Theorems

Every major theorem in the paper, ordered from the weakest hypothesis to the strongest conclusion. The first three form the central three-tier escalation; the rest extend the same machinery to discrete, multi-turn, stochastic, and pipelined settings.

The three tiers at a glance

Tier	Theorem	Extra hypothesis	Conclusion
T1	Boundary Fixation	connected + Hausdorff	∃ z with f(z)=τ and D(z)=z
T2	ε-Robust Constraint	f is L-Lipschitz, D is K-Lipschitz	f(D(x)) ≥ τ − LK·d(x,z)
T3	Persistent Unsafe Region	transversality: G > ℓ(K+1)	positive-measure set with f(D(x)) > τ

All theorems

Core impossibility

• T1 · Boundary Fixation — the pointwise result (paper Thm 4.1, Lean MoF_08).
• T2 · ε-Robust Constraint — the Lipschitz neighborhood bound (paper Thm 5.1, Lean MoF_11).
• T3 · Persistent Unsafe Region — the measure-theoretic result under transversality (paper Thm 6.3, Lean MoF_11).
• Defense Dilemma (K*) — the tradeoff in choosing the defense's Lipschitz constant (paper Thm 7.3, Lean MoF_19).

Discrete and continuous

• Discrete Impossibility — discrete IVT + non-injectivity dilemma; no topology required (paper Thm 8.2–8.3, Lean MoF_12).
• Continuous Relaxation (Tietze) — the bridge from finitely many observations to the continuous theory (paper Thm 8.1, Lean MoF_ContinuousRelaxation).

Extensions

• Multi-Turn Impossibility — the impossibility recurs at every turn (paper Thm 9.1, Lean MoF_13).
• Stochastic Impossibility — expected behavior inherits boundary fixation (paper Thm 9.2, Lean MoF_13).
• Pipeline Degradation — Lipschitz constants multiply; depth makes defense harder (paper Thm 9.4, Lean MoF_15).

Unification

• Meta-theorem — a single representation-independent statement that implies T1, the discrete dilemma, and the stochastic impossibility (Lean MoF_14).

Quantitative bounds

• Volume bounds — explicit lower bounds on $|{\mathcal{B}}_{\epsilon }|$ and the cone-based persistent region (paper Thm 7.1–7.2, Lean MoF_17, MoF_18).