Supported calculations and error guarantees

The beta prerelease scope is explicit below. General multiloop RMT evaluation remains a research objective. Inputs are squared masses; the graph APIs use positive Euclidean propagators and \(d^Dk/\pi^{D/2}\) per loop. The proposed stable-versus-experimental interface classification is in API stability and compatibility; work toward the next release is in Plan for the first stable release.

API	Supported calculation	Method	Error information
ramanujan_master_theorem	User-supplied coefficient continuation; theorem hypotheses are the caller’s responsibility	\(\Gamma(s)\varphi(-s)\)	Arithmetic precision; no automatic proof of theorem hypotheses
product_integral	Products from the documented Mellin atlas in accepted domains	Mellin residue summation	Convergence checks, tail/error estimates and cancellation control
B0, C0, bubble_3d	Named scalar one-loop functions; C0 has two lightlike external legs	Analytic functions and numerical evaluation	Working-precision agreement; independent regression data
sunset_im, sunset_subtracted, sunset_d2	Specified scheme-independent four-dimensional sunset combinations	Phase-space and dispersion integration	Precision and quadrature checks
integrate_graph	UV-convergent massive Euclidean scalar graphs, positive powers and dimension, bounded enumeration	RMT for the common scale; numerical Hepp sectors	Float64 statistical diagnostics; no deterministic error enclosure
schwinger_brackets	Exact Schwinger bracket algebra for ScalarGraph	Formal exponent matrices and Gamma-family candidates	Exact linear algebra; no convergence certificate or generic evaluator
integrate_graph_rmt	Massive unit-power D=2 sunset; \(Q>m_1+m_2+m_3\) or \(M>m_a+m_b+Q\)	Generated multivariate Gamma series with analytic pole limits	Analytic truncation bound, arithmetic estimates and precision agreement
integrate_sunset_taylor	Equal-mass unit-power D=2 sunset at finite Euclidean momentum, subject to budgets	Transformed Taylor series from an ODE and a Clausen boundary value	Analytic truncation bound, recurrence-error estimates and precision agreement

The original one_loop and sunset APIs use LoopTools-style Minkowski conventions, unlike the graph APIs. See Conventions before comparing signs or normalizations. sunset_d2 means a second derivative in the existing four-dimensional API; it does not mean dimension D=2.

Selecting a sunset evaluator

• Equal masses at zero or low Euclidean momentum: integrate_sunset_taylor.

• A justified RMT chart: compile_sunset_rmt exposes the matrix and Gamma families; integrate_graph_rmt evaluates them.

• Unequal masses outside both RMT charts: the hybrid integrate_graph may supply a statistical numerical reference. There is no automatic fallback.

An RMT boundary is a series boundary, not necessarily a singularity of the Euclidean integral. The equal-mass Taylor method crosses the RMT boundary \(Q=3m\) and overlaps the momentum chart. Large momenta may make its Taylor series too expensive; choose the RMT chart explicitly in that case.

What each error quantity means

Quantity	Meaning	What it does not establish
truncation_bound in the series APIs	Analytic bound on the omitted mathematical series, evaluated with guard precision	A rigorous interval enclosure including all floating-point operations
error_estimate in the series APIs	Tail plus estimated arithmetic loss, precision disagreement and final rounding	A formal proof of every reported digit
error_estimate in integrate_graph	Statistical diagnostics, refinement changes and a roundoff floor	A guaranteed bound or fixed confidence level after adaptive stopping
Agreement at two precisions	Evidence against unresolved arithmetic cancellation	Evidence that the underlying formula, branch or normalization is correct

RMTDomainError refuses an unsupported RMT chart. RMTSeriesConvergenceError, SunsetTaylorConvergenceError and MultiloopConvergenceError report unmet budgets or tolerances and carry partial results when available. Refusals are part of the contract. A small sequence of terms is not a convergence proof.

Coverage still open

The series backends do not implement unequal-mass continuation between the RMT charts, non-unit powers, epsilon expansions, overlapping-divergence subtraction, massless limits, tensor numerators or physical-region continuation. Topology-independent family selection and convergence proofs are also open.

The sunset backends have numerical audit evidence and a supplementary automated review with its findings addressed; see Scientific review packet. This is not human peer review. Historical audits of other APIs do not transfer automatically to these methods.