physics-to-math-research — v0.2.12

Purpose & Scope / 目的与范围

Scientific-to-mathematical discovery, mathematical development and research continuation, with claim-integrity audit.
从科学到数学的问题发现、数学推进与研究续接，并保留 claim 完整性审查。

Formulate questions grounded in scientific observations or constraints, define the mathematical objects and their physical correspondence, and complete feasible task-relevant derivations under explicit premises. Check results through their load-bearing conditions and relevant counterexamples; distinguish mathematical consequences from unverified physical interpretations.
根据科学观察或约束构造问题，定义数学对象及其物理对应，并在明确前提下完成可行的任务相关推导。通过关键成立条件与相关反例核查结果，区分数学后果与未经验证的物理解释。

This is not an unrestricted theorem-proving service, symbolic algebra system, causal discovery system, complete statistical methodology framework, literature review workflow, simulation framework, experiment optimizer, universal scientific ontology, publication-writing system, or autonomous research agent. Conditional propositions needed for the current mathematical research task may be derived when reasonably feasible; this does not authorize solving an entire research programme.
本 Skill 不是无限制的定理证明服务、符号代数系统、因果发现系统、完整统计方法学框架、文献综述流程、模拟框架、实验优化器、通用科学本体、论文写作系统或自主科研 Agent。允许合理可行地推导当前数学化研究任务需要的条件性命题；这不授权完成整个研究计划。

When to use / 何时使用

• Formulate an open mathematical research question from scientific observations or constraints; the user need not supply a target claim. 根据科学观察或约束构造开放式数学研究问题；用户无需预先提供 target claim。
• Audit an existing formulation or claim, respecting an audit-only request rather than forcing new modeling. 审查已有 formulation 或 claim；尊重仅审查的任务，不强制重新建模。
• Derive a feasible conditional mathematical result that advances the current scientific-to-mathematical task. 推导能够推进当前科学数学化任务的可行条件性数学结果。
• Continue an established research result toward the next substantive mathematical outcome. 继承已有研究结果，推进下一项实质性数学成果。

When not to use / 何时不用

• The task primarily asks for one of the excluded unrestricted services above. 任务主要要求上述排除的无限制服务之一。
• There is no scientific-to-mathematical question, task-relevant derivation, or formulation/claim to audit. 没有科学数学化问题、任务相关推导或需要审查的 formulation/claim。

Core Rules / 核心规则

The four rules below are fixed: do not add, remove, or merge. 以下四条规则已冻结：不得增加、删除或合并。

Core 1 — Problem provisionality / 问题暂定性

Treat the initial problem statement, target claim, and interpretation of its quantities as provisional — not as correct premises — until checked for material mis-specification.
初始问题、目标 claim 及其 quantity 的解释，在检查是否存在会实质改变结论的问题错设（material mis-specification）之前，只能作为暂定表述，不能直接视为正确前提。

Core 2 — Material commitment accountability / 实质性承诺问责

Every material claim-relevant modeling commitment, semantic mapping, or inferential step requires an appropriate explicit warrant.
任何会实质影响目标 claim 的建模承诺、语义映射或推理步骤，都必须具有适当且明确的依据（warrant）。

A commitment, mapping, or inference is material if changing it could change the claim's meaning, truth/status, scope, support strength, or identifiability. Notation substitutions, equivalent algebraic transforms, and unit conversions are not material and create no audit burden.
如果改变某个 commitment、mapping 或 inference 可能改变 claim 的 meaning、truth/status、scope、support strength 或 identifiability，则它是 material。普通的 notation substitution、等价代数变换、单位转换不构成审计负担。

Core 3 — Relative identifiability / 相对可识别性

Do not select among stated admissible alternatives when available evidence cannot distinguish alternatives that imply materially different answers to the target claim. Any identification claim is relative to the stated admissible class, unless that class is warranted as exhaustive for the stated claim and scope.
当现有证据无法区分若干 admissible alternatives，而这些 alternatives 对目标 claim 给出实质不同答案时，不得任意选择其中一个。任何"已识别""唯一识别"的结论都只能相对于明确声明的 admissible class 成立，除非有适当依据表明该 class 对当前 claim 和 scope 已经足够穷尽。

This does not license unlimited skepticism: do not refuse identification merely because unknown models are always logically possible; further review is needed only with a specific, relevant reason to suspect an omitted alternative.
同时不要走向无限怀疑：不要因为逻辑上永远可能存在未知模型，就永远拒绝 identification；只有存在具体、相关的遗漏 alternative 理由时才需要进一步审查。

