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 what a certified code world model can know when a sampling gate accepts it. A model can be exactly correct on everything the gate can observe yet arbitrarily wrong beyond it. Using a gate quotient formalism, the author shows that acceptance-with-certainty determines the model only on the reachable query set; beyond reach, errors are a gauge choice. On a minimal ring instrument, the paper proves an extreme case: a wrong-topology filled-disc artifact that is unfalsifiable by any sampling gate and bitwise harmless at runtime. Experiments using 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 topological relative to reach (reachable channels collapse exploitation from 1.09 to ~0 at a gamma~0.1 knee, while hidden channels with the same first Betti number retain full strength at 1.12); repairs are parameter- and sensor-limited; and mitigation must match the dimension and direction of the error, with dimension-matched persistent fences collapsing exploitation from 0.999 to 0.058 and dual-free certificates from 1.769 to 0.029.

Paper: An Enclosed Mode Is a Gauge Choice: Topology Relative to Reach in Cert... Field: Machine Learning Author: Javier Aguilar Martín Published: 2026-08-28 arXiv: 2608.28541

Abstract

A code world model accepted by a sampling gate can be exactly right on everything the gate can see and arbitrarily wrong beyond it. This 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.

Key Findings

Three principles organize the empirical results:

1. Danger is topological relative to reach. A channel that the planner can use collapses the blind model's exploitation (runtime cost drops 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. Repairs are parameter-limited and sensor-limited. No model family can recover the region from external evidence; from the interior, models propose the correct topology but cannot fix its parameters, and the proposed topology tracks an erroneous beta_1 of the guided persistent-homology summary (a sensor with geometric resolution limits) rather than the truth.

3. Mitigation must match the dimension and direction of the error. 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-free certificates symmetrically collapse invented-mode failures (1.769 to 0.029). In n dimensions, shells make misidentification nearly certain while the danger remains fully exploitable: the two axes are independent.

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

Summary

Abstract

Key Findings

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