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

Annotations already cover the safety profile (readOnly, idempotent, non-destructive). The description adds materially beyond them: it discloses a per-call cost ($0.001 USDC via x402 on Solana), an alternate HTTPS endpoint, and the input-rejection rule for perfect squares — all operationally relevant to an agent that must decide whether to spend money on a call.

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 core question is front-loaded in the first clause, and each following sentence adds something concrete: the criterion, illustrative values, the square-rejection rule, the numeric range, and pricing. The three-value example pairs are slightly padded but not wasteful.

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 decision tool with no output schema, the description supplies the decision criterion, domain limits, error condition, and cost — enough to call it correctly. The main omission is that the exact response payload (pure boolean or richer verdict) is never stated.

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 must carry the load, and it does: it identifies d as the coefficient in the Pell-type equation, restates the 1–100000 domain, and adds a validation rule absent from the schema ('A square is rejected'). What it does not say is the return shape (boolean vs. verdict plus period), leaving some semantic slack.

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?

Names a specific mathematical decision problem (does x^2 - d*y^2 = -1 have a positive integer solution) plus the exact decision criterion (odd continued-fraction period of sqrt(d)). That is far more specific than the bare name 'neg_pell'. It does not explicitly distinguish itself from the near-identical sibling neg_pell_count, so an agent must infer that this one answers yes/no while the other counts.

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 only implied: supply d and get a solvability verdict. The worked examples (d=2,5,13 succeed; d=3,6,7 fail) help an agent sanity-check inputs, but there is no statement of when to prefer this over neg_pell_count, pell_equation, or sqrt_period, and no exclusions.

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