{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261104-r05-chinese-remainder-theorem-0087", "pilot_number": "R0035-18012", "number": 18012, "family_id": "qf-mod35-crt-noncoprime-boundary", "title": { "zh-Hant": "第 18012 題：中國剩餘定理", "en": "Question 18012: A Noncoprime System Can Be Compatible or Incompatible" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261104-r05-chinese-remainder-theorem-0087", "batch_id": "batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-chinese-remainder-theorem" ], "title_zh": "中國剩餘定理", "stem_zh": "正moduli、normalized residues與exact integer congruences。操作=crt-noncoprime-boundary；資料={\"moduli\":[26,39],\"residues\":[1,14]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "([26, 39], [1, 14], 是, 53, 78, noncoprime_not_automatically_inconsistent)", "explanation_zh": "Wave5第四層核對=([26, 39], [1, 14], 是, 53, 78, noncoprime_not_automatically_inconsistent)；依完整明示資料重算；pairwise-coprime時唯一只 modulo product；noncoprime不自動inconsistent，須gcd compatibility且period一般為lcm。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "中國剩餘定理" ] }, "variant": { "family_id": "qf-mod35-crt-noncoprime-boundary", "seed": 202611045087, "parameters": { "moduli": [ 26, 39 ], "residues": [ 1, 14 ], "domain": "Exact_integer_CRT_system", "operation": "crt-noncoprime-boundary", "semantic_template": "qf-mod35-crt-noncoprime-boundary/{\"moduli\":[26,39],\"residues\":[1,14]}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_or_symbolic_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-14", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261014-advance-20261104-r05-chinese-remainder-theorem-0087", "batch_id": "batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "cc776880b60766975bec3af041c9b0dc08ea0d786b74120e8b327a5fdc6bb38e", "family_id": "qf-mod35-crt-noncoprime-boundary", "seed": 202611045087, "semantic_template": "qf-mod35-crt-noncoprime-boundary/{\"moduli\":[26,39],\"residues\":[1,14]}" }, "fields": { "title": "A Noncoprime System Can Be Compatible or Incompatible", "stem": "Use the complete integer congruence system with positive distinct moduli=[26, 39] and aligned original residues=[1, 14] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).", "answer": { "kind": "expression", "value": "([26, 39], [1, 14], Yes, 53, 78, noncoprime_not_automatically_inconsistent)", "explanation": "Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([26, 39], [1, 14], Yes, 53, 78, noncoprime_not_automatically_inconsistent)." }, "skills": [ "A Noncoprime System Can Be Compatible or Incompatible" ], "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review." } }, "givens": [ { "id": "G001", "text": { "zh-Hant": "正moduli、normalized residues與exact integer congruences。操作=crt-noncoprime-boundary；資料={\"moduli\":[26,39],\"residues\":[1,14]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use the complete integer congruence system with positive distinct moduli=[26, 39] and aligned original residues=[1, 14] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source operation=crt-noncoprime-boundary;original snapshot and mathematics retained", "after": "explicit ordered source output fields=[moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary]", "relation": "implies", "rule_id": "source_ordered_interface_explicit_without_javascript", "conditions": [], "explanation": { "zh-Hant": "原題的歷史介面標記保留在快照；本頁直接列原始欄位順序，不需JavaScript或外部表。欄位描述不自動成為數學證據，No標記須依本題的聲稱範圍解讀。", "en": "Keep the historical interface marker in the original snapshot while giving ordered fields here,without JavaScript or an external table.Labels are not mathematical evidence,and each No marker retains the operation's claim scope." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "original residue slot0 exact quotient := div(1, 26)", "after": "original residue slot0 exact quotient = 1/26", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 original residue slot0 exact quotient，不以小數近似替代。", "en": "Recompute original residue slot0 exact quotient with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "26" ], "output": "1/26", "label": "original residue slot0 exact quotient" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "original residue slot0 floor quotient multiple := mul(0, 26)", "after": "original residue slot0 floor quotient multiple = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 original residue slot0 floor quotient multiple，不以小數近似替代。", "en": "Recompute original residue slot0 floor quotient multiple with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "26" ], "output": "0", "label": "original residue slot0 floor quotient multiple" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "original residue slot0 normalized remainder := sub(1, 0)", "after": "original residue slot0 normalized remainder = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 original residue slot0 normalized remainder，不以小數近似替代。", "en": "Recompute original residue slot0 normalized remainder with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "1", "0" ], "output": "1", "label": "original residue slot0 normalized remainder" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "original residue slot0:integer value=1;positive modulus=26;exact ratio=1/26;floor quotient=0", "after": "value=q*modulus+remainder;normalized residue=1 in[0,26)", "relation": "implies", "rule_id": "crt_exact_floor_modulo_signed_input", "conditions": [], "explanation": { "zh-Hant": "對正模數用向下整數商，負原值也得到0到modulus−1的代表；不是截斷向0或改變同餘類。模1是合法平凡約束，餘數為0。", "en": "Use the floor quotient for a positive modulus,including signed inputs,to obtain the representative from0 through modulus−1.This is not truncation toward zero or a change of congruence class.Modulus1 is a valid trivial constraint with residue0." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "original residue slot1 exact quotient := div(14, 39)", "after": "original residue slot1 exact quotient = 14/39", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 original residue slot1 exact quotient，不以小數近似替代。", "en": "Recompute original residue slot1 exact quotient with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "14", "39" ], "output": "14/39", "label": "original residue slot1 exact quotient" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "original residue slot1 floor quotient multiple := mul(0, 39)", "after": "original residue slot1 floor quotient multiple = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 original residue slot1 floor quotient multiple，不以小數近似替代。", "en": "Recompute original residue slot1 floor quotient multiple with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "39" ], "output": "0", "label": "original residue slot1 floor quotient multiple" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "original residue slot1 normalized remainder := sub(14, 0)", "after": "original residue slot1 normalized remainder = 14", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 original residue slot1 normalized remainder，不以小數近似替代。", "en": "Recompute original residue slot1 normalized remainder with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "14", "0" ], "output": "14", "label": "original residue slot1 normalized remainder" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "original residue slot1:integer value=14;positive modulus=39;exact ratio=14/39;floor quotient=0", "after": "value=q*modulus+remainder;normalized residue=14 in[0,39)", "relation": "implies", "rule_id": "crt_exact_floor_modulo_signed_input", "conditions": [], "explanation": { "zh-Hant": "對正模數用向下整數商，負原值也得到0到modulus−1的代表；不是截斷向0或改變同餘類。模1是合法平凡約束，餘數為0。", "en": "Use the floor quotient for a positive modulus,including signed inputs,to obtain the representative from0 through modulus−1.This is not truncation toward zero or a change of congruence class.Modulus1 is a valid trivial constraint with residue0." