-bound","seed":202611061035,"semantic_template":"qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,2,0,2,1],\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"},"fields":{"title":"A Supplied k-Palette Is a Candidate Upper Certificate","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 0, 2, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=4. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).","answer":{"kind":"expression","value":"(4, Yes, Yes, Yes, chi_at_most_k_not_equality)","explanation":"Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (4, Yes, Yes, Yes, chi_at_most_k_not_equality)."},"skills":["A Supplied k-Palette Is a Candidate Upper Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0036","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"84cc9adf1d1db14f3a0c503248ce5d8092661323a4cdfc581c229bb1fbc71de0","family_id":"qf-mod37-chromatic-upper-bound","seed":202611061036,"semantic_template":"qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[0,3],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],\"k\":3,\"n\":6}"},"fields":{"title":"A Supplied k-Palette Is a Candidate Upper Certificate","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [0, 5], [1, 2], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=3. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).","answer":{"kind":"expression","value":"(3, Yes, Yes, Yes, chi_at_most_k_not_equality)","explanation":"Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (3, Yes, Yes, Yes, chi_at_most_k_not_equality)."},"skills":["A Supplied k-Palette Is a Candidate Upper Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0037","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c5dbb8e12077b1c18e1b3ec5088193767305cf8052f317a410b63ef0ee53646","family_id":"qf-mod37-chromatic-upper-bound","seed":202611061037,"semantic_template":"qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,2,1,0,1],\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"},"fields":{"title":"A Supplied k-Palette Is a Candidate Upper Certificate","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=4. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).","answer":{"kind":"expression","value":"(4, Yes, Yes, Yes, chi_at_most_k_not_equality)","explanation":"Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (4, Yes, Yes, Yes, chi_at_most_k_not_equality)."},"skills":["A Supplied k-Palette Is a Candidate Upper Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0038","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"464b1c4e76e1263f8a5ff84697d26e863d3d445a2548e4f8271de8a55e894556","family_id":"qf-mod37-chromatic-upper-bound","seed":202611061038,"semantic_template":"qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,0,2,0,1],\"edges\":[[0,1],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"},"fields":{"title":"A Supplied k-Palette Is a Candidate Upper Certificate","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 2, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=4. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).","answer":{"kind":"expression","value":"(4, Yes, Yes, Yes, chi_at_most_k_not_equality)","explanation":"Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (4, Yes, Yes, Yes, chi_at_most_k_not_equality)."},"skills":["A Supplied k-Palette Is a Candidate Upper Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0039","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9712c53b3325e2b2268a4e721c04138134d57f3668ba1d9b16d0b57553b46dbf","family_id":"qf-mod37-chromatic-upper-bound","seed":202611061039,"semantic_template":"qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,2,0,2,1],\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"},"fields":{"title":"A Supplied k-Palette Is a Candidate Upper Certificate","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 0, 2, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=4. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).","answer":{"kind":"expression","value":"(4, Yes, Yes, Yes, chi_at_most_k_not_equality)","explanation":"Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (4, Yes, Yes, Yes, chi_at_most_k_not_equality)."},"skills":["A Supplied k-Palette Is a Candidate Upper Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0040","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45e730ca06212f19d67ddfbc6806a45ddf2fe122c91db793f20947dbdf5f10cb","family_id":"qf-mod37-chromatic-upper-bound","seed":202611061040,"semantic_template":"qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,0,2,0,1],\"edges\":[[0,1],[0,3],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"},"fields":{"title":"A Supplied k-Palette Is a Candidate Upper Certificate","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [0, 5], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 2, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=4. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).","answer":{"kind":"expression","value":"(4, Yes, Yes, Yes, chi_at_most_k_not_equality)","explanation":"Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (4, Yes, Yes, Yes, chi_at_most_k_not_equality)."},"skills":["A Supplied k-Palette Is a Candidate Upper Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0041","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9de1c9fcd352d91e3958c45fa01647700b4c5a9bed730f99f62b2bcd266f42ea","family_id":"qf-mod37-chromatic-upper-bound","seed":202611061041,"semantic_template":"qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,2,3,0,1],\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],\"k\":5,\"n\":6}"},"fields":{"title":"A Supplied k-Palette Is a Candidate Upper Certificate","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 3, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=5. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).","answer":{"kind":"expression","value":"(5, Yes, Yes, Yes, chi_at_most_k_not_equality)","explanation":"Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (5, Yes, Yes, Yes, chi_at_most_k_not_equality)."},"skills":["A Supplied k-Palette Is a Candidate Upper Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0042","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"676d58a6aa70635599cfd719622d1a71d6df895413258a4458be3c2a947e6e99","family_id":"qf-mod37-chromatic-upper-bound","seed":202611061042,"semantic_template":"qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],\"k\":3,\"n\":6}"},"fields":{"title":"A Supplied k-Palette Is a Candidate Upper Certificate","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=3. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).","answer":{"kind":"expression","value":"(3, Yes, Yes, Yes, chi_at_most_k_not_equality)","explanation":"Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (3, Yes, Yes, Yes, chi_at_most_k_not_equality)."},"skills":["A Supplied k-Palette Is a Candidate Upper Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0043","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a4458bd81fbb061410e89d7a519322a3f80ad0ef632d99ec7ee2a86446fde64c","family_id":"qf-mod37-chromatic-upper-bound","seed":202611061043,"semantic_template":"qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,2,0,2,1],\"edges\":[[0,1],[0,2],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],\"k\":4,\"n\":6}"},"fields":{"title":"A Supplied k-Palette Is a Candidate Upper Certificate","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 0, 2, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=4. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).","answer":{"kind":"expression","value":"(4, Yes, Yes, Yes, chi_at_most_k_not_equality)","explanation":"Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (4, Yes, Yes, Yes, chi_at_most_k_not_equality)."},"skills":["A Supplied k-Palette Is a Candidate Upper Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0044","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a49132589f1d97fb81f51115a54cc678c999b0915a191648478227f8a591189","family_id":"qf-mod37-chromatic-upper-bound","seed":202611061044,"semantic_template":"qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[0,3],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],\"k\":3,\"n\":6}"},"fields":{"title":"A Supplied k-Palette Is a Candidate Upper Certificate","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [0, 5], [1, 2], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=3. