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

Annotations already declare readOnly, idempotent, non-destructive, open-world behavior, so the bar is lower. The description adds genuinely useful behavior beyond that: composite p is rejected (an error mode), the per-call price of $0.001 USDC via x402 on Solana, and an equivalent HTTPS endpoint. It does not say what error is surfaced for invalid input or the shape of the numeric result, so it stops short of full disclosure.

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, identity, example, constraint, ranges, and payment/endpoint details are ordered sensibly with little filler. The Kummer identity is slightly redundant with the opening sentence but serves as a check on interpretation. The endpoint line is logistics rather than description, but is necessary for an x402 tool.

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 must indicate the return value, and 'the number of carries' implies a non-negative integer count. Input ranges, the prime-base precondition, and payment terms are all covered. What is missing is the failure behavior for out-of-range input and explicit confirmation of the return type.

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 — and it does: a and b are the addends bounded 0 to 1e12, p is the base bounded 2 to 1e6 and must be prime. That is exactly the semantic content the bare schema lacks. It could still be clearer that a and b are non-negative integers regardless of base, but the compensation is strong.

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 computed quantity: the number of carries when a and b are added in prime base p, with Kummer's theorem restated as the exponent of p in C(a+b, a). This is unambiguous and distinguishes it from generic binom or valuation siblings, though it never names an alternative tool explicitly. It arguably assumes the reader knows what a 'carry count' means, but the worked example (5 + 3 in base 2 = 3 carries) resolves that.

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 statement of when to use this tool versus related siblings such as binom, factorial_valuation, or padic, even though those cover overlapping mathematical territory. The only usage-adjacent remark is that a composite base is rejected, which is a validation rule rather than selection guidance. An agent must infer applicability from the math alone.

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