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

Annotations already declare readOnly, idempotent, non-destructive, openWorld, and the deterministic "smallest m" definition is consistent with idempotency. Beyond that, the description adds genuinely useful context the annotations don't: the rejection behavior when no m exists through 12, the $0.001 x402/USDC cost, and the equivalent HTTPS endpoint.

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 core definition is front-loaded in the first sentence, followed by examples, domain constraint, failure mode, and pricing/endpoint. Dense but every clause carries information; only the pricing/URL tail could be considered peripheral to invocation logic.

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 single-integer lookup with no output schema, the description covers the domain, the failure mode, and enough worked examples (1→3, 3→3, 9→10) to infer the integer return. The only gap is an explicit statement of the response shape, which the examples mostly cover.

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 load, and it does explain that n is the lower bound for the search plus its 1–12 domain. But the range duplicates the schema's minimum/maximum, and nothing is said about what the returned value is beyond the three illustrative examples.

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 first sentence gives a precise mathematical definition: the smallest m ≥ n (≤12) whose alternating factorial is prime, which is a specific verb-free but exact relation. It is distinguishable from siblings like alternating_prime or alternating_factorial by the "next/smallest" semantics, though it never names an alternative explicitly.

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 constrains usage (n runs 1–12, and input is rejected if no valid m exists), which tells the agent the valid domain and failure mode. However it never says when to pick this over siblings such as alternating_prime or alternating_factorial, leaving the choice to inference from the name.

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