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

Annotations already cover readOnly/idempotent/non-destructive, so the description's added value is the commercial and transport context: $0.001 USDC per successful call via x402 on Solana and an equivalent HTTPS endpoint. That is genuinely useful for an agent that must consider payment path. It stops short of disclosing response shape or error behavior for out-of-range inputs.

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 definition and formula, followed by worked examples, then domain bounds, then pricing/transport. Every sentence adds a distinct fact, and nothing is padded.

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 two-parameter pure-math tool with no output schema, the definition is essentially complete: what it computes, valid input domains, base-case behavior, and cost. The only gap is that it does not state the return representation (bare integer vs. structured payload), which matters slightly given there is no output schema.

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 carries the burden: it defines n as the index and s as the polygon side count, and the examples map s=3→triangular, 4→square, 5→pentagonal, 6→hexagonal, which the schema cannot convey. It repeats the min/max bounds already in the schema, and the note 'The 0th number is 0 and the 1st is 1' usefully pins the indexing convention.

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 gives a precise verb-free but exact specification: 'The nth s-gonal number' with the closed-form formula n*((s-2)*n - (s-4))/2 and concrete worked values (triangular 4 = 10, square 5 = 25, pentagonal 5 = 35, hexagonal 5 = 45). That formula alone separates it from siblings such as centered_polygonal (different formula) and polygonal_index (the inverse operation), so an agent can pick it without opening the schema.

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?

Usage is only implied by the valid ranges (s from 3 to 20, n from 0 to 1000000) and by the examples; there is no explicit statement of when to choose this over polygonal_index, pentagonal, or centered_polygonal, and no mention of what happens when a value is out of the stated bounds.

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