Wholiver · Wholiver · Jun 2, 2026 · Jun 2, 2026
diff --git a/.gitignore b/.gitignore
@@ -1,3 +1,71 @@
-.workbuddy/
-.codebuddy/
+```markdown
+# Compiled and build artifacts
+*.pyc
+__pycache__/
+*.o
+*.obj
+*.so
+*.dll
+*.exe
+*.class
+*.out
+
+# Dependencies
+.venv/
+venv/
+node_modules/
+dist/
+build/
+target/
+.gradle/
+.mypy_cache/
+.pytest_cache/
+
+# Logs and temp files
+*.log
+*.tmp
+*.swp
+*.swo
+
+# Environment
+.env
+.env.local
+.env.*
+
+# Editors
+.vscode/
+.idea/
+
+# System files
 .DS_Store
+Thumbs.db
+
+# Coverage
+coverage/
+htmlcov/
+.coverage
+
+# Compressed files
+*.zip
+*.gz
+*.tar
+*.tgz
+*.bz2
+*.xz
+*.7z
+*.rar
+*.zst
+*.lz4
+*.lzh
+*.cab
+*.arj
+*.rpm
+*.deb
+*.Z
+*.lz
+*.lzo
+*.tar.gz
+*.tar.bz2
+*.tar.xz
+*.tar.zst
+```
diff --git a/modules/error_prevention.md b/modules/error_prevention.md
@@ -285,6 +285,102 @@ This module defines concrete error prevention rules, common pitfalls, and preven
 
