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

Annotations already cover the safety profile (readOnly, idempotent, non-destructive, openWorld). The description adds real context beyond them: the input-validation rule that non-coprime pairs are rejected, the per-call price of $0.001 USDC via x402 on Solana, and the alternate HTTPS endpoint. It does not state what the return type/shape is, which would have completed the picture.

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 leads with what the tool computes, then the formula and examples, then the constraint, price, and transport — a sensible front-loaded order with no filler sentences. The worked examples are slightly redundant with the stated closed form but serve as verification anchors.

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 pure read-only arithmetic tool with no output schema, the description is nearly self-sufficient: it gives the formula, validation behavior, cost, and an alternate endpoint. The remaining gap is that the return value (a single integer count) is never stated and the input bounds are left solely to the schema.

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 has to carry the burden. It does convey that a and b are denominations that must be coprime and illustrates their meaning with concrete pairs (2,5) and (4,7), which is meaningful beyond the bare integer schema. It never mentions the 2–1,000,000 bounds or that both are required, so compensation is partial.

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?

It states a precise verb+resource: the count of positive integers not representable with coprime denominations a and b, reinforced by the closed form (a-1)*(b-1)/2 and two worked examples. It does not, however, differentiate itself from closely related siblings such as coin_min, coin_ways, coin_check, mcnugget_count, or frobenius, which an agent scanning that list would need.

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 math definition and one concrete constraint is stated ('A pair that is not coprime is rejected'), so the agent knows what inputs are invalid. But there is no explicit when-to-use/when-not framing and no pointer to the sibling tools that compute the Frobenius number, representation ways, or representability checks.

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