# 全 3 冊完全形式化 — ギャップ調査 (2026-07-16)

> ⚠ **DEMOTED (hub 裁定 9154、2026-07-19) — scope の一次情報にしない**。scope の正本 =
> `git log` + `issues/` + **実測 grep**。本ノート (+ [`three_books_full_survey_2026_07_16.json`](three_books_full_survey_2026_07_16.json))
> は 2026-07-16 時点の**出発点スナップショット**として保持するが、Isaacs だけで Ch.1/2/5/8/9/10 の
> 6 章が実体と食い違い、"Confirmed missing (slim pass)" 等の検証済み文言も無効だった (Ch.8 は
> 「ディレクトリすら無い」と断言していたが実際は 14 leaf / 5,707 行)。**未/部分ラベルは着手前に必ず
> 実測で再確認する** (CLAUDE.md「スコープ」節)。
>
> (原文: FT 完成 (`feitThompson` axiom-clean, 2026-07-15) を受け、FT 経路限定を解除して 3 冊
> (Isaacs / BG / Peterfalvi) 全体を形式化するフェーズへの移行調査。教科書側の**全番号付き結果**を
> 列挙し、repo/mathlib 被覆を判定した。)

## 手法と信頼性

- multi-agent workflow (`wf_9c031527-959`): 52 unit (Isaacs 11 章 / BG §1–16+App.A–E / Pf §1–16+付録 5) それぞれに
  調査 agent (mmd 全読 + repo/mathlib/Coq grep) → missing/partial 限定の敵対的検証 agent、最後に完全性 critic。
- critic が全 unit の列挙を mmd と突合し **contiguous + endpoint 一致を確認済** (Isaacs 1.46/2.20/3.36/4.38/5.30/6.24/7.8/8.44/9.31/10.28/X.23、BG 各節、Pf (1.10)〜(14.16))。
- critic 指摘 7 件のラベル問題は inline 検証の上で本データに**補正適用済み** (JSON の `corrections` 参照)。
- **doneness 規約**: opaque-Prop scaffold (自由 Prop フィールドの Hypothesis/Conclusion 構造) は sorry の有無に関わらず **missing** と数える ([[scaffold-sorry-free-not-done]])。
- effort は agent の概算 (S<1日 / M=1–3日 / L=1–2週 / XL=数週+)。着手判断の基準ではなく規模感の記録 ([[feedback-cost-scope-not-a-criterion]])。

## 全体集計

> 🚨 **2026-07-19: Isaacs の行は全面的に信用できない (実測で全章クローズ済と判明)**。
> 下表と per-unit 表は 2026-07-16 時点の調査結果だが、Isaacs については **Ch.1–Ch.10 +
> 付録の全 349 結果が形式化済 (実 sorry 0)** であることが同 07-19 の実測で確定した。
> 特に **Ch.8 (「ディレクトリすら存在しない」と断言) と Ch.10 (「Cor 10.17 のみ」) は
> 100% 誤り** — 実体は `Ch08_PermutationGroups/` 14 leaf・`Ch10_MoreTransfer/` 6 leaf。
> 原因は Ch.8/Ch.10 の実装が調査翌日以降 (2026-07-17〜19) に入ったこと。
> **各行の「Confirmed missing (slim pass)」「refutation attempted, failed」等の検証文言も
> Isaacs については無効**。BG / Peterfalvi の行は未再検証。
> ⇒ **Isaacs のギャップを理由に作業を起こす前に、必ず番号 grep + 記述的名前 grep で
> 実物を確認すること**。この note の Isaacs 記述を一次情報として使わない。

| 冊 | 結果数 | 済 | 特殊化 | 部分 | 未 | mathlib | 実作業ギャップ (未+部分) |
|---|---|---|---|---|---|---|---|
| Isaacs | 349 | 143 | 8 | 23 | 101 | 74 | 124 (S:82 M:40 L:2) |
| BG | 228 | 177 | 20 | 14 | 13 | 4 | 25 (S:10 M:11 L:3 XL:1) |
| Peterfalvi | 238 | 149 | 26 | 13 | 50 | 0 | 63 (S:23 M:24 L:10 XL:6) |
| **計** | **815** | 469 | 54 | 50 | 164 | 78 | **212** (S:115 M:75 L:15 XL:7) |

status 定義: **済** = 教科書強度の Lean statement が sorry-free / **特殊化** = 存在するが教科書より狭い (要判定・要一般化) /
**部分** = 一部のみ or sorried / **未** = 対応物なし (反証 grep 済) / **mathlib** = mathlib が完全被覆 (wrapper 不要、対応は docstring/notes 記録)。

## Isaacs

| unit | n | 済 | 特殊化 | 部分 | 未 | mathlib |
|---|---|---|---|---|---|---|
| Isaacs Ch.1 | 46 | 25 | 0 | 1 | 0 | 20 |
| Isaacs Ch.2 | 24 | 19 | 0 | 1 | 0 | 4 |
| Isaacs Ch.3 | 39 | 18 | 2 | 5 | 10 | 4 |
| Isaacs Ch.4 | 42 | 29 | 3 | 1 | 6 | 3 |
| Isaacs Ch.5 | 34 | 21 | 1 | 1 | 2 | 9 |
| Isaacs Ch.6 | 26 | 19 | 1 | 4 | 2 | 0 |
| Isaacs Ch.7 | 9 | 9 | 0 | 0 | 0 | 0 |
| Isaacs Ch.8 | 44 | 0 → **44** | 0 | 0 | 25 → **0** | 19 |
| Isaacs Ch.9 | 34 | 0 → **34** | 1 → **0** | 4 → **0** | 29 → **0** | 0 |
| Isaacs Ch.10 | 28 | 1 → **28** | 0 | 1 → **0** | 26 → **0** | 0 |
| Isaacs Appendix (The Basics) | 23 | 8 | 0 | 0 | 0 | 15 |

### Isaacs Ch.1 — Isaacs Ch.1 Sylow Theory

> ✅ **2026-07-19: 唯一の「部分」だった Lem 1.43 の等号条件節もクローズ (issue 9153)**。
> `OddOrder/GroupTheory/ChermakDelgado.lean` の
> `chermakDelgadoMeasure_mul_eq_conditions` が書籍の 2 結論
> (`J = HK` かつ `C_G(D) = C_G(H)·C_G(K)`) を lattice 仮定なしで与える。
> ⇒ **Isaacs Ch.1 は残作業なし**。


> Isaacs Ch.1 (Sylow Theory, results 1.1-1.46) is essentially complete: 45 of 46 numbered results are either fully covered by mathlib (18 results — all of classical Sylow E/C/D theory, orbit-stabilizer, Frattini argument, nilpotency TFAE, central-series comparison) or formalized sorry-free in the repo at full book strength (27 results), with the single gap being the equality-case clause of Lemma 1.43 (inequality proved; the equality consequences exist only inline for lattice members, effort S). The repo's genuinely new contributions beyond mathlib are: the strong Sylow counting congruence 1.16, the p-group normal-series lemmas 1.19/1.23 (with the normal form of Cor 1.24 living in the BG tree as normal_subgroup_card_pow_le_of_pGroup), the O_p(G) (`opCore`) and Fitting subgroup (`fitting`) definitions with extensive API (mathlib has neither), the small-order nonsimplicity theorems 1.30-1.36 including the full G ≅ S₄ classification for order 24, Brodkey theory 1.37-1.40, and the complete Chermak-Delgado theory 1.41-1.46 (factored into OddOrder/GroupTheory/ChermakDelgado.lean, with 1.42's equality condition even strengthened to an iff). Two structural notes: the formalization is often stronger or differently factored than the book (1.36 produces an explicit normal subgroup; 1.33 gives an explicit MulEquiv to Perm (Fin 4)), and mathlib-covered results are deliberately not wrapped (per project wrapper policy), with the correspondences recorded in section docstrings whose cited mathlib names I verified all exist in the pinned mathlib. Minor cosmetic point: Cor 1.24's Ch01 docstring cites only the weak (non-normal) mathlib form and does not cross-reference the full normal version already proved in BG S01_Solvable.lean. Slim verification pass: I attempted to refute and instead CONFIRMED the sole partial label (Lem 1.43 — inequality standalone at ChermakDelgado.lean:101, but the equality-case consequences J = HK and C_G(D) = C_G(H)C_G(K) exist only inline under lattice-membership hypotheses in chermakDelgadoLattice_sup_eq_mul, the centralizer-product clause never at set level, and pinned mathlib has no ChermakDelgado file), and read the actual Lean statements of the three most load-bearing formalized results — Thm 1.16 (card_sylow_modEq_one_of_max_inter), Thm 1.19 (IsPGroup.normal_inf_center_nontrivial), and Cor 1.28 (fitting + normal/characteristic/isNilpotent instances + nilpotent_normal_le_fitting) — confirming all three at full book strength, so no status changes were made.

| 結果 | 状態 | 規模 | 内容 | メモ |
|---|---|---|---|---|
| Lem 1.43 | 部分 | S | m_G(H)·m_G(K) ≤ m_G(H∩K)·m_G(⟨H,K⟩), equality forcing J = HK and C_G(D) = C_G(H)C_G(K) | Partial label re-verified in slim pass (refutation attempted, failed). The main inequality is fully proved (ChermakDelgado.lean:101). The equality-case clause is not a standalone statement: J = HK is derived inline only … |

### Isaacs Ch.2 — Isaacs Ch.2 Subnormality

> ✅ **2026-07-19 実測: 唯一のギャップとされた Lem 2.14 は既に完全形式化済 (本節は stale)**。
> `Ch02_Subnormality/DihedralBasics.lean` (sorry-free、`Main.lean` から import 済) が
> (a) の iff (`forall_involution_iff_conj_eq_inv`)、全反転 (`conj_eq_inv_of_notMem_zpowers`)、
> `c*t` involution、生成 (`closure_mul_involution_eq_top`)、および (b) の
> `C` nontrivial (`mul_ne_one_of_ne`) / `s,t ∉ C` (`left,right_notMem_zpowers_mul`) /
> `|D:C| = 2` (`index_zpowers_mul_eq_two`) / capstone
> (`sq_eq_one_of_notMem_zpowers_mul`) を全て book 強度で持つ。有限性も仮定していない。
> ⇒ **Isaacs Ch.2 は残作業なし**。

> Isaacs Ch.2 (Subnormality, results 2.1-2.20) is essentially completely formalized and sorry-free: 16 of 20 numbered results have full-book-strength Lean statements in OddOrder/Isaacs/Ch02_Subnormality/{Basic,Theorem211Wielandt,Main}.lean, 3 more (2.4, 2.7, and the subnormality/normal-closure definitions) are covered directly by mathlib (Subgroup.IsSubnormal.inf, Subgroup.commute_of_normal_of_disjoint) per the repo's no-wrapper policy. The only gap is Lemma 2.14 (dihedral structure of a group generated by two involutions): its Matsuyama-facing essence is in Ch02 and — beyond what the original survey recorded — the DihedralGroup identification exists in Ch06 (dihedralIsoOfInverting), leaving only (a)'s easy iff direction and (b)'s explicit C≠1/s,t∉C/|D:C|=2 packaging unstated — a small (S) task. Notable factoring differences: Lucchini 2.20's full theorem lives in Ch04_Commutators/ForwardFromCh02.lean for import-order reasons (stale comments in Ch02/Ch03 still call its K=⊥ case an 'axiom', but it is now a proven theorem with zero sorries and zero axiom declarations), and the repo goes beyond the book by deriving the single-element Baer-Suzuki p-core theorem (baerSuzuki_pCore) from Baer 2.12. End-of-section Problems (2A.x-2D.x) are not formalized, consistent with the project formalizing numbered results only. Slim-pass verification: re-read Thm 2.5, Thm 2.9, and Thm 2.12 statements and confirmed full book strength; for Lem 2.14 the 'partial' label is confirmed but its notes were corrected — the repo DOES identify the inverting index-2-cyclic configuration with mathlib's DihedralGroup via Ch06_FrobeniusActions/DQSDRecognition.lean (dihedralIsoOfInverting, general finite group), so only the trivial iff direction and part (b)'s index/membership packaging remain missing.

| 結果 | 状態 | 規模 | 内容 | メモ |
|---|---|---|---|---|
| Lem 2.14 | 部分 | S | Dihedral group basics: two involutions generate a dihedral group; index-2 cyclic character… | More is present than first surveyed, but the label 'partial' stands. Ch02 (Theorem211Wielandt.lean) proves the Matsuyama-facing essence: t inverts every element of ⟨s·t⟩ and ⟨s,t⟩ = ⟨st⟩ ∪ ⟨st⟩·t. CORRECTION: the Dihedra… |

### Isaacs Ch.3 — Isaacs Ch.3 Split Extensions

> ⚠ **この節の per-item 表は 2026-07-18 時点で stale** (lane a 確認)。
> 本文の "slim verification pass (**2026-07-16**)" 以降に入った形式化が反映されていない。
> **実測で反証済みの項目**:
> - **Lem 3.6 「未」→ 実際は完了**: `Ch03_SplitExtensions/CrossedHomomorphism.lean`
>   (commit `b21ea73b5`, 2026-07-17) が (a)(b)(c)(d) 全部を**非可換係数**で実装済み。
>   表が「mathlib の `oneCocycles` は可換係数限定」と指摘したギャップは、
>   まさにその形で埋まっている。
> - **Thm 3.15 「未」→ 実際は完了**: `Ch07_ThompsonSubgroup/ForwardFromCh03.lean`
>   (commit `dcfb8f906`, 2026-07-17) が Burnside 経由で証明済み。表が言う
>   「docstring-only placeholder」はもう当てはまらない (実 theorem が入っている)。
> - **Lem 3.7 は「未」でなく意図的な no-wrapper 判断**: `CrossedHomomorphism.lean`
>   の docstring に mathlib `leftTransversals.diff` との逐条対応表がある
>   (CLAUDE.md ラッパー方針に従い Lean 実体を書かない)。
>
> → **着手前に必ず repo を直接 grep すること** (番号 + descriptive 名の両方)。
> **2026-07-18 追加検証: 残りの 未/部分 項目も全て repo にヒットした** — Ch.3 は
> 実質完了と見てよい。確認した所在:
> 3.16/3.17 = `Theorem315.lean`、3.18 = `PiSeparableSeries.lean`、
> 3.22 = `Main.lean` (**full form** `piLength_le_one_of_abelian_pi_hall` が
> AxiomsCheck 入り — 表の「commutator 含意どまり」は誤り)、
> 3.26/3.31/3.32 = `Ch04_Commutators/ForwardFromCh03.lean`、
> **3.33/3.34 = `Ch04_Commutators/HartleyTurull.lean`** (2026-07-17, 宣言 8 本 —
> 表の「Tier 2 クラスタは entirely missing」は誤り)、
> 3.35/3.36 = `CyclicExtensions.lean` (いずれも AxiomsCheck 入り)。
> → Ch.3 の残作業は **特殊化債務のみ** (Lem 3.1 / Lem 3.11)。Isaacs 全章は現在 **sorry ゼロ**なので、
> sorry 数はこの章の残作業の指標にならない (未形式化項目は sorry を生まない)。
> **2026-07-19: その 2 件とも解消済** (Lem 3.11 = `3cff7a105`、Lem 3.1 = issue 1039)。
> ⇒ **Isaacs Ch.3 は本 survey の全項目クローズ**。レーン a の次の frontier は Ch.4 Mann クラスタ。


> Ch.3 is very thoroughly covered: 27 of 36 numbered results are fully formalized or mathlib-covered, including the FT-critical Hall E/C theorems (plus Hall D, which the book leaves as Problem 3C.1), Hall-Higman 1.2.3, Schur-Zassenhaus (existence = mathlib; the conjugacy Theorem 3.12 is repo-proved at full strength for both solvability cases, with the stronger conclusion that the conjugator lies in N), and the whole Glauberman coprime-action "Tier 1" (3.23-3.25, 3.27-3.30) — which lives in OddOrder/Isaacs/Ch04_Commutators/ForwardFromCh03.lean and OddOrder/Mathlib/SchurZassenhausConj.lean, not in the Ch03 directory. The main gaps are (a) the two Burnside-gated solvability criteria 3.15 and 3.17, whose placeholder file is still intentionally empty even though Burnside p^a q^b is now proved sorry-free in Ch.7 (burnside_p_pow_q_pow), so both are unblocked; and (b) the coprime-orbit "Tier 2" cluster 3.26, 3.31-3.34 (class bijection, Hartley-Turull, orbit-size product), entirely missing though all Glauberman ingredients exist. Four results are weaker than the book: 3.22 proves only a commutator containment rather than genuine pi-length <= 1 (BG's HasPiLengthOne def exists with no theorem), 3.16 lacks the |H:H∩K|=|G:K| clause, 3.35 has only the uniqueness half, and 3.36's existence statement omits the |G/N|=m clause (degenerate witnesses satisfy it). The crossed-homomorphism machinery 3.6/3.7 was never formalized because mathlib's QuotientDiff proof route made it unnecessary; pi-Hall/pi-separable theory is repo-built (IsHallSubgroup, IsPiSeparable via an upper pi-Fitting series) and is in places stronger than the book (e.g. a Hall-subgroup generalization of Cor 3.25 in BG §1). Slim verification pass (2026-07-16): all 15 flagged missing/partial labels were re-attacked by book-number and descriptive-content greps and every one stands — no status changed (Ch07/ForwardFromCh03.lean remains a docstring-only placeholder whose theorem signatures sit inside code fences; the only near-miss is a specialized extension-closure instance isPiSeparable_of_normalPSubgroup_quotient_hasNormalPComplement noted under 3.18) — and the three most load-bearing formalized results (3.12 SZ-conjugacy, 3.13/3.14 Hall E/C, 3.21 Hall-Higman 1.2.3) were read in source and confirmed at full book strength.

| 結果 | 状態 | 規模 | 内容 | メモ |
|---|---|---|---|---|
| Lem 3.6 | 未 | S | Crossed homomorphism basics: φ(1)=1, kernel is a subgroup, fibers are cosets, /G:K/=/φ(G)/ | Verified missing (slim pass): only trace is a correspondence-table row in Ch03_SplitExtensions/Basic.lean:451 pointing at mathlib's OneCocycle (GroupCohomology.LowDegree), which covers only abelian-coefficient cocycles a… |
| Lem 3.7 | 未 | S | Transversal difference function d(S,T): cocycle identity, translation, d(S,Sn)=n^/G:N/ | Verified missing (slim pass): same situation as 3.6 — proof machinery for Thm 3.5. mathlib's SchurZassenhaus uses the equivalent smul_diff/QuotientDiff device internally but the book-form lemma is not stated anywhere in … |
| Thm 3.15 | 未 | M | Converse of Hall E: G with a p-complement for every prime p is solvable | Verified missing (slim pass): OddOrder/Isaacs/Ch07_ThompsonSubgroup/ForwardFromCh03.lean is still docstring-only — the solvable_of_pcomplement_exists signature at its line 13 sits inside the module docstring's code fence… |
| Lem 3.16 | 部分 | S | Coprime indices: G = HK and /H : H∩K/ = /G : K/ | Verified partial (slim pass): Theorem315.lean:32 proves only H ⊔ K = ⊤ from coprime indices. The index-equality clause /H:H∩K/ = /G:K/ is not stated anywhere in the repo (relIndex greps show only unrelated uses); mathlib… |
| Thm 3.17 | 未 | M | Wielandt: three solvable subgroups with pairwise coprime indices force G solvable | Verified missing (slim pass): solvable_of_three_subgroups appears only inside the module docstring of Ch07/ForwardFromCh03.lean (namespace intentionally empty). Requires Burnside p^a q^b for the simple case — now availab… |
| Lem 3.18 | 未 | M | A subnormal series with π/π' factors already makes G π-separable | Verified missing (slim pass): full inventory of IsPiSeparable lemmas shows only quotient/normal-subgroup/complement closures (quotient_isPiSeparable, normalSubgroup_isPiSeparable, Subgroup.isPiSeparable_of_isPiSeparable,… |
| Thm 3.22 | 部分 | S | π-separable with abelian Hall π-subgroup has π-length ≤ 1 | Verified partial (slim pass, statement+proof read at Main.lean:720): the Lean theorem concludes only ⁅O_{π',π}(G), O_{π',π}(G)⁆ ≤ O_π'(G) — a strictly weaker fact whose proof indeed uses no Hall-Higman (just O_π(Ḡ) ≤ abe… |
| Thm 3.26 | 未 | S | K ↦ K ∩ C_G(A) is a bijection from A-invariant classes of G onto classes of C_G(A) | Verified missing (slim pass): re-searched ConjClasses/invariant-class/class-bijection and the full declaration list of Ch04/ForwardFromCh03.lean (contains exactly the Tier-1 cluster, nothing more); OddOrder/GroupTheory/C… |
| Thm 3.31 | 未 | M | Hartley-Turull: coprime action can be replaced by an action on an abelian group with the s… | Verified missing (slim pass): Hartley/Turull/orbit-size greps hit only unrelated character-theory files; Main.lean:770 explicitly lists 'Thm 3.29-3.31 orbit structure' as not-implemented Tier 2. One proof ingredient exis… |
| Lem 3.32 | 未 | S | For an A-invariant P ∈ Syl_p(G), P ∩ C_G(A) is a Sylow p-subgroup of C_G(A) | Verified missing (slim pass): fixedSubgroup+Sylow/Hall greps return zero hits; Sylow occurrences in CoprimeAction.lean, CoprimeFixedPoints.lean and BG S01_Solvable.lean are all unrelated (Lem 1.14 helpers). Isaacs's proo… |
| Lem 3.33 | 未 | M | Equal fixed-point counts for all B ≤ A on two finite A-sets yield an A-equivariant bijecti… | Verified missing (slim pass): fixed-point-count/equivariant greps hit only unrelated BG/representation files. Purely combinatorial induction on /Ω/ (removing minimal orbits); self-contained with no group-theoretic prereq… |
| Thm 3.34 | 未 | M | Coprime action with coprime orbit sizes m, n has an orbit of size mn | Verified missing (slim pass, no counterpart under number or descriptive searches). Depends on 3.31 (abelian replacement), 3.16's index clause, 3.27, and the transfer-like τ map argument; roughly 2-3 days once 3.31-3.33 e… |
| Thm 3.35 | 部分 | S | Cyclic extension uniqueness: matching data (N≅N₀, g↦g₀, g^m, conjugation) gives a unique i… | Verified partial (slim pass, statement read at Main.lean:1156): only the uniqueness half is proved (two homomorphisms agreeing on N and at the generator lift coincide — a mild generalization of the book's uniqueness clau… |
| Thm 3.36 | 部分 | S | Cyclic extension existence: given a ∈ N, σ ∈ Aut(N) with σ(a)=a and σ^m = conj(a), there i… | Verified partial (slim pass, statement read at Main.lean:1278): the construction (N ⋊_σ ℤ quotiented by the central element (a⁻¹, m)) is fully proved with clauses zpowers(gN₀) = ⊤, g^m = ι a, conjugation = σ. But the ∃-s… |
| Def §3A (wreath) | 部分 | S | Wreath product H ≀ G (base group, regular wreath product) | Verified (slim pass): the pinned mathlib does ship Mathlib/GroupTheory/RegularWreathProduct.lean (regular wreath product only); no wreath content anywhere in OddOrder/. The book's general form over an arbitrary G-set Ω i… |

特殊化債務 (教科書より狭い形で証明済 — 一般化要否は着手時に判定):

- ~~**Lem 3.1**~~ **✅ 2026-07-19 解消 (issue 1039)** — 書籍 p.70 の two-abstract-groups 形
  `existsUnique_mulEquiv_of_isComplement'` を新 leaf `Ch03_SplitExtensions/SplitExtensionUniqueness.lean`
  に実装 (∃! 込み, AxiomsCheck 入り)。旧 internal form `mulEquivSubgroupOfComplement` は
  mathlib `SemidirectProduct.mulEquivSubgroup` の純粋リネーム (下流 0 件) ゆえ削除。
- ✅ **2026-07-19 解消** — `[Finite G]` を書籍どおり `[Finite ↥M]` に弱めた
  (`Ch03_SplitExtensions/Basic.lean` `solvable_minimal_normal_isElementaryAbelian`;
  abelian 節は元から有限性不要)。以下は stale: ~~Lem 3.11 Solvable minimal normal subgroup is abelian; if finite, elementary abe — Stated for a finite solvable ambient G (book: arbitrary G with M solvable minimal normal, plus an infinite-abelian clause). Covers every downstream use in the b…

### Isaacs Ch.4 — Isaacs Ch.4 Commutators

> ⚠ **2026-07-19 実測: 本節の per-item 表は全面 stale。Ch.4 は全項目クローズ済**
> (レーン a、番号 + descriptive 名の両方で grep 実施)。内訳:
> - **Mann クラスタ (4B Def M(G) / 4.14 / 4.15 / 4.17 / 4.18 / 4.19)** = `Ch04_Commutators/Mann.lean`
>   (2026-07-17 `3e8e319a8`、874 行、AxiomsCheck 6 件登録済)。表の「未」は全て誤り。
> - **4.16** も同 file の `centralizer_eq_self_of_maximal_abelian_normal_of_isNilpotent` で
>   **nilpotent 一般形**まで済 (表の「p-group 特殊化」は stale)。
> - **4.29** は `iterCommutator_inl_inr_two_eq_one_of_coprime` が既に**書籍印刷形 (無条件)**
>   (表の「solvability 仮定が残る」は stale — 条件付き版は engine として併存するだけ)。
> - **4.12** = 2026-07-19 に書籍形 (任意括弧付け) を `CommutatorTree` で実装
>   (`CommutatorTree.eval_le_lowerCentralSeries`、左結合形 `eval_leftTree` = lcs も併記)。
> - **4.6** = 2026-07-19 に同型節 `nonempty_quotientInfCenterEquivCommutator`
>   (`A/(A ∩ Z(G)) ≅ G'`) を追加し、cardinality 節の `[Finite G]` を除去
>   (`Nat.card` は無限を 0 に潰すので**書籍より一般**)。
> ⇒ **Isaacs Ch.4 は残作業なし**。

> Isaacs Ch.4 (Commutators, results 4.1-4.38 + Hall-Witt identity) is essentially complete: 30 of 39 surveyed items are fully formalized sorry-free in OddOrder/Isaacs/Ch04_Commutators/Main/{CommutatorBasics,ThreeSubgroups,ThreeSubgroupsCoprime,BaerTrick,ChainNilpotent}.lean, 3 are pure mathlib (4.2, 4.9 three-subgroups, Hall-Witt), and several are formalized STRONGER than the book (4.4/4.8 generalized from p-groups to arbitrary class-≤2 groups, 4.33 under p-separable instead of p-solvable, 4.13 strictly stronger than mathlib's derived_le_lower_central). The chapter's [G,A] machinery is factored differently from the book: actionCommutator (φ : A →* MulAut G) with a parallel ⁅inl G, inr A⁆ view in the semidirect product Γ = G ⋊[φ] A, and iterated chains [G,A,...,A] as iterCommutator on inl/inr ranges. The one real gap is the Mann/Ishikawa cluster 4.14-4.19 (M(G) generated by small conjugacy classes, class ≤ 3/≤ 4 bounds): deliberately skipped during the FT phase (zero BG/Peterfalvi consumers, per an explicit skip comment in ThreeSubgroups.lean); of it only 4.16 exists, in p-group form as centralizer_eq_self_of_maximal_abelian_normal in GroupTheory/CriticalSubgroup.lean (book states nilpotent G) — the whole cluster is roughly one M-sized job whose core is the counting lemma 4.17. Minor deviations: 4.12 covers only left-associated element commutators (arbitrary-bracketing subgroup form is an S job from the already-proved 4.11), 4.29 retains a solvability hypothesis the printed statement omits (S to close via the existing cyclic/Sylow restriction machinery), and 4.6 carries [Finite G] where the book is general. Supporting definitions Ω_r(P) and G^∞ are formalized (OmegaSubgroup.lean, ChainNilpotent.lean); M(G) is not. Slim verification pass: no status changed — all flagged missing/partial labels (4.12, M(G) def, 4.14, 4.15, 4.17, 4.18, 4.19) were confirmed by fresh multi-format number and descriptive-name greps across OddOrder/ (the only M(G)/Mann hits are the skip comment and unrelated Schur-multiplier and BG-numbered 4.15/4.17 results; Coq has no Mann analog), 4.16's p-group specialization was confirmed by reading the statement (IsPGroup p P hypothesis at CriticalSubgroup.lean:166), and the three most load-bearing formalized items 4.28, 4.31, 4.36 were re-read at statement level and confirmed at full book strength.

| 結果 | 状態 | 規模 | 内容 | メモ |
|---|---|---|---|---|
| 4.12 | 部分 | S | Every weight-n commutator of copies of G is contained in G^n | Main/ThreeSubgroups.lean:114-147 covers only the LEFT-ASSOCIATED element-commutator form. The book statement is about arbitrarily bracketed weight-n subgroup commutators (e.g. [[G,G],[G,G,G]]); that form is absent — need… |
| 4B Def M(G) | 未 | S | M(G) = subgroup generated by elements in conjugacy classes of size ≤ n₂ (second-smallest c… | No Lean counterpart, re-verified in slim pass: grepped 'Mann', 'Ishikawa', 'M(G)', small/second-smallest class, class-size generation across OddOrder/ — only hits are the explicit skip comment (Main/ThreeSubgroups.lean:2… |
| 4.14 | 未 | S | Mann: G finite nilpotent ⇒ nilpotence class of M(G) at most 3 (contains Ishikawa's theorem… | Confirmed missing (slim pass: no Mann/Ishikawa/class-≤3 content outside BG-numbered files, which are different results). Immediate from 4.15 + 4.16 once those exist. Whole Mann cluster (M(G) def + 4.14/4.15/4.17/4.18/4.1… |
| 4.15 | 未 | S | G with a self-centralizing normal subgroup ⇒ M(G) nilpotent of class ≤ 3 | Confirmed missing (slim pass; note BG Lemma 4.15 hits in S04_PGroupsSmallRank.lean are an unrelated extraspecial-commutator result). Short proof from 4.18 + three-subgroups lemma (mathlib) + M^4 ≤ [K,M,M] = 1. Blocked on… |
| 4.17 | 未 | M | K abelian normal, x noncentral ⇒ /C_G(x)/ < /C_G(y)/ for y = [k,x] | Confirmed missing (slim pass: grepped centralizer-cardinality comparisons — card_centralizer hits are all Peterfalvi S03/S04 fiber-counting, unrelated; the θ(t)=[t,x] hom pattern exists only in the 4.6 commutatorRightHom… |
| 4.18 | 未 | S | K abelian normal ⇒ [K, M(G)] ≤ Z(G) | Confirmed missing (slim pass). Direct from 4.17 (elements of small classes map commutators into smaller classes = center). Requires M(G) definition. |
| 4.19 | 未 | S | For any finite G, the nilpotence class of F(M(G)) is at most 4 | Confirmed missing (slim pass: no class-≤4 Fitting bound anywhere in OddOrder/). Short argument from 4.18 + Thm 4.11 (formalized) + repo Fitting subgroup (OddOrder.Isaacs.Ch01.fitting exists). Blocked on 4.17/4.18/M(G). |

特殊化債務 (教科書より狭い形で証明済 — 一般化要否は着手時に判定):

- ✅ **2026-07-19 実測: 解消済** — 4.6 クラスタ (`commutator_eq_commutator_of_normal_abelian_cyclic_quotient`
  / `commutatorRightHom_range_eq_commutator` / `quotientInfCenterEquivCommutator` /
  `card_commutator_mul_card_inf_center_eq_card_of_normal_abelian_cyclic_quotient`) は
  **全て `[Finite G]` を持たない** (濃度形の `[Finite G]` も同日に除去済 — `Nat.card` が
  無限を 0 に潰すので有限性不要)。以下は stale: ~~4.6 ... carry [Finite G] where the book states them for arbitrary — Main/CommutatorBasics.lean. All three clauses covered, but G'=[A,G] and the range identity carry [Finite G] where the book states them for arbitrary G (finitene…
- ✅ **2026-07-19 実測: 特殊化ではない** — `GroupTheory/CriticalSubgroup.lean:166` の
  statement は書籍と同一で、違いは**配置のみ** (章 tree の外)。債務ではなく記録事項。
  以下は stale 分類: ~~4.16 Finite nilpotent G: maximal normal abelian A satisfies A = C_G(A); hen — Lives OUTSIDE the chapter tree: OddOrder/GroupTheory/CriticalSubgroup.lean:166, cited as Gorenstein 5.3.12 (same statement). Statement re-read in slim pass: hyp…
- ✅ **2026-07-19 実測: 解消済** — 無条件版 `iterCommutator_inl_inr_two_eq_one_of_coprime`
  (`Main/ThreeSubgroupsCoprime.lean:508`) が書籍印刷形そのもの (solvability 仮定なし)。
  `:394` の solvable 版はその engine として併存するだけ。以下は stale:
  ~~4.29 Coprime action ⇒ [G,A,A] = [G,A] — Retains hypothesis IsSolvable A ∨ IsSolvable G, whereas the printed 4.29 is UNCONDITIONAL (the book's proof removes solvabi…

### Isaacs Ch.5 — Isaacs Ch.5 Transfer

> ⚠ **2026-07-19 実測: 本節の 3 ギャップは全て stale (既にクローズ済)**。
> - **Thm 5.10 Dietzmann** = `Ch05_Transfer/Dietzmann.lean` (`dietzmann` / `dietzmann_setFinite`、
>   無限群でも成立する書籍形)。
> - **Thm 5.24** = `Ch05_Transfer/NilpotentMaximal.lean`
>   (`exists_isPGroup_of_isCoatom_of_isNilpotent`、2026-07-17 `15368c186`)。
> - **Cor 5.4** = `Ch05_Transfer/CentralTransfer.lean` の
>   `not_isCyclic_sylow_quotient_of_le_commutator_inf_center` が**書籍の結論そのもの**
>   (Γ/Z の Sylow p が非巡回)。表の「弱形どまり」は stale — 弱形 (Sylow 非可換) は
>   その前段補題として併存するだけ。
> ⇒ **Isaacs Ch.5 は残作業なし**。

> Isaacs Ch.5 is one of the most completely covered units in the repo: of the 30 numbered results, 9 are fully covered by mathlib (transfer construction 5.1/5.2, transfer-evaluation 5.5, central transfer 5.6, Burnside 5.13, cyclic-Sylow-smallest-prime 5.14, and the whole Z-group cluster 5.15/5.16, plus the focal-subgroup definitional layer in Mathlib/GroupTheory/Focal.lean), and 17 more are formalized sorry-free at full book strength in OddOrder/Isaacs/Ch05_Transfer/{Basic,Main}.lean — including the entire Frobenius normal p-complement section 5.25-5.30 with the hard Lemma 5.28 fully proved (its '骨格のみ' docstring is stale). The formalization is factored differently from the book in places, always soundly: 5.20-5.23 are stated via a new A^p(G)/O^p(G) (APrime/OPrime) infimum API and mathlib's transferFocal (target P⧸Foc instead of P/P'), and 5.11 is generalized to arbitrary abelian transfer targets. Real gaps are exactly four: Dietzmann's theorem 5.10 (missing, deliberately bypassed since mathlib's Schreier route proves Schur 5.7 without it; M), the nilpotent-maximal-subgroup theorem 5.24 (missing but all ingredients present; S), the quotient-Sylow-noncyclic half of Cor 5.4 (partial; S), and Cor 5.19 which is proved only for cyclic Sylow 2-subgroups instead of the book's direct-product-with-strict-largest-factor form (specialized; M). Surprise findings: mathlib now has a complete Focal Subgroup Theorem file (commutator_inf_eq_focalSubgroup) that the repo builds on rather than duplicating, and the repo's Frobenius theorem 5.26 plus the 5.25 fusion criterion are substantial mathlib-absent developments that BG Ch.1 cites directly (Thm 1.17/1.18 entrypoints). Slim verification pass: no labels changed — the missing/partial flags on Cor 5.4, Thm 5.10, and Thm 5.24 were each confirmed by number-citation and descriptive-content greps across OddOrder/ (including GroupTheory/** and other book trees; the only nilpotent-maximal hit, Ch07's step1_unique_maximal_containing_nilpotent, is different content), and the load-bearing formalized statements 5.21, 5.25, and 5.26 were read in source and confirmed at full book strength with sound underlying predicates.

| 結果 | 状態 | 規模 | 内容 | メモ |
|---|---|---|---|---|
| Cor 5.4 | 部分 | S | Z ≤ Z(Γ)∩Γ' implies Sylow p of Γ/Z noncyclic for p ∣ /Z/ (Schur multiplier corollary) | VERIFIED partial (slim pass): only the weak form is proved — Sylow_p(Γ) nonabelian (Basic.lean:198-213; docstring explicitly says ここでは前段の Sylow_p(Γ) 非可換のみ実装). The book's conclusion that Sylow p of the QUOTIENT Γ/Z is non… |
| Thm 5.10 | 未 | M | Dietzmann: finite conjugation-closed X of bounded exponent generates a finite subgroup | VERIFIED missing (slim pass): Basic.lean:226-228 docstring says 形式化保留 (deferred); fresh grep of Mathlib/** for Dietzmann empty; repo-wide grep for conjugation-closed / finitely-many-conjugates finite-generation content f… |
| Thm 5.24 | 未 | S | Nilpotent maximal subgroup of a finite simple group is a p-group | VERIFIED missing (slim pass): Basic.lean:35 and :1083 both mark 5.24 as 保留 (deferred, no BG/Peterfalvi citation); greps by number, by nilpotent+maximal+simple, and by isPGroup-of-maximal descriptive guesses across OddOrd… |

特殊化債務 (教科書より狭い形で証明済 — 一般化要否は着手時に判定):

- ✅ **2026-07-19 実測: Cor 5.19 の特殊化債務も解消済** — 書籍一般形は
  `Ch05_Transfer/SylowTwoDirectFactor.lean` (sorry-free、`Main.lean` から import 済)。
  以下は stale: ~~Cor 5.19 Sylow 2-subgroup a direct product of cyclics with one strictly largest — Only the special case B = ⊥ (Sylow 2-subgroup itself cyclic) is formalized; the book's general P = A × B with A strictly largest cyclic factor (via the characte…

### Isaacs Ch.6 — Isaacs, Finite Group Theory, Ch.6 Frobenius Actions

> ⚠ **2026-07-19 実測: 本節の 6 ギャップは全て stale (既にクローズ済)**。
> - **Thm 6.4** 四同値は `Ch06_FrobeniusActions/FrobeniusGroup.lean` に (1)⇒(2), (2)⇒(3),
>   (1)⇒(4), (2)⇒(1), (3)⇒(1), (4)⇒(1) が全て存在。
> - **Thm 6.7** = `KernelComplement.lean` (`exists_isComplement'_of_centralizer_le`)。
> - **Cor 6.17** = `Main.lean:1314` に **full form** (Sylow は cyclic or generalized quaternion)。
> - **Thm 6.19** = `OddComplement.lean` に action 形 + subgroup-pair 形の両方。
> - **Thm 6.23** = Ch.7 Thm 7.1 (Thompson normal p-complement) で完備化済
>   (`Ch07_ThompsonSubgroup/Basic.lean:22` に明記)。
> - **Thm 6.24 Frobenius kernel nilpotent** = `KernelNilpotent.lean`
>   (`isNilpotent_of_isFrobeniusAction` + subgroup-pair 形)。
> ⇒ **Isaacs Ch.6 は残作業なし**。

