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

Annotations already declare read-only, idempotent and non-destructive behavior, and the description adds real context beyond them: a rejection rule for odd m, the enforced 2–40 domain, and the fact that the tool is paid ($0.001 USDC per successful call via x402 on Solana). The per-successful-call pricing detail matters because an agent may retry. It does not describe error/response shape beyond that.

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 mathematical definition, then constraints, then pricing and the HTTPS alternative. Four dense sentences with no filler, though the price and endpoint sentence is packed tight enough that a reader could miss the call cost.

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?

With no output schema, the description does the work by defining the returned value mathematically and giving two concrete outputs. Only the literal return type and whether failures are charged remain unstated, which is a minor gap for a paid single-integer tool.

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 carry the parameter, and it does: m must be even, at least 2, and no more than 40, with B2=6 and B4=30 as worked examples. It usefully clarifies that odd m is rejected even though the schema's minimum of 0 permits values the tool refuses.

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 precise verb+resource: the von Staudt denominator of B(m), defined constructively as the product of primes p with p-1 | m. That definition alone distinguishes it from siblings like bernoulli and bernoulli_sum, though it never names those siblings or says explicitly how results differ.

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?

It gives strong input-side rules (m even, at least 2, 2 to 40, odd m is rejected) and cites concrete values for B2 and B4, but there is no guidance on when to pick this over bernoulli or von_staudt, which is the actual selection question among the siblings.

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