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

Annotations already declare readOnly/idempotent/non-destructive, so the description focuses on what they cannot cover: a per-call price ($0.001 USDC), the x402/Solana payment requirement, and an equivalent HTTPS endpoint. The primality requirement is also disclosed. It stops short of describing failure behavior for invalid input.

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 definition is front-loaded, followed by worked examples that pin down the return convention, then constraint, then cost and endpoint. Every sentence carries distinct information with no redundancy.

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 compensates by giving concrete input→output examples (1 for 2, 3 for 7, 5 for 23, 6 for 41), so the return type is clear. What remains unstated is the behavior on invalid input and the latency/response envelope.

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?

The single parameter has 0% schema description coverage, so the description must carry the semantics — and it does, stating p is a prime in [2, 1000000], which adds the primality constraint the schema cannot express. It does not say what error is returned when p is composite.

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 a specific mathematical object — the least primitive root modulo a prime p — which is more precise than the title alone and implicitly distinguishes it from the sibling primitive_root by the word 'least'. It does not explicitly name that sibling or state how the two differ, so a 4 rather than 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?

Usage is implied by the domain constraint ('p must be prime and runs from 2 to 1000000'), which tells the agent the valid input space. There is no explicit statement of when to choose this over primitive_root, mult_order, or other order-related siblings, and no note on what happens for composite p.

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