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

Annotations already declare read-only, idempotent, non-destructive behavior, and the description adds valuable non-schema context: pricing at $0.001 USDC per successful call, x402/Solana payment, and an equivalent HTTPS endpoint. It still does not describe error behavior when a is not coprime.

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 description is front-loaded with the core definition, followed by examples, preconditions, and access/payment details. Every sentence adds useful information without padding.

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 mathematical tool with no output schema, the definition plus examples and preconditions are sufficient to invoke it correctly. The main remaining gap is explicit return-value and failure-mode description.

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 parameter meaning; it identifies a as the base and n as the modulus, requires gcd(a,n)=1, gives the valid n range, and provides worked examples. It does not clarify how negative a values are handled despite the schema allowing them.

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 precisely defines the mathematical operation: the least positive k such that a^k = 1 mod n, with concrete examples. The purpose is clear, but it does not explicitly differentiate this tool from nearby siblings such as primitive_root, mod_pow, or mod_inverse.

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 key preconditions, namely that a must be coprime to n and n must be in [2, 1000000], which implies when the tool is applicable. However, it gives no explicit guidance on when to choose this tool over related sibling operations.

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