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To Explore The Strange New World Beyond Data Distribution: System Behavior, Causality Tax, and Non-causal Base Model

Xianzhi Zeng, Jiangneng Li, Gao Cong

arXiv:2610.02839v1cs.LGcs.CL

TL;DR

Causal language modeling may be suboptimal when system behavior beyond the data distribution is not represented. The paper proposes SBD, which incorporates system behavior as a Bayesian factor, and reports that its formulation identifies extra structural error from enforcing a causal chain. The authors’ experiments are limited to NTK spectra analysis intended to validate the theory, not demonstrate end-to-end practical performance.

  • Problem

    The paper asks how language modeling can incorporate system behavior beyond the data distribution, which it identifies as a source of mismatches with causal formulation.

    Source: §1 · p. 2

  • Method

    SBD models system behavior as a latent first-principle factor and incorporates it into the ELBO alongside output generation.

    Source: §1 · p. 2

  • Results

    SBD’s ELBO identifies structural overhead from imposing a causal chain and remains globally consistent without one.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The paper concludes that system behavior can serve as a complementary abstraction to causality and distribution fitting in language modeling.

    Source: §7 · p. 11

    The experiments are confined to NTK spectra analysis intended to validate the theory rather than demonstrate end-to-end practical performance.

    Source: §6.1 · p. 11

Abstract

from arXiv · show

We show that the causality of language models (LMs) may not be necessary nor optimal. This is the case when system behavior (denoted as $S$) is incorporated as a first-principle Bayesian feature. Here, $S$ refers to extra dominant factors beyond the data space, and they involve coupled effects. Despite being the de facto foundation of modern architecture, recent studies indicate persistent mismatches and contradictions with causality. These issues largely stem from system behavior rather than the data distribution. We therefore propose the SBD framework, which incorporates $S$ as an irreducible component of the evidence lower bound (ELBO). SBD theoretically reveals a counter-intuitive Causality Tax phenomenon, where causality emerges as a suboptimal approximation with an additional structural error, due to the obliviousness to $S$. To address the challenge of latent variable analysis, we validate the SBD-predicted impact of $S$ via implicit measurements, theoretical-bound-guided controls, and Neural Tangent Kernel (NTK) evaluations. In particular, we construct Green Shell (GSH) to show the possibility of reducing Causality Tax. GSH is a non-causal variational family, and it replaces the sequential dependency chain of $S$ components with a divide-and-conquer partition. NTK spectra in the lazy-training regime confirm that GSH always achieves significantly tighter error bounds than causality, with $7dB+$ improvement in signal-to-noise ratio. In the relatively later stage of lazy-training, GSH further leads to superior generalization (up to $20\%$ richer multi-scale fitting capabilities). Taken together, SBD establishes system behavior as a complementary theoretical abstraction besides causality and distribution fitting, opening new research avenues such as designing and optimizing LM base models.

1 Introduction

The paper argues that mismatches with causal language modeling can arise from system behavior beyond the data distribution. It introduces SBD to model that behavior as a first-principle Bayesian factor and identifies an additional error associated with imposing a causal chain.

1 Introduction

  • Causality can overlook how S governs generation, potentially degrading achievable accuracy and adding error beyond the neural architecture’s intrinsic approximation error.The paper illustrates this concern with inaccurate or offensive outputs.Source: §1 · p. 2
  • The central question is how language modeling can explicitly incorporate S to identify, explain, and reduce systematic penalties.SBD is proposed as a Bayesian framework whose latent variables capture diverse factors without rigid manual specification.Source: §1 · p. 2
  • SBD treats system behavior S—transformation factors beyond the data distribution—as an irreducible component of the ELBO.The paper includes algorithms, hardware, runtime, natural languages, and other factors in S.Source: §1 · p. 2
  • The paper calls the extra structural error incurred by enforcing a causal chain the Causality Tax, and says SBD remains globally consistent without that chain.The authors note that validating and using SBD is challenging because latent variables are less explicit to investigate than data distributions.Source: §1 · pp. 2–3
  • The paper proposes GSH as a non-causal variational family and describes measurement, controls, and evaluations as its approach to validating SBD’s predicted effects.The introduction also presents SBD as a new abstraction of S treated as a first-principle factor.Source: §1 · p. 3

2 Motivation and Related Work

Prior work reports mismatches with autoregressive causality, but the reviewed approaches retain it as a foundational assumption. The paper situates its approach among system-aware modeling and Bayesian variational methods.

