Theme 1: Fit-Function Optimization and Model Selection for Structural Fields
Chinese title: 结构场拟合函数优化与模型选择
English title: Fit-Function Optimization and Model Selection for Structural Fields
One-line positioning
Upgrade fit functions from “it fits” to “interpretable, comparable, reproducible, and transferable.”
Core objectives
• Improve fitting stability (convergence, robustness, multi-start consistency)
• Reduce parameter redundancy and non-identifiability (identifiability)
• Establish unified model-selection standards (AIC/BIC/cross-validation/Bayesian evidence)
Key tasks
• Function-family benchmarking: tanh / logistic / arctan / piecewise-smooth / physically constrained families
• Regularization and constraints: monotonicity, boundary conditions, non-negativity of physical quantities, scale-invariance constraints
• Error modeling: measurement error, systematic error, outlier handling (robust losses)
• Reproducibility benchmark: same datasets, same initialization protocol, and standardized outputs (R²/RMSE/residual spectra)
Deliverables
• Fit-function benchmark report (tables + figures)
• Open-source scripts and a unified interface (CLI + config)
• “Recommended function families” with applicability domains (which objects suit which functions)
Quantitative success criteria
• On representative datasets, the selected best family outperforms the baseline in RMSE, residual structure, and parameter stability
• Multi-start optimization converges to the same optimum with a materially higher rate (e.g., >90%)

Theme 2: Cross-Scale Fitting of Physical Systems and Two-Constant Consistency Tests
Chinese title: 跨尺度物理系统拟合与两常数一致性检验
English title: Cross-Scale Fitting of Physical Systems and Two-Constant Consistency Tests
One-line positioning
Upgrade the “two constants” from “appearing in some objects” to a testable conclusion that remains valid across object classes, datasets, and methods.
Coverage (an expandable public list)
• Galaxy rotation curves; strong gravitational lensing systems; (as applicable) CMB cold-spot/void profiles
• Future extensions: Solar System structure, cluster dynamics, and other systems
Key tasks
• Unified data pipeline: sources, cleaning, units, uncertainty models, and versioning
• Unified fitting standard: consistent optimizer, multi-start strategy, confidence intervals, and residual diagnostics
• Constant validation:
• Distribution stability (do different object classes share the same distribution?)
• Significance and sensitivity of linear/power-law relations
• Quantitative comparisons to control models (empirical profiles, commonly used ΛCDM-related profiles, etc.)
Deliverables
• Two-Constants Validation Report v1
• Summary tables: parameters, confidence intervals, goodness-of-fit, and model comparisons per object
• Reproducibility package (data index + scripts + one-click figure/table generation)
Success criteria
• Statistically supported consistency of the two constants across multiple object classes—together with explicit failure boundaries (where it does not hold and why)

Theme 3: Tracking the Two Constants in Microscopic Systems with Control Tests
Chinese title: 微观系统中两常数的可追踪性与对照检验
English title: Tracking the Two Constants in Microscopic Systems with Control Tests
One-line positioning
Extend the “two constants” from astrophysical systems into microscopic/high-energy domains—while prioritizing controls and falsifiable predictions, avoiding purely post hoc fitting.
Research strategy (state clearly on the website)
• First, define the dimensionless invariants corresponding to the two constants and specify the observable mappings
• Then, test whether consistent scaling behavior emerges in public datasets
• Must include standard-model / mainstream-theory control fits and information-criterion comparisons
Candidate data domains (to be expanded with your materials)
• Public high-energy scattering cross sections, spectra, and structure-function datasets (e.g., CMS, HERA, LEP/OPAL public tables)
• Optional extensions: cosmic-ray spectra, astroparticle statistics, etc., depending on the mapping you establish
Deliverables
• Micro-domain Tracking Note v1 (methods and negative results published as first-class outputs)
• A unified control framework: same metrics, same model-selection workflow, and consistent uncertainty propagation
Success criteria
• In at least one microscopic domain, the constant mapping yields ex ante predictions (holdout or future data) that pass testing; or, alternatively, a clearly documented “non-validity boundary”

