formula-derivation

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Original

English

🇨🇳

Translation

Chinese

Formula Derivation: Research Theory Line Construction

公式推导：研究理论主线构建

Build an honest derivation package, not a fake polished theorem story.

构建真实可信的推导包，而非虚假修饰的定理故事。

Constants

常量

DEFAULT_DERIVATION_DOC =
```
DERIVATION_PACKAGE.md
```
in project root

STATUS =

COHERENT AS STATED | COHERENT AFTER REFRAMING / EXTRA ASSUMPTION | NOT YET COHERENT

DEFAULT_DERIVATION_DOC = 项目根目录下的
```
DERIVATION_PACKAGE.md
```

STATUS =

COHERENT AS STATED | COHERENT AFTER REFRAMING / EXTRA ASSUMPTION | NOT YET COHERENT

Context: $ARGUMENTS

上下文：$ARGUMENTS

Goal

目标

Produce exactly one of:

a coherent derivation package for the original target
a reframed derivation package with corrected object / assumptions / scope
a blocker report explaining why the current notes cannot yet support a coherent derivation

生成以下内容之一：

针对原始目标的连贯推导包
经过重构的推导包，修正了研究对象/假设/范围
阻塞报告，说明当前笔记为何无法支持连贯推导

Inputs

输入

Extract and normalize:

the target phenomenon, formula, relation, or theory line
the intended role of the derivation:
- exact identity / algebra
- proposition / local theorem
- approximation
- mechanism interpretation
explicit assumptions
notation and definitions
any user-provided formula chain, sketch, messy notes, or current draft
nearby local theory files if the request points to them
desired output style if specified:
- internal alignment note
- paper-style theory draft
- blocker report

If the target, object, notation, or assumptions are ambiguous, state the exact interpretation you are using before deriving anything.

提取并标准化以下信息：

目标现象、公式、关系或理论主线
推导的预期作用：
- 恒等式/代数运算
- 命题/局部定理
- 近似推导
- 机制解释
明确给出的假设
符号与定义
用户提供的任何公式链、草图、零散笔记或当前草稿
若请求指向相关文件，需包含本地理论文件
若指定了输出风格：
- 内部对齐说明
- 论文风格的理论草稿
- 阻塞报告

若目标、研究对象、符号或假设存在歧义，需在开始推导前明确说明你所采用的解释。

Workflow

工作流

Step 1: Gather Derivation Context

步骤1：收集推导上下文

Determine the target derivation file with this priority:

a file path explicitly specified by the user
a derivation draft already referenced in local notes
```
DERIVATION_PACKAGE.md
```
in project root as the default target

Read the relevant local context:

the chosen target derivation file, if it already exists
any local theory notes, formula drafts, appendix notes, or files explicitly mentioned by the user

Extract:

target formula / theory goal
current formula chain
assumptions
notation
known blockers
desired output mode

按以下优先级确定目标推导文件：

用户明确指定的文件路径
本地笔记中已引用的推导草稿
默认使用项目根目录下的
```
DERIVATION_PACKAGE.md
```

读取相关本地上下文：

所选的目标推导文件（若已存在）
任何本地理论笔记、公式草稿、附录笔记或用户明确提及的文件

提取信息：

目标公式/理论目标
当前公式链
假设条件
符号体系
已知阻塞问题
期望的输出模式

Step 2: Freeze the Target

步骤2：锁定目标

State explicitly:

what is being explained, derived, or supported
whether the immediate goal is:
- identity / algebra
- proposition
- approximation
- interpretation
what the derivation is expected to output in the end

Do not start symbolic manipulation before this is fixed.

明确说明：

要解释、推导或支持的内容
直接目标类型：
- 恒等式/代数运算
- 命题
- 近似推导
- 解释说明
推导最终预期输出的内容

在锁定目标前，不得开始符号运算。

Step 3: Choose the Invariant Object

步骤3：选择不变核心对象

Identify the single quantity or conceptual object that should organize the derivation.

