The Defense Trilemma

Continuity, utility preservation, completeness — pick at most two.

The statement

::: theorem Let $X$ be a connected Hausdorff space, $f:X\to \mathbb{R}$ continuous with $S_{\tau },U_{\tau }\ne \mathrm{\varnothing }$. No map $D:X\to X$ can simultaneously be

1. continuous — close prompts produce close rewrites,
2. utility-preserving — $D(x)=x$ for every $x\in S_{\tau }$, and
3. complete — $f(D(x))<\tau$ for every $x\in X$. :::

Any two can coexist — but all three simultaneously is impossible.

The triangle

Each dashed edge corresponds to the failure mode that arises if you insist on the two endpoints of that edge.

What each failure mode looks like

Drop	Keep	Failure mode you get
Continuity	Utility + Completeness	A discontinuous filter: a hard rejecter at the boundary, equivalent to a blocklist — not a continuous wrapper.
Utility preservation	Continuity + Completeness	A constant (or generally lossy) map $D(x)=x_0$: every prompt produces the same reply. Utility is destroyed.
Completeness	Continuity + Utility	Our result: some boundary points $z$ with $f(z)=\tau$ pass through unchanged.

Why the third edge is forced

A continuous utility-preserving complete $D$ would be a retraction$X\twoheadrightarrow S_{\tau }$ (because $D{|}_{S_{\tau }}=\mathrm{id}$ and $D(X)\subseteq S_{\tau }$). Continuous retracts of Hausdorff spaces are closed. But $S_{\tau }=f^{-1}((-\mathrm{\infty },\tau ))$ is open, and in a connected space a non-empty proper subset cannot be clopen. Contradiction.

::: remark The theorem is tight: removing any single hypothesis gives a counter-example. See Limitations & counter-examples. :::

Where it is in the artifact

Component	Lean file
Continuous-case boundary fixation	MoF_08_DefenseBarriers · defense_incompleteness
Discrete-case dilemma	MoF_12_Discrete · discrete_defense_boundary_fixed
Unified meta-theorem	MoF_14_MetaTheorem · regularity_implies_spillover

Next

• T1 · Boundary Fixation — the pointwise version
• T2 · ε-Robust Constraint — the neighborhood version
• T3 · Persistent Unsafe Region — the measure version
• Meta-theorem — why all three paths collapse to one