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

Forum topic · 小凯 · 2026-09-01

Summary

This arXiv paper (2608.28541) by Javier Aguilar Martín studies the epistemic limits of code world models certified by sampling gates. A gate-accepted model can be exactly correct on everything the gate observes yet arbitrarily wrong beyond it; the author frames such omissions as a gauge choice beyond the reachable query set. Using a minimal ring instrument, the paper proves an extreme case: a wrong-topology filled-disc artifact enclosing an unreachable interior is unfalsifiable by any sampling gate and bitwise harmless at runtime. Experiments with LLM synthesis across three model families show how a channel of width gamma moves the same artifact through three regimes—unfalsifiable-and-harmless, falsifiable-and-costly, and immediately falsified. Key findings: danger is topology relative to reach (usable channels collapse exploitation, runtime cost drops from 1.09 to ~0 at gamma~0.1, while hidden channels with the same first Betti number retain full strength at 1.12); repair is limited by parameter and sensor constraints; and mitigation must match the error's dimension and direction, with dimension-matched persistent fences collapsing exploitation from 0.999 to 0.058 and dual-freedom certificates reducing invented-mode failures from 1.769 to 0.029.

Overview

Field: Machine Learning Author: Javier Aguilar Martín Published: 2026-08-28 arXiv: 2608.28541

Abstract (translated summary)

A code world model accepted by a sampling gate can be exactly right on everything the gate can see and arbitrarily wrong beyond it. The paper characterizes what a certified model can know, and what its errors can cost, when the omission is an annular freeze mode enclosing an unreachable interior. The gate quotient makes the question precise: acceptance-with-certainty determines the model exactly on the reachable query set; beyond reach is gauge.

On a minimal ring instrument, the author proves the extreme case—a wrong-topology filled-disc artifact unfalsifiable by any sampling gate and bitwise harmless at play—and measures, with LLM synthesis across three model families, how one knob (a channel of width gamma) walks the same artifact through three regimes: unfalsifiable-and-harmless, falsifiable-and-costly, and immediately falsified.

Three principles organize the empirical results:

1. Danger is topology relative to reach. A channel the planner can use collapses the blind model's exploitation (runtime cost falls from 1.09 to ~0, with a knee at gamma ~ 0.1), while a hidden channel with the same first Betti number retains its full strength (1.12).

2. Repair is parameter- and sensor-limited. No model family recovers the region from external evidence; from the interior, models propose the correct topology but cannot fix its parameters, and the proposed topology tracks the wrong beta_1 of guided persistent-homology summaries (a sensor with measurement-geometric resolution limits) rather than the truth.

3. Mitigation must match the error's dimension and direction. Point fences fail against one-dimensional boundaries; dimension-matched persistent fences collapse exploitation into a two-lesson transient (0.999 to 0.058), and dual-freedom certificates symmetrically collapse invented-mode failures (1.769 to 0.029). In n dimensions, the shell makes misidentification nearly certain while danger remains fully exploitable: the two axes are independent.

Original Abstract (excerpt)

> A code world model accepted by a sampling gate can be exactly right on everything the gate can see and arbitrarily wrong beyond it. We characterize what a certified model can know, and what its errors can cost, when the omission is an annular freeze mode enclosing an unreachable interior. The gate quotient makes the question precise: acceptance-with-certainty determines the model exactly on the reachable query set; beyond reach is gauge. On a minimal ring instrument we prove the extreme case (a wrong-topology filled-disc artifact unfalsifiable by any sampling gate and bitwise harmless at play) and measure, with LLM synthesis across three model families, how one knob (a channel of width gamma) walks the same artifact through three regimes: unfalsifiable-and-harmless, falsifiable-and-costly, ...

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