# Particle Transformers: An Architectural Answer to the Incentives that Cause Hallucination
Author:** Xenonostra’
Date:** September 8, 2025
## Abstract
Large Language Models (LLMs) are known to “hallucinate”—producing plausible falsehoods instead of admitting uncertainty. A groundbreaking new paper, “Why Language Models Hallucinate” (Kalai et al., 2025), diagnoses this problem not as a deep architectural flaw, but as a behavioral artifact of the training process: LLMs are incentivized to be “good test-takers” who guess when uncertain because current evaluation benchmarks reward plausible answers over honest admissions of ignorance. While Kalai et al. propose a socio-technical solution—reforming evaluation metrics—we argue that a complementary architectural solution is also needed.
Inspired by their diagnosis, we propose the **Particle Transformer**, an architecture designed to provide a native mathematical substrate for the epistemic states—truth, contradiction, and uncertainty—that reformed benchmarks would require. The model uses three parallel attention streams operating over real (ℝ), complex (ℂ), and split-complex (𝔻) numbers, implemented via Hermitian products and Lorentz geometry to ensure well-defined gradients. We introduce α-entmax gating to handle mixed geometries and connect our approach to formal reject-option theory for principled abstention. This offers an architectural pathway to more trustworthy AI systems with native mechanisms to “know when they don’t know.”
## 1. Introduction: A New Diagnosis for Hallucination
The transformer architecture’s attention mechanism is the engine of modern AI, but it is fundamentally agnostic to truth, leading to the critical failure mode of hallucination. A new paper from OpenAI, “Why Language Models Hallucinate” by Kalai, Nachum, Vempala, and Zhang (2025), offers a compelling diagnosis: hallucinations are a learned behavior driven by the incentive structures of our training and evaluation pipelines.
Their core analogy is that LLMs are trained to be excellent “test-takers.” On an exam where there is no penalty for wrong answers, the optimal strategy is always to guess rather than leave a question blank. Current LLM leaderboards create this exact incentive, rewarding models that generate plausible-sounding guesses over those that might respond with “I do not know.”
While Kalai et al. propose changing benchmark scoring to reward epistemic humility, this raises a crucial architectural question: Do current transformers have a native capacity to gracefully represent and act upon uncertainty? The standard softmax output lacks a dedicated, explicit state for “unknown” or “contradictory.”
This paper explores an architecture with a built-in mathematical language for the epistemic states that reformed evaluation systems would reward. We ground our approach in established geometric deep learning, ensuring mathematical rigor and practical implementability.
## 2. Mathematical Foundations
### 2.1 Three Geometric Channels for Epistemic States
We project input embeddings into three distinct vector spaces, each with its own algebraic structure and geometric properties:
**Real Channel (ℝ):** Standard real-valued vectors representing factual information with Euclidean geometry.
**Complex Channel (ℂ):** Complex-valued vectors where phase relationships encode contradiction through interference. We employ Wirtinger calculus (Trabelsi et al., 2018) for well-defined gradients.
**Hyperbolic Channel (𝔻):** Split-complex numbers (x + yj where j² = +1) representing uncertainty, implemented via Lorentz geometry with proper log/exp maps following hyperbolic neural networks (Ganea et al., 2018; Nickel & Kiela, 2017).
### 2.2 Split-Complex Numbers and Lorentz Geometry
While we conceptually use split-complex (hyperbolic) numbers with the intuition that j² = +1 creates an indefinite metric suitable for uncertainty, we implement this via Lorentz geometry to ensure numerical stability:
For vectors in the Lorentz model, we define the Minkowski bilinear form:
$$langle x, y rangle_L = -x_0 y_0 + sum_{i=1}^{d} x_i y_i$$
The hyperbolic manifold is then:
$$mathcal{H}^d = {x in mathbb{R}^{d+1} : langle x, x rangle_L = -1, x_0 > 0}$$
The logarithmic and exponential maps at the origin o = (1, 0, …, 0):
$$log_o(x) = frac{cosh^{-1}(-langle o, x rangle_L)}{sqrt{langle x – o, x – o rangle_L}} (x – o)$$
$$exp_o(v) = cosh(|v|_L) o + sinh(|v|_L) frac{v}{|v|_L}$$
This preserves our split-complex intuition while providing stable, differentiable operations. The idempotent decomposition e₊ = (1+j)/2, e₋ = (1-j)/2 conceptually separates certain/uncertain components.
