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 limits of certification for code world models accepted by sampling gates. Such a model can be exactly correct on everything a gate can observe while being arbitrarily wrong beyond that reach. The gate quotient formalizes this: acceptance-with-certainty determines the model exactly on the reachable query set, while anything beyond reach is a gauge choice. On a minimal ring instrument, the author proves an extreme case—a wrong-topology filled-disc artifact that no sampling gate can falsify and that is bitwise harmless at play—and measures, via LLM synthesis across three model families, how a channel of width gamma walks the same artifact through three regimes: unfalsifiable-and-harmless, falsifiable-and-costly, and immediately falsified. Key findings: danger is topology relative to reach (planner-accessible channels collapse runtime cost from 1.09 to ~0 at a gamma~0.1 knee, while hidden channels with the same first Betti number keep full strength at 1.12); repair is limited by parameter and sensor constraints; and mitigation must match the dimension and direction of the error, with dimension-matched persistent fences collapsing exploitation (0.999 to 0.058) and dual freedom certificates collapsing invented-mode failures (1.769 to 0.029).

Paper Overview

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

English 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.

Key Findings

1. Danger is topology relative to reach: A channel a 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 keeps its full strength (1.12).

2. Repair is limited by parameters and sensors: No model family can recover the region from external evidence. From the interior, models propose the correct topology but cannot pin its parameters, and the proposed topology tracks the wrong beta_1 of the guided persistent-homology summary (a sensor with measured 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, shells make 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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