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Affine Earth Math Court Remote

by earth.affine
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project_shor_twin

QC-001b Shor TWIN projection. Post ONE half, receive the other. Post halfPow a (a^2 == 1 mod N, a != +/-1) and the court projects factor/cofactor by gcd(a-1,N) and gcd(a+1,N). Post factor and cofactor and it projects a by CRT (a == 1 mod factor, a == -1 mod cofactor). The projected half is verified before return. Bijection in polynomial time - it does not search for a period and does not derive a discrete log. Decimal strings; empty means not posted. Floats refused.

Input Schema

TableJSON Schema
NameRequiredDescriptionDefault
NYesthe modulus, decimal, e.g. 15 - carried in every shape; without it there is no question
factorNopost factor together with cofactor (e.g. 3 and 5) and the court projects halfPow by CRT
halfPowNopost this half (a with a^2 == 1 mod N, a != 1, N-1), e.g. 4, and the court projects factor and cofactor
cofactorNoN / factor, decimal, e.g. 5; posted together with factor

Schema Changelog

Changes observed during successful MCP inspections.

  1. Changed6 schema fields changed
    • addedInput schema / anyOf
      Added value: +[
+  {
+    "required": [
+      "N"
+    ]
+  },
+  {
+    "required": [
+      "N",
+      "halfPow"
+    ]
+  },
+  {
+    "required": [
+      "N",
+      "cofactor",
+      "factor"
+    ]
+  }
+]
    • addedInput schema / properties / N / description
      Added value: +"the modulus, decimal, e.g. 15 - carried in every shape; without it there is no question"
    • addedInput schema / properties / cofactor / description
      Added value: +"N / factor, decimal, e.g. 5; posted together with factor"
    • addedInput schema / properties / factor / description
      Added value: +"post factor together with cofactor (e.g. 3 and 5) and the court projects halfPow by CRT"
    • addedInput schema / properties / halfPow / description
      Added value: +"post this half (a with a^2 == 1 mod N, a != 1, N-1), e.g. 4, and the court projects factor and cofactor"
    • changedInput schema / required
      Previous value: -[]New value: +[
+  "N"
+]
  2. First observed

TDQS

A4/5.0
Behavior4/5

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

No annotations are provided, so the description carries the full burden and does so well: it discloses that the projected half is verified before return, that operation is bijective and polynomial-time, and input hygiene rules ('decimal strings; empty means not posted; floats refused'). It omits error/response-shape details, keeping it below a 5.

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?

Front-loads the core contract ('Post ONE half, receive the other') then expands the two cases efficiently. Dense but each sentence contributes; the 'court' terminology is slightly opaque jargon but not wasteful.

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?

With no output schema and no annotations, the description adequately covers both input modes and what is returned (factor/cofactor vs a), plus verification and formatting rules. Minor gap: it doesn't describe the return value's precise form (e.g., sign/ordering of factor vs cofactor).

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?

Schema coverage is 100% (baseline 3), but the description adds relational meaning beyond the schema: that factor and cofactor must be posted together, and the algebraic identities the projected values must satisfy (a^2==1 mod N; a==1 mod factor, a==-1 mod cofactor). This clarifies the anyOf shape selection.

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?

Names a specific operation (Shor twin projection) and states its core behavior precisely: 'Post ONE half, receive the other' with two concrete modes. It distinguishes itself from period-search/discrete-log tools, though it never names a specific sibling by name.

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

Usage Guidelines4/5

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

Gives clear conditions for the two input shapes (post halfPow vs post factor+cofactor) and explicitly states what the tool is NOT for ('does not search for a period and does not derive a discrete log'), which routes agents away from verify_period-like siblings. No explicit when-not-use against the verify_* family beyond that.

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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