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

Annotations already mark this read-only, idempotent, non-destructive, so safety is covered. The description adds genuinely useful behavior beyond that: the 15-digit threshold after which values become decimal strings, the paid $0.001 USDC per successful call x402/Solana model, and an 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?

Front-loaded with the formula, then examples, then domain bounds, then output format and billing details. Three dense sentences with essentially no filler; only the pricing/endpoint clause slightly crowds the mathematical definition.

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?

With no output schema and no per-parameter descriptions, the description supplies everything an agent needs: exact semantics, edge-case examples, valid input ranges, and the return-type switch to decimal strings past 15 digits, plus cost and transport details. Nothing needed to call it correctly is missing.

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 parameter meaning, and it does: n ranges 0-100000, k ranges 0-512, k is the number of descending factors, and the (5)_0=1 and (5)_6=0 examples implicitly document the k=0 and k>n edge cases. It could be slightly clearer by stating the k>n zero rule explicitly rather than only by example.

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 exactly what is computed with the formula (n)_k = n*(n-1)*...*(n-k+1) and concrete examples, so an agent knows this is the falling factorial rather than a generic factorial. It does not name or distinguish itself from the nearest sibling 'rising' (rising factorial), leaving that differentiation to the name.

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 establishes the mathematical semantics but gives no explicit when-to-use or when-not-to-use guidance and never points to alternatives like rising, alt_factorial, or double_factorial despite a dense sibling list. Usage is inferable only from the formula.

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