## 3. The Particle Transformer Architecture
### 3.1 Attention with Well-Defined Gradients
For input sequence X ∈ ℝⁿˣᵈ, we compute three projections into our geometric spaces:
“`python
X_R = W_R @ X # Real projection: ℝⁿˣᵈ
X_C = W_C @ X # Complex projection: ℂⁿˣᵈ
X_H = W_H @ X # Hyperbolic projection: on ℋᵈ
“`
**Critical:** We compute attention scores that are real-valued for all channels to ensure proper gradients:
$$s_R = frac{Q_R K_R^top}{sqrt{d_k}} in mathbb{R}^{n times n}$$
$$s_C = frac{text{Re}(Q_C K_C^dagger)}{sqrt{d_k}} in mathbb{R}^{n times n}$$
$$s_H = frac{langle log_o Q_H, log_o K_H rangle_L}{sqrt{d_k}} in mathbb{R}^{n times n}$$
Where K_C^† denotes the Hermitian conjugate transpose, ensuring s_C is real-valued through Hermitian similarity. This follows the complex-valued deep learning convention (Trabelsi et al., 2018) and enables Wirtinger calculus for backpropagation.
### 3.2 α-Entmax Gating Across Geometries
To handle the mixing of three different geometries, we use α-entmax (Peters et al., 2019) to produce sparse, calibrated channel weights:
$$tilde{s} = [omega_R s_R, omega_C s_C, omega_H s_H]$$
$$w = text{entmax}_alpha(tilde{s}), quad alpha in (1, 2]$$
Where ω_R, ω_C, ω_H are learned channel importance weights. The entmax function generalizes softmax (α=1) and sparsemax (α=2), allowing channels to completely deactivate when appropriate—crucial for epistemic abstention.
### 3.3 Channel-Specific Aggregation
Each channel aggregates values according to its geometry:
**Real Channel:** Standard weighted average
$$O_R = text{softmax}(s_R) cdot V_R$$
**Complex Channel:** Density matrix formulation for quantum-like superposition
$$rho = sum_i |a_irangle langle a_i|, quad a_i = text{softmax}(s_C)_i cdot V_{C,i}$$
$$O_C = text{Tr}(rho cdot V_C)$$
**Hyperbolic Channel:** Aggregation in tangent space
$$O_H = exp_oleft(text{softmax}(s_H) cdot log_o(V_H)right)$$
### 3.4 Final Particle Attention Output
The complete attention mechanism combines all channels with their gated weights:
$$text{ParticleAttention}(Q, K, V) = w_R cdot O_R + w_C cdot O_C + w_H cdot O_H$$
## 4. Training Framework with Reject-Option Theory
### 4.1 Formal Abstention via Selective Prediction
Following Chow’s reject option theory (Chow, 1957) and modern selective prediction (Geifman & El-Yaniv, 2017), we train an explicit abstention head that learns when to say “I don’t know”:
$$g(x) : mathcal{X} rightarrow {0, 1}$$
Where g(x) = 1 indicates the model should make a prediction and g(x) = 0 indicates abstention.
