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

Annotations already declare readOnly/idempotent/non-destructive, so the bar is lower; the description still adds materially by disclosing the paid-call model ($0.001 USDC per successful call via x402 on Solana) and an equivalent HTTPS endpoint. That is real operational context an agent cannot get from the annotations, though it says nothing about failure/refund behavior on unsuccessful calls.

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, then examples, range, price, and endpoint — no filler sentences. The four worked examples are slightly more than strictly needed and the range restates the schema bounds, but nothing is wasteful enough to hurt.

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, but the examples make the return value (a single integer) clear, and the range, cost, and transport alternative are all covered. For a one-parameter pure-computation tool, an agent has everything needed to call it correctly and predict the result.

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% and the schema only carries min/max, so the description must carry the meaning of n. It does: n is the index into the complement sequence, with four concrete input→output mappings that make the semantics unambiguous. The 1–1000000 range is duplicated from the schema, which keeps it from a 5.

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 gives the exact closed-form definition, floor(n*(2+sqrt(2))), plus a naming of the sequence as the Beatty complement of sqrt(2), so an agent knows precisely what value is computed. It does not explicitly distinguish itself from the sibling beatty_sqrt2 (the non-complement), which is the one obvious near-neighbor in this very large sibling set.

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?

Worked examples (n=1→3, n=2→6, n=3→10, n=5→17) and the valid range 1–1000000 implicitly tell the agent when the tool applies and how to sanity-check output. However, there is no explicit statement of when to prefer this over beatty or beatty_sqrt2, nor any exclusion or prerequisite beyond the price note.

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