Research Artifact — Not Production-Ready

This model verifies arithmetic responses using structural binary features, not semantic understanding. It achieves AUROC 0.997 on the held-out test set (Exp 89 self-bootstrap reference: 1.0). Factual or freeform answers are outside its scope.

constraint-propagation-arithmetic

Learned Ising constraint model for arithmetic response verification.

This model was trained via discriminative Contrastive Divergence (Exp 62/89) on 400 verified (question, correct answer, wrong answer) triples for arithmetic problems: addition, multiplication, and modular arithmetic. It assigns a scalar energy to any binary feature vector encoding a response. Lower energy means the response is more likely correct.

What It Is

An Ising Energy-Based Model (EBM) with quadratic energy:

E(x) = -(b^T s + s^T J s)    s = 2x - 1 ∈ {-1, +1}^200

Where:

• x is a 200-dimensional binary feature vector encoding structural properties of an LLM response (presence of numbers, operators, answer patterns — see Feature Encoding below).
• J is a 200×200 coupling matrix (symmetric, zero diagonal) — encodes which feature pairs co-occur in arithmetic-correct responses.
• b is a 200-dim bias vector — encodes which features tend to be active in correct responses.

Lower energy = response looks structurally like a correct arithmetic answer.

How It Was Trained

Algorithm: Discriminative Contrastive Divergence (full-batch)

Hyperparameter	Value
Training pairs	400 (80% of 500 generated)
Feature dimension	200 binary features
Learning rate	0.01
L1 regularization	0.0
Weight decay	0.005
Epochs	300
Source	Exp 62 (domain CD) + Exp 89 (self-bootstrap)

Training data: Programmatically generated (no LLM) triples of the form (question, correct_answer, wrong_answer) for three sub-types:

• Addition: "What is A + B?" — wrong answers off-by-one, off-by-ten, carry errors.
• Multiplication: "What is A * B?" — wrong answers from adjacent products.
• Modular arithmetic: "What is A mod B?" — wrong answers ±1 or ±2.

Benchmark Results

Metric	This export	Exp 89 reference
AUROC (test)	0.9970	1.0000
Accuracy (test)	99.0%	100.0%
Test set size	100	29
Baseline AUROC	0.5	0.5

The slight gap from Exp 89 is expected: Exp 89 used pipeline-extracted constraint features (245 dims) while this export uses the simpler 200-dim structural encoding from Exp 62.

Feature Encoding

Each response is encoded as a 200-dimensional binary vector. The encode_answer function in scripts/experiment_62_domain_constraint_learning.py generates these features:

Group	Features	Examples
Numeric (20)	digit/operator presence	has_number, has_plus, has_equals
Structural (40)	word/char counts, punctuation	short_answer, has_period
Domain-specific (80)	arithmetic/logic/code patterns	has_equation, has_if_then
Consistency (60)	question-answer agreement	q_numbers_in_answer

Usage

import numpy as np
from carnot.inference.constraint_models import ConstraintPropagationModel

# Load model (local directory or HuggingFace Hub)
model = ConstraintPropagationModel.from_pretrained(
    "exports/constraint-propagation-models/arithmetic"
)

# Encode a response as a 200-dim binary feature vector
from scripts.export_constraint_models import encode_answer
x = encode_answer("What is 47 + 28?", "The answer is 75.")

# Verify
energy = model.energy(x)    # lower = more likely correct
score = model.score(x)      # in (0,1); higher = more likely correct
print(f"Energy: {energy:.2f}, Score: {score:.3f}")

Inspect Coupling Matrix

from safetensors.numpy import load_file
tensors = load_file("model.safetensors")
J = tensors["coupling"]   # shape (200, 200), float32
b = tensors["bias"]       # shape (200,), float32
print(f"Sparsity: {(J == 0).mean():.1%}")
print(f"Max |J_ij|: {abs(J).max():.4f}")

Limitations

1. Structural features only: Verifies whether a response looks like a correct arithmetic answer (contains the right numbers, uses the right phrasing). Does not parse or evaluate arithmetic expressions.
2. Template-generated training data: Training pairs come from programmatic generators, not real LLM outputs. May not generalize to unusual phrasings.
3. 200-dim binary features: Fine-grained semantic nuances are not captured. A plausible wrong answer with the same structural features as a correct one will have similar energy.
4. Feature encoder required: You must encode responses with the same 200-dim encoder used during training. Using a different encoder will produce garbage.
5. factual/scheduling not supported: AUROC near 0.5 for those domains in Exp 89 — the structural features don't discriminate factual correctness.

Files

File	Description
model.safetensors	Coupling matrix J (200×200) and bias b (200,) as float32
config.json	Training metadata, benchmark results, limitations
README.md	This file

Citation

@misc{carnot2026constraint_arith,
  title  = {Carnot Constraint Propagation Model: Arithmetic},
  author = {Carnot-EBM},
  year   = {2026},
  url    = {https://github.com/ianblenke/carnot}
}

Spec

• REQ-VERIFY-002: Energy-based verification of LLM responses.
• REQ-VERIFY-003: Correctness score derived from model energy.
• FR-11: Self-bootstrapped improvement of the verifier from pipeline outputs.