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

Annotations already establish readOnly, idempotent, and non-destructive behavior, so the description's job is to add context, which it does: the computational formula, an explicit rejection path for non-coprime input, and the cost model ($0.001 USDC per successful call via x402 on Solana) with an HTTPS fallback endpoint. The only gap is that it never explicitly states the output is a single integer value.

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 definition and formula come first, followed by examples, validation rule, and pricing — a sensible front-loaded order with no filler. The second worked example (4 and 7 gives 17) is arguably redundant after the first, and the pricing/endpoint sentence is dense, but nothing is wasted.

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 pure-computation tool with no output schema and full annotation coverage, the description covers the formula, valid input domain, failure mode, and payment mechanics. It could be marginally more complete by naming the return type or paying the 1e6 bound, but an agent has everything needed to call it correctly.

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 parameter meaning, and it does add a constraint absent from the schema: the pair must be coprime, otherwise the call is rejected. It also restates the lower bound of 2, but omits the 1,000,000 upper bound and does not say what the integers represent beyond being inputs to ab-a-b.

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 mathematical resource and defines it exactly: 'The Frobenius number of two coprime integers a and b... It is ab-a-b, the largest integer that cannot be made with non-negative combinations.' The formula plus worked examples leave no ambiguity about what is returned, and the domain is narrow enough to be separable from siblings like mcnugget or coin_count.

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 description states the preconditions for a valid call (coprime, each at least 2) and what happens when they fail ('A pair that is not coprime is rejected'), which is real usage guidance. However, it never says when to pick this tool over adjacent siblings such as mcnugget, coin_check, or coin_ways, so the only usage rules are input constraints rather than tool selection guidance.

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