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

Annotations already cover safety (readOnly, idempotent, non-destructive), and the description adds real context beyond them: invalid non-polygonal inputs are rejected rather than returning null, plus a per-call price ($0.001 USDC via x402 on Solana) and an HTTPS alternative endpoint. The rejection behavior is the only functional behavior disclosed; nothing is said about error format or response shape, but pricing/endpoint disclosure is meaningful extra context.

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?

Front-loaded with the core definition and examples, then constraints, then commercial/transport details. Every sentence carries information, though the range restatement duplicates the schema and the pricing/URL tail is somewhat boilerplate.

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?

No output schema exists, and the description compensates by showing what the return looks like through examples (index 4, index 5) and by stating the edge case (0 is index 0) and the failure mode. A caller knows the inputs, the return meaning, and the rejection condition; only error response format is unspecified, which is minor here.

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 meaning of the two parameters, and it does: 's' is the number of sides of the polygon and 'value' is the candidate polygonal number whose index is sought, both illustrated with concrete examples. It also restates the numeric ranges already present in the schema (redundant but harmless). The distinction between s and value is clear enough to call correctly.

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 operation: returning the index n of an s-gonal number, i.e. the inverse of the polygonal-number function. Worked examples ('Triangular 10 is index 4, and square 25 is index 5') pin down the semantics without ambiguity. It does not explicitly name the sibling polygonal (the forward direction), so differentiation is by inference rather than declaration.

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 examples and the rejection rule ('A value that is not s-gonal is rejected') give an implicit sense of when the tool applies, and the input bounds constrain callers. However, there is no explicit statement of alternatives (e.g. use polygonal to go from index to value) and no guidance on what to do after a rejection, so usage must be inferred.

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