On the Inseparability of Instructions and Data in Shared-Embedding Sequence Models

Prompt injection is the top security risk for LLM-integrated applications, yet every defense proposed so far has been broken. We prove this is not a coincidence: in shared-embedding architectures that lack enforced control-data separation, perfect prompt-injection prevention is mathematically impossible. We formalize prompted systems as Prompted Action Models whose outputs include control-authoritative actions: refusal decisions, tool authorization, policy routing, and memory writes. We define Semantic-Faithful Control (SFC), the property that such behavior depends only on the meaning of untrusted input, not on how it is encoded. We then prove SFC is unachievable within the shared pipeline, via three results: a provenance-recovery impossibility (shared representations make trusted and untrusted content statistically inseparable, bounded by total variation distance); control-path exposure (untrusted tokens enter control-relevant computation through the same attention value-aggregation that determines outputs); and a finite-coverage invariance gap (finite training cannot certify invariance over infinite semantic-equivalence classes). We ground each quantity in measurements on production tokenizers and models. The result is structural, not a gap in current defenses. It mirrors the code-data confusion in Von Neumann machines that gives rise to buffer overflows, a vulnerability class that took decades of layered defenses (DEP, Write-XOR-Execute, ASLR, stack canaries, and ultimately memory-safe languages) to contain, because no single mechanism sufficed. The implication is the same: prompt injection cannot be eliminated by better in-pipeline classification or alignment alone. It requires architectural separation of instruction and data channels. We identify the root cause and the class of solution it demands.