2 Motivation and Related Work

  • Reported mismatches include inference nondeterminism, an information-symmetry paradox, denoising-based shortening of token chains, and decoding methods that relax strict token-by-token causality.The passage attributes these findings to prior works [5], [47], [4], [42], and [26], respectively.Source: §2 · p. 3
  • Despite these indications, the reviewed works retain causality as a foundational assumption, whereas this paper revisits it from first principles.Source: §2 · p. 3
  • The paper extends system-aware modeling to language modeling, using Bayesian latent variables and ELBO-based variational inference as its framing.The passage contrasts Bayesian modeling with cost analysis and Markov chains, and notes PECJ’s use of variational inference.Source: §2 · pp. 3–4

3 System Behavior in Bayesian Modelling

The section distinguishes latent variables used to fit data from latent variables representing system behavior. It formulates SBD around joint consistency of input, output, and system behavior, then interprets the ELBO’s terms.

3.1 A Shift of Abstraction: Ascending the Latent Variables

  • In autoregressive and block-diffusion formulations, latent variables support sequential or structured causal generation while observed data serve as fixed evidence for reconstruction.The paper contrasts this with contradictions attributed to runtime, hardware, algorithms, language, and other factors beyond the data distribution.Source: §3.1 · p. 4
  • In PECJ, latent Z represents environmental factors beyond data, but is marginalized to compensate for stream oscillations rather than explicitly explain their effects.Source: §3.1 · p. 4
  • SBD instead treats S as a first principle and approximates the posterior over output Y and system behavior S given input X, targeting their global consistency.Its ELBO uses the prior log p(X) as overall evidence.Source: §3.1 · pp. 4–5

3.2 Detailed Meanings

  • Token Alignment reduces to cross-entropy during training but retains an expectation over latent S during stochastic inference.The paper uses systematic changes in logits and hidden states associated with this term to probe S.Source: §3.2 · p. 5
  • System Response to Input/Output represents coherence between S and generated Y, including inductive biases about structural dynamics; the paper treats it as constant for a frozen LM without external memory.Source: §3.2 · p. 5
  • Approximation’s Internal Semantics is the entropy of the variational distribution, whose structural organization determines the underlying approximation error bound.It is fixed during inference but implicitly optimized through model weights during training.Source: §3.2 · p. 5

4 Discovery of the Causality Tax

The section argues that a causal chain is a structural choice rather than a requirement of the variational approximation. It further derives an error-bound penalty for causal constraints relative to a general variational family.

4.1 Causality is Not Necessary

  • Causality is not required by the variational family: treating q(Y, S) as a free design choice allows flexible approximation, provided the joint system converges to a high-probability configuration.Diffusion and autoregressive formulations are presented as examples whose factorization imposes sequential constraints on S.Source: §4.1 · pp. 5–6

4.2 Causality is Not Optimal

  • The UAT bound for the causal family is strictly looser when residual terms are nonzero, because enforcing conditional independence across the causal chain adds accumulated error.The residual terms integrate over possible input-distribution realizations; T denotes chain length, such as denoising steps or generated tokens.Source: §4.2 · pp. 6–7
  • The causal variational family is a strict subset of the general family, so the monotonicity-of-supremum argument bounds its expressivity no better than the general family.Source: §4.2 · p. 6

5 Validation and Analysis of SBD

The section validates SBD through perturbation measurements, a non-causal variational family, and NTK evaluations of GSH. The reported results show regularities beyond random noise and favorable NTK comparisons for GSH, with evaluation bounds tied to the lazy-training regime.

