What question did this study set out to answer?

The aim is to formalize a dual-mode operational architecture for computation and transputation in physical systems.

June 19, 2026Open Access

The GTE-Möbius Architecture: Arithmetic Unification of Computation and Transputation in the Universal Generative Principle

Key Points

The aim is to formalize a dual-mode operational architecture for computation and transputation in physical systems.
Developed the Generative Triple Evolution (GTE)-Möbius architecture as a mathematical framework.
Proved the relationship between the T(2,3) torus knot parameters and GTE.
Defined the Turing-PSC computability class and established its hierarchy.
Demonstrated the TPC class is strictly between Turing-decidable and hypercomputation.
Proved that the hierarchy depth of the TPC is 3.
Established connections between emergent gravity and trajectories in the GTE framework.

Abstract

The Universal Generative Principle requires a physical substrate capable of two operationally distinct but inseparable modes: computation (deterministic evolution under) and transputation (PSC-forced, non-computable selection among undecidable alternatives). We formalize this as the Generative Triple Evolution (GTE) -M? bius architecture-the triple (A, e, ), where A is the GTE arithmetic carrier (winding dynamics), e is the self-encoding map conferring Turing completeness, and is the class of PSC-consistent coherence measures constrained by D1-D5. The M? bius name reflects a single-surface architecture: computation and transputation share a common substrate without a sharp syntactic boundary; the G? del-Turing boundary of (A, e) is the computation/transputation boundary of the physics. The GTE-M? bius architecture is not a physical substrate independent of MDL; it is the computational-architectural description of the same physical field. MDL is the physical substrate of the universe; the M? bius structure characterizes how that field processes information and adjudicates quantum events. We prove that the T (2, 3) torus knot has both parameters GTE-derived (, zero sorry), define the Turing-PSC (TPC) computability class as strictly intermediate between Turing-decidable and hypercomputation (13 Lean theorems, ), and show its three-level hierarchy depth equals = 3 (). Conjecture C1 (GTE as the final FPSC-coalgebra) is fully proved (, zero axiom) ; four consistency conjectures (C2-C5) remain open, with C3 and C4 the most tractable. The abstract carrier A realizes as the PSC-admissible kink sector of the MDL Klein-Gordon field; binary coarse-graining recovers in the continuum limit. Three Lean-certified lifting theorems connect beable, discrete, and physical scales () ; the GTE equivalence principle and geodesic theorem () connect emergent gravity to -weighted trajectories. Quantum field-theoretic and completeness consequences are developed in companion papers SpivackCMCA, SpivackThreeTapeCMCA, SpivackPhiMDLField, SpivackCompleteness, SpivackQGR.

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