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

With no annotations at all, the description must carry the behavioral load; the 'Is X' framing does imply a pure read-only boolean predicate, and the free-trial/signed-wallet line usefully discloses the payment/auth model. It does not state the return shape (boolean vs. object), behavior for non-integer or out-of-range strings, or any cost per call beyond the trial.

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?

One dense sentence that front-loads the check, then the definition, then examples and the one tricky edge case; little waste. The emoji-laden free-trial tail is promotional but does convey pricing context.

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 one-param predicate with no annotations and no output schema, the description covers what the tool decides and which inputs are in scope. It leaves the return format and input-format constraints unstated, which matters given the string-typed parameter and the known finiteness of Fermat primes.

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% and the single parameter 'value' is typed only as an unconstrained string with no description. The description partially compensates by giving concrete example values and calling out that 2 is not a Fermat prime, but it never clarifies format (decimal digit string?), magnitude limits, or handling of leading zeros/whitespace.

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 verb+resource ('Checks whether value is a Fermat prime') and pins down the exact mathematical meaning with the form 2^(2^k)+1 plus the enumerated set (3, 5, 17, 257, 65537) and the explicit exclusion of 2. The definition implicitly separates it from the sibling x402-is-fermat (Fermat numbers) and x402-is-prime, though neither is named.

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?

No statement of when to reach for this versus x402-is-fermat, x402-is-prime, x402-is-mersenne-prime, or x402-is-safe-prime. An agent must infer routing from the math definition alone; there are no exclusions or prerequisites beyond the marketing line about free calls.

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