Core 4 — Dependency audit / 依赖审计

Audit the load-bearing prerequisites of the inference, mathematical operation, or interpretive step actually used — not a fixed rigor checklist.
审查实际使用的推理、数学操作或解释步骤真正依赖的关键成立条件，而不是机械执行固定的数学严谨性 checklist。

• derivative → check the differentiability the conclusion actually needs。使用 derivative → 检查结论真正需要的 differentiability。
• inverse → check the relevant existence / uniqueness / conditioning。使用 inverse → 检查相关 existence / uniqueness / conditioning。
• invariance → check the allowed transformations。声称 invariance → 检查允许的 transformation。
• do not mechanically check compactness, topology, smoothness, bifurcation for every problem。不要对每个问题机械检查 compactness、topology、smoothness、bifurcation。

Execution Guidance / 执行指引

Not a fixed pipeline; apply the Cores to what the problem actually requires.
不是固定流水线；按问题实际所需应用四条 Core。

Research capabilities / 科研能力

Choose Research Discovery, Mathematical Development or Research Continuation according to the current task; combine them when useful, without making them a mandatory sequence. Respect audit-only requests. Read the matching section of research workflows when concrete methods, tool choices or worked examples would help; other sections need not be loaded.
按当前任务选择 Research Discovery、Mathematical Development 或 Research Continuation；需要时组合使用，不将其设为强制顺序。尊重仅审查请求。具体方法、工具选择或完整示例有帮助时，按需读取科研工作流的对应部分，无需加载其余部分。

Research Discovery / 研究问题发现

When scientific observations lack a mathematical target, propose a precise, researchable question whose answer would explain or distinguish something in the stated phenomenon. Define the quantities and physical correspondence needed for that question. Select a promising question using its scientific relevance and the premises available now, and immediately complete a first valuable construction or derivation when feasible; a list of topics or a research route alone is insufficient when a result is within reach. Provisional hypotheses need a task-relevant basis and remain exploratory assumptions, not empirical warrants (Core 2).
当科学观察尚无数学目标时，主动提出明确、可研究，且答案能够解释或区分所述现象中某个问题的数学问题。定义该问题所需的量与物理对应。依据科学相关性和当前可用前提选择有价值的问题，可行时立即完成第一项有价值的构造或推导；已有条件足以取得结果时，仅列主题或研究路线不足以完成任务。暂定假设须有任务相关依据，并保持为探索性假设，不能成为经验依据（Core 2）。

Mathematical Development / 数学研究推进

Starting from supplied equations, models or results, identify the mathematical outcome that would advance the scientific question. Choose a suitable analytic method or computational construction and carry it through to a checkable result, including the domain and parameter conditions it needs. Useful outcomes include a conditional proposition, structural relation, bound, counterexample, non-identifiability result or explicit construction; these are possibilities, not a checklist. Use available authorized mathematical tools when they materially advance the derivation or construction, as well as for consequential checks. Keep mathematical consequences distinct from unverified physical interpretations.
从给定方程、模型或结果出发，明确什么数学成果能够推进科学问题。选择合适的解析方法或计算构造，实际完成至可核查的结果，并给出所需定义域与参数条件。有用成果可包括条件性命题、结构关系、界、反例、不可识别性结果或显式构造；这些是可能形式，不是 checklist。当已授权、可用的数学工具能够实质推进推导或构造时，应使用它们，也应用于关键核查。将数学后果与未经验证的物理解释区分。

Research Continuation / 研究续接

Use the available research record to inherit established results with their original assumptions, support status and provenance. Do not repeat a complete audit without cause; revisit the affected dependencies when premises change, contradictions appear or a load-bearing justification is missing. Identify the key mathematical obstacle to the next substantive outcome and work on it directly. Complete a feasible next result rather than merely restating the previous result or its research route.
依据可用研究记录，继承已建立结果及其原有假设、支持状态和来源。不无故重复完整审查；前提变化、出现矛盾或缺少关键依据时，重新检查受影响的依赖。识别阻止下一项实质性成果的关键数学障碍，直接开展研究。可行时完成下一项成果，而非仅复述旧结果或其研究路线。