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).","answer":{"kind":"expression","value":"(3, Yes, Yes, Yes, chi_at_most_k_not_equality)","explanation":"Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (3, Yes, Yes, Yes, chi_at_most_k_not_equality)."},"skills":["A Supplied k-Palette Is a Candidate Upper Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0045","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da0f6966e4f506a851334395fd97b56bb432f1c1490e8447d07d3221afdf41e8","family_id":"qf-mod37-chromatic-upper-bound","seed":202611061045,"semantic_template":"qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,2,1,0,1],\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],\"k\":4,\"n\":6}"},"fields":{"title":"A Supplied k-Palette Is a Candidate Upper Certificate","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=4. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).","answer":{"kind":"expression","value":"(4, Yes, Yes, Yes, chi_at_most_k_not_equality)","explanation":"Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (4, Yes, Yes, Yes, chi_at_most_k_not_equality)."},"skills":["A Supplied k-Palette Is a Candidate Upper Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0046","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eeaf664e1a2e88aa46ba2acd13376dcb83f00b703712cea68bdcfb7c4340f795","family_id":"qf-mod37-chromatic-upper-bound","seed":202611061046,"semantic_template":"qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,0,2,0,1],\"edges\":[[0,1],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],\"k\":4,\"n\":6}"},"fields":{"title":"A Supplied k-Palette Is a Candidate Upper Certificate","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [1, 3], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 2, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=4. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).","answer":{"kind":"expression","value":"(4, Yes, Yes, Yes, chi_at_most_k_not_equality)","explanation":"Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (4, Yes, Yes, Yes, chi_at_most_k_not_equality)."},"skills":["A Supplied k-Palette Is a Candidate Upper Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0047","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1218d01f67d8d273cc612612956fde92eabb20febb4b689657e16bb0955eb750","family_id":"qf-mod37-chromatic-upper-bound","seed":202611061047,"semantic_template":"qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,2,0,2,1],\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],\"k\":4,\"n\":6}"},"fields":{"title":"A Supplied k-Palette Is a Candidate Upper Certificate","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 3], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 0, 2, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=4. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).","answer":{"kind":"expression","value":"(4, Yes, Yes, Yes, chi_at_most_k_not_equality)","explanation":"Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (4, Yes, Yes, Yes, chi_at_most_k_not_equality)."},"skills":["A Supplied k-Palette Is a Candidate Upper Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0048","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"acaa68dd3ddc10ceeacb099b8565b902dcd92a84f8227ef2c5725e2bd7fd2cfb","family_id":"qf-mod37-chromatic-upper-bound","seed":202611061048,"semantic_template":"qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,0,2,0,1],\"edges\":[[0,1],[0,3],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],\"k\":4,\"n\":6}"},"fields":{"title":"A Supplied k-Palette Is a Candidate Upper Certificate","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [0, 5], [1, 2], [1, 3], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 2, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=4. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).","answer":{"kind":"expression","value":"(4, Yes, Yes, Yes, chi_at_most_k_not_equality)","explanation":"Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (4, Yes, Yes, Yes, chi_at_most_k_not_equality)."},"skills":["A Supplied k-Palette Is a Candidate Upper Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0049","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1e4ce0906d4104e645e6074ed7e8500dab1370b851191afa8cbd0ce01988bd7","family_id":"qf-mod37-chromatic-upper-bound","seed":202611061049,"semantic_template":"qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,2,3,0,1],\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],\"k\":5,\"n\":6}"},"fields":{"title":"A Supplied k-Palette Is a Candidate Upper Certificate","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 3], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 3, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=5. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).","answer":{"kind":"expression","value":"(5, Yes, Yes, Yes, chi_at_most_k_not_equality)","explanation":"Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (5, Yes, Yes, Yes, chi_at_most_k_not_equality)."},"skills":["A Supplied k-Palette Is a Candidate Upper Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0050","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4f56efb8507f956b5ab7ddd646519c85ba68ad6bc2936064a82393a576bd5d88","family_id":"qf-mod37-chromatic-upper-bound","seed":202611061050,"semantic_template":"qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[0,5],[1,2],[1,4],[2,3],[2,5],[3,4],[4,5]],\"k\":3,\"n\":6}"},"fields":{"title":"A Supplied k-Palette Is a Candidate Upper Certificate","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [1, 4], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=3. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).","answer":{"kind":"expression","value":"(3, Yes, Yes, Yes, chi_at_most_k_not_equality)","explanation":"Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (3, Yes, Yes, Yes, chi_at_most_k_not_equality)."},"skills":["A Supplied k-Palette Is a Candidate Upper Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0051","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34472d8819547e570bc70b428734e8099d1b78f5d0f3dbc17eba1fa39d67cf83","family_id":"qf-mod37-chromatic-exact-number","seed":202611061051,"semantic_template":"qf-mod37-chromatic-exact-number/{\"edges\":[[0,1],[0,5],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Exact Chromatic Number Needs Both Lower and Upper Evidence","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate).","answer":{"kind":"expression","value":"(2, [0, 1, 0, 1, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)","explanation":"Search increasing k completely, retaining the first lexicographic vertex-order coloring found with colors tried increasingly. A lower palette is exhausted and the displayed proper coloring supplies the upper certificate. The empty carrier has chromatic number0; a nonempty edgeless graph has number1. The exact result, retaining the requested order, is (2, [0, 1, 0, 1, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)."},"skills":["Exact Chromatic Number Needs Both Lower and Upper Evidence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0052","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb03d861c1462eb972d064e1e0e884e560a71b0dae83d3cf0c6f437a11cde809","family_id":"qf-mod37-chromatic-exact-number","seed":202611061052,"semantic_template":"qf-mod37-chromatic-exact-number/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Exact Chromatic Number Needs Both Lower and Upper Evidence","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate).","answer":{"kind":"expression","value":"(3, [0, 1, 2, 0, 1, 2], Yes, exhaustive_lower_and_constructive_upper_certificates)","explanation":"Search increasing k completely, retaining the first lexicographic vertex-order coloring found with colors tried increasingly. A lower palette is exhausted and the displayed proper coloring supplies the upper certificate. The empty carrier has chromatic number0; a nonempty edgeless graph has number1. The exact result, retaining the requested order, is (3, [0, 1, 2, 0, 1, 2], Yes, exhaustive_lower_and_constructive_upper_certificates)."