kbound

Inductive rule recovery toolkit. Give it K observations, get back the rule plus a sample-complexity certificate.

pip install kbound

from kbound import recover_rule

# Five observations from a deterministic system
observations = [(1, 15), (2, 18), (3, 12), (4, 16), (5, 11)]

r = recover_rule(observations)

r.rule          # 'decision(x) = (5·x² + 7·x + 3) mod 19'
r.confidence    # 1.0
r.verified      # True
r.predict(42)   # 16  (extrapolates over the recovered rule)
r.certificate   # {'k_used': 5, 'k_lower_bound': 4, 'margin': 1.25, ...}

That is the whole API for the common case. No prompt tuning, no training, no LLM call.

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What it does

Given K ≥ 4 input/output pairs from any deterministic system — an AI agent, a smart contract, a legacy decision rule, a math problem — kbound:

1. Classifies the geometric structure of the rule (spatial-separable vs. algebraically-coupled).
2. Detects which family it belongs to from a 16-operator catalog (linear residual, polynomial threshold, modular exponentiation, Boolean gate, k-bit parity, linear recurrence, …).
3. Recovers the parameters via the matching solver — Boyar's attack for LCGs, Vandermonde solve over F_p for polynomials, GF(2) elimination for additive-feedback generators, and so on.
4. Verifies the recovered rule reproduces every observation, then emits a certificate with K_used, the family-specific K_lower_bound, and the safety margin.

If nothing in the catalog matches, it returns RecoveryResult(no_recovery, …) — honest failure, no hallucination.

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How it compares

	What it does	When kbound wins
SymPy	Manipulates a formula you give it (solve, simplify, integrate).	When you do not have the formula yet — only observations.
DreamCoder / general program synthesis	Open-ended program search via library learning.	When the rule fits a known algebraic family — kbound is O(K) to seconds, DreamCoder is minutes to hours.
Z3 / SMT	Constraint satisfaction over a manually-encoded theory.	When you want push-button recovery without writing the encoding. (kbound uses Z3 internally for the bitwise non-linear class.)
LLM K-shot induction	Few-shot prompt: "given these examples, what's f(x)?"	When the rule is algebraically coupled: frontier LLMs hit ~0.02 accuracy on lcg_mod50 at K=16 in-context (Dovzak 2026). kbound hits 1.00.

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Use cases

• Build deterministic copilots for AI agents: drop kbound-mcp in as an MCP server; agents (Claude Desktop, Cursor, Cline, …) now answer K-shot induction questions correctly instead of hallucinating. See the MCP integration section below.
• Audit deployed AI agents: recover the rule the agent is actually following from a trace log; compare against the vendor-declared specification (kbound.check_compliance); emit auditor-ready Markdown + CSV + remediation SLA.
• Cryptanalysis & security research: recover LCG / glibc / Java / MT19937 / TGFSR-additive-feedback / 2-LCG-XOR-composition state from observed outputs, with a sample-complexity bound on the recovery.
• Behavioral diff between system versions: kbound.diff_traces(v1, v2) surfaces silent rule changes between deployments.
• Reverse-engineer legacy decision systems: feed (input, output) pairs from production logs, get back the executable rule.

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Operator catalog (16 families, today)

Linear Residual Policy             y = (a·x + c) mod m            small modulus
Linear Residual (Large-Modulus)    y = (a·x + c) mod m            unknown m, Boyar 1989
Linear Residual w/ Truncation      java.util.Random — top-32 bits of m=2⁴⁸ state
Polynomial Threshold Rule          y = (a·x² + b·x + c) mod p     small modulus
Polynomial Threshold (Large-Mod)   y = polynomial mod arbitrary m, deg 2-3
Cubic Threshold Rule               y = (a·x³ + b·x² + c·x + d) mod p
Scaling Residual Rule              y = (a·x) mod p
Inverse Residual Rule              y = x⁻¹ mod p
Exponential Pattern Rule           y = a^x mod p
Additive Composition Rule          y = (x₁ + x₂) mod p
Linear Recurrence Policy           y_n = (a·y_{n-1} + b·y_{n-2}) mod m
Linear Recurrence (Large-Modulus)  order-2 / order-3, arbitrary m
Binary Decision Gate               2-input boolean truth table
k-Bit Parity Rule                  XOR over a subset of bits
Stateful Pattern (Large State)     MT19937 — recovered from K ≥ 624 outputs
Stateful Pattern (Additive Fbk)    glibc TYPE_3 — recovered from K ≥ 100 outputs

Each operator has a typical K_lower_bound. The certificate every recovery emits cites it.

