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

No annotations are provided, so the description carries the full burden. It discloses the computational method (checking g^phi over prime factors) and a billing/auth constraint (5 free trial calls per signed wallet), which are useful behavioral cues. However, it does not state that the operation is read-only, what the return type is (boolean), or how invalid inputs are handled.

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 functional content is front-loaded in a single efficient sentence, and the description is appropriately sized for a simple predicate tool. The trailing free-trial promotional note is somewhat off-topic but still brief and potentially billing-relevant.

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 simple two-parameter math predicate with no annotations and no output schema, the description covers purpose, method, and parameter roles. It omits expected return format, input validation requirements, and error behavior, and the schema itself marks zero required parameters, leaving an agent with notable gaps.

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 clearly maps both parameters to their mathematical roles: g is the candidate root and p is the modulus. This adds meaningful semantics beyond the bare string types in the schema, though format details (e.g., decimal strings) are absent.

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 a specific verb ('Tests whether') and resource ('g is a primitive root modulo p'), and even outlines the algorithm used. It is immediately distinguishable from the dozens of sibling is-* predicates by naming the exact mathematical property being checked.

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 purpose statement—call it when you need to know whether g is a primitive root modulo p—but there is no explicit when-to-use guidance, no prerequisites (e.g., p must be prime), and no mention of alternative tools. The description leaves selection context to inference.

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