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "CRT modulus product prefix0 := mul(1, 26)", "after": "CRT modulus product prefix0 = 26", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 CRT modulus product prefix0，不以小數近似替代。", "en": "Recompute CRT modulus product prefix0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "26" ], "output": "26", "label": "CRT modulus product prefix0" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "LCM prefix0 gcd:positive original gcd inputs=[1, 26]", "after": "Euclidean rows[a,b,q,r] with a=q*b+r=[[1, 26, 0, 1], [26, 1, 26, 0]];positive gcd=1", "relation": "implies", "rule_id": "crt_pair_gcd_exact_euclidean_witness", "conditions": [], "explanation": { "zh-Hant": "每次餘數替換保同一公因數集，最後非零值才是gcd；逐列列出q與r供來源核對。", "en": "Each remainder replacement preserves common divisors;the last nonzero value is the gcd.List every quotient/remainder row for source-bound checking." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "CRT LCM prefix0 quotient := div(1, 1)", "after": "CRT LCM prefix0 quotient = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 CRT LCM prefix0 quotient，不以小數近似替代。", "en": "Recompute CRT LCM prefix0 quotient with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "1" ], "output": "1", "label": "CRT LCM prefix0 quotient" } }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "CRT LCM prefix0 value := mul(1, 26)", "after": "CRT LCM prefix0 value = 26", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 CRT LCM prefix0 value，不以小數近似替代。", "en": "Recompute CRT LCM prefix0 value with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "26" ], "output": "26", "label": "CRT LCM prefix0 value" } }, { "id": "S014", "uses": [ "G001", "S013" ], "before": "CRT modulus product prefix1 := mul(26, 39)", "after": "CRT modulus product prefix1 = 1014", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 CRT modulus product prefix1，不以小數近似替代。", "en": "Recompute CRT modulus product prefix1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "26", "39" ], "output": "1014", "label": "CRT modulus product prefix1" } }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "LCM prefix1 gcd:positive original gcd inputs=[26, 39]", "after": "Euclidean rows[a,b,q,r] with a=q*b+r=[[26, 39, 0, 26], [39, 26, 1, 13], [26, 13, 2, 0]];positive gcd=13", "relation": "implies", "rule_id": "crt_pair_gcd_exact_euclidean_witness", "conditions": [], "explanation": { "zh-Hant": "每次餘數替換保同一公因數集，最後非零值才是gcd；逐列列出q與r供來源核對。", "en": "Each remainder replacement preserves common divisors;the last nonzero value is the gcd.List every quotient/remainder row for source-bound checking." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S016", "uses": [ "G001", "S015" ], "before": "CRT LCM prefix1 quotient := div(26, 13)", "after": "CRT LCM prefix1 quotient = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 CRT LCM prefix1 quotient，不以小數近似替代。", "en": "Recompute CRT LCM prefix1 quotient with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "26", "13" ], "output": "2", "label": "CRT LCM prefix1 quotient" } }, { "id": "S017", "uses": [ "G001", "S016" ], "before": "CRT LCM prefix1 value := mul(2, 39)", "after": "CRT LCM prefix1 value = 78", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 CRT LCM prefix1 value，不以小數近似替代。", "en": "Recompute CRT LCM prefix1 value with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "39" ], "output": "78", "label": "CRT LCM prefix1 value" } }, { "id": "S018", "uses": [ "G001", "S017" ], "before": "original modulus pair0,1:positive original gcd inputs=[26, 39]", "after": "Euclidean rows[a,b,q,r] with a=q*b+r=[[26, 39, 0, 26], [39, 26, 1, 13], [26, 13, 2, 0]];positive gcd=13", "relation": "implies", "rule_id": "crt_pair_gcd_exact_euclidean_witness", "conditions": [], "explanation": { "zh-Hant": "每次餘數替換保同一公因數集，最後非零值才是gcd；逐列列出q與r供來源核對。", "en": "Each remainder replacement preserves common divisors;the last nonzero value is the gcd.List every quotient/remainder row for source-bound checking." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S019", "uses": [ "G001", "S018" ], "before": "CRT original pair0,1 residue difference := sub(1, 14)", "after": "CRT original pair0,1 residue difference = -13", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 CRT original pair0,1 residue difference，不以小數近似替代。", "en": "Recompute CRT original pair0,1 residue difference with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "1", "14" ], "output": "-13", "label": "CRT original pair0,1 residue difference" } }, { "id": "S020", "uses": [ "G001", "S019" ], "before": "CRT original pair0,1 difference divided by gcd := div(-13, 13)", "after": "CRT original pair0,1 difference divided by gcd = -1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 CRT original pair0,1 difference divided by gcd，不以小數近似替代。", "en": "Recompute CRT original pair0,1 difference divided by gcd with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "-13", "13" ], "output": "-1", "label": "CRT original pair0,1 difference divided by gcd" } }, { "id": "S021", "uses": [ "G001", "S020" ], "before": "original constraint slots=[0, 1];moduli=[26, 39];gcd=13;normalized residue difference=-13", "after": "difference/gcd=-1;pair coprime=否;pair compatible=是", "relation": "implies", "rule_id": "crt_every_pair_compatibility_not_automatic_inconsistency", "conditions": [], "explanation": { "zh-Hant": "任意共同解會使兩餘數差被gcd整除，所以先核每對；不互質不等於無解，差也可能是gcd倍數。", "en": "A common solution forces each residue difference to be divisible by the corresponding gcd.Check every pair;noncoprimality is not automatic inconsistency." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S022", "uses": [ "G001", "S021" ], "before": "original aligned moduli=[26, 39];original signed residues=[1, 14];normalized residues=[1, 14];all pair audits=[[0, 1, 13, 是]]", "after": "pairwise coprime=否;gcd compatible=是;product=1014;LCM period=78", "relation": "implies", "rule_id": "crt_original_constraint_order_product_vs_lcm", "conditions": [], "explanation": { "zh-Hant": "模數與餘數對齊保原順序，不拿semantic排序各自重排。互質主題用product，非互質邊界保lcm；modulus1不增加限制。", "en": "Retain original aligned constraint order,not separate semantic sorting.Main coprime operations use the product;the noncoprime boundary retains the lcm.Modulus1 adds no restriction." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S023", "uses": [ "G001", "S022" ], "before": "merge original slot0 Euclidean step0 quotient fraction := div(1, 26)", "after": "merge original slot0 Euclidean step0 quotient fraction = 1/26", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge original slot0 Euclidean step0 quotient fraction，不以小數近似替代。", "en": "Recompute merge original slot0 Euclidean step0 quotient fraction with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "26" ], "output": "1/26", "label": "merge original slot0 Euclidean step0 quotient fraction" } }, { "id": "S024", "uses": [ "G001", "S023" ], "before": "merge original slot0 Euclidean step0 divisor multiple := mul(0, 26)", "after": "merge original slot0 Euclidean step0 divisor multiple = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge original slot0 Euclidean step0 divisor multiple，不以小數近似替代。", "en": "Recompute merge original slot0 Euclidean step0 divisor multiple with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "26" ], "output": "0", "label": "merge original slot0 Euclidean step0 divisor multiple" } }, { "id": "S025", "uses": [ "G001", "S024" ], "before": "merge original slot0 Euclidean step0 remainder := sub(1, 0)", "after": "merge original slot0 Euclidean step0 remainder = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge original slot0 Euclidean step0 remainder，不以小數近似替代。", "en": "Recompute merge original slot0 Euclidean step0 remainder with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "1", "0" ], "output": "1", "label": "merge original slot0 Euclidean step0 remainder" } }, { "id": "S026", "uses": [ "G001", "S025" ], "before": "merge original slot0 Euclidean step0 first coefficient multiple := mul(0, 0)", "after": "merge original slot0 Euclidean step0 first coefficient multiple = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge original slot0 Euclidean step0 first coefficient multiple，不以小數近似替代。", "en": "Recompute merge original slot0 Euclidean step0 first coefficient multiple with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "0" ], "output": "0", "label": "merge original slot0 Euclidean step0 first coefficient multiple" } }, { "id": "S027", "uses": [ "G001", "S026" ], "before": "merge original slot0 Euclidean step0 first coefficient