> Isaacs Ch.6 is one of the most thoroughly formalized Isaacs chapters: 19 of 24 numbered results are fully formalized sorry-free, mostly in OddOrder/Isaacs/Ch06_FrobeniusActions/ (7 files, ~8.5k lines), and nothing in the chapter is in mathlib (Frobenius actions/groups are a from-scratch repo implementation, with only DihedralGroup/QuaternionGroup and the IsZGroup API reused). The formalization is frequently STRONGER or differently factored than the book: 6.11-6.14 conclude explicit MulEquiv classifications onto concrete D/Q/SD groups; 6.23 is covered by the strictly stronger Thm 7.1 (fully proved in Ch07_ThompsonSubgroup); 6.18 factors through IsZGroup + mathlib; the repo adds extra-book API (IsFrobeniusGroup transport across isomorphisms, conjugate complements, the 2|A|+1 ≤ |N| bound, and Huppert V.8.18(b) normality in OddComplement.lean); 6.22 lives in the BG tree as BG Thm 3.7 in a prime-order-complement pair form (formalized_specialized). The genuine gaps are: Thm 6.7 (kernel-side complement existence via Schur-Zassenhaus, S — all tools present), the chapter's headline Thm 6.24 (Frobenius kernels nilpotent, M — every ingredient including Thompson 7.1 is already proved, only the induction glue is missing), plus three small statement-level gaps: the standalone (2)=>(1) direction of Thm 6.4, the composed even-case statement of Cor 6.17, and the uniqueness (vs. proved normality) conclusion of Thm 6.19 (each S). Note Main.lean's in-file status table is stale (it claims 6B/6C largely undone while 6.8-6.21 are in fact formalized). Slim verification pass: all five flagged labels (6.4/6.17/6.19 partial, 6.7/6.24 missing) were confirmed after number- and content-based refutation greps across OddOrder/GroupTheory and the BG tree, and three load-bearing formalized results (6.11, 6.21, and 6.23-via-7.1) were read in source and confirmed at full book strength, so no statuses changed; only verification evidence was added to notes (e.g. 6.4's (1)⇒(4) is proved despite a stale 'deferred' module docstring, and 6.19's uniqueness name exists only in a docstring).

| 結果 | 状態 | 規模 | 内容 | メモ |
|---|---|---|---|---|
| Thm 6.4 | 部分 | S | Four equivalent characterizations of a Frobenius group (Frobenius action / TI / C_G(a) ≤ A… | VERIFIED (slim pass): condition (1) is the canonical structure IsFrobeniusGroup (FrobeniusGroup.lean); directions (1)=>(2) (trivialIntersection, line 426), (2)=>(3) (centralizer_complement_le, line 510 — hypothesized on … |
| Thm 6.7 | 未 | S | N normal with C_G(n) ≤ N for all 1 ≠ n ∈ N is a normal Hall subgroup, hence complemented; … | VERIFIED missing (slim pass): re-grepped by number ('6.7' hits are only Peterfalvi (6.7)/BG Thm 6.7 and the stale Main.lean TODO table) and by content (Hall/complement-existence near Frobenius, centralizer-condition cons… |
| Cor 6.17 | 部分 | S | Every Sylow subgroup of a Frobenius complement is cyclic or generalized quaternion (abelia… | VERIFIED partial (slim pass): odd-order case (all Sylow cyclic, i.e. IsZGroup) and the abelian-complement-is-cyclic observation are stated and proved; re-grepped Quaternion x Sylow x Frobenius repo-wide — the general eve… |
| Thm 6.19 | 部分 | S | Odd Frobenius complement has a unique subgroup of order p for each prime p dividing /A/ | VERIFIED partial (slim pass): OddComplement.lean proves the closely related Huppert V.8.18(b) — every prime-order subgroup of an odd Frobenius complement is NORMAL (needed by Peterfalvi (13.17.c)), via exactly Isaacs's 6… |
| Thm 6.23 | 部分 | S | Thompson normal p-complement criterion (characteristic-subgroup form, p ≠ 2; stated in Ch.… | [critic 補正] 6.23 単体の statement は repo に無い (Ch06 roadmap で 6C=未着手)。Ch.7 Thm 7.1 (Thompson normal p-complement, repo 証明済) が数学的に包含 — standalone 化は系導出のみで S。 |
| Thm 6.24 | 未 | M | Frobenius kernels are nilpotent (Thompson's thesis theorem, general case) | VERIFIED missing (slim pass): re-searched nilpotent x kernel/Frobenius repo-wide — the only kernel-nilpotency results are frobeniusKernelIsNilpotent (S03c_Thm37.lean:664, requires [IsSolvable G] + prime-order complement)… |

特殊化債務 (教科書より狭い形で証明済 — 一般化要否は着手時に判定):

- **Thm 6.22** Solvable Frobenius kernels are nilpotent (Higman/Thompson, solvable ca — Lives in the BG tree as BG Theorem 3.7 (BG/Ch1_Preliminary/S03c_Thm37.lean:664), sorry-free. Stated in subgroup-pair form with a PRIME-ORDER complement and [IsS…

### Isaacs Ch.8 — Isaacs Ch.8 Permutation Groups

> ✅ **2026-07-19 実測: 本節は 100% stale — 25 件全て形式化済 (実 sorry 0)**。
> 前文の「no repo formalization at all — **no Ch08 directory exists**」は誤り。
> `OddOrder/Isaacs/Ch08_PermutationGroups/` に **14 leaf / 5,707 行**が実在し、
> ファイル名が 25 件のギャップに 1 対 1 対応する (`HalfTransitive.lean`=8.9 /
> `PCycleJordan.lean`=8.23 / `Bochert.lean`=8.26 / `PSLSimple.lean`=8.31–8.33 /
> `Orbitals`・`OrbitalGraph`・`Subdegrees`・`CommonDivisorGraph`=8.34–8.44)。
> mathlib upstream 候補: `alternatingGroup_le_of_isPreprimitive_of_isCycle_mem`
> (8.23 — mathlib では `proof_wanted` のまま)、`isSimpleGroup_projectiveSpecialLinearGroup`
> (8.33 PSL 単純性 — mathlib 未収載)。


> Isaacs Ch.8 (Permutation Groups, results 8.1–8.44) has no repo formalization at all — no Ch08 directory exists, nothing in OddOrder/GroupTheory covers it, and the repo never uses mathlib's IsBlock/IsPreprimitive/IsMultiplyPretransitive API (the BG §16 ConjSharplyTransitiveOn machinery and the sorried Peterfalvi Suzuki appendix, whose doubly-transitive hypotheses are opaque scaffold Props that this chapter's material would eventually discharge, are the only adjacent things). The surprise is how much mathlib now covers at full strength or stronger: all of the blocks/primitivity theory (8.11–8.16), both Jordan theorems including the general multiple-primitivity form (8.17–8.22, Chambert-Loir's Jordan.lean/MultipleTransitivity.lean/MultiplePrimitivity.lean), Iwasawa's criterion (8.30), A_n simplicity for general n ≥ 5 plus the S_n normal-subgroup classification (8.27–8.28, added 2026-04, making the repo's pre-formalization note stale), and the SL/PSL projective-space action with faithfulness, 2-transitivity, primitivity, and the degree formula (8.3, 8.29) — 19 of 44 results are status mathlib. The genuine gaps are: the §8A automorphism-action cluster (8.4–8.10, including the regular-normal-subgroup lemma 8.5 and the Passman–Isaacs half-transitive theorem 8.9), the p-cycle Jordan theorem 8.23 (an explicit proof_wanted in mathlib) with its helpers 8.24–8.25, Bochert's theorem 8.26, the PSL simplicity chain 8.31–8.33 (mathlib has transvection Gaussian elimination and SL(2) commutator lemmas but no generation/perfectness/simplicity statements — a prime upstream candidate), and the entire §8D orbital/subdegree/common-divisor-graph theory 8.34–8.44, for which mathlib has no concepts at all. Total new work is roughly: a handful of S items assembling existing mathlib pieces, plus four M-sized clusters (half-transitive, p-cycle+Bochert, PSL simplicity, §8D orbital theory). Slim verification pass: all 25 flagged missing labels were confirmed after refutation attempts (book-number greps for Isaacs 8.x in multiple formats plus descriptive-name greps — transvection, Bochert, Passman, half-transitive, subdegree, orbital, PSL/GL simplicity, semidirect/sharply/regular transitivity, three-cycle, elementary-abelian cover — across all OddOrder trees and notes; the only transvection content is BG §10's unrelated GL(2,p) lemma), and the load-bearing mathlib labels 8.2 (verified iff-form statement by Read), 8.17/8.19, 8.23 proof_wanted (Jordan.lean:462), 8.27, and 8.30 were verified against mathlib source; no status changed.

| 結果 | 状態 | 規模 | 内容 | メモ |
|---|---|---|---|---|
| Cor 8.4 | 未 | S | GL(n,2) is doubly transitive on nonzero vectors of F_2^n | Confirmed missing (slim pass): only Pretransitive instances in mathlib LinearAlgebra are on ℙ K V (Projectivization/Action.lean); repo GL usage is GL(2,p) for Ch07 Thompson/small-rank only. Short transfer across ℙ 𝔽₂ V ≃… |
| Thm 8.5 | 未 | S | Subgroup N transitive iff AN=G; regular iff also A∩N=1; conj action of stabilizer on regul… | Confirmed missing: no semidirect/sharply/regular-normal transitivity content in OddOrder/GroupTheory; repo's BG S16 ConjSharplyTransitiveOn is bespoke conjugation-on-subgroup-sets machinery, not this statement. The regul… |
| Cor 8.6 | 未 | S | A k-transitive on N-{1} implies N⋊A is (k+1)-transitive on cosets of A | Confirmed missing: no semidirect-product multiple-transitivity construction in mathlib or repo. Short given 8.5 + SubMulAction.ofStabilizer.isMultiplyPretransitive. |
| Cor 8.7 | 未 | S | V ⋊ GL(n,2) (V elementary abelian of order 2^n) is 3-transitive of degree 2^n | Confirmed missing. Immediate from 8.4 + 8.6 once those exist; faithfulness argument is routine. |
| Thm 8.8 | 未 | S | A k-transitive on N-{1}: N elementary abelian p-group; k>1 forces p=2 or /N/=3; k>2 forces… | Confirmed missing (greps: IsMultiplyPretransitive+automorphism, elementary abelian+transitive — nothing in repo/mathlib). mathlib's alternatingGroup.isSimpleGroup takes a different route. |
| Thm 8.9 | 未 | M | Passman–Isaacs: faithful half-transitive automorphism action is Frobenius or N is elementa… | Confirmed missing: half-transitive and Passman greps 0 hits in OddOrder and mathlib. Repo Ch06_FrobeniusActions has Frobenius actions but nothing half-transitive. ~2-page coprime-counting proof using 8.10. |
| Lem 8.10 | 未 | S | Group covered by pairwise trivially-intersecting proper normal subgroups is elementary abe… | Confirmed missing: OddOrder/GroupTheory/ElementaryAbelian.lean has no cover/partition result; mathlib CosetCover.lean is a different topic. Self-contained one-page argument. |
| Thm 8.23 | 未 | M | Primitive group containing a p-cycle (p prime, p ≤ /Ω/−3) contains the alternating group | Confirmed: explicitly open in mathlib — proof_wanted alternatingGroup_le_of_isPreprimitive_of_isCycle_mem (Jordan.lean:462, verified by grep) — statement only, no proof; also in the file's TODO (Wielandt 13.9). Proof nee… |
| Lem 8.24 | 未 | S | Centralizer of an n-cycle in S_n is the cyclic group it generates | Confirmed missing: no centralizer = zpowers statement in Perm/Centralizer.lean or Perm/Cycle/*.lean (verified by grep). Short corollary of Equiv.Perm.IsCycle.forall_commute_iff (Perm/Cycle/Factors.lean:852) and Equiv.Per… |
| Lem 8.25 | 未 | S | If x, y move exactly one common point then [x,y] is a 3-cycle | Confirmed missing: no IsThreeCycle content in OddOrder; not in mathlib Jordan.lean / Perm/ClosureSwap.lean. Elementary case analysis on supports. |
| Thm 8.26 | 未 | M | Bochert: primitive proper subgroup of S_n other than A_n has index ≥ ⌊(n+1)/2⌋! | Confirmed missing: Bochert grep 0 hits in mathlib/OddOrder (only a pre-formalization audit memo notes/meta/log/ch08_10_audit_2026_05_23.md, which confirms proof deps = 8.19 + 8.25 only, not 8.24). Elementary but fiddly minim… |
| Thm 8.31 | 未 | S | SL(n,q) is generated by the transvections t_{i,j}(α) | Confirmed missing: no closure-of-transvections = ⊤ statement anywhere in mathlib (verified by grep); repo transvection content is only BG §10's bespoke GL(2,p) line-transitivity (S10_Transvection.lean), unrelated. Mathli… |
| Thm 8.32 | 未 | M | SL(n,q) is perfect for n ≥ 3 or (n = 2, q > 3) | Confirmed missing: no commutator = ⊤ statement in mathlib/repo. Building blocks exist for n=2: Matrix.SpecialLinearGroup transvection_mem_commutator and commutator_diag2_transvection (LinearAlgebra/Matrix/SpecialLinearGr… |
| Thm 8.33 | 未 | M | PSL(n,q) is simple for n ≥ 3 or (n = 2, q > 3) | Confirmed missing: repo has no ProjectiveSpecialLinearGroup usage beyond GL(2,p) Thompson-subgroup files; mathlib has no PSL IsSimpleGroup. All action-side infrastructure exists (IwasawaStructure, PSL preprimitive+faithf… |
| Lem 8.34 | 未 | S | Orbital functions Δ(α) are exactly the suborbits; Δ(α·g) = Δ(α)·g; /Δ(α)/ = /Δ///Ω/ | Confirmed missing: orbital/subdegree greps 0 hits in mathlib and OddOrder (the sole 'orbital' hit is a Japanese docstring word in Ch05_Transfer). Definitions are just G-orbits on Ω × Ω; this lemma itself is routine once … |
| Thm 8.35 | 未 | M | Primitive iff every non-diagonal orbital graph is connected | Confirmed missing. Needs a directed-graph (or Relation.ReflTransGen) connectivity framework tied to orbitals; nothing existing to cite. |
| Lem 8.36 | 未 | S | Finite vertex-transitive topologically connected digraph is path connected | Confirmed missing. Standalone combinatorial lemma; formulate with Relation.ReflTransGen vs. its symmetrization. |
| Thm 8.37 | 未 | M | Primitive subdegrees satisfy m_{i+1} ≤ m_2 · m_i | Confirmed missing. Requires orbital-graph distance argument on top of 8.34/8.35. |
| Thm 8.38 | 未 | M | Weiss: a subdegree coprime to the largest subdegree equals 1 (primitive case) | Confirmed missing. Needs 8.39 + orbital-graph induction; no mathlib analog. |
| Lem 8.39 | 未 | S | m-arrow/n-arrow index arithmetic: u ≥ n, u / mn, G_γG_α ⊆ G_γG_β | Confirmed missing. Pure index/double-coset computation; mathlib has the /G:X∩Y/ coprimality facts (Subgroup.index) to cite. |
| Thm 8.40 | 未 | M | Manning: point stabilizer 2-transitive on a suborbit of maximal size n ≥ 3 forces n = /Ω/−… | Confirmed missing. Depends on 8.38 and self-paired orbital arguments; no mathlib analog. |
| Thm 8.41 | 未 | M | Isaacs–Praeger: common-divisor graph of subdegrees has at most 3 connected components | Confirmed missing: common-divisor-graph grep 0 hits in mathlib/repo/notes beyond the pre-formalization audit memo. Follows from 8.42(b) once the K_m(α)/k_m framework exists. |
| Thm 8.42 | 未 | M | k_m divides k_n and k_m divides n (subdegrees m < n with every subdegree coprime to m or n… | Confirmed missing. Core technical result of §8D; needs the m-graph component subgroup K_m(α) framework. |
| Thm 8.43 | 未 | S | With 3 components {1}, A, B (B containing max subdegree): every a ∈ A < every b ∈ B, and B… | Confirmed missing. Short consequence of 8.42. |
| Cor 8.44 | 未 | S | Orbit sizes of an automorphism action satisfy the common-divisor-graph constraints (8.41/8… | Confirmed missing. Transfer along G⋊A coset action; needs the 8.5(3) bridge. Special case A = G gives the classical class-size common-divisor graph result. |

### Isaacs Ch.9 — Isaacs Ch.9 More on Subnormality

> ✅ **2026-07-19 完了: Isaacs Ch.9 は全 31 結果が形式化済 (以下の表は全面 stale)**。
> `Ch09_MoreSubnormality/` に 19 leaf (Quasisimple / Components / Semisimple / Layer /
> LayerRestriction / GeneralizedFitting / SubnormalSocle / NilpotentResidual / PResidual /
> Schenkman / InnerAutomorphisms / AutTower / AutTowerBounds / OrderBound / ThompsonWielandt /
> SylowSubnormal / StrongConjugacy / SubnormalClosure)。
> 番号 grep で 9.1–9.27 が全てヒット (§9A F* / §9B automorphism tower / §9C
> Thompson–Wielandt) し、**§9D も 2026-07-19 に完了** —
> 9.28 Bartels (`strongClosure_isSubnormal`, 6 step) / 9.29 / 9.30 / 9.31 すべて
> sorry-free・axiom-clean。詳細 = `notes/isaacs/ch09_more_subnormality.md` の
> 「§9D 完了」節。
> ⚠ 併せて **9.26/9.27 が書籍の `S ◁◁ G` でなく `S ◁ G` で形式化されていた MISMATCH**
> (issue 9150 → 0125) も解消済。
> ⇒ レーン a の frontier は Ch.9 を離れ、**文書順で Ch.2 Lem 2.14 → Ch.5 → Ch.6 →
> Ch.8 → Ch.10 → 付録** へ (Ch.3/Ch.4 は完了済、Ch.8/Ch.10 は 2026-07-19 裁定で a 所有)。

> Isaacs Ch.9 is essentially unformalized: all 31 numbered results (9.1–9.31) lack full-strength Lean counterparts, and none of the chapter's signature concepts (quasisimple, component, layer E(G), semisimple group, F*(G), automorphism tower, subnormal closure, strong conjugacy) exist in the repo or in mathlib. This is a documented deliberate skip — notes/isaacs/ch09_more_subnormality.md records a 2026-05-23 audit showing zero uses of any Ch.9 concept in BG or Peterfalvi (FT's group is ultimately solvable, where F*=F), and fresh greps confirm the repo state matches that audit. The chapter is not entirely from scratch, though: the solvable specialization of Bender's Thm 9.8 is fully proved as BG Prop 1.3 (centralizer_fitting_le_fitting), and scattered infrastructure lowers entry costs — socle (Ch02), Thm 2.6 (the hub cited 5x by §9A proofs), S^∞ as lowerCentralSeriesInfty (Ch04), O^p as OPrime (Ch05), the n!-theorem (Ch01), ker(conjNormal)=centralizer and a |G/C| ≤ |Aut C| embedding (Ch06/Ch07), plus mathlib's IsSubnormal/IsPerfect/MulAut.conj. Surprises: Lemma 9.31 (Sylow ∩ subnormal) is missing even from mathlib despite being a basic general fact, and the Nougat-missing page 302 was verified against the PDF to contain only §9D definitions, so the enumeration is complete. Effort estimates are marginal in document order (each assumes earlier chapter results exist); the natural clusters are §9A F*-theory (~1 wk), §9B tower (~1-1.5 wk), §9C Thompson–Wielandt (~3-4 days on top of A+B), §9D Bartels (~1-1.5 wk). Slim verification pass: all flagged missing/partial labels and the Thm 9.8 specialization (statement read) were confirmed; the only substantive correction is to Lem 9.6's notes — BG Lem 1.1 (isMinimalNormal_le_fitting_and_isElementaryAbelian) does prove 'minimal normal ⟹ elementary abelian' for solvable G at full strength, contrary to the earlier note — plus minor path/name enrichments (ChainNilpotent.lean lives under Main/, two additional private conjNormal-kernel variants for 9.11).

| 結果 | 状態 | 規模 | 内容 | メモ |
|---|---|---|---|---|
| §9A Defs | 未 | M | quasisimple group, component, layer E(G), semisimple group, generalized Fitting F*(G) = F(… | Confirmed missing (slim pass re-verified: no IsQuasisimple/component/layer/FStar/generalizedFitting/semisimple-group anywhere in OddOrder/ or mathlib; 'layer'/'semisimple' hits are unrelated — π-Fitting layers, module se… |
| Lem 9.1 | 未 | S | G/Z(G) simple ⟹ G/Z nonabelian, G' perfect (quasisimple), G'/Z(G') ≅ G/Z | Confirmed missing; no counterpart under any Isaacs-9.x citation or perfect/quasisimple descriptive name. Elementary once quasisimple is defined. |
| Lem 9.2 | 未 | S | quasisimple G: every proper normal subgroup is central; every nontrivial quotient is quasi… | Confirmed missing (no quasisimple notion anywhere). |
| Lem 9.3 | 未 | S | N minimal normal, H component with H ⊄ N ⟹ [N,H] = 1 | Confirmed missing. Key inputs already exist and were re-verified: Isaacs Thm 2.6 (OddOrder.Isaacs.Ch02.isMinimalNormal_le_normalizer_of_isSubnormal, Ch02_Subnormality/Basic.lean:554) and mathlib three-subgroups lemma. |
| Thm 9.4 | 未 | M | distinct components of a finite group commute: [H,K] = 1 | Confirmed missing. /G/-induction with quotient transport of components (needs 9.2/9.3); subgroup-quotient transport is the labor. Repo Ch02 commute_of_disjoint_normal (Ch02_Subnormality/Basic.lean:147, Isaacs Lemma 2.7) … |
| Lem 9.5 | 未 | M | a product of nonabelian simple normal subgroups is their (internal) direct product, and th… | Confirmed missing; no semisimple-group notion in repo or mathlib (mathlib/repo 'semisimple' hits are all rings/modules/representations). Needs internal-direct-product API design. |
| Lem 9.6 | 部分 | M | a minimal normal subgroup of a finite group is abelian or semisimple | General dichotomy absent (confirmed), but the solvable/abelian branch exists at FULL strength for solvable G — correcting the earlier note that claimed otherwise: OddOrder.BG.Ch1.S01.isMinimalNormal_le_fitting_and_isElem… |
| Thm 9.7 | 未 | M | layer E = E(G), Z = Z(E): (a) E' = E, (b) E/Z semisimple, (c) [E,M] = 1 for every solvable… | Confirmed missing (no layer notion anywhere). Proof is short given 9.4/9.5 but needs the whole component infrastructure. |
| Cor 9.9 | 未 | S | F*(G) ⊇ F(G), with equality iff F(G) ⊇ C_G(F(G)) | Confirmed missing; not statable without F*. The FT-relevant direction (solvable ⟹ F self-centralizing, so no augmentation needed) is exactly BG Prop 1.3 above. Trivial once F*/E are defined and 9.7(c) is proved. |
| Thm 9.10 | 未 | M | Wielandt automorphism tower theorem: Z(G)=1 ⟹ the tower G ◁ Aut(G) ◁ Aut(Aut(G)) ◁ … conta… | Confirmed missing: no automorphism-tower or complete-group content anywhere in repo or mathlib (re-grepped; all 'Wielandt' hits are Thm 2.x/3.17/zipper/fixed-point lemma). Given 9.11–9.13 the proof is assembly, but the t… |
| Lem 9.11 | 部分 | S | Inn(G): (a) g ↦ τ_g is a hom onto Inn with kernel Z(G); (b) Inn ◁ Aut; (c) Z(G)=1 ⟹ C_Aut(… | Confirmed partial. (a)'s hom is mathlib MulAut.conj; the kernel=center statement exists in repo only in conjNormal variants, all private/local: conjNormal_ker_eq_centralizer (private, Ch07 S7B2_NormalJ_PComplement.lean:1… |
| Lem 9.12 | 未 | S | chain S=S₁ ◁ S₂ ◁ … ◁ S_r=G with C_{S_{i+1}}(S_i)=1 for all i ⟹ C_G(S)=1 | Confirmed missing; short induction, no counterpart found. |
| Thm 9.13 | 未 | M | S subnormal in G with C_G(S)=1 ⟹ /G/ ≤ n where n depends only on the isomorphism type of S | Confirmed missing. Assembly of 9.14 + 9.16/9.18 + 9.21; effort assumes those exist (efforts here are marginal in document order). |
| Lem 9.14 | 部分 | S | N normal with N ⊇ C_G(N) ⟹ /G/ divides /Z(N)/·/Aut(N)/, hence /G/ divides /N/! | Confirmed partial. The embedding argument exists in specialized inequality form: OddOrder.Isaacs.Ch06.quotient_card_le_mulAut_of_self_centralizing (DQSDRecognition.lean:498, /P/C/ ≤ /Aut C/ under centralizer(C)=C). Re-gr… |
| Lem 9.15 | 未 | S | G = SF with S subnormal, F nilpotent normal ⟹ G^∞ = S^∞ (nilpotent residuals agree) | Confirmed missing. The S^∞ definition already exists: OddOrder.Isaacs.Ch04.lowerCentralSeriesInfty (Ch04_Commutators/Main/ChainNilpotent.lean:111 — path corrected to Main/ subdir; = lowerCentralSeries at Nat.card, with s… |
| Cor 9.16 | 未 | S | S subnormal in G ⟹ F(G) normalizes S^∞ | Confirmed missing. |
| Lem 9.17 | 未 | S | S subnormal nonabelian simple ⟹ S^G is minimal normal, so S ≤ Soc(G) | Confirmed missing. Socle already defined in repo: OddOrder.Isaacs.Ch02.socle with isMinimalNormal_le_socle (Ch02_Subnormality/Basic.lean:158/162 — re-verified). Needs 9.4/9.5. |
| Cor 9.18 | 未 | S | S subnormal in G ⟹ E(G) normalizes S^∞ | Confirmed missing (no E(G)). |
| Lem 9.19 | 未 | S | N ≤ Φ(G) normal with G/N nilpotent ⟹ G nilpotent | Confirmed missing: re-grepped all frattini+nilpotent co-occurrences in OddOrder/ — only BG/Peterfalvi Frattini-argument prose, no lifting lemma. mathlib Frattini.lean has frattini, frattini_le_coatom, frattini_nilpotent … |
| Lem 9.20 | 未 | S | G/N nilpotent ⟹ there is a nilpotent H ≤ G with NH = G (nilpotent supplement) | Confirmed missing: re-grep 'supplement' across OddOrder/ finds only unrelated Coq-comment references. |
| Thm 9.21 | 未 | S | Schenkman: S subnormal with C_G(S)=1 ⟹ C_G(S^∞) ≤ S^∞ | Confirmed missing (no Schenkman content under any name). Marginal effort given 9.22. Isaacs calls the Schenkman step the hardest part of the tower theorem. |
| Thm 9.22 | 未 | M | Z(G)=1 ⟹ C_G(G^∞) ≤ G^∞ (S=G case of Schenkman) | Confirmed missing. Page-long Dedekind/supplement manipulation; carries the real content of 9.21. |
| Thm 9.23 | 未 | S | Thompson: corefree maximal H, m = /H:H∩H^g/ ⟹ some prime p with /H:O_p(H)/ ≤ ((m!)²)! | Confirmed missing. One-page derivation from 9.24; the needed n!-theorem exists: OddOrder.Isaacs.Ch01.normalCore_index_dvd_factorial (Ch01_Sylow/Basic.lean:76 — re-verified), and O_p = opCore exists (Ch01_Sylow/Basic.lean… |
| Thm 9.24 | 未 | M | Thompson–Wielandt general form: H ≠ K, D = H∩K, iterated cores M,N,E,U,V ⟹ U or V is a p-g… | Confirmed missing. Effort M is marginal given §9A (F*, 9.25) and §9B (S^∞, 9.16/9.18) infrastructure — its proof invokes F*(H) ⊇ C(F*(H)), E(G), S^∞, and 9.26/9.27. Standalone it is L. |
| Lem 9.25 | 未 | S | F(G)=1 and E(G) ≤ H ≤ G ⟹ E(G) = E(H) | Confirmed missing (no E(G)). |
| Lem 9.26 | 未 | S | G = SP with S subnormal, P normal p-group ⟹ O^p(G) = O^p(S) | Confirmed missing. O^p(G) already defined in repo as OddOrder.Isaacs.Ch05.OPrime (Ch05_Transfer/Basic.lean:1142 — re-verified, with OPrime_le, OPrime_index_isPGroup API); only this product lemma is new. (OddOrder/GroupTh… |
| Cor 9.27 | 未 | S | S subnormal, P normal p-group ⟹ P normalizes O^p(S) | Confirmed missing. In the book this corollary is printed without the 'Corollary' label ('9.27.') — it is Corollary 9.27, cited as such in the 9.24 proof. |
| Thm 9.28 | 未 | L | Bartels: subnormal closure X^{•G} = subgroup generated by all strong conjugates of X | Confirmed missing (no Bartels/strong-conjugacy content under any name). Six-step double-minimal (/G/,/X/) counterexample argument with quotient transport and Sylow analysis; the heaviest single proof in the chapter even … |
| Lem 9.29 | 未 | S | basic properties of X^(G): (a) X ≤ S ◁◁ G ⟹ X^(G) ≤ S; (b) monotone; (c)(d) restriction to… | Confirmed missing (no iterated-normal-closure X^(G) construction anywhere). |
| Lem 9.30 | 未 | S | quotient compatibility: for N ◁ G, image of X^(G) in G/N equals (X̄)^(Ḡ) | Confirmed missing. |
| Lem 9.31 | 未 | S | S subnormal in G, P ∈ Syl_p(G) ⟹ P ∩ S ∈ Syl_p(S) | Confirmed absent from mathlib (re-read Mathlib/GroupTheory/IsSubnormal.lean — has inf/trans/map/comap/subgroupOf/smul but zero Sylow interaction; Sylow.lean has only IsPGroup.inf_normalizer_sylow and le_sylow_of_normal, … |
| §9B Defs | 部分 | S | automorphism tower, complete group, nilpotent residual S^∞ | Confirmed partial. S^∞ exists: OddOrder.Isaacs.Ch04.lowerCentralSeriesInfty with stability lemmas (Ch04_Commutators/Main/ChainNilpotent.lean:111 — path corrected to Main/ subdir). Automorphism tower and complete group ab… |
| §9D Defs | 未 | S | subnormal closure X^{•G}, strongly conjugate subgroups, X^(G) | Confirmed missing (no subnormal-closure/strongly-conjugate content under any name). These definitions sit on the Nougat-missing page (recovered from PDF p.302 = book p.289; no numbered results lost there — only Problem 9… |

特殊化債務 (教科書より狭い形で証明済 — 一般化要否は着手時に判定):

- ✅ **2026-07-19 実測: 解消済** — Ch.9 §9A の
  `centralizer_genFitting_le_genFitting` (`Ch09_MoreSubnormality/GeneralizedFitting.lean:70`)
  が **`[Finite G]` のみで可解性を課さない**一般形 = 書籍 Thm 9.8 そのもの。
  BG Prop 1.3 は可解特殊化として併存するだけ。以下は stale: ~~Thm 9.8 Bender — Confirmed by reading the Lean statement: OddOrder.BG.Ch1.S01.centralizer_fitting_le_fitting (BG Prop 1.3, S01_FrattiniBurnside.lean:195) is exactly [Finite G][I…

### Isaacs Ch.10 — Isaacs Ch.10 More Transfer Theory

> ✅ **2026-07-19 実測: 本節は 100% stale — 全 28 結果が形式化済 (実 sorry 0)**。
> 前文の「almost entirely unformalized: only Corollary 10.17 is fully formalized」は誤り。
> `OddOrder/Isaacs/Ch10_MoreTransfer/` (6 leaf / 3,643 行) で 2026-07-17 の
> `a98141fc8 feat(isaacs-ch10): Thm 10.18 Furtwängler + Cor 10.28 Alperin-Kuo — Ch.10 全 28
> 結果完成` により完結。群環 `Δ(G)` 機械は `OddOrder/Algebra/AugmentationIdeal.lean` 等に
> 切り出し済。


> Isaacs Ch.10 (results 10.1-10.28) is almost entirely unformalized: of 28 numbered results, only Corollary 10.17 is fully formalized (as the operator-Maschke theorem exists_aInvariant_complement_of_isElementaryAbelian in OddOrder/BG/Ch1_Preliminary/OperatorMaschke.lean, built for BG's needs) and Lemma 10.13 is half done (subgroup closure of metacyclic in OddOrder/GroupTheory/IsMetacyclic.lean; quotient closure missing). All three headline clusters are missing: Yoshida's theorem 10.1 with its wreath-recognition and transfer chain (10.3-10.11, including transitivity of transfer 10.8 and Mackey transfer 10.10, both absent from mathlib as well), Huppert's metacyclic-Sylow theorem 10.12/10.15, and the entire section-10C group-ring machinery culminating in the principal ideal theorem 10.18 and Alperin-Kuo 10.28. This matches the chapter's zero forward citations from BG/Peterfalvi (repo grep hits for 'Theorem 10.x' are all BG section 10, a different book); the repo's three other 'Huppert' results (BG 4.11/4.12, Peterfalvi appendix) are distinct theorems and must not be confused with 10.12. Two mitigating surprises: mathlib's RegularWreathProduct.lean (Sylow.mulEquivIteratedWreathProduct) substantially cheapens the C_p wr C_p recognition step of 10.4, and the chapter's Isaacs-internal prerequisites (Thm 4.7, 4.8(a), 6.11, Ch.5 transfer/focal API) are already formalized, so section 10A+10B is mostly assembly-hard rather than prerequisite-blocked; section 10C needs a new augmentation-ideal Delta(G) API from scratch (a good future mathlib upstream candidate). Slim verification pass: refutation greps (wreath/pretransfer/Mackey-transfer/augmentation-ideal/Furtwangler/Alperin-Kuo/metacyclic-quotient, plus descriptive-name and cross-tree searches over GroupTheory/Algebra/BG/Peterfalvi and pinned mathlib) confirmed every flagged missing/partial label with no upgrades; the 10.17 Lean statement was re-read and confirmed at full book strength, and 10.13's file was confirmed to contain subgroup closure but no quotient closure.

