SageMath MCP Server

verify_claim

Idempotent

Independently re-check a mathematical claim in Sage syntax and get a verdict: proved, refuted, supported, or undecided. Use it to confirm your algebra before presenting it.

Instructions

Independently re-check a stated mathematical claim and report how far the evidence goes: proved, refuted, supported or undecided.

Use this to verify your own algebra before presenting it. The claim is a single comparison in Sage syntax -- an equality, an inequality, or anything that evaluates to True/False:

integral(x^2/(e^x-1), x, 0, oo) == 2*zeta(3) sin(x)^2 + cos(x)^2 == 1 pi < 22/7 e^pi != pi^e

The check climbs a ladder: Sage's symbolic prover, the exact difference ((lhs-rhs).simplify_full().is_zero()), exact arithmetic over QQbar/AA when the claim is constant, then certified interval arithmetic and numeric sampling over the free variables. Verdicts are honest by construction: 'proved' and 'refuted' are exact decisions ('refuted' always exhibits its counterexample or certified enclosure); 'supported' means the numeric evidence is consistent with the claim without proving it, and says how many samples at what precision; 'undecided' means every rung was inconclusive -- it never means false.

Exactness is never assumed. Decimal literals are read exactly (0.1 means 1/10, never the 53-bit double), and a comparison whose operands are machine floats (RR/RDF/CC, an .n() result) is reported as 'supported' over inexact numbers, never as an exact proof -- state it over ZZ/QQ/QQbar or symbolically for an exact verdict. The session's active assumptions (assume(x > 0), assume(x, 'integer')) are honored: a sampled counterexample must lie inside the stated domain, and any verdict that relied on an assumption names it in the evidence.

Input Schema

TableJSON Schema

Name	Required	Description	Default
`claim`	Yes	The claim to check, as a single comparison, e.g. 'sin(x)2 + cos(x)2 == 1'
`samples`	No	Sample points per free variable sweep when the claim cannot be decided exactly
`session`	No	Workspace to use, as a name or a portable handle. Workspaces have independent variables. A name is scoped to this MCP session; a handle returned by start_sage_session (workspace_token) reaches the same workspace across reconnects and is a bearer credential -- keep it secret. Omit for 'default'.	default
`timeout`	No	Override the evaluation timeout in seconds
`precision_bits`	No	Precision of the certified interval arithmetic behind numeric verdicts

Output Schema

TableJSON Schema

Name	Required	Description
`claim`	Yes	The claim that was checked, whitespace-folded.
`method`	No	The rung that decided: exact_comparison, symbolic_prover, exact_difference, exact_algebraic, certified_interval, numeric_sampling, float_comparison (operands were machine floats, so the result is only supported, never exact) or exhausted.
`samples`	No	Number of sample points supporting a numeric-sampling verdict.
`verdict`	Yes	proved/refuted are exact; supported is evidence short of proof; undecided means every rung of the ladder was inconclusive.
`evidence`	No	What the deciding rung actually established, including the counterexample for a sampled refutation.
`assumptions`	No	The session's active assumptions in force during the check (e.g. "x is integer"). A verdict is only valid under these; they are also named in the evidence so no branch can conceal them.
`precision_bits`	No	Interval-arithmetic precision behind a numeric verdict.

Schema Changelog

Changes observed during successful MCP inspections.

Addedv0.7.0Sep 12, 2026

TDQS

A4.6/5.0

Behavior5/5

Does the description disclose side effects, auth requirements, rate limits, or destructive behavior?

With annotations present, the description adds substantial context beyond them: the verification 'ladder', the meaning of each verdict including that 'refuted' exhibits a counterexample and 'undecided' never means false, how decimal literals are read exactly, that float comparisons are reported as 'supported' not proof, and that session assumptions are honored and named. This is rich behavioral disclosure an agent needs and cannot infer from readOnlyHint/idempotentHint.

Agents need to know what a tool does to the world before calling it. Descriptions should go beyond structured annotations to explain consequences.

Conciseness4/5

Is the description appropriately sized, front-loaded, and free of redundancy?

Front-loaded with the purpose and verdict taxonomy, then examples, then the ladder and exactness caveats. Dense and slightly long, but for a tool with nuanced semantics nearly every sentence carries needed information; minor verbosity keeps it from a 5.

Shorter descriptions cost fewer tokens and are easier for agents to parse. Every sentence should earn its place.

Completeness5/5

Given the tool's complexity, does the description cover enough for an agent to succeed on first attempt?

Given an output schema exists and return values need not be explained, the description covers everything an agent needs: verdict semantics, exactness caveats, assumption handling, sampling, and precision. Nothing critical is missing for correct invocation or interpretation.

Complex tools with many parameters or behaviors need more documentation. Simple tools need less. This dimension scales expectations accordingly.

Parameters4/5

Does the description clarify parameter syntax, constraints, interactions, or defaults beyond what the schema provides?

Schema coverage is 100%, so the baseline is 3. The description still adds meaning: it specifies the claim is a single comparison in Sage syntax with worked examples, describes sampling behavior ('says how many samples at what precision'), and explains precision_bits as controlling certified interval arithmetic, extending beyond the raw schema text.

Input schemas describe structure but not intent. Descriptions should explain non-obvious parameter relationships and valid value ranges.

Purpose5/5

Does the description clearly state what the tool does and how it differs from similar tools?

States a specific verb+resource ('Independently re-check a stated mathematical claim') and its output behavior (reporting how far evidence goes: proved, refuted, supported, undecided). This is clearly distinct from siblings like calculate_expression, simplify_expression, and solve_equation, so an agent can select it without opening the schema.

Agents choose between tools based on descriptions. A clear purpose with a specific verb and resource helps agents select the right tool.

Usage Guidelines4/5

Does the description explain when to use this tool, when not to, or what alternatives exist?

Explicitly states when to use it: 'Use this to verify your own algebra before presenting it.' That gives clear context for invocation. However, it names no alternatives and gives no explicit when-not-to-use or exclusions relative to the many sibling compute tools.

Agents often have multiple tools that could apply. Explicit usage guidance like "use X instead of Y when Z" prevents misuse.