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

Annotations already declare a read-only, idempotent, non-destructive operation, so the safety profile is covered. The description still adds real value annotations cannot: the k = 0 sentinel for non-hyperperfect inputs and the $0.001 USDC x402/Solana pricing per successful call. It stops short of describing output shape beyond k or any latency/rate limits.

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 defining formula and returned value, then examples, edge-case convention, domain, and finally pricing/transport. Every sentence carries information; nothing is repeated from the schema except the domain bound, which is used as a usage constraint.

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 predicate with no output schema, the description supplies everything needed to call and interpret it: exact semantics, domain bounds, a sentinel return value for negatives, four worked examples, cost, and an alternate HTTPS endpoint.

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%, but the domain is trivially recoverable: the single parameter n is an integer, and the description states 'n runs from 1 to 100000', which matches the schema's min 1 / max 100000. With one plain integer parameter and no enums or nesting, nothing further is needed, so the description adequately compensates without adding nuance beyond the schema.

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?

It gives a precise mathematical definition (n = 1 + k*(sigma(n) - n - 1)) plus the returned quantity k, which leaves no ambiguity about what the tool computes. It does not, however, contrast itself with near-neighbours such as perfect_check, multiperfect, almost_perfect or abundancy, so an agent must already know the term to pick it over those siblings.

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: 'n runs from 1 to 100000' plus the note that k is 0 for non-hyperperfect n tells the agent how to interpret a miss, and the worked examples (6, 28, 21, 12, 120) anchor expected behaviour. There is no explicit statement of when to choose this over the sibling perfect/abundancy tools, so routing must be inferred.

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