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

With no annotations, the description must carry behavioral disclosure and it does: verification before return, polynomial-time bijection, no period search/discrete log, decimal-only input, empty-means-not-posted, and floats refused. It does not explicitly discuss side effects or edge-case behavior when both halves are supplied, but the main behavioral profile is clearly conveyed.

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 dense but not bloated; the front-loaded 'Post ONE half, receive the other' immediately communicates the contract. The only slight deduction is for cryptic jargon such as 'QC-001b' and 'court', which carry a little less value than the mathematical content.

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 four-parameter tool with no output schema and no annotations, the description covers the transformation, verification, complexity, and input format. It could be more explicit about the required presence of N in both modes and about what happens if both halves are posted, but an agent can infer the intended call pattern from the stated math.

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%, yet the description gives every parameter concrete meaning: N is the modulus in the congruences, halfPow is a nontrivial square root a, and factor/cofactor are the CRT pair satisfying a == 1 mod factor and a == -1 mod cofactor. It also defines value constraints and the empty-string convention, far exceeding the bare string 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?

The description states a concrete bidirectional projection between halfPow a and factor/cofactor modulo N, with explicit math formulas for both directions. It also distinguishes itself from period-finding/discrete-log tools by saying it 'does not search for a period and does not derive a discrete log.'

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?

It gives two explicit call modes: post halfPow to get factor/cofactor, or post factor/cofactor to get a, and explains the exact congruence conditions for each. It does not name sibling alternatives explicitly, but the mode-based usage and the 'does not search for a period' exclusion give the agent enough contextual guidance.

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