difference := sub(1, 0)", "after": "merge original slot0 Euclidean step0 first coefficient difference = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge original slot0 Euclidean step0 first coefficient difference，不以小數近似替代。", "en": "Recompute merge original slot0 Euclidean step0 first coefficient difference with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "1", "0" ], "output": "1", "label": "merge original slot0 Euclidean step0 first coefficient difference" } }, { "id": "S028", "uses": [ "G001", "S027" ], "before": "merge original slot0 Euclidean step0 second coefficient multiple := mul(0, 1)", "after": "merge original slot0 Euclidean step0 second coefficient multiple = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge original slot0 Euclidean step0 second coefficient multiple，不以小數近似替代。", "en": "Recompute merge original slot0 Euclidean step0 second coefficient multiple with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "1" ], "output": "0", "label": "merge original slot0 Euclidean step0 second coefficient multiple" } }, { "id": "S029", "uses": [ "G001", "S028" ], "before": "merge original slot0 Euclidean step0 second coefficient difference := sub(0, 0)", "after": "merge original slot0 Euclidean step0 second coefficient difference = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge original slot0 Euclidean step0 second coefficient difference，不以小數近似替代。", "en": "Recompute merge original slot0 Euclidean step0 second coefficient difference with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "0", "0" ], "output": "0", "label": "merge original slot0 Euclidean step0 second coefficient difference" } }, { "id": "S030", "uses": [ "G001", "S029" ], "before": "merge original slot0:step=0;current remainders=[1, 26];coefficient pairs=[[1, 0], [0, 1]];exact ratio=1/26;floor q=0", "after": "next remainder=1;next signed coefficient pair=[1, 0];a*x+b*y equals next remainder", "relation": "implies", "rule_id": "crt_extended_euclid_invariant_signed_coefficients", "conditions": [], "explanation": { "zh-Hant": "同一q同時更新餘數與兩個有號係數，保每列a·x+b·y=餘數；非只算gcd再猜inverse。", "en": "Use the same quotient for remainders and both signed coefficient recurrences,preserving a*x+b*y=remainder.Do not compute only a gcd and guess an inverse." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S031", "uses": [ "G001", "S030" ], "before": "merge original slot0 Euclidean step1 quotient fraction := div(26, 1)", "after": "merge original slot0 Euclidean step1 quotient fraction = 26", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge original slot0 Euclidean step1 quotient fraction，不以小數近似替代。", "en": "Recompute merge original slot0 Euclidean step1 quotient fraction with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "26", "1" ], "output": "26", "label": "merge original slot0 Euclidean step1 quotient fraction" } }, { "id": "S032", "uses": [ "G001", "S031" ], "before": "merge original slot0 Euclidean step1 divisor multiple := mul(26, 1)", "after": "merge original slot0 Euclidean step1 divisor multiple = 26", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge original slot0 Euclidean step1 divisor multiple，不以小數近似替代。", "en": "Recompute merge original slot0 Euclidean step1 divisor multiple with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "26", "1" ], "output": "26", "label": "merge original slot0 Euclidean step1 divisor multiple" } }, { "id": "S033", "uses": [ "G001", "S032" ], "before": "merge original slot0 Euclidean step1 remainder := sub(26, 26)", "after": "merge original slot0 Euclidean step1 remainder = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge original slot0 Euclidean step1 remainder，不以小數近似替代。", "en": "Recompute merge original slot0 Euclidean step1 remainder with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "26", "26" ], "output": "0", "label": "merge original slot0 Euclidean step1 remainder" } }, { "id": "S034", "uses": [ "G001", "S033" ], "before": "merge original slot0 Euclidean step1 first coefficient multiple := mul(26, 1)", "after": "merge original slot0 Euclidean step1 first coefficient multiple = 26", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge original slot0 Euclidean step1 first coefficient multiple，不以小數近似替代。", "en": "Recompute merge original slot0 Euclidean step1 first coefficient multiple with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "26", "1" ], "output": "26", "label": "merge original slot0 Euclidean step1 first coefficient multiple" } }, { "id": "S035", "uses": [ "G001", "S034" ], "before": "merge original slot0 Euclidean step1 first coefficient difference := sub(0, 26)", "after": "merge original slot0 Euclidean step1 first coefficient difference = -26", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge original slot0 Euclidean step1 first coefficient difference，不以小數近似替代。", "en": "Recompute merge original slot0 Euclidean step1 first coefficient difference with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "0", "26" ], "output": "-26", "label": "merge original slot0 Euclidean step1 first coefficient difference" } }, { "id": "S036", "uses": [ "G001", "S035" ], "before": "merge original slot0 Euclidean step1 second coefficient multiple := mul(26, 0)", "after": "merge original slot0 Euclidean step1 second coefficient multiple = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge original slot0 Euclidean step1 second coefficient multiple，不以小數近似替代。", "en": "Recompute merge original slot0 Euclidean step1 second coefficient multiple with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "26", "0" ], "output": "0", "label": "merge original slot0 Euclidean step1 second coefficient multiple" } }, { "id": "S037", "uses": [ "G001", "S036" ], "before": "merge original slot0 Euclidean step1 second coefficient difference := sub(1, 0)", "after": "merge original slot0 Euclidean step1 second coefficient difference = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge original slot0 Euclidean step1 second coefficient difference，不以小數近似替代。", "en": "Recompute merge original slot0 Euclidean step1 second coefficient difference with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "1", "0" ], "output": "1", "label": "merge original slot0 Euclidean step1 second coefficient difference" } }, { "id": "S038", "uses": [ "G001", "S037" ], "before": "merge original slot0:step=1;current remainders=[26, 1];coefficient pairs=[[0, 1], [1, 0]];exact ratio=26;floor q=26", "after": "next remainder=0;next signed coefficient pair=[-26, 1];a*x+b*y equals next remainder", "relation": "implies", "rule_id": "crt_extended_euclid_invariant_signed_coefficients", "conditions": [], "explanation": { "zh-Hant": "同一q同時更新餘數與兩個有號係數，保每列a·x+b·y=餘數；非只算gcd再猜inverse。", "en": "Use the same quotient for remainders and both signed coefficient recurrences,preserving a*x+b*y=remainder.Do not compute only a gcd and guess an inverse." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S039", "uses": [ "G001", "S038" ], "before": "merge original slot0 final first Bezout term := mul(1, 1)", "after": "merge original slot0 final first Bezout term = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge original slot0 final first Bezout term，不以小數近似替代。", "en": "Recompute merge original slot0 final first Bezout term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "merge original slot0 final first Bezout term" } }, { "id": "S040", "uses": [ "G001", "S039" ], "before": "merge original slot0 final second Bezout term := mul(26, 0)", "after": "merge original slot0 final second Bezout term = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge original slot0 final second Bezout term，不以小數近似替代。", "en": "Recompute merge original slot0 final second Bezout term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "26", "0" ], "output": "0", "label": "merge original slot0 final second Bezout term" } }, { "id": "S041", "uses": [ "G001", "S040" ], "before": "merge original slot0 final Bezout identity := sum(1, 0)", "after": "merge original slot0 final Bezout identity = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge original slot0 final Bezout identity，不以小數近似替代。", "en": "Recompute merge original slot0 final Bezout identity