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

With no annotations, the description carries the full burden; it does disclose the pricing/auth model ('5 free trial calls per signed wallet') and the internal algorithm, which is useful. However, it omits return format, behavior for edge inputs (n=0, n=1, negative, non-integer), and failure modes for the string-typed parameter.

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?

Front-loaded with the core definition and the parenthetical formula detail is compact and earns its place; the pricing suffix is short. Slightly dense but no wasted sentences.

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?

No output schema and no annotations mean the description must cover behavior, yet it only explains the math, not the response shape or input constraints. Adequate for understanding what is computed, insufficient for reliably invoking it.

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% for the single parameter 'n', and the description only embeds it in the lambda(n) notation without stating type semantics, accepted format, size limits, or that it is a string-encoded integer. It does not compensate for the undocumented schema.

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?

States a specific verb ('Computes') and named resource ('Carmichael function lambda(n)'), and explicitly distinguishes itself from Euler's totient, a close sibling tool in this catalog. An agent can identify the operation 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.

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

The 'not Euler's totient' clause gives an implicit disambiguation, but there is no explicit guidance on when to pick this over x402-is-carmichael or x402-euler-phi, nor on valid input ranges. Usage is only inferable from the mathematical definition.

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