The Principia Machina/Tools/SageMath

SageMath and SageManifolds: Open-Source Differential Geometry and Tensor Calculus

Added to the survey on 2026-10-06 by the Research Toolkit's research pass. The scores below are that pass's assessment from the evidence cited under References, on the survey's four axes and scale; they are not the original survey's (2026-09-26), and no rule of the Principia Machina reads them. See the additions on the front page.

Overview & Architectural Fit

SageMath is an umbrella computer algebra system that integrates many open-source mathematics libraries behind one Python interface. SageManifolds, now fully part of SageMath, adds differential geometry and tensor calculus on manifolds: Lorentzian metrics, the Riemann, Ricci, Weyl, Schouten and Cotton tensors, Lie derivatives, differential forms and geodesics. Because it is free, unlike the commercial host xAct needs, anyone can run it to reproduce a geometric result.

Mathematical Evaluation for TPM

• Lagrangian / Action (6/10): Symplectic structures, p-forms and Lie derivatives support geometric Lagrangian and Hamiltonian formulations; no packaged pipeline varies an action.
• Field / PDE (5/10): Tensor fields are handled symbolically; there is no high-performance solver for partial differential equations.
• Relativity (8.5/10): A full open-source coordinate workflow for general relativity: curvature of a given metric, invariants, geodesics, and the Killing equation ${\mathrm{\nabla }}_{\mu }{\xi }_{\nu }+{\mathrm{\nabla }}_{\nu }{\xi }_{\mu }=0$, the condition that ${ℒ}_{\xi }g=0$.
• Quantum / Data (3/10): Its strengths are algebra and number theory, not quantum dynamics or learning from data.

Integration Pipeline

1. Declare the manifold and a chart, then the metric in components.
2. Compute the connection, the curvature tensors and their invariants.
3. Test a symmetry: compute the Lie derivative of the metric, or of the Lagrangian, along a candidate generator; zero is the precondition for a Noether charge.
4. Integrate geodesics numerically for comparison with an analytic orbit.
5. Cross-check any component result against xAct or Cadabra2.

Status, Licence and Limits

• Status: Active. Stable 10.9 (tag of 5 May 2026), announced on sagemath.org; GitHub also shows 10.10 dated 28 September 2026, not yet announced on the site when checked (not confirmed).
• Licence: GPLv2 or later for the repository; components GPLv3-compatible.
• Home: https://www.sagemath.org and https://sagemanifolds.obspm.fr
• Limits: Abstract-index manipulation, such as canonicalizing large Riemann monomials, is weaker than in xAct. Very large component computations can be slow. The full installation is heavy (a build of about 10 GB).

On the Principia Machina

A tool runs only where the sponsor installed and bound it, and what it returns is untrusted input, vetted before use; its numbers are never values of the site's calculator, TPC. An equation it derives is transcribed into the kernel's form grammar before anything checks it, and lands as a RESULT of regime L with EQUATION and ROLE. See Agent/Tools and the operator's guide.

References

• SAGEMANIFOLDS2015 – Gourgoulhon E., Bejger M., Mancini M. (2015). Tensor calculus with open-source software: the SageManifolds project. J. Phys.: Conf. Ser. 600, 012002. doi:10.1088/1742-6596/600/1/012002.
• SAGEREPO – The Sage Developers (2026). SageMath 10.9 / 10.10 (GitHub releases). github.com/sagemath/sage/releases.
• SAGEMANIFOLDSSITE – SageManifolds project. sagemanifolds.obspm.fr.