5.1 Measurement on the Impacts of S

  • E–1/r correlation exceeds 0.6 for all models and 0.9 for some longer prompts, while E–r correlation stays below 0.4.The best lag typically differs from −1, indicating structured temporal propagation; Figure 2 reports the measurement results.Source: §5.1 · p. 8, Figure 2 · p. 8

5.2 Optimized Approximation Error about S

  • GSH partitions S into shells with independent q(si) approximations, preserving temporal ordering and spatially disjoint hidden-state and output regions.The design avoids causal-chain dependencies and localizes approximation of each output partition.Source: §5.2 · pp. 8–9
  • GSH is a proof of concept: complete causality removal conflicts with standard KV caching, and shell partitioning and auxiliary components use practical heuristics.The implementation instead uses a block-wise strategy.Source: §5.2 · p. 9

5.3 Evaluations about S under NTK

  • NTK spectral SNR serves as an approximately monotonic proxy for the UAT error bound in lazy training, but that connection breaks in full feature learning.A lower MLC is treated as preferable because signal energy is spread across multiple significant eigenvalues.Source: §5.3 · pp. 9–10
  • Across all datasets, GSH consistently achieves the best comparison; in later lazy-training it has the highest SNR and lowest MLC.The models share a 150M-parameter backbone and are evaluated on four datasets.Source: Table 1 · p. 10, §5.3 · p. 10
  • Transformer-layer gradient cutoffs retain about 22 dB SNR and 0.8 MLC, while an LM-head cutoff yields MLC = 0.94.The authors connect the possibility of backward-propagation-free training to GSH's layer-decoupling nature and suggest it may extend to larger LMs.Source: §5.3 · pp. 10–11

6 Discussions

The discussion identifies limits in the paper’s scope and outlines future directions, including broader benchmarking and possible GSH efficiency benefits. It also flags risks from deliberate manipulation of system behavior.

6.1 Limitations and Impact Statement

  • The analysis leaves detailed decomposition of system-behavior components for future work; its lemmas hold only within their stated validity regimes, and GSH remains partly analytically opaque.Experiments assess NTK spectra to validate theory, not end-to-end practical performance.Source: §6.1 · p. 11
  • Manipulating system behavior could steer model weights, training vulnerabilities, or infrastructure toward attack planes such as emergent misalignment.Source: §6.1 · p. 11

6.2 Future Work

  • Future GSH work includes testing whether it can reduce gradient-graph memory overhead and benchmarking knowledge, reasoning, spatial-structural tasks, long-context processing, safety, and robustness.The paper says comprehensive empirical exploration of this vision is beyond its scope.Source: §6.2 · p. 11
  • The authors suggest exploring intra-head decoding parallelism, semantic approximation during prefilling, and possible joint optimization across AI and database systems.Source: §6.2 · p. 11

7 Conclusion

The paper frames system behavior as a Bayesian latent variable in language modeling and presents GSH as a non-causal variational approximation. Its discussion connects this framing to reported mismatches with causality and argues that GSH can avoid additional UAT residual error under stated conditions.

7 Conclusion

  • The conclusion presents system behavior as Bayesian latent variables and GSH as a realizable non-causal variational approximation to reduce the Causality Tax.Source: §7 · p. 11

A.1 Detailed Mismatches with Causality

  • Reported mismatches include inference non-determinism [5], retroactive changes to past representations, and the Arrows of Time paradox concerning information symmetry [47].Source: §A.1 · p. 16

A.2 Cognitive and Language Semantic Investigations

  • Studies cited by the paper describe system-level findings such as metacognitive reasoning [21], targeted representation interventions, and reasoning failures under complexity shifts.The authors characterize these findings as justification for modeling beyond token-distribution alignment, not as phenomena SBD fully explains.Source: §A.2 · pp. 16–17

A.3 Modelling Behaviors and Strategies of LMs

  • Prior work examines emergent abilities [76], contingency planning [22], and competing in-context learning mechanisms [57]; this paper instead treats system behavior itself as part of the model.It distinguishes that approach from [67], where the system is primarily an analytical metaphor.Source: §A.3 · p. 17

B.1 Mean Field from Mathematical Physics

  • GSH replaces temporal dependency assumptions with spatial partitioning into shells, while retaining layerwise information directionality rather than reducing the model to a bag of words.The paper describes the shell family as a mean-field instantiation and argues that causality is subsumed rather than simply discarded.Source: §B.1 · pp. 17–18

B.2 GSH is Free From Additional UAT Residual Loss

  • GSH’s temporal execution order adds no approximation error, while its spatial partition constraint is absorbed by the UAT bound when the variational family is sufficiently expressive.The paper also states that localized shell approximations typically attenuate prior-induced discrepancies faster during iterative updates.Source: §B.2 · p. 19

B.3 Mean Field into Training Loss

The section develops GSH as a field-estimation variational family with discrete and continuous training-loss instantiations. It then describes a concrete implementation that combines a continuous field constraint with black-box renormalization.

