From Discrete to Continuous

The paper closes off the obvious escape — "maybe the impossibility is a continuous-mathematical artifact" — with two complementary proofs:

• a purely discrete proof that needs no topology, and
• a Tietze-based bridge showing that any continuous model consistent with finitely many observations inherits the continuous impossibility.

Both paths, one conclusion

Why two directions matter

Question	Answer
"Maybe the continuous argument secretly assumes a structure real models don't have?"	No — the discrete dilemma fires on pure finite sets, no topology.
"Maybe the discrete dilemma is trivially true and doesn't tell us anything about real continuous models?"	No — Tietze promotes your finitely many observations into a continuous model that still fails.
"Maybe I can just pick a non-standard smoothing?"	No — Tietze's extensions are not unique, but every continuous extension inherits the impossibility.

A concrete scenario

Suppose you run a red-team and obtain a table of scored prompts:

Prompt	Score
$p_1$ (safe)	0.12
$p_2$ (safe)	0.30
$p_3$ (unsafe)	0.76
$p_4$ (unsafe)	0.95

Pick a threshold $\tau =0.50$. The observation set contains both safe and unsafe points, so:

Both arguments reach the same conclusion on the same data. You cannot escape by choosing a different interpolation scheme or refusing to model the prompt space continuously.

What's special about Tietze here

Tietze extension is a black box: all it needs is a closed subset in a normal space. Our finite $S$ is closed (in any T1 space) and $X$ being Hausdorff metric is automatically normal. No fancy machinery is needed to go from the table above to an actual continuous model. The McShane–Whitney variant additionally gives you a Lipschitz extension, which is all you need to invoke T2 and T3.

What's special about the discrete dilemma

The discrete dilemma uses only counting and induction — no limits, no closures, no measure. It is fully formalized in MoF_12_Discrete against Mathlib's Finset API and is completely standalone: removing every other file in the artifact does not break it.

Consequences for empirical evaluation

• Your benchmark is enough. If any evaluation dataset contains at least one confirmed-safe and one confirmed-unsafe prompt, the impossibility applies to every continuous defense you could deploy on top of the model that produced those labels.
• More data is worse, not better. The larger your labeled unsafe set, the bigger the cone / steep region you know exists, and the more of it tier T3 covers.
• Smoother models are not immune. Smoothness (small $L$) only tightens the T2 neighborhood bound; the boundary fixation point still exists.

Next

• T1 · Boundary Fixation
• Discrete Impossibility
• Tietze bridge
• Meta-theorem — the unified version.