How the Atlas is organised, what the curation categories mean, how convergence rate (δ) is measured, and how to interpret path-independence evidence for 2D and 3D CMFs.
In particular, numerical agreement between several trajectories is evidence, not a formal proof, unless it is supported by an exact cocycle identity or another symbolic certification method.
The Atlas assigns each entry to one of three curation categories. These are editorial metadata used for browsing and prioritisation — they are not mathematical theorems. The classification is based on a combination of dimension, polynomial degree or operator complexity, identification status of the limiting value, and certification level where available.
Low-complexity entries used as a baseline for testing, calibration, and comparison. Typical examples include small 2D families, rational limits, and simple gauge-equivalent families.
Entries whose limiting value appears mathematically nontrivial — either because it is identified with a known constant, or because it appears irrational but has not yet been fully identified. These are the core objects in the Atlas.
Higher-dimensional entries, structurally richer families, and entries under active investigation. Intended for objects that may warrant further symbolic analysis, proof attempts, or comparison with known CMF constructions.
The self-delta δ is a coarse empirical estimate of convergence speed. It measures the average gain in binary digits of accuracy per step, using the final computed value as a proxy for the limit. It is defined from three points along the walk:
\[ \delta \;=\; \frac{\log_2\!\bigl(|v_{n_1} - v_N|\bigr) \;-\; \log_2\!\bigl(|v_{n_2} - v_N|\bigr)}{n_2 - n_1} \quad \text{bits per step} \]where \(n_1 = \lfloor N/3 \rfloor\), \(n_2 = \lfloor 2N/3 \rfloor\), \(v_k\) is the partial convergent at step \(k\), and \(v_N\) is the last value (used as a proxy for the true limit).
δ is displayed in the Explorer after every walk, with a quality indicator and a link back to this page.
A Conservative Matrix Field (CMF) is called flat or path-independent when its matrix transport depends only on the endpoints, not on the monotone lattice path used between them. In dimension 2, this is encoded by the local cocycle condition (also called a zero-curvature condition) on each elementary lattice square:
\[ K_x(k,\,m) \cdot K_y(k+1,\,m) \;=\; K_y(k,\,m) \cdot K_x(k,\,m+1) \]When this identity holds at every lattice point, products along different monotone paths between the same lattice points are equal — not merely equal in the limit, but exactly equal at every finite step. Limit-independence of the associated continued fraction is a consequence of this finite structural property.
The terminology is analogous to conservative vector fields in calculus: path-independence means that the line integral depends only on the endpoints. In the CMF setting, the analogous statement is that matrix transport between lattice points depends only on the endpoints when the cocycle condition holds. The finite path-independence of matrix products is the structural property; limit-independence of the associated continued fraction is one of its consequences.
For 2D CMFs with an explicit polynomial form, the 2D Flatness Residual Test page evaluates the residual
\[ R(k,m) \;=\; K_x(k,m)\,K_y(k{+}1,m) \;-\; K_y(k,m)\,K_x(k,m{+}1) \]at every lattice point in a finite test grid. If the field is exactly flat, this residual vanishes identically. The test reports the maximum Frobenius norm of \(R\) across the grid. Values close to machine precision provide strong numerical evidence of flatness on that grid, but finite-grid numerical testing should not be confused with an exact symbolic proof.
In dimensions \(d \ge 3\), the rigorous notion is still given by the relevant cocycle identities among the generating matrices or operators. For rigorous certification, the Atlas should eventually expose either the full generating operator data or an exact symbolic identity implying the relevant cocycle relations.
A practical numerical test is to observe whether multiple trajectories with different fixed shifts converge to the same value. For a flat CMF, Kx walks with different fixed coordinates \(m_0\) should yield the same limit. Observing this numerically at large depth is empirical evidence consistent with path-independence. It is not a formal proof unless accompanied by a symbolic verification, and it may be inconclusive when convergence is slow. The demo in Section 4 illustrates this approach.
This demo compares three Kx trajectories for the same 3D CMF with m = 0, 1, 2 (different fixed rows). Its purpose is to test whether the sampled trajectories appear to approach a common value.
A trajectory is obtained by multiplying a sequence of \(2\times 2\) matrices along one chosen lattice direction:
\[ P_N \;=\; K(s_0)\cdot K(s_0+1)\cdots K(s_0+N-1) \]The convergent at step \(N\) is read from the resulting Möbius action as
\[ v_N \;=\; \frac{P_N[0,1]}{P_N[1,1]} \]whenever the denominator is nonzero. For 2D CMFs, \(K = K_x(k,\, m_\text{fixed})\) or \(K = K_y(k_\text{fixed},\, m)\). For 3D CMFs a third direction \(K_z\) is also available.
For 2D entries arising from a conjugate polynomial pair \((f, \bar f)\), the standard conservative-matrix-field construction defines
\[ M_X(x,y) \;=\; \begin{pmatrix} 0 & b(x) \\ 1 & a(x,y) \end{pmatrix}, \qquad M_Y(x,y) \;=\; \begin{pmatrix} \bar f(x,y) & b(x) \\ 1 & f(x,y) \end{pmatrix} \]where the coefficient functions are
\[ a(x,y) \;=\; f(x,y) - \bar f(x+1,y), \qquad b(x) \;=\; f(x,0)\cdot\bar f(x,0). \]In this construction \(b\) depends only on the walk variable \(x\), not on the fixed coordinate \(y\). The horizontal trajectory at fixed \(y = y_0\) uses \(M_X(k, y_0)\); the vertical trajectory at fixed \(x = x_0\) uses \(M_Y(x_0, m)\).
For higher-dimensional entries (3D CMF Hunter, RamanujanTools), the Atlas may use an operator-level or source-specific internal representation. In those cases the displayed polynomial should be understood as metadata or a projection unless the full generating matrices are explicitly provided. RamanujanTools entries store their matrix expressions directly and the walk evaluates them by symbolic substitution at each step.
Precision is maintained at 30 decimal digits (mpmath). The product is normalised by a nonzero scalar every 20 steps to prevent overflow; this does not change the convergent \(v_N\). The self-delta δ is computed from three widely-separated trajectory points; a separate local convergence-rate indicator is computed from the final window of convergents.