Theme 4: Dual Tensor Invariance Principle—Axiomatization, Derivation, and Testable Implications
Chinese title: 对偶张量不变性原理：公理化、推导与可检验推论
English title: Dual Tensor Invariance Principle: Axiomatization, Derivation, and Testable Implications
One-line positioning
Upgrade the core theory from explanatory narrative to a reviewable mathematical structure with falsifiable implications.
Key tasks
• Axiomatic formulation: define objects, transformation groups, conserved quantities/invariants, domains of validity, and boundary conditions
• Derivation chain: from invariance → observable relations → necessary/sufficient conditions under which the two constants emerge (prefer explicit propositions)
• Connection to established theory: symmetry–conservation links (Noether-style structure as a reference point, while keeping your framework primary)
• Testable implications: list at least 3–5 observable predictions spanning astrophysical and microscopic directions
Deliverables
• Principle Paper v1 (a submission-ready theoretical-paper framework)
• Formal notation glossary; theorem/proposition list; proof sketches; and a versioned review log
• Prediction Table: each implication mapped to data sources and test protocols
Success criteria
• A peer-reviewable formal text (definitions → propositions → proofs/derivations → implications)
• Implications can be independently tested and replicated using public data
Theme 1：优化拟合函数
中文标题：结构场拟合函数优化与模型选择
English: Fit-Function Optimization and Model Selection for Structural Fields
一句话定位：把拟合函数从“能拟合”升级为“可解释、可比较、可复现、可推广”。
核心目标
• 提升拟合稳定性（收敛、鲁棒、多起点一致）
• 降低参数冗余与不可辨识性（identifiability）
• 建立统一模型选择标准（AIC/BIC/交叉验证/贝叶斯证据）
关键任务
• 函数族对比：tanh / logistic / arctan / 分段光滑 / 物理约束型函数
• 正则化与约束：单调性、边界条件、物理量非负、尺度不变约束
• 误差模型：测量误差、系统误差、外点处理（robust loss）
• 复现基准：同一数据、同一初始化策略、统一输出指标（R²/RMSE/残差谱）
交付物
• Fit-function benchmark 报告（表格 + 图）
• 开源脚本与统一接口（CLI + config）
• “推荐函数族”与适用域（何种对象适合哪种函数）
成功标准（可量化）
• 在代表性数据集上，最优函数族在 RMSE、残差结构、参数稳定性上显著优于基线
• 多起点拟合收敛到同一最优解的比例提升（例如 >90%）

Theme 2：物理系统拟合与两个常数验证
中文标题：跨尺度物理系统拟合与两常数一致性检验
English: Cross-Scale Fitting of Physical Systems and Two-Constant Consistency Tests
一句话定位：把“两常数”从“在某些对象上出现”升级为“跨对象、跨数据源、跨方法仍成立”的可检验结论。
覆盖对象（建议官网列为可扩展清单）
• 星系旋转曲线、强引力透镜系统、（如适用）CMB 冷斑/空洞剖面
• 后续可扩展：太阳系结构、星系团动力学等
关键任务
• 统一数据流水线：数据来源、清洗、单位、误差、版本号
• 统一拟合标准：同一优化器、多起点、置信区间、残差诊断
• 常数检验：
• 分布稳定性（不同对象是否同一分布）
• 线性/幂律关系的显著性与敏感性分析
• 与对照模型（经验模型、ΛCDM 常用剖面等）定量比较
交付物
• “Two-Constants Validation Report v1”
• 汇总表：每个对象的参数、置信区间、拟合优度、模型对照
• 可复现包（数据索引 + 脚本 + 一键生成图表）
成功标准
• 两常数在多个类别对象上保持一致的统计证据（并给出失败边界：在哪些对象上不成立、为什么）

Theme 3：微观系统对两个常数的追踪研究
中文标题：微观系统中两常数的可追踪性与对照检验
English: Tracking the Two Constants in Microscopic Systems with Control Tests
一句话定位：把“两常数”从天体系统推进到微观/高能数据域，但必须以“对照组与可证伪预测”为核心，避免只做“后验拟合”。
研究策略（建议在官网写清楚）
• 先定义：两常数对应的无量纲组合（dimensionless invariants）与可观测映射
• 再验证：在公开数据上是否出现一致标度行为
• 必须包含：标准模型/主流理论的对照拟合与信息准则比较
可能数据域（按你后续资料补齐即可）
• 公开高能散射截面、谱分布、结构函数等数据（如 CMS、HERA、LEP/OPAL 等公开表）
• 也可包含：宇宙线谱、天体粒子统计等（取决于你要建立的“同一常数”映射）
交付物
• “Micro-domain Tracking Note v1”（方法与负结果同样发布）
• 统一对照框架：同一指标、同一模型选择流程、同一不确定性传播
成功标准
• 至少在一个微观数据域中，常数映射能提出事前预测（holdout/未来数据）并通过检验；或明确给出“不成立边界”