},"skills":["Exact Chromatic Number Needs Both Lower and Upper Evidence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0053","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8991050796fd8ceae688e24c72c8548a46a9297ce2f9d42bbcdba869bd1fefca","family_id":"qf-mod37-chromatic-exact-number","seed":202611061053,"semantic_template":"qf-mod37-chromatic-exact-number/{\"edges\":[[0,1],[0,3],[0,5],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Exact Chromatic Number Needs Both Lower and Upper Evidence","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [0, 5], [1, 2], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate).","answer":{"kind":"expression","value":"(2, [0, 1, 0, 1, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)","explanation":"Search increasing k completely, retaining the first lexicographic vertex-order coloring found with colors tried increasingly. A lower palette is exhausted and the displayed proper coloring supplies the upper certificate. The empty carrier has chromatic number0; a nonempty edgeless graph has number1. The exact result, retaining the requested order, is (2, [0, 1, 0, 1, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)."},"skills":["Exact Chromatic Number Needs Both Lower and Upper Evidence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0054","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"03e657db273cd576a38d50b42e3b1ff602e5241d9206f06f1437b2f1494adbf2","family_id":"qf-mod37-chromatic-exact-number","seed":202611061054,"semantic_template":"qf-mod37-chromatic-exact-number/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Exact Chromatic Number Needs Both Lower and Upper Evidence","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate).","answer":{"kind":"expression","value":"(3, [0, 1, 2, 1, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)","explanation":"Search increasing k completely, retaining the first lexicographic vertex-order coloring found with colors tried increasingly. A lower palette is exhausted and the displayed proper coloring supplies the upper certificate. The empty carrier has chromatic number0; a nonempty edgeless graph has number1. The exact result, retaining the requested order, is (3, [0, 1, 2, 1, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)."},"skills":["Exact Chromatic Number Needs Both Lower and Upper Evidence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0055","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5294a3f3b8db86a2c78662a163eafd7c618fa7b1cdff48ee1622936363c28670","family_id":"qf-mod37-chromatic-exact-number","seed":202611061055,"semantic_template":"qf-mod37-chromatic-exact-number/{\"edges\":[[0,1],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Exact Chromatic Number Needs Both Lower and Upper Evidence","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [1, 3], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate).","answer":{"kind":"expression","value":"(3, [0, 1, 0, 2, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)","explanation":"Search increasing k completely, retaining the first lexicographic vertex-order coloring found with colors tried increasingly. A lower palette is exhausted and the displayed proper coloring supplies the upper certificate. The empty carrier has chromatic number0; a nonempty edgeless graph has number1. The exact result, retaining the requested order, is (3, [0, 1, 0, 2, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)."},"skills":["Exact Chromatic Number Needs Both Lower and Upper Evidence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0056","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c1a639716df76b38fd711f2885bd8dae0eb1ee49a830587f203c7f6fe99bd505","family_id":"qf-mod37-chromatic-exact-number","seed":202611061056,"semantic_template":"qf-mod37-chromatic-exact-number/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Exact Chromatic Number Needs Both Lower and Upper Evidence","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 3], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate).","answer":{"kind":"expression","value":"(3, [0, 1, 2, 0, 1, 2], Yes, exhaustive_lower_and_constructive_upper_certificates)","explanation":"Search increasing k completely, retaining the first lexicographic vertex-order coloring found with colors tried increasingly. A lower palette is exhausted and the displayed proper coloring supplies the upper certificate. The empty carrier has chromatic number0; a nonempty edgeless graph has number1. The exact result, retaining the requested order, is (3, [0, 1, 2, 0, 1, 2], Yes, exhaustive_lower_and_constructive_upper_certificates)."},"skills":["Exact Chromatic Number Needs Both Lower and Upper Evidence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0057","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b60d9384ce11366545185a8c29821eb42635012e154123e28f6cb5dfabb09a23","family_id":"qf-mod37-chromatic-exact-number","seed":202611061057,"semantic_template":"qf-mod37-chromatic-exact-number/{\"edges\":[[0,1],[0,3],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Exact Chromatic Number Needs Both Lower and Upper Evidence","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [0, 5], [1, 2], [1, 3], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate).","answer":{"kind":"expression","value":"(3, [0, 1, 0, 2, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)","explanation":"Search increasing k completely, retaining the first lexicographic vertex-order coloring found with colors tried increasingly. A lower palette is exhausted and the displayed proper coloring supplies the upper certificate. The empty carrier has chromatic number0; a nonempty edgeless graph has number1. The exact result, retaining the requested order, is (3, [0, 1, 0, 2, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)."},"skills":["Exact Chromatic Number Needs Both Lower and Upper Evidence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0058","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"04c04d97f679669487f51b9bf96ef2c35390d78672cd6b78b08e0a21416f99dc","family_id":"qf-mod37-chromatic-exact-number","seed":202611061058,"semantic_template":"qf-mod37-chromatic-exact-number/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Exact Chromatic Number Needs Both Lower and Upper Evidence","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 3], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate).","answer":{"kind":"expression","value":"(4, [0, 1, 2, 3, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)","explanation":"Search increasing k completely, retaining the first lexicographic vertex-order coloring found with colors tried increasingly. A lower palette is exhausted and the displayed proper coloring supplies the upper certificate. The empty carrier has chromatic number0; a nonempty edgeless graph has number1. The exact result, retaining the requested order, is (4, [0, 1, 2, 3, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)."},"skills":["Exact Chromatic Number Needs Both Lower and Upper Evidence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0059","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f15b9e27b4e28ed83ea56928033ef017815c6a6b19eab27082f357a1c01aad3a","family_id":"qf-mod37-chromatic-exact-number","seed":202611061059,"semantic_template":"qf-mod37-chromatic-exact-number/{\"edges\":[[0,1],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Exact Chromatic Number Needs Both Lower and Upper Evidence","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate).","answer":{"kind":"expression","value":"(2, [0, 1, 0, 1, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)","explanation":"Search increasing k completely, retaining the first lexicographic vertex-order coloring found with colors tried increasingly. A lower palette is exhausted and the displayed proper coloring supplies the upper certificate. The empty carrier has chromatic number0; a nonempty edgeless graph has number1. The exact result, retaining the requested order, is (2, [0, 1, 0, 1, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)."