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

No annotations are provided, so the description carries the burden. It does disclose two non-obvious traits: that input can come from either 'n' or 'value' (parameter aliasing) and that there is a pricing/trial boundary tied to a signed wallet. It does not explain precedence between the two inputs or behavior on invalid/non-numeric input.

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 mathematical definition is front-loaded in a single clause, which is efficient. The trailing promo fragment ('🆓 5 free trial calls per signed wallet') is mildly noisy but practically informative rather than padding.

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 pure, no-side-effect computation with no output schema and no annotations, the description states the algorithm but leaves input format, param selection, and the result type unstated. Adequate for a trivial function, but not fully self-contained given 0% schema coverage.

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 coverage is 0% with two undocumented string parameters, so the description must compensate. It partially does by revealing that 'n' and 'value' are alternative names for the same input, which the schema never states, but it omits the expected numeric format (the schema types are strings) and which name is canonical.

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 precise verb+resource: it computes the count of integers below n that are coprime to n, which is an unambiguous statement of the Euler totient. It does not differentiate from the very close sibling x402-euler-phi, nor from x402-euler, so an agent could easily confuse the three.

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?

No when-to-use guidance, no conditions, and no mention of the near-identical siblings (x402-euler-phi, x402-euler) that would be the obvious alternatives. The agent is left to infer selection from the math definition alone.

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