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

Annotations already declare a safe, idempotent read, and the description adds context those annotations cannot convey: a per-call price of $0.001 USDC, an x402/Solana payment rail, charging only on success, and an identical HTTPS endpoint. This is exactly the kind of cost/payment disclosure an agent needs before calling. It stops short of describing output shape or failure behavior.

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?

Three dense sentences with the mathematical definition front-loaded, followed by domain constraint and then cost/transport. Every sentence carries information an agent cannot get from the schema or annotations.

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?

With no output schema, the description does the work of defining the return values (x = half the companion Pell-Lucas number, y = the Pell number) and the cost model. The one gap is it never states the literal response fields or what happens when norm mismatches, which matters for a tool that charges per successful call.

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 parameters, and it does: n is bounded to 0-8192 and norm is explained as a value that must equal the sign (-1)^n, a validation rule absent from the schema. The norm explanation is terse and could be clearer about whether it is required, but it adds genuine meaning.

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 mathematical object (the nth solution of x^2 - 2y^2 = (-1)^n) and pins down what x and y are in terms of the Pell and companion Pell-Lucas families, which implicitly separates it from the sibling pell/pell_lucas tools. It is specific, though it never explicitly contrasts itself with pell_compose or neg_pell.

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?

It states the valid domain (n from 0 to 8192) and the precondition that a supplied norm must equal (-1)^n, which is real invocation guidance. However, it gives no when-to-use guidance relative to the many adjacent Pell-family siblings, so the agent must infer selection from the names alone.

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