CMF Sources & Families

RamanujanTools — Generalized Hypergeometric pFq CMFs

This family consists of CMFs imported directly from the RamanujanTools open-source library, developed by the Ramanujan Machine research group at the Technion — Israel Institute of Technology.

These entries represent Conservative Matrix Fields built from generalized hypergeometric series \({}_pF_q\), including 3F2, 4F3, 5F4, and 6F5 families. Their construction is grounded in the deep mathematical framework connecting hypergeometric recurrence relations to path-independent matrix products. Each entry has been formally verified by the Ramanujan Machine team and is assigned A+ certification in the CMF Atlas.

The target constants include \(\zeta(3)\) (Apéry's constant), \(\zeta(5)\), \(\pi\), Catalan's constant \(G\), and other named constants of high mathematical interest.

Key characteristics

• Higher-dimensional structures: 4D, 5D, 7D, 9D, 11D (pFq order determines dimension)
• Built from generalized hypergeometric \({}_pF_q\) series (3F2 through 6F5)
• Targets: \(\zeta(3)\), \(\zeta(5)\), \(\pi\), Catalan's constant \(G\)
• All carry A+ certification — verified by the Ramanujan Machine team

Gauge Transformed — Extended Coordinate Systems

What is a gauge transformation in this context?

A gauge transformation — as used here — refers to the process of re-expressing a known one-dimensional number series or recurrence relation as a two-dimensional (or higher-dimensional) matrix system on an integer lattice. The term is borrowed from physics, where gauge transformations change the local description of a system without altering its physical content.

In the CMF framework, a known scalar recurrence \(a_{n+1} = r(n) \cdot a_n\) can often be "lifted" into a 2D matrix product \(K_1(k,m)\cdot K_2(k+1,m) = K_2(k,m)\cdot K_1(k,m+1)\) by choosing matrix entries that reproduce the original scalar recurrence along the diagonal. The resulting CMF is then path-independent by construction.

What gauge transformation does — and does not — provide

Gauge transformation is a powerful tool for studying a number series. By embedding a 1D recurrence into a 2D coordinate system, researchers can analyze:

• Convergence speed: How quickly the lattice walk converges to the limiting constant
• Arithmetic structure: The GCD, height, and complexity of matrix entries
• Irrationality measures: Lower bounds on how "far" the limit is from rationals
• Families of related series: Systematic variations via the free parameter \(m\)

Important note: Gauge transformation does not create genuinely new mathematical constants or identities. The limit of a gauge-transformed CMF is the same constant as the original 1D series. The 2D embedding widens the perspective — offering richer tools to study the series — but the underlying mathematics is inherited from the source recurrence.

Sources in this family include D-finite recurrences, Ore algebra constructions, OEIS-bank series, training seed series, and known mathematical families lifted to 2D.

Browse Gauge Transformed entries →

CMF Hunter — Proprietary Discovery Engine

The CMF Hunter family contains Conservative Matrix Fields discovered by David Vesterlund's proprietary CMF generation algorithm — an ongoing research project to build a robust, systematic method for finding genuinely new CMFs that cannot be derived from known series by simple gauge transformation.

Core methodology

The central idea is to exploit the path-independence criterion as a system of nonlinear equations. Concretely:

1. Seed with recurring functions drawn from open databases such as LMFDB, OEIS, and known polynomial families
2. Parameterize the matrix entries \(K_1(k,m)\) and \(K_2(k,m)\) with unknown coefficients
3. Impose the flatness condition \(K_1 K_2(k{+}1,m) = K_2 K_1(k,m{+}1)\) as a constraint system
4. Solve the resulting polynomial equations to find coefficient combinations that satisfy path-independence across the full integer lattice
5. Verify flatness numerically and symbolically; identify the limiting constant if possible

The process is highly iterative — hundreds of candidate families are generated and rejected before a valid CMF with interesting convergence properties is found. This is not a brute-force search: the seed selection and parameterization strategy are guided by structural insights about which polynomial families are likely to yield flat matrix fields.

The CMF Hunter also includes results from an ML-assisted variant (the ml_loop pipeline), which uses a trained model to propose candidate polynomial families for the flatness equation solver.

Mathematical significance

Unlike gauge-transformed CMFs, Hunter entries represent genuinely new structures: the CMFs discovered here encode number series and lattice recurrences that are not known to arise from standard scalar recurrences. Many Hunter CMFs have unidentified limiting constants — making them active research targets for PSLQ-based constant recognition and irrationality proofs.

Browse CMF Hunter entries →

Verification Levels

Each CMF in the atlas carries a certification level indicating the strength of its mathematical verification.