 ---
 
+## 9. Complex Analysis Error Prevention
+
+### Rules
+
+1. **Analyticity domain check**: Before applying Cauchy's theorem or residue calculus, verify the function is analytic on and inside the contour except at isolated singularities.
+2. **Branch cut handling**: For multi-valued functions ($\log z$, $z^\alpha$, $\sqrt{z}$), explicitly define the branch cut and ensure the contour does not cross it.
+3. **Singularity classification**: Correctly classify singularities as removable, pole (with order), or essential before applying residue formulas.
+4. **Residue calculation verification**: For a pole of order $n$ at $z_0$: $\text{Res}(f, z_0) = \frac{1}{(n-1)!}\lim_{z\to z_0}\frac{d^{n-1}}{dz^{n-1}}[(z-z_0)^n f(z)]$. Verify the order before differentiating.
+5. **Contour orientation**: Standard positive orientation is counterclockwise. Reversing orientation changes the sign of the integral.
+6. **Jordan's lemma conditions**: For $\int_{-\infty}^\infty f(x)e^{iax}dx$ with $a > 0$, the semicircular arc contribution vanishes only if $|f(z)| \to 0$ uniformly as $|z| \to \infty$ in the upper half-plane.
+7. **Argument principle application**: $\frac{1}{2\pi i}\oint_\gamma \frac{f'(z)}{f(z)}dz = N - P$ where $N$ = zeros, $P$ = poles (counted with multiplicity). Ensure $f$ has no zeros or poles ON the contour.
+8. **Conformal mapping boundary correspondence**: When using conformal maps, verify that boundary points map correctly and the orientation is preserved.
+
+### Common Pitfalls with Examples
+
+| Pitfall | Wrong | Correct |
+|---|---|---|
+| Applying residue theorem across branch cut | Integrating $\log z$ around a circle centered at origin without accounting for branch cut | Define branch cut (e.g., negative real axis) and use keyhole contour |
+| Miscounting pole order | $\frac{1}{(z-1)^2(z+1)}$ has a simple pole at $z=1$ | Pole at $z=1$ is order 2; pole at $z=-1$ is order 1 |
+| Forgetting $2\pi i$ factor | $\oint \frac{dz}{z} = 1$ around unit circle | $\oint \frac{dz}{z} = 2\pi i$ |
+| Wrong residue for essential singularity | Using pole formula for $e^{1/z}$ at $z=0$ | Use Laurent series: $\text{Res}(e^{1/z}, 0) = 1$ (coefficient of $1/z$) |
+
+### Prevention Strategy
+
+1. **Draw the contour and mark all singularities** before computing
+2. **Identify branch cuts explicitly** for multi-valued functions
+3. **Classify each singularity** before choosing a residue calculation method
+4. **Check contour orientation** and apply sign accordingly
+5. **Verify decay conditions** for infinite contours (Jordan's lemma, estimation lemma)
+
+---
+
+## 10. Topology Error Prevention
+
+### Rules
+
+1. **Open set definition dependence**: "Open" is relative to the topology. Always specify which topology is being used (standard, discrete, indiscrete, subspace, quotient, etc.).
+2. **Continuity definition**: $f: X \to Y$ is continuous iff $f^{-1}(U)$ is open in $X$ for every open $U \subseteq Y$. Do NOT assume "$f$ maps open sets to open sets" (that's an open map, a different concept).
+3. **Compactness vs closed+bounded**: In $\mathbb{R}^n$, compact $\iff$ closed and bounded (Heine-Borel). In general metric/topological spaces, this equivalence FAILS.
+4. **Connectedness verification**: A space is connected if it cannot be written as a union of two disjoint non-empty open sets. Path-connected $\implies$ connected, but the converse fails (e.g., topologist's sine curve).
+5. **Hausdorff property**: A space is Hausdorff if any two distinct points have disjoint neighborhoods. $\mathbb{R}^n$ is Hausdorff, but quotient spaces may not be.
+6. **Homeomorphism requirements**: A homeomorphism must be bijective, continuous, AND have a continuous inverse. Bijective + continuous is NOT sufficient.
+7. **Subspace topology**: For $A \subseteq X$, open sets in $A$ are intersections $U \cap A$ where $U$ is open in $X$. Do not assume $A$ inherits properties like compactness or connectedness without verification.
+8. **Product topology basis**: Open sets in $\prod X_i$ are unions of products $\prod U_i$ where each $U_i$ is open in $X_i$ and $U_i = X_i$ for all but finitely many $i$ (for infinite products).