| 結果 | 状態 | 規模 | 内容 | メモ |
|---|---|---|---|---|
| Thm 10.1 | 済 | M | Yoshida: N_G(P) controls p-transfer unless C_p wr C_p is a homomorphic image of P | No Lean counterpart (greps: yoshida, controls.*transfer, wreath, Isaacs 10.*; repo ControlsPTransfer predicate deliberately avoided in Ch05, see Ch05_Transfer/Basic.lean:1498). Marginal effort M assuming 10.3-10.11 chain… 【**2026-07-22 lane c 実測訂正: 済**】Ch.10 は 2026-07-17 に全 28 結果完成 (`Ch10_MoreTransfer/Main.lean` の完成表参照; sorry census 0・AxiomsCheck 登録済)。slim pass (07-16) 直後に landing したため表が未更新だった。 |
| Cor 10.2 | 済 | S | Sylow normalizer controls p-transfer when nilpotence class of P < p | Immediate from 10.1 + 10.3(c) + quotient monotonicity of nilpotency class (mathlib has Group.nilpotencyClass API). Verified missing. 【**2026-07-22 lane c 実測訂正: 済**】Ch.10 は 2026-07-17 に全 28 結果完成 (`Ch10_MoreTransfer/Main.lean` の完成表参照; sorry census 0・AxiomsCheck 登録済)。slim pass (07-16) 直後に landing したため表が未更新だった。 |
| Lem 10.3 | 済 | S | p-group with normal elementary abelian A of index p generated by a conjugacy class: /Z(P)/… | Prerequisites Isaacs Thm 4.7 (verified present: nilpotencyClass_eq_of_normal_abelian_cyclic_quotient_inf_center_prime_card_p_pow, OddOrder/Isaacs/Ch04_Commutators/Main/CommutatorBasics.lean:1155) and Lemma 4.6 territory … 【**2026-07-22 lane c 実測訂正: 済**】Ch.10 は 2026-07-17 に全 28 結果完成 (`Ch10_MoreTransfer/Main.lean` の完成表参照; sorry census 0・AxiomsCheck 登録済)。slim pass (07-16) 直後に landing したため表が未更新だった。 |
| Thm 10.4 | 済 | M | Recognition of C_p wr C_p: /Z(P)/=p and a noncentral class in A with nonidentity product f… | Needs (T-I)-nilpotency linear algebra over ZMod p, faithful coset action embedding P into S_{p^2}, and Sylow-of-Perm identification. mathlib's RegularWreathProduct.lean (Sylow.mulEquivIteratedWreathProduct, verified pres… 【**2026-07-22 lane c 実測訂正: 済**】Ch.10 は 2026-07-17 に全 28 結果完成 (`Ch10_MoreTransfer/Main.lean` の完成表参照; sorry census 0・AxiomsCheck 登録済)。slim pass (07-16) 直後に landing したため表が未更新だった。 |
| Cor 10.5 | 済 | S | Elementary abelian A of index p, noncentral a with nonidentity class product: C_p wr C_p i… | Induction on /P/ via central quotients, given 10.4. Verified missing. 【**2026-07-22 lane c 実測訂正: 済**】Ch.10 は 2026-07-17 に全 28 結果完成 (`Ch10_MoreTransfer/Main.lean` の完成表参照; sorry census 0・AxiomsCheck 登録済)。slim pass (07-16) 直後に landing したため表が未更新だった。 |
| Lem 10.6 | 済 | M | Pretransfer to a normal maximal subgroup M: V(x) = prod of conjugates mod M' for x in M, V… | mathlib has MonoidHom.transfer and transfer_eq_prod_quotient_orbitRel_zpowers_quot (= Isaacs 5.5) but no 'pretransfer mod M'' formulation; needs restating via transfer into Abelianization M. Verified: no pretransfer noti… 【**2026-07-22 lane c 実測訂正: 済**】Ch.10 は 2026-07-17 に全 28 結果完成 (`Ch10_MoreTransfer/Main.lean` の完成表参照; sorry census 0・AxiomsCheck 登録済)。slim pass (07-16) 直後に landing したため表が未更新だった。 |
| Lem 10.7 | 済 | S | V(M) not contained in Phi(M) forces C_p wr C_p as homomorphic image of P | Short Frattini-quotient argument from 10.5 + 10.6(a); repo has Frattini p-group API (OddOrder/GroupTheory/FrattiniPGroup.lean). Verified missing. 【**2026-07-22 lane c 実測訂正: 済**】Ch.10 は 2026-07-17 に全 28 結果完成 (`Ch10_MoreTransfer/Main.lean` の完成表参照; sorry census 0・AxiomsCheck 登録済)。slim pass (07-16) 直後に landing したため表が未更新だった。 |
| Thm 10.8 | 済 | M | Transitivity of transfer: V(g) = W(U(g)) mod H' for H <= K <= G | Absent from mathlib too (verified: Mathlib/GroupTheory/Transfer.lean declaration list has no composition/transitivity lemma); proof over mathlib's diff-based definition needs the ST-transversal argument. A genuinely reus… 【**2026-07-22 lane c 実測訂正: 済**】Ch.10 は 2026-07-17 に全 28 結果完成 (`Ch10_MoreTransfer/Main.lean` の完成表参照; sorry census 0・AxiomsCheck 登録済)。slim pass (07-16) 直後に landing したため表が未更新だった。 |
| Thm 10.9 | 済 | M | Technical criterion: S < P, V(x) outside R*Phi(S) with R generated by S-elements of smalle… | Assembly of 10.6-10.8 plus transfer-evaluation (Isaacs 5.5 = mathlib transfer_eq_prod_quotient_orbitRel_zpowers_quot). Verified missing. 【**2026-07-22 lane c 実測訂正: 済**】Ch.10 は 2026-07-17 に全 28 結果完成 (`Ch10_MoreTransfer/Main.lean` の完成表参照; sorry census 0・AxiomsCheck 登録済)。slim pass (07-16) 直後に landing したため表が未更新だった。 |
| Thm 10.10 | 済 | L | Mackey transfer: V(k) = prod over (H,K)-double coset reps x of x W_x(k) x^{-1} mod H' | Not in mathlib (Mathlib/GroupTheory/DoubleCoset.lean is thin: definitions + partition only). Needs double-coset transversal machinery (T = union of x S_x) and conjugated pretransfers; the priciest single item in S10A. Ve… 【**2026-07-22 lane c 実測訂正: 済**】Ch.10 は 2026-07-17 に全 28 結果完成 (`Ch10_MoreTransfer/Main.lean` の完成表参照; sorry census 0・AxiomsCheck 登録済)。slim pass (07-16) 直後に landing したため表が未更新だった。 |
| Lem 10.11 | 済 | S | Failure of p-transfer control yields normal M < N of index p containing U(G) for every pre… | Correspondence-theorem argument; requires first fixing a Lean-side ControlsPTransfer formulation (repo Ch05 uses APrime-equality instead of a predicate, confirmed at Ch05_Transfer/Basic.lean:1498). Verified missing. 【**2026-07-22 lane c 実測訂正: 済**】Ch.10 は 2026-07-17 に全 28 結果完成 (`Ch10_MoreTransfer/Main.lean` の完成表参照; sorry census 0・AxiomsCheck 登録済)。slim pass (07-16) 直後に landing したため表が未更新だった。 |
| Thm 10.12 | 済 | S | Huppert: p > 2, nonabelian metacyclic Sylow p-subgroup implies p divides /G:G'/ | Three-line assembly from 10.1 + 10.14 + 10.15 once those exist. NOT the same as the repo's 'Huppert' results: BG Prop 4.11 / Thm 4.12 (S04c_Prop411.lean, S04b_Thm412.lean) are Endliche Gruppen I Satz III.11.6/III.13.7, a… 【**2026-07-22 lane c 実測訂正: 済**】Ch.10 は 2026-07-17 に全 28 結果完成 (`Ch10_MoreTransfer/Main.lean` の完成表参照; sorry census 0・AxiomsCheck 登録済)。slim pass (07-16) 直後に landing したため表が未更新だった。 |
| Lem 10.13 | 済 | S | Metacyclic groups are closed under quotients (a) and subgroups (b) | (b) is formalized sorry-free in OddOrder/GroupTheory/IsMetacyclic.lean at full generality; (a) quotient closure has no Lean counterpart (verified: file has exactly of_isCyclic, isCyclic_commutator, subgroup, isSolvable; … 【**2026-07-22 lane c 実測訂正: 済**】(a) 商閉性は commit `af8b3dad1` (2026-07-17) の `IsMetacyclic.of_surjective` (全射像で閉じる一般形) で完成。 |
| Lem 10.14 | 済 | S | For p > 2, C_p wr C_p is not a homomorphic image of a metacyclic group | Short given 10.13(a) and 10.3(b); repo already has IsMetacyclic.isCyclic_commutator, the key ingredient. Verified missing. 【**2026-07-22 lane c 実測訂正: 済**】Ch.10 は 2026-07-17 に全 28 結果完成 (`Ch10_MoreTransfer/Main.lean` の完成表参照; sorry census 0・AxiomsCheck 登録済)。slim pass (07-16) 直後に landing したため表が未更新だった。 |
| Thm 10.15 | 済 | M | P normal Sylow, nonabelian metacyclic, p > 2 implies p divides /N:N'/ | Induction on /P/ using Maschke (10.17, formalized), Isaacs Thm 4.8(a) (formalized, CommutatorBasics.lean:1495) and Thm 6.11 (unique-subgroup-of-order-p, available via isCyclic_of_subgroups_card_prime_unique in Ch06). All… 【**2026-07-22 lane c 実測訂正: 済**】Ch.10 は 2026-07-17 に全 28 結果完成 (`Ch10_MoreTransfer/Main.lean` の完成表参照; sorry census 0・AxiomsCheck 登録済)。slim pass (07-16) 直後に landing したため表が未更新だった。 |
| Thm 10.16 | 済 | M | Maschke, group form: K acts on V = U x W, U abelian K-invariant, u -> u^m bijective on U, … | The book's general form (V an arbitrary group, U possibly infinite with bijective m-power map) has no Lean counterpart. mathlib's Maschke (Mathlib/RepresentationTheory/Maschke.lean) is the module version; the repo's Oper… 【**2026-07-22 lane c 実測訂正: 済**】Ch.10 は 2026-07-17 に全 28 結果完成 (`Ch10_MoreTransfer/Main.lean` の完成表参照; sorry census 0・AxiomsCheck 登録済)。slim pass (07-16) 直後に landing したため表が未更新だった。 |
| Thm 10.18 | 済 | S | Principal ideal theorem (Furtwangler): the transfer G -> G'/G'' is trivial | Immediate corollary of 10.25; the real cost is the S10C group-ring chain 10.19-10.27. Absent from mathlib (no group-ring augmentation-ideal transfer theory). Verified: zero hits for Furtwangler/principal-ideal; all repo … 【**2026-07-22 lane c 実測訂正: 済**】Ch.10 は 2026-07-17 に全 28 結果完成 (`Ch10_MoreTransfer/Main.lean` の完成表参照; sorry census 0・AxiomsCheck 登録済)。slim pass (07-16) 直後に landing したため表が未更新だった。 |
| Lem 10.19 | 済 | S | The augmentation ideal Delta(G) of Z[G] has Z-basis {g - 1 : g /= 1} | No Delta(G) construct anywhere in repo (verified; MonoidAlgebra is used only for character theory); straightforward over mathlib MonoidAlgebra/Finsupp basis API. 【**2026-07-22 lane c 実測訂正: 済**】Ch.10 は 2026-07-17 に全 28 結果完成 (`Ch10_MoreTransfer/Main.lean` の完成表参照; sorry census 0・AxiomsCheck 登録済)。slim pass (07-16) 直後に landing したため表が未更新だった。 |
| Thm 10.20 | 済 | M | G/G' isomorphic to Delta(G)/Delta(G)^2 via g -> (g-1) | Standard two-directional construction (homomorphism phi plus the Z-basis-defined theta); no counterpart in repo or mathlib. Verified missing. 【**2026-07-22 lane c 実測訂正: 済**】Ch.10 は 2026-07-17 に全 28 結果完成 (`Ch10_MoreTransfer/Main.lean` の完成表参照; sorry census 0・AxiomsCheck 登録済)。slim pass (07-16) 直後に landing したため表が未更新だった。 |
| Lem 10.21 | 済 | S | Components of elements of Delta(K)Delta(G) w.r.t. a transversal lie in Delta(K), with comp… | Generator-wise computation once the t-component additive maps are defined. Verified missing. 【**2026-07-22 lane c 実測訂正: 済**】Ch.10 は 2026-07-17 に全 28 結果完成 (`Ch10_MoreTransfer/Main.lean` の完成表参照; sorry census 0・AxiomsCheck 登録済)。slim pass (07-16) 直後に landing したため表が未更新だった。 |
| Cor 10.22 | 済 | S | Delta(K)^2 = Delta(K)Delta(G) inter Delta(K) = Delta(K)Delta(G) inter Z[K] | Immediate from 10.21. Verified missing. 【**2026-07-22 lane c 実測訂正: 済**】Ch.10 は 2026-07-17 に全 28 結果完成 (`Ch10_MoreTransfer/Main.lean` の完成表参照; sorry census 0・AxiomsCheck 登録済)。slim pass (07-16) 直後に landing したため表が未更新だった。 |
| Cor 10.23 | 済 | S | Image of Delta(K) in Delta(G)/Delta(K)Delta(G) is isomorphic to K/K' | Composition of 10.22 with 10.20 applied to K. Verified missing. 【**2026-07-22 lane c 実測訂正: 済**】Ch.10 は 2026-07-17 に全 28 結果完成 (`Ch10_MoreTransfer/Main.lean` の完成表参照; sorry census 0・AxiomsCheck 登録済)。slim pass (07-16) 直後に landing したため表が未更新だった。 |
| Thm 10.24 | 済 | M | For K normal in G, the transfer image v(G) is isomorphic to Xi(Delta(G)-bar), Xi = left mu… | The bridge between mathlib's MonoidHom.transfer and the group-ring model; requires the pretransfer computation mod Delta(K)Delta(G). Verified missing. 【**2026-07-22 lane c 実測訂正: 済**】Ch.10 は 2026-07-17 に全 28 結果完成 (`Ch10_MoreTransfer/Main.lean` の完成表参照; sorry census 0・AxiomsCheck 登録済)。slim pass (07-16) 直後に landing したため表が未更新だった。 |
| Thm 10.25 | 済 | M | For G' <= K <= G, v(g)^{/K:G'/} = 1 for the transfer v : G -> K/K' | Assembly of 10.24 + 10.26 + 10.27 with the /A:UA/ = /G:G'/ index computation; generalizes 10.18. Verified missing. 【**2026-07-22 lane c 実測訂正: 済**】Ch.10 は 2026-07-17 に全 28 結果完成 (`Ch10_MoreTransfer/Main.lean` の完成表参照; sorry census 0・AxiomsCheck 登録済)。slim pass (07-16) 直後に landing したため表が未更新だった。 |
| Thm 10.26 | 済 | M | Module over commutative ring: /A : UA/ = m finite implies some r in R with rA = 0 and r = … | Classical-adjoint determinant trick; mathlib has Matrix.adjugate and adjugate_mul plus Nakayama-style relatives (Submodule.exists_sub_one_mem_and_smul_eq_zero_of_fg_of_le_smul) but not this exact statement. Verified miss… 【**2026-07-22 lane c 実測訂正: 済**】Ch.10 は 2026-07-17 に全 28 結果完成 (`Ch10_MoreTransfer/Main.lean` の完成表参照; sorry census 0・AxiomsCheck 登録済)。slim pass (07-16) 直後に landing したため表が未更新だった。 |
| Lem 10.27 | 済 | S | epsilon annihilating Delta(G)-bar has augmentation m divisible by /G:K/, and (m//G:K/) Xi … | Component analysis using 10.21 plus transitivity of the dot action on the transversal. Verified missing. 【**2026-07-22 lane c 実測訂正: 済**】Ch.10 は 2026-07-17 に全 28 結果完成 (`Ch10_MoreTransfer/Main.lean` の完成表参照; sorry census 0・AxiomsCheck 登録済)。slim pass (07-16) 直後に landing したため表が未更新だった。 |
| Cor 10.28 | 済 | S | Alperin-Kuo: g^{/G : G' inter Z(G)/} = 1 for all g in G | Short given 10.8 + 10.18 + mathlib MonoidHom.transferCenterPow (= Isaacs 5.6, verified present in pinned mathlib Transfer.lean). Verified: zero Alperin/Kuo hits in repo. Repo's nearby-looking Ch05 lemmas (pow_index_cente… 【**2026-07-22 lane c 実測訂正: 済**】Ch.10 は 2026-07-17 に全 28 結果完成 (`Ch10_MoreTransfer/Main.lean` の完成表参照; sorry census 0・AxiomsCheck 登録済)。slim pass (07-16) 直後に landing したため表が未更新だった。 |

### Isaacs Appendix (The Basics) — Isaacs Appendix: The Basics

> The Isaacs appendix (pp. 325-343, results X.1-X.23, fully present in the mmd at lines 5915-6253) is the elementary-group-theory compendium, and as expected coverage is nearly total: 17 of 23 results are fully covered (14 by mathlib at full strength with verified declaration names, 3 by the repo — the product formula X.2 in OddOrder/Mathlib/Subgroup.lean, the cyclic-subgroup uniqueness X.7 in BG/S10_LocalLemmasCore, and X.12's join form in Isaacs/Ch03). The 6 remaining entries are all partial/missing on inessential clauses only: X.1's converse direction (its useful direction exists but is private in Ch02), X.4's direct-diamond bijection (missing entirely, the only wholly absent result), X.5's element-level nongenerator converse, X.11's equality-iff-HK=G clause, X.12's set-product strengthening, and X.22's literal prefix-intersection iff-criterion (mathlib factors internal direct products through iSupIndep/noncommPiCoprod instead); every gap is S-effort and none was ever needed on the FT path. Surprising finds: mathlib is stronger here than notes/meta/mathlib_coverage.md suggests — it now has Frattini nongenerating (frattini_nongenerating), the correspondence theorem as a lattice OrderIso (QuotientGroup.comapMk'OrderIso), and the N/C theorem (Subgroup.normalizerMonoidHom with computed kernel). The repo never cites appendix numbers (no 'Isaacs X.n' docstrings), so all matching was by content; OddOrder/Mathlib/Subgroup.lean is effectively the appendix's upstream-candidate home and already flags X.2 for a future mathlib PR. Closing the whole unit to 100% would be at most 2-3 days of routine work. Slim verification pass (2026-07-16): all six flagged labels (X.1/X.5/X.11/X.12/X.22 partial, X.4 missing) survived active refutation — grepped OddOrder/** (all book trees, GroupTheory, Algebra, Mathlib dirs) by citation formats and content-derived names, and read mathlib's Frattini.lean, Order/Radical.lean, Index.lean, and Subgroup/Pointwise.lean directly; the load-bearing formalized entries X.2 and X.7 were confirmed at full book strength by reading their Lean statements, so no status changed and only note-level enrichments were added (X.22 gains two repo partial-support lemmas; X.11's mathlib inequality carries a benign ≠0 side condition).

| 結果 | 状態 | 規模 | 内容 | メモ |
|---|---|---|---|---|
| X.1 | 済 | S | HK is a subgroup iff HK = KH (permutable subgroups) | ✅ 2026-07-18 `Appendix/DirectDiamond.lean` `coe_sup_eq_mul_iff_mul_comm` (public 化)。 |
| X.4 | 済 | S | Direct-diamond correspondence: X ↦ X∩K injects {X : H≤X≤HK} into {W : D≤W≤K}, image = {W : WH subgroup} | ✅ 2026-07-18 `Appendix/DirectDiamond.lean` `directDiamond_bijOn`/`injOn`/`image` (Dedekind + X.1)。 |
| X.5 | 済 | S | Frattini subgroup = set of nongenerators | ✅ 2026-07-18 `Appendix/SubgroupBasics.lean`: converse `mem_frattini_of_nongenerating` (finiteness 不要)、full iff `mem_frattini_iff_nongenerating [IsCoatomic]`。forward half = mathlib `frattini_nongenerating`。 |
| X.11 | 済 | S | /K : H∩K/ ≤ /G:H/, with equality iff HK = G | ✅ 2026-07-18 `Appendix/SubgroupBasics.lean` `relIndex_eq_index_iff_mul_eq_univ [Finite G]`。⚠ "HK=G" は set product `↑H*↑K=univ` (H⊔K=⊤ より真に強い、S₃ 反例)。inequality は mathlib `relIndex_le_of_le_right`。証明は repo Lem X.2 経由。 |
| X.12 | 済 | S | Coprime indices /G:H/, /G:K/ => HK = G | ✅ 既済 (再確認 2026-07-18): set-product 形 = `Ch03.set_mul_eq_univ_of_coprime_index` (Theorem315.lean)、join 形 = `Ch03.sup_eq_top_of_coprime_index`。SubgroupBasics で docstring 注記。 |
| X.22 | 済 | S | Internal direct product criterion: G = M₁⋯M_r with M_i ⊴ G is direct iff (M₁⋯M_{k-1}) ∩ M_k = 1 | ✅ 2026-07-18 `Appendix/SubgroupBasics.lean` `iSupIndep_iff_disjoint_biSup_lt [Finite ι]` (正規族の iSupIndep ⟺ prefix-disjoint)。hard 方向 `supIndep_of_prefix`、modular law instance `disjoint_sup_of_normal`。 |

## BG

> ## ⚠⚠ BG / Peterfalvi 節も実測監査で 42% が stale と判明 (hub, 2026-07-19、issue 9154 §2)
>
> **未/部分/特殊化 ラベルの 111 件を実測再検証した結果: 47 件 (42%) が実体と食い違った。**
> Isaacs 節と同じ失敗モードで、本 note は **scope 正本から降格済** (hub 裁定 9154)。
> **着手前に必ず repo 実測で再確認すること。**
>
> | 判定 | 件数 | 内容 |
> |---|---|---|
> | `STALE_ALREADY_DONE` | 37 | note は 未/部分/特殊化 だが、実際は教科書強度で形式化済 |
> | `STALE_PARTIAL` | 10 | ラベルと実体の程度が食い違う (**うち 2 件は「済」なのに実際は部分** = 危険側) |
> | `CONFIRMED` | 64 | ラベルどおり = 本物の残作業 |
>
> **特に大きい塊**: Pf App Suzuki の I.1/I.2/I.3 Props **13 件が「未」だが実際は完了**
> (lane b が構築済)。BG §1-§4 の特殊化債務 bullet 5 件も本文では解消済と書きながら
> 見出しが `特殊化債務` のままで、見出しだけ読むと未解決に見える。
>
> **⚠ 危険側のズレ (「済」なのに実際は部分)**:
> - **BG Thm A** — A(6) の `M_F ≠ 1` / `M' ⊊ M` / `M'/M_F nilpotent` が欠落。
>   特に `M_F ≠ ⊥` を主張する宣言はリポジトリに**存在しない** → **issue 0126**。
> - **BG §16 引用ブロック**「genuine residual gaps 4 件」 — 3 件は landing 済で散文が stale、
>   残る 1 件 (Thm A の一般 `M'/M_F` nilpotent) は実在するギャップ。
>   同ブロックの「S14 に real sorry が 2 件残る」も無効 (`OddOrder/BG/Ch4_FamilyOfMaximal/**`
>   全 35 ファイルの実 sorry = **0**)。
>
> **リポジトリ側の stale も 1 件**: `S15_MF/OpicoreCentralizer.lean:410-413` の
> `fitting_not_ti_cases` docstring が「(d)(e) は未達、15.7 の完全形式化と読むな」と書いているが、
> (d)(e) とも別宣言として landing 済 (issue 3022 close) → issue 0126。
>
> **✅ 2026-07-19 行単位の反映を実施**: 表の行に反映できた **21 件を訂正** (未/部分 → 済 が 20、
> 済 → 部分 が 1 = BG Thm A)。各行のメモ末尾に `【2026-07-19 hub 監査で訂正: … 】` と
> repo 上の所在を追記した。BG App.E の集計行 (旧 未5/5) も 部分3/未2 に訂正。
> **残り 26 件は表の行でなく箇条書き・散文ブロック**で、その多くは**本文に既に訂正が入っていて
> 見出しラベルだけが古い**タイプ (BG §1-§4 の特殊化債務 bullet 5 件、Pf §4 の (2.6)-(2.10.3)
> 7 件は lane c が 2026-07-18 に注記済)。影響が小さいため優先度を下げて残置する。
>
> 監査 workflow = `wf_27b12223-fd5` (7 バッチ並列 read-only)。


| unit | n | 済 | 特殊化 | 部分 | 未 | mathlib |
|---|---|---|---|---|---|---|
| BG §1 | 27 | 20 | 3 | 1 | 0 | 3 |
| BG §2 | 7 | 2 | 2 | 2 | 1 | 0 |
| BG §3 | 10 | 9 | 1 | 0 | 0 | 0 |
| BG §4 | 21 | 14 | 2 | 3 | 1 | 1 |
| BG §5 | 8 | 8 | 0 | 0 | 0 | 0 |
| BG §6 | 7 | 4 | 1 | 1 | 1 | 0 |
| BG §7 | 8 | 8 | 0 | 0 | 0 | 0 |
| BG §8 | 1 | 1 | 0 | 0 | 0 | 0 |
| BG §9 | 6 | 6 | 0 | 0 | 0 | 0 |
| BG §10 | 16 | 16 | 0 | 0 | 0 | 0 |
| BG §11 | 8 | 8 | 0 | 0 | 0 | 0 |
| BG §12 | 20 | 20 | 0 | 0 | 0 | 0 |
| BG §13 | 14 | 14 | 0 | 0 | 0 | 0 |
| BG §14 | 15 | 13 | 1 | 1 | 0 | 0 | ⚠ 2026-07-23 hub 監査で訂正: Lem 14.13(b) は 07-18 landing 済 (partial→済)。残る唯一の "部分" = Cor 14.12 M*-pinned surfacing (残 2 clause は内部で証明済・consumer 0 の effort-S packaging 債務、genuine 新規数学でない)。issue 9318 HUB 裁定参照
| BG §15 | 10 | 6 | 4 | 0 | 0 | 0 |
| BG §16 | 11 | 4 | 3 | 3 | 1 | 0 | ⚠ 2026-07-23 hub 監査で訂正: 全 11 結果 (Thm A/B/C/D/E, Prop 16.1, Thm I/II, tamely-imbedded 定義) sorry-free 完成。Thm A A(6)/(7) は 0126 close 済、Thm II は無条件形。旧「3 部分 + 1 未」は 07-18〜22 landing で全解消の stale snapshot
| BG App.A | 6 | 6 | 0 | 0 | 0 | 0 |
| BG App.B | 6 | 6 | 0 | 0 | 0 | 0 |
| BG App.C | 19 | 12 | 3 | 0 | 4 | 0 |
| BG App.D | 3 | 0 | 0 | 3 | 0 | 0 | ⚠ 2026-07-23 hub 監査で訂正 (旧: 全 3 partial): 全 3 結果 (Lem D.1 Sylow-TI / Lem D.2 P≤N(P)′ / 3-step dichotomy) 完成・sorry-free・axiom-clean (lane c 自作 issue 3020/9133、非 contested)。partial は stale
| BG App.E | 5 | 0 | 0 | 3 | 2 | 0 | ⚠ 2026-07-19 hub 監査で訂正 (旧: 部分0/未5)。E.1/E.2/E.3 は部分 (E.1 は class≤2, ≤3 が全 n で sorry-free)、未は E.4/E.5 のみ |

### BG §1 — BG §1 Elementary Properties of Solvable Groups

> BG §1 is essentially complete: all 22 numbered results (Lem 1.1–Lem 1.22) have sorry-free Lean counterparts, spread across OddOrder/BG/Ch1_Preliminary/S01* + PLength* files, the Isaacs Ch03–Ch06 tree (Prop 1.5(d), 1.6(a)(b)(d)(e), Thm 1.11, 1.17, 1.18, Prop 1.16(1)), OddOrder/GroupTheory (Frattini, critical subgroup, TI, narrow), and mathlib (§1F: Cor 1.19 and Maschke are deliberately left to mathlib per the no-wrapper policy). [2026-07-18: Lemma 1.21(d) CLOSED — `PLengthPComplement.hasPLengthOne_iff_hasNormalPComplement_pElementsSubgroup`, axiom-clean, so **BG §1 is now fully complete (all 22 numbered results)**. Note the survey's "effort S" was optimistic: `U = ⟨p-elements⟩ = O^{p'}(G)` was NOT in the repo (the existing `OPrime` is O^p, the p-residual — a different object) and had to be defined (`pElementsSubgroup`), plus a new reusable `hasNormalPComplement_of_normal_pPrime_index` bridge; actual effort was moderate.] The former only gap was Lemma 1.21(d) (p-length one ⇔ ⟨p-elements⟩ has a normal p-complement), never consumed downstream. Three results are stated in equivalent-but-not-literal forms worth noting: Lem 1.1 assumes IsSolvable G where the book only needs M solvable, Lem 1.9 exists as the coprime chain-stabilizer engine rather than the 'A/C_A(G) is a π-group' packaging, and Prop 1.15(a) is stated as C_G(O_{p',p}) ≤ O_{p',p} / its O_{p'}=1 specialization rather than via a Sylow subgroup T of O_{p',p}(G) (a 5-line corollary of formalized pieces). In several places the formalization is stronger than the book: Prop 1.16 works for any coprime noncyclic abelian A (book: p-group on p'-group), Hall–Higman is proved in general π-separable form with no O_{π'}=1 hypothesis, the Isaacs-side coprime-action lemmas (Prop 1.5(d), Prop 1.6) need only IsSolvable A ∨ IsSolvable G, and Lemma 1.21 has an extra dual transfer lemma. All §1 preamble definitions (narrow, TI-subset, p-length one, prime/regular manner, Z-group) are formalized or covered by mathlib. Verification pass: confirmed the Lem 1.21 partial label by refutation attempts against the Coq analog p_elt_gen_length1 (no p-element-generated-subgroup/O^{p'} content anywhere in the repo; OpResidual.lean is opPi, not the residual); re-read and confirmed full book strength for the load-bearing Prop 1.3, Prop 1.5, and Thm 1.13 statements, correcting only the Prop 1.5 note (parts (a)(b)(c)(e) use exactly the book's IsSolvable G hypothesis; only part (d) has the more general IsSolvable A ∨ IsSolvable G).

| 結果 | 状態 | 規模 | 内容 | メモ |
|---|---|---|---|---|
| Lem 1.21 | 済 | S→M | p-length one transfer: (a) subgroups, (b) mod normal p'-subgroup, (c) mod normal p-subgrou… | [2026-07-18 (d) CLOSED ⟹ Lem 1.21 全 (a)-(e) 完成] PLengthTransfer.lean: (a)(b)(c)(e). Part (d) = PLengthPComplement.hasPLengthOne_iff_hasNormalPComplement_pElementsSubgroup (new pElementsSubgroup def + normal-Hall-p'⟹normal-p-complement bridge), axiom-clean. |

特殊化債務 (教科書より狭い形で証明済 — 一般化要否は着手時に判定):

- **Lem 1.1** solvable minimal normal subgroup lies in Z(F(G)) and is elementary abe — S01_FrattiniBurnside.lean, sorry-free. Conclusion is full (M ≤ F(G) ∧ M ≤ C_G(F(G)) ∧ elementary abelian = M ≤ Z(F(G))), but hypothesis is [IsSolvable G] where … **[2026-07-22 lane c: 特殊化解消 — `[IsSolvable ↥M]` に一般化 (book 強度)、call site 無変更で green、axiom-clean]**
- **Lem 1.9** stabilizer of a normal series of a π-group: A/C_A(G) is a π-group — S01_Solvable.lean, sorry-free. Formalized as the coprime core: a coprime operator group stabilizing a normal chain acts trivially — both the 2-step instance and… **[2026-07-22 lane c: 逐語 packaging 追加 — `primeFactors_card_quotient_ker_subset_of_stabilizes_chain` (S01_Solvable.lean、A/C_A(K) = A⧸ψ.ker が π-群)、axiom-clean。§1 の特殊化債務は全て解消]**
- **Prop 1.15** (a) Hall–Higman Lemma 1.2.3: C_G(T) ≤ O_{p',p}(G) for T Sylow-p in O_{ — (b) fully formalized sorry-free at book strength (general G solvable, no O_{p'}=1 hypothesis). (a) is present as the O_{p'}(G)=1 specialization C_G(O_p(G)) ≤ O_… **[2026-07-18: (a) 一般形 CLOSED ⟹ 特殊化債務解消]** `S01_Solvable.centralizer_le_oPiPrimePiCore_of_cover` が O_{p'}(G)=1 仮定を除去した一般形 (π-separable G, 任意の π-層) を axiom-clean で証明。Sylow 仮定は covering 仮定 `O_{π',π}(G) ≤ T ⊔ O_{π'}(G)` に弱められており (book より強い)、book の literal Sylow 形への橋渡しは `sup_oPiCore_compl_eq_oPiPrimePiCore_of_isSylow` (同 file、axiom-clean)。証明は Ḡ=G/O_{π'}(G) へ落として `Isaacs.Ch03.hall_higman_1_2_3` を適用するだけ (Lem 1.14 は不要 — centralizer の push-forward は易方向で足りる)。

### BG §2 — BG §2 General Results on Representations

> BG §2 contains 7 numbered results and is in strong shape: the two hard theorems that the FT critical path actually consumes — Prop 2.4 (all ten eigenspace/block parts, pure linear algebra at full generality) and Thm 2.6 (odd-order 2-dim representations, both parts at full book strength) — are completely formalized and sorry-free, as is a fully-grounded Thm 2.5 (specialized to algebraically closed fields; the book's general-field descent bridge exists in BaseChange.lean but was never assembled into a wrapper). The formalization is factored very differently from the book: almost everything lives in ~15 shared modules under OddOrder/GroupTheory/RepresentationTheory/ (EigenspaceUnderCyclicAction, EigenspaceBlockDecomp, CyclicEndConjCount, ExtraspecialThm25*, Clifford*, AbsolutelyIrreducible), with only Thm 2.6 in the BG tree itself, and the S02_RepresentationsBasic planning docstrings are badly stale. The main gaps are foundational-generality ones with zero FT-path consumers: no absolute-irreducibility predicate exists (so Prop 2.1's iff/(c) for general G and the positive-characteristic half of Lemma 2.3 are open) and Lemma 2.7 is entirely missing (cited nowhere else in BG and unused in the Coq FT proof). [2026-07-18 UPDATE: two of the originally-flagged §2 gaps are now CLOSED at full book strength. (i) Prop 2.2(a)'s spurious char ∤ |H| hypothesis was removed — CliffordAlgClosed.isSemisimpleModule_resRep_of_isIrreducible now supplies restriction semisimplicity by the classical Clifford argument in ANY characteristic, so restriction_isSimpleModule/restriction_isIrreducible dropped [Finite ↥H] [NeZero (Nat.card ↥H : k)]. (ii) Prop 2.2(b) was generalized off ℂ into the new CyclicExtension.lean: exists_extension_of_nonempty_equiv_conjRep extends an irreducible rep of H⊴K along a cyclic quotient over ANY algebraically closed field in any characteristic (the ℂ character-orthogonality step is now only in the CyclicCharacterExtension.lean Isaacs-11.22 layer, which imports the generic core). Both axiom-clean. (iii) [2026-07-18 later same day] Lem 2.3 (Fong-Swan) CLOSED at full book strength: FongSwan.finrank_dvd_card_of_isAbsolutelyIrreducible (general field, solvable G, absolutely irreducible ⟹ finrank F V ∣ |G|), via the elementary BG Clifford-induction proof (not Brauer lifting), any characteristic; axiom-clean. (iv) [2026-07-18 later same day] Lem 2.7 CLOSED at full book strength (ElemAbelianAutAction.elemAbelian_aut_action; statement recovered from a Nougat MISSING page via PDF); axiom-clean. **§2 is now essentially complete: all 7 numbered results (2.1-2.7) formalized** — the only residual is re-verifying whether Prop 2.1's general-G iff/(c) is fully closed (lane c added the IsAbsolutelyIrreducible predicate + general-field Burnside 2026-07-17; the alg-closed shape BG uses downstream is done).] Slim verification pass: all three flagged labels CONFIRMED after refutation attempts (Prop 2.1 partial, Lem 2.3 partial, Lem 2.7 missing — with one new finding folded into Prop 2.1's notes: SemilinearField.lean's endField/finrank_end_eq_one proves the 2.1(c) finite-field-endomorphism content for commutative groups only), and both 'formalized' results (Prop 2.4, Thm 2.6) had their actual Lean statements read and confirmed at full book strength.

| 結果 | 状態 | 規模 | 内容 | メモ |
|---|---|---|---|---|
| Prop 2.1 | 済 | M | Schur/absolute irreducibility: (a) abs. irred. iff Hom_FG(M,M)=F; (b) faithful + Hom_FG=F … | VERIFIED partial. GroupTheory/RepresentationTheory/AbsolutelyIrreducible.lean proves, sorry-free, the algebraically-closed instance (Burnside: E(G) = End_F V for f.d. irreducible), which is the only shape BG uses downstr… 【**2026-07-19 hub 監査で訂正: 部分 → 済**】実測で教科書強度・sorry-free と確認。所在: OddOrder/GroupTheory/RepresentationTheory/AbsolutelyIrreducible.lean — (a) `isAbsolutelyIrreducible_iff_bijective_algebraMap` (L672), (b) `asAlgebraHom_surjective_of_algebraMap_intertwiningMap_surjective` (L137) / `span_ |
| Lem 2.3 | 済 | M | Fong–Swan corollary: G solvable, M absolutely irreducible FG-module implies dim M divides … | [2026-07-18 CLOSED at full book strength] `FongSwan.finrank_dvd_card_of_isAbsolutelyIrreducible` (general field F, solvable finite G, absolutely irreducible ⟹ finrank F V ∣ |G|), axiom-clean, via alg-closed core `finrank_dvd_card_of_isAlgClosed_of_irreducible` (strong induction on |G|: prime-index H◁G, char-free Clifford, case (i) restriction_isSimpleModule / case (ii) finrank_eq_index_mul_finrank_of_not_linearEquiv_conj) + base change to AlgebraicClosure F. Uses the elementary BG proof (Clifford + Prop 2.2), NOT Brauer lifting, so any characteristic. Prior note: char-0 content was covered by finrank_dvd_card over ℂ; now the general-field statement is proved. |
| Lem 2.7 | 済 | M | p ≠ q primes, P ≅ (Z/p)², Q ≅ (Z/q)², Q ⊆ Aut(P): (a) q ∣ p−1; (b) some α ∈ Q^# acts as a … | [2026-07-18 CLOSED at full book strength] `ElemAbelianAutAction.elemAbelian_aut_action` (P 2-dim 𝔽_p space, Q elem.abelian order q² faithful 𝔽_p-linear = Q⊆Aut(P) ⟹ q∣p−1 ∧ ∃ α≠1 scalar of order q), axiom-clean, AxiomsCheck-registered. Statement recovered from PDF (Nougat MISSING_PAGE_FAIL:29). Reusable crux `isCyclic_of_faithful_isIrreducible` = Gorenstein Thm 3.2.3 (finite abelian faithful irreducible ⟹ cyclic, via F[Q] comm + annihilator-maximal + field-units-cyclic). Reuses LineScalarCharacter + Maschke. |

特殊化債務 (教科書より狭い形で証明済 — 一般化要否は着手時に判定):

- **Prop 2.2** Clifford for cyclic quotient: M irred FH-module, M ≅ M^x for all x, F  — [2026-07-18: NO LONGER specialization debt — both parts now at full book strength over any algebraically closed field in any characteristic.] (a) = restriction_isIrreducible (CliffordMultiplicityOne.lean), char-free after removing the Maschke hypotheses; (b) = exists_extension_of_nonempty_equiv_conjRep (CyclicExtension.lean), char-free. (a) is fully assembled and sorry-free: CliffordAlgClosed.lean (conjugates permute simple constituents, conjugate span, isotypic structure) + SkolemNoether.lean …
- **Thm 2.5** Extraspecial P of order p^{2n+1} ⋊ cyclic H of order h with C_P(x)=Z(P — Fully-grounded group-level version, sorry-free, in ExtraspecialThm25Final.lean (assembled from ExtraspecialThm25.lean/Thm25Group.lean eigenspace-keystone machin…

### BG §3 — BG §3 Actions of Frobenius Groups

> BG §3 (Actions of Frobenius Groups, mmd L795-1359) contains exactly 10 numbered results (Lemmas 3.1-3.3, Theorems 3.4-3.8, Proposition 3.9, Theorem 3.10), formalized across 15 sorry-free files OddOrder/BG/Ch1_Preliminary/S03*.lean (~13,250 lines). Nine of the ten are formalized at full book strength, including the two hardest: Theorem 3.6 (p-length one, a 3,822-line minimal-counterexample proof through equations (3.6)-(3.38) with a separate orbit-parity terminal file) and Theorems 3.4/3.5, which even include general-field versions via base change and faithfully-flat descent exactly as the book's remark on algebraic closure requires. Two results are strictly STRONGER than the book: Theorem 3.7 (Frobenius kernel nilpotency) is proved without the odd-order hypothesis, and Proposition 3.9 drops the p'-group hypothesis on the acted-on group. [2026-07-18: Theorem 3.10's general packaging CLOSED ⟹ **BG §3 is now fully complete (all 10 results)**.] The former gap was Theorem 3.10's general-nilpotent-M packaging (only the FT-consumed elementary-abelian forms were assembled). Now `OddOrder.BG.Ch1.S03g.bgThm310_nilpotent` (`S03g_Thm310Nilpotent.lean`) gives the full book statement — general nilpotent M, general kernel, all three parts — and is in fact STRONGER than the book (the group-level Case-1 induction, in the `φ : H →* MulAut M` framework, uses only that M is solvable, so it holds for any solvable M). Assembled from three new pieces: `S03g_Thm310FixedPointSplit` (the coprime fixed-point order split, (b)-glue), `S03g_Thm310GroupForm` (the elementary-abelian base in group form, `invariants ↔ fixedSubgroup` bridge), and `S03g_Thm310Nilpotent` (the induction, gluing via the order split + Lemma 1.9). Axiom-clean, AxiomsCheck-registered.

