Golden-Ratio Logical Ladders: A φ-Scaled Hyperdoctrine Linking Classical and Quantum Logics
Yoshida, Kazuyuki
We propose a categorical framework in which different logical systems, such as intuitionistic, linear, and quantum logic, are not treated as mutually incompatible "boxes" but as different levels of a single self-similar ladder. Technically, we introduce the notion of a Fibonacci logical ladder over a base category C, that is, a sequence of indexed logics P_n whose internal degrees of freedom d_n satisfy a Fibonacci type recurrence d_{n+1} = d_n + d_{n-1}. The asymptotic growth ratio d_{n+1}/d_n then converges to the golden ratio phi, which we interpret as a scale parameter governing the self similarity of the logical hierarchy. Within this scheme, low rungs of the ladder can be instantiated by Heyting or topos based logics, intermediate rungs by resource sensitive systems, and high rungs by orthomodular logics arising from Hilbert space projections. We provide a minimal toy model in which a finite Heyting algebra and a finite orthomodular lattice generate a three step Fibonacci ladder, and we discuss how more sophisticated examples connect to known appearances of the golden ratio in quantum theory, such as Fibonacci anyons and subfactor indices. Conceptually, the framework suggests a shift from a static view of "the" correct logic to a dynamical, scale dependent picture: the logical form of propositions depends on the observational level at which systems are distinguished, and this observational hierarchy is itself governed by a universal self similarity ratio phi.
•Dec 12, 2025•0 citations