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

Annotations already declare the tool as read-only, idempotent, open-world, and non-destructive. The description adds meaningful operational context beyond annotations: a per-call price ($0.001 USDC), an HTTPS alternative endpoint, and the exact valid input range. It does not contradict the annotations.

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

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

The description is compact and front-loads the formula and examples. The first sentence is awkwardly headless, but subsequent sentences are efficient. Pricing and endpoint information, while slightly tangential, are clearly separated at the end.

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

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

For a simple one-parameter read-only math function with no output schema, the description supplies enough to call the tool correctly: range constraint and example outputs. It could be improved by explicitly stating the return type (integer cofactor), but the examples largely fill that gap.

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

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

Schema description coverage is 0%, so the description must compensate. It does so partially by stating the valid range (n=2–8) and giving output examples for n=5 through n=8, which helps clarify the parameter's effect. However, it does not define what n semantically represents (e.g., index in the sequence) nor explain the outputs for n=2, 3, or 4.

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

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

The description gives a computational formula and concrete examples (n=5 gives 139, etc.), so the tool's mathematical purpose is inferable. However, it never explicitly states that the tool returns the cofactor for a given index n, and it omits a clear subject in the first sentence. It also fails to distinguish itself from closely related siblings like euclid_mullin, euclid_mullin_prod, and euclid_mullin_sum.

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

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

The description provides a hard constraint (n runs from 2 to 8) but gives no guidance on when to use this tool versus alternative siblings. There is no mention of when not to use it or which sibling handles related computations, leaving selection entirely to inference.

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