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

Annotations already declare readOnly, idempotent, openWorld and non-destructive, so the safety profile is covered. The description adds genuinely non-obvious behavior: mandatory odd-prime validity for p, a per-call price of $0.001 USDC via x402 on Solana, and an HTTPS endpoint alternative — details no annotation or schema conveys.

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-loaded with the definition, followed by examples, then constraints, then pricing/transport. Short and mostly waste-free; the range sentence is partly redundant with the schema's minimum/maximum fields.

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?

No output schema exists, so the description must convey return values — it does so through the three examples (1 and -1), though it omits the 0 case when p divides a, leaving one return value undefined. Pricing and endpoint details cover otherwise invisible invocation costs.

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 coverage is 0%, so the description must carry the load. It does state the semantic constraint that p must be odd prime (narrower than the schema's p >= 0) and that a may be negative, but the numeric ranges it cites merely restate the schema's min/max, and it never explains what happens when p divides a.

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?

Names the exact mathematical object and its definition condition ('The Legendre symbol (a/p) for an odd prime p'), with worked examples that pin down the sign convention. It does not distinguish itself from related siblings such as jacobi or kronecker, which compute the same symbol in more general settings.

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?

There is no guidance on when to use this rather than jacobi (which generalizes to composite odd denominators), kronecker (which allows even denominators), or quadratic_residue. The only implicit usage cue is the 'p must be an odd prime' constraint.

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