ScorpTech x402 API

Cyclotomic Coeff

cyclotomic_coeff

Read-onlyIdempotent

The coefficient of x^k in Phi_n. Phi_6 has coefficients 1, -1, 1, so k=1 is -1. The coefficient of x^7 in Phi_105 is -2. k past the degree is rejected. n runs from 1 to 120. Price: $0.001 USDC per successful call (x402, Solana). Same tool over HTTPS: POST https://scorptech.it.com/api/x402/v1/cyclotomic-coeff

Tool call template

Replace each placeholder with a value matching the input schema. Only required parameters are included; add optional parameters as needed.

{
  "id": 1,
  "jsonrpc": "2.0",
  "method": "tools/call",
  "params": {
    "arguments": {
      "n": "<n>",
      "k": "<k>"
    },
    "name": "cyclotomic_coeff"
  }
}

Input Schema

Name	Required	Description	Default
`k`	Yes
`n`	Yes

Schema Changelog

Changes observed during successful MCP inspections.

AddedOct 23, 2026

TDQS

A3.6/5.0

Behavior4/5

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

Annotations already declare the operation as read-only, idempotent, non-destructive, and open-world. The description adds valuable behavioral context beyond that: it states the error behavior ('k past the degree is rejected'), the valid range for n, and the pricing/payment model (x402, Solana, $0.001 USDC per successful call).

Agents need to know what a tool does to the world before calling it. Descriptions should go beyond structured annotations to explain consequences.

Conciseness4/5

Is the description appropriately sized, front-loaded, and free of redundancy?

The description is front-loaded with the core definition and examples, then moves to constraints and payment details. It is reasonably concise, though the pricing and HTTPS endpoint sentences are somewhat tacked on and could be separated for cleaner structure.

Shorter descriptions cost fewer tokens and are easier for agents to parse. Every sentence should earn its place.

Completeness4/5

Given the tool's complexity, does the description cover enough for an agent to succeed on first attempt?

Given no output schema, the description sufficiently explains what is computed, the valid input ranges, the rejection condition, and the payment model. The return value is implied by the examples (an integer coefficient), but the description does not explicitly state the output type; adding that would make it fully complete.

Complex tools with many parameters or behaviors need more documentation. Simple tools need less. This dimension scales expectations accordingly.

Parameters4/5

Does the description clarify parameter syntax, constraints, interactions, or defaults beyond what the schema provides?

With 0% schema description coverage, the description carries the full burden and does well: it defines n as the cyclotomic polynomial index (1–120) and k as the exponent, clarifies that k beyond the polynomial's degree is rejected, and gives examples that map parameter values to outputs. It could explicitly state k's lower bound (0), but otherwise the meaning is clear.

Input schemas describe structure but not intent. Descriptions should explain non-obvious parameter relationships and valid value ranges.

Purpose4/5

Does the description clearly state what the tool does and how it differs from similar tools?

The description precisely states that the tool returns the coefficient of x^k in the cyclotomic polynomial Phi_n, with concrete examples that make the operation unambiguous. It does not explicitly distinguish this from sibling tools like cyclotomic, cyclotomic_height, or cyclotomic_one, so it falls short of full sibling differentiation.

Agents choose between tools based on descriptions. A clear purpose with a specific verb and resource helps agents select the right tool.

Usage Guidelines2/5

Does the description explain when to use this tool, when not to, or what alternatives exist?

No guidance is given about when to choose this tool over alternatives such as cyclotomic (which likely returns the full polynomial) or related sibling tools. Constraints like n from 1 to 120 and k past the degree being rejected are stated, but they are input limits, not usage guidance.

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

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