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

Annotations already cover read-only/idempotent/no-destructive, so the description's job is to add operational context, and it does: $0.001 USDC per successful call via x402 on Solana, plus an equivalent HTTPS POST endpoint. Output shape is demonstrated ('for 2 it is 3/1') rather than merely asserted, though the exact payload encoding is left implicit.

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 and worked examples are front-loaded, and the cost/endpoint sentence is short and useful rather than padding. Every sentence contributes, though the pricing clause sits mid-paragraph rather than being clearly segregated.

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 one-parameter, non-destructive computation with no output schema, the definition supplies formula, domain, sample outputs, cost, and an alternate transport — close to everything needed to call it. The remaining gap is the precise return encoding (string fraction vs. numerator/denominator fields).

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 meaning of n. It restates the 1..200 range (already in the schema) and shows three sample values of n, but never explicitly says n is the index into the central binomial coefficients — the reader infers that from the ratio formula.

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 names the exact quantity computed and supplies the closed form C(n)/C(n-1) = (4n-2)/n plus concrete worked values, so an agent knows precisely what comes back. It does not differentiate itself from nearby siblings such as central_binomial or central_sum, which is the only thing separating it from a 5.

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?

There is no statement of when to reach for this versus central_binomial, central_mod, or central_sum, and no exclusions or prerequisites beyond the numeric domain. The agent must infer selection from the formula alone.

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