Running summarizing tools on a new article

Abstract: Modern AI architectures operate natively in continuous high-dimensional geometric spaces (R^N, N >= 10,000), yet current programming paradigms force them to communicate via discrete sequential text channels — a mismatch we call the impedance gap. Recent empirical work has demonstrated that eliminating this gap substantially improves multi-agent coordination: Interlat [Du et al. 2025] shows that transmitting hidden states instead of tokens carries approximately 40,000 bits of information per state versus ~15 bits per language token — a 2,667× semantic bandwidth gain — while LatentMAS [ICML 2026 Spotlight] achieves superior multi-agent performance and major wall-clock speedups via training-free latent alignment. We investigate whether a formal programming language can be specified directly over the geometric space of transformer and SSM activations, without serialization to text. We propose POLYDIM, a language whose fundamental program unit is a geometric transformation T: R^N -> R^N. We prove four algebraic theorems establishing POLYDIM's formal foundations: associativity of composition (T1), linearity of superposition (T2), functoriality of projection to executors (T3, proved unconditionally for three contracts — COMPILE and EXPORT via the Subspace Commutativity Lemma, RENDER via the Flutter Algebraic Isomorphism), and uniqueness of fixed points (T4). We formalize a two-layer architecture separating the invariant algebraic core (COMPOSE, MIX, FIXPOINT, PROJECT) from architecture-specific implementation primitives (ATTEND for transformers, RECUR for SSMs such as Mamba-3 [ICLR 2026]), guaranteeing that the language specification is robust to architectural transitions. A Python bootstrap with 29/29 tests and +64.5% semantic improvement validates the conceptual framework. POLYDIM fills the gap between empirical latent communication research and formal language design: the specification that systems like Interlat and LatentMAS implicitly require but do not provide.