特殊化債務 (教科書より狭い形で証明済 — 一般化要否は着手時に判定):

- **Thm 3.10** Solvable Frobenius G=KR acting on nonidentity nilpotent M coprimely, C — Split over S03g_Thm310{,Core,Module,General,ElemAbelian}.lean, all sorry-free. Coverage: (a)+(b)+(c) fully proved for M an ELEMENTARY ABELIAN p-group with gener… **[2026-07-18: (a) の nilpotency 仮定 CLOSED]** `prime_card_complement_of_frobenius_conj` (S03g_Thm310.lean) が担いでいた `_hMnilp : Group.IsNilpotent ↥M` は証明本体で**一度も使われていなかった** (signature に 1 回出るのみ) ため除去。U abelian 版では minimal E-invariant normal への reduction に nilpotency が不要ゆえ、(a) の群形は **book より強い** (M は nonidentity E-invariant + coprime のみ)。唯一の caller ElemAbelianNeighbor.lean:417 も追従済 (build green, axiom-clean)。~~**(b)(c) の M elementary abelian 制限は未解消** (nilpotent M への G-invariant normal series 帰納が残作業)。~~ **[2026-07-22 lane c 実測訂正: この注記は同日 (07-18) の `bgThm310_nilpotent` landing (unit summary 参照) より古く stale — (b)(c) は一般 nilpotent (実は solvable) M で完成済み。§3 の特殊化債務はゼロ]**

### BG §4 — BG §4 p-Groups of Small Rank

> BG §4 (Lemma 4.1–Theorem 4.20, 20 numbered results + the section's definition block) is one of the most completely formalized BG units: all files are sorry-free, and every apex result — Prop 4.11 (Huppert metacyclic), Thm 4.12, Lem 4.13/4.14 (via the Gorenstein 4.15(ii) precursor chain in S04d/S04e), Thm 4.16 (Blackburn classification, blackburnRankTwoClassification), Lem 4.17, Thm 4.18, and Thm 4.20(a)(c) — is proved, spread across S04*-leaves in BG/Ch1_Preliminary plus shared leaves (SCN, PRank, IsMetacyclic, IsExtraspecial, CentralProduct, NarrowPGroup). Several statements are strictly stronger than the book: Lem 4.15 and Prop 4.4(a) are p-general, Lem 4.17 drops the book's solvability hypothesis, and Thm 4.16 needs no faithfulness of the action. The gaps are concentrated in small "normal-refinement" utility results the FT path never needed: Prop 4.4(b) (C_G(A)=A×H for A∈SCN(Sylow)), the normal half of Lem 4.5(a) and the noncyclic half of 4.5(c) in general (only abelian-center resp. rank≥3 cases exist), Prop 4.6 (whose content is 4.5(c)), and Thm 4.20(b); additionally Thm 4.12(b)(c) carry a removable extra IsSolvable A hypothesis, Cor 4.19 exists only in its post-O_{p'}-reduction form, and 4.20(c)'s series package records factor cardinality rather than Sylow isomorphism and does not record the ascending-prime ordering. [2026-07-18: (b) CLOSED = `S05_Thm420b.characteristic_le_derived_normal_of_rank_fitting_le_two` (Frattini G=F·N_G(S) + Fitting iSup 分解), axiom-clean ⟹ Thm 4.20 全 (a)(b)(c) 済。] Closing the whole unit to full book strength is roughly one M-sized task (Gorenstein 5.4.10 for 4.5/4.6) plus three S-sized ones (4.4(b), 4.19 wrapper, 4.20(b)/packaging). **[2026-07-18: ALL of these are now CLOSED — BG §4 is fully complete (Prop 4.4(b), Lem 4.5(c)/Prop 4.6, Thm 4.20(b), Thm 4.12(b)(c) generalization, Cor 4.19 general form all formalized axiom-clean; and Lem 4.5(a) was found already-done, correcting a stale survey entry).]** Slim verification pass: no corrections needed — all four flagged labels (Prop 4.4 partial, Lem 4.5 partial, Prop 4.6 missing, Thm 4.20 partial) survived targeted refutation attempts (number and descriptive-content greps across OddOrder/GroupTheory, Isaacs, and BG trees found nothing beyond the already-recorded special cases, and the source docstrings explicitly confirm the deferrals), while the three load-bearing formalized rows checked in source (Lem 4.7, Thm 4.16, Thm 4.18) were confirmed at full book strength.

| 結果 | 状態 | 規模 | 内容 | メモ |
|---|---|---|---|---|
| Prop 4.4 | 済 | S | SCN(R) = maximal abelian normal subgroups; C_G(A) = A × H (H p'-group) for A ∈ SCN(Sylow) | [2026-07-18 (b) CLOSED ⟹ Prop 4.4 完成] (a) `isSCN_iff_isMaximalAbelianNormal` (GroupTheory/SCN.lean). (b) = `GroupTheory.centralizer_eq_dprod_of_isSCN_of_sylow` (S04_Prop44b.lean, Gorenstein 7.6.5): C_G(A) = A × H internal direct product, H p'-group. Proof: A central normal Sylow-p of C_G(A) (via new Sylow-in-normal helper `sylow_relIndex_normal_not_dvd`) + Schur-Zassenhaus. axiom-clean. |
| Lem 4.5 | 済 | S | noncyclic odd p-group: (a) normal E_{p²} exists; (b) cyclic index-p subgroup ⇒ Ω₁(R)=E_{p²… | [2026-07-18 CORRECTION: survey was STALE.] (a) の **normal 結論は一般形で済** — `exists_normal_isElementaryAbelian_card_prime_sq_of_not_isCyclic` (S04_SmallRankBasic §4E "Lemma 4.5(a) unconditional = Gorenstein 5.4.10"、axiom-clean、maximal-abelian-normal A の cyclic/noncyclic 2-case、metacyclic 経由)。(b) 済。**残 = (c)** (Ω₁(Z₂(R)) noncyclic exponent p) — standalone 未、ただし (a) + Prop 4.3(a) (§4A、Ω₁ exponent p for cl≤2) から S-sized 導出。**(c) も 2026-07-18 完成** (`omega1UpperCentralTwo_not_isCyclic_and_card_prime_sq_le_of_not_isCyclic`) ⟹ Lem 4.5 全 (a)(b)(c) 済。 |
| Prop 4.6 | 済 | S | noncyclic normal S ⊴ R contains an R-normal elementary abelian subgroup of order p² | [2026-07-18 CLOSED] `S04_Lem45c_Prop46.exists_normal_isElementaryAbelian_card_prime_sq_le_of_normal_not_isCyclic` (+ Lem 4.5(c) `omega1UpperCentralTwo_not_isCyclic_and_card_prime_sq_le_of_not_isCyclic`), axiom-clean. Ω₁(Z₂(S)) noncyclic exp-p (4.5(a)+Prop 4.3(a)) + Lem 1.22 で order-p² 抽出。survey が 4.5(a) を stale に未記載していたのを訂正した副産物。 |
| Thm 4.20 | 済 | M | G solvable odd with r(G) ≤ 2 or r(F(G)) ≤ 2: (a) G' nilpotent; (b) T char S Sylow, T ⊆ S' … | (a) formalized at full strength (r(F(G))≤2 branch subsumes r(G)≤2 since r(F)≤r(G)); conclusion G' ≤ F(G) in S05_NarrowPGroups.lean (proved via the stronger Thm 5.7 rather than Cor 4.19). (c) formalized: existence of Char… |

特殊化債務 (教科書より狭い形で証明済 — 一般化要否は着手時に判定):

- **Thm 4.12** Huppert: metacyclic R under coprime action — [2026-07-18: (b)(c) の removable [IsSolvable A] 除去 ⟹ 特殊化債務解消、全 (a)(b)(c) が operator group A の solvability 仮定なしの book strength] S04b_Thm412.lean, all sorry-free。R が p-群 (nilpotent⟹solvable) ゆえ下層 Prop 1.6 の IsSolvable A∨IsSolvable R を Or.inr で供給。…
- **Cor 4.19** G solvable odd, G* ⊴ G with r_p(G*) ≤ 2 ⇒ G' centralizes every p-chief — [2026-07-18: 一般形 CLOSED ⟹ 特殊化債務解消] `S04g_Cor419.commutator_le_chiefFactorCentralizer_of_rank_le_two_of_le_normal` (general G*, U⊆G*), axiom-clean。O_{p'}(G*)=1 reduction を Ḡ=G/O_{p'}(G*) への passing (linchpin U⊓W≤V + chief-factor transfer + collapse) で reduced endpoint に乗せる。旧 reduced form も S04g_Thm418 に残存 (FT path 消費)。…

### BG §6 — BG §6 Additional Results

> BG §6 contains 7 numbered results; 5 are fully formalized and sorry-free, mostly in OddOrder/BG/Ch1_Preliminary/S06_Additional.lean. The formalization is factored differently from the book in two notable ways: Thm 6.1 lives in AppA_PStability.lean as thmA4b (the book itself identifies Thm 6.1 with Thm A.4(b)), and Thm 6.2 follows the book's own remark (and the Coq formalization) by proving the general form with Puig's L(S) instead of Thompson's J(S) — Z(L(S))·O_{p'}(G) ⊴ G in AppB_Thm62.lean — with the literal J(S) form only in a reduced case via Isaacs Thm 7.6. The gaps are Lemma 6.3(b) (missing part; part (a) is proved in both conclusions, the first even without the coprimality the book's Hall hypothesis supplies) and Theorem 6.4 (entirely missing; the Coq odd-order formalization also deliberately skips it as unneeded for the revised proof, and nothing in the repo consumes it). The S06 file's header comparison table is stale — it marks 6.3(b)-6.4 and 6.7 all as TODO, but Lemma 6.6 (all four parts) and Thm 6.7 are in fact fully proved later in the same file. Verification pass: both flagged labels (6.3 partial, 6.4 missing) were confirmed after refutation attempts (number, descriptive-content, and Coq-name greps across OddOrder/ including GroupTheory/**, all negative — every 'Thm 6.4' hit is Isaacs Thm 6.4 on Frobenius groups, and OpResidual.lean's mention is motivational only); statement-level reads confirmed full book strength for Thm 6.1, Lem 6.5, Lem 6.6, and Thm 6.7, with one note refinement — the Lem 6.6 foundation equality is actually proved WITHOUT solvability (omit [IsSolvable G]), matching Coq rather than merely the book.

| 結果 | 状態 | 規模 | 内容 | メモ |
|---|---|---|---|---|
| Lem 6.3 | 済 | S | (a) normal Hall H ≤ G' with complement K ⟹ H = [H,K] and C_H(K) ≤ H'; (b) G' nilpotent, /G… | [2026-07-18: (b) CLOSED ⟹ Lem 6.3 全済] (b) = `S06_Lem63b.lemma63b` (G' nilpotent + |G/G'| prime ⟹ G' Hall + G'=[G,K]), axiom-clean; Q=G/O_{p'}(G') が p-群 derived-index-p ⟹ Burnside cyclic ⟹ G'=O_{p'}(G')。Part (a) fully formalized in S06_Additional.lean, in two declarations (L307, L394): first conclusion ⁅H,K⁆ = H proved WITHOUT coprimality (stronger than the book, whose Hall hypothesis supplies it); sec… |
| Thm 6.4 | 済 | M | Conjugation of H-invariant π-subgroups: H π'-subgroup normalizing π-subgroups J₁, J₂ (G₀ n… | Verified missing — refutation attempted and failed: every 'Thm 6.4'/'Theorem 6.4' hit in OddOrder/ is Isaacs Thm 6.4 (the Frobenius-group four-condition equivalence in Isaacs/Ch06_FrobeniusActions/FrobeniusGroup.lean), a… 【**2026-07-22 lane c 実測訂正: 済**】issue 3026 close (2026-07-19)。`S06_Thm64Case2.lean` の `exists_centralizing_conj_sup_isPiGroup_of_normalHall` = BG Thm 6.4 無条件形、AxiomsCheck 済。survey の「missing」は stale。 |

特殊化債務 (教科書より狭い形で証明済 — 一般化要否は着手時に判定):

- **Thm 6.2** Normal-J: Z(J(S))·O_{p'}(G) ⊴ G — [2026-07-18 精査] L(S) 一般形 (book 推奨代替 = AppB.zCenter_lOdd_sup_oPiCore_normal) は**済+FT 消費**、role 満たす。literal J(S): reduced case (normalJ_normal_of_odd) 済 + **O_{p'} reduction を conditional lemma で landing** (S06_Thm62JS.zCenterThompsonJ_sup_oPiCore_normal_of_reduced, sorry-free/axiom-clean, hZJ 仮定)。残 = hZJ = **Glauberman Z(J) 定理** (Isaacs 意図的省略の major result; C_G(Z(P))=P は非 abelian P で導出不可, 反例 extraspecial)。issue 3017 pending (major upstream 待ちの正当な繰延, FT gate 無)。The literal book statement with Thompson's J(S) is proved only in a reduced case: normalJ_normal_of_odd (S06_Additional.lean, via Isaacs Thm 7.6 normal_J) gives…

### BG §14 — BG §14 Maximal Subgroups of Type P and Counting

> BG §14 (mmd L3787–4158) is essentially complete: all 13 numbered results are formalized sorry-free at full book strength, distributed over ~10 topic leaves in OddOrder/BG/Ch4_FamilyOfMaximal/S14_TypePCounting/ plus S16_PairIntersection/S16_Lemma1413/S16_MainResults (Thm 14.4(b) and parts of 14.7(d)/14.9 deliberately live in the S16 layer). The formalization is much more finely factored than the book — Prop 14.2 and Thm 14.7 are each split into 10+ named part-lemmas following the Coq BGsection14 decomposition, and includes faithfulness improvements over naive readings (a false conjunct of Prop 14.2 detected and removed; Cor 14.3's scaffold transposition repaired; τ₂-freeness in 14.13(a) stated prime-wise because the repo tau2 admits composite exponents). The only genuine gaps: Lemma 14.13(b) (a 5-line consequence of Thm 14.4, absent — Coq omits it too; effort S) and the M*-pinned form of two Cor 14.12 clauses (delivered in equivalent existential E-setup form, M*-identification proved internally but not surfaced; effort S). The 2 remaining S14 sorries (GlobalCounting.lean:808 sigmaLength_one_frobenius_type; TypePDuality.lean:1291 nonidentity_covered_by_sigma_pieces) block NO book result: both are explicitly frozen, consumer-free mis-encoded historical surfaces — the former a vacuous-hypothesis variant of Lemma 14.13(a) (faithful version proved as S16.non_disjoint_signalizer_frobenius), the latter a false-relative-to-BG variant of Cor 14.9 covering by M_σ^# instead of M̃ (faithful covering proved via the Mtilde/zTilde APIs and theoremE) — deleting or leaving them frozen is a housekeeping choice, not a formalization gap. SLIM VERIFICATION PASS (2026-07-16): no status changes — the Lem 14.13 'partial' label was confirmed after attempted refutation (14.13(b) absent from all of OddOrder/ under number-citation and descriptive-content greps, and Coq BGsection14.v:2409 formalizes only (a)), and the three most load-bearing results (Thm 14.4 sigmaLength_one_centralizer_structure, Thm 14.7 typeP_duality, Cor 14.9 sharpSubgroup_top_cover_reps_dichotomy/theoremE with an unconditional reps transversal) were confirmed at full book strength by reading their Lean statements.

| 結果 | 状態 | 規模 | 内容 | メモ |
|---|---|---|---|---|
| Lem 14.13 | 済 (2026-07-16 後始末) | none | (a) σ(N(x)) ∩ π(M) ≠ ∅ forces M ∈ 𝓜_F, τ₂(M) = ∅, M Frobenius with kernel M_σ; (b) N(y)-co… | (a) fully proved in S16_Lemma1413.lean (sorry-free), faithful to Coq non_disjoint_signalizer_Frobenius (BGsection14:2412): the second maximal is the signalizer neighbour N[x] = FT_signalizerBase x, τ₂-freeness stated pri… |

特殊化債務 (教科書より狭い形で証明済 — 一般化要否は着手時に判定):

- **Cor 14.12** type-P₂ signalizer neighbour: H ∈ 𝓜(N_G(R)) is type F, U ⊆ H_σ, M∩H =  — GlobalCounting.lean, sorry-free. Four of six clauses verbatim (H type F, U ≤ H_σ, M ⊓ H = U ⊔ K, N_H(U) ⊄ M; hypotheses were tightened to BG in 2026-06-22 faith…

### BG §15 — BG §15 The Subgroup M_F

> BG §15 is essentially completely formalized and sorry-free: all nine numbered results (15.1–15.9) have Lean counterparts in OddOrder/BG/Ch4_FamilyOfMaximal/S15_MF/ (8 topic leaves, ~11.5k lines, no sorries; the two remaining S14 sorries are frozen no-consumer mis-encodings, so nothing here depends on a sorry). Lemma 15.1, Corollaries 15.3/15.4/15.6, and Theorem 15.8 are at full book strength; Theorem 15.2, Corollary 15.5, Theorem 15.7, and Corollary 15.9 are proven in substance but their assembled Lean statements expose less than the book prints, with the missing conjuncts either living in helper/S16-bridge lemmas or deliberately dropped following the authoritative Coq formalization. Two deviations are documented book errors, not repo gaps: 15.7(c)'s printed equality M' = F(M) is an overstatement (weakened to M' ⊆ F(M), exactly as Coq's nonTI_Fitting_structure), and 15.9(c) is dropped (as in Coq, "not used later"; an earlier Lean draft of it was found unsound). Surprising factoring: the deep 15.7(e) trichotomy is fully proven but lives in §16 bridge form (isTypeI_of_isTypeF for type F, isTypeV_of_isTypeP1_mf_eq_msigma with the Singer/SL2 |W₁| ∣ p±1 divisibilities for type P1), and Theorem 15.8 required three "keystone" clauses of Coq's Ptype_structure/Fcore_structure absent from the repo's §14 API — all landed sorry-free. Several docstrings are stale (still saying "§14 gated/sorried" or "(8.8) not yet formalized" although everything cited is now clean).

特殊化債務 (教科書より狭い形で証明済 — 一般化要否は着手時に判定):

- **Thm 15.2** 1 ⊂ M_F ⊆ M_σ ⊆ M' ⊂ M; if M_F ≠ M_σ: M type P₁, q=|K*| ∈ π(M_F)∩β(M), — Main assembly mf_ne_msigma_typeP1_structure (S15_MF/Corollary155.lean) is sorry-free and carries the hard content: type P₁, p,q prime, q ∈ π(M_F)∩β(M), K* ≤ M_F…
- **Cor 15.5** F(M) decomposition: Y = O_{σ(M)'}(F(M)) cyclic τ₂-subgroup; M'' ⊆ F(M) — fitting_decomposition (S15_MF/Corollary155.lean, sorry-free) covers (a),(b),(d), M_F ≤ M', the internal direct product F(M)=F(M_σ)×Y (sup/trivial-inf/commuting …
- **Thm 15.7** F(M) not TI ⟹ (a) M ∈ ℳ_F ∪ ℳ_{P₁} and M_F = M_σ; (b) X = F(M)∩F(M)^g  — All substantive content proven sorry-free, but factored differently from the book. (a): complete (rank-theoretic core piSet_mf_inf_beta_disjoint + type-P₂ exclu… **[2026-07-18 精査 ⚠ 逐条照合] bundled statement は book 5 条項のうち実質 (a) のみ運搬** (issue 3022、初版の (e) 限定から scope 拡大)。`fitting_not_ti_cases` (S15_MF/OpicoreCentralizer.lean:361) を mmd L4249 と逐条対照した結果: **(a)** faithful / **(b)** 準恒真 — book は `X = F(M)∩F(M)^g` を固定して `X ⊆ H かつ X cyclic` を述べるが、Lean は `∃ X, X ≤ MF M ∧ X ≠ ⊥ ∧ IsCyclic X` で、これは `M_F ≠ 1` と同値 (証明 :406-412 は実際に任意の位数 q の元を渡すだけ) / **(c)** `=`→`≤` の弱化は**正当** (book の等式が type-F で overstatement、MathComp `nonTI_Fitting_structure` も包含; 直さない) / **(d)** 4 条項中 3 件欠落 — `E₃=1` は `E3_eq_bot_of_not_fittingIsTI` (:263) に在るが bundled 結論に入れておらず、`E₂⊴E`・`E/E₂≅E₁`・cyclic は未形式化 / **(e)** 恒真 — 末尾 conjunct が `A ∨ (¬A ∧ (a))` で (a) が第2枝を与えるため排中律に潰れる。根本原因は **`∃ X` decoupling** ((c)(e) が `X` に言及しないので束縛が効かず、book の `p = |X|` 結合も失われた)。定理自体は真で unsound ではないが (b)(e) を book content として数えてはいけない。⚠ **下流は block していない**: `isTypeI_of_isTypeF` / `isTypeV_of_isTypeP1_mf_eq_msigma` は vacuous な (e) を経由せず非 TI 分岐を自前で再導出して sorry-free・axiom-clean で完結 (2026-07-18 検証)。よって修正は「下流の重複再導出を (e) の cite に置換できる」strict improvement。 **[2026-07-19 ✅ issue 3022 完了]** 5 条項すべて book strength に到達。**(b)** `X = F(M)∩F(M)^g` を束縛し `∀ g ∉ M` 形へ / **(d)** `sigmaComplement_structure_of_not_fittingIsTI` + `quotientE2MulEquivE1` で 4 条項 / **(e)** 新 leaf `S16_MainResults/FittingNonTITrichotomy.lean` の **`fitting_not_ti_structure_e`** が Coq `nonTI_Fitting_structure` (:947-950) と同形 — 共通 conjunct (`p = |X|` 素数・`O_p(M_F)` 非可換・`O_{p'}(M_F)` cyclic) を括り出し (e2) ∨ (e3) を内側 disjunction に。⚠ **(e2) に型の制約は無い** (BG 原文でも Coq でも exponent 条件そのもので type `P₁` でも成り立ちうる; 「(e2)=type F」は誤読)。恒真性チェック済 — 第2枝は `¬可換 O_p(M_F)` であって `¬可換 M_F` ではないので (e1) の否定にならない。新規形式化は 3 件のみ (`|K*| = p` / 冪零群で `O_{π'}` 可換なら `K' ≤ O_π` / type `P₁` の (e2) ∨ (e3))、他は既存 axiom-clean 補題の assembly。⚠ 上の「(e) は §16 bridge form にしか無い」という記述も本 commit で解消 (§15(e) 自体が独立の定理として存在するようになった)。
- **Cor 15.9** (Sibley; Feit–Thompson) x ∈ M_σ# with C_G(x) ⊄ M, N ∈ 𝓜(C_G(x)) not ty — centralizer_escape_final_local (S16_MainResults/TaxonomyOutput.lean, sorry-free) proves parts (a)+(b) at full strength and additionally exposes ¬FittingIsTI M: …

### BG §16 — BG §16 The Main Results

> BG §16 is one of the most thoroughly formalized units in the repo: all of Theorems A–E, Proposition 16.1, and Theorems I–II have sorry-free Lean counterparts under OddOrder/BG/Ch4_FamilyOfMaximal/S16_MainResults/ (~7.7k lines across 6 leaves plus S16_PairIntersection and S16_Lemma1413), with the key monoliths (theoremA_maximal_structure_faithful, theoremC_paired_structure, proposition_type_classification, theoremII_tame_embedding, typeP_pair_inf_eq, non_disjoint_signalizer_frobenius) explicitly asserted axiom-clean in AxiomsCheck.lean; several docstrings still saying \"sorry-bearing/not axiom-clean\" are stale. The formalization is differently factored than the book: the Type I–V definitions are the Peterfalvi (8.1)–(8.8) data-structure variants shared with the Peterfalvi tree (OddOrder/GroupTheory/MaximalSubgroupType.lean), and Theorem I's full clause data (N_G(W₀)=W, S=W₁S′ factorizations) is delivered through the Peterfalvi-facing Section16MaximalPair/Section16TypePStructure bundles rather than the single dichotomy monolith. The genuine residual gaps are small sub-clauses: Theorem A's general \"M′/M_F nilpotent\" (only the hard type-P₁ case is a standalone lemma; type-P generally is carried by TypePData's U_nilpotent+derived_complement fields, leaving the type-F assembly), the reverse (⊇) half of Theorem C(9)'s identity A₀(M)−A(M)=𝒞_M(Ẑ), and — the largest — Theorem II's (Tii)(a)–(e) \"system of supporting subgroups\" packaging plus the tamely-imbedded-subset definition, whose raw ingredients are all proved but never assembled (Lemma 14.13(b)'s M-conjugation choice exists in neither Lean nor Coq); notably the Coq formalization also only formalizes \"the part of Theorem II relevant to the proof\" and drops (Tii)(a), so this gap mirrors math-comp. Two frozen mis-encoded surfaces with real sorries sit nearby in S14 (sigmaLength_one_frobenius_type, nonidentity_covered_by_sigma_pieces) but are documented no-consumer records and do not gate anything in §16. SLIM-PASS CORRECTIONS: Theorem E upgraded partial→formalized — its flagged-missing E(3) disjointness of 𝒞_G(Ẑ) from the M̃-cover is in fact proved sorry-free (S14.conjClassSet_T_Mtilde_disjoint + family_inf_msigma_union_eq, packaged as exceptional_disjoint_thickened in Peterfalvi.S10.nonTypeICovering_of_isTypeP); the partial/missing labels on Thm A, Thm C, Thm II, and Def tamely imbedded were re-attacked by number- and content-based greps (including the Coq analogs) and all confirmed, and load-bearing spot-reads of Thm B, Thm I, and Prop 16.1 confirmed full book strength.

| 結果 | 状態 | 規模 | 内容 | メモ |
|---|---|---|---|---|
| Thm A | 部分 | S | Structure of maximal M: M = KUM_σ, Hall/cyclic/centralizer conjuncts A(1)-(8) | ✅ **完了 2026-07-18** (book A(1)-(8) 全)。monolith `theoremA_maximal_structure_faithful` が A(1)-(8) を証明済 (AxiomsCheck)。M'/M_F nilpotent (T2) は type-P (`TypePData.U_nilpotent`)/P₁ (`isNilpotent_complement_of_isTypeP1_mf_ne_msigma`)/F (`isNilpotent_complement_of_isTypeF`) 全型で成立。⚠ 「∀ maximal M, M'/M_F nilpotent」統一 capstone + Cor 15.5(c) quotient wiring は **book 外の repo 内 packaging follow-on** (book gap でない、低優先)。 【⚠ **2026-07-19 hub 監査で訂正: 済 → 部分**】完了と記録されていたが実測では未達部分がある。The note's stated grounds are stale in every particular: it cites 'Sorry at Suzuki.lean:72' and 'Conclusion has free Prop fields (classification : Prop, classification_holds)'. That declaration no longer exists — the old scaffold was deleted. The conclusion is 【**2026-07-22 lane c: A(6)/A(7) essential gap を landing (issue 0126 close)**】monolith が book 逐条 16 conjuncts へ。A(6) の欠落 3 条項 `M_F ≠ ⊥`/`M' ⊊ M`/`M'/M_F nilpotent` (Cor 15.5(c) quotient 形、`derivedInG_quotient_maxNilpotentNormalHall_isNilpotent`) と A(7) の `F(M)=C_M(M_F)M_F`/`K≠1→F(M)⊆M'` を追加、axiom-clean 維持 (commit `6a0e6be17`+`112988882`)。A(3) `U ⊴ UK` は arbitrary-U monolith で unsound ゆえ意図的に除外 (docstring 明記)。⟹ **Thm A は essential content 完了**。 |
| Thm C | 済 | S | K ≠ 1 case: paired maximal M*, cyclic Z = K×K*, Ẑ TI, conjugacy classification C(1)-(11) | ✅ **完了 2026-07-18** (C(1)-C(11) 全)。`theoremC_paired_structure` = C(1)(2)(3)(4)(7)(8)(10)(11)+C(9)-TI半分、C(6)=typeP_pair_inf_eq。**C(9) 完全恒等式** = `a0_minus_a_eq_conj_zTilde` (逆包含追加)。**C(5)** = `theoremC_5_overgroup_unique` (TheoremC5.lean、prime-order X⊆K*→𝓜(C_G(X))={M} / Y⊆K→{M*}、Prop 14.2(c)+Thm 14.7 経由、axiom-clean、AxiomsCheck)。全 axiom-clean。 |
| Thm II | 済 | M | A(M), A₀(M) tamely imbedded: (Ti) fusion, D ⊆ A(M), /ℳ(C_G(x))/=1, (Tii) supporting subgro… | ✅ **完了・unconditional 2026-07-18**: `theoremII_tamelyImbedded` (TheoremIIPackaging.lean) が `A(M) は TamelyImbedded` を **仮説なし**で証明 (axiom-clean [propext/Classical.choice/Quot.sound]、AxiomsCheck)。(Ti)+full (Tii)(a)-(e)+(Tiii) 全て internal 証明済。**clause (d) を `clause_d_of_neighbour` で discharge** (旧 hypothesis `hd` 削除): (d)-Type-I = Thm B(5)+type-F 恒等式 `A₀(Mᵢ)=A(Mᵢ)` (`Kᵢ=⊥`)、(d)-**Type-II** = Thm B(5) + Thm C(9) (`theoremC_paired_structure` conjunct 10) + **order-determined cross-piece exclusion** (`aSet_mem_conj_iff_of_hatMsigma`=BG"distinct orders"、`κ∩τ₂=∅`) の合体。⚠ **issue 3019 Gap 2c の「Thm C(5) が必要」判断は over-cautious だった** — C(5) 不要、既存 B(5)+C(9)+順序分離で Type-II も閉じる。⚠ Coq は (Tii)(a) drop、本 repo は full scope で (a)-(e)+(Tiii) 達成。 |
| Def tamely imbedded | 済 | S | Tamely imbedded subset of G + system of supporting subgroups (Remark after Theorem II) | ✅ 済 2026-07-18: `TamelyImbedded` + `SystemOfSupportingSubgroups` + `FrobeniusTypeIWithNonTIFitting` (S16_MainResults/TamelyImbedded.lean、mmd:4575 に faithful、(Ti)-(Tiii)+(a)-(e))。既存 object のみ (新数学なし)。⚠ D = `escapingSharpSet` (x≠1 込み、book 忠実; `escapingCentralizerSet` は x≠1 欠で誤)。FT Def 9.1(ii)(e) との差を docstring 注記。 |

特殊化債務 (教科書より狭い形で証明済 — 一般化要否は着手時に判定):

- **Thm D** M_σ fusion control, cyclic M_σ ∩ M^g, R(x) normal complement + sharply — TaxonomyOutput.lean:933, sorry-free. D(3)'s 'C_M(x) Hall in C_G(x)' is encoded σ-agnostically as card/index coprimality (documented fix matching Coq 14.4(b)/(e)…
- **Def Types I–V** π*, Frobenius type, and the five maximal-subgroup types I–V (condition — Deliberately formalized as the Peterfalvi (8.1)–(8.8) data-structure variants (OddOrder/GroupTheory/MaximalSubgroupType.lean), shared verbatim between the BG an…
- **§16 notation** Z, Ẑ, M̂_σ, A(M), A₀(M), M̃ = ⋃xR(x) and the per-type A(M)/A₀(M) chara — All notation defined (S16_MainResults/Notation.lean; Peterfalvi-side in GroupTheory/MaximalSubgroupType.lean). Two documented deviations: A0Set subtracts the G-…

### BG App.C — BG App.C The Final Contradiction

> **⚠ 2026-07-18 判断 (lane c)**: App.C 本体 (Theorem C / Lem C.1-C.3 / final_contradiction) は既に
> sorry-free・axiom-clean で完了。残る Rem(II) (SL(2,2^q) の具体例) と Rem(V) (奇素数 wlog) は
> **illustrative であって番号付き定理でない**ため **低優先繰延** (⚠ 恒久対象外ではない —
> CLAUDE.md「文献引用のみ/例示も低優先繰延」)。判断理由: Rem(II) の形式化は SL(2,2^q) の群構成と
> Thm C 配置の instance 化が必要で、例示目的に対し労力が大きい。かつ lane b が
> `GroupTheory/SpecificGroups/{Suzuki,ProjectiveUnitary}/**` で類似の有限群構成を進行中のため、
> **その資産が揃ってから着手する方が安上がり** (重複構築の回避)。Rem(IV)/Prob 1 は従来どおり低優先繰延。

> BG Appendix C is essentially complete: Theorem C and Lemmas C.1-C.3 are all machine-checked, sorry-free, and axiom-clean, with the final bridge (final_contradiction) wiring Theorem C against Peterfalvi Section 16's field-normalizer data to kill the minimal counterexample. Lemmas C.1 and C.2 are at full (or slightly better than) book strength on the concrete field GaloisField p q — notably the q>=5 branch of C.2 is a complete class-sum/character-theoretic proof (Frobenius character classification, structure constants, column orthogonality, Cauchy-Schwarz error bound), and the Lean C.2 never even uses condition (A). The factoring is inverted relative to the book and is the one surprise: the finite-field and character halves live in OddOrder/BG/AppC_*.lean, while the generators-and-relations group argument of Lemma C.3 (Steps 1-4, Remarks (X)/(XI) via Isaacs Thm 4.34 coprime decomposition) is proved on the Peterfalvi side (S16_AppendixC3/S16_CoreSetup/S16_CoreBounds/S16_NonExistenceGCore) on the transported configuration, with FieldNormalizerData genuinely constructed (fieldNormalizerData_of_repr), not posited. The only specialization: Theorem C, hypothesis (B), and Lemma C.3 are stated for the S16 configuration (Q elementary abelian q-group, p,q odd, inside the odd minimal counterexample) rather than the book's abstract standalone form with an arbitrary finite abelian p'-group Q — full strength for FT, but the abstract Theorem C (e.g. the SL(2,2^q) instance) is not expressible. Genuine gaps are marginal: the illustrative SL(2,2^q) example (Remark II), the trivial odd-primes wlog (Remark V), and the out-of-scope literature items (Remark IV / Problem 1). Slim verification pass: all four flagged missing labels (Rem II, IV, V, Prob 1) were confirmed by content-based grep (the repo's only SL(2) material is the unrelated Isaacs Ch07 Lem 7.4), and the load-bearing statements theoremC/final_contradiction, NormSet.lemmaC1 (AppC_NormSet.lean:1459), and NormSet.lemmaC2 (actually housed in AppC_LemmaC2.lean:25, with (A) literally an unused argument) were read and confirmed at the claimed strength — no status changes, only note refinements (final_contradiction's extra S16 standing hypotheses hnoV/hncH0C, and the AppC_LemmaC2.lean location).

| 結果 | 状態 | 規模 | 内容 | メモ |
|---|---|---|---|---|
| Rem (II) | 未 | M | Peterfalvi's example: hypothesis of Theorem C is satisfied by p=2, G=SL(2,2^q) with explic… | CONFIRMED missing after content-based refutation attempt: the only SL(2)/SpecialLinearGroup material in the repo is Isaacs Ch07 S7A1_JpGL2p.lean (Lem 7.4 unique involution in SL(2,F), over ZMod p) — no SL(2,2^q), no Galo… |
| Rem (IV) **※低優先繰延** | 未 | XL | Norton-Glauberman extension: Theorem C conclusion can be strengthened to p <= 3 | CONFIRMED missing (grep for Norton / p <= 3 strengthening: zero hits repo-wide). Pure literature pointer to external work [12]; the book gives no proof and it is not needed for Theorem C or FT. Not a formalization target… |
| Rem (V) | 未 | S | By (A) one may assume p and q are odd | CONFIRMED missing: no p=2/q=2 wlog reduction lemma in any AppC file or elsewhere. The trivial reduction (p=2 gives p<=q outright; q=2 with (A) forces p=2) is not formalized because it is unnecessary in the repo factoring… |
| Prob 1 **※低優先繰延** | 未 | XL | Can p = 3 in Theorem C? | CONFIRMED missing (no hits repo-wide). Open research problem stated at the end of App.C (see Remark IV); no proof exists in the literature, so it is not a formalization target. Listed only for completeness; effort rating… |

特殊化債務 (教科書より狭い形で証明済 — 一般化要否は着手時に判定):

- **Thm C** Main theorem: primes p,q with (A) and embedding hypothesis (B) into G  — VERIFIED by reading AppC_FinalContradiction.lean:154/164. Sorry-free and axiom-clean (AxiomsCheck.lean:3820 theoremC, :8294 final_contradiction). Proved by asse…
- **Hyp (B)** Hypothesis (B): monomorphism sigma: H=PU -> G, abelian p'-group Q, y i — FieldNormalizerData (S16_CoreLemmas.lean:634) carries sigma injective, sigma(P)=P / sigma(P0)=W2 / sigma(U)=U, W2 normalizes Q, y in Q, W2^y normalizes U — i.e.…
- **Lem C.3** E = E^{-1} (generators-and-relations argument under hypothesis (B)) — The entire hard argument is genuinely proved, sorry-free: Steps 1-3 (decomposition x = u s1 v, the s1 u s2 in U dichotomy, (PU) cap (PU)^{t1} = U) exist both as…

### BG App.D — BG App.D CN-Groups of Odd Order

> **⚠ 2026-07-19 訂正 (lane c, issue 9133)**: 下記散文の「the one genuinely missing mathematical
> input is Gorenstein §14.1 solvable-CN structure theory (Cor 14.1.6, 3-step groups), **absent
> from the entire repo**」は**もう正しくない**。`OddOrder/GroupTheory/CNGroupStructure.lean` に
> 3-step 群の定義 + Cor 1.6 (Thm 1.5 から sorry-free で導出) が在る。残 sorry は Thm 1.5 本体のみ
> (step 1/2/3 は証明済)。また同 issue の実測で、当初「不在」とされた前提のうち **3 件が誤り**
> だった (Gorenstein Thm 5.3.5 は repo に在り証明も使わない / `C_G(F)≤F` は配置問題 /
> Thm 10.3.1(v) は Isaacs Thm 6.9 の対偶として repo に在る)。最新は issue 9133 / 3020 が正本。

> **⚠ 2026-07-18 更新 (lane c)**: opaque-Prop scaffold を de-opacify し、**偽だった 2 文を修正**。
> 旧 `MinimalSimpleCNHypothesis` は minimal-simplicity/nonsolvability を自由 `Prop`+`_holds` で持ち
> 任意の奇数位数 CN 群で充足可能だったため、D.1/D.2 は「全奇数位数 CN 群」の主張になり **両方偽**
> (D.2 は `G=C₃` で機械検査反証、D.1 は `F_{3⁶}⋊(C₇⋊C₃)` で反証)。仮説を `IsCNGroup` +
> `IsMinimalSimpleOdd` に置換して honest 化、vacuous な `cnTheorem_reduction`/`CNTheoremReductionData`
> は削除 (sorry 3→2、opaque 0、build green、下流 consumer 無)。**残ブロッカーは Gorenstein §14.1
> (3-step 群 + Cor 14.1.6) のみ** = issue 3020。⚠ FT 完成による vacuous discharge は禁止 (依存順序が逆)。

> BG Appendix D (pp. 153-156, mmd lines 5132-5199) is a 4-page reading guide for the Feit-Hall-Thompson CN-theorem containing exactly two numbered results, Lemmas D.1 (Sylow-TI in a minimal simple CN-group of odd order) and D.2 (P ≤ N_G(P)′); the tags (D.1)-(D.3) in the text are display equations inside D.1's proof, not results. The repo file OddOrder/BG/AppD_CNGroups.lean is a statement-level scaffold carrying all 3 of the unit's sorries with zero proof content, and its standing hypothesis MinimalSimpleCNHypothesis encodes minimal-simplicity/nonsolvability as opaque free Prop fields, so Lemma D.1 as literally stated asserts Sylow-TI for every odd-order CN group (false: an odd solvable CN group such as F_{3^6} ⋊ (C7 ⋊ C3) violates it) and none of the sorries is honestly dischargeable without first de-opacifying the hypothesis; the third sorry (cnTheorem_reduction) is even trivially dischargeable with True-instantiations, i.e. pure packaging. Supporting infrastructure is good: the ZJ input is sorry-free via the book-sanctioned L(P)/Thm B.4 substitute (AppB_Thm62.zCenter_lOdd_sup_oPiCore_normal) and the Focal Subgroup Theorem (BG Thm 1.17) is in mathlib (Subgroup.commutator_inf_eq_focalSubgroup); the one genuinely missing mathematical input is Gorenstein §14.1 solvable-CN structure theory (Cor 14.1.6, 3-step groups), absent from the entire repo and the dominant cost, making D.1 an L and D.2 an S conditional on it. Two structural caveats: math-comp/odd-order has no Coq analog of App.D, and since feitThompson is complete the honest minimal-simple-CN hypothesis is vacuous, so everything could be discharged vacuously — logically sound but defeating the appendix's expository purpose. Slim verification pass: all three flagged partial labels CONFIRMED after refutation attempts (no alternative formalization of CN-groups, Sylow-TI, focal-control, or 3-step exclusion found under any book tree or OddOrder/GroupTheory/**; mathlib and AppB citations verified by reading the sources); one strengthening — D.2 as stated is definitively false (G = C3 counterexample under True-instantiation), not merely 'likely false'.

| 結果 | 状態 | 規模 | 内容 | メモ |
|---|---|---|---|---|
| Lem D.1 | 済 | L | Sylow p-subgroups of a minimal simple CN-group of odd order form a TI family (P ∩ Q ≠ 1 ⇒ … | VERIFIED partial. SORRY 1 of 3 (OddOrder/BG/AppD_CNGroups.lean:48). Statement-only skeleton, zero proof content; refutation search found no alternative formalization anywhere (no other CN-group definition, no Sylow-TI-fa… 【**2026-07-22 lane c 実測訂正: 済**】issues 3020/9133 close。AppD_CNGroups/ (Basic/MaximalSylowIntersection/SylowTI) に実証明で landing、AxiomsCheck「BG Appendix D — COMPLETE」節参照。旧 opaque scaffold は削除済、BG 実 sorry 0。 |
| Lem D.2 | 済 | S | P ≤ N_G(P)′ for every nontrivial Sylow subgroup P of a minimal simple CN-group of odd orde… | VERIFIED partial. SORRY 2 of 3 (OddOrder/BG/AppD_CNGroups.lean:57). Statement matches the book (P ≤ derivedInG (normalizer P)) but inherits the opaque MinimalSimpleCNHypothesis defect — and as literally stated it is DEFI… 【**2026-07-22 lane c 実測訂正: 済**】同上 (issues 3020/9133 close、AxiomsCheck「BG Appendix D — COMPLETE」)。 |
| App.D (guide prose) | 済 | M | Reduction package: D.1/D.2 replace Gorenstein Ch.7/8/10 passages; consequences = no 3-step… | VERIFIED partial. SORRY 3 of 3 (OddOrder/BG/AppD_CNGroups.lean:75). Not a numbered book result — packages App.D's unnumbered closing prose (confirmed in mmd lines 5193-5196: (a) no 3-step subgroup ⇒ G Thm 14.2.2; (b) D.1… 【**2026-07-22 lane c 実測訂正: 済**】同上 + 3-step dichotomy = `GroupTheory.CNGroupStructure` COMPLETE (issue 9133)。 |

### BG App.E — BG Appendix E: Further Results of Feit and Thompson

> **⚠ 2026-07-18 追記 (lane c) — Hall collecting process 着手 (issue 9132)**: 新 leaf
> `GroupTheory/HallCollection.lean` (sorry 0、全 axiom-clean) に**収集過程の枠組み**を landing —
> 1 ステップ漸化式 `pow_succ_collect` (公理 propext のみ)、深さ管理 (`[γᵢ,γⱼ] ≤ γ_{i+j}`)、
> 尾部の崩壊/吸収補題、および `exists_hallCollection_of_residue` (固定 `n` の E.1 は `γ_n` を法とする
> 合同そのもの = 隠れた exactness を仮定していない保証)。さらに `AppE.hallCollection_of_class_le_three`
> で **`γ₃=1` なら全 `n` について E.1** を証明 (既存 class≤2 を真に包含、Pascal 分割
> `C(n+1,3)=C(n,2)+C(n,3)` が weight 3 の二項係数を出す鍵)。⚠ 一般形 `hallCollection` の statement は
> **逐語不変** (弱化なし)、sorry 22→22。残る唯一の障害 = 自由冪零群の basic-commutator 基底 +
> 収集係数の多項式性 (Hall 多項式/Lazard) — mathlib に不在。詳細 = issue 9132。
>
> **⚠ 2026-07-18 更新 (lane c)**: 163 行の opaque-Prop scaffold を **482 行の book-strength 記述に
> de-opacify** (opaque フィールド 0 / `_holds` 0 をコメント除去後に確認)。**sorry-free で証明**:
> `hallCollection_of_class_le_two` (E.1 class≤2)、`pow_mul_of_class_le_two` (E.2(b) class≤2)、
> **`RegularOperatorSetup.card_A_dvd_half_p_sub_one` = E.3(a) 完成** (`q ∣ (p−1)/2`)、collectionTail 補題群
> (全 axiom-clean、E.1class≤2 と E.3(a) は AxiomsCheck 登録)。残 9 sorry は全て honest statement の下
> (「opaque で sorry-free」より「honest + sorry」を優先)。**依存グラフは 1 根に収束: 一般 class≤p−1 の
> Hall collecting process が付録 E 全体の唯一の unlock** (他の §4/§5/§14/§15/§16 前提は全て repo に在り)
> = issue 3021。

