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

Annotations already cover the read-only/idempotent/non-destructive profile, and the description adds genuinely non-annotated facts: a $0.001 USDC per-call price, an x402/Solana payment path, a parallel HTTPS endpoint, and the invariant that the residue is strictly less than the modulus. It doesn't state failure behavior for payment or whether retries are billed, which keeps it from a 5.

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 computation and the constraint set are front-loaded, with pricing and the HTTPS alternative correctly deferred to the end. Sentences are telegraphic but each carries a constraint or a fact; nothing is filler.

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 3-required-parameter arithmetic tool with no output schema, the description covers the formula, all domain constraints, the return value bound, and the payment/transport context. Only the exact return type and payment-failure semantics are unaddressed, which are minor for this complexity.

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 compensate, and it largely does: mod is 'a positive decimal integer' with zero rejected, k must be a valid ascent of n, and n is bounded to 1..60. It doesn't note that mod is passed as a digit string (up to 600 characters per the pattern), which is an easy mistake an agent could make from the integer-typed siblings.

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?

Opens with the exact computation, 'A(n, k) mod m for the Eulerian numbers', which is a specific verb+resource and distinct from the exact-value siblings eulerian/eulerian_poly/eulerian_row. It never explicitly names those siblings, so an agent must infer the modular-vs-exact split from the name alone.

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?

Preconditions are stated clearly (modulus must be positive, zero rejected, k must be an ascent of n, n in 1..60), which tells the agent when a call will succeed. It gives no guidance on when to prefer this over eulerian or eulerian_row, so alternative selection is left implicit.

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