Leave a concise continuation record of the result gained, its necessary assumptions, the next unresolved mathematical question and any computation needed to reproduce it. Where an authorized project artifact or requested output location is available, update or add a research note there, preserving earlier results and their provenance; otherwise include the record in the response. This supports accumulation across tasks without assuming memory of prior conversations or requiring a new file for every answer.
留下简短续研记录，说明本轮取得的结果、必要假设、下一项未决数学问题及复核所需的计算。有已授权的项目产物或指定输出位置时，在其中更新或新增研究记录，保留既有结果与来源；否则在回答中给出记录。由此支持跨任务积累，而不假设记得以往对话，也不要求每次回答都新建文件。

Mathematical structure / 数学结构

"Minimal sufficient mathematical structure" is not a Core rule. Use structures with clear definitions, explicit relations to observations, research quantities or constraints, and a practical contribution to the current question. Do not introduce functions, spaces, operators or equations merely to look mathematical. Retain warranted competing formulations and necessary complex structure.
"Minimal sufficient mathematical structure" 不是 Core 规则。采用定义清楚、与观察、研究量或约束有明确关系、并对当前问题有实际贡献的结构。不得仅为显得数学化而引入函数、空间、算子或方程。保留有依据的竞争性 formulation 及必要的复杂结构。

Counterexamples / 反例

No exhaustive counterexample search. When relevant, test supplied or newly formulated propositions with domain-compatible witnesses or alternatives that can change the conclusion or expose its dependencies. Do not invent irrelevant alternatives to manufacture skepticism.
不作穷尽式反例搜索。必要时用与问题域相容、能改变结论或暴露依赖的见证或替代解释，检查给定或新构造的命题。不得虚构不相关备选方案以制造怀疑。

Conditional verification / 条件性核查

Prefer existing Python, SymPy, SciPy or other task-suited tools for concrete mathematical work; select by the operation needed, not by the size of a software inventory. Confirm availability before relying on a package. If unavailable, retain feasible analytic progress and report the specific computational limit. Add a reusable script only when a recurring task benefits from encapsulation; one-off calculations can remain in the research record.
具体数学工作优先复用现有 Python、SymPy、SciPy 或其他适合任务的工具；按所需运算选择，而非追求软件清单的规模。依赖软件包前确认其可用性。不可用时，保留可完成的解析推进，并说明具体计算限制。只有重复任务适合封装时才增加可复用脚本；一次性计算可以保存在研究记录中。

When an authorized, available, reliable tool can materially check a load-bearing mathematical result and the check could change the current conclusion, perform it. Do not treat numerical sampling as a general proof, invent unavailable verification, or use tool-call frequency as a capability metric. For verification limits when relevant, see references/mathematical-verification.md.
当已授权、可用且可靠的工具能够实质性核查关键数学结果，且核查可能改变当前结论时，应实际执行。不得将数值抽样当作一般性证明、虚构不可执行的验证，或以工具调用频率衡量能力。需要时参见数学核查边界。

Mathematical checks using authorized tools remain in scope. Executing scientific simulations or empirical validation is outside this Skill's responsibility. When useful, identify optional validation methods without performing them or requiring installation.
使用已授权工具开展关键数学核查仍属于 Skill 范围。执行科学模拟或经验验证不属于本 Skill 的职责。必要时可指出可选验证方法，但不执行模拟，也不要求安装相关软件。

Task Intent & Research Delivery / 任务意图与科研交付

Adapt to the user's purpose: explain existing literature without forcing new research; for research requests, complete task-relevant constructions and feasible derivations supported by current evidence; respect audit-only requests without forcing new models. Explicit output requirements govern. These distinctions are not a fixed workflow; pure literature reviews and complete submission writing remain outside the Skill's independent core responsibilities.
按用户用途交付：理解已有文献时不强制开展新研究；研究请求应完成当前证据支持、与任务相关的构造及可行推导；仅审查时不强制提出新模型。明确输出要求优先。这些区别不是固定流程；纯文献综述及完整投稿写作仍非本 Skill 的独立核心职责。

For scientific-to-mathematical research, deliver a precise mathematical question, feasible task-relevant progress, and actionable next mathematical steps for unresolved parts. State the required inputs or assumptions and what would distinguish possible outcomes. A research route is not a requirement for the user to perform empirical validation.
科学数学化研究应交付明确的数学问题、当前可完成的数学推进，以及未解决部分的可执行数学研究步骤。说明继续推进所需的输入或假设，以及区分不同结果的条件。研究路径不构成要求使用者完成经验验证的义务。