> Appendix E (5 numbered results: Thm E.1 Hall collection, Prop E.2 regular p-group consequences, Thm E.3/Prop E.4 Feit–Thompson 1991 regular-operator bounds, Cor E.5 Type I/II application) is essentially unformalized: AppE_FurtherResults.lean is a 163-line opaque-Prop scaffold whose 5 sorried theorems map 1:1 to E.1–E.5 but state nothing at book strength — every hypothesis and conclusion is an arbitrary `Prop` field with a self-carried `_holds` proof, so each `∃ data, ...` statement is trivially satisfiable. Substantial special cases of E.1/E.2 were already proved sorry-free for BG Prop 4.3: class-≤2 and class-≤3 collection formulas with pinned binomial exponents and the corresponding Ω₁-exponent-p / p-power-multiplicativity theorems (S04_SmallRankBasic.lean, CriticalSubgroup.lean, Isaacs Ch04 CommutatorBasics.lean) — so E.1/E.2 are partial, missing only the general-class (≤ p−1) versions, which require formalizing Hall's collecting process in full (L, the hardest single gap). E.3–E.5 have zero genuine content anywhere in the repo and no Coq analog (math-comp/odd-order formalizes only appendices A–C); E.3 is the big-ticket item (XL), E.4 is M given E.3's internals, and E.5 is mostly wiring (L) since its §12/§14/§15/§16 prerequisites are formalized. Closing this unit honestly requires rewriting all five statements, not just filling the sorries. Slim verification pass: all five flagged labels (partial/partial/missing/missing/missing) were confirmed against refutation greps (book-number citations, descriptive names, mathlib Hall–Petrescu search, Coq appendix listing) and the E.1/E.2 special-case and E.5 prerequisite claims were spot-read in the Lean sources; the only corrections are lean_name namespace fixes (OddOrder.BG.Ch1.S04 not OddOrder.BG.S04; Omega.exponent_eq_of_class_le_two lives in CriticalSubgroup.lean) plus adding the class-≤2 Hall–Petrescu special case in OddOrder/GroupTheory/CriticalSubgroup.lean to E.1.

| 結果 | 状態 | 規模 | 内容 | メモ |
|---|---|---|---|---|
| Thm E.1 | 済 | L | Philip Hall's commutator collection formula: x^n y^n = (xy)^n c_2^{C(n,2)} ... c_n^{C(n,n)… | [critic 補正] AppE_FurtherResults.lean の 5 結果は全て同型の opaque-Prop scaffold — missing に統一 (E.1 Philip Hall lower central)。 【**2026-07-22 lane c 実測訂正: 済**】issues 9132/3021 close。`AppE.hallCollection` = **E.1 全文** (AppE_CollectionFormula.lean)、AxiomsCheck 済 (2026-07-20)。 |
| Prop E.2 | 済 | M | p-group of nilpotence class ≤ p−1: (a) Ω₁(R) has exponent 1 or p; (b) if R′ ⊆ Ω₁(R) then x… | [critic 補正] 同上 (E.2 φ(x)=x^p 準同型; BG Prop 4.3 が一部内容を実証明済という部分クレジットは notes のみに残す)。 【**2026-07-22 lane c 実測訂正: 済**】`AppE.omega_pow_eq_one_of_lowerCentralSeries_eq_bot` (E.2(a)) + `AppE.pow_mul_of_commutator_le_omega` (E.2(b))、AxiomsCheck 済 (2026-07-20)。 |
| Thm E.3 | 済 | XL | Feit–Thompson 1991 regular-operator bound: /A/ = q acting regularly on p-group R fixing R₀… | Sorry #3 (AppE_FurtherResults.lean:103). VERIFIED missing: the only Lean counterpart is a fully vacuous scaffold — `OperatorHypothesis` carries 10 opaque Prop fields (regular action, /A/ = q, /R₀/ = p, C_R(R₀) split, ...… 【**2026-07-22 lane c 実測訂正: 済**】issue 3021 キャンペーン完遂: Step 1–4 全て landing、「E.3(b) Step 3, (c), and (d) — Steps 3 and 4 complete」節 (AxiomsCheck)。 |
| Prop E.4 | 済 | M | Under Thm E.3 hypotheses with /Ω₁(R)/ ≥ p⁴, B regular on R and B not fixing R₀: C_S(Z₂(S))… | Sorry #4 (AppE_FurtherResults.lean:124). VERIFIED missing, same situation as E.3: vacuous opaque-Prop scaffold only (`CentralizerUpperCenterData` = 5 opaque Prop fields), no genuine content found by refutation greps, no … 【**2026-07-22 lane c 実測訂正: 済**】`AppE_BetaSupply`+`AppE_PropE4` (「Proposition E.4, corrected」節、AxiomsCheck 済)。 |
| Cor E.5 | 済 | L | In the Cor 15.9 situation (x ∈ M_σ of prime order p, C_G(x) ⊄ M, N ∈ M(C_G(x)), N ∉ M_F), … | Sorry #5 (AppE_FurtherResults.lean:161). VERIFIED missing: vacuous opaque-Prop scaffold only (single opaque conclusion field `all_maximal_typeI_or_typeII`); no genuine counterpart; no Coq analog. Unlike E.3/E.4, its book… 【**2026-07-22 lane c 実測訂正: 済**】issue 3028 close (2026-07-21)。`AppE_E5Counting` (「Corollary E.5 … the closing」節、AxiomsCheck 済)。 |

## Peterfalvi

> 🎯 **2026-07-27: Peterfalvi 特殊化債務キャンペーンの棚卸し完了**。全 bullet を実測し直した結果、
> **残るのは (6.4)/(6.6) の coherence 半分 1 件だけ**になった (規模 L で繰延、下記 §8 の注記参照)。
> 内訳:
> * **本キャンペーンで実際に一般化したもの** — (6.2)/(6.3) を一般 solvable kernel + oracle 無し +
>   Dade 非依存へ (issues 0153/0154)、(6.6) X-characterization を一般 kernel へ (しかも `K` の
>   正規性が不要と判明し書籍より強い)、(8.18.c) を型 I-or-II ペアへ、加重 (5.6) engine の両 break を
>   抽象 τ へ、(11.4) から不要な型仮説を除去。
> * **実測で stale と判明したもの (作業不要)** — (2.6)-(2.10.3) の `hconj` threading (binder ゼロ)、
>   (7.8) (書籍ページ画像と 1 対 1 照合済)、(8.15) (claim 1 の `A₁` 節と claim 3 が既に型 uniform、
>   `A₀`/`A` が型ごとなのは台集合の定義がそうだから)、(8.18)(a)(b) (元から型 generic)、
>   (9.7)(b) (`caseB_exists_galoisField_repr_withAut` が書籍の三層同型そのもの)、
>   (9.11) (`S11.nineEleven_coherent_A0` が Hypothesis (9.2) 上の generic 宣言)、
>   (10.11)、(11.8)、(13.8) (書籍逐語の S 側が在る)。
> * **副産物: 見せかけの opaque scaffold を 2 件除去** — `CliffordCaseBData.field_model`
>   (唯一の producer が `True`、消費者ゼロ) と `OddOrderSpecialization` (消費者ゼロ、書籍の
>   「K/M nilpotent」を自由 Prop に潰していた)。
>
> ⚠ 教訓: 本 note の未/部分/特殊化ラベルは **2026-07-16 の一度きりのスナップショット**で、
> その後の実装で大半が陳腐化している。**着手前に必ず実測する** (CLAUDE.md「スコープ」節)。


| unit | n | 済 | 特殊化 | 部分 | 未 | mathlib |
|---|---|---|---|---|---|---|
| Pf §1-2 | 4 | 4 | 0 | 0 | 0 | 0 |
| Pf §3 | 10 | 10 | 0 | 0 | 0 | 0 |
| Pf §4 | 14 | 7 | 7 | 0 | 0 | 0 |
| Pf §5 | 9 | 9 | 0 | 0 | 0 | 0 |
| Pf §6 | 10 | 10 | 0 | 0 | 0 | 0 |
| Pf §7 | 9 | 9 | 0 | 0 | 0 | 0 |
| Pf §8 | 8 | 3 | 5 | 0 | 0 | 0 |
| Pf §9 | 11 | 7 | 4 | 0 | 0 | 0 |
| Pf §10 | 18 | 16 | 2 | 0 | 0 | 0 |
| Pf §11 | 11 | 8 | 3 | 0 | 0 | 0 |
| Pf §12 | 11 | 10 | 1 | 0 | 0 | 0 |
| Pf §13 | 9 | 8 | 1 | 0 | 0 | 0 |
| Pf §14 | 17 | 17 | 0 | 0 | 0 | 0 |
| Pf §15 | 19 | 18 | 1 | 0 | 0 | 0 |
| Pf §16 | 16 | 16 | 0 | 0 | 0 | 0 |
| Pf App: Suzuki | 32 | 0 | 0 | 0 | 32 | 0 |
| Pf App: Huppert | 3 | 1 | 0 | 2 | 0 | 0 | ⚠ 2026-07-22 lane c 訂正: **3/3 済** — Lemma/Prop 1 は 07-19 hub 監査で済、Prop 2(b) の C_U(s)↪Aut(F) 節も `exists_injective_semilinear_companion` (SemilinearField.lean:520, commit 60314f15e, AxiomsCheck 登録済) で解消済み |
| Pf App: NearFields | 6 | 2 | 0 | 2 | 2 | 0 |
| Pf App: Suzuki2Groups | 8 | 2 | 0 | 2 | 4 | 0 |
| Pf App: FeitSibley | 13 | 0 | 0 | 1 | 12 | 0 |

### Pf §3 — Peterfalvi §3 Preliminary Results from Character Theory, results

> Peterfalvi §3 (results (1.1)–(1.10)) is essentially complete: 8 of 10 results are fully formalized sorry-free, and the two remaining ((1.5), (1.7)) are partial only in specific sub-parts. The formalization is heavily factored away from the S03 files themselves into OddOrder/GroupTheory/RepresentationTheory/** (a bespoke ClassFunction character theory — BrauerPermutation, SupportedSpanOrthogonality, IsometryDifferencePair, InducedIrreducible, CliffordCorrespondence/Decomposition, InducedInvariantConstituent, CyclotomicGaloisAction, CyclotomicCharacterCongruence), and is frequently STRONGER than the book: (1.1) comes with a full unconditional Brauer permutation lemma, (1.9.b) produces one automorphism uniform over all virtual characters, (1.10.a) needs only x^p=1, (1.7.c) needs only a weaker coprimality, and (1.8) — hiding under an S08 '(6.2) θ-bound' label — weakens the kernel hypothesis. The genuine gaps: (1.5.d)'s verbatim formula and the general-normal-subgroup packaging of (1.5.e) are unstated (though compositions of proved general lemmas, effort S), and for (1.7) only the numeric count n = |T:H|/e² and the general (1.7.a) distinctness packaging lack named statements (effort S). Nothing in this unit is covered by mathlib, which lacks induced-character/Clifford theory at this level. One stale docstring in S03_PreliminaryCharacter.lean claims the (1.6.a) converse 'is not formalised here' while a later section of the same file proves it. SLIM-PASS CORRECTIONS: refuted the earlier draft's claim that general (1.7.b) is missing — it IS formalized coprimality-free in InducedInvariantConstituent.lean (e-scaled decomposition into irreducibles, equal degrees |G:T|·e·θ(1), uniform multiplicity e, via the book's own regular-character argument), so (1.7) stays partial only for the e² count and general (a)-distinctness (effort M→S); the (1.5) partial label was confirmed (general pieces of (e) all exist, named composition only Frobenius-specialized); and (1.2)/(1.3)/(1.4) were verified at full book strength by reading the Lean statements. **2026-07-19 CORRECTION (lane c): the above gap list is now stale — Pf §3 は 10/10 完了。** (1.5.d) の逐語形と (1.7.b) の e² count は commit 594ecf652 (2026-07-18) で、(1.7.a) の distinctness (= Clifford 対応の単射性、Clifford.lean:30 に TODO と明記されていた唯一の真のギャップ) は 2026-07-19 に形式化された。詳細は下の表の各行の更新注記。

| 結果 | 状態 | 規模 | 内容 | メモ |
|---|---|---|---|---|
| (1.5) | 済 | S | Induction from a normal subgroup (Clifford suite): (a) Res Ind θ = r·Σθ^g; (b) ‖χ‖² = r, i… | (a)-(c) formalized at full strength in InducedIrreducible.lean: (a) as /H/•Res(Ind θ) = /I_G(θ)/•Σ_{distinct conjugates} (equivalent, and stated for any class function); (b) as /H/·‖Ind θ‖² = /I_G(θ)/ plus isIrreducibleC… 【2026-07-18 lane c 更新: (1.5.e) の一般形 `inner_induce_conj_eq_zero_of_odd` (H ⊴ L 一般、Frobenius 仮定不要) を CliffordDecomposition.lean に追加。残る gap は (1.5.d) の逐語式のみ。】【2026-07-19 lane c 更新: **この「残る gap」も既に解消済** — (1.5.d) の逐語形 `inner_self_inv_smul_apply_one_smul_restrict_induce` (InducedIrreducible.lean:296、`χ(1)·Res_H^G χ / ‖χ‖² = |G:H|·∑_β θ^β(1)·θ^β`) が commit 594ecf652 (2026-07-18 17:41) で追加されていた。(a)-(e) 全て本文強度・sorry 無し・axiom-clean。部分→済。】 |
| (1.7) | 済 | S | Clifford correspondence: (a) Ind_T^G ψᵢ distinct irreducible, Ind_H^G θ = Σeᵢχᵢ; (b) T/H a… | CORRECTED (slim pass): the earlier claim that general (b) is missing and 'all repo versions route through the canonical extension theorem' is WRONG. (b) at full book strength (T/H abelian only, e possibly > 1, NO coprima…【2026-07-19 lane c 更新: 残っていた 2 gap を両方閉じて **済**。(i) e² count は `index_eq_card_induce_constituents_mul_sq_of_invariant` / `card_induce_constituents_eq_index_div_sq_of_invariant` (InducedInvariantConstituent.lean:137/225) で commit 594ecf652 (2026-07-18) に解消済 (survey 未反映だった)。(ii) 一般 (1.7.a) の distinctness = Clifford 対応の単射性 (Clifford.lean:30 に TODO として明記されていた真のギャップ) を新規形式化: `eq_of_induce_eq_induce_of_liesOver_of_inertia_eq` (CliffordCorrespondence.lean:550) — ψ,ψ' ∈ Irr(T) が共に θ の上にあり Ind_T^G ψ = Ind_T^G ψ' なら ψ = ψ'。θ-part counting を 2 項に拡張した `sum_restrictionMultiplicity_mul_le_restrictionMultiplicity` (:294) が要。本文逐語の包装 `induce_eq_sum_smul_induce_of_inertia_eq` (:694) = 「χᵢ は相異なる既約 + Ind_H^G θ = ∑ eᵢχᵢ」。全て sorry 無し・axiom-clean。】 |

### Pf §4 — Peterfalvi §4 The Dade Isometry, results

> Peterfalvi §4 (The Dade Isometry, results (2.1)-(2.11)) is essentially complete and entirely sorry-free: every numbered result including the three sub-lemmas (2.10.1)-(2.10.3) has a genuine Lean counterpart, spread over OddOrder/GroupTheory/CoprimeConjugacy.lean (for (2.1)) and the ~4,500-line S04 tree, and the main theorem (2.6) is an honest construction (Hypothesis.fullDadeIsometryData) including the full (2.10) Möbius/inclusion-exclusion proof of virtual-character preservation. The one real gap is (2.4.a): H(a^x) = H(a)^x is never derived from Hypothesis (2.2) in general — it is carried as an extra predicate HConjInvariant threaded as a hypothesis through (2.6)-(2.10) and discharged per-instance downstream (trivial-H TI cases and constructed §12/§14/§15 data); since the repo already has both needed halves (mem_H_of_mem_centralizer_coprime, card_centralizer_eq), closing it is well under a day and would mechanically de-specialize all the hconj-carrying statements. (2.4.c) (N_G(aH(a)) = C_G(a)) is absent as a normalizer equation, but its functional single-coset content and its only book use — the orbit count in (2.7)/(2.10.3) — are proved (card_conj_fiber / card_conjFiber_coset_eq_card_centralizer / sum_card_centralizerIn_eq), so nothing depends on it. Notable differences from the book: (2.1) is factored as a general coprime-conjugacy primitive with the pi-part argument replaced by a single CRT exponent, the repo adds uniqueness statements beyond the book ((2.5) IsDadeMap.unique; every (2.2) datum over a TI pair has H = ⊥), and the Dade map is constructed over any commutative coefficient ring rather than just ℂ. Slim verification pass: the (2.4) partial label was confirmed against source after refutation attempts (the S04 docstring itself records that the general (2.4.a) derivation awaits π-core API; no (2.4.c) normalizer statement exists, though its coset-form content card_conjFiber_coset_eq_card_centralizer was found and added to the notes), the three most load-bearing formalized entries ((2.2), (2.5), (2.11)) were read at source and confirmed full book strength, and all spot-checked Lean names exist — no status changes were needed.

| 結果 | 状態 | 規模 | 内容 | メモ |
|---|---|---|---|---|
| (2.4) | 済 | S | (a) H(a^x)=H(a)^x for x∈L; (b) conjugate cosets aH(a), bH(b) force a ~_L b; (c) N_G(aH(a))… | Label verified against source (refutation attempted, failed). (b) fully proved (isConj_in_L_of_mul_H, via the CRT cross-rigidity isConj_of_isConj_mul). (a) is NOT derived from (2.2) in general: the S04_DadeIsometryBasic.… 【2026-07-18 lane c 更新: (2.4.a) を Hypothesis (2.2) から導出 (`Hypothesis.hConjInvariant` / `mem_H_iff_coprime_orderOf`, S04_DadeIsometryBasic.lean)。O_{π'} API 不要の局所版で解決。`HConjInvariant` 引数は全て `hyp.hConjInvariant` で discharge 可能 (signature 無変更)。残る gap は (2.4.c) の normalizer 等式のみ。】【**2026-07-19 hub 監査で訂正: 部分 → 済**】【2026-07-19 lane a 更新: **その (2.4.c) も既に形式化済** — `conj_coset_H_eq_iff_mem_centralizer` (S04_DadeIsometryBasic.lean:884) が `g` が coset `H(a)·a` を共役で安定化する ⟺ `g ∈ C_G(a)` を elementwise 形で証明 (= `N_G(aH(a)) = C_G(a)`)。sorry 無し。(a)(b)(c) 全て本文強度ゆえ 部分→済。⟹ **Pf §4 は 14/14 完了**。残るのは `hconj` 明示引数の signature cleanup のみで、これは強度の債務でない (数学的には全て `hyp.hConjInvariant` で discharge 可能)。】 |

特殊化債務 (教科書より狭い形で証明済 — 一般化要否は着手時に判定):

- **(2.6)** Dade isometry: (a) (α^τ,β^τ)_G = (α,β)_L; (b) τ maps ℤ[Irr L, A] into  — Both parts honestly PROVED (sorry-free), bundled as fullDadeIsometryData — (a) via the (2.7) adjoint formula, (b) via the full (2.10) inclusion-exclusion. Sole … **[2026-07-18 精査]** §4 の 7 件が共通で担いでいた narrowing = threaded `hconj : hyp.HConjInvariant`。これは `S04_DadeIsometryBasic.hConjInvariant (hyp : Hypothesis G A L) : hyp.HConjInvariant` (:833) として**無条件定理**になったので、数学的には全 7 件が book strength で dischargeable。~~**残るのは signature cleanup のみ** (S04_InduceConjFinset.lean 等で `hconj` が明示引数のまま ~20 箇所)。~~ **[2026-07-27 実測: この注記は stale — 債務ゼロ]** `HConjInvariant` の全 27 出現を数えたところ **binder は 1 つも残っていない** (25 件は docstring/コメント、実コードは定義 2 件と `S14_MaximalI/FrobeniusStructureBasic.lean:178` の局所証明のみで、後者も既に無条件定理 `Hypothesis.hConjInvariant` 経由)。挙げられていた `S04_InduceConjFinset.lean` には `HConjInvariant` が 1 度も現れない。⟹ §4 の特殊化債務は解消済。
- **(2.7)** Adjoint formula: (α^τ, χ)_G = (α, ψ)_L where ψ(a) = |H(a)|⁻¹ Σ_{x∈H(a) — Main equality fully proved for any IsDadeMap, with ψ an explicit input matching the averaging on A (so the book's 'in particular ψ = Res χ' case is a direct ins…
- **(2.8)** M(B) = H(B) ⋊ N_L(B) for nonempty B ⊆ A — Complete semidirect package: H(B)=⋂H(a) (hIntersection), N_L(B) built by hand as the L-set-stabilizer, disjointness H(B) ∩ N_L(B) = 1 (no hconj needed, matching…
- **(2.9)** α_B = α ∘ f_B ∈ CF(M(B)), f_B : M(B) → L with kernel H(B); preserves v — f_B realized as M(B) → M(B)/H(B) ≅ N_L(B) ↪ L via the (2.8) IsComplement'; ker f_B = H(B) proved; the defining equation α_B(hx) = α(x) is alphaB_apply_mul; virt…
- **(2.10)** Inclusion-exclusion identity α^τ = -Σ_{B∈ℬ} (-1)^{|B|} Ind_{M(B)}^G α_ — The full pointwise identity is proved (dadeMap_eq_neg_sum_mobiusTermCF): support side via a 5-step pipeline (orbit-averaging the transversal to the powerset, pe…
- **(2.10.1)** L-conjugation invariance: Ind_{M(B^x)}^G α_{B^x} = Ind_{M(B)}^G α_B — Fully proved including the constituent conjugations H(B^x)=H(B)^x, N_L(B^x)=N_L(B)^x, M(B^x)=M(B)^x and the α_B transport. Takes hconj (the book proof cites (2.…
- **(2.10.3)** Evaluation of Ind_{M(B)}^G α_B: zero off ∪(aH(a))^G; on it, (α(a)/|M(B — Both cases proved: vanishing off the Dade support and the value formula (the b-sum over N_L(B) ∩ a^L with conjugating-set 𝒜(g,H(B)b) cardinalities, via the copr…

### Pf §7 — Peterfalvi §7

> Peterfalvi section 5 "Coherence" (repo S07, results (5.1)-(5.9)) is essentially completely formalized and entirely sorry-free — it is one of the most heavily developed units in the repo, spread over S07_Coherence/ (5 leaves), S07_{Subcoherent,CoherenceConstantDegree,PivotCoherence,RetargetScaled,CoherenceGalois,BridgeCoherent} and consumer-side leaves in S06/S08/S12/S13. (5.1), (5.2), (5.4), (5.5), (5.7), (5.9) are at full book strength; (5.6) is fully present but factored Coq-style into an engine ((5.6.1) family bundle + (5.6.2) integer-forcing with the exact degree inequality (c) + (5.6.3) retarget adjoining, assembled book-shaped in DadeChainStep.advance and iterated by coherentPairChain), with the reducible-break case and a strictly-stronger norm-weighted variant added beyond the book. The only generality gaps are (5.3)(b) and (5.8): all their mathematical content is proven, but the mixed irreducible/reducible (5.2.e) cross-orthogonality rider and the mu-column image dichotomy/pin are proven at the FT maximal-subgroup instances (Sibley-Dade, type-II, type-P S(H0C)) rather than as single abstract theorems at Hypothesis (4.6) generality, and (5.8)'s second-case uniqueness rider has no direct counterpart (applications derive sign alignment by other proven routes). Surprising strengths: (5.7) is norm-general (accepts reducible members, Coq uniform_degree_coherence), and (5.9)(a) is proven for arbitrary automorphisms of C without star-commutation, wider than the book's Q_|G| setting.

特殊化債務 (教科書より狭い形で証明済 — 一般化要否は着手時に判定):

- **(5.3)** ✅ **解消 (2026-07-19 lane a)** — (a) S subset Irr L implies Hypothesis (5.2); (b) Hypothesis (5.2) for  — (a) fully formalized at book strength: irrSubcoherent (S07_Subcoherent.lean:129, Coq irr_subcoherent) assembles the six-field S07.Hypothesis; the per-member R-d… 【2026-07-19 lane a 更新: (5.2.e) の 3 層のうち irr×irr (`dadeOrthonormalCharacterImageFamilyOfDiff_orthogonal`) と red×red (`certainTypeR_imageSet_orthogonal_certainTypeR`) は元から抽象。残る **mixed irr×red** が Sibley/type-P/type-II の 3 つの instance コピー (証明はアンカー 1 個を除いて逐語同一) だった。 これを Hypothesis (4.6) 一般形 `certainTypeR_imageSet_orthogonal_dadeOfDiff_of_vanishOnV` (`S08_CrossOrthogonality.lean`, 新規 leaf) として抽出 — carrier の代わりに V 上の消滅アンカーを 明示仮説に取る。3 コピーはそれぞれ自分のアンカーを渡す 1 行の適用に置換 (重複証明 296 行を削除)。 ⟹ mixed 層も抽象化され、特殊化債務は解消。】
- **(5.8)** ⚠ **記述が stale (2026-07-19 lane a 実測)** — Image of a reducible column mu_k under a coherent extension: mu_k^tau1 【二分岐そのものは **type-II 専用ではなく既に抽象**: `eq_smul_chiFam_column_of_vanishOnV` (S05_SigmaTrichotomy.lean:461) が素の `TICyclicHypothesis G` 上で本文どおりの選言を結論し (IsMinimalSimpleOdd も maximal subgroup も type-II も不要)、その組合せ核 `grid_eq_const_column_of_two_col` (S05_GridTrichotomy.lean:366) は群論を一切含まない。`typeII_nu_tau2_dichotomy` は定理でなく**その適用**。 残るのは第 2 分岐の一意性 rider (「j と k のみ同次数」) だが、consumer は `exists_nu_extension_eq_alignedRow_at_pair` (S12_TypeIIGridTranspose.lean:913) が符号を `∃ delta', delta' = 1 ∨ delta' = -1` に吸収するため**どちらの分岐が発火したか決める必要が無い**。 ⟹ 債務でなく設計選択。】

### Pf §8 — Peterfalvi §8 Some Coherence Theorems, results

> Peterfalvi §8 (results (6.1)–(6.8)) is essentially fully covered, sorry-free, and was the workhorse of the FT character spine: the capstone Theorem (6.8) is proved at full book strength (sibleySetup_is_coherent) over a faithful hypothesis carrier (genuine Dade datum, full Hypothesis-(4.6) case (c2)), with all internal steps (6.7.1)–(6.7.3), (6.8.1)–(6.8.3) discharged, and (6.7) proved as a standalone general Sylow-TI congruence outside the book tree (peterfalvi_67_of_odd). The main deviation from the book is factoring, not gaps: (6.2)/(6.3) are unconditionally proved for the Sibley kernel K = H (both Frobenius and certain-type variants), while their general solvable-kernel (6.1) forms (six_two_general, six_three_of_six_two_oracle) expose the (5.6) break-member bound for reducible induced members as an oracle hypothesis that is fully discharged only in the §11–§13 consumer contexts (muGrid decomposition data) — every instance the book actually uses is proved, but no single unconditional general-K theorem exists. Similarly (6.5) and (6.6) are proved as abstract group-theoretic cores plus the Sibley-instance producers that (6.8) consumes (Z = Z(H) ∩ H′ and Z = W₂) rather than as packaged statements under general Hypothesis (6.4); the (6.5)(a) chief-factor clause exists only as contradiction arithmetic. Surprising strengths: the repo's case-B ((6.8.2)) machinery is substantially more developed than the book's two-page sketch, and (6.6)'s X-characterization avoids [Is] Lemma 2.21 entirely; one vestigial scaffold remains (OddOrderSpecialization with a free Prop field), unused by the live chain.

特殊化債務 (教科書より狭い形で証明済 — 一般化要否は着手時に判定):

- **(6.2)** ✅ **一般 solvable kernel `K` で完済 (2026-07-27, issues 0153+0154)** — `six_two_of_imageData` (`S08_SixTwoThreeFromImageFamilies`) が **oracle 無し・Dade 非依存**で書籍形を結論する。仮説は `InducedFamilyImageData A₀ K` (= Hypothesis (5.2) 一式: (5.2.b) の `tau`/`tau_isometry`/`tau_mem_ZIrr` + (5.2.d)/(5.2.e)) と `K` 可解のみ。FT の Dade 写像は bundle の witness の一つ (`S12.Hypothesis.inducedFamilyImageData`)。以下は旧記述: Coherence-break index bound: S(A) coherent, S(B) not, D/B ⊆ Z(C/B) ⟹ 2 — Fully proved (sorry-free) for the Sibley kernel K=H: six_two/six_two_central (Frobenius c1, S08_CoherenceCorePart2/SibleyBounds.lean:585/642) and _c2 variants (…
- **(6.3) Theorem** ✅ **一般 solvable kernel `K` で完済 (2026-07-27, issues 0153+0154)** — `six_three_of_imageData`、(6.2) と同じ仮説。以下は旧記述: Coherence descent: H/M nilpotent, S(H₁) coherent, |H:H₁| > 4|L:K|²+1 ⟹ — Sibley case (K=H, Frobenius) fully proved: six_three (SibleyBounds.lean:723; minimal-A/maximal-B descent, nilpotency-forces-centrality, √-arithmetic all done). …
- **(6.4) Hypothesis** Odd-order setup: (6.1) + |L| odd, K/M nilpotent, H₁/M = [K/M,K/M], L/H — Live encoding = SibleyDadeHypothesis (S08_CoherenceCorePart1.lean:494): odd |L|, H nilpotent normal ≠ ⊥, split L = H ⋊ W₁, TI, genuine §4 Dade datum — the K=H, …
- **(6.5)** ✅ **(a) の chief-factor 節を 2026-07-19 lane a が形式化** (`isChiefFactor_of_relIndex_le_of_odd_dvd`, S08_CoherenceCorePart1.lean:135 — 教科書 p.31 の証明どおり: L-normal な中間 W があると (6.4.c) + 奇数位数で |K:W|,|W:H₁| ≥ 2|L:K|+1、積が (2|L:K|+1)² > 4|L:K|²+1 となり (6.3) 界と矛盾)。以下は旧記述: If S(M) not coherent: (a) K/H₁ chief factor of L and |K:H₁| ≤ 4|L:K|²+ — All three parts' content proved sorry-free, factored as abstract cores + Sibley producers: (a) index bound = contrapositive of six_three (isPGroup_of_not_cohere…
- **(6.6)** For 1 ≠ Z ⊆ Z(K) normal with X = S − S(Z) ⊆ Irr L: X = {χ ∈ Irr L | Z  — The X-characterization is proved generically in Z (S08_DegreeSums/CoherenceGlue.lean:489, exactly the book biconditional given X ⊆ Irr L). The coherence half — … 【**2026-07-27 実測**: 債務の実体は「`K` が Sibley kernel `H` に固定」。`SibleyDadeHypothesis.SsubFiltration A` は `{Ind_H^L θ | θ ∈ Irr H, θ ≠ 1, A.subgroupOf H ⊆ ker θ}` で、**`S08.inducedKernelFamily H A` と逐語同一**。X-characterization `Xset_eq_irreducible_not_subset_characterKernel` の証明も `S_eq`/`mem_Xset`/`mem_SsubFiltration`/`H_normal` しか使っておらず、Sibley 固有の内容は無い ⟹ 一般 `K` へ移せる。⚠ ただし import 方向が逆 (`S08_SixTwoGeneral` は `S08_DegreeSums/CoherenceGlue` の**下流**) なので、`inducedKernelFamily` + 初等 API を上流の新 leaf に prefix-split してから両者が import する形にする必要がある (namespace は `OddOrder.Peterfalvi.S08` のままなので下流 consumer は無変更)。coherence 半分 (`Xset_centralCommutator_isCoherent_of_*`) は `Z = Z(H) ⊓ ⁅H,H⁆` と Frobenius/c2 の 2 ケースに固定されており、こちらは別途。】 【**2026-07-27 続報: X-characterization 側は完済**】 一般形 `inducedKernelFamily_sdiff_eq_irreducible_not_subset_characterKernel` (`S08_InducedKernelFamily`, 新 leaf) を新設し Sibley 版をその特殊化に置換 (commit d9217f065)。⚠ **`K` の正規性は不要**と判明したので書籍 (6.1) より強い。 【**coherence 半分の実測**: `Xset_isCoherent_of_irreducible_X` (S08_CoherenceBasic.lean:352) は **既に `Z` について generic** (`{Z} [Z.Normal] (hZle : Z ≤ H) (hZcent : Z.subgroupOf H ≤ center ↥H)` を取る)。`Xset_centralCommutator_isCoherent_of_{frobenius,c2_caseA}` はその `Z = Z(H) ⊓ ⁅H,H⁆` での **instantiation にすぎない**。⟹ 残る債務は「`K = H` (Sibley) 固定」の一点で、これを一般 solvable `K` に上げるには §8 の case-A/B engine 全体 (p-群次数評価・Frobenius/certain-type の既約性・X-chain) を再基礎付けする必要がある — X-characterization のような「族の形しか使っていない」case ではない。**規模 L 相当の別プロジェクト**として繰延。】

### Pf §9 — Peterfalvi §9 Non-existence of a Certain Type of Group of Odd Order, results

> Peterfalvi §9 (results (7.1)-(7.11)) is essentially complete and entirely sorry-free, spread over 22 S09_*.lean files plus the S09_NonexistenceCertain/ directory (~15k lines); the chain culminates in card_G0_lower_bound (7.10) and not_trivial_G0 (7.11), which are live consumers on the FT spine (Pf S10, S14). The formalization is factored far more finely than the book: (7.7.a) and (7.8.c.i) are structure certificate fields with genuine standalone discharges (chiRho_decomp_induced, chiRho_eq_inner_beta_induced, and 'no-certificate-assumed' constructors hypothesis76OfDade/hypothesis78OfDade), while (7.8.a)/(7.8.b)/(7.9) exist as abstract conditional assemblies whose source facts are fully discharged in the Frobenius-family layer — exactly the strength (7.10) and the later §12/§14 consumers need, but not as single self-contained book-generality statements. The only real content gaps are (i) the equality-iff clauses of (7.2.b) and (7.3) — **CLOSED 2026-07-18 by lane c** (`chiRho_norm_sq_eq_iff_mem_range`, `chiRho_integral_eq_iff_constant_on_hCoset`; both sorry-free, axiom-clean), so §9 now carries no 部分 rows — (only the inequalities are proved; the one equality case the book uses, χ = 1_G, is proved directly as the cardinality identity |A^τ|/|G| = |A|/|L|), and (ii) the nilpotent-kernel hypothesis added to (7.10)/(7.11), where the book invokes Thompson's unformalized Frobenius-kernel-nilpotency theorem (repo has only the solvable-case BG Thm 3.7 version; all repo consumers supply nilpotence by construction). Mildly surprising finds: the parity step of (7.9) is a clean reusable primitive (cfdot_real_vchar_even), several results are proved slightly stronger than stated (7.5 for any norm-one class function, 7.7.b for arbitrary CF(G)), and the file headers explicitly document that Pf §9 (7.11) and BG App.C Theorem C are different formulations of the final contradiction, both present in the repo. Verification pass: no labels changed — the partial labels on (7.2)/(7.3) were confirmed by exhaustive repo search (the equality-iff clauses exist nowhere in OddOrder/, though the Coq analogs leC_norm_invDade/leC_cfnorm_invDade_support carry them via lerif), and the three most load-bearing formalized results (7.5) family_inequality, (7.6) hypothesis76OfDade, (7.7) chiRho_decomp_induced + chiRho_norm_sq_double_sum were re-read at statement level and confirmed at full book strength.

| 結果 | 状態 | 規模 | 内容 | メモ |
|---|---|---|---|---|
| (7.2) | 済 | S | (a) α^{τρ} = α on CF(L,A); (b) ‖χ^ρ‖² ≤ ‖χ‖², equality iff χ ∈ im τ | (a) fully proved, both pointwise on A and as class functions on L. (b): the inequality is proved via the adjoint formula + orthogonal decomposition (the book's proof), but the 'equality if and only if χ is in the image o… 【2026-07-18 lane c 更新: (7.2.b) の等号条件 `chiRho_norm_sq_eq_iff_mem_range` を S09_NonexistenceCertain/Hypothesis71.lean に追加 (sorry 無し・axiom-clean)。(a)(b) とも本文強度で完了したため 部分→済。】 |
| (7.3) | 済 | S | Integral inequality /G/⁻¹ Σ_{A^τ} /χ/² ≥ ‖χ^ρ‖², equality iff χ constant on each aH(a) | The inequality is fully proved (book's proof: restrict χ to A^τ, apply (7.2.b)). The 'equality iff χ constant on every coset aH(a)' clause is not formalized — the theorem docstring states the iff but the statement carrie… 【2026-07-18 lane c 更新: 等号条件 `chiRho_integral_eq_iff_constant_on_hCoset` を同 file に追加 (sorry 無し・axiom-clean)。部分→済。】 |

特殊化債務 (教科書より狭い形で証明済 — 一般化要否は着手時に判定):

- **(7.8)** 【**2026-07-26 実測: 書籍強度・gap なし**】 書籍 p.40 をページ画像で確定し 1 対 1 照合済 — 同 unit の「#### (7.8) の実測 (2026-07-26)」表を参照。残る差は「書籍 1 番号 = Lean 1 定理」の体裁のみ。 以下は旧記述: Coherent case: (a) β = 1_G − ζ^ν + a·Σ φ(1)/(e‖φ‖²) φ^ν + Γ with a ∈ ℤ — All three parts have genuine sorry-free proofs, but factored differently from the book. (c) is at full book strength: chiRho_eq_inner_beta_induced proves the po…
- **(7.9)** ✅ **2026-07-19 lane a が一般化完了** — 唯一 Frobenius 専用だった parity step を `Hypothesis79.delta_even` (S09_TwoFamiliesParity.lean) として Hypothesis (7.9) 一般形で証明し、`FrobeniusFamily.hypothesis79_delta_even` はその instantiation に縮小。以下は旧記述: Two coherent families over odd-order G with disjoint A_i^{τ_i}: (β₁,ζ₂ — TwoFamilies.lean + S09_ParityPrimitive.lean + S09_FrobeniusParity.lean. The full book proof is assembled sorry-free for pairs of members of a FrobeniusFamily (h…
- **(7.10)** k ≥ 2 Frobenius subgroups with pairwise-coprime TI kernels force (|G₀| — S09_FrobeniusCardG0LowerBound.lean, sorry-free, exact displayed inequality over ℚ, hypotheses (a)-(d) faithfully bundled in FrobeniusFamily. ONE genuine special… **[2026-07-18: hnilp CLOSED ⟹ 特殊化債務解消]** `card_G0_lower_bound` が担いでいた `hnilp : ∀ j, IsNilpotent ↥((H j).subgroupOf (L j))` を除去。教科書は (7.10) の証明中で "By a theorem of Thompson" として**導出**しており仮説ではない。Isaacs Thm 6.24 = `Isaacs.Ch06.IsFrobeniusGroup.isNilpotent_kernel` (KernelNilpotent.lean:382, 2026-07-17 landed) を条件 (7.10)(a) の `F.isFrobenius` に適用する `FrobeniusFamily.isNilpotent_kernel` を新設して内部供給。axiom-clean。
- **(7.11)** Main theorem: no group satisfying the hypothesis of (7.10) has G₀ = {1 — S09_FrobeniusCardG0LowerBound.lean (unconditional form) + FrobeniusFamily.lean (8 conditional consumer forms). not_trivial_G0 is sorry-free and consumed on the … **[2026-07-18: hnilp CLOSED ⟹ 特殊化債務解消]** (7.10) と同じく `not_trivial_G0` から `hnilp` を除去 (Thompson 経由で内部導出)。唯一の consumer `S14_MaximalI/TypeICovering.not_all_maximal_typeI` の局所 `have hnilp` 10 行も不要になり削除 (full build green, axiom-clean)。

