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

Beyond the annotations (readOnly, idempotent), the description adds genuinely useful operational context: the accepted input range 1 to 1e12, the per-call price of $0.001 USDC via x402 on Solana, and an equivalent HTTPS endpoint. It does not discuss failure modes or latency, which keeps it short of a 5.

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 definition, followed by examples, domain, pricing, and endpoint. Every sentence adds information, though the endpoint URL and four worked examples make it slightly denser than strictly necessary.

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 exists, so the description supplies the return semantics via concrete values (1, 2, p-1, 4), implying an integer result. Combined with the range and pricing, an agent has enough to call it correctly; only the explicit return type is unstated.

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%, so the description must carry the parameter meaning, and it does: n is an integer in [1, 1000000000000], with the special cases λ(1)=1, λ(8)=2, λ(p)=p-1, λ(15)=4 illustrating boundary and prime behavior. It could be clearer that n is the only argument, but the constraint and edge semantics are well conveyed.

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 mathematical object and verb: 'Carmichael's lambda of n, the exponent of the multiplicative group mod n.' This distinguishes it from the many sibling variants (carmichael_number, carmichael_count, carmichael_next, carmichael_sum, carmichael_factors), which are enumeration/predicate tools rather than a single evaluation.

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?

Usage is only implied: the reader infers you call it to evaluate λ(n) for one integer. There is no explicit statement of when to prefer it over carmichael_number or carmichael_next. The sample values hint at its single-value evaluation role but do not spell out alternatives.

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