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

Beyond the readOnly/idempotent annotations, the description discloses the substantive behavioural limits: chains longer than 20 steps or larger than 12 digits are rejected, and billing is $0.001 USDC per successful call via x402 on Solana. The alternative HTTPS endpoint is also given. It stops short of saying what the response body looks like or what error a rejected chain returns.

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, then examples, then constraints, then pricing/endpoint. Every sentence carries information, with only the 'n runs from 2 to 10000' clause being redundant against the schema bounds.

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 single-integer pure-computation tool with no output schema, the description supplies the definition, illustrative examples, failure thresholds, input domain, cost model and an HTTP fallback. An agent has everything needed to decide and call correctly.

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 is the only prose about n, and it does restate the valid domain (2 to 10000) — but that duplicates the schema's minimum/maximum rather than adding new meaning. No detail on types, performance cliffs at large n, or what happens for n where the home prime is not reached.

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 measurable quantity — the digit count of the home prime of n — with worked examples (2 -> 1, 4 and 10 -> 3, 16 -> 8) that pin down the definition. It is distinguishable from generic siblings, though it never explicitly contrasts with home_prime or home_prime_steps, which compute neighbouring quantities of the same chain.

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 by the narrow domain (you call it when you want the length of a home-prime chain for a given n), and the rejection rule tells you when the call will fail. But there is no explicit statement of when to prefer this over home_prime_steps or home_grows, so the agent must infer the choice from name alone.

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