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SageMath and Scientific Computation

SageMath 10.9 ran a focused exact symbolic check in a pinned Docker image, while the engine still has no Sage adapter or claim-bound result.

The engine’s host probe says SageMath is absent:

sagemath available: False | reason: none of sage was found on PATH

And yet this chapter shows SageMath output. That is not a contradiction. The probe checks the host PATH; the run below happened inside the official Docker image. Being available in a container is not the same as being integrated into the engine. The engine’s sagemath.probe.v1 registry entry still says “SageMath capability only; no evidence”, and the manual sagemath-lane remains unrun in the hosted record.

The failure to keep in view is subtler than a missing tool. A tool can run and produce a correct-looking expression while the claim, the domain and the evidence record remain unbound. A successful Sage process is not a successful investigation.

SageMath is a candidate computer-algebra system for the book. This chapter records a reproducible SageMath 10.9 run in a pinned container, and what it does and does not add. There is still no Sage EvidenceResult, no claim ID and no cross-system binding. The chapter builds on the status ladder (Chapter 3) and the SymPy chapter (Chapter 9).

The run

The script is ch14_sagemath_demo.sage. It repeats the algebra already shown in Chapter 9 and Chapter 20, namely the gamma residual

$$ \left(1/\sqrt{1-v^2}\right)^2(1-v^2) - 1 = 0 $$
and the false-shear residual
$$ t^2 - (x+vt)^2 - (t^2-x^2) = -vt(vt+2x). $$
Run on 2026-11-09 in `sagemath/sagemath:10.9`, it printed: 
sagemath 10.9
ch14 gamma residual (symbolic v): 0
ch14 same residual at v=2 (outside |v|<1): 0
ch14 gamma^2(1-v^2)-1 at v=2 directly: 0
ch14 algebraic residual, factored: -(g^2*v^2 - g^2 + 1)*(t + x)*(t - x)
ch14 shear residual, factored: -(tt*vv + 2*xx)*tt*vv
ch14 shear residual at t=1, x=0, v=1/2: -1/4
ch14 exact rational check: 5/6

No new mathematical claim is introduced.

What the output means

The gamma residual simplifies to $0$ for symbolic $v$. The algebraic residual factors, up to Sage’s sign and ordering, as

$$ \bigl(\gamma^2(1-v^2) - 1\bigr)(t^2-x^2). $$
The false shear has residual $-vt(vt+2x)$, and the witness $t=1$, $x=0$, $v=1/2$ gives $-1/4$. These are exact symbolic computations, not kernel proofs.

The $v=2$ line is the important one. Sage returns $0$ there, even though $\lvert v\rvert<1$ is false and the real square root is not defined. Like SymPy, the simplification works over a complex square root and does not enforce the physical side condition. The domain warning from Chapter 9 is not hypothetical.

What is still missing

  • The sagemath ExternalToolSpec has no source path, command or claim ID, so it cannot produce an EvidenceResult.
  • The script is an illustrative teaching example. It is not a versioned claim and does not appear in a case manifest.
  • The false-shear residual is a negative control at the level of an expression, not an adapter’s negative-control record.
  • No claim has a Sage binding, so there is no L0–L4 correspondence audit.
  • No human review of the script or its mathematical interpretation is recorded.

A future Sage adapter would be a computational system, not a kernel. Its strongest appropriate status would be computation_verified_within_stated_domain, never formally_verified.

Where Sage would matter

The book’s longer-term plan needs tools beyond a handful of real variables. None of these topics is in this edition:

Planned topic (old outline chapter) Why SymPy alone may not be enough
Lorentz group and rotations in 3+1 dimensions (23) Matrix groups, exact linear algebra
Tensors and index notation (24) Tensor algebra at scale
Curvature of a metric (25) Symbolic differential geometry
Numerical comparison with data (28, 34) Floating point, with error control

Only the scalar residual check has been tried. Whether Sage is the right tool for tensor, geometric or numerical work remains open, and the decision whether to reuse Physlib (the MATH04 milestone) is a separate gate.

What this establishes

Sage and SymPy agree on the gamma residual and on the shape of the algebraic and shear factorizations at this scale. That is an informal comparison of two teaching scripts, not a cross-system claim contract; no L0–L4 result is claimed.

A computer-algebra system is most dangerous exactly where it is most helpful: it returns confident closed forms for expressions whose side conditions it never checked. Sage’s $v=2$ result shows the SymPy lesson again, concretely. That is why the book treats Sage as a computational tool, not as physical validation or a proof kernel. The script is short enough to check by hand, but no human review is recorded.

Try it yourself

docker run --rm -v "$PWD:/work" -w /work \
  sagemath/sagemath:10.9 \
  sage /work/content/examples/math/tool-chapters/ch14_sagemath_demo.sage

Run it from the repository root. The host probe can be rerun too:

PYTHONPATH=src uv run python content/examples/math/tool-chapters/ch13_16_probe_and_audit_demo.py

It will still report sagemath available: False, because Docker does not put sage on the host PATH. That is a fact about the host, not the container.

Turning this into an engine result, rather than an illustrative run, would take five steps:

  1. Add a Sage adapter with a source path, command, claim ID and encoded domain.
  2. Add a negative control that the adapter must reject or mark as a mismatch.
  3. Run it through the case runner (Chapter 16) and record the EvidenceResult.
  4. Compare the factorization with SymPy under the cross-system contract (Chapter 15).
  5. Decide how numerical error bounds would be represented if Sage is used for floating-point work.

Open questions.

  1. Does Sage add a kind of evidence the engine lacks, or only a faster path to computation_verified_within_stated_domain?
  2. Is one exact scalar example enough to justify the chapter, or should it wait for an adapter and a claim-bound result?
  3. How should a Sage result record its ring or field (QQ, RR, AA, the symbolic ring) alongside the computation?

Evidence record

Executed, illustrative. The recorded command, run on 2026-11-09, was:

docker run --rm -v "C:/Projects/math:/work" -w /work \
  sagemath/sagemath:10.9 \
  sage /work/content/examples/math/tool-chapters/ch14_sagemath_demo.sage

The image was pulled as sagemath/sagemath:10.9, digest sha256:e068670ae5863b54b2550e72437ec637b0283acb0dc712c8584c124dbf44e667, SageMath version 10.9. This is not an engine evidence bundle. The host probe still reports SageMath as not_installed, and the manual sagemath-lane remains skipped in hosted runs.

Sources

In the repository: src/math_engine/adapters/external.py (the sagemath spec), docs/architecture/adapter-registry.md, docs/research/formalization-landscape.md and the example script. The image tag and digest are part of the execution record, not a mathematical citation.