An open mathematical research database of Conservative Matrix Fields — 2D and higher-dimensional lattice recurrences with path-independent matrix products, encoding continued fractions, hypergeometric series, and named constants.
The CMF Atlas is currently in its beta stage. This platform is designed to facilitate mathematical research by making Conservative Matrix Fields (CMF) more accessible to the global community.
Please be aware that this is a live, ongoing project. Our current efforts are focused on two primary areas:
As this is a work in progress, errors in data may occur. We highly value community involvement — if you identify any discrepancies or bugs, please let us know via the Contact Page.
This is an independent project dedicated to highlighting the research conducted by the Ramanujan Machine Team. The structures presented here are built upon their discovery of the Conservative Matrix Field.
For further technical details, please refer to their official resources:
A Conservative Matrix Field is a pair of matrix-valued functions \(K_1(k,m)\) and \(K_2(k,m)\) on a 2D integer lattice satisfying the flatness condition:
This means the matrix product along any lattice path from \((0,0)\) to \((K,M)\) is path-independent. The limiting value — as \(K,M\to\infty\) — converges to constants such as \(\zeta(3)\), \(\pi\), \(\ln 2\), and \(\zeta(5)\).
For the telescope family with polynomial form \(f(x,y)\):
where \(g(k,m)=f(k,m)\), \(\bar{g}(k,m)=\bar{f}(k,m)\), \(b(k)=g(k,0)\cdot\bar{g}(k,0)\), \(a(k,m)=g(k,m)-\bar{g}(k{+}1,m)\).
CMF Atlas v2.3 contains entries from multiple discovery pipelines, including pFq CMFs from RamanujanTools.
All public CMF entries expose the following canonical fields via GET /cmfs/browse and GET /cmfs/{id}/full:
idUnique integer identifier (stable across releases)dimensionLattice dimension (integer ≥ 2)f_polySymPy polynomial f(x,y) — present for telescope-polynomial CMFsfbar_polyConjugate polynomial f̄(x,y)degreeTotal polynomial degree of fprimary_constantCanonical label of the identified limit (e.g. “ζ(3)”, “π”)identified_constantPSLQ-identified exact form, if foundflatness_verifiedBoolean — true if symbolic flatness proof existscanonical_fingerprintHash-based unique key for the representationentry_uriCanonical URL for the entry pagerelease_versionAtlas version in which this entry is includedcertification_level
A_plus
A_certified
B_verified_numeric
C_scouting
proof_statussymbolically_certified | verified | numeric_only | unverifiedidentification_statuspslq_identified | matched | unidentifiedconstruction_typetelescope_polynomial | hypergeometric_pfq | matrix_explicitsource_familyRamanujanTools | Gauge Transformed | CMF Hunter | …irrationality_statusproven_transcendental | proven_irrational | open_question | algebraictranscendence_statusproven_transcendental | open_question | algebraicFull schema: /docs · Machine-readable release manifest: /release
If you use CMF Atlas in research, please cite it. Data licensed under CC BY 4.0. A DOI is pending registration — use the BibTeX below in the interim.
@misc{vesterlund2026cmfatlas,
author = {Vesterlund, David},
title = {{CMF Atlas}: A Database of Conservative Matrix Fields},
year = {2026},
url = {https://davidvesterlund.com/cmf-atlas/},
note = {Version 2.3, open research dataset. DOI pending.},
license = {CC BY 4.0}
}
@misc{vesterlund2026cmf1,
author = {Vesterlund, David},
title = {{CMF Atlas} Entry \#1: $f(x,y) = (x+y)^3$},
year = {2026},
url = {https://davidvesterlund.com/cmf-atlas/entry.html?id=1},
note = {CMF Atlas v2.3, limit = $\zeta(3)$},
license = {CC BY 4.0}
}
Each entry page provides a pre-filled BibTeX block with the correct formula and metadata.
CMF Atlas is served by a FastAPI backend at cmfatlas-production.up.railway.app. Interactive Swagger docs: /docs.
All endpoints are available at both /endpoint and /api/v1/endpoint (stable versioned alias, redirects 307).
Versioned alias example: GET /api/v1/cmfs/browse?dimension=2
CMF Atlas includes CMFs imported directly from the
RamanujanTools repository.
Currently imported: e, pi, symmetric_pi, zeta3,
hypergeometric_derived_2F1 (3D), hypergeometric_derived_3F2 (5D, at z=½),
and five FFbar parametric families. All carry A+ certification.
# Browse pFq CMFs in the atlas
GET /cmfs/browse?source_type=pfq
# Individual entry with hypergeometric parameters
GET /cmfs/{id}/full
The CMF Atlas dataset and documentation are released under Creative Commons Attribution 4.0 (CC BY 4.0).
You are free to share, adapt, and use the data for any purpose including commercial and academic research, provided you give appropriate credit.