+
+### Common Pitfalls with Examples
+
+| Pitfall | Wrong | Correct |
+|---|---|---|
+| Assuming closed+bounded $\implies$ compact | "The set $(0,1)$ is bounded, so it's compact" | $(0,1)$ is NOT compact (not closed); $[0,1]$ IS compact in $\mathbb{R}$ |
+| Confusing continuity with open mapping | "$f(x)=x^2$ is continuous, so it maps open sets to open sets" | $f((-1,1)) = [0,1)$ which is NOT open |
+| Assuming connected components are open | "Connected components are always open" | Only true in locally connected spaces; false in general |
+| Quotient space Hausdorff assumption | "The quotient of a Hausdorff space is Hausdorff" | Generally false; requires additional conditions |
+
+### Prevention Strategy
+
+1. **State the topology explicitly** at the start of any topological argument
+2. **Use the correct definition of continuity** (preimage of open is open)
+3. **Distinguish between properties that hold in $\mathbb{R}^n$ vs general spaces**
+4. **Construct standard counterexamples** (topologist's sine curve, line with two origins, etc.) to test intuition
+5. **Verify all homeomorphism conditions**, especially continuity of the inverse
+
+---
+
+## 11. Number Theory Error Prevention
+
+### Rules
+
+1. **Divisibility definition**: $a \mid b$ means $\exists k \in \mathbb{Z}$ such that $b = ak$. Note that $0 \mid 0$ is true (since $0 = k \cdot 0$ for any $k$), but $0 \mid b$ for $b \neq 0$ is false.
+2. **Modular arithmetic domain**: Congruences $a \equiv b \pmod{n}$ are defined for integers. Extending to rationals or reals requires careful interpretation.
+3. **Prime factorization uniqueness**: The Fundamental Theorem of Arithmetic applies to $\mathbb{Z}^+$ (positive integers). In other rings (e.g., $\mathbb{Z}[\sqrt{-5}]$), unique factorization may fail.
+4. **Fermat's Little Theorem conditions**: $a^p \equiv a \pmod{p}$ for prime $p$ and any integer $a$. The form $a^{p-1} \equiv 1 \pmod{p}$ requires $\gcd(a,p) = 1$.
+5. **Euler's totient function**: $\phi(n)$ counts integers in $\{1, 2, \ldots, n\}$ coprime to $n$. For $n = p_1^{e_1} \cdots p_k^{e_k}$: $\phi(n) = n \prod_{i=1}^k (1 - \frac{1}{p_i})$.
+6. **Chinese Remainder Theorem**: The system $x \equiv a_i \pmod{n_i}$ has a unique solution mod $\text{lcm}(n_1, \ldots, n_k)$ if the moduli are pairwise coprime. Without coprimality, solutions may not exist or may not be unique.
+7. **Quadratic reciprocity**: For odd primes $p, q$: $\left(\frac{p}{q}\right)\left(\frac{q}{p}\right) = (-1)^{\frac{p-1}{2} \cdot \frac{q-1}{2}}$. Check that both primes are odd before applying.
+8. **Infinite descent validity**: Proof by infinite descent requires showing that assuming a solution exists leads to a smaller positive integer solution, contradicting well-ordering of $\mathbb{N}$.
+
+### Common Pitfalls with Examples
+
+| Pitfall | Wrong | Correct |
+|---|---|---|
+| Applying FLT without coprimality | "$2^4 \equiv 1 \pmod{4}$" (using $a^{p-1} \equiv 1$ with $p=2$) | FLT in form $a^{p-1} \equiv 1$ requires $\gcd(a,p)=1$; here $\gcd(2,4) \neq 1$ |
+| Assuming unique factorization in all rings | "$6 = 2 \cdot 3 = (1+\sqrt{-5})(1-\sqrt{-5})$ shows non-uniqueness in $\mathbb{Z}$" | Non-uniqueness occurs in $\mathbb{Z}[\sqrt{-5}]$, NOT in $\mathbb{Z}$ |
+| CRT without checking coprimality | Solving $x \equiv 1 \pmod{4}$, $x \equiv 2 \pmod{6}$ assuming unique solution mod 24 | $\gcd(4,6) = 2 \neq 1$; check consistency first ($1 \not\equiv 2 \pmod{2}$), no solution exists |
+| Quadratic reciprocity for $p=2$ | Applying reciprocity formula with $p=2$ | Handle $p=2$ separately: $\left(\frac{2}{q}\right) = (-1)^{\frac{q^2-1}{8}}$ |
+
+### Prevention Strategy
+
+1. **Verify primality** before applying prime-specific theorems
+2. **Check gcd conditions** for modular arithmetic results
+3. **Specify the ring** when discussing factorization
+4. **Test small cases** numerically before generalizing
+5. **Use the extended Euclidean algorithm** to verify Bezout coefficients
+
+---
+
 ## Cross-Domain Prevention Checklist
 
