Separators codimension-one manifolds that partition phase space appear in fluid dynamics, phase-field thermodynamics, spectral band theory, and chaotic iteration. Existing treatments are tied to the specific physics of each domain, preventing a unified analysis of robustness and symmetry. We introduce a carrier-free separator grammar that defines (i) a divider set D⊂X, (ii) an optional Z2 lift ι acting on a lifted diagnostic space, and (iii) a veto-gate comprising lens-stability, no-peeking, and null-bite conditions. The grammar is anchored by a local hyperbolic saddle normal form, guaranteeing a four-sector separatrix skeleton. We demonstrate the framework on four distinct carriers: a two-dimensional streamfunction yielding Lagrangian coherent structures, a drift-null ridge in a phase-field effective potential, the fractal band-gap boundary of the Harper–Hofstadter model, and an invariant basin boundary of an iterated map. In each case the divider, an even diagnostic I, and an odd marker P satisfy the exact parity algebraI ∘ ι=I, P ∘ ι=−P and survive all veto tests. Our results establish a single invariant skeleton that transcends carrier languages, opening a pathway to systematic, falsifiable analysis of transport barriers across physics. Transport barriers—structures that separate distinct dynamical or thermodynamic regimes—play a central rôle in many areas of mathematical physics. In incompressible fluids they appear as Lagrangian coherent structures (LCS) that delineate regions of distinct particle itineraries 1; in phase-field models they manifest as drift-null ridges that separate metastable phases 2; in condensed-matter physics the Hofstadter butterfly displays a fractal network of spectral gaps that separate admissible from forbidden energies 3; and in discrete dynamical systems invariant basin boundaries separate distinct attractor basins 4. Despite their ubiquity, current analyses are largely domain-specific: each community develops its own language, invariants, and validation protocols. Consequently, a rigorous statement such as “the separator is robust under smooth reparameterisations” is proved separately for each case, and there is no common framework to compare results across disciplines. In this paper we propose a carrier-free separator grammar that abstracts away the underlying physics while retaining the essential geometric and algebraic structure. The grammar consists of three ingredients: Divider set D – a codimension-one hypersurface that partitions an analysis space X. Optional Z2 lift ι – an involution on a lifted diagnostic space Σ~ that induces an even/odd split of observables. Veto-gate – a collection of falsifiability conditions (lens-stability, no-peeking, null-bite) that guarantee the separator is not an artefact of representation. The core mathematical engine is the hyperbolic saddle normal form, which guarantees a universal four-sector separatrix skeleton around any non-degenerate saddle. Building on this skeleton, we construct a universal template for separators, even/odd diagnostics, and stability indices that is independent of the carrier language. We illustrate the grammar with four carriers (Sections 6.1–6.4). The third carrier—fractal parameter-space separatrices arising from the Harper–Hofstadter model—is presented as a novel optional module (Section 6.3). Its inclusion demonstrates that the grammar applies not only to spatial manifolds but also to synthetic parameter spaces. Finally, we describe the Bilax-Return certification protocol (Section 7), an empirical pipeline that tests the parity algebra and veto-gate on data. The protocol has already been validated on the four carriers (Table 1) and provides a reproducible, statistically rigorous falsifier.
kirjonen et al. (Fri,) studied this question.