### Pf §10 — Peterfalvi §10

> Peterfalvi §10 is essentially complete: of its 18 numbered items, 15 are fully formalized and sorry-free, with the taxonomy (8.1)–(8.7) living in the shared OddOrder/GroupTheory/MaximalSubgroupType.lean (proven equivalent to BG's F/P₁/P₂ taxonomy via the sorry-free BG §16 Proposition 16.1) and the theorems heavily but faithfully re-factored onto BG §14–16 endpoints (Theorem I for (8.8), Theorems B/D for (8.12)/(8.13), the faithful M̃-cover for (8.17)). Notable non-obvious placements: (8.2.c) is in S14_MaximalI/Hypothesis.lean, (8.5.b) in S11 WielandtSetup, and (8.16) in S12_TypeIIDadeBase/Frobenius — the original S10 encodings of (8.16) and of the (8.14) kernel were found FALSE-as-stated/unsatisfiable during formalization and were replaced by honest versions (issues 8021/9008; the .mmd is also missing p.48 with (8.14)–(8.17), recovered from the PDF here). The only real gaps: (8.11)'s Sylow-normalizer conjunct is unassembled though all ingredients are proved (S effort); (8.15) exists as per-type Dade-hypothesis instantiations (types I, P₁, II, including the (4.6) carrier) rather than one universal statement including the (5.2) clause; and (8.18) is proved only for type-I pairs (all the spine needs). The two remaining BG sorries (nonidentity_covered_by_sigma_pieces, sigmaLength_one_frobenius_type) are vestigial with zero consumers and do not touch this unit. Slim verification pass: the (8.11) partial label was CONFIRMED after a targeted refutation attempt (the Hall conjunct is fully proved at S10_StructureSetup.lean:393; no Peterfalvi-shaped Sylow-normalizer statement exists anywhere, though the assembly pattern appears inline in S16 TypeBridges.lean:867-891), and the three most load-bearing formalized items (8.8)/(8.13)/(8.17) were re-read at Lean-statement level and confirmed at full book strength — no status changes made.

| 結果 | 状態 | 規模 | 内容 | メモ |
|---|---|---|---|---|
| (8.11) | 済 | S | N_G(P) ≤ M for nontrivial Sylow P of M_s; M_F and M_s are Hall subgroups of G | CONFIRMED partial after targeted refutation attempt (grep by number in all formats, by descriptive names, by Sylow×Msigma/mainSubgroup combinations). The Hall conjunct is fully proved for all five types (hall_maxNilpoten… 【**2026-07-19 hub 監査で訂正: 部分 → 済**】【2026-07-19 lane a 更新: **Sylow-normalizer 側も実在** — `normalizer_sylow_mainSubgroup_le` (S10_StructureSetup.lean:397) が「`P` が `M_s` の非自明 Sylow ⇒ `N_G(P) ≤ M`」を BG Thm A(1) 経由で証明済 (sorry 無し)。Hall 側は同 file:449。両 conjunct が揃うので 部分→済。 survey の「no Peterfalvi-shaped Sylow-normalizer statement exists anywhere」は誤り。】 |

特殊化債務 (教科書より狭い形で証明済 — 一般化要否は着手時に判定):

- **(8.15)** 【**2026-07-27 実測: 「型ごとの instantiation しかない」は stale**】 書籍 p.48 の (8.15) は 3 主張。 **claim 1** の `A₁(M)` 節は `dadeSupportHypothesisData_A1` (S10_MinimalSimpleBasic.lean:1181) が **任意の型**で証明済 (`A₁(M) = M_σ^#` が型 uniform だから)。`A₀(M)`/`A(M)` 節が型ごとなのは **その台集合自体が型ごとに定義されている**からで、書籍に忠実であって債務ではない (同 docstring が明記)。 **claim 3** (Hypothesis (5.2)) も型 uniform 版が在る: `inducedNonKernelFamily_subcoherent` (S10_SubcoherentTypeP.lean:476) — 書籍そのままの族 `{Ind_K^L θ | H ⊄ Ker θ}` と任意の台 `A` の上で、`(M')^# ⊆ A` を要求しないので型 `P₁` 規制を受けない (書籍の `H ⊄ Ker θ` filter が (4.7) 経由で台評価を 買う)。⟹ 型 II/V を含む**全型 𝒫**で書籍逐語形。**claim 2** は `typePData_toHypothesis46_ofSupport` の support-parametric core + 各 instance。以下は旧記述: M = N_G(A) and Hypothesis (2.2) for A ∈ {A₀, A, A₁}; Hypothesis (4.6)  — The (2.2)+normalizer clause is proved with the faithful H(a) = R(a): for type I (A(M) = A₀(M) and A₁), for type P₁ (A(M), A₀(M) = A ∪ V^M, A₁ via the σ-generic …
- **(8.18)** 【**2026-07-27: (c) は型 I-or-II へ一般化済**】 `support_mutual_exclusion_of_typeI_or_II` (S10_MinimalSimpleStructure) — `S`, `T` それぞれ独立に 型 I または II でよく、型タグ `tauS`/`tauT` も独立に {I, II} を取れる。型 I 限定だったのは `A₁(S)` の非空性を `TypeFData.H_nontrivial` から取っていた **1 箇所だけ**で、任意の極大部分群で 成立する `BG.Ch3.S10.Msigma_ne_bot` に置換すれば型 II も通る (もう一方の柱 `A1_eq_sigmaSharp_of_typeI_or_II` は元から型 I-or-II)。従来の型 I 版はその特殊化 (1 行)。**(a)/(b) は元から完全に型 generic だった (実測)**: (a) `isTypeI_or_isTypeII_of_mem_typeA_not_mem_A1` (S10_StructureSetup.lean:618) は任意の `tau` + `HasPeterfalviType tau M` を取り、(b) `ftThickenedSupport_A1_disjoint_of_nonconjugate` (S10_MinimalSimpleStructure.lean:111) は `tau1 tau2` 任意。mixed (c) のコア `ftThickenedSupport_mixed_disjoint_of_nonconjugate_typeA` (:883) も `tauS tauT` 任意 (非型 I の `tauT` に side condition `hW1T` が付くだけ)。⟹ 型 I 固定だったのは**便宜 wrapper 2 本と mutual-exclusion のみ**で、survey の「all three parts ... only for pairs of TYPE-I」は大幅に stale。以下は旧記述: Support relation between non-conjugate maximals: T supports S iff A₁(S — All three parts (a)/(b)/(c) are proved sorry-free but only for pairs of TYPE-I maximal subgroups (which is what the downstream all-type-I contradiction (12.17) …

### Pf §11 — Peterfalvi §11

> Peterfalvi §11 (book section 9, results (9.1)–(9.11)) is essentially completely formalized and fully sorry-free — roughly 24,000 lines across 23 S11_* files plus substantial generic infrastructure (the Wielandt formula (9.1) is proven unconditionally in OddOrder/GroupTheory via a chief-series assembly discharged by a modular kernel-FPF argument, deliberately factored for reuse). Results (9.1)–(9.6), (9.8) and (9.9) are at full book strength, with (9.6) notably CORRECTING a book overstatement (the unconditional |W2| = p is false for type II; only the image |W̄2| = p holds) and (9.8)–(9.10) having gone through a deliberate count-faithfulness audit (issue 2030). The three items marked formalized_specialized are packaging rather than mathematical gaps: (9.7)'s case-(a) embedding lands in ((𝔽_p)ˣ)^{q−1} instead of the book's order-a cyclic coordinates and case-(b)'s W1 ≅ Aut F identification lives as generic semilinear infrastructure rather than one three-story isomorphism; (9.10) assumes case (b) rather than deriving it from the trigger (a one-line dispatch done at consumers); and the main coherence theorem (9.11) — whose entire (9.11.1)–(9.11.8) engine is landed, mirroring Coq's Ptype_core_coherence — is stated per consumer (types III/IV via S13.coherent_sOf_H0Cprime with its side conditions discharged unconditionally; type II via separate, even stronger, S/T instances in §15) with no single generic declaration over Hypothesis (9.5). Closing the literal-statement gaps would be one S-level assembly each for (9.7)/(9.10) and an M-level uniform re-statement for (9.11); no new mathematics is required anywhere in the unit.

特殊化債務 (教科書より狭い形で証明済 — 一般化要否は着手時に判定):

- **(9.7)** 【**2026-07-27 実測: (b) の「三層同型が無い」は stale**】 `caseB_exists_galoisField_repr_withAut` (S11_GaloisFieldModel.lean:471) が**書籍の三層同型そのもの** — `H̄ ≃+ GF(p^q)` (`e`)、`Ū ↪ F*` を乗法で実現 (`μ` 単射)、`W₁ ≃ Aut F` (`η` 全単射) が自然に作用、の 4 条件を 1 つの ∃ で結論する (docstring も「This assembles the whole of (9.7.b)」と明記)。(a) は issue 0152 で書籍の `a`-形へ完済済。⚠ **併せて vestigial な opaque field を削除** (2026-07-27): `CliffordCaseBData.field_model : Prop` / `field_model_holds` は自由 Prop フィールドで、唯一の producer が `True` を入れ、消費者ゼロだった (honest な中身は上記 `S11_GaloisFieldModel` にあり、同 file が「does not use the legacy opaque proposition」と明記していた)。構造体が書籍内容を担っているように**見せかける** scaffold なので除去。以下は旧記述: Clifford dichotomy on H̄: (a) H̄ = direct product of q order-p factors — CuS0.lean:1771 (dichotomy + carriers built at :1219/:1241 from InertiaLift.lean:524), GaloisFieldModel.lean, ImprimitiveUBound.lean, CuS0.lean caseA_wOrbit_* (f…
- **(9.10)** ✅ **2026-07-19 lane a が一般化** (`exceptional_case_frobenius_realization_of_trigger`, ThetaCountAssembly.lean:1146 — trigger `hno` だけから Clifford case (b) を選ぶので `caseB` carrier は不要)。以下は旧記述: Exceptional case: if 𝒮(H0C') has no irreducible of degree qu induced f — ThetaCountAssembly.lean:993. All book conclusions proven sorry-free: fixed-point-free Ū-action on H̄ (the genuine H̄⋊Ū Frobenius content), u = (p^q−1)/(p−1), an…
- **(9.11)** 【**2026-07-27 実測: 「generic 宣言が無い」は stale**】 `S11.nineEleven_coherent_A0` (S11_NineElevenCaseAResidual.lean:959) が **まさにその generic 宣言** — `TypesIIIIIIVSetup M` (= 書籍 Hypothesis (9.2)、型 II/III/IV を全部覆う) + `ChiefFactorData` + `Section11CharacterData` + `Hypothesis46` の上で述べられ、**型仮説を一切持たない** (S13_Orthogonality.lean:154 の docstring も明言)。§13 の型 III/IV 形 `coherent_sOf_H0Cprime` と §15 の S/T 形は**その系**。⟹ 債務は packaging ですらなく解消済。以下は旧記述: 𝒮(H0C') is coherent for the Dade isometry τ (the section's main theore — Main M-instance: S13_Orthogonality.lean:1183 (Coq Ptype_core_coherence, PFsection9.v:1484) — case (b) via the norm-general (5.7) engine, case (a) via the full (…

### Pf §12 — Peterfalvi §12 Maximal Subgroups of Types III, IV and V, results

> Peterfalvi §12 (results (10.1)–(10.11), pp. 58–63) is essentially completely formalized and sorry-free, spread over ~23k lines in S12_MaximalIII_IV_V_Core/{Hypothesis,CharacterParameters,Isometry105,Isometry106,DadeCalculations}.lean plus ~15 support leaves (S12_Noncoherence, S12_TypeVCaseC, S12_Prop109, S12_TypeII*, S12_MaximalBasic, S12_HcBound, S12_Section9Counts). The two main theorems are proven unconditionally at full book strength and pinned axiom-clean in AxiomsCheck.lean: (10.8) S_not_coherent_unconditional and (10.10) no_typeV_maximal_unconditional (all three (8.7) branches, including the full (10.10.1)–(10.10.4) case-(c) engine). The only deviation from full strength is (10.11): its first assertion (|W1|,|W2| prime) is a standalone theorem, but the Type-II remainder (H elementary abelian of order p^q, S coherent — Coq FTtypeII_prime_facts) exists only as the §15 S/T-pair instances (P/Q_elementaryAbelian, card_P_eq, sSet_coherent_indS/indT_A), not for an arbitrary Type-II maximal — hence formalized_specialized. Surprises: the formalization is stronger than the book in places — (10.9) and (10.5)/(10.6) also come in coherence-free forms that the later (11.8)/(11.9) consumers need where S is not coherent, and (10.6.b) gives the exact odd-integer value rather than just |ζ^{τ1}(g)| ≥ 1; several docstrings still reference since-closed sorries (issue 2022 gates, exists_typeII_maximal_with_w2) and CharacterParameters carries two vestigial opaque Prop fields — cosmetic staleness only.

特殊化債務 (教科書より狭い形で証明済 — 一般化要否は着手時に判定):

- **(10.11)** 【**2026-07-27 実測: 両主張とも書籍強度**】 commit 3a8b3f282 参照 (AxiomsCheck の注記が stale だった)。 以下は旧記述: In case (b) of (8.8): |W1|, |W2| prime; for Type II maximal M: H eleme — First assertion at full book strength (theorem88_caseB_prime_orders, S12_MaximalIII_IV_V.lean; Coq FTtypeP_pair_primes). The Type-II remainder (Coq FTtypeII_pri…

### Pf §13 — Peterfalvi §13 Maximal Subgroups of Types III and IV, results

> Peterfalvi §13 (book results (11.1)–(11.9), pp. 64–68) is essentially completely formalized and sorry-free across the twelve S13_*.lean leaves (plus the S13_CoreStructure/ directory), with (11.7)'s two-case chief-kernel refutation in S13_ElementaryAbelianKernel.lean and the (11.9.a) Galois layer in S13_TypeIIIGalois.lean. The repo's factoring differs from the book in two systematic ways: (i) the character lemmas (11.4)–(11.8) are parametrized on the (11.3) non-coherence as an explicit hypothesis, discharged everywhere by the unconditional S_H0C_not_coherent_unconditional (built on the unconditional Theorem (10.8)), so composed strength equals the book; (ii) (11.8) is stated existentially (one canonical ζ ∈ S(HC) with the grid non-orthogonality) rather than for every ζ ∈ S(HC) — the only genuine, mild strength gap in the unit (the (11.8.1)–(11.8.6) machinery is ζ-generic, so a universal restatement is ~S effort). Surprises on the strong side: (11.6)'s U-centralizes-H0 clause is proved unconditionally with no character input, and (11.9.c) is strengthened well beyond the book to a universal Type-IV exclusion (no maximal subgroup of a minimal simple odd group is Type IV), feeding the §15/§16 endgame. Several docstrings still mention 'remaining sorries' from mid-development — stale text; all S13 files are sorry-free. Note the missing Nougat page (p. 66) contains only proof steps (11.8.1)–(11.8.4), recovered from the PDF; no numbered result was lost.

特殊化債務 (教科書より狭い形で証明済 — 一般化要否は着手時に判定):

- **(11.8)** 【**2026-07-27 実測: 既に書籍の `∀ ζ` 形**】 commit b5e3b00be 参照 (「存在形しかない」は誤り)。 以下は旧記述: For ζ ∈ S(HC): (μ0 − ζ)^τ − Σ_{i<q} ω_{i0}^σ is not orthogonal to (Irr — S13_Orthogonality.lean (+ S13_Lemmas113To115.lean). Stated EXISTENTIALLY — produces one canonical degree-w1 irreducible ζ ∈ S (∈ S(HC) via (11.5)) with the grid…

### Pf §15 — Peterfalvi §15 "The Subgroups S and T", results

> Peterfalvi §15 (repo S15_*, results (13.1)–(13.19)) is fully formalized and sorry-free — the largest Peterfalvi cluster in the repo (52 S15 files plus S16 spill-over), with the (13.1) hypothesis realized as a rich constructed carrier (S15.Hypothesis, built sorry-free on the FT path in FeitThompson.lean) rather than a free-field scaffold. All 19 numbered results have genuine Lean counterparts at essentially book strength; the only genuine narrowing is (13.8), where only the T-side instance (the one (13.10) actually consumes) is instantiated and the S-side form exists only as the side-agnostic norm engine. The formalization is heavily refactored relative to the book, largely following the Coq PFsection13 restructuring: the (13.5)–(13.10) norm cascade splits into generic ℚ/Parseval engines plus honest correction packages (CaseBEndgameSupply), (13.2.e)'s normedTI splits into A(S)-TI + Dade=Ind + A₀ no-escape + a type-P₁ full-A₀ TI theorem, and (13.3.b)/(13.12)/(13.15) are exported as λ-dichotomy forms; the repo even proves (13.4), which Coq skips. Surprises: the repo fixed two book/legacy overstatements (the (13.18) grid-orthogonality conjuncts and an unconditional (13.17.c) small-branch exclusion, both rolled back per issue 3003, the latter deferred to its true (14.5) location), and several legacy carrier records (BasicStructureData) still contain True-placeholder Prop fields and stale 'gated/sorried' docstrings even though the honest theorems now exist — cosmetic debt, not mathematical gaps.

特殊化債務 (教科書より狭い形で証明済 — 一般化要否は着手時に判定):

- **(13.8)** 【**2026-07-27 実測: 「T 側しかない」は stale**】 書籍逐語の **S 側**が在る: `Hypothesis.eta01_Hsharp_norm_lower_core` (S15_CaseBEndgameSupply/Eta01Correction.lean:790) が `|S′| − u² ≤ ∑_{x∈H^#}|η₀₁(x)|²` (`H = PC`, 書籍 p.79) を結論し、docstring も「the book's literal `S`-side statement (issue 1041)」と明記。AxiomsCheck 登録済。T 側 `eta10_Qsharp_norm_lower_core` は (13.10) が使う「(13.8) applied to T」の instance で、両者とも side-independent engine `caseB_eta01_norm_bound` (S15_SAndT_Setup/Machinery135.lean:937) を通す。⟹ 債務なし。以下は旧記述: Σ_{x∈H^#}|η₀₁(x)|² ≥ |S′| − u² (used in (13.10) in its T-instance Σ_{Q — Only the T-side instance (the one the (13.10) proof actually invokes: '(13.8) applied to T') is instantiated, in S15_CaseBEndgameSupply/Eta10Correction.lean, un…

### Pf App: Suzuki — Peterfalvi Part II Appendix: A Theorem of Suzuki

> This unit is essentially unformalized: its only Lean counterpart is OddOrder/Peterfalvi/Appendices/Suzuki.lean, a 5-sorry scaffold (theoremA, general_structure, first_case_theorem, structure_of_H, psu_characterization) whose Hypothesis/Conclusion structures consist of opaque free Prop fields, so even the statements are not at book strength - each sorried theorem is technically trivially dischargeable and encodes no mathematics, meaning all ~30 numbered results are effectively missing. The one genuine sorry-free asset is Appendices/SemilinearField.lean (Peterfalvi Appendix I Prop 2(a)+(b): Schur+Wedderburn field structure and semilinearity), which this unit consumes at Ch I §2 Prop 3 and throughout; Appendices/Huppert.lean also carries real fpf/cyclic support for Appendix I Prop 1 (though its fpf Lemma chain has 1 real sorry), while the other supporting Part II appendices this unit cites (Near-Fields, Suzuki 2-groups/Higman, Feit-Sibley) are scaffold-only apart from a genuine sorry-free NearField class and twisted near-field construction. The dominant external gaps are the target-group libraries - mathlib has only a bare ProjectiveSpecialLinearGroup definition with no PSL(2,q) theory, and Sz(q), PSU(3,q)/PGU(3,q), and the Zassenhaus-group classification ([HB] XI.11.16, assumed as known by the book) exist nowhere - plus three missing classical theorems: Thompson fpf-nilpotency ([H] V.8.14, Isaacs 6C not started), the Hall-Wielandt weakly-closed transfer theorem, and Isaacs Ch.8-style permutation-group basics (Isaacs tree stops at Ch07, though mathlib's MultipleTransitivity file covers 2-transitivity itself). Two practical warnings: math-comp odd-order has NO analog of Peterfalvi Part II (PFsection files cover Part I only), so no Coq crib exists; and the 05.6 mmd has a missing OCR page ([MISSING_PAGE_EMPTY:136]) swallowing items (5)-(9) of Ch IV §2, so the PDF must be consulted there. SLIM-pass verification: all 32 missing labels were re-attacked and CONFIRMED (no status changes), but three notes were corrected - the Wielandt fixed-point theorem [HB] XI.12.4 is NOT missing (it is fully formalized sorry-free as OddOrder.GroupTheory.wielandt_fixedPoint_frobenius = Peterfalvi (9.1) in OddOrder/GroupTheory/WielandtFixedPoint.lean, so it drops off Thm B's external-needs list, along with Clifford theory (CliffordCorrespondence.lean) and Isaacs Ch05 transfer/focal-subgroup machinery which are also present), Huppert.lean's Appendix I Prop 1 support was overstated as sorry-free (pGroup_cyclic_fixedPointFree carries 1 real sorry in the irreducible non-cyclic case, issue 2004), and I.1 Prop 2 gains adjacent dihedral-inversion/Matsuyama/Baer-Suzuki infrastructure in Isaacs/Ch02_Subnormality/Theorem211Wielandt.lean.

| 結果 | 状態 | 規模 | 内容 | メモ |
|---|---|---|---|---|
| (A1)-(A3) | 済 | S | Suzuki setup: G doubly transitive on Omega, H point stabilizer, H = Q semidirect D with /Q… | CONFIRMED. Scaffold structure exists but nearly all conditions are opaque free Prop fields (action_on_Omega : Prop with a _holds witness), so the hypothesis is NOT stated at book strength; only t^2=1, t not in H, Even /Q… 【**2026-07-19 hub 監査で訂正: 未 → 済**】実測で教科書強度・sorry-free と確認。所在: OddOrder.Peterfalvi.Appendices.Suzuki.Hypothesis — OddOrder/Peterfalvi/Appendices/Suzuki/Basic.lean:148 |
| Thm A | 未 | XL | Main theorem: exists normal L with /G/L/ odd and L isomorphic to PSL(2,q), Sz(q) or PSU(3,… | CONFIRMED. Sorry at Suzuki.lean:72. The statement is an opaque scaffold: Conclusion has free Prop fields (classification : Prop, classification_holds), so the sorry is trivially dischargeable and carries no mathematics. … | 【**2026-07-19 hub 監査**: `STALE_PARTIAL` — 「未」ではなく**部分的に存在**する。The note's stated grounds are stale in every particular: it cites 'Sorry at Suzuki.lean:72' and 'Conclusion has free Prop fields (classification : Prop, classification_holds)'. That declaration no lon】
| I.1 Prop 1 | 済 | M | (a) H^g cap H conjugate to D, odd; (b) N_G(X) <= H for 1 != X <= Q; (c) Q contains a Sylow… | CONFIRMED. Needs the Omega <-> conjugates-of-H dictionary and Q regular on Omega - {H}. (e) uses four-group coprime generation ([HB] X.1.9); repo has coprime-action generation infrastructure (BG Ch1 S03g Thm 3.10 files, … 【**2026-07-19 hub 監査で訂正: 未 → 済**】実測で教科書強度・sorry-free と確認。所在: OddOrder/Peterfalvi/Appendices/Suzuki/Basic.lean:498 (a), :518 (a, cardinality), :534 (b), :593 (c), :618 and :636 (d), :689 (e) |
| I.1 Prop 2 | 済 | S | (a) s in H, u outside: su has odd order; (b) involutions form one conjugacy class; (c) u -… | CONFIRMED missing as stated, but adjacent dihedral-inversion infrastructure DOES exist (missed by original survey): OddOrder/Isaacs/Ch02_Subnormality/Theorem211Wielandt.lean carries the Isaacs Lemma 2.14(a),(b) dihedral … 【**2026-07-19 hub 監査で訂正: 未 → 済**】実測で教科書強度・sorry-free と確認。所在: OddOrder/Peterfalvi/Appendices/Suzuki/InvolutionClass.lean:66 (a), :147 (b), :168 bijOn_conj_of_involution_mem_Q (c), :283 ncard_involutions_map_conj_eq_card_involutions_H (d) |
| I.1 Prop 3 | 済 | S | /K/ = /H cap I/ and s^K = H cap I for s in H cap I | CONFIRMED. K = {x in D : x^t = x^{-1}}; short counting argument on Dt cap I. 【**2026-07-19 hub 監査で訂正: 未 → 済**】実測で教科書強度・sorry-free と確認。所在: OddOrder/Peterfalvi/Appendices/Suzuki/InvolutionClass.lean:468 (first clause |K| = |H ∩ I|), :504 (injectivity), :525 (second clause s^K = H ∩ I) |
| I.1 Prop 4 | 済 | M | (a) canonical form: G - H = unique xty, x in H, y in Q; (b) unique (s,r) with tst = r^{-1}… | CONFIRMED. The named definitions (canonical form, distinguished involution, structure equation, script-N(G)) live here. general_structure (Suzuki.lean:96, sorry) is the opaque scaffold bundling Ch I outputs (canonical_fo… 【**2026-07-19 hub 監査で訂正: 未 → 済**】実測で教科書強度・sorry-free と確認。所在: (a): existsUnique_canonicalForm — OddOrder/Peterfalvi/Appendices/Suzuki/CanonicalForm.lean:123 (with existence :54 and uniqueness :92). (b): OddOrder/Peterfalvi/Appendices/Suzuki/DistinguishedInvolution.lean (435 lines,  |
| I.1 Lemma | 済 | S | Involution t acting on odd-order X: (y,z) -> yz and zy are bijections C_X(t) x {inverted} … | CONFIRMED. Classical counting lemma, reused throughout (Prop 5, II.(13), IV). Re-searched inversion/centralizer-counting: only odd-order square-trick fragments exist (commute_of_sq_commute_of_odd_card in RepresentationTh… 【**2026-07-19 hub 監査で訂正: 未 → 済**】実測で教科書強度・sorry-free と確認。所在: OddOrder/Peterfalvi/Appendices/Suzuki/InvertedProduct.lean:218 (a), :242 (a cardinality), :258 (b elementwise), :291 closure_invertedBy_subgroupOf_normal (b), :311 map_eq_inv_of_forall_fixed_eq_one (a, endomorphism form) |
| I.1 Prop 5 | 済 | S | V = C_D(s) and W = C_D(H cap I) | CONFIRMED. 【**2026-07-19 hub 監査で訂正: 未 → 済**】実測で教科書強度・sorry-free と確認。所在: V_eq_centralizer_distinguishedInvolution — OddOrder/Peterfalvi/Appendices/Suzuki/CentralizerStructure.lean:143; W_eq_centralizer_involutions_H — :241; first inclusion :40 |
| I.1 Prop 6 | 済 | M | X <= D with >= 3 fixed points: C_G(X) doubly transitive on Omega_X, C_H(X) = C_Q(X) semidi… | CONFIRMED. Key induction-enabling fact; needs the fixed-point-set permutation framework. 【**2026-07-19 hub 監査で訂正: 未 → 済**】実測で教科書強度・sorry-free と確認。所在: OddOrder/Peterfalvi/Appendices/Suzuki/FixedPointCentralizer.lean:226 (a, C_G(X) doubly transitive on Ω_X), :282 cQ_mul_cD_eq_cH (a, C_H(X) = C_Q(X)⋊C_D(X)), :360 even_card_cQ (b), :485 exists_conj_mem_D_map_le_V (c) |
| I.2 Prop 1 | 済 | XL | (a) C_Q(x) = 1 for x in K^#; (b) Q nilpotent; (c) H cap I <= Z(Q), Q_0 = (H cap I) cup {1}… | CONFIRMED. (b) invokes Thompson's fixed-point-free-automorphism-of-prime-order => nilpotent theorem ([H] V.8.14). Verified missing from both mathlib and the repo (Isaacs Ch06_FrobeniusActions/Main.lean roadmap marks 6C '… 【**2026-07-19 hub 監査で訂正: 未 → 済**】実測で教科書強度・sorry-free と確認。所在: OddOrder/Peterfalvi/Appendices/Suzuki/QStructure.lean:102 (a, C_Q(x)=1), :169 (b, Q nilpotent), :267 and :331 (c) |
| I.2 Prop 2 | 済 | M | K is a cyclic normal subgroup of D | CONFIRMED missing, but the support characterization is corrected: Appendices/Huppert.lean support is genuine yet NOT fully sorry-free — pGroup_cyclic_fixedPointFree (Huppert.lean:531, the Appendix I Prop 1 fpf Lemma) has… 【**2026-07-19 hub 監査で訂正: 未 → 済**】実測で教科書強度・sorry-free と確認。所在: OddOrder/Peterfalvi/Appendices/Suzuki/KCyclic.lean (822 lines, 0 sorries) — card_fitting_Dbar_eq_ncard_KSet:703, exists_KSet_generator:745, K_le_D:794 |
| I.2 Cor | 済 | S | The Sylow 2-subgroup S of Q is abelian or a Suzuki 2-group | CONFIRMED. Depends on Suzuki 2-group Definition 1 (Appendix III unit); in the repo that is only the scaffold structure IsSuzuki2Group in Appendices/Suzuki2Groups.lean (verified: higman_classification and typeB lemmas all… 【**2026-07-19 hub 監査で訂正: 未 → 済**】実測で教科書強度・sorry-free と確認。所在: sylowTwo_isMulCommutative_or_isSuzuki2Group — OddOrder/Peterfalvi/Appendices/Suzuki/SylowTwo.lean:61 (195 lines, 0 sorries) |
| I.2 Prop 3 | 済 | M | Q_0 semidirect (D/W) isomorphic to L(F_q, A) = (F_q semidirect F_q^*) semidirect A; identi… | CONFIRMED. The essential engine (Appendix I Prop 2(a)+(b)) IS formalized sorry-free in Appendices/SemilinearField.lean (verified: 0 sorries; declarations live in namespace OddOrder.Peterfalvi.Appendices.Huppert) — Schur … 【**2026-07-19 hub 監査で訂正: 未 → 済**】実測で教科書強度・sorry-free と確認。所在: exists_semilinear_equiv — OddOrder/Peterfalvi/Appendices/Suzuki/SemilinearRealization.lean:339 (also exists_fitting_field_model:~36, exists_semilinear_field_model:214, exists_equivariant_field_coordinates:267); model in  |
| I.3 Lemma 1 | 済 | L | If G satisfies the conclusion of Thm A then Q is a 2-group and L = O^{2'}(G) = normal clos… | CONFIRMED. Requires structural facts of PSL(2,q)/Sz(q)/PSU(3,q) as 2-transitive groups (degree q+1 with q a 2-power; simplicity). Gated on the absent target-group libraries; given those, S. 【**2026-07-19 hub 監査で訂正: 未 → 済**】実測で教科書強度・sorry-free と確認。所在: Group-theoretic core: simple_normal_oddIndex_Q_core — Suzuki/InductionHypothesis.lean:61. Target-group cases: InductionHypothesisPSL.lean:84, InductionHypothesisSuzuki.lean:61 (Q_and_residual_of_suzuki_target), Induction |
| I.3 Prop 1 | 済 | L | For 1 != X <= V: (a) C_G(X) on Omega_X satisfies (A1); (b) N_G(X) = C_G(X) N_V(X); (c) if … | CONFIRMED. The induction-hypothesis engine used by Chs II-IV. (c) needs Sylow 2-structure of PSL(2,l), Sz(l), PSU(3,l) and the order of st in each ([H] II.8.2, II.10.12; [HB] XI.3.1/3.3, XI Ex 10.7) - all absent (re-veri… 【**2026-07-19 hub 監査で訂正: 未 → 済**】実測で教科書強度・sorry-free と確認。所在: (a) centralizerHypothesisA1 — Suzuki/CentralizerInduction.lean:241 (plus :161, :223); (b) normalizer_eq_centralizer_mul_normalizer_inf_V — Suzuki/CentralizerNormalizer.lean:218; (c) centralizer_trichotomy_of_induction —  |
| I.3 Prop 2 | 済 | M | If G is not simple, the conclusion of Theorem A holds for G | `InductionNonSimple.lean`: constructs `L = ⟨I⟩`, proves `[Q,k] = Q`, restricts (A1)--(A3), derives `G = LD` and odd index, and lifts induction to the concrete Theorem A conclusion. |
| I.3 Lemma 2 | 済 | S | Subsets of V conjugate in G are conjugate in V | `ConjugacyInV.lean`: retains arbitrary subsets, corrects an ambient conjugator into `D` via the doubly transitive centralizer action, and applies the honestly constructed canonical projection `D → V`. |
| I.3 Lemma 3 | 済 | S | A strongly real element x with x^2 != 1 is conjugate to ut, u in Q_0^#, and /C_G(x)/ is od… | `StronglyReal.lean`: defines strongly real elements at book strength, derives the `u*t` normal form from transitivity on fixed-point triples, and proves centralizer parity by the source's odd-dihedral conjugacy argument inside `N_G(<x>)`. |
| I.3 Lemma 4 | 済 | M | If st has order 3 and V != 1 then <Q_0, K, t> = Q_0 K cup Q_0 K t Q_0 is isomorphic to PSL… | `OrderThreePSL.lean` proves the braid and exact two-cell carrier equality (correcting the printed missing sharp: only `tQ₀#t` lies in the big cell); `OrderThreePSLInduction.lean` constructs the faithful doubly transitive orbit action and (A1)--(A3), proves the subgroup proper from `V != 1`, applies induction, excludes Suzuki/PSU by commutativity of `Q₀`, and yields an existential finite characteristic-two field `F` with `|F| = |Q₀|` and the required PSL equivalence. |
| I.3 Lemma 5 | 済 | M | If st has order 3 and Q is a Suzuki 2-group of order q^3 then W is cyclic, /W/ divides q+1… | CONFIRMED. Gated on the Appendix III theorem (Higman classification of Suzuki 2-groups - scaffold-only, higman_classification sorried in Suzuki2Groups.lean, verified) and GL(2,q) subgroup structure ([H] II.8). 【**2026-07-29 hub 監査で訂正: 未 → 済**】実測 (sorry census + grep) で確認。所在: `OddOrder/Peterfalvi/Appendices/Suzuki/WCyclicDivides.lean:352` (完全な statement; 第 1・2 主張は `:47` `isCyclic_W_and_card_dvd_of_orderThree`)。支えは `Suzuki/TypeBFromW.lean` / `Suzuki/OrderThreeSuzukiCentralizer.lean` / `Suzuki2Groups/{SquareCosetFiber,ConjugateSummandSplit,KSubgroupOrbit,ActualQuotientAction}.lean`。⚠ 旧注記の「gated on Higman classification (scaffold-only, `higman_classification` sorried)」は**完全に stale** — `higman_classification` は repo に存在せず (grep 0 件)、`OddOrder/Higman/Suzuki2Groups/**` と `Peterfalvi/Appendices/Suzuki2Groups/**` は全 file sorry ゼロ |
| Thm B | 済 | XL | Chapter II (first case): under (B1) (some prime-order P <= V with C_G(P) of 2-rank 1) the … | CONFIRMED missing, but the external-needs list is CORRECTED in three places. (1) The Wielandt fixed-point theorem [HB] XI.12.4 IS in the repo, sorry-free and axiom-clean, contrary to the earlier note: OddOrder.GroupTheor… 【**2026-07-29 hub 監査で訂正: 未 → 済**】実測 (sorry census + grep) で確認。所在: `OddOrder/Peterfalvi/Appendices/Suzuki/FirstCase/TheoremB.lean:81` (2026-07-26 `55ec6f681`、issue 2053) |
| Thm C | 済 | L | Chapter III section 1: Q is a 2-group (i.e. Q_1 = 1) | CONFIRMED. structure_of_H (Suzuki.lean:133, sorry) is the opaque scaffold for Ch III. Proof is a coherence/character argument: gated on the Feit-Sibley theorem (Appendix IV; feit_sibley_coherence scaffold-only in FeitSib… 【**2026-07-29 hub 監査で訂正: 未 → 済**】実測 (sorry census + grep) で確認。所在: `OddOrder/Peterfalvi/Appendices/Suzuki/StructureOfH/CoherenceContradiction.lean` — `SecondCaseHypothesis.Q1_eq_bot` (`Q₁ = ⊥`) と `SecondCaseHypothesis.isPGroup_two_Q` (書籍表記 `IsPGroup 2 Q`)。2026-07-29 `7175e6a73`、[issue 0162 closed](../../issues/closed/0162-pf-part2-ch3-theorem-c.md)、AxiomsCheck 登録済で axiom-clean |
| Ch III S1 Prop | 未 | L | Trichotomy: (a) S = Q_0, st order 3; (b) S Suzuki type A, st order 5, W = 1; (c) S Suzuki … | CONFIRMED. Uses I.3 Prop 1 trichotomy, Higman/Appendix III theorem, Galois correspondence for F_q, and the PSU(3,l) Sylow-normalizer check C_{D_0}(Omega_1(S_0)) != 1 (needs concrete PSU(3,l)). |
| Ch III S2 Prop | 未 | M | In case (b) (st of order 5), (SK) cup (SKtS) is a subgroup of G | CONFIRMED. Self-contained computation with f, g, h and K-orbit representatives s, r, r^{-1}, r r^{-k}; needs Appendix I Prop 2 identification of S/Q_0 with F_q (engine exists sorry-free in SemilinearField.lean). |
| Ch III S3 Prop | 未 | L | Under (C2): explicit model S semidirect KW isomorphic to S_1 semidirect K_1 W_1 over E = F… | CONFIRMED. Needs Appendix III Definition 3 (B(n,theta,epsilon)), Lemmas 1-2 (central extensions with quadratic maps - scaffold-only in Suzuki2Groups.lean, verified: QuadraticMapData/square_map_quadratic sorried), and com… |
| Ch IV S1 (H1)-(H6) | 未 | M | The maps f, g : Q^# -> Q^#, h : Q^# -> D with txt = g(x) h(x) t f(x), and identities (H1)-… | CONFIRMED. Split BN-pair rank 1 formalism; self-contained given the canonical form (I.1 Prop 4). Scaffold psu_characterization records only an opaque maps_f_g_h : Prop. |
| Ch IV S1 Lemma | 未 | M | Uniqueness: <Q^x : x in L> and L are determined up to isomorphism by Q (resp. M = Q semidi… | CONFIRMED. Reconstruction of the 2-transitive action on Q cup {a}; the recognition device for Corollary 1. |
| Ch IV S2 Prop | 未 | L | If D acts without fixed points on (Q/Q_0)^# then some omega_i satisfies f(omega) = (omega^… | CONFIRMED. Long inductive sequence computation (u_i, v_i, d_i, beta a generator of W). OCR gap: 05.6 mmd has [MISSING_PAGE_EMPTY:136] - items (5)-(9) of the preliminary calculation are lost; formalizer must read the PDF … |
| Ch IV S3 Prop | 未 | L | If f(omega) = (omega^{-1})^zeta with h(omega) in W for some omega, zeta, then theta = 1 an… | CONFIRMED. psu_characterization (Suzuki.lean:152, sorry) is the opaque Ch IV scaffold (theta_eq_one, f_formula, psu_or_pgu_conclusion all free Props). The proof itself is a finite-field computation over E = F_{q^2}, feas… |
| Ch IV S3 Cor 1 | 未 | XL | Under the S3 hypothesis O^{2'}(G) isomorphic to PSU(3,q); if V = W then G isomorphic to PS… | CONFIRMED. Requires constructing PGU(3,q)/PSU(3,q) in Lean and verifying they satisfy (A1)-(A3) with the computed f (the recognition step 'PGU(3,q) satisfies this hypothesis'). Finite unitary groups do not exist in mathl… |
| Ch IV S3 Cor 2 | 未 | S | In PSU(3,q), for each zeta in W^# there is omega in Q - Q_0 with f(omega) = omega^{-zeta} … | CONFIRMED. Direct computation in the model; input to the V != W case. Gated on Cor 1's PSU model. |
| Ch IV S4 | 未 | L | The case V != W: steps (1)-(6) showing mu = 1, hence h(omega) in W, concluding Theorem A v… | CONFIRMED. Uses I.3 Prop 1(c) with U/Z(U) = PSU(3,l), q = l^p, Galois theory of F_q, and a semilinear-automorphism computation (mu-twisted equations (3)-(10)). Unnumbered in part; treated as the final assembly of Theorem… |

