{ "schema_version": "ocme-derivation-reader-v1", "guide_id": "derivation-mko-orbit-stabilizer-theorem-01", "mko_id": "mko-orbit-stabilizer-theorem", "title": { "zh-Hant": "fibers確為stabilizer左陪集", "en": "Fibers as left cosets of the stabilizer" }, "contract": { "schema_version": "ocme-derivation-guide-v0.1", "guide_id": "derivation-mko-orbit-stabilizer-theorem-01", "source": { "mko_id": "mko-orbit-stabilizer-theorem", "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f", "sha256": "0ae3563c72f6bb6e1702aa5a64f490a1f2b0ca0741f19d17ac3790775e66c38f", "english_sha256": "d40f8220b027fb6f66d44947a84f3e7b76317d3a768f913574f6cb6651f5bae7" }, "mode": "worked_example", "audience_prerequisites": [ "calc" ], "goal": { "zh-Hant": "point0的orbit大小3、stabilizer大小2；三個fiber均為左陪集，所以6=3·2。", "en": "At point 0 the orbit has size 3 and stabilizer size 2; all three fibers are left cosets, giving 6=3*2." }, "givens": [ { "id": "G1", "statement": { "zh-Hant": "G={e,r,r²,s,rs,r²s}，依序編碼(0,0),(1,0),(2,0),(0,1),(1,1),(2,1)。乘法(i,j)(k,l)=(i+(−1)^j k mod3,j+l mod2)。點載體X={0,1,2,3}；(i,j)作用x<3為i+(−1)^j x mod3，並固定3。", "en": "G={e,r,r2,s,rs,r2s}, encoded respectively (0,0),(1,0),(2,0),(0,1),(1,1),(2,1), with (i,j)(k,l)=(i+(-1)^j*k mod3,j+l mod2). Points X={0,1,2,3}; (i,j) acts on x<3 by i+(-1)^j*x mod3 and fixes 3." } } ], "steps": [ { "id": "S01", "uses": [ "G1" ], "before_tex": "G=\\{e,r,r^2,s,rs,r^2s\\}", "after_tex": "\\begin{array}{c|cccccc}*&e&r&r^2&s&rs&r^2s\\\\\\hline e&e&r&r^2&s&rs&r^2s\\\\r&r&r^2&e&rs&r^2s&s\\\\r^2&r^2&e&r&r^2s&s&rs\\\\s&s&r^2s&rs&e&r^2&r\\\\rs&rs&s&r^2s&r&e&r^2\\\\r^2s&r^2s&rs&s&r^2&r&e\\\\\\end{array}", "relation": "implies", "rule": "calc", "conditions": [], "explanation": { "zh-Hant": "將給定配對公式逐36對展開，全部輸出在同一六元素載體，沒有缺格。", "en": "Expand the pair formula for all 36 products; every output belongs to the same six-element carrier and no cell is missing." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1" ], "explanation": { "zh-Hant": "labels依序綁定六個pair，故同名r²不代表額外第七元素。", "en": "The six labels bind the six pairs; r^2 is not an additional seventh element." } }, "leaf_basis": "calc" }, { "id": "S02", "uses": [ "G1", "S01" ], "before_tex": "(i,j)(k,l)=(i+(-1)^jk,j+l)\\quad(\\bmod(3,2))", "after_tex": "((i,j)(k,l))(m,n)=(i+(-1)^jk+(-1)^{j+l}m,j+l+n)=(i,j)((k,l)(m,n));\\quad e=(0,0);\\quad(i,j)^{-1}=(-(-1)^ji,j)", "relation": "implies", "rule": "calc", "conditions": [], "explanation": { "zh-Hant": "兩種括號的座標公式相同，證全部216 triples結合律；(0,0)兩側中性，列出的逆元兩側乘回(0,0)。", "en": "Both bracketings have identical coordinates, proving associativity for all 216 triples; (0,0) is a two-sided identity and the inverse formula multiplies back on both sides." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S01" ], "explanation": { "zh-Hant": "指數j+l只依mod2，因(−1)²=1，公式對所有代表一致；逆元清單e,r²,r,s,rs,r²s完整。", "en": "The sign depends only on j+l modulo 2 since (-1)^2=1; the full inverse list is e,r^2,r,s,rs,r^2s." } }, "leaf_basis": "calc" }, { "id": "S03", "uses": [ "G1", "S02" ], "before_tex": "X=\\{0,1,2,3\\}", "after_tex": "\\begin{array}{c|cccc}\\cdot&0&1&2&3\\\\\\hline e&0&1&2&3\\\\r&1&2&0&3\\\\r^2&2&0&1&3\\\\s&0&2&1&3\\\\rs&1&0&2&3\\\\r^2s&2&1&0&3\\\\\\end{array}", "relation": "implies", "rule": "calc", "conditions": [], "explanation": { "zh-Hant": "全部24個作用值由同一公式展開。對x<3，g·(h·x)=i+(−1)^j k+(−1)^{j+l}x=(gh)·x；3兩邊皆固定。e逐點固定。", "en": "Expand all 24 action values. For x<3, g*(h*x)=i+(-1)^j*k+(-1)^(j+l)*x=(gh)*x; both sides fix 3, and e fixes every point." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S02" ], "explanation": { "zh-Hant": "表格對每個g與每個point恰有一值；相容式涵蓋全部6²·4=144 triples，不是只核每列像排列。", "en": "Every group-point pair has one value; the formula covers all 144 triples, not merely permutation rows." } }, "leaf_basis": "calc" }, { "id": "S04", "uses": [ "G1", "S03" ], "before_tex": "\\phi_0:G\\to X,\\quad\\phi_0(g)=g\\cdot0", "after_tex": "\\operatorname{im}\\phi_0=O=\\{0,1,2\\},\\quad H=\\phi_0^{-1}(\\{0\\})=\\{e,s\\}", "relation": "implies", "rule": "calc", "conditions": [], "explanation": { "zh-Hant": "完整0-column計orbit與stabilizer，兩者綁同一point0。", "en": "The full 0-column computes orbit and stabilizer, both bound to point zero." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S03" ], "explanation": { "zh-Hant": "point3雖也在X，不能混用其stabilizer G。", "en": "Point 3 also lies in X, but its stabilizer G cannot be substituted." } }, "leaf_basis": "calc" }, { "id": "S05", "uses": [ "G1", "S04" ], "before_tex": "g=e,r,r^2,s,rs,r^2s", "after_tex": "\\phi_0^{-1}(0)=\\{e,s\\},\\quad\\phi_0^{-1}(1)=\\{r,rs\\},\\quad\\phi_0^{-1}(2)=\\{r^2,r^2s\\}", "relation": "implies", "rule": "calc", "conditions": [], "explanation": { "zh-Hant": "六個group元素全部分到三個fiber，沒有遺漏或重複。", "en": "All six group elements are partitioned among three fibers without omissions or overlap." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S04" ], "explanation": { "zh-Hant": "每個目標在orbit內都有前像，所以φ₀對orbit滿射。", "en": "Every orbit point has a preimage, so phi0 is onto its orbit." } }, "leaf_basis": "calc" }, { "id": "S06", "uses": [ "G1", "S01", "S05" ], "before_tex": "H=\\{e,s\\}", "after_tex": "eH=\\{e,s\\},\\quad rH=\\{r,rs\\},\\quad r^2H=\\{r^2,r^2s\\}", "relation": "implies", "rule": "calc", "conditions": [], "explanation": { "zh-Hant": "用完整乘法表逐元素計三個左陪集，正好等於三個fibers。", "en": "Compute all three left cosets from the full table; they exactly match the fibers." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S01", "S05" ], "explanation": { "zh-Hant": "這是g₀H左陪集，乘法順序不能改成Hg₀。", "en": "These are left cosets g0*H; changing to H*g0 would change the order." } }, "leaf_basis": "calc" }, { "id": "S07", "uses": [ "G1", "S02", "S03", "S04", "S06" ], "before_tex": "g_0\\cdot0=y", "after_tex": "g\\cdot0=y\\iff(g_0^{-1}g)\\cdot0=0\\iff g_0^{-1}g\\in H\\iff g\\in g_0H", "relation": "equivalent", "rule": "calc", "conditions": [], "explanation": { "zh-Hant": "作用公理與逆元給fiber=左陪集的雙向證明，不是只碰巧清單相等。", "en": "Action laws and