B.3 Mean Field into Training Loss

  • GSH’s mean-field objective can be instantiated with discrete categorical fields or continuous fields, but each choice trades tractable token alignment against preserving hidden-state geometry.Cross-entropy follows from an exponential scoring choice, while MSE follows from a quadratic energy; both are particular instantiations rather than the full field-estimation framework.Source: §B.3 · pp. 19–20
  • The softmax/cross-entropy instantiation is convenient but limited; the paper leaves richer discretization, non-softmax normalizers, and geometry-aware mappings for future work.Source: §B.3 · p. 20

B.4 Detailed Implementation of GSH

  • The continuous-field implementation uses a screened Green’s function for pairwise interactions, then aggregates them into a layer-level energy that promotes internal consistency and diversity without token-level causal ordering.The distance decay attenuates long-range couplings, while the similarity term discourages overly similar token states.Source: §B.4 · p. 21
  • Black-box renormalization approximates the intractable partition function by interpolating the empirical data distribution with a uniform prior, providing a discrete-token alignment step.The mixing coefficient trades output diversity against fitting accuracy, and setting it to zero recovers cross-entropy.Source: §B.4 · pp. 21–22
  • Applying renormalization only at the output head can enable backpropagation-free operation but may leave inconsistencies; applying it across the network is more consistent but costlier.Omitting the continuous-field step removes the backpropagation-free possibility, while omitting renormalization risks divergence.Source: §B.4 · p. 22
  • The block-wise variant retains causal masks between blocks but uses non-causal GSH reconstruction within each block, using surrounding unmasked blocks as boundary conditions.Source: §B.4 · p. 22

C SNR as a Proxy for UAT Error Bound

The section uses NTK spectral properties to connect lazy-training behavior with approximation bounds. It defines SNR as a proxy for the UAT error bound and argues that the proxy supports comparisons under controlled evaluation conditions.

C SNR as a Proxy for UAT Error Bound

D Other Experimental Settings

The experiments examine GSH against causal baselines under lazy-training conditions and test sensitivity to data scale and run variability. Across the tested data scales, GSH has the highest SNR, while MLC trends differ between non-causal and autoregressive models.

D Other Experimental Settings

  • Evaluations use lazy training to isolate architecture-induced system behavior from data-driven feature learning and preserve the NTK-based theoretical comparison.The regime keeps updates near initialization; the paper notes that NTK connections to bounds break down in full feature learning.Source: D Other Experimental Settings · p. 23
  • The controlled comparison trains 150M-parameter autoregressive, block-diffusion, and GSH models with shared configurations; GSH and block diffusion decode five tokens per step.The models use the same backbone and optimizer settings; autoregressive decoding emits one token per step.Source: D Other Experimental Settings · pp. 23–24
  • GSH removes within-block causality, relying on latent-field directionality and ROPE's spatial inductive bias to preserve contextual coherence.The comparison places GSH beyond block diffusion, which retains causality across blocks but not within each block.Source: D Other Experimental Settings · pp. 23–24
  • At the same random seed, relative error stays within ±4% on the same machine, ±7% across NVIDIA CUDA machines, and ±12% across CUDA and ROCm platforms.Across 50 random seeds on TinyStories, the paper reports 1σ error bars and says other cases are similar.Source: D Other Experimental Settings · p. 24

E.1 Additional Sensitivity Study

  • Across 100–10,000 training passages, GSH has the highest SNR and growth speed; GSH and block diffusion MLC decrease as autoregressive MLC increases.The examined data range keeps token count far below parameter count so lazy-training NTK spectra remain a reliable proxy for the theoretical bound; the sharp curve near 2,000 is attributed mainly to proof-of-concept implementation choices.Source: §E.1 · p. 25

E.2 Additional Ablation Study

The ablation study tests discrete and continuous components of GSH and their interaction. Its results indicate that both are needed to combine noise suppression with richer peripheral features.