The risk-coverage trade-off is:
$$text{Risk}@text{Coverage}(c) = frac{mathbb{E}[ell(f(x), y) cdot g(x)]}{mathbb{E}[g(x)]} quad text{s.t.} quad mathbb{E}[g(x)] geq c$$
### 4.2 Comprehensive Loss Function
“`python
def particle_transformer_loss(outputs, labels, epistemic_labels, alpha=1.5):
“””
Rigorous loss function with geometric awareness and abstention
Args:
outputs: (O_R, O_C, O_H, w, abstain_logits)
labels: Ground truth including facts, contradictions, uncertainties
epistemic_labels: {TRUE, CONTRADICT, UNKNOWN}
alpha: entmax parameter
“””
# Channel-specific losses
L_true = cross_entropy(decode_R(outputs.O_R), labels.facts)
# Complex contradiction via phase contrast
L_contra = phase_contrast_loss(outputs.O_C, labels.contradictions)
# Abstention loss with selective prediction
L_abstain = binary_cross_entropy(
sigmoid(outputs.abstain_logits),
labels.should_abstain
)
# Risk-coverage trade-off
L_selective = selective_risk_loss(
outputs.predictions,
labels.targets,
outputs.abstain_probs,
target_coverage=0.85
)
# Calibration loss (Expected Calibration Error)
L_calibration = ece_loss(outputs.predictions, labels.targets)
# Channel orthogonality regularization
R_orthogonal = orthogonality_penalty(
outputs.O_R, outputs.O_C, outputs.O_H
)
# Channel balance regularization (replacing “information conservation”)
R_balance = (variance(outputs.w) – tau).clamp(min=0)
# Entropy regularization for entmax sparsity
R_entropy = -alpha * torch.sum(outputs.w * torch.log(outputs.w + 1e-8))
return (α * L_true + β * L_contra + γ * L_abstain +
δ * L_selective + κ * L_calibration +
λ₁ * R_orthogonal + λ₂ * R_balance + λ₃ * R_entropy)
“`
### 4.3 Phase Contrast Loss for Contradictions
“`python
def phase_contrast_loss(complex_outputs, contradiction_pairs):
“””
Encourages contradictory statements to have opposing phases
Uses Hermitian products to ensure real-valued loss
“””
z1, z2 = complex_outputs[contradiction_pairs[:, 0]], complex_outputs[contradiction_pairs[:, 1]]
# Hermitian inner product (real-valued)
similarity = torch.real(torch.sum(z1.conj() * z2, dim=-1))
# Contradictions should have negative similarity (π phase difference)
target = -torch.ones_like(similarity)
return F.mse_loss(similarity, target)
“`
### 4.4 Training Algorithm with Gradient Handling
“`python
def train_particle_transformer(model, loader, optimizer, config):
“””
Training loop with proper gradient handling for mixed geometries
“””
for epoch in range(config.epochs):
for batch in loader:
# Project to three geometric spaces
Q_R, K_R, V_R = model.project_real(batch.x)
Q_C, K_C, V_C = model.project_complex(batch.x) # Complex gradients via Wirtinger
Q_H, K_H, V_H = model.project_hyperbolic(batch.x) # On Lorentz manifold
# Compute real-valued attention scores
s_R = torch.matmul(Q_R, K_R.T) / math.sqrt(config.d_k)
s_C = torch.real(torch.matmul(Q_C, K_C.conj().T)) / math.sqrt(config.d_k)
s_H = lorentz_dot(model.logmap(Q_H), model.logmap(K_H)) / math.sqrt(config.d_k)
# α-entmax gating
s_stack = torch.stack([
config.omega_R * s_R,
config.omega_C * s_C,
config.omega_H * s_H
], dim=-1)
w = entmax_alpha(s_stack, alpha=config.alpha)
# Channel-specific aggregation
O_R = torch.softmax(s_R, dim=-1) @ V_R
O_C = density_matrix_aggregate(torch.softmax(s_C, dim=-1), V_C)
O_H = model.expmap(torch.softmax(s_H, dim=-1) @ model.logmap(V_H))
# Combine with gating
output = w[…, 0:1] * O_R + w[…, 1:2] * O_C + w[…, 2:3] * O_H
# Abstention head
abstain_logits = model.abstention_head(output)
# Compute loss
outputs = ParticleOutput(
O_R=O_R, O_C=O_C, O_H=O_H,
w=w, predictions=output,
abstain_logits=abstain_logits,
abstain_probs=torch.sigmoid(abstain_logits)
)
loss = particle_transformer_loss(
outputs, batch.labels, batch.epistemic_labels,
alpha=config.alpha
)
# Backpropagation with gradient clipping
loss.backward()
# Clip gradients (especially important for hyperbolic)
torch.nn.utils.clip_grad_norm_(
model.parameters(),
config.grad_clip,
norm_type=2
)
optimizer.step()
optimizer.zero_grad()
# Log metrics
if batch_idx % config.log_interval == 0:
log_metrics(outputs, batch, epoch, batch_idx)
“`
## 5. Entropy Measures and Regularization
### 5.1 Channel-Specific Entropy
Instead of ad-hoc entropy definitions, we use principled measures:
**Real Channel:** Standard Shannon entropy
$$H_R = -sum_i p_i log p_i$$
**Complex Channel:** Von Neumann entropy via density matrix
$$H_C = -text{Tr}(rho log rho)$$
where ρ is the density matrix constructed from attention weights.