},"skills":["Exact Chromatic Number Needs Both Lower and Upper Evidence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0060","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3874e5b136982277a3f569020057ece3df5d2abbc6f10493353e84db6288dfb6","family_id":"qf-mod37-chromatic-exact-number","seed":202611061060,"semantic_template":"qf-mod37-chromatic-exact-number/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Exact Chromatic Number Needs Both Lower and Upper Evidence","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate).","answer":{"kind":"expression","value":"(3, [0, 1, 2, 0, 2, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)","explanation":"Search increasing k completely, retaining the first lexicographic vertex-order coloring found with colors tried increasingly. A lower palette is exhausted and the displayed proper coloring supplies the upper certificate. The empty carrier has chromatic number0; a nonempty edgeless graph has number1. The exact result, retaining the requested order, is (3, [0, 1, 2, 0, 2, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)."},"skills":["Exact Chromatic Number Needs Both Lower and Upper Evidence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0061","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"948d1d09e49a946de1f1e78cd42494f1426e4328bb18d088187174d1a04ecb50","family_id":"qf-mod37-chromatic-exact-number","seed":202611061061,"semantic_template":"qf-mod37-chromatic-exact-number/{\"edges\":[[0,1],[0,3],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Exact Chromatic Number Needs Both Lower and Upper Evidence","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [0, 5], [1, 2], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate).","answer":{"kind":"expression","value":"(2, [0, 1, 0, 1, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)","explanation":"Search increasing k completely, retaining the first lexicographic vertex-order coloring found with colors tried increasingly. A lower palette is exhausted and the displayed proper coloring supplies the upper certificate. The empty carrier has chromatic number0; a nonempty edgeless graph has number1. The exact result, retaining the requested order, is (2, [0, 1, 0, 1, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)."},"skills":["Exact Chromatic Number Needs Both Lower and Upper Evidence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0062","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c891865f2bf52e67b6e24b6cf296228db0bcc563c56a633e6cf04aaa7dab48d5","family_id":"qf-mod37-chromatic-exact-number","seed":202611061062,"semantic_template":"qf-mod37-chromatic-exact-number/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Exact Chromatic Number Needs Both Lower and Upper Evidence","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate).","answer":{"kind":"expression","value":"(3, [0, 1, 2, 1, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)","explanation":"Search increasing k completely, retaining the first lexicographic vertex-order coloring found with colors tried increasingly. A lower palette is exhausted and the displayed proper coloring supplies the upper certificate. The empty carrier has chromatic number0; a nonempty edgeless graph has number1. The exact result, retaining the requested order, is (3, [0, 1, 2, 1, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)."},"skills":["Exact Chromatic Number Needs Both Lower and Upper Evidence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0063","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2fd591e67c91c0faa2dbb8f861b05aad53f022e02e013bffcfeb4cf333e562b5","family_id":"qf-mod37-chromatic-exact-number","seed":202611061063,"semantic_template":"qf-mod37-chromatic-exact-number/{\"edges\":[[0,1],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Exact Chromatic Number Needs Both Lower and Upper Evidence","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate).","answer":{"kind":"expression","value":"(3, [0, 1, 0, 2, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)","explanation":"Search increasing k completely, retaining the first lexicographic vertex-order coloring found with colors tried increasingly. A lower palette is exhausted and the displayed proper coloring supplies the upper certificate. The empty carrier has chromatic number0; a nonempty edgeless graph has number1. The exact result, retaining the requested order, is (3, [0, 1, 0, 2, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)."},"skills":["Exact Chromatic Number Needs Both Lower and Upper Evidence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0064","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b49d5d52de343e9bc138fbf4e07682d0f1bfc3df906df6b585499b3216d533e6","family_id":"qf-mod37-chromatic-exact-number","seed":202611061064,"semantic_template":"qf-mod37-chromatic-exact-number/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Exact Chromatic Number Needs Both Lower and Upper Evidence","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate).","answer":{"kind":"expression","value":"(3, [0, 1, 2, 0, 2, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)","explanation":"Search increasing k completely, retaining the first lexicographic vertex-order coloring found with colors tried increasingly. A lower palette is exhausted and the displayed proper coloring supplies the upper certificate. The empty carrier has chromatic number0; a nonempty edgeless graph has number1. The exact result, retaining the requested order, is (3, [0, 1, 2, 0, 2, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)."},"skills":["Exact Chromatic Number Needs Both Lower and Upper Evidence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0065","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dba6b29def8f54546bc8188c052c450ff5d9b88fb7e68923e83065d385914173","family_id":"qf-mod37-chromatic-exact-number","seed":202611061065,"semantic_template":"qf-mod37-chromatic-exact-number/{\"edges\":[[0,1],[0,3],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Exact Chromatic Number Needs Both Lower and Upper Evidence","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [0, 5], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate).","answer":{"kind":"expression","value":"(3, [0, 1, 0, 2, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)","explanation":"Search increasing k completely, retaining the first lexicographic vertex-order coloring found with colors tried increasingly. A lower palette is exhausted and the displayed proper coloring supplies the upper certificate. The empty carrier has chromatic number0; a nonempty edgeless graph has number1. The exact result, retaining the requested order, is (3, [0, 1, 0, 2, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)."},"skills":["Exact Chromatic Number Needs Both Lower and Upper Evidence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0066","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e82f462a319b9c1ece13e50268f809cf71318b20bac8566895f173887797b3c7","family_id":"qf-mod37-chromatic-exact-number","seed":202611061066,"semantic_template":"qf-mod37-chromatic-exact-number/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Exact Chromatic Number Needs Both Lower and Upper Evidence","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate).","answer":{"kind":"expression","value":"(4, [0, 1, 2, 3, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)","explanation":"Search increasing k completely, retaining the first lexicographic vertex-order coloring found with colors tried increasingly. A lower palette is exhausted and the displayed proper coloring supplies the upper certificate. The empty carrier has chromatic number0; a nonempty edgeless graph has number1. The exact result, retaining the requested order, is (4, [0, 1, 2, 3, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)."},"skills":["Exact Chromatic Number Needs Both Lower and Upper Evidence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0067","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"afb2978ad5b15c8cf15e3f0be0b5c117b4f0536ad2b9579ccab05fb69f99f416","family_id":"qf-mod37-chromatic-exact-number","seed":202611061067,"semantic_template":"qf-mod37-chromatic-exact-number/{\"edges\":[[0,1],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Exact Chromatic Number Needs Both Lower and Upper Evidence","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate).","answer":{"kind":"expression","value":"(2, [0, 1, 0, 1, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)","explanation":"Search increasing k completely, retaining the first lexicographic vertex-order coloring found with colors tried increasingly. A lower palette is exhausted and the displayed proper coloring supplies the upper certificate. The empty carrier has chromatic number0; a nonempty edgeless graph has number1. The exact result, retaining the requested order, is (2, [0, 1, 0, 1, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)."