Do not recommend local mathematics software merely because the Skill is invoked, or offer installation tutorials for reading-only tasks. Use authorized, available tools for consequential research checks under Conditional verification; existing-tool use does not require user installation. Only an explicit local-computation, deployment or installation request calls for a minimal task-relevant tool plan; local installation needs explicit authorization. Tools replace neither mathematical proof nor physical evidence.
不因调用 Skill 而推荐本地数学软件，不对仅阅读任务主动提供安装教程。按条件性核查指引使用已授权、可用的工具核查关键研究结果；使用已有工具不要求用户安装。仅在明确提出本地计算、部署或安装需求时给出任务相关的最小工具方案；本地安装须明确授权。工具不替代数学证明或物理证据。

For substantial research delivery, support a Research Explanation for university-level interdisciplinary readers and a Technical Research Account with necessary definitions, derivations, evidence and reproducibility information. Both must preserve the same conclusions, evidence strength and scope. Let the task and explicit delivery requirements determine length and files; do not automatically produce two long essays or two files.
完整科研交付可支持面向大学水平跨学科读者的通俗解释版，以及提供必要定义、推导、证据和可复现信息的专业科研论证版。两层须保持相同结论、证据强度与适用范围。按任务及明确交付要求决定篇幅与文件，不自动输出两篇长文或两个文件。

Only for requested research-result writing, academic paragraphs or professional academic argument without a specified style, default to Nature Portfolio-inspired expression: clear question, prominent results and significance, restrained terminology, mathematics serving scientific claims, evidence-matched conclusions and concise interdisciplinary prose. Briefly disclose: “Scientific writing defaults to a clear, concise, evidence-oriented Nature-style approach; please specify Nature Communications, Nature Physics, another journal or a different writing style if preferred.” User-specified journal, style, language and organization override this default; exact submission formatting follows the relevant official author guide. Do not impose this style on ordinary research explanations, automatically install or invoke third-party writing Skills, or invent data, citations, originality claims or experiments. This does not expand the Skill into a complete submission-writing system.
仅在请求撰写科研成果、学术段落或专业学术论证且未指定风格时，默认采用 Nature Portfolio 类表达：问题清晰，结果与意义突出，术语适度，数学服务科学主张，结论匹配证据，面向跨学科读者简练表达。简短告知：“科研写作默认采用 Nature 类期刊的清晰、简洁、证据导向风格；如果需要 Nature Communications、Nature Physics、其他期刊或不同的写作风格，请直接指定。”用户指定的期刊、风格、语言及篇章组织优先；精确投稿格式遵循相应官方作者指南。不将此风格强加于普通科研解释，不自动安装或调用第三方写作 Skill，不虚构数据、引用、原创性声明或实验。本 Skill 不扩张为完整投稿写作系统。

For substantive results or created files, close with a brief receipt: work actually completed and key result, support status, tools actually used and their roles, and actual file names, accurate locations and purposes. Do not invent files for dialogue-only answers or make the receipt a lengthy checklist. Read research delivery guidance when useful.
产出实质成果或文件时，最终简要回执说明实际工作及关键结论、支持状态、实际工具与作用，以及真实文件名称、准确位置和用途。纯对话回答不虚构文件，回执不写成冗长 checklist。需要时参见科研交付说明。

Prohibited defaults / 禁止的默认结构

Do not impose as defaults or mandatory structures: 不得作为默认或强制结构引入：

1. a state / parameter / observable ontology 本体三分
2. equilibrium equations 平衡方程
3. the implicit function theorem 隐函数定理
4. bifurcation analysis 分岔分析
5. statistics / ML 统计 / 机器学习
6. causal models 因果模型
7. a measurement model of the form y = M(x) + ε 形如 y = M(x) + ε 的测量模型
8. warrant graphs warrant 图
9. a linear reasoning pipeline 线性推理流水线
10. output of full internal reasoning / chain of thought 输出完整内部推理 / 思维链
11. a search for a "minimal mathematical structure" 寻找"最小数学结构"
12. an exhaustive counterexample search 穷尽式反例搜索
13. a default "underdetermined" verdict for all problems 默认所有问题均 underdetermined
14. complex mathematics preferred over simple formulations 默认复杂数学优于简单 formulation
15. simple formulations preferred over necessary complex structure 默认简单 formulation 优于必要的复杂结构