Typical possibilities include:

objective / utility / loss
total cost / energy / welfare
conserved quantity / state variable
expected metric / effective rate / effective cost

If the current notes start from a narrower quantity, decide explicitly whether it is:

the true top-level object
a proxy
a local slice
an approximation

Do not let a convenient proxy silently replace the actual conceptual object.

确定用于组织推导过程的单一量或概念对象。

典型选择包括：

目标函数/效用函数/损失函数
总成本/能量/福利
守恒量/状态变量
期望指标/有效速率/有效成本

若当前笔记从较窄的量出发，需明确判断该量是否为：

真正的顶层研究对象
替代指标
局部切片
近似值

不得让便利的替代指标悄然取代实际的核心概念对象。

Step 4: Normalize Assumptions and Notation

步骤4：标准化假设与符号

Restate:

all assumptions
all symbols
regime boundaries or special cases
which quantities are fixed, adaptive, or state dependent

Identify:

hidden assumptions
undefined notation
scope ambiguities
whether the current formula chain already mixes exact steps with approximations

Preserve the user's original notation unless a cleanup is necessary for coherence. If you adopt a cleaner internal formulation, keep that as a derivation device rather than silently replacing the user's target.

重新表述：

所有假设条件
所有符号
适用范围边界或特殊情况
哪些量是固定的、自适应的或依赖状态的

识别：

隐含假设
未定义的符号
范围歧义
当前公式链是否已混合了精确步骤与近似步骤

除非为保证连贯性必须清理，否则保留用户原始符号。若采用更清晰的内部表述，需将其作为推导工具，而非悄然替换用户的目标内容。

Step 5: Classify the Derivation Steps

步骤5：分类推导步骤

For every nontrivial step, determine whether it is:

identity: exact algebraic reformulation
proposition: a claim requiring conditions
approximation: model simplification or surrogate
interpretation: prose-level meaning of a formula

Never merge these categories without signaling the transition. If one part is only interpretive, do not present it as if it were mathematically proved.

对于每个非平凡步骤，确定其类型：

恒等式：精确的代数重述
命题：需要特定条件的论断
近似：模型简化或替代
解释：公式的文字层面含义

未经明确说明，不得合并这些类别。若某部分仅为解释性内容，不得将其表述为已通过数学证明的内容。

Step 6: Build a Derivation Map

步骤6：构建推导图谱

Choose a derivation strategy, for example:

definition -> substitution -> simplification
primitive law -> intermediate variable -> target expression
global quantity -> perturbation -> decomposition
exact model -> approximation -> interpretable closed form
general dynamic object -> simplified slice -> local theorem -> return to general case

Then write a derivation map:

target formula or theory line
required intermediate identities or lemmas
which assumptions each nontrivial step uses
where approximations enter
where special-case and general-case regimes diverge or collapse

If the derivation needs a decomposition, derive it from the chosen global quantity. Do not make a split appear magically from one local variable itself.

选择推导策略，例如：

定义 -> 代入 -> 简化
基本定律 -> 中间变量 -> 目标表达式
全局量 -> 扰动 -> 分解
精确模型 -> 近似推导 -> 可解释的闭式解
通用动态对象 -> 简化切片 -> 局部定理 -> 回归通用情况

然后编写推导图谱：

目标公式或理论主线
所需的中间恒等式或引理
每个非平凡步骤使用的假设
近似步骤的引入位置
特殊情况与通用情况的分歧或重合点

若推导需要分解，需从选定的全局量出发进行分解。不得让拆分从某个局部变量凭空产生。

Step 7: Write the Derivation Document

步骤7：编写推导文档

Write to the chosen target derivation file.

If the target derivation file already exists:

read it first
update the relevant section
do not blindly duplicate prior content

If the user does not specify a target, default to

DERIVATION_PACKAGE.md

in project root.