Theme 4：对偶张量不变性原理的论证
中文标题：对偶张量不变性原理：公理化、推导与可检验推论
English: Dual Tensor Invariance Principle: Axiomatization, Derivation, and Testable Implications
一句话定位：把核心理论从“解释性叙述”升级为“可审阅的数学结构 + 可证伪推论”。
关键任务
• 公理化表述：定义对象、变换群、守恒量/不变量、适用域与边界条件
• 推导链：从不变性 → 可观测量关系 → 两常数出现的必要/充分条件（尽量给出明确命题）
• 与现有理论的连接：对称性/守恒（可参考 Noether 风格的结构，但以你的框架为主）
• 可检验推论：至少列出 3–5 条可观测预测（对应天体/微观两个方向）
交付物
• “Principle Paper v1”（可投稿的理论论文框架）
• 形式化符号表、定理/命题列表、证明草案与审阅版本记录
• “Prediction Table”：每条推论对应数据源与检验方法
成功标准
• 给出可同行审阅的形式化文本（定义—命题—证明/推导—推论）
• 推论可被独立团队用公开数据重复检验
Sponsorship Proposal (Short Form)
1. Project Title and Positioning
Project Title: Structural-Field Fitting and Two-Constant Validation: Reproducible Methods and Falsifiable Conclusions (2026 Research Themes)
Applicant: AEEA Research Team (Avoid Earth Extinction Association)
Core Engine: MEST-AI and MEST-TPC (Mass–Energy–Spacetime Turning-Point Tensor Computation)
Mission Statement: To raise structural-field fitting from “it fits” to “interpretable, comparable, reproducible, and transferable,” and to test the “two constants” as a cross-object, cross-dataset, cross-method hypothesis—explicitly documenting both validation evidence and failure boundaries.

2. Why This Is Worth Supporting
Cross-scale modeling efforts often face recurring bottlenecks:
• Fit functions may match data yet remain non-interpretable and non-comparable across studies.
• Non-identifiability and parameter redundancy can create “apparent laws” that are difficult to falsify.
• Differences in datasets, uncertainty treatment, optimizers, and initialization strategies undermine reproducibility and lead to non-aligned conclusions.
This project addresses these gaps by delivering:
1. Standardized benchmarks (same data, same protocols, same metrics)
2. Formal model selection (AIC/BIC/cross-validation/Bayesian evidence)
3. Falsifiability-first outputs (negative results and failure boundaries are first-class deliverables)
4. Reproducibility packages (data indices, versioning, scripts, and one-click figure/table generation)

3. Research Themes and Deliverables
Theme 1: Fit-Function Optimization and Model Selection for Structural Fields
Goal: Improve fitting stability and interpretability, reduce redundancy and non-identifiability, and establish unified model-selection standards.
Primary deliverables:
• Fit-function benchmark report (tables + figures)
• Open-source scripts and a unified interface (CLI + config)
• “Recommended function families” and applicability guidance (object-to-function mapping)

Theme 2: Cross-Scale Fitting of Physical Systems and Two-Constant Consistency Tests
Goal: Upgrade the “two constants” from “appearing in some objects” to a testable conclusion that remains valid across object classes, datasets, and methods—or yields explicit failure boundaries.
Primary deliverables:
• Two-Constants Validation Report v1
• Summary tables: parameters, confidence intervals, goodness-of-fit, and control-model comparisons per object
• Reproducibility package (data index + scripts + one-click figure/table generation)

Theme 3: Tracking the Two Constants in Microscopic Systems with Control Tests
Goal: Extend the two-constant hypothesis into microscopic/high-energy public datasets with controls and ex ante predictions as the core—avoiding purely post hoc fitting.
Primary deliverables:
• Micro-domain Tracking Note v1 (methods and negative results published as first-class outputs)
• Unified control framework: consistent metrics, identical model-selection workflow, and uncertainty propagation