**Hyperbolic Channel:** Tsallis entropy compatible with entmax
$$H_H^{(alpha)} = frac{1}{alpha – 1} left(1 – sum_i p_i^alpha right)$$
This connects directly to α-entmax normalization, providing consistency between the attention mechanism and entropy regularization.
### 5.2 Channel Balance Regularization
Instead of claiming “information conservation” as an unproven proposition, we implement it as a regularizer:
$$mathcal{R}_{text{balance}} = max(0, text{Var}[w_R, w_C, w_H] – tau)$$
This encourages balanced use of channels while allowing task-specific specialization.
## 6. Theoretical Analysis
### 6.1 Gradient Flow Analysis
**Real Channel:** Standard backpropagation through real-valued operations.
**Complex Channel:** Wirtinger calculus ensures well-defined gradients:
$$frac{partial L}{partial z} = frac{1}{2}left(frac{partial L}{partial x} – ifrac{partial L}{partial y}right)$$
$$frac{partial L}{partial z^*} = frac{1}{2}left(frac{partial L}{partial x} + ifrac{partial L}{partial y}right)$$
**Hyperbolic Channel:** Riemannian gradients on the Lorentz manifold:
$$text{grad}_{mathcal{H}} f(x) = text{proj}_{T_xmathcal{H}}(nabla_E f(x))$$
where proj is the tangent space projection.
### 6.2 Computational Complexity
Per attention head, the Particle Transformer requires:
– **Real Channel:** O(n²d) standard attention
– **Complex Channel:** O(n²d) with 4× arithmetic ops (complex multiplication)
– **Hyperbolic Channel:** O(n²d) + O(nd) for log/exp maps
– **Gating:** O(n²) for entmax computation
Total: ~3-4× the cost of standard attention, which can be optimized through:
1. Sparse routing based on entropy signals
2. Mixed precision (FP16 for confident predictions)
3. Caching log/exp map computations
### 6.3 Connection to Reject-Option Theory
**Theorem (Selective Prediction Guarantee):** Under the particle transformer with abstention head g(x), for any target coverage c ∈ [0,1], the selective risk satisfies:
$$text{Risk}@text{Coverage}(c) leq frac{1}{c} cdot text{Risk}_{text{full}} + (1 – c) cdot text{Risk}_{text{abstain}}$$
This provides a formal guarantee on the model’s ability to abstain on uncertain inputs.
## 7. Experimental Design
### 7.1 Evaluation Metrics
Following Kalai et al.’s recommendations and formal calibration theory:
1. **Expected Calibration Error (ECE):**
$$text{ECE} = sum_{m=1}^M frac{|B_m|}{n} |text{acc}(B_m) – text{conf}(B_m)|$$
2. **Risk-Coverage Curves:** Plot risk vs. coverage for varying abstention thresholds
3. **Contradiction Detection F1:** Precision/recall for identifying contradictory pairs
4. **Epistemic AUROC:** Area under ROC for distinguishing known/unknown
### 7.2 Benchmarks
1. **TruthfulQA-Abstain:** Extended with “I don’t know” as rewarded response
2. **ContradictionQA:** Paired contradictory statements requiring detection
3. **CalibratedQA:** Questions with ground-truth uncertainty levels
4. **SelectiveWikiQA:** Wikipedia QA with coverage requirements
### 7.3 Baselines
– Standard Transformer
– Ensemble Methods (Deep Ensembles)
– MC Dropout
– Hyperbolic Transformer (Hypformer)
– Complex Transformer (Trabelsi et al.)
