An Enclosed Mode Is a Gauge Choice: Topology Relative to Reach in Certified Code World Models

Certified code world models are characterized by the principle that acceptance-with-certainty determines the model exactly only on the reachable query set, rendering all topology and behavior beyond reach as gauge (arbitrary and unfalsifiable by the sampling gate).

Key findings from the research include: Topology Relative to Reach: Danger is defined not by the mode's intrinsic topology, but by its topology relative to what the planner can reach; a "facing channel" accessible to the planner collapses exploitation, while a "hidden channel" maintains full danger despite identical topology. Unfalsifiable Artifacts: A wrong-topology artifact (e.g., a filled disc instead of an annulus) can be unfalsifiable by any sampling gate and bitwise harmless if the interior is unreachable, yet become costly if the planner can access it. Repair Limitations: Repair is parameter-bound and sensor-bound; models cannot recover the region from outside evidence, and from inside, they fail to pin parameters to the gate’s precision due to geometric resolution limits. Dimensional Rarity: In higher dimensions, the rarity of encountering the enclosed mode in samples collapses geometrically, making misidentification near-certain while the danger remains fully exploitable.

The study concludes that safety questions should shift from "is the model right?" to "does the place where it is wrong intersect the operative reach of a competent planner?"

The paper frames an “enclosed mode” in certified code world models not as a physical or semantic boundary, but as a gauge choice that partitions representation space into a reachable region and a beyond-reach region. Under this view, an annular freeze mode creates a topological separation: states inside the reachable annulus are those that can be reached from valid code or specification states, while states outside are not. The central mathematical device is a gate quotient, which identifies representations related by gauge-equivalent transformations—i.e., different encodings that carry the same operational or certified meaning. The paper’s main insight is that the relevant topology is therefore not absolute, but topology relative to reach: semantic distinctions should be made only after quotienting by gauge and restricting attention to the reachable component.

Its key contribution is a characterization of when certified code world models can make exact acceptance-with-certainty claims on reachable sets. The paper shows that, under the annular freeze/gate quotient structure, acceptance decisions are well-defined and invariant across gauge-equivalent reachable states, so a certified “accept” or “reject” is not an artifact of a particular encoding. By contrast, beyond reach, the model’s outputs are shown to be gauge-dependent: they may vary with the chosen representation without corresponding to any stable semantic property of the code. In other words, the work separates the part of the model’s state space where certification has invariant meaning from the part where apparent confidence or acceptance is merely a coordinate or encoding effect.

This matters because it gives a principled way to diagnose and avoid spurious overconfidence in certified world models for code. Many learned or symbolic certifiers can appear decisive everywhere, but the paper suggests that such decisiveness is meaningful only relative to the reachable, gauge-quotiented part of the space. The result is a useful bridge between certification theory, representation learning, and code semantics: it argues that robust code world models should be built around explicit reach-aware quotienting rather than treating all latent or encoded states as equally interpretable. More broadly, the framework helps clarify when a model’s “certainty” is a semantic invariant and when it is merely a gauge artifact, which is important for trustworthy verification, safety-critical code reasoning, and the design of certified AI systems.