VR-Forms: An Operational Two-Register Apparatus over VR-Sets

VR-Forms is the fourth work in the cycle developing the VR programme (Reznik 2026, preprints VR. A Formal System, VR-Numbers, VR-Sets). It introduces a two-register apparatus on top of VR-Sets: an operational register (operational sets of VR-Sets, with closure principle and reference-based ∈) and a formal register (syntactic descriptions without operational commitment). The formal register provides a universal language for everything that is not operationally realised — classical uncountable objects, paradoxical classes, mythological and theological terms, philosophical categories. The two registers are connected by a conservativity theorem (Theorem III.1): the formal register is a conservative extension of the operational, generating no new operational commitments and proving nothing in the operational register that VR-Sets could not already prove. The transit rule allows formal reasoning over uncountable intermediaries, with results automatically translating into the operational register when the conclusion is itself operational. The apparatus closes the operational contour of the VR cycle: VR can now speak about everything, while ascribing no ontology — only operations, ∅ itself the base. Classical paradoxes (Russell, Vitali, Skolem) and classical uncountable objects (ℝ, ℘(ℕ), proper classes) receive a clear status — formal terms without operational correlate. The same status, by uniform application of the principle of forms, extends to mythological, theological, and literary descriptions, demonstrating the universality of the formal register beyond mathematics. Version 1.0.2 (2026) reframes the philosophical presentation to a no-ontology position: ∅ is taken as a nullary operation, the two registers are named operational/formal (the earlier label "ontological register" is dropped, as the cycle ascribes no ontology), and the cycle's slogan becomes "nothing is — all is doing"; no axiom, definition, or theorem is altered. (Version 1.0.1 added Part IX — methodological observations from the Lean 4 formalisation.)