E.2 Additional Ablation Study

  • The six ablations replace or remove GSH components, including Green-kernel parameterization, continuous renormalization, discrete setup, and cross-entropy or MSE objectives.The comparisons include MSE+S2, S1+CE, S1-only, S2-only, MSE-only, and CE-only.Source: §E.2 · p. 25
  • Discrete-only S1-only or CE-only variants achieve MLC < 0.7 but SNR < 16 dB, while the study concludes both continuous setup and discrete renormalization are required.The passage also reports that continuous-only variants have high noise suppression but concentrated NTK energy, with MLC > 0.95.Source: §E.2 · pp. 25–26
  • Standard GSH reaches 22.5 dB SNR versus 20.3 dB for MSE+S2, although S1-only has lower SNR than MSE-only (34.1 dB vs. 36.2 dB).The reported reversal after continuous renormalization is described as a complex interaction between continuous and discrete components.Source: §E.2 · p. 26

F Detailed Discussions of Future Works

The discussion sketches future directions for system-behavior-based model families, inference infrastructure, and closer integration of AI with databases. It also raises interpretability and probabilistic-modeling connections, while flagging some ideas as speculative.

F Detailed Discussions of Future Works

  • Future non-causal foundation-model families could extend system-behavior modeling, generalized continuous and discrete losses, and field-theoretic optimization beyond GSH.The paper suggests exploring backward-propagation-free training at large scale, drawing on GSH's stated possibility.Source: F Detailed Discussions of Future Works · p. 26
  • For decoding, intra-head parallelism could let one head emit multiple tokens through field reconstruction; for prefilling, strict token matching could give way to semantic approximation.The proposed changes may allow more flexible reuse of intermediate states once causal constraints are lifted.Source: F Detailed Discussions of Future Works · p. 26
  • The authors argue that causal assumptions make LLM data reuse all-or-nothing, motivating possible co-optimization of data flow and models as interacting processes.They propose field-theoretic integration in which storage, retrieval, and inference could become continuous operations of one system field.Source: F Detailed Discussions of Future Works · p. 26
  • The paper conjectures that Arrows of Time asymmetries may be interpreted as directional derivatives of system behavior with respect to token order.It presents this as a possible bridge between computational asymmetry and non-causal variational modeling, not an established result.Source: F Detailed Discussions of Future Works · p. 26
  • The authors suggest that hallucinations and repetitions may be viewed as manifestations of system behavior, supporting Bayesian rather than post-hoc interpretability accounts.This connection is framed in relation to recent interpretability studies.Source: F Detailed Discussions of Future Works · p. 27
  • The paper speculates about parallels between misconfigured language models and narcissistic traits, then asks whether system behavior could be a dominant, controllable component of intelligence across domains.The proposed human analogy is explicitly tentative.Source: F Detailed Discussions of Future Works · p. 27

G Potential Risks of Misuse

The discussion identifies potential misuse risks from making system behavior explicit and from applying field-based formulations. It also argues that open publication may help the community anticipate vulnerabilities and develop countermeasures.

G Potential Risks of Misuse

  • Explicitly modeling system behavior could expose attack surfaces, including exploitation for deceptive outputs, self-reinforcing dependency loops, or covert triggers.The passage compares these potential risks to manipulation of model weights, adversarial-training vulnerabilities, and malicious infrastructure components.Source: G Potential Risks of Misuse · p. 27
  • Pathological or adversarial renormalizations could induce misleading behavior even on well-curated datasets, raising model-trustworthiness concerns.The authors present this as a potential misuse of the field-formulation principles.Source: G Potential Risks of Misuse · p. 27
  • The proposed parallel between misconfigured models and pathological human traits is tentative; the authors raise possible risks of unstable or manipulative internal dynamics.They describe a conceptual connection to emergent misalignment, not an established finding.Source: G Potential Risks of Misuse · p. 27
  • The authors argue that open publication may help the community anticipate vulnerabilities, design countermeasures, and develop diagnostics for early signs of degeneracy.They conclude that transparency's benefits outweigh the associated risks.Source: G Potential Risks of Misuse · p. 27
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