inverses give the two-way fiber/left-coset proof, not merely coincident lists." }, "reverse_audit": { "kind": "inverse_equivalence", "required_ids": [ "G1", "S02", "S03", "S04", "S06" ], "explanation": { "zh-Hant": "逆向g=g₀h使g·0=g₀·(h·0)=g₀·0，h須在同一H。", "en": "Conversely g=g0*h gives g*0=g0*(h*0)=g0*0, with h in the same H." } }, "leaf_basis": "calc" }, { "id": "S08", "uses": [ "G1", "S02", "S06", "S07" ], "before_tex": "L_{g_0}:H\\to g_0H,\\quad h\\mapsto g_0h", "after_tex": "L_{g_0}^{-1}(k)=g_0^{-1}k\\Rightarrow|g_0H|=|H|=2", "relation": "implies", "rule": "calc", "conditions": [], "explanation": { "zh-Hant": "左平移有明示逆函數，故每個fiber確有相同大小2。", "en": "Left translation has an explicit inverse, so every fiber has the same size two." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S02", "S06", "S07" ], "explanation": { "zh-Hant": "單純看到三組兩元素不解釋一般機制，逆映射補足雙射理由。", "en": "Three observed pairs alone would not explain the mechanism; the inverse map supplies the bijection." } }, "leaf_basis": "calc" }, { "id": "S09", "uses": [ "G1", "S04", "S05", "S08" ], "before_tex": "G=\\phi_0^{-1}(0)\\sqcup\\phi_0^{-1}(1)\\sqcup\\phi_0^{-1}(2)", "after_tex": "|G|=3\\cdot2=|G\\cdot0|\\,|\\operatorname{Stab}_G(0)|=6", "relation": "implies", "rule": "mko_definition", "conditions": [], "explanation": { "zh-Hant": "不交完整fibers與等大小雙射證乘法計數式。", "en": "Disjoint complete fibers and equal-size bijections prove the counting identity." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S04", "S05", "S08" ], "explanation": { "zh-Hant": "並不要求作用free或transitive：s固定0且point3不在orbit0。", "en": "Neither freeness nor transitivity is needed: s fixes 0 and point 3 is outside its orbit." } }, "leaf_basis": "mko_definition" } ], "result": { "zh-Hant": "point0的orbit大小3、stabilizer大小2；三個fiber均為左陪集，所以6=3·2。", "en": "At point 0 the orbit has size 3 and stabilizer size 2; all three fibers are left cosets, giving 6=3*2." }, "negative_controls": [ { "claim": { "zh-Hant": "改用Stab(3)=G仍可和orbit0相乘。", "en": "Use Stab(3)=G with orbit 0 in the product." }, "expected": "reject", "reason": { "zh-Hant": "不同point混綁會得3·6=18≠6。", "en": "Mixing points gives 3*6=18, not 6." } }, { "claim": { "zh-Hant": "此定理要求free。", "en": "The theorem requires a free action." }, "expected": "reject", "reason": { "zh-Hant": "s≠e固定0，但fiber證書與6=3·2均成立。", "en": "Nonidentity s fixes 0, yet the fiber certificate and 6=3*2 hold." } }, { "claim": { "zh-Hant": "只有數值6=3·2就夠證任意action。", "en": "Numerical 6=3*2 alone certifies any action." }, "expected": "reject", "reason": { "zh-Hant": "還須同一作用與point的完整fibers及coset雙射；任意數字不給這些。", "en": "The same action/point needs full fibers and coset bijections, which arbitrary numbers do not supply." } } ], "status": "candidate", "gaps": [], "verification": { "structure": "checked_local_script", "mathematics": "candidate_argument", "formalization": "not_formalized" }, "rule_glossary": [ { "id": "calc", "kind": "prerequisite", "statement": { "zh-Hant": "假設精確模3、模2整數算術、有限集合外延相等與群／作用的定義；不預設此運算已為群。", "en": "Assume exact arithmetic modulo 3 and 2, extensional finite-set equality, and definitions of groups/actions; this operation is not pre-assumed to be a group." } }, { "id": "mko_definition", "kind": "cited", "statement": { "zh-Hant": "固定同一完整有限作用G作用X與x。