Do NOT write directly into paper sections or appendix

.tex

files unless the user explicitly asks for that target.

The derivation package must include:

target
status
invariant object
assumptions
notation
derivation strategy
derivation map
main derivation steps
remarks / interpretations
boundaries and non-claims

Writing rules:

do not hide gaps with words like "clearly", "obviously", or "similarly"
define every symbol before use
mark approximations explicitly
separate derivation body from remarks
if the true object is dynamic or state dependent but a simpler slice is analyzed, say so explicitly
if a formula line is only heuristic, label it honestly

将内容写入选定的目标推导文件。

若目标推导文件已存在：

先读取现有内容
更新相关章节
不得盲目重复已有内容

若用户未指定目标文件，默认使用项目根目录下的

DERIVATION_PACKAGE.md

。

除非用户明确要求，否则不得直接写入论文章节或附录的

.tex

文件。

推导包必须包含：

目标
状态
不变核心对象
假设条件
符号体系
推导策略
推导图谱
主要推导步骤
备注/解释
适用边界与非论断内容

写作规则：

不得使用“显然”“易见”“同理”等词汇掩盖逻辑缺口
所有符号需在使用前定义
明确标记近似步骤
将推导主体与备注分开
若真实研究对象是动态或依赖状态的，但仅分析了简化切片，需明确说明
若公式仅为启发式内容，需如实标注

Step 8: Final Verification

步骤8：最终验证

Before finishing the target derivation file, verify:

the target is explicit
the invariant object is stable across the derivation
every assumption used is stated
each formula step is correctly labeled as identity / proposition / approximation / interpretation
the derivation does not silently switch objects
special cases and general cases still belong to one theory line
boundaries and non-claims are stated

If the derivation still lacks a coherent object, stable assumptions, or an honest path from premises to result, downgrade the status and write a blocker report instead of forcing a clean story.

在完成目标推导文件前，需验证：

目标明确
不变核心对象在推导过程中保持一致
所有使用的假设均已说明
每个公式步骤均正确标记为恒等式/命题/近似/解释
推导过程中未悄然切换研究对象
特殊情况与通用情况仍属于同一理论主线
适用边界与非论断内容已明确说明

若推导仍缺乏连贯的研究对象、稳定的假设，或从前提到结果的合理路径，需降级状态并编写阻塞报告，而非强行构建看似完整的推导故事。

Required File Structure

要求的文件结构

Write the target derivation file using this structure:

undefined

使用以下结构编写目标推导文件：

undefined

Derivation Package

推导包

Target

目标

[what is being derived or explained]

[要推导或解释的内容]

Status

状态

COHERENT AS STATED / COHERENT AFTER REFRAMING / NOT YET COHERENT

Invariant Object

不变核心对象

[top-level quantity organizing the derivation]

[用于组织推导的顶层量]

Assumptions

假设条件

Notation

符号体系

Derivation Strategy

推导策略

[chosen route and why]

[选择的路径及原因]

Derivation Map

推导图谱

Target depends on ...
Intermediate step A uses ...
Approximation enters at ...

目标依赖于...
中间步骤A使用...
近似步骤引入于...

Main Derivation

主要推导

Step 1. ... Step 2. ... ...

步骤1. ... 步骤2. ... ...

Remarks and Interpretation

备注与解释

Boundaries and Non-Claims

适用边界与非论断内容

Open Risks

未解决风险

undefined

undefined

Output Modes

输出模式

If the derivation is coherent as stated

若推导按原始要求即可连贯

Write the full structure above with a clean derivation package.

按照上述完整结构编写清晰的推导包。

If the notes are close but not coherent yet

若笔记接近连贯但尚未达成

Write:

the exact mismatch
the corrected invariant object, assumption, or scope
the reframed derivation package

需编写：

具体的不匹配点
修正后的不变核心对象、假设或范围
重构后的推导包

If the derivation cannot be made coherent honestly

若无法真实构建连贯推导