--- name: formula-derivation description: Structures and derives research formulas when the user wants to 推导公式, build a theory line, organize assumptions, turn scattered equations into a coherent derivation, or rewrite theory notes into a paper-ready formula document. Use when the derivation target is not yet fully fixed, the main object still needs to be chosen, or the user needs a coherent derivation package rather than a finished theorem proof. argument-hint: [problem-goal-current-formulas-or-notes] allowed-tools: Read, Write, Edit, Grep, Glob --- # Formula Derivation: Research Theory Line Construction Build an honest derivation package, not a fake polished theorem story. ## Constants - DEFAULT_DERIVATION_DOC = `DERIVATION_PACKAGE.md` in project root - STATUS = `COHERENT AS STATED | COHERENT AFTER REFRAMING / EXTRA ASSUMPTION | NOT YET COHERENT` ## Context: $ARGUMENTS ## Goal Produce exactly one of: 1. a coherent derivation package for the original target 2. a reframed derivation package with corrected object / assumptions / scope 3. a blocker report explaining why the current notes cannot yet support a coherent derivation ## Inputs 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. ## Workflow ### Step 1: Gather Derivation Context Determine the target derivation file with this priority: 1. a file path explicitly specified by the user 2. a derivation draft already referenced in local notes 3. `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 ### Step 2: Freeze the Target 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. ### Step 3: Choose the Invariant Object 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. ### Step 4: Normalize Assumptions and Notation 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. ### Step 5: Classify the Derivation Steps 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. ### Step 6: Build a Derivation Map 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. ### Step 7: Write the Derivation Document 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 ### Step 8: Final Verification 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. ## Required File Structure Write the target derivation file using this structure: ```md # Derivation Package ## Target [what is being derived or explained] ## Status COHERENT AS STATED / COHERENT AFTER REFRAMING / NOT YET COHERENT ## Invariant Object [top-level quantity organizing the derivation] ## Assumptions - ... ## Notation - ... ## Derivation Strategy [chosen route and why] ## Derivation Map 1. Target depends on ... 2. Intermediate step A uses ... 3. Approximation enters at ... ## Main Derivation Step 1. ... Step 2. ... ... ## Remarks and Interpretation - ... ## Boundaries and Non-Claims - ... ## Open Risks - ... ``` ## Output Modes ### If the derivation is coherent as stated Write the full structure above with a clean derivation package. ### If the notes are close but not coherent yet Write: - the exact mismatch - the corrected invariant object, assumption, or scope - the reframed derivation package ### If the derivation cannot be made coherent honestly Write: - `Status: NOT YET COHERENT` - the exact blocker: - missing object - unstable assumptions - notation conflict - unsupported approximation - theorem-level claim without enough conditions - what extra assumption, reframe, or intermediate derivation would be needed ## Relationship to `proof-writer` Use `formula-derivation` when the user says things like: - “我不知道怎么起这条推导主线” - “这个公式到底该从哪个量出发” - “帮我把理论搭顺” - “把说明文档变成可写进论文的公式文档” - “这几段公式之间逻辑不通” Use `proof-writer` only after: - the exact claim is fixed - the assumptions are stable - the notation is settled - and the task is now to prove or refute that claim rigorously ## Chat Response After writing the target derivation file, respond briefly with: - status - whether the target survived unchanged or had to be reframed - what file was updated ## Key Rules - Never fabricate a coherent derivation if the object, assumptions, or scope do not support one. - Prefer reframing the derivation over overclaiming. - Separate assumptions, identities, propositions, approximations, and interpretations. - Keep one invariant object across special and general cases whenever possible. - Treat simplified constant-parameter cases as analysis slices, not as the conceptual main object. - If uncertainty remains, mark it explicitly in `Open Risks`; do not hide it in polished prose. - Coherence matters more than elegance. --- Source attribution: pinkpixel-dev Original project: https://github.com/pinkpixel-dev/skills-collection-1/blob/dc709653773653302e333a45d3b6f3b70fdff8f8/SKILLS/formula-derivation/SKILL.md License: Apache-2.0 License information: https://www.apache.org/licenses/LICENSE-2.0 Original upstream instructions; downloaded from Mega AI Hub. ## Upstream license notice Apache License Version 2.0, January 2004 http://www.apache.org/licenses/ TERMS AND CONDITIONS FOR USE, REPRODUCTION, AND DISTRIBUTION 1. Definitions. "License" shall mean the terms and conditions for