},"skills":["Exact Chromatic Number Needs Both Lower and Upper Evidence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0068","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"493f81ad51757664e5acdd066dfcd813eb137eac3e873fd7d80886f84c2d936c","family_id":"qf-mod37-chromatic-exact-number","seed":202611061068,"semantic_template":"qf-mod37-chromatic-exact-number/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Exact Chromatic Number Needs Both Lower and Upper Evidence","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate).","answer":{"kind":"expression","value":"(3, [0, 1, 2, 0, 2, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)","explanation":"Search increasing k completely, retaining the first lexicographic vertex-order coloring found with colors tried increasingly. A lower palette is exhausted and the displayed proper coloring supplies the upper certificate. The empty carrier has chromatic number0; a nonempty edgeless graph has number1. The exact result, retaining the requested order, is (3, [0, 1, 2, 0, 2, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)."},"skills":["Exact Chromatic Number Needs Both Lower and Upper Evidence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0069","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af9da6d36a4ff053ff8f64d41ef8afc5767726edcf4d657a756929d08da31e82","family_id":"qf-mod37-chromatic-exact-number","seed":202611061069,"semantic_template":"qf-mod37-chromatic-exact-number/{\"edges\":[[0,1],[0,3],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Exact Chromatic Number Needs Both Lower and Upper Evidence","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [0, 5], [1, 2], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate).","answer":{"kind":"expression","value":"(2, [0, 1, 0, 1, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)","explanation":"Search increasing k completely, retaining the first lexicographic vertex-order coloring found with colors tried increasingly. A lower palette is exhausted and the displayed proper coloring supplies the upper certificate. The empty carrier has chromatic number0; a nonempty edgeless graph has number1. The exact result, retaining the requested order, is (2, [0, 1, 0, 1, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)."},"skills":["Exact Chromatic Number Needs Both Lower and Upper Evidence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0070","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef6c3ff7b41691de4f3aade9dadf475cf5b4a5b1bc2589f7c78a840cb84df0d3","family_id":"qf-mod37-chromatic-exact-number","seed":202611061070,"semantic_template":"qf-mod37-chromatic-exact-number/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Exact Chromatic Number Needs Both Lower and Upper Evidence","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate).","answer":{"kind":"expression","value":"(3, [0, 1, 2, 1, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)","explanation":"Search increasing k completely, retaining the first lexicographic vertex-order coloring found with colors tried increasingly. A lower palette is exhausted and the displayed proper coloring supplies the upper certificate. The empty carrier has chromatic number0; a nonempty edgeless graph has number1. The exact result, retaining the requested order, is (3, [0, 1, 2, 1, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)."},"skills":["Exact Chromatic Number Needs Both Lower and Upper Evidence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0071","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"29d66a8b897fac4acb078a6f641f425ce26622b9eb8d7238c60e4048a59fb63b","family_id":"qf-mod37-chromatic-exact-number","seed":202611061071,"semantic_template":"qf-mod37-chromatic-exact-number/{\"edges\":[[0,1],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Exact Chromatic Number Needs Both Lower and Upper Evidence","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [1, 3], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate).","answer":{"kind":"expression","value":"(3, [0, 1, 0, 2, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)","explanation":"Search increasing k completely, retaining the first lexicographic vertex-order coloring found with colors tried increasingly. A lower palette is exhausted and the displayed proper coloring supplies the upper certificate. The empty carrier has chromatic number0; a nonempty edgeless graph has number1. The exact result, retaining the requested order, is (3, [0, 1, 0, 2, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)."},"skills":["Exact Chromatic Number Needs Both Lower and Upper Evidence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0072","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92b364deff488bc060222a639c6d7e722ee929444b7296aa5c62d29dd35c773e","family_id":"qf-mod37-chromatic-exact-number","seed":202611061072,"semantic_template":"qf-mod37-chromatic-exact-number/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Exact Chromatic Number Needs Both Lower and Upper Evidence","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 3], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate).","answer":{"kind":"expression","value":"(3, [0, 1, 2, 0, 2, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)","explanation":"Search increasing k completely, retaining the first lexicographic vertex-order coloring found with colors tried increasingly. A lower palette is exhausted and the displayed proper coloring supplies the upper certificate. The empty carrier has chromatic number0; a nonempty edgeless graph has number1. The exact result, retaining the requested order, is (3, [0, 1, 2, 0, 2, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)."},"skills":["Exact Chromatic Number Needs Both Lower and Upper Evidence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0073","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a0b0af2064d5dff1156db2dee383459ddad50d23874c61476fb9cb7dc55a8428","family_id":"qf-mod37-chromatic-exact-number","seed":202611061073,"semantic_template":"qf-mod37-chromatic-exact-number/{\"edges\":[[0,1],[0,3],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Exact Chromatic Number Needs Both Lower and Upper Evidence","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [0, 5], [1, 2], [1, 3], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate).","answer":{"kind":"expression","value":"(3, [0, 1, 0, 2, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)","explanation":"Search increasing k completely, retaining the first lexicographic vertex-order coloring found with colors tried increasingly. A lower palette is exhausted and the displayed proper coloring supplies the upper certificate. The empty carrier has chromatic number0; a nonempty edgeless graph has number1. The exact result, retaining the requested order, is (3, [0, 1, 0, 2, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)."},"skills":["Exact Chromatic Number Needs Both Lower and Upper Evidence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0074","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39211fbf9762cb28635601cec9738720f04548ea6c6546a41b4605c904b40780","family_id":"qf-mod37-chromatic-exact-number","seed":202611061074,"semantic_template":"qf-mod37-chromatic-exact-number/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Exact Chromatic Number Needs Both Lower and Upper Evidence","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 3], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate).","answer":{"kind":"expression","value":"(4, [0, 1, 2, 3, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)","explanation":"Search increasing k completely, retaining the first lexicographic vertex-order coloring found with colors tried increasingly. A lower palette is exhausted and the displayed proper coloring supplies the upper certificate. The empty carrier has chromatic number0; a nonempty edgeless graph has number1. The exact result, retaining the requested order, is (4, [0, 1, 2, 3, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)."