VR-Forms: A Formal Register for Non-Operational Discourse over the VR Cycle

VR-Forms is the fourth work of the VR cycle, following VR. A Formal System (Zenodo DOI 10.5281/zenodo.20212092), VR-Numbers (Zenodo DOI 10.5281/zenodo.20272743), and VR-Sets (Zenodo DOI 10.5281/zenodo.20303536). It closes the ontological contour of the cycle by introducing a formal register — a syntactic layer for speech about what is not operational. The slogan of the cycle, here in its completed form: "Only ∅ is. Everything else either acts or is said. Action is operational and relates to ∅; speech is formal and refers to nothing by necessity." The preceding three works built mathematics on the single ontological primitive ∅: arithmetic (VR), numerical extensions ℤ, ℚ, ℝ, ℂ (VR-Numbers), and an operational set theory with a countable universe (VR-Sets). VR-Sets revealed the boundary of operational ontology — uncountable collections, paradoxical classes, and other non-operational descriptions have no place in it. VR-Forms provides the language for speaking about everything that lies beyond this boundary, without violating the minimalism of the operational ontology. Principal results: (1) Two registers. The ontological register (VR-Sets) and the formal register (the present extension). The first speaks of operational sets that "are" as actions upon ∅; the second speaks of formulas — formal terms — that refer to nothing by necessity. A formal term is the syntactic record of a description, considered without ontological commitments. It is form without content. (2) The principle of forms. Parallel to the closure principle of VR-Sets: every syntactically correct record of a description specifies a formal term. Unconditional, in contrast to the conditional closure principle. Paradoxicality, non-realisability, contradictoriness are properties of descriptions, not prohibitions. (3) Conservativity theorem. VR-Forms is a conservative extension of VR-Sets in the ontological register. The formal register adds no new theorems about operational sets; it only provides a language for the non-operational. Consistency of VR-Forms relative to ZF is established. (4) Transits. Methodological rule: a derivation in the formal register ending in an ontological formula counts as a proof in VR-Sets. Direct and problematic transits are distinguished — the latter indicating places where classical mathematics essentially leans on uncountability. (5) Universality of the formal register. The apparatus is demonstrated on mathematical formal terms (classical ℝ, ℘(ℕ), the Vitali set, the Russell class, proper classes of NBG) and on non-mathematical ones (dragons, gods, philosophical categories, literary worlds). Skolem's paradox is removed as a difference of registers. Classical paradoxes are reinterpreted as properties of form. The work shows that operational ontology can speak about everything — including what is not in its ontology — without losing minimalism. (6) A precise form of nominalism. VR-Forms provides a contemporary formal articulation of the medieval nominalist position: universals are names, not entities; only the concrete is real. The work replaces "the concrete" with "the operational" — what has an operational correlate in the VR-ontology. Eight parts. The preprint closes with technical open questions (machine formalisation in Lean, precise range of the translation π, alternative logics of the formal register), substantive extensions (VR-Audit as the natural fifth work of the cycle, connections with reverse mathematics, topoi, homotopy type theory, foundations of physics), and philosophical open directions.