diff --git a/.gitignore b/.gitignore index 6f223dc..0eb8fe1 100644 --- 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 +``` \ No newline at end of file diff --git a/modules/error_prevention.md b/modules/error_prevention.md index 123196b..0c87989 100644 --- 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 index 7e3fa0c..4a1fafc 100644 --- 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 index e10d420..6e859b9 100644 --- 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