use, reproduction, and distribution as defined by Sections 1 through 9 of this document. "Licensor" shall mean the copyright owner or entity authorized by the copyright owner that is granting the License. "Legal Entity" shall mean the union of the acting entity and all other entities that control, are controlled by, or are under common control with that entity. For the purposes of this definition, "control" means (i) the power, direct or indirect, to cause the direction or management of such entity, whether by contract or otherwise, or (ii) ownership of fifty percent (50%) or more of the outstanding shares, or (iii) beneficial ownership of such entity. "You" (or "Your") shall mean an individual or Legal Entity exercising permissions granted by this License. "Source" form shall mean the preferred form for making modifications, including but not limited to software source code, documentation source, and configuration files. "Object" form shall mean any form resulting from mechanical transformation or translation of a Source form, including but not limited to compiled object code, generated documentation, and conversions to other media types. "Work" shall mean the work of authorship, whether in Source or Object form, made available under the License, as indicated by a copyright notice that is included in or attached to the work (an example is provided in the Appendix below). "Derivative Works" shall mean any work, whether in Source or Object form, that is based on (or derived from) the Work and for which the editorial revisions, annotations, elaborations, or other modifications represent, as a whole, an original work of authorship. For the purposes of this License, Derivative Works shall not include works that remain separable from, or merely link (or bind by name) to the interfaces of, the Work and Derivative Works thereof. "Contribution" shall mean any work of authorship, including the original version of the Work and any modifications or additions to that Work or Derivative Works thereof, that is intentionally submitted to Licensor for inclusion in the Work by the copyright owner or by an individual or Legal Entity authorized to submit on behalf of the copyright owner. For the purposes of this definition, "submitted" means any form of electronic, verbal, or written communication sent to the Licensor or its representatives, including but not limited to communication on electronic mailing lists, source code control systems, and issue tracking systems that are managed by, or on behalf of, the Licensor for the purpose of discussing and improving the Work, but excluding communication that is conspicuously marked or otherwise designated in writing by the copyright owner as "Not a Contribution." "Contributor" shall mean Licensor and any individual or Legal Entity on behalf of whom a Contribution has been received by Licensor and subsequently incorporated within the Work. 2. Grant of Copyright License. Subject to the terms and conditions of this License, each Contributor hereby grants to You a perpetual, worldwide, non-exclusive, no-charge, royalty-free, irrevocable copyright license to reproduce, prepare Derivative Works of, publicly display, publicly perform, sublicense, and distribute the Work and such Derivative Works in Source or Object form. 3. Grant of Patent License. Subject to the terms and conditions of this License, each Contributor hereby grants to You a perpetual, worldwide, non-exclusive, no-charge, royalty-free, irrevocable (except as stated in this section) patent license to make, have made, use, offer to sell, sell, import, and otherwise transfer the Work, where such license applies only to those patent claims licensable by such Contributor that are necessarily infringed by their Contribution(s) alone or by combination of their Contribution(s) with the Work to which such Contribution(s) was submitted. If You institute patent litigation against any entity (including a cross-claim or counterclaim in a lawsuit) alleging that the Work or a Contribution incorporated within the Work constitutes direct or contributory patent infringement, then any patent licenses granted to You under this License for that Work shall terminate as of the date such litigation is filed. 4. Redistribution. You may reproduce and distribute copies of the Work or Derivative Works thereof in any medium, with or without modifications, and in Source or Object form, provided that You meet the following conditions: (a) You must give any other recipients of the Work or Derivative Works a copy of this License; and (b) You must cause any modified files to carry prominent notices stating that You changed the files; and (c) You must retain, in the Source form of any Derivative Works that You distribute, all copyright, patent, trademark, and attribution notices from the Source form of the Work, excluding those notices that do not pertain to any part of the Derivative Works; and (d) If the Work includes a "NOTICE" text file as part of its distribution, then any Derivative Works that You