## 8. Implementation Details
### 8.1 Initialization Strategy
Given three different geometries, careful initialization is crucial:
“`python
def initialize_particle_transformer(model, config):
# Real channel: Xavier/He initialization
for p in model.real_params:
if len(p.shape) >= 2:
nn.init.xavier_normal_(p)
# Complex channel: Complex Xavier (Trabelsi et al., 2018)
for p in model.complex_params:
std = np.sqrt(2.0 / (p.shape[0] + p.shape[1]))
real_part = torch.randn_like(p.real) * std
imag_part = torch.randn_like(p.imag) * std
p.data = torch.complex(real_part, imag_part)
# Hyperbolic channel: Initialize near origin in Lorentz model
for p in model.hyperbolic_params:
p.data = project_to_lorentz_manifold(
torch.randn_like(p) * config.hyperbolic_init_scale
)
# Channel gates: Balanced initialization
model.omega_R.data.fill_(1.0)
model.omega_C.data.fill_(1.0)
model.omega_H.data.fill_(1.0)
“`
### 8.2 Optimization Details
“`python
optimizer = torch.optim.AdamW(
[
{‘params’: model.real_params, ‘lr’: 1e-3},
{‘params’: model.complex_params, ‘lr’: 5e-4}, # Lower LR for complex
{‘params’: model.hyperbolic_params, ‘lr’: 1e-4}, # Even lower for hyperbolic
{‘params’: model.gate_params, ‘lr’: 1e-3}
],
weight_decay=1e-5
)
scheduler = torch.optim.lr_scheduler.CosineAnnealingWarmRestarts(
optimizer, T_0=1000, T_mult=2
)
“`
## 9. Discussion and Future Work
### 9.1 Broader Impact
The Particle Transformer addresses the fundamental problem of epistemic uncertainty in AI systems. By providing native mathematical substrates for different epistemic states, we enable:
1. **Safer AI deployment:** Models that abstain when uncertain
2. **Better human-AI collaboration:** Clear signals of model confidence
3. **Improved scientific discovery:** Distinguishing known/unknown/contradictory
### 9.2 Limitations and Future Directions
1. **Computational overhead:** 3-4× cost needs optimization for production
2. **Hyperparameter sensitivity:** Many parameters to tune (α, τ, coverage targets)
3. **Theoretical gaps:** Formal convergence guarantees for mixed-geometry optimization
Future work should explore:
– Learned geometry selection (let the model choose its number system)
– Dynamic channel allocation based on input complexity
– Extension to other architectures (CNNs, GNNs)
## 10. Conclusion
The work of Kalai et al. (2025) diagnoses hallucination as a behavioral artifact of test-taking incentives. Our Particle Transformer offers an architectural prescription that provides native mathematical substrates for epistemic states through parallel attention over real, complex, and hyperbolic geometries. By grounding our approach in established geometric deep learning, complex-valued neural networks, and formal reject-option theory, we provide a rigorous path toward AI systems that can genuinely “know when they don’t know.”
The key innovations—Hermitian attention scoring for complex channels, Lorentz geometry for hyperbolic uncertainty, α-entmax gating across geometries, and formal abstention heads—work together to create an architecture aligned with the reformed evaluation metrics Kalai et al. propose. This represents a crucial step toward trustworthy AI that combines behavioral incentives with architectural capabilities for epistemic honesty.
# References for Particle Transformers Paper
## Primary References
– Kalai, A., Nachum, I., Vempala, S., & Zhang, Y. (2025). Why Language Models Hallucinate. OpenAI Technical Report. Available at: https://openai.com/research/hallucination
## Complex-Valued Deep Learning
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## Hyperbolic Transformers
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## Selective Prediction and Reject Option
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## Calibration and Uncertainty Quantification
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## Sparse Attention and Entmax
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## Entropy Measures and Information Theory
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## Gradient Methods for Non-Euclidean Spaces
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## Wirtinger Calculus
– Wirtinger, W. (1927). Zur formalen theorie der funktionen von mehr komplexen veränderlichen. *Mathematische Annalen*, 97(1), 357-375.