},"skills":["Exact Chromatic Number Needs Both Lower and Upper Evidence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0075","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"03a2f07f801d39923bc621c2b48656416e25c4ae57466e757f844bb7be7c8141","family_id":"qf-mod37-chromatic-exact-number","seed":202611061075,"semantic_template":"qf-mod37-chromatic-exact-number/{\"edges\":[[0,1],[0,5],[1,2],[1,4],[2,3],[2,5],[3,4],[4,5]],\"n\":6}"},"fields":{"title":"Exact Chromatic Number Needs Both Lower and Upper Evidence","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [1, 4], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate).","answer":{"kind":"expression","value":"(2, [0, 1, 0, 1, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)","explanation":"Search increasing k completely, retaining the first lexicographic vertex-order coloring found with colors tried increasingly. A lower palette is exhausted and the displayed proper coloring supplies the upper certificate. The empty carrier has chromatic number0; a nonempty edgeless graph has number1. The exact result, retaining the requested order, is (2, [0, 1, 0, 1, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)."},"skills":["Exact Chromatic Number Needs Both Lower and Upper Evidence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0076","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"de4ff9713787f1462907a8ed52c5b116df000d80c040ccdc64170f36aa700e0f","family_id":"qf-mod37-chromatic-minimum-boundary","seed":202611061076,"semantic_template":"qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[0,5],[1,2],[2,3],[3,4],[4,5]],\"k\":3,\"n\":6}"},"fields":{"title":"Equality of k with the Minimum Is Not Validity of This Coloring","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Compare k=3. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).","answer":{"kind":"expression","value":"(3, Yes, 2, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)","explanation":"Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (3, Yes, 2, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)."},"skills":["Equality of k with the Minimum Is Not Validity of This Coloring"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0077","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e88aecacb5206244a7c7e2a3467292c822c5e85b850c9e9a6161153c7e6ad55","family_id":"qf-mod37-chromatic-minimum-boundary","seed":202611061077,"semantic_template":"qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,2,0,1,2],\"edges\":[[0,1],[0,2],[0,5],[1,2],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"},"fields":{"title":"Equality of k with the Minimum Is Not Validity of This Coloring","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 0, 1, 2] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Compare k=4. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).","answer":{"kind":"expression","value":"(4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)","explanation":"Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)."},"skills":["Equality of k with the Minimum Is Not Validity of This Coloring"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0078","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3563100c29e59d340edff0c8db01a6697f58957df873d3d2ccb82c1ffc083028","family_id":"qf-mod37-chromatic-minimum-boundary","seed":202611061078,"semantic_template":"qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[0,3],[0,5],[1,2],[2,3],[3,4],[4,5]],\"k\":3,\"n\":6}"},"fields":{"title":"Equality of k with the Minimum Is Not Validity of This Coloring","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [0, 5], [1, 2], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Compare k=3. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).","answer":{"kind":"expression","value":"(3, Yes, 2, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)","explanation":"Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (3, Yes, 2, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)."},"skills":["Equality of k with the Minimum Is Not Validity of This Coloring"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0079","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11cc7afef30958bc4cd41d22a37205b30823681115f508902e118c1005bb2aff","family_id":"qf-mod37-chromatic-minimum-boundary","seed":202611061079,"semantic_template":"qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,2,1,0,1],\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"},"fields":{"title":"Equality of k with the Minimum Is Not Validity of This Coloring","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Compare k=4. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).","answer":{"kind":"expression","value":"(4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)","explanation":"Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)."},"skills":["Equality of k with the Minimum Is Not Validity of This Coloring"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0080","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb86df3ac68c8749ed0545807e7ebd35afab9eccc184eb05891f8c3e8ded5139","family_id":"qf-mod37-chromatic-minimum-boundary","seed":202611061080,"semantic_template":"qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,0,2,0,1],\"edges\":[[0,1],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"},"fields":{"title":"Equality of k with the Minimum Is Not Validity of This Coloring","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [1, 3], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 2, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Compare k=4. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).","answer":{"kind":"expression","value":"(4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)","explanation":"Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)."},"skills":["Equality of k with the Minimum Is Not Validity of This Coloring"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0081","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd1af32d6d673bdb5449edc3ade6532a724091374c7f962c4e2e3e67a90abb07","family_id":"qf-mod37-chromatic-minimum-boundary","seed":202611061081,"semantic_template":"qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,2,0,1,2],\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"},"fields":{"title":"Equality of k with the Minimum Is Not Validity of This Coloring","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 3], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 0, 1, 2] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Compare k=4. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).","answer":{"kind":"expression","value":"(4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)","explanation":"Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)."},"skills":["Equality of k with the Minimum Is Not Validity of This Coloring"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0082","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9c86b705a2adc1e06ac14b82882e0362c8347f4bccfa8a2928024d04d43ff78","family_id":"qf-mod37-chromatic-minimum-boundary","seed":202611061082,"semantic_template":"qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,0,2,0,1],\"edges\":[[0,1],[0,3],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"},"fields":{"title":"Equality of k with the Minimum Is Not Validity of This Coloring","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [0, 5], [1, 2], [1, 3], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 2, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Compare k=4. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).","answer":{"kind":"expression","value":"(4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)","explanation":"Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)."},"skills":["Equality of k with the Minimum Is Not Validity of This Coloring"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0083","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f810a687af40a7580503177077ac3f033f6b3814bb69c18e9a365b4cb63e5fde","family_id":"qf-mod37-chromatic-minimum-boundary","seed":202611061083,"semantic_template":"qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,2,3,0,1],\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],\"k\":5,\"n\":6}"},"fields":{"title":"Equality of k with the Minimum Is Not Validity of This Coloring","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 3], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 3, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Compare k=5. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).","answer":{"kind":"expression","value":"(5, Yes, 4, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)","explanation":"Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (5, Yes, 4, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)."},"skills":["Equality of k with the Minimum Is Not Validity of This Coloring"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0084","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b8fbcc9036a5c84fb080604c4ba7dd38b2386b512db7a6013dc30c6d80b25b5","family_id":"qf-mod37-chromatic-minimum-boundary","seed":202611061084,"semantic_template":"qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],\"k\":3,\"n\":6}"},"fields":{"title":"Equality of k with the Minimum Is Not Validity of This Coloring","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Compare k=3. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).","answer":{"kind":"expression","value":"(3, Yes, 2, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)","explanation":"Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (3, Yes, 2, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)."},"skills":["Equality of k with the Minimum Is Not Validity of This Coloring"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0085","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"56ee72cbb8852f957949debb529c3f849d3686c3b92bac220941ab31c5070ec4","family_id":"qf-mod37-chromatic-minimum-boundary","seed":202611061085,"semantic_template":"qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,2,0,2,1],\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"},"fields":{"title":"Equality of k with the Minimum Is Not Validity of This Coloring","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 0, 2, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Compare k=4. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).","answer":{"kind":"expression","value":"(4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)","explanation":"Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)."},"skills":["Equality of k with the Minimum Is Not Validity of This Coloring"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0086","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"013727025ea89b3d75d3a17fd1303fbe75d4f124a0a4230754917da9000755b0","family_id":"qf-mod37-chromatic-minimum-boundary","seed":202611061086,"semantic_template":"qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[0,3],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],\"k\":3,\"n\":6}"},"fields":{"title":"Equality of k with the Minimum Is Not Validity of This Coloring","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [0, 5], [1, 2], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Compare k=3. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).","answer":{"kind":"expression","value":"(3, Yes, 2, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)","explanation":"Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (3, Yes, 2, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)."},"skills":["Equality of k with the Minimum Is Not Validity of This Coloring"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0087","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6eea04a2d114937523ca08b28e5656819ce9e63ceb5bcb75247e9064c9d030aa","family_id":"qf-mod37-chromatic-minimum-boundary","seed":202611061087,"semantic_template":"qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,2,1,0,1],\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"},"fields":{"title":"Equality of k with the Minimum Is Not Validity of This Coloring","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Compare k=4. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).","answer":{"kind":"expression","value":"(4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)","explanation":"Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)."},"skills":["Equality of k with the Minimum Is Not Validity of This Coloring"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0088","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"917587c4292717093a43ab64756253bc7139462774395d57591a7f6befdc79bc","family_id":"qf-mod37-chromatic-minimum-boundary","seed":202611061088,"semantic_template":"qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,0,2,0,1],\"edges\":[[0,1],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"},"fields":{"title":"Equality of k with the Minimum Is Not Validity of This Coloring","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 2, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Compare k=4. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).","answer":{"kind":"expression","value":"(4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)","explanation":"Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)."},"skills":["Equality of k with the Minimum Is Not Validity of This Coloring"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0089","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6d442287f572920efcb506669f2287f8a995545eeed23159c7684cbbb98ccdaa","family_id":"qf-mod37-chromatic-minimum-boundary","seed":202611061089,"semantic_template":"qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,2,0,2,1],\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"},"fields":{"title":"Equality of k with the Minimum Is Not Validity of This Coloring","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 0, 2, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Compare k=4. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).","answer":{"kind":"expression","value":"(4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)","explanation":"Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)."},"skills":["Equality of k with the Minimum Is Not Validity of This Coloring"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0090","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"93fca8880fac8dc1f98f9c9eb296e4eec1acb0f3f3e42047babc1806a2567609","family_id":"qf-mod37-chromatic-minimum-boundary","seed":202611061090,"semantic_template":"qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,0,2,0,1],\"edges\":[[0,1],[0,3],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"},"fields":{"title":"Equality of k with the Minimum Is Not Validity of This Coloring","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [0, 5], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 2, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Compare k=4. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).","answer":{"kind":"expression","value":"(4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)","explanation":"Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)."},"skills":["Equality of k with the Minimum Is Not Validity of This Coloring"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0091","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b753d0db1c1d07e354cbc08d45058400b254bfba082535200f3491ad434912a6","family_id":"qf-mod37-chromatic-minimum-boundary","seed":202611061091,"semantic_template":"qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,2,3,0,1],\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],\"k\":5,\"n\":6}"},"fields":{"title":"Equality of k with the Minimum Is Not Validity of This Coloring","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 3, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Compare k=5. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).","answer":{"kind":"expression","value":"(5, Yes, 4, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)","explanation":"Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (5, Yes, 4, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)."},"skills":["Equality of k with the Minimum Is Not Validity of This Coloring"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0092","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb559d59bf61f0af2a559bfab2072949b9f7a28449dc7fb808aacb10cca6540b","family_id":"qf-mod37-chromatic-minimum-boundary","seed":202611061092,"semantic_template":"qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],\"k\":3,\"n\":6}"},"fields":{"title":"Equality of k with the Minimum Is Not Validity of This Coloring","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Compare k=3. