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

Goes well beyond the readOnly/idempotent annotations by disclosing paid access ($0.001 USDC per successful call via x402 on Solana), the equivalent HTTPS endpoint, and the edge-case behavior that a zero modulus is rejected. Cost and payment rail are exactly the kind of operational context an agent needs before calling.

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 and formula, then constraints, then pricing and the HTTPS alternative. The illustrative C0–C3 values are mildly redundant given the formula but serve as a quick sanity check; nothing else is 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?

There is no output schema, so the description should say what comes back — a plain number, a decimal string, and how the output looks when mod is applied (residue vs. full term) is left unstated, which matters for values with hundreds of digits. Everything else an agent needs to invoke it correctly is present.

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 burden. It does explain that mod optionally reduces the term and that zero is rejected, but it does not mention that mod is supplied as a digit string up to 200 characters, nor does its range note for n add anything the schema's min/max doesn't already state.

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 exactly what is computed (the nth Catalan number) and backs it with the closed form C_n = binom(2n, n) / (n+1) plus the first few values. An agent can immediately tell this apart from neighbors like binom, central_binomial, or fuss_catalan.

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 implied rather than stated: it gives the valid domain (n from 0 to 2048) and rejects mod=0, but never says when to prefer this over fuss_catalan (generalized Catalan) or central_binomial, nor when the modulus variant is appropriate.

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