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## Related Transformer Architectures
– Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, L., Gomez, A. N., Kaiser, Ł., & Polosukhin, I. (2017). Attention is all you need. In *Advances in Neural Information Processing Systems (NeurIPS 2017)* (Vol. 30).
– Kitaev, N., Kaiser, Ł., & Levskaya, A. (2020). Reformer: The efficient transformer. In *International Conference on Learning Representations (ICLR 2020)*.
– Zaheer, M., Guruganesh, G., Dubey, K. A., Ainslie, J., Alberti, C., Ontanon, S., Pham, P., Ravula, A., Wang, Q., Yang, L., & Ahmed, A. (2020). Big bird: Transformers for longer sequences. In *Advances in Neural Information Processing Systems (NeurIPS 2020)* (Vol. 33).
## Uncertainty in Neural Networks
– Gal, Y., & Ghahramani, Z. (2016). Dropout as a Bayesian approximation: Representing model uncertainty in deep learning. In *International Conference on Machine Learning (ICML 2016)* (pp. 1050-1059).
– Lakshminarayanan, B., Pritzel, A., & Blundell, C. (2017). Simple and scalable predictive uncertainty estimation using deep ensembles. In *Advances in Neural Information Processing Systems (NeurIPS 2017)* (Vol. 30).
– Kendall, A., & Gal, Y. (2017). What uncertainties do we need in Bayesian deep learning for computer vision? In *Advances in Neural Information Processing Systems (NeurIPS 2017)* (Vol. 30).
– Malinin, A., & Gales, M. (2018). Predictive uncertainty estimation via prior networks. In *Advances in Neural Information Processing Systems (NeurIPS 2018)* (Vol. 31).
## Benchmarks and Evaluation
– Lin, S., Hilton, J., & Evans, O. (2022). TruthfulQA: Measuring how models mimic human falsehoods. In *Proceedings of the 60th Annual Meeting of the Association for Computational Linguistics (ACL 2022)* (pp. 3214-3252).
– Hendrycks, D., Burns, C., Basart, S., Zou, A., Mazeika, M., Song, D., & Steinhardt, J. (2021). Measuring massive multitask language understanding. In *International Conference on Learning Representations (ICLR 2021)*.
– Wang, A., Singh, A., Michael, J., Hill, F., Levy, O., & Bowman, S. R. (2018). GLUE: A multi-task benchmark and analysis platform for natural language understanding. In *Proceedings of the 2018 EMNLP Workshop BlackboxNLP: Analyzing and Interpreting Neural Networks for NLP* (pp. 353-355).
## Additional Mathematical Background
– Lee, J. M. (2018). *Introduction to Riemannian manifolds* (Vol. 176). Springer.
– Do Carmo, M. P. (2016). *Riemannian geometry*. Birkhäuser.
– Absil, P. A., Mahony, R., & Sepulchre, R. (2008). *Optimization algorithms on matrix manifolds*. Princeton University Press.
– Boumal, N. (2023). *An introduction to optimization on smooth manifolds*. Cambridge University Press.
Disclaimer: This paper presents theoretical and speculative work that has not been peer-reviewed or experimentally validated. The Particle Transformer architecture proposed here is a conceptual framework with mathematical formulations and pseudocode, but:No actual implementation has been built or testedNo experiments have been conducted to verify the claimsNo empirical results or performance metrics are availableThe mathematical frameworks, while rigorous in formulation, have not been proven to work in practiceThis work should be considered a theoretical contribution proposing a potential architectural solution to the hallucination problem in large language models. The ideas presented require:Actual implementation in a deep learning frameworkExtensive experimental validation on benchmark datasetsComparison with existing uncertainty quantification methodsPeer review by experts in geometric deep learning and uncertainty estimationVerification that the three-channel architecture can be trained effectivelyReaders should treat this as a conceptual proposal that may inspire future research, not as a validated solution. The mathematical soundness of combining real, complex, and hyperbolic geometries in parallel attention mechanisms, while theoretically interesting, remains to be demonstrated empirically.Any use or implementation of these ideas should be undertaken with the understanding that this is untested, speculative research that may not perform as theorized when put into practice.