with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "0" ], "output": "1", "label": "merge original slot0 final Bezout identity" } }, { "id": "S042", "uses": [ "G001", "S041" ], "before": "merge slot0 new residue minus old class := sub(1, 0)", "after": "merge slot0 new residue minus old class = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge slot0 new residue minus old class，不以小數近似替代。", "en": "Recompute merge slot0 new residue minus old class with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "1", "0" ], "output": "1", "label": "merge slot0 new residue minus old class" } }, { "id": "S043", "uses": [ "G001", "S042" ], "before": "merge slot0 difference divided by gcd := div(1, 1)", "after": "merge slot0 difference divided by gcd = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge slot0 difference divided by gcd，不以小數近似替代。", "en": "Recompute merge slot0 difference divided by gcd with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "1" ], "output": "1", "label": "merge slot0 difference divided by gcd" } }, { "id": "S044", "uses": [ "G001", "S043" ], "before": "merge slot0 reduced old modulus := div(1, 1)", "after": "merge slot0 reduced old modulus = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge slot0 reduced old modulus，不以小數近似替代。", "en": "Recompute merge slot0 reduced old modulus with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "1" ], "output": "1", "label": "merge slot0 reduced old modulus" } }, { "id": "S045", "uses": [ "G001", "S044" ], "before": "merge slot0 reduced new modulus := div(26, 1)", "after": "merge slot0 reduced new modulus = 26", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge slot0 reduced new modulus，不以小數近似替代。", "en": "Recompute merge slot0 reduced new modulus with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "26", "1" ], "output": "26", "label": "merge slot0 reduced new modulus" } }, { "id": "S046", "uses": [ "G001", "S045" ], "before": "merge slot0 new LCM period := mul(1, 26)", "after": "merge slot0 new LCM period = 26", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge slot0 new LCM period，不以小數近似替代。", "en": "Recompute merge slot0 new LCM period with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "26" ], "output": "26", "label": "merge slot0 new LCM period" } }, { "id": "S047", "uses": [ "G001", "S046" ], "before": "merge slot0 reduced-unit inverse exact quotient := div(1, 26)", "after": "merge slot0 reduced-unit inverse exact quotient = 1/26", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge slot0 reduced-unit inverse exact quotient，不以小數近似替代。", "en": "Recompute merge slot0 reduced-unit inverse exact quotient with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "26" ], "output": "1/26", "label": "merge slot0 reduced-unit inverse exact quotient" } }, { "id": "S048", "uses": [ "G001", "S047" ], "before": "merge slot0 reduced-unit inverse floor quotient multiple := mul(0, 26)", "after": "merge slot0 reduced-unit inverse floor quotient multiple = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge slot0 reduced-unit inverse floor quotient multiple，不以小數近似替代。", "en": "Recompute merge slot0 reduced-unit inverse floor quotient multiple with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "26" ], "output": "0", "label": "merge slot0 reduced-unit inverse floor quotient multiple" } }, { "id": "S049", "uses": [ "G001", "S048" ], "before": "merge slot0 reduced-unit inverse normalized remainder := sub(1, 0)", "after": "merge slot0 reduced-unit inverse normalized remainder = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge slot0 reduced-unit inverse normalized remainder，不以小數近似替代。", "en": "Recompute merge slot0 reduced-unit inverse normalized remainder with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "1", "0" ], "output": "1", "label": "merge slot0 reduced-unit inverse normalized remainder" } }, { "id": "S050", "uses": [ "G001", "S049" ], "before": "merge slot0 reduced-unit inverse:integer value=1;positive modulus=26;exact ratio=1/26;floor quotient=0", "after": "value=q*modulus+remainder;normalized residue=1 in[0,26)", "relation": "implies", "rule_id": "crt_exact_floor_modulo_signed_input", "conditions": [], "explanation": { "zh-Hant": "對正模數用向下整數商，負原值也得到0到modulus−1的代表；不是截斷向0或改變同餘類。模1是合法平凡約束，餘數為0。", "en": "Use the floor quotient for a positive modulus,including signed inputs,to obtain the representative from0 through modulus−1.This is not truncation toward zero or a change of congruence class.Modulus1 is a valid trivial constraint with residue0." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S051", "uses": [ "G001", "S050" ], "before": "merge slot0 parameter congruence product := mul(1, 1)", "after": "merge slot0 parameter congruence product = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge slot0 parameter congruence product，不以小數近似替代。", "en": "Recompute merge slot0 parameter congruence product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "merge slot0 parameter congruence product" } }, { "id": "S052", "uses": [ "G001", "S051" ], "before": "merge slot0 normalized parameter exact quotient := div(1, 26)", "after": "merge slot0 normalized parameter exact quotient = 1/26", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge slot0 normalized parameter exact quotient，不以小數近似替代。", "en": "Recompute merge slot0 normalized parameter exact quotient with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "26" ], "output": "1/26", "label": "merge slot0 normalized parameter exact quotient" } }, { "id": "S053", "uses": [ "G001", "S052" ], "before": "merge slot0 normalized parameter floor quotient multiple := mul(0, 26)", "after": "merge slot0 normalized parameter floor quotient multiple = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge slot0 normalized parameter floor quotient multiple，不以小數近似替代。", "en": "Recompute merge slot0 normalized parameter floor quotient multiple with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "26" ], "output": "0", "label": "merge slot0 normalized parameter floor quotient multiple" } }, { "id": "S054", "uses": [ "G001", "S053" ], "before": "merge slot0 normalized parameter normalized remainder := sub(1, 0)", "after": "merge slot0 normalized parameter normalized remainder = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge slot0 normalized parameter normalized remainder，不以小數近似替代。", "en": "Recompute merge slot0 normalized parameter normalized remainder with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "1", "0" ], "output": "1", "label": "merge slot0 normalized parameter normalized remainder" } }, { "id": "S055", "uses": [ "G001", "S054" ], "before": "merge slot0 normalized parameter:integer value=1;positive modulus=26;exact ratio=1/26;floor quotient=0", "after": "value=q*modulus+remainder;normalized residue=1 in[0,26)", "relation": "implies", "rule_id": "crt_exact_floor_modulo_signed_input", "conditions": [], "explanation": { "zh-Hant": "對正模數用向下整數商，負原值也得到0到modulus−1的代表；不是截斷向0或改變同餘類。模1是合法平凡約束，餘數為0。", "en": "Use the floor quotient for a positive modulus,including signed inputs,to obtain the representative from0 through modulus−1.This is not truncation toward zero or a change of congruence class.Modulus1 is a valid trivial constraint with residue0." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S056", "uses": [ "G001", "S055" ], "before": "merge slot0 old modulus times parameter := mul(1, 1)", "after": "merge slot0 old modulus times parameter = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge slot0 old modulus times parameter，不以小數近似替代。", "en": "Recompute merge slot0 old modulus times parameter with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "merge slot0 old modulus times parameter" } }, { "id": "S057", "uses": [ "G001", "S056" ], "before": "merge slot0 unnormalized common solution := sum(0, 1)", "after": "merge slot0 unnormalized common solution = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge slot0 unnormalized common solution，不以小數近似替代。", "en": "Recompute merge slot0 unnormalized common solution with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "1" ], "output": "1", "label": "merge slot0 unnormalized common solution" } }, { "id": "S058", "uses": [ "G001", "S057" ], "before": "merge slot0 least class representative exact quotient := div(1, 26)", "after": "merge slot0 least class representative exact quotient = 1/26", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge slot0 least class representative exact quotient，不以小數近似替代。", "en": "Recompute merge slot0 least class representative exact quotient with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "26" ], "output": "1/26", "label": "merge slot0 least class representative exact quotient" } }, { "id": "S059", "uses": [ "G001", "S058" ], "before": "merge slot0 least class representative floor quotient multiple := mul(0, 26)", "after": "merge slot0 least class representative floor quotient multiple = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge slot0 least class representative floor quotient multiple，不以小數近似替代。", "en": "Recompute merge slot0 least class representative floor quotient multiple with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "26" ], "output": "0", "label": "merge slot0 least class representative floor quotient multiple" } }, { "id": "S060", "uses": [ "G001", "S059" ], "before": "merge slot0 least class representative normalized remainder := sub(1, 0)", "after": "merge slot0 least class representative normalized remainder = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge slot0 least class representative normalized remainder，不以小數近似替代。", "en": "Recompute merge slot0 least class representative normalized remainder with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "1", "0" ], "output": "1", "label": "merge slot0 least class representative normalized remainder" } }, { "id": "S061", "uses": [ "G001", "S060" ], "before": "merge slot0 least class representative:integer value=1;positive modulus=26;exact ratio=1/26;floor quotient=0", "after": "value=q*modulus+remainder;normalized residue=1 in[0,26)", "relation": "implies", "rule_id": "crt_exact_floor_modulo_signed_input", "conditions": [], "explanation": { "zh-Hant": "對正模數用向下整數商，負原值也得到0到modulus−1的代表；不是截斷向0或改變同餘類。模1是合法平凡約束，餘數為0。", "en": "Use the floor quotient for a positive modulus,including signed inputs,to obtain the representative from0 through modulus−1.This is not truncation toward zero or a change of congruence class.Modulus1 is a valid trivial constraint with residue0." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S062", "uses": [ "G001", "S061" ], "before": "old FULL solution class=0 mod1;new original constraint=1 mod26;gcd=1;Bezout coefficients=[1, 0];reduced coprime moduli=[1, 26]", "after": "write old x=a+M*t;reduced equation gives t=1 mod26;ALL merged solutions x=1 mod26", "relation": "implies", "rule_id": "crt_constructive_merge_all_solutions_not_one_witness", "conditions": [], "explanation": { "zh-Hant": "代回舊解類後除gcd，Bézout除g給互質係數，t恰落在一個模m/g的類。乘回M得到lcm週期；這包含所有解，不僅驗算找到一點。", "en": "Substitute into the old solution class and divide by the gcd.Bezout divided by that gcd supplies the coprime inverse,so t lies in exactly one class modulo m/g.Multiplying back by M gives the lcm period and all solutions,not merely one checked point." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S063", "uses": [ "G001", "S062" ], "before": "merge original slot1 Euclidean step0 quotient fraction := div(26, 39)", "after": "merge original slot1 Euclidean step0 quotient fraction = 2/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge original slot1 Euclidean step0 quotient fraction，不以小數近似替代。", "en": "Recompute merge original slot1 Euclidean step0 quotient fraction with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "26", "39" ], "output": "2/3", "label": "merge original slot1 Euclidean step0 quotient fraction" } }, { "id": "S064", "uses": [ "G001", "S063" ], "before": "merge original slot1 Euclidean step0 divisor multiple := mul(0, 39)", "after": "merge original slot1 Euclidean step0 divisor multiple = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge original slot1 Euclidean step0 divisor multiple，不以小數近似替代。", "en": "Recompute merge original slot1 Euclidean step0 divisor multiple with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "39" ], "output": "0", "label": "merge original slot1 Euclidean step0 divisor multiple" } }, { "id": "S065", "uses": [ "G001", "S064" ], "before": "merge original slot1 Euclidean step0 remainder := sub(26, 0)", "after": "merge original slot1 Euclidean step0 remainder = 26", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge original slot1 Euclidean step0 remainder，不以小數近似替代。", "en": "Recompute merge original slot1 Euclidean step0 remainder with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "26", "0" ], "output": "26", "label": "merge original slot1 Euclidean step0 remainder" } }, { "id": "S066", "uses": [ "G001", "S065" ], "before": "merge original slot1 Euclidean step0 first coefficient multiple := mul(0, 0)", "after": "merge original slot1 Euclidean step0 first coefficient multiple = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge original slot1 Euclidean step0 first coefficient multiple，不以小數近似替代。", "en": "Recompute merge original slot1 Euclidean step0 first coefficient multiple with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "0" ], "output": "0", "label": "merge original slot1 Euclidean step0 first coefficient multiple" } }, { "id": "S067", "uses": [ "G001", "S066" ], "before": "merge original slot1 Euclidean step0 first coefficient difference := sub(1, 0)", "after": "merge original slot1 Euclidean step0 first coefficient difference = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge original slot1 Euclidean step0 first coefficient difference，不以小數近似替代。", "en": "Recompute merge original slot1 Euclidean step0 first coefficient difference with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "1", "0" ], "output": "1", "label": "merge original slot1 Euclidean step0 first coefficient difference" } }, { "id": "S068", "uses": [ "G001", "S067" ], "before": "merge original slot1 Euclidean step0 second coefficient multiple := mul(0, 1)", "after": "merge original slot1 Euclidean step0 second coefficient multiple = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge original slot1 Euclidean step0 second coefficient multiple，不以小數近似替代。", "en": "Recompute merge original slot1 Euclidean step0 second coefficient multiple with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "1" ], "output": "0", "label": "merge original slot1 Euclidean step0 second coefficient multiple" } }, { "id": "S069", "uses": [ "G001", "S068" ], "before": "merge original slot1 Euclidean step0 second coefficient difference := sub(0, 0)", "after": "merge original slot1 Euclidean step0 second coefficient difference = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge original slot1 Euclidean step0 second coefficient difference，不以小數近似替代。", "en": "Recompute merge original slot1 Euclidean step0 second coefficient difference with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "0", "0" ], "output": "0", "label": "merge original slot1 Euclidean step0 second coefficient difference" } }, { "id": "S070", "uses": [ "G001", "S069" ], "before": "merge original slot1:step=0;current remainders=[26, 39];coefficient pairs=[[1, 0], [0, 1]];exact ratio=2/3;floor q=0", "after": "next remainder=26;next signed coefficient pair=[1, 0];a*x+b*y equals next remainder", "relation": "implies", "rule_id": "crt_extended_euclid_invariant_signed_coefficients", "conditions": [], "explanation": { "zh-Hant": "同一q同時更新餘數與兩個有號係數，保每列a·x+b·y=餘數；非只算gcd再猜inverse。", "en": "Use the same quotient for remainders and both signed coefficient recurrences,preserving a*x+b*y=remainder.Do not compute only a gcd and guess an inverse." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S071", "uses": [ "G001", "S070" ], "before": "merge original slot1 Euclidean step1 quotient fraction := div(39, 26)", "after": "merge original slot1 Euclidean step1 quotient fraction = 3/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge original slot1 Euclidean step1 quotient fraction，不以小數近似替代。", "en": "Recompute merge original slot1 Euclidean step1 quotient fraction with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "39", "26" ], "output": "3/2", "label": "merge original slot1 Euclidean step1 quotient fraction" } }, { "id": "S072", "uses": [ "G001", "S071" ], "before": "merge original slot1 Euclidean step1 divisor multiple := mul(1, 26)", "after": "merge original slot1 Euclidean step1 divisor multiple = 26", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge original slot1 Euclidean step1 divisor multiple，不以小數近似替代。", "en": "Recompute merge original slot1 Euclidean step1 divisor multiple with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "26" ], "output": "26", "label": "merge original slot1 Euclidean step1 divisor multiple" } }, { "id": "S073", "uses": [ "G001", "S072" ], "before": "merge original slot1 Euclidean step1 remainder := sub(39, 26)", "after": "merge original slot1 Euclidean step1 remainder = 13", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge original slot1 Euclidean step1 remainder，不以小數近似替代。", "en": "Recompute merge original slot1 Euclidean step1 remainder with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "39", "26" ], "output": "13", "label": "merge original slot1 Euclidean step1 remainder" } }, { "id": "S074", "uses": [ "G001", "S073" ], "before": "merge original slot1 Euclidean step1 first coefficient multiple := mul(1, 1)", "after": "merge original slot1 Euclidean step1 first coefficient multiple = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge original slot1 Euclidean step1 first coefficient multiple，不以小數近似替代。", "en": "Recompute merge original slot1 Euclidean step1 first coefficient multiple with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "merge original slot1 Euclidean step1 first coefficient multiple" } }, { "id": "S075", "uses": [ "G001", "S074" ], "before": "merge original slot1 Euclidean step1 first coefficient difference := sub(0, 1)", "after": "merge original slot1 Euclidean step1 first coefficient difference = -1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge original slot1 Euclidean step1 first coefficient difference，不以小數近似替代。", "en": "Recompute merge original slot1 Euclidean step1 first coefficient difference with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "0", "1" ], "output": "-1", "label": "merge original slot1 Euclidean step1 first coefficient difference" } }, { "id": "S076", "uses": [ "G001", "S075" ], "before": "merge original slot1 Euclidean step1 second coefficient multiple := mul(1, 0)", "after": "merge original slot1 Euclidean step1 second coefficient multiple = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge original slot1 Euclidean step1 second coefficient multiple，不以小數近似替代。", "en": "Recompute merge original slot1 Euclidean step1 second coefficient multiple with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "0" ], "output": "0", "label": "merge original slot1 Euclidean step1 second coefficient multiple" } }, { "id": "S077", "uses": [ "G001", "S076" ], "before": "merge original slot1 Euclidean step1 second coefficient difference := sub(1, 0)", "after": "merge original slot1 Euclidean step1 second coefficient difference = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge original slot1 Euclidean step1 second coefficient difference，不以小數近似替代。", "en": "Recompute merge original slot1 Euclidean step1 second coefficient difference with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "1", "0" ], "output": "1", "label": "merge original slot1 Euclidean step1 second coefficient difference" } }, { "id": "S078", "uses": [ "G001", "S077" ], "before": "merge original slot1:step=1;current remainders=[39, 26];coefficient pairs=[[0, 1], [1, 0]];exact ratio=3/2;floor q=1", "after": "next remainder=13;next signed coefficient pair=[-1, 1];a*x+b*y equals next remainder", "relation": "implies", "rule_id": "crt_extended_euclid_invariant_signed_coefficients", "conditions": [], "explanation": { "zh-Hant": "同一q同時更新餘數與兩個有號係數，保每列a·x+b·y=餘數；非只算gcd再猜inverse。", "en": "Use the same quotient for remainders and both signed coefficient recurrences,preserving a*x+b*y=remainder.Do not compute only a gcd and guess an inverse." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S079", "uses": [ "G001", "S078" ], "before": "merge original slot1 Euclidean step2 quotient fraction := div(26, 13)", "after": "merge original slot1 Euclidean step2 quotient fraction = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge original slot1 Euclidean step2 quotient fraction，不以小數近似替代。", "en": "Recompute merge original slot1 Euclidean step2 quotient fraction with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "26", "13" ], "output": "2", "label": "merge original slot1 Euclidean step2 quotient fraction" } }, { "id": "S080", "uses": [ "G001", "S079" ], "before": "merge original slot1 Euclidean step2 divisor multiple := mul(2, 13)", "after": "merge original slot1 Euclidean step2 divisor multiple = 26", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge original slot1 Euclidean step2 divisor multiple，不以小數近似替代。", "en": "Recompute merge original slot1 Euclidean step2 divisor multiple with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "13" ], "output": "26", "label": "merge original slot1 Euclidean step2 divisor multiple" } }, { "id": "S081", "uses": [ "G001", "S080" ], "before": "merge original slot1 Euclidean step2 remainder := sub(26, 26)", "after": "merge original slot1 Euclidean step2 remainder = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge original slot1 Euclidean step2 remainder，不以小數近似替代。", "en": "Recompute merge original slot1 Euclidean step2 remainder with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "26", "26" ], "output": "0", "label": "merge original slot1 Euclidean step2 remainder" } }, { "id": "S082", "uses": [ "G001", "S081" ], "before": "merge original slot1 Euclidean step2 first coefficient multiple := mul(2, -1)", "after": "merge original slot1 Euclidean step2 first coefficient multiple = -2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge original slot1 Euclidean step2 first coefficient multiple，不以小數近似替代。", "en": "Recompute merge original slot1 Euclidean step2 first coefficient multiple with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "-1" ], "output": "-2", "label": "merge original slot1 Euclidean step2 first coefficient multiple" } }, { "id": "S083", "uses": [ "G001", "S082" ], "before": "merge original slot1 Euclidean step2 first coefficient difference := sub(1, -2)", "after": "merge original slot1 Euclidean step2 first coefficient difference = 3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge original slot1 Euclidean step2 first coefficient difference，不以小數近似替代。", "en": "Recompute merge original slot1 Euclidean step2 first coefficient difference with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "1", "-2" ], "output": "3", "label": "merge original slot1 Euclidean step2 first coefficient difference" } }, { "id": "S084", "uses": [ "G001", "S083" ], "before": "merge original slot1 Euclidean step2 second coefficient multiple := mul(2, 1)", "after": "merge original slot1 Euclidean step2 second coefficient multiple = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge original slot1 Euclidean step2 second coefficient multiple，不以小數近似替代。", "en": "Recompute merge original slot1 Euclidean step2 second coefficient multiple with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "1" ], "output": "2", "label": "merge original slot1 Euclidean step2 second coefficient multiple" } }, { "id": "S085", "uses": [ "G001", "S084" ], "before": "merge original slot1 Euclidean step2 second coefficient difference := sub(0, 2)", "after": "merge original slot1 Euclidean step2 second coefficient difference = -2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge original slot1 Euclidean step2 second coefficient difference，不以小數近似替代。", "en": "Recompute merge original slot1 Euclidean step2 second coefficient difference with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "0", "2" ], "output": "-2", "label": "merge original slot1 Euclidean step2 second coefficient difference" } }, { "id": "S086", "uses": [ "G001", "S085" ], "before": "merge original slot1:step=2;current remainders=[26, 13];coefficient pairs=[[1, 0], [-1, 1]];exact ratio=2;floor q=2", "after": "next remainder=0;next signed coefficient pair=[3, -2];a*x+b*y equals next remainder", "relation": "implies", "rule_id": "crt_extended_euclid_invariant_signed_coefficients", "conditions": [], "explanation": { "zh-Hant": "同一q同時更新餘數與兩個有號係數，保每列a·x+b·y=餘數；非只算gcd再猜inverse。", "en": "Use the same quotient for remainders and both signed coefficient recurrences,preserving a*x+b*y=remainder.Do not compute only a gcd and guess an inverse." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S087", "uses": [ "G001", "S086" ], "before": "merge original slot1 final first Bezout term := mul(26, -1)", "after": "merge original slot1 final first Bezout term = -26", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge original slot1 final first Bezout term，不以小數近似替代。", "en": "Recompute merge original slot1 final first Bezout term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "26", "-1" ], "output": "-26", "label": "merge original slot1 final first Bezout term" } }, { "id": "S088", "uses": [ "G001", "S087" ], "before": "merge original slot1 final second Bezout term := mul(39, 1)", "after": "merge original slot1 final second Bezout term = 39", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge original slot1 final second Bezout term，不以小數近似替代。", "en": "Recompute merge original slot1 final second Bezout term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "39", "1" ], "output": "39", "label": "merge original slot1 final second Bezout term" } }, { "id": "S089", "uses": [ "G001", "S088" ], "before": "merge original slot1 final Bezout identity := sum(-26, 39)", "after": "merge original slot1 final Bezout identity = 13", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge original slot1 final Bezout identity，不以小數近似替代。", "en": "Recompute merge original slot1 final Bezout identity with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-26", "39" ], "output": "13", "label": "merge original slot1 final Bezout identity" } }, { "id": "S090", "uses": [ "G001", "S089" ], "before": "merge slot1 new residue minus old class := sub(14, 1)", "after": "merge slot1 new residue minus old class = 13", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge slot1 new residue minus old class，不以小數近似替代。", "en": "Recompute merge slot1 new residue minus old class with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "14", "1" ], "output": "13", "label": "merge slot1 new residue minus old class" } }, { "id": "S091", "uses": [ "G001", "S090" ], "before": "merge slot1 difference divided by gcd := div(13, 13)", "after": "merge slot1 difference divided by gcd = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge slot1 difference divided by gcd，不以小數近似替代。", "en": "Recompute merge slot1 difference divided by gcd with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "13", "13" ], "output": "1", "label": "merge slot1 difference divided by gcd" } }, { "id": "S092", "uses": [ "G001", "S091" ], "before": "merge slot1 reduced old modulus := div(26, 13)", "after": "merge slot1 reduced old modulus = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge slot1 reduced old modulus，不以小數近似替代。", "en": "Recompute merge slot1 reduced old modulus with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "26", "13" ], "output": "2", "label": "merge slot1 reduced old modulus" } }, { "id": "S093", "uses": [ "G001", "S092" ], "before": "merge slot1 reduced new modulus := div(39, 13)", "after": "merge slot1 reduced new modulus = 3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge slot1 reduced new modulus，不以小數近似替代。", "en": "Recompute merge slot1 reduced new modulus with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "39", "13" ], "output": "3", "label": "merge slot1 reduced new modulus" } }, { "id": "S094", "uses": [ "G001", "S093" ], "before": "merge slot1 new LCM period := mul(26, 3)", "after": "merge slot1 new LCM period = 78", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge slot1 new LCM period，不以小數近似替代。", "en": "Recompute merge slot1 new LCM period with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "26", "3" ], "output": "78", "label": "merge slot1 new LCM period" } }, { "id": "S095", "uses": [ "G001", "S094" ], "before": "merge slot1 reduced-unit inverse exact quotient := div(-1, 3)", "after": "merge slot1 reduced-unit inverse exact quotient = -1/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge slot1 reduced-unit inverse exact quotient，不以小數近似替代。", "en": "Recompute merge slot1 reduced-unit inverse exact quotient with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "-1", "3" ], "output": "-1/3", "label": "merge slot1 reduced-unit inverse exact quotient" } }, { "id": "S096", "uses": [ "G001", "S095" ], "before": "merge slot1 reduced-unit inverse floor quotient multiple := mul(-1, 3)", "after": "merge slot1 reduced-unit inverse floor quotient multiple = -3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge slot1 reduced-unit inverse floor quotient multiple，不以小數近似替代。", "en": "Recompute merge slot1 reduced-unit inverse floor quotient multiple with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-1", "3" ], "output": "-3", "label": "merge slot1 reduced-unit inverse floor quotient multiple" } }, { "id": "S097", "uses": [ "G001", "S096" ], "before": "merge slot1 reduced-unit inverse normalized remainder := sub(-1, -3)", "after": "merge slot1 reduced-unit inverse normalized remainder = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge slot1 reduced-unit inverse normalized remainder，不以小數近似替代。", "en": "Recompute merge slot1 reduced-unit inverse normalized remainder with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "-1", "-3" ], "output": "2", "label": "merge slot1 reduced-unit inverse normalized remainder" } }, { "id": "S098", "uses": [ "G001", "S097" ], "before": "merge slot1 reduced-unit inverse:integer value=-1;positive modulus=3;exact ratio=-1/3;floor quotient=-1", "after": "value=q*modulus+remainder;normalized residue=2 in[0,3)", "relation": "implies", "rule_id": "crt_exact_floor_modulo_signed_input", "conditions": [], "explanation": { "zh-Hant": "對正模數用向下整數商，負原值也得到0到modulus−1的代表；不是截斷向0或改變同餘類。模1是合法平凡約束，餘數為0。", "en": "Use the floor quotient for a positive modulus,including signed inputs,to obtain the representative from0 through modulus−1.This is not truncation toward zero or a change of congruence class.Modulus1 is a valid trivial constraint with residue0." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S099", "uses": [ "G001", "S098" ], "before": "merge slot1 parameter congruence product := mul(1, 2)", "after": "merge slot1 parameter congruence product = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge slot1 parameter congruence product，不以小數近似替代。", "en": "Recompute merge slot1 parameter congruence product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "2" ], "output": "2", "label": "merge slot1 parameter congruence product" } }, { "id": "S100", "uses": [ "G001", "S099" ], "before": "merge slot1 normalized parameter exact quotient := div(2, 3)", "after": "merge slot1 normalized parameter exact quotient = 2/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge slot1 normalized parameter exact quotient，不以小數近似替代。", "en": "Recompute merge slot1 normalized parameter exact quotient with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "2", "3" ], "output": "2/3", "label": "merge slot1 normalized parameter exact quotient" } }, { "id": "S101", "uses": [ "G001", "S100" ], "before": "merge slot1 normalized parameter floor quotient multiple := mul(0, 3)", "after": "merge slot1 normalized parameter floor quotient multiple = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge slot1 normalized parameter floor quotient multiple，不以小數近似替代。", "en": "Recompute merge slot1 normalized parameter floor quotient multiple with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "3" ], "output": "0", "label": "merge slot1 normalized parameter floor quotient multiple" } }, { "id": "S102", "uses": [ "G001", "S101" ], "before": "merge slot1 normalized parameter normalized remainder := sub(2, 0)", "after": "merge slot1 normalized parameter normalized remainder = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge slot1 normalized parameter normalized remainder，不以小數近似替代。", "en": "Recompute merge slot1 normalized parameter normalized remainder with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "2", "0" ], "output": "2", "label": "merge slot1 normalized parameter normalized remainder" } }, { "id": "S103", "uses": [ "G001", "S102" ], "before": "merge slot1 normalized parameter:integer value=2;positive modulus=3;exact ratio=2/3;floor quotient=0", "after": "value=q*modulus+remainder;normalized residue=2 in[0,3)", "relation": "implies", "rule_id": "crt_exact_floor_modulo_signed_input", "conditions": [], "explanation": { "zh-Hant": "對正模數用向下整數商，負原值也得到0到modulus−1的代表；不是截斷向0或改變同餘類。