distribute must include a readable copy of the attribution notices contained within such NOTICE file, excluding those notices that do not pertain to any part of the Derivative Works, in at least one of the following places: within a NOTICE text file distributed as part of the Derivative Works; within the Source form or documentation, if provided along with the Derivative Works; or, within a display generated by the Derivative Works, if and wherever such third-party notices normally appear. The contents of the NOTICE file are for informational purposes only and do not modify the License. You may add Your own attribution notices within Derivative Works that You distribute, alongside or as an addendum to the NOTICE text from the Work, provided that such additional attribution notices cannot be construed as modifying the License. You may add Your own copyright statement to Your modifications and may provide additional or different license terms and conditions for use, reproduction, or distribution of Your modifications, or for any such Derivative Works as a whole, provided Your use, reproduction, and distribution of the Work otherwise complies with the conditions stated in this License. 5. Submission of Contributions. Unless You explicitly state otherwise, any Contribution intentionally submitted for inclusion in the Work by You to the Licensor shall be under the terms and conditions of this License, without any additional terms or conditions. Notwithstanding the above, nothing herein shall supersede or modify the terms of any separate license agreement you may have executed with Licensor regarding such Contributions. 6. Trademarks. This License does not grant permission to use the trade names, trademarks, service marks, or product names of the Licensor, except as required for reasonable and customary use in describing the origin of the Work and reproducing the content of the NOTICE file. 7. Disclaimer of Warranty. Unless required by applicable law or agreed to in writing, Licensor provides the Work (and each Contributor provides its Contributions) on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied, including, without limitation, any warranties or conditions of TITLE, NON-INFRINGEMENT, MERCHANTABILITY, or FITNESS FOR A PARTICULAR PURPOSE. You are solely responsible for determining the appropriateness of using or redistributing the Work and assume any risks associated with Your exercise of permissions under this License. 8. Limitation of Liability. In no event and under no legal theory, whether in tort (including negligence), contract, or otherwise, unless required by applicable law (such as deliberate and grossly negligent acts) or agreed to in writing, shall any Contributor be liable to You for damages, including any direct, indirect, special, incidental, or consequential damages of any character arising as a result of this License or out of the use or inability to use the Work (including but not limited to damages for loss of goodwill, work stoppage, computer failure or malfunction, or any and all other commercial damages or losses), even if such Contributor has been advised of the possibility of such damages. 9. Accepting Warranty or Additional Liability. While redistributing the Work or Derivative Works thereof, You may choose to offer, and charge a fee for, acceptance of support, warranty, indemnity, or other liability obligations and/or rights consistent with this License. However, in accepting such obligations, You may act only on Your own behalf and on Your sole responsibility, not on behalf of any other Contributor, and only if You agree to indemnify, defend, and hold each Contributor harmless for any liability incurred by, or claims asserted against, such Contributor by reason of your accepting any such warranty or additional liability. END OF TERMS AND CONDITIONS APPENDIX: How to apply the Apache License to your work. To apply the Apache License to your work, attach the following boilerplate notice, with the fields enclosed by brackets "[]" replaced with your own identifying information. (Don't include the brackets!) The text should be enclosed in the appropriate comment syntax for the file format. We also recommend that a file or class name and description of purpose be included on the same "printed page" as the copyright notice for easier identification within third-party archives. Copyright 2026 Pink Pixel Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. # Mega AI Hub resource metadata Creator / publisher: pinkpixel-dev Original project: https://github.com/pinkpixel-dev/skills-collection-1/blob/dc709653773653302e333a45d3b6f3b70fdff8f8/SKILLS/formula-derivation/SKILL.md Attribution basis: Source repository record Fields of work: Data & Analytics, Education & Research Roles: Academic Writer, Data Scientist, Researcher Specialties: API Tools: Derivation Package Country applicability: Country not specified Classification basis: Source catalog labels Instruction SHA-256: 3f1ce5b87c4b2d561efa2f3262471a7ad158fe8d2fd02e57cdf77aa27e0764d2