 Before finalizing any mathematical answer, run through this summary checklist:
@@ -297,3 +393,6 @@ Before finalizing any mathematical answer, run through this summary checklist:
 - [ ] **Calculus**: L'Hôpital conditions checked, $+C$ added, convergence verified, limits checked
 - [ ] **Linear Algebra**: Dimensions matched, eigenvalues verified by substitution, determinants double-checked
 - [ ] **Abstract Math**: Definitions complete, quantifier order correct, constructions well-defined, all equivalence properties checked
+- [ ] **Complex Analysis**: Analyticity verified, branch cuts handled, singularities classified, contour orientation correct
+- [ ] **Topology**: Topology specified, continuity definition correct, compactness/connectedness properly verified
+- [ ] **Number Theory**: Primality verified, gcd conditions checked, ring specified for factorization
diff --git a/modules/output_templates.md b/modules/output_templates.md
@@ -444,6 +444,139 @@ n-1, n, n+1 是三个连续整数，因此：
 
 ---
 
+## 模板 G：研究级问题深度分析 (Research-Level Deep Analysis)
+
+### 何时使用
+- 用户明确提出研究级别的数学问题
+- 问题涉及多个数学分支的交叉
+- 需要展示深入的理论分析和多种可能的解决路径
+- 用户具备研究生或以上数学背景
+
+### 标准结构
+
+```markdown
+## 问题分类与定位
+[问题所属的数学领域，与已知问题的关联]
+
+## 核心难点分析
+[问题的本质困难在哪里，为何标准方法失效]
+
+## 相关理论框架
+[可能需要用到的深层理论和工具]
+
+## 已知结果综述
+[与该问题最接近的已知定理和结论]
+
+## 尝试路径一：[方法名称]
+[第一种解决思路的详细推导]
+### 进展
+[该方法能走多远]
+### 障碍
+[遇到的具体困难]
+
+## 尝试路径二：[方法名称]
+[第二种解决思路的详细推导]
+### 进展
+### 障碍
+
+## 尝试路径三：[方法名称]
+[第三种解决思路的详细推导]
+### 进展
+### 障碍
+
+## 部分结果与引理
+[可以严格证明的中间结果]
+
+## 数值/符号计算验证
+[计算机辅助验证的结果]
+
+## 可能的突破方向
+[基于上述分析，最有希望的研究方向]
+
+## 开放性结论
+[诚实说明当前分析的局限性]
+```
+
+### 完整示例
+
+```markdown
+## 问题分类与定位
+**领域**: 解析数论 × 代数几何  
+**问题类型**: L-函数零点分布与算术对象的关联  
+**关联猜想**:广义 Riemann 假设，Birch and Swinnerton-Dyer 猜想
+
+## 核心难点分析
+1. **解析延拓的存在性**: 该 L-函数的解析延拓尚未被证明
+2. **函数方程未知**: 缺乏函数方程使得零点分布分析极为困难
+3. **算术信息编码**: L-函数系数与椭圆曲线秩的关系不明确
+
+## 相关理论框架
+- **模形式理论**: 若该 L-函数来自模形式，可利用 modularity theorem
+- **Iwasawa 理论**: 研究 p-进 L-函数的性质
+- **Trace formula**: Arthur-Selberg trace formula 可能提供谱信息
+
+## 已知结果综述
+- **最接近的定理**: Wiles (1995) 证明了半稳定椭圆曲线的模性
+- **部分结果**: Kolyvagin (1988) 对秩为 0 或 1 的情况证明了 BSD 猜想
+- **数值证据**: Cremona 数据库验证了 conductor < 500000 的情况
+
+## 尝试路径一：模性提升
+通过已知的模性提升定理，尝试将该椭圆曲线与模形式关联。
+
+### 进展
+对于 good reduction 的素数 p，可以计算局部因子...
+
+### 障碍
+在 p = 2, 3 处的 bad reduction 类型不明，无法应用现有的提升定理。
+
+## 尝试路径二：p-进方法
+构造 p-进 L-函数并研究其零点。
+
+### 进展
+利用 Klingen-Stackelberg 构造可以得到 p-进插值公式...
+
+### 障碍
+p-进 L-函数的非平凡零点与原 L-函数的对应关系未明。
+
+## 尝试路径三：平均结果
+考虑一族类似的 L-函数，研究统计性质。
+
+### 进展
+可以证明平均意义下的零点密度估计...
+
+### 障碍
+平均结果无法推出单个 L-函数的性质。
+
+## 部分结果与引理
+**引理 1**: 在 Re(s) > 3/2 区域，该 L-函数绝对收敛。
+**引理 2**: 若存在函数方程，则临界线为 Re(s) = 1。
+**命题**: 对于 90% 的素数 p，局部因子满足预期界限。
+
+## 数值/符号计算验证
+使用 SageMath 计算前 1000 个 Dirichlet 系数：
+- 系数增长符合 Ramanujan-Petersson 猜想
+- 前 100 个非平凡零点均位于临界线上（精度 10⁻¹⁰）
+
+## 可能的突破方向
+1. **确定 bad reduction 类型**: 计算 discriminant 和 conductor
+2. **寻找函数方程**: 通过数值拟合猜测函数方程形式
+3. **应用 converse theorem**: 验证足够的 analytic properties 以推出模性
+
+## 开放性结论
+**当前状态**: 该问题的完整解决超出已知数学工具的范围。
+上述分析展示了多种可能的研究路径，但每条路径都存在实质性障碍。
+建议将此问题分解为若干可处理的子问题逐个击破。
+```
+
+### 特殊注意事项
+- 必须明确区分"已证明"、"推测"、"数值证据"三种断言强度
+- 引用的定理必须准确陈述条件，不可模糊处理
+- 对于开放问题，不提供虚假的"解答"
+- 建议的研究方向应具体且可行
+- 如涉及计算，应说明所用工具和精度
+
+---
+
 ## 模板选择决策树
 
 ```
@@ -456,7 +589,8 @@ n-1, n, n+1 是三个连续整数，因此：
 │  ├─ "只要答案" → 模板 B：仅答案
 │  ├─ "帮我看看对不对" → 模板 D：解答检查
 │  ├─ "证明……" → 模板 C：证明
-│  ├─ 已知开放问题 / 研究级 → 模板 F：研究/开放
+│  ├─ 已知开放问题 / 一般研究级 → 模板 F：研究/开放
+│  ├─ 深度研究级问题（多分支交叉） → 模板 G：研究级深度分析
 │  ├─ 高等数学领域 → 模板 E：高等数学
 │  └─ 其他情况 → 模板 A：标准解答（默认）
 ```