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Compliance primitive

When the rule is declared by a vendor and you want to audit how often the agent honors it:

from kbound import check_compliance, ClaimedSpec

claimed = ClaimedSpec(
    family="linear_residual",
    params={"a": 3, "c": 5, "m": 11},
    source="vendor system prompt v7.0",
)

report = check_compliance(observations, claimed)

report.agreement_pct       # 0.73 — the agent honors the spec 73% of the time
report.counterexamples     # first 10 deviations: input, observed, expected
report.is_compliant()      # False at default 95% threshold

Render to Markdown, legal-grade CSV, remediation SLA, or AI Bill-of-Materials section:

from kbound import (
    render_compliance_report_md,
    render_legal_counterexample_csv,
    render_remediation_sla,
    render_ai_bom_section,
)

The compliance check is the primitive that converts "no rule recovered" into a positive audit finding when you have a declared spec to test against.

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MCP integration (Claude Desktop, Cursor, Cline)

kbound ships an MCP server that lets any MCP-aware AI agent call rule recovery as a tool. This closes the gap reported in Dovzak 2026: frontier LLMs hit ~0.02 in-context accuracy on algebraically-coupled K-shot induction; with the MCP server they delegate to a deterministic solver.

pip install kbound[mcp]

Claude Desktop

Add to ~/Library/Application Support/Claude/claude_desktop_config.json (macOS) or %APPDATA%\Claude\claude_desktop_config.json (Windows):

{
  "mcpServers": {
    "kbound": {
      "command": "kbound-mcp"
    }
  }
}

Restart Claude Desktop. Five tools become available: recover_rule_from_observations, predict_under_recovered_rule, list_supported_rule_families, verify_compliance_against_claimed_rule, classify_observation_geometry.

Cursor / Cline

Add to ~/.cursor/mcp.json (Cursor) or your Cline config:

{
  "mcpServers": {
    "kbound": {
      "command": "kbound-mcp",
      "args": []
    }
  }
}

Manual test

You can run the server directly (it speaks JSON-RPC over stdio):

kbound-mcp
# or:
python -m kbound.mcp

For interactive testing, install the MCP Inspector:

npx @modelcontextprotocol/inspector kbound-mcp

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Foundations

kbound is grounded in four papers (Dovzak):

1. The Compositional Depth Decay is a Dispatch Tax — exponential-decay law fingerprinting architectures' compositional generalization.
2. The Geometric Boundary of Few-Shot Rule Induction in Frontier LLMs — measures where commercial LLMs cross from 0.88 accuracy to chance, depending on rule geometry.
3. Geometry of Few-Shot Rule Induction — the spatial-separable vs. algebraically-coupled distinction; defines the Relational Consistency Executor (RCE) that kbound implements for the algebraic side.
4. A Universal Sample-Complexity Framework for Black-Box Rule Recovery — the master inequality K ≥ (dim − log|Sym(r)|)/b that yields the K-bound certificate every recovery cites.

You do not need to read them to use kbound. They are the answer to "is this principled?" rather than "how do I use it?"

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Status

v0.1.0 — alpha. The catalog ships 16 operator families, of which 13 are validated end-to-end against real-world trace batteries; 3 are experimental (scaling_residual_rule, inverse_residual_rule, k_bit_parity_rule — recovery works but customer-fit on real data has not been measured). Each operator's status is exposed at kbound.OPERATORS[<key>].validation_status. The MCP server adapter and the full documentation site are in progress. Public API surface (recover_rule, check_compliance, classify_geometry, diff_traces, the renderers) is stable for v0.x.

Roadmap (short):

• kbound/mcp.py — MCP server packaging, registered in the public catalog.
• kbound diff CLI — behavioral diff between trace versions.
• Documentation site at originalkazdov.github.io/kbound.
• Continuous benchmark of frontier LLMs vs. kbound across the operator catalog.

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Contributing

Issues and PRs welcome. New operators are first-class: see kbound/solvers/ for the solver template plus the operator metadata pattern in kbound/operators.py.

License

MIT.