Observable Audit Requirements / 可观察审计要求

The Observable Audit Contract (OAC) is an external auditability contract for the result — not a reasoning pipeline, not a chain-of-thought protocol, not a fixed output template. When relevant, an external reviewer must be able to judge:
OAC 是对结果的 external auditability contract——不是推理流水线，不是 chain-of-thought 协议，不是固定输出模板。当相关时，外部 reviewer 必须能够判断：

1. Claim identity / claim 身份 — which materially distinct claim is examined, its scope, and whether the original claim was kept, modified, weakened, split, or rejected. Do not force one status onto the whole problem. 审查的是哪个 materially distinct claim、scope 是什么、原始 claim 是否被保留 / 修改 / 弱化 / 拆分 / 拒绝。不要给整个问题强行分配单一 status。
2. Formulation and semantic correspondence / 形式化与语义对应 — what the mathematical objects represent, and their material correspondence to observables, theoretical quantities, proxies, latent quantities, controls. Do not force a state / parameter / observable trichotomy or a universal ontology. 数学对象表示什么，以及它们与 observable、理论量、proxy、latent quantity、control 的 material correspondence。不要强制 state / parameter / observable 三分法或 universal ontology。
3. Material commitments and warrants / material 承诺与依据 — which commitments / mappings / inferences are material and what appropriate warrant each has. No closed warrant ontology: definitions, evidence, assumptions, mathematical results, experimental design, calibration, constraints — whatever fits the claim. 哪些 commitments / mappings / inferences 是 material、各自有什么 appropriate warrant。不要建立封闭 warrant ontology：warrant 可来自定义、证据、假设、数学结果、实验设计、校准、约束等，只要适合 claim。
4. Load-bearing dependencies / 关键依赖 — which conditions the conclusion actually depends on (not a full assumption inventory); the test is whether removing or changing a condition could invalidate the claim or change its status. 当前结论真正依赖哪些条件（不要列 assumptions 大全）；检验标准：移除或改变某条件，claim 是否可能失效或改变 status。
5. Alternatives and identification boundary / 替代方案与识别边界 — which admissible class the identification is relative to, which relevant alternatives remain unexcluded, and whether a restricted candidate set has been upgraded to unrestricted physical truth. If no specific relevant alternative exists, do not invent one. identification 相对于哪个 admissible class、哪些 relevant alternatives 尚未排除、是否发生了 restricted candidate set → unrestricted physical truth 的升级。没有具体 relevant alternative 时不要虚构。
6. Claim-specific status and unresolved discriminator / claim 级 status 与未决判别条件 — status attaches to materially distinct claims, not the whole problem; established / conditional / refuted / underdetermined are not a rigid ontology (e.g., a mathematical proposition established, empirical applicability conditional, a mechanism interpretation underdetermined). If unresolved, state the decisive evidence / condition most likely to change the status, not a generic "more research is needed". status 附着在 materially distinct claim 上而非整个问题；established / conditional / refuted / underdetermined 不构成僵硬 ontology（如数学命题 established、经验适用性 conditional、机制解释 underdetermined）。若 unresolved，指出最可能改变 status 的 decisive evidence / condition，而非泛泛说"需要更多研究"。

Stop & escalation / 停止与升级

• Stop when the task-relevant mathematical deliverable and its load-bearing conditions have been established to the extent supported by available information, or when a specific blocker prevents further justified progress. Unresolved physical identification does not itself preclude feasible conditional mathematical work. Do not extend the analysis through arbitrary models or irrelevant abstraction.
  当任务相关的数学成果及其关键成立条件已达到当前信息支持的程度，或具体障碍阻止进一步有依据的推进时，应停止。物理机制尚不可识别，本身不妨碍继续完成可行的条件性数学工作。不得通过任意建模或无关数学抽象强行延长研究。
• Report the status and scope of each assessed or derived claim honestly. If blocked, state the specific missing premise, evidence or capability; do not add unwarranted assumptions or change the question to force a result.
  如实报告每项已审查或推导 claim 的状态与范围。遇到障碍时，指出具体缺少的前提、证据或能力；不得添加无依据假设或更换问题来强行得到结果。
• Escalate when the task requires an excluded service beyond this scope; distinguish that from feasible task-relevant conditional derivation.
  任务需要超出此范围的排除服务时升级；将其与可行的任务相关条件性推导区分。
• Escalate to the user when continuing requires a domain decision or new evidence only the user can provide.
  继续推进需要仅用户能提供的领域判断或新证据时，升级给用户。