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

With no annotations, the description carries the full burden and does disclose two real behavioral traits: float matrices are refused and n is capped at 3. It says nothing about the return value, determinism, or cost, which matters for a '#P exact' computation, so the disclosure is partial.

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 description is short with no filler, but the compression comes at the cost of clarity: the terse jargon ('#P', 'Same law as') obscures meaning, and the size constraint and refusal condition are stacked without clear structure.

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-param compute tool with full schema coverage and no output schema, the description covers the input format and two constraints, making it usable. It still omits what the tool returns (the integer value) and any authentication or cost context, leaving a modest gap.

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 100% and the single matrix param is fully documented in the schema with a format example. The description adds 'matrix is a decimal-string grid,' reinforcing the encoding but contributing little beyond that baseline.

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 names the computed object ('exact integer permanent') and a size constraint (n≤3), so an agent can infer it computes permanents of small matrices. However, there is no explicit verb, the '#P' prefix is unexplained jargon, and it does not clearly differentiate itself from the many verify_* siblings beyond a passing reference to the math_court domain.

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?

'Float permanent refused' implies a constraint on valid inputs, but the description never states when to use this tool versus the sibling verify_* or math_court tools. No prerequisites, no alternatives, no positive usage condition are given.

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