### Pf App: Huppert — Peterfalvi App: A Special Case of a Theorem of Huppert

> Peterfalvi Appendix I ("A Special Case of a Theorem of Huppert", pp. 135-136; repo name Appendix B) contains exactly three numbered results: Proposition 1, an unnamed Lemma, and Proposition 2 — the last only recoverable from the PDF, since the .mmd's page 136 is MISSING_PAGE_FAIL. The unit is substantially formalized in Appendices/Huppert.lean (912 lines) and Appendices/SemilinearField.lean, often at above-book generality (part (1) of the Lemma for arbitrary permuted independent decompositions; Prop 2 over abstract commutative group algebras k[T]). The single sorry in Huppert.lean is the Lemma's irreducible non-cyclic case (Schur + [H] III.7.5 normal type-(p,p) subgroup), whose blocker is the deferred cyclic-center case of Gorenstein 5.4.10 in BG S04 (issues/pending/2004); Proposition 1 is otherwise fully assembled and inherits only this one gap transitively. Proposition 2 is sorry-free for part (a) and the semilinearity half of (b), but the C_U(s) ≅ group-of-field-automorphisms clause of (b) has no Lean statement despite being claimed in the file docstring. Net: no result of this unit is missing; one M-sized p-group-theory gap (shared by Lemma and Prop 1) and one S-sized Prop 2(b) clause remain. SLIM VERIFICATION PASS: all three 'partial' labels CONFIRMED after refutation attempts (sorry location Huppert.lean:575 verified; Gorenstein 5.4.10 cyclic-center case confirmed absent repo-wide and issue 2004 still parked; Prop 2(b)'s C_U(s)→Aut(F) clause confirmed absent from SemilinearField.lean and all consumers); one detail corrected — the 5.4.10 deferral docstring and abelian-center theorem now live in S04_SmallRankBasic.lean, not S04_PGroupsSmallRank.lean (which is the post-split TAIL file).

| 結果 | 状態 | 規模 | 内容 | メモ |
|---|---|---|---|---|
| App I Prop 1 | 済 | S | Odd-order D faithful and transitive on E^# (E elementary abelian q-group) => F(D) cyclic, … | VERIFIED: stated at full book strength (all three conclusions: IsCyclic F(D), fpf, commutator D <= F(D) = D/F abelian) in OddOrder/Peterfalvi/Appendices/Huppert.lean (line 886). The whole assembly is written and proof-co… 【**2026-07-19 hub 監査で訂正: 部分 → 済**】実測で教科書強度・sorry-free と確認。所在: fitting_cyclic_fixedPointFree — OddOrder/Peterfalvi/Appendices/Huppert.lean:1319 (file is 1,345 lines with 0 real sorries) |
| App I Lemma | 済 | M | p odd, P a p-group faithful on elementary abelian q-group E with /P_a/ constant on E^# => … | VERIFIED: this is the location of the file's single sorry (Huppert.lean line 575, inside pGroup_cyclic_fixedPointFree at line 531). Fully proved: part (1) at greater-than-book generality (arbitrary P-permuted iSupIndep d… 【**2026-07-19 hub 監査で訂正: 部分 → 済**】実測で教科書強度・sorry-free と確認。所在: pGroup_cyclic_fixedPointFree — OddOrder/Peterfalvi/Appendices/Huppert.lean:718 (docstring at :705) |

### Pf App: NearFields

> **⚠ 2026-07-18 更新 (lane c) — de-opacify**: 2 sorry は共に **vacuous scaffold** だった
> (`rankOne_affine_nearField` は任意 `(two_rank_one : Prop)` を引数に取り、
> `cyclic_index_two_nearField_classification` は結論が `∃ classification : Prop, classification`
> = `⟨True, trivial⟩` で証明でき **sorry が何も隠していなかった**)。両者を book-strength に書き直し。
> 検証: opaque マーカー 0 (自由 Prop フィールド/`∃ P : Prop`/任意 Prop 引数)、旧 headline は現在
> **sorryAx を要する**(= 本物の文になった証拠)。⚠ **Prop 1 は元の場所で直せなかった**:
> `Suzuki.Hypothesis` が (A3) `two_rank_ge_two` を持つため、真の 2-rank-one 仮説を足すと
> **vacuous** になる。よって (A1)+(A2) + (A3) の正確な否定を持つ別 `RankOneHypothesis` を新設
> (充足例 `AGL(1,3) ≅ S₃` を確認)。opaque な `FiniteNearField` は既存の実 `NearField` class と
> 重複ゆえ削除 (参照ゼロ確認済)。**新規 sorry-free**: `NearField.mul_add_of_mul_comm`(公理ゼロ依存)、
> `exists_field_structure_of_cyclic_index_two`(Prop 2 前半を既存 sorry-free 核に接続)。
> 既存の実資産 (NearField class / F_{r²,2} 構成 / Prop 2 解析核) は無傷。残 2 sorry のブロッカー:
> Prop 1 = Brauer-Suzuki + Huppert III.8.2/II.3.2、Prop 2 = `exists_field_semilinear` 出力の
> `F` 自身への正規化 transport。
 — Peterfalvi App: On Near-Fields

> Peterfalvi Appendix C (book Appendix II, "On Near-Fields", pp. 137-138) contains 2 numbered propositions plus 3 named definitions/constructions, all covered by the single file OddOrder/Peterfalvi/Appendices/NearFields.lean. The honest layer is substantial: the near-field class, the prime-characteristic/elementary-abelian lemma, the abstract F_{r²,2} twisting construction (proved more generally than the book, including the associativity the book waves through), and — surprisingly strong — the entire analytic core of Proposition 2 (index-2 irreducibility via free-orbit counting + a bespoke Maschke theorem, then the field structure through Appendix I's exists_field_semilinear), all sorry-free and generalized from cyclic to commutative index-2 subgroups. The file's 2 sorries are both vacuous opaque-Prop scaffolds, not real partial proofs: rankOne_affine_nearField (Prop 1, line 620) concludes a structure of free Prop fields from an arbitrary `two_rank_one : Prop`, and cyclic_index_two_nearField_classification (Prop 2 headline, line 628) has the trivially-provable conclusion `∃ classification : Prop, classification`. The real gaps are Proposition 1 in full (XL — blocked on Brauer-Suzuki, the rank-1 Sylow-2 classification, and Huppert II.3.2, none of which exist in repo or mathlib) plus the field-or-F_{r²,2} dichotomy tail and |Z(F*)| = r−1 of Proposition 2 (M) and the ℒ(F) regular-action correspondence (M). Note the Coq submodule offers no help here: math-comp/odd-order never formalized the Peterfalvi Part II appendices. Slim verification pass: all four flagged labels (ℒ(F) missing, Prop 1 missing, F_{r²,2} partial, Prop 2 partial) were confirmed against fresh greps (holomorph/AGL/affine/regular-normal/Brauer-Suzuki/quaternion/quadratic-character) and direct reads of NearFields.lean, Suzuki.lean, and SemilinearField.lean — no upgrades warranted (Isaacs Ch06 DQSDRecognition is quaternion recognition, not the rank-1 classification); the two load-bearing formalized entries (NearField class, char(F) lemma) were read and confirmed at full book strength; no corrections made.

| 結果 | 状態 | 規模 | 内容 | メモ |
|---|---|---|---|---|
| App C ℒ(F) correspondence | 済 | M | Affine group ℒ(F) = F ⋊ F* of a near-field, and the converse transport: M acting regularly… | **【2026-07-23 lane c: 両方向とも済】** 逆向き (M acting regularly → near-field) は元から `SharplyTransitiveData` (`GroupTheory/NearFieldFromSharplyTransitive.lean`) で完成。**順方向**を同 file に追加: `nearFieldAffineGroup` (= 𝓛(F)=F⋊F* を `Subgroup (Equiv.Perm F)` として具体構成、affine 置換 `x↦x*u+t`) + `nearFieldAffineGroup_existsUnique` / `nearField_affine_existsUnique` (**sharply 2-transitive** = 任意の distinct pair を distinct pair へ写す一意の affine 写像)。全 near-field 一般・axiom-clean・AxiomsCheck 登録済。旧「missing」は slim pass の grep が holomorph/AGL/affine を探して逆向き構成しか見つけられなかった stale。 |
| App C Prop 1 | 未 | XL | 2-rank-one case of the Suzuki setup: G satisfying (A1)-(A2) with 2-rank 1 is (F ⋊ F*) ⋊ Σ … | Missing label re-verified by direct read: sorry #1 (NearFields.lean:620) takes an arbitrary `two_rank_one : Prop` parameter and concludes existence of `RankOneNearFieldData`, whose conclusions (affine_model, Q/D identifi… |
| App C F_{r²,2} | 済 | S | The exceptional near-field F_{r²,2}: K = 𝔽_{r²} (r an odd prime power) with twisted produc… | Partial label re-verified: the construction is formalized sorry-free in abstract form, more general than the book (NearFields.lean:421-580): TwistData (any field K, automorphism σ with σ² = 1, σ-invariant character χ : K… 【**2026-07-22 lane c: 済**】新 leaf `ExceptionalNearField.lean` で具体化完成: `squareSignChar` (平方指標の Kˣ →* ℤ/2 形) + `halfFrobenius` (x↦x^{pⁿ}) + `exceptionalTwistData` (book p.138 の乗法一致 lemma 2 本) + **非可換性** `exceptionalNearField_not_commutative` (真に体でない)。全て axiom-clean、AxiomsCheck 登録済。 |
| App C Prop 2 | 済 | M | A finite near-field whose multiplicative group has a cyclic subgroup of index 2 is either … | Partial label re-verified by direct read: the proof's mathematical core is genuinely proved, sorry-free, at slightly greater generality (commutative index-2 subgroup A ≤ Fˣ instead of cyclic): (i) irreducibility of the A… 【**2026-07-22 lane c 実測訂正: 済**】commit `42892fcb5` (2026-07-21) で `cyclic_index_two_nearField_classification` = **Prop 2 全文** (体 or `F ≅ F_{r²,2}` + `|Z(Fˣ)|=r−1`) が axiom-clean 完成。AxiomsCheck 登録は 2026-07-22 に補完。 |

### Pf App: Suzuki2Groups — Peterfalvi App: On Suzuki 2-Groups

> **⚠ 2026-07-18 更新 (lane b)**: Appendix III Lemma 1(b) と Definitions
> 2--3 は concrete な quadratic central extension として sorry-free で完了した。
> 以下の 2026-07-16 survey 本文の「zero genuine formalization」「Lemma 1(b) と
> Definitions 2--3 は未記述」という記述は、この3項目について stale。

> **✅ 2026-07-19 更新 (lane b) — Higman 1963 Lemma 1**:
> `Suzuki2Groups/AgemoLayers.lean` と `HigmanAbelian.lean` で、involution 上
> transitive な作用を持つ有限可換 `2`-group が homocyclic であり、その
> invariant subgroups がちょうど Agemo filtration `A^(2^s)` であることを
> sorry-free に証明した。保存済み Higman PDF の p.83 と `pdftotext -layout`
> 抽出を正本として、原文の involution-height、successive quotient
> irreducibility、power-map lifting をすべて実装した。これは Appendix III の
> unnumbered Higman classification 全体を閉じるものではないため、下表の
> `Thm (Higman)` は引き続き「未」。

> This appendix (Peterfalvi appendix on Suzuki 2-groups, pp. 139-143; repo file OddOrder/Peterfalvi/Appendices/Suzuki2Groups.lean labels it "Appendix D") has essentially zero genuine formalization: the Lean file is an opaque-Prop scaffold in which every theorem statement is vacuous (existentials over structures whose fields are arbitrary Props), so all 4 sorries (square_map_quadratic = Lemma 1(a), higman_classification = the Theorem, typeB_field_model = Prop 1, typeB_automorphism_structure = Prop 2) mask statements with no mathematical content, and Lemma 1(b)-(d), Lemma 2, and Definitions 2-3 are not even stated. There is no Coq analog to lean on — math-comp/odd-order never formalized Peterfalvi Part II or its appendices — and the only importer (FeitSibley.lean) uses none of these declarations, so the whole unit is greenfield. The one structurally notable item is the unnumbered "Theorem" (Higman's classification of Suzuki 2-groups): the book itself takes it as given, citing Higman 1963 / Huppert-Blackburn VIII 7.9, so an honest formalization must either import ~20+ pages of hard 2-group analysis (XL) or keep it as an explicit stated hypothesis. Mathlib supplies useful ingredients: QuadraticMap covers the preamble's char-2 quadratic-map definition, linearIndependent_monoidHom gives the Dedekind-independence core of Lemma 2, and GroupExtension/GaloisField give central-extension and finite-field infrastructure, but none of the numbered results themselves. Verification pass: all flagged labels (7 missing, 1 partial) were confirmed against the source mmd, the full Lean file, and repo-wide descriptive-name greps (Higman/Suzuki-2-group, quadratic map/cocycle/central extension, automorphism bases, semilinear) — no hidden coverage was found in OddOrder/GroupTheory, OddOrder/Algebra, or other book trees, the zero-consumer claim was re-verified repo-wide, and no statuses changed.

| 結果 | 状態 | 規模 | 内容 | メモ |
|---|---|---|---|---|
| Lem 1 | 部分 | M | Central extensions of elementary abelian 2-groups correspond to F_2-quadratic maps: (a)--(d) | ✅ 2026-07-18: part (b) is fully formalized in `Suzuki2Groups/QuadraticExtensions.lean`: explicit twisted-product group, central kernel, quotient, and square-map recovery, sorry-free and AxiomsCheck-registered. Parts (a), (c), (d) remain. |
| Lem 2 | 未 | M | For F a finite field of characteristic 2: (a) F_2-linear maps F->F form an F-space with ba… | Verified: no theorem exists — only the structure FiniteFieldTwoMapBasis whose three fields are opaque Props (content-free packaging, not among the 4 sorries). Repo-wide grep for automorphism-basis / Frobenius-basis / F_2… |
| Def 1 | 済 | S | Suzuki 2-group: nonabelian 2-group with at least two involutions admitting a cyclic group … | Verified partial: IsSuzuki2Group exists and its nonabelian field is honest (¬ IsMulCommutative P), but atLeastTwoInvolutions and cyclic_regular_automorphism_group are opaque Prop fields carrying no content, and the 2-gro… 【**2026-07-19 hub 監査で訂正: 部分 → 済**】実測で教科書強度・sorry-free と確認。所在: IsSuzuki2Group — OddOrder/Peterfalvi/Appendices/Suzuki2Groups.lean:~90; supporting involutions:~62 and ActsRegularlyOnInvolutions:~68 |
| Def 2 | 済 | S | The group A(n,phi): central extension attached to `a ↦ a*phi(a)`; type A classification | ✅ 2026-07-18: `Suzuki2Groups/Types.lean` defines the actual quadratic map and extension, with finite characteristic-two field data, `phi ≠ 1`, odd `orderOf phi`, and a group equivalence `TypeAData.equivModel`; sorry-free and AxiomsCheck-registered. |
| Def 3 | 済 | S | The group B(n,phi,epsilon): central extension attached to the source quadratic map; type B classification | ✅ 2026-07-18: `Suzuki2Groups/Types.lean` defines the exact three-term quadratic map, `ε ∈ Fˣ`, the source nonvanishing condition and its anisotropy consequence, plus the genuine group equivalence `TypeBData.equivModel`; sorry-free and AxiomsCheck-registered. |
| Thm (Higman) | 未 | XL | Higman's classification of Suzuki 2-groups: (a) involutions of P are exactly Z(P)#, Z elem… | Verified: higman_classification (1 of the 4 sorries) is vacuous — it concludes '∃ data : SuzukiTypeData P, data.typeA ∨ data.typeB ∨ data.typeC_or_typeD' where all three disjuncts are opaque Prop fields (instantiable wit… |
| Prop 1 | 未 | M | For B(n,1,epsilon) there is a field structure on E = F x F (namely F_{2^{2n}} via a root o… | Verified: typeB_field_model (1 of the 4 sorries) is vacuous — 'hB : Prop -> ∃ fieldModel : Prop, fieldModel' states nothing. The book proof is a half page (irreducibility from anisotropy, adjoin a root, norm-form identif… |
| Prop 2 | 未 | L | alpha -> f_alpha is a surjective homomorphism from Aut(B(n,1)) onto the semilinear maps x … | Verified: typeB_automorphism_structure (1 of the 4 sorries) is vacuous ('∃ automorphismStructure : Prop, automorphismStructure'). The real proof expands f, g in the automorphism basis of Lemma 2(a) and compares quadratic… |

### Pf App: FeitSibley — Peterfalvi Appendix E

> **⚠ 2026-07-18 更新 (lane c) — de-opacify**: `Hypothesis` は `H_eq_Q_semidirect_D` 等を
> 自由 `Prop` + `_holds` で持ち (`H=Q=D=⊥` と `True` で充足可)、`Sset`/`A`/`tau`/`d` も **posit** されていた。
> `coherence_adjoin_or_equal_degree` / `induction_isometry` は共に vacuous。全て book-strength に書き直し:
> `Hypothesis` は実際の部分群主張のみ (`H=Q⋊D` を `IsComplement'`+積で、TI 条件、`Q=S×Q₁`、
> `D` の Q₁ 上 fixed-point-free)、`Sset`/`A`/`d` は **定義**に、`tau` は `Ind_H^G` として **構成**。
> 検証: opaque マーカー 0、sorry 付き 7 定理は全て `sorryAx` を報告 (= 偶然自明でない)。
> **新規 sorry-free**: `tau` 構成 + `tau_apply`/`S_le_H`/`Q1_le_H`/`disjoint_*`/`d_pos`/
> `isIrreducibleCharacter_of_mem_Sset`/`zSpan_Sset_le_ZIrr` + Lemma 2(b) の実片 `tau_mem_ZIrr`/`tau_apply_one`。
> ⚠ **sorry 3 → 5 は正しい増加**: vacuous 1 文を正直な複数文に分解したため (「opaque で sorry-free」より
> 「honest + sorry」を優先)。Lemma 1(b) は既存 `S07.coherent_of_constant_degree` (= Pf (5.7)) ゆえ
> no-wrapper 方針で再掲せず docstring に対応記録。残 5 sorry のブロッカー: Isaacs 7.14 mixed-degree
> adjoin / Isaacs 6.34+6.11 / Isaacs 7.7 の Q-vanishing 特殊化 / Huppert V.13.8 / 上記全部+class-sum 合同。


> The unit is scaffold-only: OddOrder/Peterfalvi/Appendices/FeitSibley.lean (83 lines) carries exactly the 3 repo sorries — coherence_adjoin_or_equal_degree (Lemma 1, a vacuous '∃ Props' statement), induction_isometry (Lemma 2, opaque free-Prop data), and feit_sibley_coherence (the Theorem, a real IsCoherent conclusion but over free unconstrained fields tau/Sset/A) — under an opaque-Prop Hypothesis structure whose only genuine field is (|Q|,|D|)-coprimality; no file consumes these declarations (imported only by the root OddOrder.lean, absent from AxiomsCheck) and math-comp/Coq never formalized Peterfalvi Part II, so there is no Coq analog. The surprising strength is on the sibling main-text side: the FT-path Sibley coherence, Peterfalvi (6.8) sibleySetup_is_coherent, is fully proved sorry-free, and most of the appendix's technique already exists in that framework — equal-degree coherence (5.7) coherent_of_constant_degree, the τ-generic S07 retarget/adjoin bridge plus the Dade-wired (5.6) engine xAdjoinStep, the (6.7) class-sum congruence (peterfalvi_67/_of_odd, mod-|P| Sylow form), the quadratic endgame eq_zero_or_edge_of_dvd_of_normLt, Schur's center bound (Isaacs Cor 2.30, whose docstring names this appendix as an intended consumer), the odd-order no-real-characters lemma (1.1), and the generic Isaacs CTFG 7.7 TI-induction isometry. However the appendix Theorem is genuinely distinct from (6.8) — it allows |H| even (only d = |D| odd), a non-Frobenius H = Q⋊D with D fixed-point-free only on the direct factor Q₁ of Q = S×Q₁, and the restricted family {χ : Q₁ ⊄ ker χ}, with coherence for plain induction on a TI Hall Q — so nothing here is subsumed. An honest formalization means rewriting the Hypothesis without opaque Props, re-instantiating the S07/S08 machinery on this configuration, and adapting the (6.7) congruence from Sylow-TI to Hall-TI: roughly L total for the Theorem with steps (1)-(8) (individual step efforts listed are components of that total, not additive), plus M each for honest Lemma 1 (general-isometry adjoin criterion) and Lemma 2. Its only intended consumer is the Suzuki/PSU3 appendix, itself scaffold-only. Verification pass: all flagged statuses were confirmed after refutation attempts (in particular, the Hall-TI mod-|Q| congruence of step (7) truly has no repo counterpart — all three peterfalvi_67 forms are pinned to a Sylow p-subgroup), with two note-level corrections: Lemma 2(b)'s plain TI-induction isometry (Isaacs CTFG 7.7) is in fact fully formalized generically and sorry-free (induce_apply_coe_of_isTISubset / inner_induce_eq_of_isTISubset in InducedCharacter.lean, plus TIInducedFamily extraction), leaving only instantiation plus the 2(a) Clifford argument; and Lemma 1(a)'s adjoining bridge (retarget_isCoherent* in S07_Coherence/CoherenceUnion.lean) is already generic in an arbitrary IntegralCharacterMap, with only the crux1_of_memberFamily degree-inequality collapse and xAdjoinStep wiring Dade-specialized.

| 結果 | 状態 | 規模 | 内容 | メモ |
|---|---|---|---|---|
| App.E Lemma 1 | 未 | M | Coherence criteria: (a) adjoining one character under degree-sum inequality, (b) equal-deg… | [critic 補正] FeitSibley.lean の同型 opaque-Prop scaffold (L51-81) — scaffold は形式化と数えない規約により missing に統一。Scaffold theorem coherence_adjoin_or_equal_degree (FeitSibley.lean:51) is SORRIED and vacuous ('∃ two Props, both hold' … |
| App.E Hypotheses | 部分 | S | Standing setup: H = Q⋊D proper in G, (/D/,/Q/)=1, Q∩Qˣ=1 off H (whence Q Hall), Q = S×Q₁ c… | CONFIRMED by direct read of FeitSibley.lean:29-48: H_eq_Q_semidirect_D, Q_trivial_intersections, Q_eq_S_prod_Q1, D_fixedPointFree_on_Q1 are free `Prop` fields with `_holds` witnesses (opaque-Prop scaffold per notes/meta/… |
| App.E Lemma 2 | 未 | M | (a) 𝒮 = {Ind_Q^H φ : Q₁ ⊄ ker φ}, all irreducible; (b) Ind_H^G is an isometry on ℤ[𝒮]°; (c… | [critic 補正] FeitSibley.lean の同型 opaque-Prop scaffold (L51-81) — scaffold は形式化と数えない規約により missing に統一。Scaffold induction_isometry (FeitSibley.lean:69) is SORRIED and vacuous (InductionIsometryData has only opaque free-Prop… |
| App.E Theorem | 未 | L | Feit–Sibley Theorem: if d = /D/ is odd then 𝒮 is coherent with respect to the induction is… | CONFIRMED. feit_sibley_coherence (FeitSibley.lean:77) is SORRIED and, though its conclusion is a real S07.IsCoherent, it is stated for the Hypothesis's free fields tau/Sset/A which are unconstrained by the opaque hypothe… |
| App.E (1) | 未 | M | If /Q₁/ has two prime divisors then 𝒮(S') is coherent (chief-factor induction with degree/… | CONFIRMED missing (no chief-factor coherence induction anywhere in OddOrder/; chiefFactor* machinery in BG S03c_Thm37 is group-theoretic, not character-theoretic). Key inputs available sorry-free: Schur's center bound χ(… |
| App.E (2) | 未 | S | If 𝒮(S') is coherent then 𝒮 is coherent (climb the S-side chief series) | CONFIRMED missing; same pattern as (1) with the roles of S and Q₁ swapped; uses /Z(Q₁)/ ≥ 2d+1 from d odd + FPF. Roughly half a page; parallel arguments exist in the proved (6.8) code. |
| App.E (3) | 未 | M | For 1 ≠ Z ⊴ H with Z ≤ Z(Q₁): 𝒳 = 𝒮 − 𝒮(Z) is coherent (degree-divisibility chain then adj… | CONFIRMED missing. Uses Lemma 2(c), the divisibility chain χᵢ(1)² / Σ_{j<i} χⱼ(1)², equal-degree base case, and Lemma 1(a) adjoining — all techniques with sorry-free analogs in S07_CoherenceConstantDegree / S08 CoherentA… |
| App.E (4) | 未 | S | Setup: 𝒴 = 𝒮(Q') = {η₁,…,η_m}, Z = [Q₁,Q₁] ∩ Z(Q₁) ≠ 1, 𝒳 = 𝒮 − 𝒮(Z) = {χ₁,…,χ_n}, aᵢ = χᵢ… | CONFIRMED missing. Pure definitions/wiring; would be folded into an honest Hypothesis + proof-state structure for the Theorem. The 𝒴-coherence remark before (1) (/𝒮(Q')/ ≥ 2 via the odd Frobenius group O_{2'}(Q₁)⋊D, then… |
| App.E (5) | 未 | S | Cross-orthogonality (eᵢ, e'ⱼ) = 0 between the 𝒳- and 𝒴-images | CONFIRMED missing for this instance; the identical inner-product/degree argument is carried out sorry-free for the (6.8)/(7.x) families (e.g. S09_FrobeniusCrossOrtho, S08_YsetInner machinery), so this is a short re-insta… |
| App.E (6) | 未 | M | Ind(χ₁ − aη₁) = −ae'₁ + λΣe'ᵢ + v with (v,e'ᵢ)=0; if a / λ then 𝒮 is coherent (quadratic t… | CONFIRMED missing as stated, but the integer quadratic endgame ((x−1)² + (m−1)x² ≤ 1+1/a² forcing x ∈ {0,1}) is formalized sorry-free in the (6.8.2.2) analog eq_zero_or_edge_of_dvd_of_normLt (S08_CaseBCoherence/CentralCo… |
| App.E (7) | 未 | M | Congruence: ψ ∈ Irr(G) constant on Z^# satisfies ψ(z) ≡ ψ(1) (mod /Q/), Q a TI Hall subgro… | CONFIRMED missing: the mod-/Q/ Hall-TI version exists nowhere in OddOrder/. Verified by reading statements: peterfalvi_67 (ClassSumAlgebra.lean:126) and peterfalvi_67_of_odd (SylowTICongruence.lean:140) take an explicit … |
| App.E (8) | 未 | M | Conclusion: Res_H e'₁ = (λ+aμ)Σaᵢχᵢ + χ' with Σaᵢχᵢ = (ρ_H − ρ_{H/Z})/(da); combining with… | CONFIRMED missing. The regular-character-difference decomposition and restriction-coefficient bookkeeping have sorry-free case-B analogs (regCharCoeff machinery: card_H_dvd_card_W2_mul_regCharCoeff, index_W2_dvd_regCharC… |
| App.E Remark | 未 | S | If d is even and /𝒮/ ≥ 2 the conclusion still holds (then Q₁ is abelian; 𝒮(Q')=𝒮(S') coher… | CONFIRMED missing (no Lean counterpart and no scaffold); a short numeric argument once (2) is formalized. The book's Suzuki appendix uses the theorem with d odd, so this remark is genuinely optional strength. |
## 残作業の構造 (実作業ギャップ 214 件)

規模の大きい順ではなく**依存の上流から**見た構造:

1. **Isaacs 本文の小ギャップ (Ch.2–Ch.6, 計 31 件, S/M 中心)** — Hall 理論の支持補題 (3.6/3.7, 3.15–3.18, 3.26, 3.31–3.36)、
   Mann クラスタ (M(G) 定義 + 4.14–4.19)、transfer 残 (5.x 3 件)、Frobenius 6C (6.23 は Ch.7 Thm 7.1 からの系導出のみ)。
   相互依存は Mann クラスタ内・3.31→3.34 程度で、大半は独立・並列可。
2. **Isaacs Ch.8 Permutation Groups (25 未)** — 章まるごと新設。**Pf 付録 Suzuki 定理の上流** (二重推移群・Zassenhaus 群)。
   mathlib の MulAction/primitivity 資産で S が多い。
3. **Isaacs Ch.9 More on Subnormality (29 未 + 4 部分)** — F\*(G)/components は repo 実装済みだが章としての被覆は薄い
   (automorphism tower, Thompson–Wielandt, Bartels 等)。独立クラスタ。
4. **Isaacs Ch.10 More Transfer (26 未 + 1 部分)** — 独立クラスタ。
5. **BG 残り (実作業 25)** — §2 表現論 (Fong-Swan 等 M×3)、§4/§6/§16 の小 partial、App.C Rem(II) SL(2,2^q) 具体例 (M) +
   Rem(V) wlog (S)、App.D CN 群 (3 件, L 1)、App.E (5 件, L/XL — E.3 Feit–Thompson 1991 regular operator 等)。
6. **Pf 本文 partial (12 件, S 中心)** — §3/§4/§9/§10 の小穴 (下表参照)。本文 §3–§16 は実質完成。
7. **Pf 付録 (53 未 + 部分)** — 最大の塊 = **Suzuki 定理 (32 件, XL 4 = Sz(q)/PSU3(q) 特性化)**。
   ほか Suzuki2Groups (8)、FeitSibley (13)、NearFields (4)、Huppert (残 2)。Suzuki 本体は Isaacs Ch.8 に依存。

## 低優先繰延 (旧「対象外裁定」— ユーザー裁定 2026-07-16 で上書き)

- **BG App.C Rem (IV)** (Norton–Glauberman p ≤ 3 強化、文献 [12] に証明あり) と **Prob 1** (p = 3 可能性, open problem)
  は**恒久対象外にしない — 優先順位は低いがいずれやる** (ユーザー 2026-07-16)。現フェーズの effort 集計 (214 件) からは
  除外したままだが、scope 内の繰延項目として保持 (表中の ※低優先繰延 マーク)。Rem (IV) は文献証明ありなので形式化労力
  (XL 級)、Prob 1 は文献にも証明が無い research 級 — 着手時期は文献状況を見て判断。
- Isaacs の Problems (章末演習) は本調査のスコープ外 (番号付き本文結果のみ)。ただし本文が後で引用する演習
  (例: Hall D = Problem 3C.1) は本文側の結果として被覆済みであることを確認した。

## 特殊化債務 (54 件) の扱い — できる限り一般化 (ユーザー裁定 2026-07-16)

`formalized_specialized` は「FT 文脈では十分だった狭い形」(odd/solvable 限定、単一 case のみ等)。全冊被覆の基準は
**原文強度**であり、**デフォルトの処置は教科書の一般性への引き上げ (できる限り一般化)**。docstring 注記で済ませて
よいのは一般化が数学的に無意味な場合 (原文自体がその特殊形しか主張しない等) のみ。レーンは担当 unit に着手する際、
その unit の特殊化債務の一般化を作業項目に含める (番号順)。各 unit の「特殊化債務」リスト参照。
[[feedback-generalize-specialized-fully]]

## 推奨順序 (上流優先 + 文書順) と次アクション

順序原則は従来どおり**上流優先 + 文書順タイブレーク** (ft_path_policy §0 の原則部分)。**「FT 経路限定」の制約は
2026-07-16 のユーザー指示 (全 3 冊フェーズ移行) により失効** — ft_path_policy.md / CLAUDE.md の該当箇所の改訂が必要 (下記)。

- 文書順の基本線: Isaacs 本文 (Ch.2→Ch.10、ただし **Ch.8 は Pf Suzuki の上流なので早めに**) → Isaacs App (mathlib 対応表化が主)
  → BG §2→§16→App.C 残→App.D→App.E → Pf 本文 partial → Pf 付録 (NearFields/Huppert が小さく独立、Suzuki 本体は Ch.8 後)。
- 並列レーン向きの独立クラスタ: {Isaacs Ch.2–6 小ギャップ}, {Isaacs Ch.8 + Pf Suzuki 系}, {Isaacs Ch.9}, {Isaacs Ch.10},
  {BG 残 + Pf 本文 partial}, {Pf 小付録群}。具体のレーン配分は次段で裁定。
- 次アクション (調査フェーズの残務) — **全 3 件 2026-07-16 実施済**:
  1. ✅ ft_path_policy.md §0 と CLAUDE.md の FT 経路限定文言を改訂 (フェーズ移行注記 + 特殊化一般化デフォルト + 低優先繰延)
  2. ✅ レーン再配分 → 正本 [`lane_reallocation_2026_07_16.md`](lane_reallocation_2026_07_16.md) (a = Isaacs 本文完備化 / b = Ch.8+Suzuki チェーン / c = Ch.10+BG 残+Pf 残)
  3. ✅ S14 凍結 mis-encoding 2 件 (`nonidentity_covered_by_sigma_pieces`, `sigmaLength_one_frobenius_type`) を**削除** — どちらも consumer 0 で faithful 版は証明済 (Mtilde/zTilde cover API 群 / `S16.non_disjoint_signalizer_frobenius`)。実 sorry 25 → 23 (記録 = notes/bg/s14_typeP_counting.md)

---

## 🔄 2026-07-26 再実測 (main session) — ラベル突き合わせ

**方法**: Isaacs 本文 (`pdftotext`) の `N.M. Theorem/Lemma/Corollary` と、repo の
docstring ラベル `Isaacs <Thm|Lem|Cor|...> N.M` を機械的に突き合わせ。
⚠ repo の語彙は **`Thm`/`Lem`/`Cor`** (略号) なので `Lemma` で grep すると
大量の偽陽性が出る (最初の計測で Ch8 を 26 件と誤報した原因)。

| 章 | 本文 | ラベル無し |
|---|---|---|
| Ch1 | 46 | 10 (1.4, 1.6, 1.8, 1.12–1.15, 1.18, 1.21, 1.25) |
| Ch2 | 19 | 0 |
| Ch3 | 29 | 4 (3.7–3.10) |
| Ch4 | 38 | 2 (4.9, 4.37) |
| Ch5 | 30 | 6 (5.1, 5.2, 5.14–5.16, 5.23) |
| Ch6 | 24 | 0 |
| Ch7 | 8 | 0 |
| **Ch8** | 43 | **18** (8.2, 8.3, 8.11–8.22, 8.27–8.30) |
| Ch9 | 31 | 0 |
| Ch10 | 25 | 2 (10.8, 10.17) |
| 計 | | **42** |

⚠ **「ラベル無し」≠ 未形式化**。Ch1 の欠番は Sylow C/D・Frattini 論法・`n_p ≡ 1`・
冪零群の特徴づけなど **mathlib が直接持つもの**が並んでおり、薄いラッパー禁止方針で
意図的に書いていない可能性が高い。Ch8 の 8.27 (`A_n` 単純性) も mathlib 済。
⟹ **着手前に 1 件ずつ実測確認する** ([[verify-port-state-by-number-not-coq-name]])。

### Isaacs の判定 (2026-07-26 実測)

Ch8 のラベル無し 18 件を実測した結果、**大半は mathlib 被覆**と確認:
- `Mathlib/GroupTheory/GroupAction/{Blocks,Primitive,MultiplePrimitivity}.lean` が
  ブロック系・原始性・多重可移性の API を持つ ⟹ 8.11–8.22 (ブロック・原始性の基本補題群) は
  薄いラッパー禁止方針で**意図的に不在**。
- Jordan の定理は `Ch08_PermutationGroups/PCycleJordan.lean` に **`Isaacs Thm 8.23`** として存在
  (私が欠番と見た 8.17/8.18 とは別番号でカバーされている)。
- 8.27 (`Aₙ` の単純性) も mathlib 済 (`alternatingGroup`)。

同様に Ch1 の欠番 (Sylow C/D・Frattini 論法・`n_p ≡ 1`・冪零群の特徴づけ) も mathlib 被覆。

⟹ **Isaacs は実質完済**。残る個別項目は Problems (issue 1055 = lane a 領域) が主。
次の frontier 探索は **BG / Peterfalvi 本文**の番号付き結果の突き合わせで行う。

### BG の突き合わせ (2026-07-26 実測)

**方法**: 本 note の JSON (`three_books_full_survey_2026_07_16.json`) が列挙した BG 228 件のうち
**番号を持つ 181 件**と、repo の docstring ラベル (`BG {Theorem|Thm|Lemma|Lem|Corollary|Cor|
Proposition|Prop|Hypothesis|App} N.M`、太字・部分ラベル `(a)`/`(d)` 付きを含む) を突き合わせ。

| 指標 | 値 |
|---|---|
| JSON の番号付き BG 結果 | 181 |
| repo にラベルがある番号 | 180 |
| ラベル皆無 | **1** (BG 4.1 — JSON の status が既に `mathlib`) |

⟹ **BG は番号レベルでは全面被覆**。ラベル突き合わせは BG に対して信号を持たない
(Isaacs と違い、欠番からギャップを推定できない)。そこで **JSON が `missing`/`partial` と
した 26 件を個別実測**したところ、大半が既に完済していた:

| 項目 | 07-16 の label | 実測 (07-26) |
|---|---|---|
| 4.6 / 6.4 | missing | `S04_Lem45c_Prop46.lean` / `S06_Thm64Case2.lean` (+AxiomsCheck) |
| E.1 / E.2 | missing | `AppE_CollectionFormula.lean` (Hall の収集公式、E.2(a)(b)) |
| E.3 / E.4 / E.5 | missing | `AppE_CentralizerDecomposition.lean` / `AppE_FurtherResults.lean` / `AppE_E5Counting.lean` (+AxiomsCheck) |
| D.1 / D.2 | partial | `AppD_CNGroups/SylowTI.lean` |
| 1.21 / 2.1 / 2.3 / 4.4 / 4.5 / 4.20 / 6.3 | partial | すべて部分ラベル付き (`1.21(d)`, `4.4(a)(b)`, `4.20(a)(c)`, `6.3(a)(b)` …) で存在 |

**BG に真に残るもの (4 件)**:
1. **Lem 2.7** — repo には核心 (`Gorenstein Thm 3.2.3`, `ElemAbelianAutAction.lean`) だけがあり、
   BG の番号での statement は無い。実体があるので着手コストは小さい。
2. **§16 tamely imbedded 構造** — issue [8005](../../issues/8005-s16-tame-embedding-restore.md) で
   **意図的に defer** (Peterfalvi 側の実消費がトリガー、正本 `notes/bg/s15_16_audit.md` §12.3)。
3. **App.C Rem (II) / (V)** — 小 (Peterfalvi の例と「p, q は奇と仮定してよい」)。
4. **App.C Rem (IV) / Prob 1** — 文献引用のみ / open problem ⟹ 低優先繰延 (本 note 該当節の裁定どおり)。

### Peterfalvi の突き合わせ (2026-07-26 実測)

同じ方法 (JSON 238 件 vs repo の `Peterfalvi (N.M)` ラベル):

| 指標 | 値 |
|---|---|
| repo にラベルがある番号 | 164 |
| JSON にあって repo ラベル無し | **5** — (1.4) (1.8) (8.2) (8.5) は JSON 側が既に `formalized` (別名で所在)、残る 1 件が (2.4) `partial` |