Theme 4: Dual Tensor Invariance Principle—Axiomatization, Derivation, and Testable Implications
Goal: Elevate the core theory from explanatory narrative to a reviewable mathematical structure with falsifiable implications.
Primary deliverables:
• Principle Paper v1 (a submission-ready theoretical-paper framework)
• Formal notation glossary; theorem/proposition list; proof sketches; and a versioned review log
• Prediction Table: each implication mapped to data sources and test protocols

4. 2026 Work Plan (Proposed Milestones)
Q1 — Standards and pipelines
• Finalize dataset and uncertainty conventions; define fitting protocols (optimizer, multi-start, output metrics)
• Theme 1: initial function-family comparisons; reproducibility interface (CLI) prototype
Q2 — Benchmarking and controls
• Theme 1: publish Benchmark v1 (stability, residual structure, parameter stability)
• Theme 2: begin systematic multi-class validation and compile midterm summary tables
Q3 — Cross-domain extension and ex ante testing
• Theme 2: complete cross-object consistency tests and sensitivity analyses
• Theme 3: complete at least one micro-domain mapping + control comparisons + holdout testing
Q4 — Annual deliverables and publication-grade outputs
• Two-Constants Validation Report v1 + reproducibility package
• Principle Paper v1 (formal text + implication table)
• Annual Methods & Integrity Report (including negative results and failure boundaries)

5. Budget Usage (High-Level)
Sponsorship support will primarily fund four categories:
1. Data curation and versioning: source indexing, unit/uncertainty standardization, archival publishing
2. Compute and engineering: multi-start optimization, cross-validation, model selection, batch pipelines
3. Review and integrity process: third-party review, reproducibility verification, and versioned documentation
4. Publication and dissemination: reproducibility package releases, figures/reports, public briefs/workshops
(If requested, we can provide a tiered budget with quarterly acceptance criteria and line items for compute, labor, and data-publication costs.)

6. Integrity, Compliance, and Research Standards
• Mandatory control models: all key claims must be benchmarked against controls that do not rely on our framework (empirical models, widely used profiles, etc.).
• Falsifiability-first: we will explicitly state what observations would refute each hypothesis or implication.
• Reproducible by design: reports will ship with data indices, version numbers, scripts, and full configuration settings.
• No over-claims: improved fits are not treated as proof of physical truth; decisions rely on information criteria, robustness, and controls.

7. Sponsor Value (Non-Equity, Research-Oriented Returns)
• Early access: quarterly progress briefs, annual reports, and reproducibility-package updates
• Transparent reviewability: access to protocols, control designs, version logs, and negative results
• Acknowledgment: sponsor recognition in public reports and publications (aligned with sponsor preference)
• Engagement option: sponsors may nominate a representative as a reviewer/observer for milestone reviews (without influencing conclusions)