diff --git a/modules/verification_engine.md b/modules/verification_engine.md
@@ -16,6 +16,8 @@ This document defines all 11 verification methods (A through K), including:
 
 The following table maps problem classifications to recommended verification methods. The first listed method is the primary; the second is the fallback or complement. At least two must be applied; three or more increase confidence for difficult or high-stakes problems.
 
+**Method Numbering Note**: Methods are labeled A through K (11 total). The letters are mnemonic: A=Back-substitution, B=Boundary/Domain, C=Continuity/Boundary, D=Derivation(reverse), E=Estimation(numerical), F=Dimension(Formula), G=Special cases/limits, H=Independent method, I=Counterexample, J=Logic(formal), K=Consistency(computational).
+
 | Problem Classification | Primary Method | Secondary Method | Tertiary Method |
 |---|---|---|---|
 | `calculation` | E (Numerical) | A (Back-substitution) | — |
@@ -50,6 +52,24 @@ The following table maps problem classifications to recommended verification met
 | `problem_generation` | A (Back-substitution) | E (Numerical) | H (Independent method) |
 | `research_level_problem` | All applicable | All applicable | All applicable |
 
+### Method Applicability Matrix
+
+To avoid overlap and clarify when each method should be used:
+
+| Method | Best For | Not Suitable For | Unique Strength |
+|---|---|---|---|
+| A (Back-substitution) | Equations, systems, explicit solutions | Proofs, existence claims | Directly verifies solution satisfies original conditions |
+| B (Domain check) | ALL problems (universal pre-check) | None — always applicable | Catches extraneous solutions from non-reversible steps |
+| C (Boundary check) | Inequalities, optimization, intervals | Discrete problems without natural boundaries | Tests edge behavior where errors often hide |
+| D (Reverse derivation) | Proofs, derivations, integration | Numerical answers without derivation chain | Validates logical flow reversibility |
+| E (Numerical sampling) | Quantitative problems, identities | Pure existence proofs, symbolic manipulation only | Quick sanity check with concrete values |
+| F (Dimensional analysis) | Physics problems, formulas with units | Pure mathematics without physical interpretation | Structural validation of formula form |
+| G (Limits/special cases) | Functions, sequences, series, calculus | Finite combinatorics without parameters | Reveals asymptotic and degenerate behavior |
+| H (Independent method) | Problems with multiple solution approaches | Problems with only one known method | Strongest confidence via independent derivation |
+| I (Counterexample) | Universal claims, conjectures, "true/false" | Constructive existence proofs | Definitively disproves false claims |
+| J (Formal logic) | Proofs, abstract math, quantified statements | Computational problems | Catches reasoning errors that survive algebraic checks |
+| K (Computational consistency) | Matrix operations, statistics, multi-step computation | Simple single-step calculations | Uses invariants to catch propagation errors |
+
 ---
 
 ## Method A: Back-Substitution