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

Annotations already cover the safety profile (readOnly, idempotent, non-destructive), and the description adds genuinely non-obvious operational context: a per-call price of $0.001 USDC via x402 on Solana, plus an equivalent HTTPS endpoint. It also states the accepted input range. It does not say what a failed payment or error response looks like, but for a pure arithmetic function this is solid added context.

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, then examples, then constraints and pricing — a sensible order with no filler. The four examples take space but each demonstrates a distinct case (residue, non-residue, non-coprime, zero modulus), so they earn their place.

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 showing return values explicitly (1, -1, 0, 1), so the agent knows an integer is returned. Ranges, cost, and an alternate transport are all covered. The only missing piece is a pointer to the sibling symbol tools for agents unsure which variant to call.

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 the semantic load, and it does: the notation (a/n) establishes that a is the numerator argument and n the modulus/denominator, and the examples pin down edge behavior such as n=0 and non-coprime cases. The stated ±10^12 bound duplicates the schema's minimum/maximum, so it adds no new information there.

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 the specific mathematical object ('The Kronecker symbol (a/n)') and states it works for any integers a and n, so an agent knows exactly what is computed. The worked examples ((2/7)=1, (3/7)=-1, (2/4)=0, (1/0)=1) make the operation unambiguous. It does not, however, distinguish itself from sibling tools jacobi and legendre_symbol, which compute closely related symbols.

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 explicit when-to-use statement and no mention of the obvious alternatives (jacobi, legendre_symbol) or the conditions that select one over the other. Usage is only implied by the phrase 'for any integers a and n' plus the (2/4)=0 example, which hints that even/composite n is handled — a contrast with Legendre/Jacobi that the agent must infer rather than read.

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