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

Annotations already declare readOnly, idempotent, and non-destructive behavior. The description adds meaningful context beyond them: the domain restriction that p must be prime in 2..1000000, and the x402 payment model ($0.001 per successful call) plus an HTTPS endpoint. It does not say what happens when p is composite or out of range, so it falls short of a 5.

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-loads the core predicate and examples in the first sentence, then adds constraints and pricing. The endpoint and price sentences are useful but slightly operational clutter relative to the pure mathematical definition.

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 two-parameter boolean predicate with no output schema, the definition covers the core condition, examples, critical primality constraint, pricing, and endpoint. Remaining gaps are edge-case behavior for invalid or composite p and full semantics for g.

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 compensate. It clarifies that p must be prime and bounded by 2..1000000, but only names g as the candidate root, leaving g's allowed range (-1e12..1e12) and behavior for negative g unmentioned.

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 predicate: 'Whether g is a primitive root modulo the prime p,' with concrete examples. It is clearly a boolean check, but it never explicitly distinguishes itself from the sibling least_primitive_root, which could be confused with it.

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?

The examples and the 'p must be prime and runs from 2 to 1000000' constraint imply when the tool applies. However, it never names alternatives or states when not to use it (e.g., use least_primitive_root to compute a root rather than test one).

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