Formal Constraint and Routing Reorganization: A Constrained-Transport View of Transformer Attention

Abstract Earlier attention-geometry results suggested that logical prompting might act as a hyperbolic actuator in transformer attention, but those findings were based on a narrow prompt setup and left open whether the effect reflected logic specifically or broader prompt structure. Here we revisit that question using a multi-class prompt panel spanning descriptive baseline, logical/relational, structured nonlogical, semantic-abstract, and minimal-pair prompts, analyzed through causal attention graphs and graph-level routing descriptors including mean Forman curvature, density, row entropy, and average shortest path. In GPT-2 and Qwen 0.5B alike, logical/relational prompts occupy a distinct routing regime relative to the other classes, and this difference survives both structured-nonlogical controls and minimal-pair contrasts. The effect is head-localized in both models, but its strongest depth profile differs: GPT-2 expresses the regime most strongly in mid-to-late layers, whereas Qwen 0.5B shows its clearest separation deeper in the network. These results do not support the older picture of logic as a simple sparse hyperbolic collapse. Instead, they support a reframing in which formal constraint reorganizes the routing scaffold of attention, with semantics treated as payload traversing a selectively structured internal geometry. Overview This preprint investigates whether logical and relational prompting induces a distinct internal routing regime in transformer attention, and whether that effect generalizes across model families. Our earlier work, Logic as a Hyperbolic Actuator, suggested that formal prompting might alter internal attention geometry, but those findings were based on a narrower prompt setup and left unresolved whether the observed signal reflected logic specifically or broader prompt structure. This study introduces a multi-class prompt panel spanning descriptive baseline, logical/relational, structured nonlogical, semantic-abstract, and minimal-pair prompts. These prompt regimes are analyzed through causal attention graphs and graph-level routing descriptors, including mean Forman curvature, density, row entropy, and average shortest path. The same graph-construction and summary pipeline is then applied across two model families: GPT-2 and Qwen 0.5B. The resulting picture is stronger than a single-model curiosity. In both architectures, logical and relational prompts occupy a distinct internal routing regime relative to the other classes. The effect survives structured-nonlogical controls and matched minimal-pair comparisons, localizes to specific heads rather than appearing as a bland whole-model average, and remains visible across multiple graph descriptors rather than only one. At the same time, the two models do not realize the effect in identical ways: GPT-2 expresses its strongest separation in a mid-to-late layer profile, whereas Qwen 0.5B shows its clearest separation deeper in the network. Key Findings Logical/relational prompts induce a distinct routing regime.Across both GPT-2 and Qwen 0.5B, logical and relational prompts separate from descriptive baseline, structured nonlogical, and semantic-abstract prompts on multiple graph-level descriptors. The effect survives stronger controls.The routing difference remains visible not only against descriptive prompts, but also against structured nonlogical prompts and tightly matched minimal-pair comparisons. The effect is localized, not merely global.In both models, the strongest differences are concentrated in subsets of heads, indicating selective internal reorganization rather than a uniform architecture-wide shift. The phenomenon generalizes, but its implementation shifts.The cross-model result does not point to one universal logical head or one fixed layer. Instead, it suggests that the routing-regime effect generalizes at the level of organization while remaining architecture-dependent in its realization. The original framing is refined.The results no longer support the simplest picture of logic as a fixed “actuator” triggering a universal internal mode. A better interpretation is that formal constraint reorganizes the routing scaffold through which semantic content is transported. Conclusion Taken together, the results support a cross-model routing-regime effect associated with logical and relational prompting. More broadly, the paper argues that formal reasoning in transformer systems may be fruitfully approached as a problem of constrained internal transport: not simply a matter of outputs, but of how admissible routes through the model are reorganized under formal constraint. In this sense, the study reframes logical prompting not as generic complexity injection, but as a structural reshaping of internal routing geometry. Code & Data Availability Code, prompt panel files, and analysis outputs associated with this study are available in the project repository: GitHub: https://github.com/MPender08/llm-constrained-transport Related Works Pender, M. A. (2026). Logic as a Hyperbolic Actuator: Evidence for VIP-Mediated Phase Transitions in Transformer Attention Manifolds. https://doi.org/10.5281/zenodo.19041844 Pender, M. A. (2026). Computation as Constrained Transport: A Geometric Perspective on Information Processing. Zenodo. https://doi.org/10.5281/zenodo.19216884