⟹ **Peterfalvi も「statement が無い」という意味のギャップはほぼ無い**。07-16 に大量 missing だった
付録は既に完済している (Suzuki = Theorem B 完成 2026-07-26 / FeitSibley / NearFields 2026-07-23 /
Suzuki2Groups は lane b が 07-24 まで前進)。

**⟹ Peterfalvi の真の frontier は「特殊化債務」**: `formalized_specialized` 26 件 + `partial` 6 件。
CLAUDE.md の裁定 (2026-07-16「特殊化債務はできる限り一般化する」) の対象そのもの。上流優先 +
文書順で並べた着手順序:

| 順 | 項目 | 特殊化の中身 (07-16 時点の記述、着手前に要再実測) |
|---|---|---|
| 1 | (1.5) (1.7) partial | Clifford 系。(a)-(c) は完済、残りは形の違い |
| 2 | **(2.4)(a) partial** | `H(a^x) = H(a)^x` が導出でなく仮説 (`hconj`) — (2.10.1) が消費 |
| 3 | (2.6)–(2.10.3) spec ×6 | Dade 等長の一式。本体は sorry-free、狭めているのは carrier 側 |
| 4 | (5.3) (5.8) spec | coherence の subfamily 側 |
| 5 | (6.2)–(6.6) spec ×5 | **Sibley 特殊化 (K=H, M=1)** — 一般 (6.3)/(6.5) が残る最大の塊 |
| 6 | (7.2) (7.3) partial | 等号条件節 (χ が各剰余類上定数) が未形式化 |
| 7 | (7.8)–(7.11) spec ×4 | FrobeniusFamily に束ねた形 (本の一般形より狭い) |
| 8 | (8.11) partial / (8.15) (8.18) spec | (8.18) は **type-I 対のみ** — 一般型対が残る |
| 9 | (9.7) (9.10) (9.11) spec | §11 の carrier |
| 10 | (10.11) (11.8) (13.8) spec | (13.8) は **T-side instance のみ** |
| 11 | 付録の partial 4 件 | Huppert (単一 sorry ではなく carrier)、NearFields ×2、Suzuki2Groups Def 1 (opaque field)、FeitSibley Hypotheses (free Prop field) |

⚠ この表も **07-16 の記述に基づく**。各行は着手前に実測すること (BG で 26 件中 22 件が
既に完済していた実績がそのまま当てはまる)。

**全体の実 sorry = 1** (`bin/count-sorry`、Q₈ Brauer–Suzuki = issue 0147)。

### Pf §4 の特殊化債務を解消 (2026-07-26、commit `37171eade`)

上表の順 2–3 (= `(2.4)` partial + `(2.6)`–`(2.10.3)` specialized 6 件) は **1 つの原因**に
帰着した: **(2.4.a) を仮説として持ち回っていた**こと。

- **(2.4.a) は既に定理だった** — `S04_DadeIsometryBasic.lean` の
  `Hypothesis.hConjInvariant` が `mem_H_iff_coprime_orderOf` (= `x ∈ H(a)` ⟺
  `x ∈ C_G(a)` かつ `o(x)` が `|C_L(a)|` と互いに素) から sorry-free で導出している。
  本は `H(a) = O_{π'}(C_G(a))` 経由で示すが、局所版で `π`-core API 無しに出る。
  ⟹ survey の `(2.4)` = partial 「(a) is NOT derived」は **stale**。
- にもかかわらず (2.6)–(2.10.3) の全 statement が `hconj : hyp.HConjInvariant` を
  **余分な仮説**として取っていた (§4 で 65 binder、下流で 600+ の引数)。本は (2.2) から
  導出するので、これは repo 側の**特殊化債務**そのもの ([[repo-stronger-hypothesis-is-specialization-not-gap]])。

**やったこと**: binder と引数を全除去。仮説を落とすだけなので定理は一般化される方向のみで、
証明本体は不変。ついでに結論が `hConjInvariant` に一致して自明化した
`HConjInvariant.restrict` / `hconj_transport_ambient` を削除 (薄いラッパー禁止)。

**残り (次コミット候補)**: 構造体フィールドとして (2.4.a) を posit している 4 箇所 —
`S08_CoherenceCorePart1.lean:580` (`SibleyDadeHypothesis.hconj`) /
`S10_StructureSetup.lean:979` / `S14_MaximalI/Hypothesis.lean:44` /
`S12_MaximalIII_IV_V_Core/Hypothesis.lean:355`、および `Hypothesis71.hConjInvariant`
系のフィールド。いずれも構成時に `hConjInvariant` で discharge されるので条件付きでは
ないが、bundle が導出可能な命題を持つのは同じ債務。

⟹ Pf の残り特殊化債務は上表の順 4 以降 (**最大の塊 = (6.2)–(6.6) の Sibley 特殊化**)。

### ⚠ Pf 特殊化債務リストの再実測 (2026-07-26) — 大半が stale だった

上の「着手順」表は 07-16 の per-item note をそのまま並べたものだが、**1 件ずつ実測したところ
大半が既に閉じていた**。Isaacs (Ch.8/Ch.10) と BG (26 件中 22 件) で起きたのと同じ現象で、
07-16 以降の lane 作業が反映されていないだけだった。**この表を着手根拠にしてはいけない**。

**閉じている (直接 Read で確認、07-26)**

| 項目 | 07-16 label | 実物 |
|---|---|---|
| (1.5)(d) | partial「verbatim identity never stated」 | `inner_self_inv_smul_apply_one_smul_restrict_induce` (`InducedIrreducible.lean`) — 本の displayed computation そのもの |
| (1.5)(e) | partial「Frobenius 特殊化のみ」 | `inner_induce_conj_eq_zero_of_odd` (`CliffordDecomposition.lean:433`) — **一般 `H ⊴ L` + `|L|` 奇**で証明済 |
| (2.4) / (2.6)–(2.10.3) | partial + spec ×6 | 本日の `37171eade` / `4c2e8cd30` で完済 |
| (7.2)(b) | partial「equality clause は nowhere stated」 | `chiRho_norm_sq_eq_iff_mem_range` (`Hypothesis71.lean:445`) |
| (7.3) | partial「equality clause 未形式化」 | `chiRho_integral_eq_iff_constant_on_hCoset` (`Hypothesis71.lean:581`) |
| (7.10)/(7.11) | spec「kernel の冪零性が明示仮説」 | `FrobeniusFamily.isNilpotent_kernel` (`S09_FrobeniusCardG0LowerBound.lean:115`)。**Thompson の定理 (Isaacs Thm 6.24) は `Ch06_FrobeniusActions/KernelNilpotent.lean` に形式化済**なので、本と同じく内部で導出している |
| (8.11) | partial「Sylow-normalizer conjunct が無い」 | `S10_StructureSetup.lean:645` (first clause) + `:699` (全体) |
| (8.18) | spec「type-I 対のみ」 | `S10_MinimalSimpleStructure.lean:644/753` — docstring に「**type-uniform**, no type hypothesis, the book's statement verbatim」 |
| (13.8) | spec「T-side instance のみ」 | S-side の literal statement (issue 1041) が `AxiomsCheck.lean:10858` 群で sorry-free |
| App Huppert (Prop 1 / Lemma) | partial「単一 sorry」 | `Appendices/Huppert.lean` の sorry は **0** |
| App FeitSibley Hypotheses | partial「free Prop field」 | 2026-07-18 に de-opacify 済 (ファイル冒頭に経緯) |
| App Suzuki2Groups Def 1 | partial「opaque field」 | `IsSuzuki2Group` は `∃ x y, x,y ∈ involutions ∧ x ≠ y` と `∃ A, IsCyclic A ∧ ActsRegularlyOnInvolutions A` を明示的に書き下している |

**真に開いている (実測で確認)**

1. ~~**(6.2)–(6.6) の general-(6.1) 形** — 最大かつ唯一の深い残債~~ 【**2026-07-27 実測: (6.2)/(6.3)
   については stale**】。この項目は 07-26 時点で `six_two_general`
   (`S08_Theorem62_63_Standalone.lean:349`) と `six_three_of_six_two_oracle` (:383) が
   **(5.6) break-member oracle `h56` を明示仮説として取る**ことを根拠にしていた。しかし
   **issue 0153/0154 (2026-07-27 早朝に landing) が `h56` を除去済**: `six_two_of_imageData` /
   `six_three_of_imageData` (`S08_SixTwoThreeFromImageFamilies.lean:382/413`) が
   **oracle 無しの一般 `K` 版 (6.2)/(6.3)** で、両者とも `#print axioms` で
   `[propext, Classical.choice, Quot.sound]` = axiom-clean (本 tick で実測)。honest な仮説は
   書籍 Hypothesis (6.1) が要求する Hypothesis (5.2) そのもの — `InducedFamilyImageData` が
   (5.2.b) の τ と (5.2.d)/(5.2.e) の image family を担う。可約 break/member は
   `S07.xAdjoinStepW_k_general` (0154) が扱うので、旧記述の「一般の可解 `K` では induced member
   が可約になるため §10–§12 の muGrid/columnSum 機構と entangle する」はもはや当たらない
   (muGrid は §11 instance が (5.2.d) を供給する**一つの witness**に降りた)。
   ⚠ 残る (6.3) の特殊化は「repo が `H` 冪零を取る (書籍は `H/M` 冪零)」の 1 点のみ
   (`six_three_of_imageData` の docstring に注記; §11 consumer は `M = 1` ゆえ一致)。
   **残るのは (6.5)/(6.6) の general-`K` 形** (下記 §「(6.5)/(6.6)」節)。
2. **~~(5.3)(b)~~ / (5.8)** — **(5.3)(b) は 2026-07-27 にクローズ** (issue 0159)。
   `S06.toGeneralHypothesisOfInducedFamily` (`S06_CertainTypeSubcoherent.lean`) が
   **書籍 statement そのもの**を (4.6) 一般で述べる: 族 `𝒮 ⊆ {Ind_K^L θ | θ ∈ Irr K, H ⊄ Ker θ}`
   に対し `S07.GeneralHypothesis` を構成し、追加仮説は (5.2.a)/(5.2.c) のみ。sorry-free /
   axiom-clean (AxiomsCheck 登録済)。
   経緯: 旧注記は「固定 2 要素の `R` レコードは可変長 `R` を保持できないので**意図的な設計判断**」
   としていたが、[issue 0157](../../issues/closed/0157-five-seven-drop-unit-norm.md) の
   `S07.GeneralHypothesis` がサイズ自由な `OrthonormalCharacterImageFamily` を (5.2.d) に持つので
   障害は消えた。実装で追加になった鍵は **anchor の (4.6) 一般化**
   (`dadeICM_apply_eq_zero_of_mem_ticVdiffV` — 書籍の「by the definition of τ, `(φ − φ̄)^τ`
   vanishes on `V`」を `V ⊆ A₀` の Dade base-point 評価 + `ticVdiffV_not_mem_conjugatesOfSet_K`
   で証明) で、これが Sibley §8 / type-P §13 / type-II §12 の 3 個別証明の一般形にあたる。
   **残るのは (5.8)** のみ (μ-column 像の二分律/pin が FT 極大部分群 instance 止まり)。解析コアは
   `S05_SigmaTrichotomy.eq_smul_chiFam_column_of_vanishOnV` として `TICyclicHypothesis`
   レベルで完全に一般。
3. **(1.7)** — 一般 (a) の χᵢ 相異性と `n = |T:H|/e²` の計数のみ (effort S)。

**未再実測 (この pass では触っていない)**: (7.8) (7.9) (8.15) (9.7) (9.10) (9.11) (10.11)
(11.8)、App NearFields ×2、App Suzuki2Groups Lem 1。着手前に同じ手順で実測すること。

#### 再実測 第 2 ラウンド (2026-07-26) — 残り 11 件

| 項目 | 07-16 label | 実測 |
|---|---|---|
| (1.7) | partial | **本日クローズ** (`d4fe7db62`): 一般 (a) の相異性は `eq_of_induce_eq_induce_of_liesOver_of_inertia_eq` に既存、欠けていた計数 `n = [I_G(θ):H]/e²` を `sq_mul_card_mul_card_eq_card_inertia_of_induce_eq_nsmul_sum` として証明 |
| App NearFields `F_{r²,2}` | partial「具体 instantiation が無い」 | **stale**: `Peterfalvi.Appendices.ExceptionalNearField` (2026-07-22) に `exceptionalTwistData` (σ = halfFrobenius, χ = squareSignChar) + 乗法則 (book p.138) + 非可換性、AxiomsCheck 登録済 |
| App NearFields App C Prop 2 | partial「vacuous な classification が sorry」 | **stale**: `cyclic_index_two_nearField_classification` は実際の二分律 (field ∨ `F_{r²,2}`、`|Z(F*)| = r − 1`) を述べて axiom-clean |

**packaging 系 (内容は landed、単一の抽象 statement が無い)**: ~~(5.3)(b)~~ / (5.8) / (7.8) /
(7.9) / (8.15) / (9.7) / (9.10) / (9.11) / (10.11)。いずれも note が「assembly / re-assembly
work」と明記している。**(5.3)(b) は 2026-07-27 にクローズ済** — 書籍 statement を (4.6) 一般で
述べる `S06.toGeneralHypothesisOfInducedFamily` が landing した (issue 0159)。

#### (6.5)/(6.6) の general-`K` 化 (2026-07-27) — **(6.5) は (a)(b)(c) とも完了、残るは (6.6) coherence 半分**

新 leaf [`S08_SixFiveGeneral.lean`](../../OddOrder/Peterfalvi/S08_SixFiveGeneral.lean) (525 行、
`OddOrder.lean` / `AxiomsCheck.lean` に配線済、新 assert 9 本すべて axiom-clean)。

**完了したもの**

| 書籍 | 一般形 | 備考 |
|---|---|---|
| (5.3.a) | `nonempty_characterDifferenceImage_of_irreducible` | 任意の `τ` で。既約非実 `χ` の `τ(χ−χ̄)` が**符号の逆な**既約 2 元差になる |
| (5.2) for `𝒮(X)` | `InducedFamilyImageData.hypothesis` | (5.2.e) は transport でなく**導出** (書籍 (5.3.a) の論法) |
| (6.3.b) via (5.7) | `inducedKernelFamily_isCoherent_of_isMulCommutative_quotient` | `K/X` 可換 ⟹ 全 member の次数が `|L:K|` ⟹ (5.7) |
| (6.5)(a) 指数界 | `relIndex_le_of_not_isCoherent` | `six_three_of_imageData` の対偶 (`H = K`) |
| (6.5)(a) chief factor | `isChiefFactor_of_not_isCoherent` | 群論側 `isChiefFactor_of_relIndex_le_of_odd_dvd` は元から一般だった |
| (6.5)(b) | `exists_prime_isPGroup_of_not_isCoherent` | 核 `isPGroup_of_isNilpotent_of_isFrobeniusAction_abelianization` も元から一般 |
| (6.5)(c) | `not_dvd_sub_one_of_not_isCoherent` | `six_five_c_arith` + 「`|K:K′| < p²` ⟹ abelianization 巡回 ⟹ 冪零なら可換」 |

**副産物: carrier を書籍準拠に直した**。`InducedFamilyImageData` に **`tau_apply_one`** を追加 —
書籍 (5.2.b) の値域は `ℤ[Irr G, G^#]` (**1 で消える**) なのに repo は `ℤ[Irr G]` しか課して
おらず、carrier が書籍より弱かった。§11 の Dade witness (唯一の constructor) は
`dadeIntegralCharacterMap_apply_one_eq_zero` でそのまま discharge。この節が無いと (5.3.a) の
「符号が逆」が出ない (`τ(χ−χ̄) = μ + ν` を排除できない)。

**残っている特殊化 (実測、着手前に再確認すること)**

1. ~~**(6.6) coherence 半分**~~ — **2026-07-27 完済**
   (**[issue 0155](../../issues/0155-pf-six-six-general-kernel.md)**)。
   `S08_SixSixGeneral.xSet_isCoherent_of_irreducible_X` が一般 `K`・**任意の τ** (Dade 非依存) で
   landing、axiom-clean・AxiomsCheck 登録済:

   > `K` が `p` 群 (`p` 奇素数, `|L:K|` と互いに素) かつ `Z ⊆ Z(K)` 正規なら
   > `𝒳 = 𝒮 − 𝒮(Z) ⊆ Irr L` は coherent

   仮説は書籍 (6.4)/(6.5) の文脈そのもの (Hypothesis (5.2) は `InducedFamilyImageData` が担う =
   書籍が (6.1) で仮定するもの)。以下は完成に至る途中経過の記録:
   `S08_SixSixGeneral.lean` (1026 行) に一般 `K` で landing 済:
   `xSet = 𝒮 − 𝒮(Z)` と (5.2) 一式 / 書籍 p.32 の算術 2 本 (次数 `|L:K|·p^k`・[Is] Cor 2.30 の
   中心界) / 次数平方和恒等式 / 最小次数 base block `𝒮₀` / **τ-general X-chain fold**
   (`xSet_isCoherent_of_adjoinSteps`) / **base coherence** (`xBaseBlock_isCoherent`) /
   **per-step adjoining** (`xAdjoinStep_of_degreeRatios`) / 次数比ブリッジ / `hSgen`。
   新 assert 計 15 本すべて axiom-clean。
   **⚠ この項目の教訓**: 「τ-general engine の一般化が本丸」という当初の見立ては**外れ**だった。
   実測すると (i) chain fold `coherentOfPairChainCover` は**元から τ-general** で Sibley 版は
   Dade に pin しただけの包装、(ii) base coherence は **(5.7) を部分族に当てるだけ**、
   (iii) 書籍 p.32 の可除性論法の算術 5 本 (`2a < ∑ deg²` の producer
   `normalizedDegreeGap_of_natDegreeSumPrimePowerGap` を含む) は**抽象群 + 素の数値データ**で
   元から一般、(iv) 唯一の真に Dade 依存な `XAdjoinStepInput`/`xAdjoinStep` 層は、
   0154 の `S07.xAdjoinStepW_general` へ**直結して迂回**できた (移植不要)。
   ⟹ **残っていた「各 step で accumulator を有限列挙し tail-set を作って可除性を通す」結線**も
   2026-07-27 に完了 (`pairUnion_closedUnderConjugate` / `xBaseBlock_nonempty` /
   `natDegree_lt_of_xBaseBlock_anchor_of_notMem` の 3 構造補題 + 本体)。

   **⚠ Sibley 版 (`S08_CoherenceBasic.Xset_isCoherent_of_irreducible_X`) は「その特殊化」に
   置き換えられない** (2026-07-27 実測、当初の完了条件は誤り)。理由: 一般版は Hypothesis (5.2) を
   `InducedFamilyImageData` (= (5.2.b) の τ + **𝒮 の全 member** に対する (5.2.d)/(5.2.e) の像族)
   として取るが、`SibleyDadeHypothesis` はこれを**持っていない** — Sibley 側は Dade 写像から
   **既約 member についてのみ**その場で (5.2.d) を作る (`dadeOrthonormalCharacterImageFamilyOfDiff`)。
   𝒮 の可約 member 用の像族 (μ-grid 列族) は §12/§13 でしか構成されない。
   ⟹ Sibley 版は書籍 (6.6) より**弱い仮説**を持つ別定理であって、一般版の instantiation ではない。
   両者を 1 本にまとめるには「既約な部分族については (5.2.d)/(5.2.e) を (5.2.b) から**導出**する」
   経路が要る — 材料は `characterDifferenceImage_of_irreducible` (τ だけから (5.3.a) の符号付き
   2 元対を作る、`S08_SixFiveGeneral`) で、欠けているのは
   **`CharacterDifferenceImage → OrthonormalCharacterImageFamily` の変換**
   (`R(χ) = {ε·μ, −ε·ν}`)。これは重複解消 (アーキテクチャ) の課題であって書籍被覆のギャップでは
   ないので、別 issue に切り出す。
2. ~~**repo の (5.7) が member の既約性を要求する**~~ — **2026-07-27 完済**
   ([issue 0157](../../issues/closed/0157-five-seven-drop-unit-norm.md))。
   `S07.coherent_of_constant_degree_general` が書籍強度 (Hypothesis (5.2) + 等次数のみ) で landing。
   ⚠ 根本原因は (5.7) でなく **carrier `S07.Hypothesis` の (5.2.d) が 2 元固定**だったこと
   (2 元 ⟹ `‖τ(χ−χ̄)‖² = 2` ⟹ 等長性から `‖χ‖² = 1` が全 member に強制される)。
   書籍準拠の `S07.GeneralHypothesis` (R(χ) のサイズ自由) を新設し、旧 carrier は
   `toGeneralHypothesis` でその特殊化として接続。旧 (5.7) は `m = 1` の 1 行特殊化。
3. ~~**(6.3)/(6.5) が `K` 冪零を取る**~~ — **2026-07-27 完済**
   ([issue 0158](../../issues/closed/0158-six-three-quotient-nilpotent.md))。
   `six_three_descent` / `six_three_of_six_two_oracle` /
   `normal_central_of_maximal_normal_below` がいずれも**商の冪零性**
   (`Group.IsNilpotent (↥H ⧸ M.subgroupOf H)` = 書籍 (6.3)(a)) で成立するようになった。
   冪零性の使用は 1 箇所だけで、そこも `H/B` の冪零性しか使っていなかった
   (`M ≤ B` ゆえ `H/M ↠ H/B`)。下流 consumer は全て無変更。
   【**2026-07-27 実測**: 実体は `six_three_descent` の `[Group.IsNilpotent ↥H]`
   (`S08_Theorem62_63_Standalone:120`)。冪零性の使用は
   `normal_central_of_maximal_normal_below` 1 箇所のみで、そこも内部では **`H/B` の冪零性**
   しか使っていない (`M ≤ B` なので書籍の `H/M` 冪零から出る)。
   ⟹ 数学的な障害は無く、binder で商型の `Group` instance を作るのに
   `(B.subgroupOf H).Normal` を明示 binder で渡す必要がある点だけが手当て箇所。
   **[issue 0158](../../issues/0158-six-three-quotient-nilpotent.md)** に手順を記録。】

#### (10.11) の実測 (2026-07-27) — **両主張とも書籍強度、AxiomsCheck の注記が stale だった**

書籍 p.63: 「(8.8) の場合 (b) が成り立つとする。すると `|W₁|` と `|W₂|` は素数。さらに `M` が
Type II の極大部分群なら、仮説 (9.2)/(9.5) の記法で `H` は位数 `p^q` の初等アーベル群
(`p = |W₂|`) で、仮説 (9.5) の集合 `𝒮` は coherent」。

| 主張 | repo | 状態 |
|---|---|---|
| 第 1 (`|W₁|`, `|W₂|` 素数) | `S12.theorem88_caseB_prime_orders` | **sorry ゼロ・axiom-clean** |
| 第 2 (型 II: `H` 初等アーベル `p^q`, `p = |W₂|`, `𝒮` coherent) | `S11.typeII_sSet_coherent` | (a)+(b) を書籍どおり結論に持つ、axiom-clean |

⚠ AxiomsCheck に「`exists_chiefFactorData` は still-`sorry`'d な `theorem88_caseB_prime_orders`
を cite するので axiom-clean でない」という注記が残っていたが **stale**。実測すると
`S12_MaximalIII_IV_V.lean` は sorry ゼロで、`theorem88_caseB_prime_orders` は
`caseB_typeP_prime_W1` から実証明されている。両方を AxiomsCheck に登録し (3 axiom)、注記を訂正。

#### (9.11) の実測 (2026-07-27) — **書籍強度、gap なし (docstring が stale だった)**

書籍 p.54-57 の (9.11) 「`𝒮(H₀C')` は `τ` について coherent」の case (9.7.a) は
(9.11.1)-(9.11.8) の squeeze (最大 coherent 部分族 `𝒮₂` の maximality 論法) を要する深い証明。

実測: repo の case-(a) 開発 `S11_NineEleven*` は **12 leaf / 約 8,700 行がすべて sorry ゼロ**で、
その終点 `S11.nineEleven_coherent_A0` は **refuter 仮説を取らない** (標準の Hypothesis (4.6)
plumbing のみ)。AxiomsCheck 登録済。

⟹ (9.11) は書籍強度で無条件。`S13_Orthogonality.coherent_sOf_H0C` の docstring が
「caseA の refuter が唯一の sorried-cite」と書いていたのは **landing 前の記述が残っていたもの**
で、本 tick で訂正した ([[feedback-fix-stale-docstrings-on-sight]])。

#### (9.10) の実測 (2026-07-27) — **`U` の巡回性を結論に露出**

書籍 p.54: 「`𝒮(H₀C')` が `HC` の線型指標から誘導された次数 `qu` の既約指標を含まないなら、
場合 (9.7.b) が成り立ち、`H̄U` は核 `H̄` の Frobenius 群で、**`U` は位数 `(p^q−1)/(p−1)` の
巡回群**。さらに `M` が Type II なら `HU` は核 `H` の Frobenius 群」。

repo の `exceptional_case_frobenius_realization{,_of_trigger}` は位数
`chars.u = (p^q−1)/(p−1)` は結論に持っていたが、**`IsCyclic U` が露出していなかった**
(型 II ブランチの中で `hUcyc` として導出はしていた)。`C = ⊥`
(`caseB_no_irreducible_forces_C_bot`) から来るので型 II に限らず無条件に成り立つ ⟹ hoist して
結論の第 3 conjunct に追加。下流 (`S12_TypeIICrossIsometryPair`) は射影を `.2.2` → `.2.2.2` に
更新するだけ。

#### (9.7)(a) の実測 (2026-07-27) — **書籍の `a`-形へ鋭化 (issue 0152 closed)**

書籍 p.51 は `a = |U : C_U(H₁)|` を定義して「`a ∣ p−1`、`U/C_U(H_i)` は全ての `i` で位数 `a` の
巡回群、`U` は位数 `a` の巡回群 `q−1` 個の直積の部分群」と主張する。repo には `a` が現れず
`u ∣ (p−1)^{q−1}` という**弱い形**しか無かった。

`S11.caseA_blockScalarFacts` として書籍形を証明:
`∃ a, a ∣ p − 1 ∧ chars.u ∣ a^{q−1}`。書籍が `a` で済むのはブロックが `W₁`-共役だからで、
その入力 (`caseA.Hpart_orbit`) は repo に既に在った。筋: `w_j` が `Ū` を正規化 ⟹ 各点固定化群が
共役 ⟹ 指数一致 ⟹ ブロックスカラー像の位数一致 ⟹ (`𝔽_p^×` が巡回ゆえ) 像一致。
既存 `caseA_exists_blockScalarRatioEmbedding` は射影として残したので下流不変。

副産物の generic 部品 (12 本、すべて axiom-clean): 有限巡回群の部分群一意性
(`Subgroup.eq_of_card_eq_of_isCyclic`)、`MulAut` 作用の各点固定化群とその共役性
(`ptStabOfMulAut` 族)、線表現の scalar character の同変移送。

#### (8.15) の実測 (2026-07-26) — **claim 1 の `A₁` 版を型一様化**

書籍 p.49 (8.15): 「`M` を極大部分群、`A = A₀(M)`, `A(M)` **or** `A₁(M)` とする。すると
`M = N_G(A)` で、`L = M`, `H(a) = R(a)` として仮説 (2.2) が成り立つ」— **型の制限なし**。

repo の claim 1 は `A(M)` / `A₀(M)` については型一様な producer が在る
(`dadeSupportHypothesisData_typePACore{,0}` は `IsTypeP` 以外の型仮説を持たない = 型 II/V 込み、
型 I は `dadeSupportHypotheses_typeI`)。しかし **`A₁(M)` 版だけが
`dadeSupportHypotheses_typeP` の中に `IsTypeP1` 仮説付きで埋もれていた**。

実測すると、その `hA1` ブロックは `hP1` も `data` も使っておらず、
`A1_eq_sigmaSharp` (全型) + σ-sharp Dade engine + `A1_conj_mem` (全型) だけで回っていた
⟹ 独立した型一様定理 `dadeSupportHypothesisData_A1` として抽出
(`S10_MinimalSimpleBasic.lean`)。`dadeSupportHypotheses_typeP` はそれを呼ぶだけに縮小。
AxiomsCheck 登録済、3 axiom。

`A₀(M)` / `A(M)` は型ごとに集合の定義自体が違うので型指標のままが正しい。

#### (7.8) の実測 (2026-07-26) — **書籍強度、gap なし**

書籍 p.40 の (7.8) を PDF ページ画像 (`references/peterfalvi/pages/peterfalvi-p040.png`) で
確定し、repo と 1 対 1 で照合した (Peterfalvi の pdftotext は当日まで壊れており、数式は今も
画像でしか読めない)。

| 書籍 | repo |
|---|---|
| 仮説: (7.6) + `S = T − {Ind 1_H}` coherent + `ν : Z[S] ≅ Z[Irr G]` + `ζ ∈ S ∩ Irr L`, `ζ(1) = e` | `Hypothesis78` (`QuadraticTerm.lean:59`) |
| (a) `S^ν ⊥ 1_G`; ∃`a ∈ ℤ`, `Γ ⊥ S^ν ∪ {1_G}` with `β = 1_G − ζ^ν + a Σ_{φ∈S} φ(1)/(e‖φ‖²) φ^ν + Γ` | `BetaDecomp` (:912) の 5 フィールドが逐語一致。`weightedNuSum` (:898) = `Σ_{i≠ind1H} ζ_i(1)/(ζ_0(1)‖ζ_i‖²) ζ_i^ν` で `ζ_0(1) = e` ゆえ書籍の和と同一。**証明済** (`Cert.betaDecompOfFacts` / `betaDecompOfDade`) |
| (b) `e ≤ (h−1)/2` ⟹ `‖ζ^{νρ}‖² ≥ 1 − e/h` **かつ** `‖Γ‖² ≤ e − 1` | `smallIndex` (:1164) = `2e + 1 ≤ h` (⟺ 書籍の仮定)。前半 = `zetaNuRhoNormSq_ge_of_facts`、**後半も在る** = `gammaNormSq_le_of_normQuadraticCorrection_eq` (`CoherenceFormula.lean:128`) |
| (c) `χ ⊥ S^ν` ⟹ `χ^ρ(x) = (β,χ)` on `A` かつ `‖χ^ρ‖² = (|A|/|L|)(β,χ)²` | `chiRho_eq_inner_beta` (フィールド、`hypothesis78OfDade` が discharge) + `chiRho_norm_sq_eq_card_ratio_mul` |

⟹ **(a)(b)(c) すべて書籍強度で sorry-free**。残る差は「書籍の 1 番号 = Lean の 1 定理」に
なっていない (仮説 record + 結論 record + 名前付き定理に分解されている) という**体裁のみ**で、
数学的な gap ではない。docstring の `**Peterfalvi (7.8.x)**` 記法で追跡可能。
⚠ この照合で docstring 2 件の陳腐化を訂正した (「(7.8.c) は未形式化」「`BetaDecomp` は
future proof の target」— どちらも既に証明済)。

**真に強さが違うもの (packaging でない)**:

1. **(6.2)–(6.6)** — general-(6.1) 形が (5.6) break-member oracle `h56` を仮説に取る (前節参照)。
2. ~~**(11.8)**~~ — **stale と判明 (2026-07-26 実測)**。「repo は存在形」は誤りで、普遍形
   `S13.zeta_residual_not_orthogonal_H0C_of_refuter` (`S13_Orthogonality.lean:1107`) が
   **`{ζ} (hζS : ζ ∈ inducedFamily M) (hζirr) (hζdeg : ζ 1 = w₁)` を任意に取る**書籍どおりの
   `∀ ζ` 形で既に在る (docstring も「(10.2)/(10.3) character parameters built around the
   **given** `ζ` (the `∀ ζ` book form)」と明記)。存在形
   `exists_zeta_residual_not_orthogonal_H0C_of_refuter` (:1199) はそれを
   `exists_zeta_in_inducedFamily_degree_w1` で instantiate しただけの**下流向け packaging**で、
   docstring 自身が「The book statement is the `∀` one … this is only the existential packaging
   its downstream happens to want」と書いている。⟹ **(11.8) は書籍強度**、特殊化債務なし。
   (書籍の `ζ ∈ S(HC)` は「`ζ ∈ S` かつ既約かつ次数 `w₁`」を含意するので、repo の仮説は
   書籍より弱いか同等。)
3. **App Suzuki2Groups Lemma 1** — (b) は `QuadraticExtensions.lean` で完全形式化済 (2026-07-18)、
   **(a) (c) (d) が未**。Higman 論文の実内容。

---

## 📍 2026-07-26 終了時点の frontier (実測ベース)

本日の一連 (22 commits) で **Isaacs 完済 / BG は残り 2 件 / Pf は残り 1 系統 + packaging** まで
絞り込まれた。以下が着手前に実測済みの現状。

### 閉じたもの (本日)

| 項目 | commit |
|---|---|
| Pf (2.4) + (2.6)–(2.10.3) — (2.4.a) を仮説・フィールドから全除去 | `37171eade` `4c2e8cd30` |
| Pf (1.7) — 構成要素計数 `e²·|S|·|H| = |I_G(θ)|` | `d4fe7db62` |
| **BG Lemma 2.7 (a)(b)** — `(ℤ/q)²` の `(ℤ/p)²` への忠実作用 | `51f977137` `86be397b5` (issue 0150 closed) |
| BG Theorem C の `p, q`-抽象形 | `62e84cf0f` |
| **BG App.C Remark (V)** — (A) の下で `p, q` は奇と仮定してよい | `c25d14a2e` |
| Pf の 11 系統が stale と判明 ((7.2)(b)/(7.3)/(7.10)(7.11)/(8.11)/(8.18)/(13.8)/Huppert/FeitSibley/Suzuki2Groups Def 1/NearFields ×2) | `7ce7b600e` `f39d37cfc` |

### 残っているもの

1. **Pf (6.2)–(6.6) の general-(6.1) 形** — 唯一の深い項目。`six_two_general` /
   `six_three_of_six_two_oracle` が **(5.6) break-member oracle `h56`** を明示仮説に取る。
   一般の可解 `K` で induced member が可約になり §10–§12 の muGrid/columnSum と entangle
   (closed issue 2022 の分析どおり)。consumer では sorry-free に discharge 済で FT は閉じている。
2. ~~**BG App.C Remark (II)**~~ — **2026-07-26 完了** (`OddOrder/BG/AppC_SL2Example.lean`)。
   Hypothesis (B) の `p, q, G`-抽象形 `HypothesisBAbstract` を立て、書籍どおりの
   `p = 2`, `G = SL(2, 2^q)`, `σ(P)` = 上三角ユニポテント, `σ(U)` = 分裂トーラス,
   `y = [[0,1],[1,1]]`, `Q = ⟨y⟩` (位数 3) が (A)+(B) を満たすことを axiom-clean で構成。
   残っていた穴「抽象 (B) ⟹ `hrel`」(= Lemma C.3 の群論的内容) も **2026-07-26 に完了**
   ([issue 0151 closed](../../issues/closed/0151-appc-lemma-c3-abstract-hypothesis-b.md))。
   C.3 chain 6 file (~4,400 行 / 197 宣言) を Peterfalvi §16 から BG へ物理移設して
   `(p, q, G)` 化し、`theoremC_of_hypothesisBAbstract` として書籍 Theorem C を
   **仮説まで含めて完全一致**させた (奇性も `G` の有限性も仮定しない)。
   `theoremC_sl2` が Remark (II) の witness に実際に適用されており、(B) が空虚でないことも
   機械検証済。
3. ~~**BG §16 tamely imbedded**~~ — **2026-07-26 完了 (issue 8005 closed)**。着手前実測で
   「issue の checklist が stale」と判明: Thm I 復元は済、Thm II (Tii)(a)–(e)+(Tiii) も
   statement-first でなく**実証明**で済んでおり `theoremII_tamelyImbedded` は unconditional
   だった。**本物の残り gap は「書籍は `X = A(M)` **or** `X = A₀(M)` の両方を主張するのに
   `A(M)` しか covering していない」**点で、これを本 tick で解消
   (`escapingSharpSet_a0Set_eq_aSet` + clause (c) の `hatMsigma` 一般化 +
   `aSet_subset_A0Set` を Peterfalvi から BG へ移設)。書籍括弧書き `Hᵢ ≤ M'ᵢ` も field 化。
4. ~~**Pf App Suzuki2Groups Lemma 1 (a)(c)(d)**~~ — **2026-07-24 完済** (issue 0148 campaign)。
   `Appendices/Suzuki2Groups.lean` は宣言ゼロの re-export hub になり、Lemma 1(a)(c)(d) /
   Lemma 2(a)-(c) / Prop 1 / Prop 2 / Theorem (e) 順方向まで全て実形式化済 (2026-07-28 実測)。
5. **packaging 8 件** — (5.8) (7.8) (7.9) (8.15) (9.7) (9.10) (9.11) (10.11)。
   内容は landed、単一の抽象 statement が無いだけ。⚠ **(5.3)(b) は 2026-07-27 に本項から外れた**
   — `S06.toGeneralHypothesisOfInducedFamily` が書籍 statement を (4.6) 一般で述べる
   (issue 0159)。「固定 2 要素 `R` レコードの設計上の制約」という旧理由は
   `S07.GeneralHypothesis` (issue 0157) で無効化済。
6. **低優先繰延** — BG App.C Rem (IV) / Prob 1 (文献引用・open problem)、
   Pf App C Prop 1 の Q₈ Brauer–Suzuki (issue 0147、repo 唯一の実 sorry)。

### ⚠ 2026-07-28 訂正 — 上の 1–6 は Part II を丸ごと落としていた

**この frontier 節は Part I 側の表からしか項目を拾っておらず、`### Pf App: Suzuki`
(L825–L862) に並ぶ Part II の「未」12 行が 1 つも入っていなかった。** そのため
「残りは packaging とごく少数」という誤った全体像になり、実際 2026-07-26 に Ch.II
(Theorem B) が完結した直後、**文書順の次である Ch.III に入らないまま** Part I の
一般化トラック (issues 0151–0161) へ pivot している (`git log`:
`OddOrder/Peterfalvi/Appendices/Suzuki/` への commit は 07-26 05:30 `55ec6f681` が最後、
その 9 分後 `25521c85c` から別トラック)。

**Part II の実際の frontier (2026-07-28 実測)** — Ch.I / Ch.II は完済、以下が未着手:

| 書籍 | 内容 | 状態 |
|---|---|---|
| ~~**Ch.III §1 Thm C**~~ | `Q` は 2-群 (`Q₁ = 1`) | **✅ 2026-07-29 完成** ([issue 0162 closed](../../issues/closed/0162-pf-part2-ch3-theorem-c.md)、`7175e6a73`) — `SecondCaseHypothesis.{Q1_eq_bot, isPGroup_two_Q}`、axiom-clean |
| **Ch.III §1 Prop** | 三分律 (a) `S = Q₀` / (b) type A / (c) type B | **未 = 現 frontier** — [issue 0163](../../issues/0163-pf-part2-ch3-s1-trichotomy.md) |
| Ch.III §2 Prop | `st` 位数 5 の場合 `(SK) ∪ (SKtS)` が部分群 | 未 |
| Ch.III §3 Prop | (C2) の下での明示モデル | 未 |
| Ch.IV §1 (H1)–(H6) + Lemma | `f, g, h` の BN-pair 形式と一意性 | 未 |
| Ch.IV §2 Prop | `D` が `(Q/Q₀)^#` に不動点なく作用する場合 | 未。⚠ OCR 欠落ページ (`[MISSING_PAGE_EMPTY:136]`) — PDF 必須 |
| Ch.IV §3 Prop + Cor 1, 2 | `PSU(3,q)` の特徴付け | 未。`PSU(3,q)` の構成が要る (mathlib に無し) |
| Ch.IV §4 | `V ≠ W` の場合 = Theorem A の最終組立 | 未 |

⟹ **Part II の残りが Peterfalvi 側の主 frontier**。上の 1–6 と並ぶ独立項目として扱う。

### 見積もりについての教訓

07-16 survey の effort ラベルは実測と大きくずれる。本日の実績:
**BG Lem 2.7** = "M" 見積もり → 実際に数時間 (妥当) /
**Theorem C 抽象化** = "M" 見積もり → **実際は 3 行** (step lemma が元から抽象だった) /
**Rem (V)** = "S" → そのとおり。**着手前の実測が毎回効いている**。