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This paper describes first how Euclidian- and Minkowskian–Banach spaces are related via the definition of a metric or signature vector. Also, it is discussed later on how these spaces can be generated using homothecies of the unit sphere or shell. Such possibility allows for proposing a process aiming at the dimension condensation in such spaces. The condensation of dimensions permits the account of the incompleteness of classical QSPR procedures, independently of whether the algorithm used is statistical bound or AI-neural network related. Next, a quantum QSPR framework within Minkowskian vector spaces is discussed. Then, a well-defined set of general isometric vectors is proposed, and connected to the set of molecular density functions generating the quantum similarity metric matrix. A convenient quantum QSPR algorithm emerges from this Minkowskian mathematical structure and isometry.

By establishing a dictionary between the QM harmonic oscillator and the Collatz process, it reveals very important clues as to why the Collatz conjecture most likely is true. The dictionary requires expanding any integer n into a binary basis (bits) $$n = \sum a_{nl} 2^l$$ (l ranges from 0 to $$N - 1$$ ) that allows to find the correspondence between every integer n and the state $$|\Psi _n \rangle$$ , obtained by a superposition of bit states $$|l \rangle$$ , and which are related to the energy eigenstates of the QM harmonic oscillator. In doing so, one can then construct the one-to-one correspondence between the Collatz iterations of numbers $$n \rightarrow { n \over 2 }$$ (n even); $$n \rightarrow 3 n + 1$$ (n odd) and the operators $$\mathbf{L}_{{ n \over 2}}; \mathbf{L}_{ 3 n + 1 }$$ , which map $$\Psi _n$$ to $$\Psi _{{ n \over 2 }}$$ , or to $$\Psi _{ 3 n + 1 }$$ , respectively, and which are constructed explicitly in terms of the creation $$\mathbf{a}^{\dagger }$$ , annihilation $$\mathbf{a }$$ , and unit operator $$\mathbf{1 }$$ of the QM harmonic oscillator. A rigorous analysis reveals that the Collatz conjecture is most likely true, if the composition of a chain of $$\mathbf{L}_{{ n \over 2}}; \mathbf{L}_{ 3 n + 1 }$$ operators (written as $$L_*$$ in condensed notation) leads to the null-eigenfunction conditions $$(\mathbf{L_* L_* \ldots L_* } - {{\mathcal {P}}} ) \Psi _n = 0$$ , where $${{\mathcal {P}}}$$ is the operator that projects any state $$\Psi _n$$ into the ground state $$\Psi _1 \equiv | 0 \rangle$$ representing the zero bit state $$|0 \rangle$$ (since $$2^0 = 1$$ ). In essence, one has a realization of the integer/state correspondence typical of QM such that the Collatz paths from n to 1 are encoded in terms of quantum transitions among the states $$\Psi _n$$ , and leading effectively to an overall downward cascade to $$\Psi _1$$ . The QM oscillator approach explains naturally why the Collatz conjecture fails for negative integers because there are no states below the ground state.

A generalized way to define Minkowski spaces as Banach spaces permits to define with the aid of a metric vector generalized scalar products and hence generalized norms. The possible nullity of some norms in Minkowski spaces leads to define Minkowski hypersurfaces, which can be connected to multidimensional energy surfaces.