模1是合法平凡約束，餘數為0。", "en": "Use the floor quotient for a positive modulus,including signed inputs,to obtain the representative from0 through modulus−1.This is not truncation toward zero or a change of congruence class.Modulus1 is a valid trivial constraint with residue0." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S104", "uses": [ "G001", "S103" ], "before": "merge slot1 old modulus times parameter := mul(26, 2)", "after": "merge slot1 old modulus times parameter = 52", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge slot1 old modulus times parameter，不以小數近似替代。", "en": "Recompute merge slot1 old modulus times parameter with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "26", "2" ], "output": "52", "label": "merge slot1 old modulus times parameter" } }, { "id": "S105", "uses": [ "G001", "S104" ], "before": "merge slot1 unnormalized common solution := sum(1, 52)", "after": "merge slot1 unnormalized common solution = 53", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge slot1 unnormalized common solution，不以小數近似替代。", "en": "Recompute merge slot1 unnormalized common solution with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "52" ], "output": "53", "label": "merge slot1 unnormalized common solution" } }, { "id": "S106", "uses": [ "G001", "S105" ], "before": "merge slot1 least class representative exact quotient := div(53, 78)", "after": "merge slot1 least class representative exact quotient = 53/78", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge slot1 least class representative exact quotient，不以小數近似替代。", "en": "Recompute merge slot1 least class representative exact quotient with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "53", "78" ], "output": "53/78", "label": "merge slot1 least class representative exact quotient" } }, { "id": "S107", "uses": [ "G001", "S106" ], "before": "merge slot1 least class representative floor quotient multiple := mul(0, 78)", "after": "merge slot1 least class representative floor quotient multiple = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge slot1 least class representative floor quotient multiple，不以小數近似替代。", "en": "Recompute merge slot1 least class representative floor quotient multiple with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "78" ], "output": "0", "label": "merge slot1 least class representative floor quotient multiple" } }, { "id": "S108", "uses": [ "G001", "S107" ], "before": "merge slot1 least class representative normalized remainder := sub(53, 0)", "after": "merge slot1 least class representative normalized remainder = 53", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 merge slot1 least class representative normalized remainder，不以小數近似替代。", "en": "Recompute merge slot1 least class representative normalized remainder with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "53", "0" ], "output": "53", "label": "merge slot1 least class representative normalized remainder" } }, { "id": "S109", "uses": [ "G001", "S108" ], "before": "merge slot1 least class representative:integer value=53;positive modulus=78;exact ratio=53/78;floor quotient=0", "after": "value=q*modulus+remainder;normalized residue=53 in[0,78)", "relation": "implies", "rule_id": "crt_exact_floor_modulo_signed_input", "conditions": [], "explanation": { "zh-Hant": "對正模數用向下整數商，負原值也得到0到modulus−1的代表；不是截斷向0或改變同餘類。模1是合法平凡約束，餘數為0。", "en": "Use the floor quotient for a positive modulus,including signed inputs,to obtain the representative from0 through modulus−1.This is not truncation toward zero or a change of congruence class.Modulus1 is a valid trivial constraint with residue0." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S110", "uses": [ "G001", "S109" ], "before": "old FULL solution class=1 mod26;new original constraint=14 mod39;gcd=13;Bezout coefficients=[-1, 1];reduced coprime moduli=[2, 3]", "after": "write old x=a+M*t;reduced equation gives t=2 mod3;ALL merged solutions x=53 mod78", "relation": "implies", "rule_id": "crt_constructive_merge_all_solutions_not_one_witness", "conditions": [], "explanation": { "zh-Hant": "代回舊解類後除gcd，Bézout除g給互質係數，t恰落在一個模m/g的類。乘回M得到lcm週期；這包含所有解，不僅驗算找到一點。", "en": "Substitute into the old solution class and divide by the gcd.Bezout divided by that gcd supplies the coprime inverse,so t lies in exactly one class modulo m/g.Multiplying back by M gives the lcm period and all solutions,not merely one checked point." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null } ], "math_model": { "result": [ [ 26, 39 ], [ 1, 14 ], true, 53, 78, "noncoprime_not_automatically_inconsistent" ] }, "conclusion": { "canonical_value": "([26, 39], [1, 14], 是, 53, 78, noncoprime_not_automatically_inconsistent)", "explanation": { "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。", "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged." } }, "definition_note": { "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。", "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate." }, "source": { "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db", "question_binding": { "batch_id": "batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations/questions.jsonl", "batch_sha256": "8a6832a58fe05b8550feab8b85b98a43c311d5e8ea9b3a32c16c1d0fbf3064b3", "original_line": 87, "original_line_sha256": "056c49b014ab705958cc92abc30bf3106209113bd3d195888dd1177b132048c9", "question_compact_sha256": "cc776880b60766975bec3af041c9b0dc08ea0d786b74120e8b327a5fdc6bb38e", "english_row_compact_sha256": "16f9d16d6946fa9b8d2efc539e8ca5f077c98774f229fd438e26992dd01f107e", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations.en.jsonl" }, "mko_bindings": { "mko-chinese-remainder-theorem": { "sha256": "97d98262a4c1d1774983902029b0e964992d7472332d99d3c22e80f77a1a44cc", "english_sha256": "68fb5d62577d9598790d36a1aa4c65339ef78bfc9567c23162560b6e15029f1c", "original_path": "public/data/mko/chinese-remainder-theorem.json", "english_path": "public/data/i18n/editions/mko-chinese-remainder-theorem.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "9adff0bbee39d9c3dec346304161ea9633576ccd5ca954f6b64ad482d2533a8a", "original_candidate_sha256": "2e021e3494140ed5dc02336772deff25313bf81f2f778a322b400af330b326ee", "original_candidate_line_sha256": "2e021e3494140ed5dc02336772deff25313bf81f2f778a322b400af330b326ee" }, "historical_source_state": { "review_state": { "derivation": "candidate_authored_and_locally_checked", "source_question_state": "candidate", "independent_AI_or_human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN" }, "scope_note": { "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。", "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged." }, "label": "Frozen producer metadata; current per-question review is separate below." }, "revision": { "id": "review-20261026-008-p36", "review_date": "2026-10-26", "owner_authorized_reader_publication": true, "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.", "review_receipts": { "semantic_review_sha256": "713db6309b22bf8b31f0253d4aaa7aca9011ab908587ae0e9d4a365edce1f9bc", "question_first_witness_sha256": "0338a3db344ab237113fc63be29153c3684da96e388109e9dcd913a0a693aed2" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 110, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "aa8132e9d04a8e1d4a2e4bb9fc2abe9ad0899964d1a4273769eaec92ebc8a101", "arithmetic_steps": 89 }, "uses_answer_as_premise": false, "human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN", "Canonical_truth_promotion": false, "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged." } }