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

With no annotations to lean on, the description carries the full burden and delivers: it discloses the computation method, the two independent expansions and what happens if they disagree (MISS naming both), the deterministic output fields (energy_num, energy_den), and three distinct refusal codes for bound, float, and non-string inputs. That is unusually rich behavioral disclosure for a mutation-free compute tool.

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?

It is front-loaded with purpose, then format, return shape, and refusal modes in that order, so scanning yields the essentials fast. It is dense with domain tokens ('QMACourtLaw.fixturePermanent', 'per=P_want_6') and semicolon-packed, which costs some readability, but almost every clause carries signal.

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?

There is no output schema, so the description must explain return values itself, and it does: energy_num/energy_den, verdict WIN/MISS with the equality condition, and the detail string. Combined with the refusal taxonomy and the n≤3 bound, an agent has everything needed to call and interpret the tool.

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 already 100% (the matrix format 'rows | cells , integer entries' is in the schema), so the baseline is 3. The description adds genuine meaning beyond that: the integer/decimal-string constraint, the refusal of floats (REFUSED_FLOAT), and the refusal of non-string JSON values (REFUSED_NOT_A_STRING), which clarifies the parameter's accepted domain.

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 opening clause states a specific verb and resource: '#P exact integer permanent, n≤3', and immediately names the algorithm (direct expansion, cross-checked against Ryser's formula). It is distinguishable from the verify_* and math_court siblings because it computes a permanent rather than verifying a circuit or phase.

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 implies usage bounds by stating 'n>3 is REFUSED_INSTANCE_BOUND' and referencing 'the same law as math_court domain qma_permanent', which gives the agent a rough sense of scope. However, it never explicitly says when to pick this tool over the many sibling verifiers, nor names an alternative.

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