action map phi_x(g)=g·x對orbit滿射；若g0·x=y，fiber精確等於左陪集g0 Stab_G(x)，故|G|=|G·x||Stab_G(x)|。不要求action free、faithful或transitive；proper sample、不同point綁定或只碰巧數值相等皆不足。", "en": "Fix the same fully certified action of G on X and point x. The action map phi_x(g)=g*x is surjective onto the orbit. If g0*x=y, its fiber over y is exactly the left coset g0*Stab_G(x); hence |G|=|G*x|*|Stab_G(x)|. The action need not be free, faithful or transitive. A proper group sample, a different point binding or a coincidentally equal numerical product is insufficient." }, "source": { "mko_id": "mko-orbit-stabilizer-theorem", "sha256": "0ae3563c72f6bb6e1702aa5a64f490a1f2b0ca0741f19d17ac3790775e66c38f", "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f" } } ], "revision_log": [], "numeric_checks": [ { "id": "count", "lhs": "3*2", "rhs": "6", "step": "S09" } ] }, "presentation": { "md_scope": { "zh-Hant": "同一完整有限作用與同一point0，明示fiber-coset雙向與平移雙射。", "en": "The same complete finite action and point zero, with a two-way fiber/coset proof and translation bijections." }, "md_symbols": [ [ "G", "六元素群載體", "six-element group carrier" ], [ "r,s", "r³=e,s²=e,sr=r²s的labels", "labels with r^3=e,s^2=e,sr=r^2s" ], [ "\\phi_0", "point0的action map", "action map at point zero" ], [ "H", "同point的stabilizer", "stabilizer at the same point" ] ], "md_error_controls": [ { "title": "同一群表 / Same group table", "zh": "全部積、逆元與cosets綁定這個完整Cayley表。", "en": "Bind every product, inverse and coset to this complete Cayley table." }, { "title": "完整枚舉 / Complete enumeration", "zh": "任何全稱有限性質都需全載體或全稱公式，不用proper sample。", "en": "Every universal finite property needs the full carrier or a universal formula, not a proper sample." } ], "md_open_items": [] }, "revision": { "id": "ocme-new142-reader-adoption-20261021", "collection": "coverage-20261021", "source_commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f", "adopted_source_sha256": "f78007176051f8883326c27cdfa614a2deb1ac3401d22f708d72381b178efd39", "original_source_sha256": "f78007176051f8883326c27cdfa614a2deb1ac3401d22f708d72381b178efd39", "owner_adopted": true, "bilingual_sync": "complete_source_bound", "formal_Lean": "NOT_RUN", "reader_review": "142_of_142_source_bound", "wording_review": "original_Lead_decisions_resolved", "glossary_reader_paraphrases": [], "independent_AI_concurrence": "SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW", "issue_ids": [], "parent_manifest_sha256": "d29f4cae8229ebd24f60d496cb1d3cd4ad8c0a4dca9d44daab0c3245c544283a", "scope": "Owner-adopted reader revision. Lead and bilingual review concern the recorded text and premises; no new universal or Lean proof, canonical promotion or historical-label rewrite." } }