EML-MCP

# EML-MCP MCP server for EML (Exp-Minus-Log) mathematical notation. `eml_mcp_server.py` is a single-file MCP server focused on symbolic EML, pure-mode EML, SymPy comparison, and a compact set of generalized EML family tools derived from the follow-up algebraic-structure paper. ## Why this project came to fruition The EML methods paper inspired me to develop a new MCP server so that I could test out the EML methods. The follow-up paper on the algebraic structure behind EML then motivated a second layer of tooling: compact MCP functions for exploring generalized EML families, their hidden abelian-group structure, their constructive recovery chain, and the open constant-free generator question. ## What is included The server exposes thirteen MCP tools: - `eml_compile` - `eml_eval` - `eml_simplify` - `eml_stability_check` - `eml_fit` - `sympy_eval` - `sympy_simplify` - `eml_family_library` - `eml_extract_group_structure` - `eml_recover_core_family` - `eml_generate_from_addition_formula` - `eml_constant_free_scan` - `eml_explore_family` ## Practical mode and pure mode The EML route supports two compilation styles: - **practical mode** — the default mode, designed for compact trees and efficient evaluation - **pure mode** — an opt-in mode that rewrites numeric constants into EML trees built from the distinguished constant `1` and the `eml(...)` operator Pure mode is intended for paper-style structural inspection and experimentation. Variables are still represented as variables, but numeric constants are rewritten into pure EML-style constant constructions. The EML tools that support the `pure` flag are: - `eml_compile` - `eml_eval` - `eml_simplify` - `eml_stability_check` - `eml_fit` ## Generalized EML family tools The `eml_family_*` tools are curated structural tools based on the generalized operator view from the follow-up paper. They are intentionally concise and work with a built-in family library rather than attempting to prove arbitrary user-defined operators. ### Built-in families - `original_eml` - `cosine_arccos` - `arccot_cot` - `tanh_artanh` - `elliptic_pair` - `involutive_piecewise` ### What the family tools do - `eml_family_library` — list the built-in family catalogue or inspect one family - `eml_extract_group_structure` — expose the neutral element, inverse map, and induced abelian group law - `eml_recover_core_family` — show the six-step constructive recovery chain - `eml_generate_from_addition_formula` — report the family-specific addition formula and derived operations - `eml_constant_free_scan` — summarise the open constant-free-generator question and curated candidates - `eml_explore_family` — return an aggregated view of one family ## Main files - `eml_mcp_server.py` — single-file MCP server, direct examples, and test harness - `docs/FUNCTION_CATALOG.md` — tool catalogue and argument overview - `docs/ARCHITECTURE.md` — component-level design - `docs/METHOD.md` — mathematical and computational approach - `docs/TEST.md` — how to run the MCP tests and call the server from Python - `SKILLS.md` — practical usage guide for humans and LLMs ## Run modes ```bash python3 eml_mcp_server.py server python3 eml_mcp_server.py test python3 eml_mcp_server.py examples python3 eml_mcp_server.py all ``` ## Semi-complex but readable examples ### Example 1: identity through both routes ```text sin(x)**2 + cos(x)**2 ``` - compile it to EML with `eml_compile` - simplify the resulting tree with `eml_simplify` - confirm the same identity with `sympy_simplify` ### Example 2: inspect pure mode for constants ```text pi + e + 1/2 ``` Use `eml_compile` with `pure: true` to inspect how the constants are rewritten into an EML tree whose constant leaves reduce to the distinguished constant `1`. ### Example 3: evaluate a familiar geometric form ```text sqrt(a^2 + b^2) ``` Use `eml_eval` and `sympy_eval` with bindings such as `a=5`, `b=12` to confirm that both routes give `13`. ### Example 4: inspect a mild branch-risk case ```text sin(x) + log(x) ``` Use `eml_compile` followed by `eml_stability_check` over a region such as `x in [0.2, 2.0]` to see how the logarithm affects the numerical risk profile. ### Example 5: inspect the built-in family library Use `eml_family_library` to list the curated generalized EML families, then request one by name such as `original_eml` or `tanh_artanh`. ### Example 6: inspect one family structurally Use `eml_explore_family` with a built-in family name to see: - the operator formula - the induced group structure - the recovery chain - the addition formula - the derived operations ### Example 7: fit a compact law from data Use `eml_fit` on a small and interpretable dataset, for example `y = log(x)` sampled at `x = [0.5, 1.0, 2.0, 4.0]`, and compare the best candidate families. ## Requirements The script uses: - Python 3.10+ - `numpy` - `sympy` - `scipy` - `mcp` for `server` and `test` modes ## MCP transport The server uses the official MCP Python SDK over stdio, which makes it easy to test from Python subprocesses and LLM tool wrappers. ## References - Andrzej Odrzywołek, *All elementary functions from a single operator* (EML methods) https://arxiv.org/html/2603.21852v2 - Tomasz Stachowiak, *Algebraic structure behind Odrzywołek’s EML operator* https://arxiv.org/abs/2604.23893