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

Annotations already cover read-only, idempotent, and non-destructive behavior, so the description's extra value is the invalid-input rejection rule, the $0.001 per-success payment, and the alternate HTTPS endpoint. It does not describe auth flow or rate limits, but the pay-per-call model is disclosed.

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?

It is front-loaded with the definition and examples, followed by constraints, payment, and endpoint details in a logical order. The length is justified for a one-parameter tool because each part supports correct invocation.

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-parameter math tool with no output schema, it gives output examples, valid range, invalid-input behavior, and cost, so an agent can call it. The omission of a definition of Higgs prime means an agent may not know the full valid domain without another tool or trial-and-error.

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 n; it supplies the valid numeric range and concrete input-output pairs, showing n is the prime value rather than an index. The one gap is that it never explicitly defines what qualifies as a Higgs prime, leaving the exact valid input set to be discovered by rejection.

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 opening sentence defines the exact operation and output: distance from a Higgs prime to the next one. The concrete examples lock in the interpretation and distinguish it from sibling operations such as higgs_next or higgs_index.

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?

It gives a key precondition—non-Higgs-prime inputs are rejected—and the n range, which helps avoid invalid calls. However, it does not say when to choose this over prime_gap or the other Higgs siblings, so usage guidance remains implied.

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