8. Sponsorship Request
We invite sponsors to support the 2026 research themes and their public, reproducible deliverables through:
• Annual sponsorship: full coverage of Themes 1–4 and annual deliverables
• Directed sponsorship: focus on one theme (e.g., Theme 2 validation or Theme 4 formalization)
• In-kind support: compute credits, data services, or reproducibility-review support
Upon request, we can provide:
• A one-page sponsor brief (ultra-condensed)
• A tiered budget with quarterly deliverables and acceptance criteria
• Sample table of contents for public vs technical report editions
项目名称（中文）：结构场拟合与两常数验证：从可复现方法到可证伪结论（2026 研究主题集）
Project Title (EN): Structural-Field Fitting and Two-Constant Validation: Reproducible Methods and Falsifiable Conclusions (2026 Research Themes)
申请单位：AEEA Research Team（Avoid Earth Extinction Association）
研究引擎：MEST-AI 与 MEST-TPC（Mass–Energy–Spacetime Turning-Point Tensor Computation） 统一方法平台
核心价值：以“可复现、可对照、可证伪”为硬标准，将结构场拟合从“能拟合”升级为“可解释、可比较、可推广”，并对“两常数”给出跨对象、跨数据源、跨方法的一致性检验与失败边界。
当前跨尺度建模研究面临普遍问题：
拟合函数可拟合但不可解释，结果难以比较与复核；
参数不可辨识（identifiability）与过拟合造成“看似规律、难以证伪”；
不同研究在数据、误差、优化器与初始化策略上差异巨大，导致结论不可对齐。
本项目以方法学为主轴，明确提供：
统一基准（Benchmark）：同数据、同协议、同指标；
模型选择（Model Selection）：AIC/BIC/交叉验证/贝叶斯证据等标准化流程；
可证伪输出：不仅给“成功案例”，也发布负结果与失败边界；
可复现包：数据索引、版本号、脚本与一键出图，便于第三方复核。
目标：提升拟合稳定性与可解释性，降低参数冗余与不可辨识性，建立统一模型选择标准。
主要产出：
Fit-function benchmark 报告（表格 + 图）
开源脚本与统一接口（CLI + config）
“推荐函数族”与适用域指南（对象—函数映射）
目标：将“两常数”从“部分对象出现”升级为**跨对象、跨数据源、跨方法仍成立（或明确不成立边界）**的可检验结论。
主要产出：
Two-Constants Validation Report v1
汇总表：参数、置信区间、拟合优度、对照模型比较
可复现包（数据索引 + 脚本 + 一键生成图表）
目标：把两常数映射到微观/高能公开数据域，强调对照组与事前预测，避免纯后验拟合。
主要产出：
Micro-domain Tracking Note v1（方法与负结果同样发布）
统一对照框架：同指标、同模型选择流程、同不确定性传播
目标：把核心理论升级为可审阅的数学结构，产出可证伪推论与数据检验路线。
主要产出：
Principle Paper v1（可投稿的理论论文框架）
符号表、命题/定理列表、证明草案与审阅记录
Prediction Table（每条推论对接数据源与检验方法）
Q1：基准与管线成型
数据与误差口径统一；拟合协议（优化器、多起点、输出指标）定版
Theme 1 函数族对比与约束机制初版；复现接口（CLI）雏形
Q2：系统化 benchmark 与对照框架
Theme 1 发布第一版 benchmark（含稳定性、残差结构、参数稳定性）
Theme 2 开始多类别对象批量验证，形成中期汇总表
Q3：跨域扩展与事前预测
Theme 2 完成跨对象一致性检验与敏感性分析
Theme 3 完成至少一个微观数据域的“映射定义 + 对照比较 + holdout 检验”
Q4：年度交付与可发布版本
Two-Constants Validation Report v1 + 复现包
Principle Paper v1（理论文本 + 推论表）
年度 Methods & Integrity Report（包括负结果与失败边界）
赞助资金将主要用于四类成本：
数据整理与版本化：数据源索引、单位与误差模型规范、存档与可追溯发布
算力与工程化：多起点拟合、交叉验证、模型选择与批处理管线
研究与审阅机制：第三方审阅、复现验证、方法文档与审阅记录维护
成果发布与传播：可复现包发布、图表与报告制作、公开简报/研讨会
（如赞助方需要，我们可提供分档预算与资源明细：算力、人员工时、数据发布成本等。）
对照模型强制：所有核心结论必须与不依赖本框架的对照模型比较（经验模型、主流剖面等）。
可证伪优先：报告中将明确列出“哪些观测会否定我们的假说/推论”。
可复现交付：数据索引、版本号、脚本与参数设置随报告发布；负结果与失败边界同等披露。
不做夸大承诺：不将“拟合更好”直接等同于“物理真实”，以信息准则、稳健性与对照为裁决标准。
优先获取：季度进展简报、年度报告、复现包更新
透明可审阅：方法定义、对照设计、版本记录与负结果同步披露
署名与致谢：公开报告与论文中的赞助致谢（按赞助等级与意愿）
参与机制：赞助方可指定代表作为 Reviewer/Observer 参与阶段评审（不干预结论）
我们诚挚邀请赞助方支持 2026 年度研究主题集的推进。
可选支持方式：
年度赞助：覆盖全主题（Theme 1–4）与年度交付物
定向赞助：指定某一主题（如 Theme 2 两常数验证 或 Theme 4 原理公理化）
资源赞助：算力/数据服务/复现审阅支持
如需，我们可以提供：
1 页 Sponsor Deck（更短版）
分档预算与具体交付清单（按季度验收）
公开版与技术版报告样例目录