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).","answer":{"kind":"expression","value":"(3, Yes, 2, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)","explanation":"Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (3, Yes, 2, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)."},"skills":["Equality of k with the Minimum Is Not Validity of This Coloring"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0093","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c8003b65d8de78556bb8fb232287d7001484f9692d9deb70a53e0c0162843e0e","family_id":"qf-mod37-chromatic-minimum-boundary","seed":202611061093,"semantic_template":"qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,2,0,2,1],\"edges\":[[0,1],[0,2],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],\"k\":4,\"n\":6}"},"fields":{"title":"Equality of k with the Minimum Is Not Validity of This Coloring","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 0, 2, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Compare k=4. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).","answer":{"kind":"expression","value":"(4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)","explanation":"Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)."},"skills":["Equality of k with the Minimum Is Not Validity of This Coloring"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0094","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ba49401ad40e3476957e5114e5ce493bded0586e0e748297c9cb9d9ac144550","family_id":"qf-mod37-chromatic-minimum-boundary","seed":202611061094,"semantic_template":"qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[0,3],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],\"k\":3,\"n\":6}"},"fields":{"title":"Equality of k with the Minimum Is Not Validity of This Coloring","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [0, 5], [1, 2], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Compare k=3. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).","answer":{"kind":"expression","value":"(3, Yes, 2, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)","explanation":"Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (3, Yes, 2, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)."},"skills":["Equality of k with the Minimum Is Not Validity of This Coloring"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0095","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39cf408ac3ec06547f027c509511350e48df0f524e88b7be0a32f93189a8624d","family_id":"qf-mod37-chromatic-minimum-boundary","seed":202611061095,"semantic_template":"qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,2,1,0,1],\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],\"k\":4,\"n\":6}"},"fields":{"title":"Equality of k with the Minimum Is Not Validity of This Coloring","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Compare k=4. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).","answer":{"kind":"expression","value":"(4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)","explanation":"Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)."},"skills":["Equality of k with the Minimum Is Not Validity of This Coloring"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0096","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce1bb6a6700550b41a24beb99422d986796f4f9e8428d13e9578f75705785e39","family_id":"qf-mod37-chromatic-minimum-boundary","seed":202611061096,"semantic_template":"qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,0,2,0,1],\"edges\":[[0,1],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],\"k\":4,\"n\":6}"},"fields":{"title":"Equality of k with the Minimum Is Not Validity of This Coloring","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [1, 3], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 2, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Compare k=4. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).","answer":{"kind":"expression","value":"(4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)","explanation":"Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)."},"skills":["Equality of k with the Minimum Is Not Validity of This Coloring"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0097","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d46a3e2377feb1008b84165c38b5bd9b2e1978176ca44afe57be625ce125db6","family_id":"qf-mod37-chromatic-minimum-boundary","seed":202611061097,"semantic_template":"qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,2,0,2,1],\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],\"k\":4,\"n\":6}"},"fields":{"title":"Equality of k with the Minimum Is Not Validity of This Coloring","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 3], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 0, 2, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Compare k=4. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).","answer":{"kind":"expression","value":"(4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)","explanation":"Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)."},"skills":["Equality of k with the Minimum Is Not Validity of This Coloring"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0098","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"408f739acfe3462306f5bba41ee1302b4b19ab8b2bde1162d9fa6704e9f89876","family_id":"qf-mod37-chromatic-minimum-boundary","seed":202611061098,"semantic_template":"qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,0,2,0,1],\"edges\":[[0,1],[0,3],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],\"k\":4,\"n\":6}"},"fields":{"title":"Equality of k with the Minimum Is Not Validity of This Coloring","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [0, 5], [1, 2], [1, 3], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 2, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Compare k=4. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).","answer":{"kind":"expression","value":"(4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)","explanation":"Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)."},"skills":["Equality of k with the Minimum Is Not Validity of This Coloring"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0099","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0fe2ebfd0ff3076cc81ad580f5266ce3f942db69fed1d92ba65f63cf02bf1f20","family_id":"qf-mod37-chromatic-minimum-boundary","seed":202611061099,"semantic_template":"qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,2,3,0,1],\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],\"k\":5,\"n\":6}"},"fields":{"title":"Equality of k with the Minimum Is Not Validity of This Coloring","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 3], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 3, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Compare k=5. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).","answer":{"kind":"expression","value":"(5, Yes, 4, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)","explanation":"Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (5, Yes, 4, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)."},"skills":["Equality of k with the Minimum Is Not Validity of This Coloring"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r01-chromatic-number-0100","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1f961faa47c2ec7680acb966645e077035fd40706c9334044f7009518730ff17","family_id":"qf-mod37-chromatic-minimum-boundary","seed":202611061100,"semantic_template":"qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[0,5],[1,2],[1,4],[2,3],[2,5],[3,4],[4,5]],\"k\":3,\"n\":6}"},"fields":{"title":"Equality of k with the Minimum Is Not Validity of This Coloring","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [1, 4], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Compare k=3. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).","answer":{"kind":"expression","value":"(3, Yes, 2, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)","explanation":"Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (3, Yes, 2, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)."},"skills":["Equality of k with the Minimum Is Not Validity of This Coloring"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}