Selected Publications

• Assembly theory: optimal joint assembly spaces and assembly space energy

  Szymon Łukaszyk, Piotr Masierak (AT)

  

  We showed that an ensemble of elements can be assembled in a joint space in three non-equivalent ways, differing in size and complexity class, with individual optimality unattainable for some ensembles unless multiple copies of the same element are allowed, differing only by assembly history. We introduced a discrete Dirichlet energy of the assembly space that equals its size and a secondary energy independent of the assembly index and depth.

• Assembly theory: formalizing assembly spaces, discovering patterns and bounds

  Wawrzyniec Bieniawski, Piotr Masierak, Andrzej Tomski, Szymon Łukaszyk, Szymon Tworz (AT)

  

  We derived analytical bounds on the minimum and maximum assembly indices (ASI), as well as relations between the ASI, assembly depth, depth index, Shannon entropy, and expected waiting times. Comparing ASI with dictionary-based and grammar-based compression schemes, we showed that ASI provides supreme compression by exploiting global combinatorial patterns.

• On the quantum separability of qubit registers

  Szymon Łukaszyk (QT)

  

  The free energy principle qualifies every bipartitionally separable quantum system as a Bayesian agent. However, in the study we show that panpsychism can be avoided — and, thus, life distinguished from non-life — by a taxonomy of such quantum systems within a unique quantum state of the universe.

• Black hole merger as an event converting two qubits into one

  Szymon Łukaszyk (QT)

  

  The work provides the black hole Hamiltonian, closed forms of black hole energy eigenstates, and shows that black hole mergers, the most frequent gravitational events observed by LIGO/VIRGO interferometers, are congruent with the no-deleting and no-hiding quantum theorems.

• Computational Complexity of Determining the Assembly Index

  Piotr Masierak (AT)

  

  The work proves that the assembly index problem is NP-complete, through an explicit correspondence between assembly plans and straight-line grammars.

• On the 'Assembly Theory and its Relationship with Computational Complexity'

  Szymon Łukaszyk (AT)

  

  We defined the assembly space as an acyclic, 2-in-regular digraph of strings, where edges preserve the commutativity of an assembly step and define the order of concatenation. The uniqueness of a vertex suffices to introduce the notion of an assembly pool and determine if an assembly step is allowed: inaccessible unit-length strings that cannot be assembled from shorter strings form the initial assembly pool, while strings present in the assembly space can not be assembled again (using different pathways) as they would not be unique.

• Assembly Theory of Binary Messages

  Szymon Łukaszyk, Wawrzyniec Bieniawski (AT)

  

  We applied assembly theory to bitstrings, showing that the assembly index is bounded from below by the length of the shortest addition chain (OEIS A003313). We defined a degree of causation for the minimum-assembly-index bitstrings that exhibits regularities useful for determining the shortest addition chain.

• Life as the Explanation of the Measurement Problem

  Szymon Łukaszyk (ED, QT)

  

  We argue that only living agents process quantum information through their triangulated, holographic spheres of perception, where spherical Planck triangles correspond to bits of information. We also show that a black hole is a qubit in an equal superposition of its two energy eigenstates, with the ground state corresponding to a vanishing energy and the excited state corresponding to the black hole’s energy eigenvalue, that saturates three bounds of the quantum orthogonalization interval. We hypothesize that BHs are capable of evolving information within a range bounded by their energy and according to the principles of assembly theory.

• Shanon Entropy of Chemical Elements

  Szymon Łukaszyk (ED, AT)

  

  Hund’s rule is the rule of minimum Shannon entropy. We extend it to the Aufbau rule providing a monotonically increasing entropy function. We show that some elements that violate the Aufbau rule have the same entropies in the actual and Aufbau electron configurations. On the other hand, lower entropies of all actual elements’ configurations are associated with higher or equal numbers of unpaired electrons in these configurations compared to the Aufbau configurations. The only exception to this rule is palladium.

• Omnidimensional Convex Polytopes

  Szymon Łukaszyk, Andrzej Tomski (ED)

  

  We showed that the volumes and surfaces of convex polytopes present in all natural dimensions are holomorphic functions, making these objects omnidimensional. Application of these relations to polytopes inscribed in and circumscribed on n-spheres revealed their previously unknown geometric properties.

• Black Hole Horizons as Patternless Binary Messages and Markers of Dimensionality

  Szymon Łukaszyk (ED)

  

  Based on entropic gravity and the Bekenstein entropy we have shown that a black hole is a patternless binary string having a maximum Shannon binary entropy.

• Four Cubes: On the Necessity of Four-Dimensional Perception

  Szymon Łukaszyk (ED)

  

  The aim of the work was to show that the perception of reality in three real spatial dimensions and imaginary time is necessary due to the properties of the four-dimensional Euclidean space, which is the only one admitting a continuum of differentiable manifolds that are homeomorphic but not diffeomorphic to it.

• Reply to "Various issues around the L₁-norm distance"

  Andrzej Tomski, Szymon Łukaszyk

  

  A short note refuting the objections raised about the probability metric: that it is identical to the mean absolute difference and thus ill-defined. They are false, as the mean absolute difference is defined solely for independent and identically distributed random variables, whereas the probability metric does not impose such a restriction.

• Probability Metric,
  examples of approximation applications in experimental mechanics

  Szymon Łukaszyk

  

  The Probability Metric (now known as Łukaszyk–Karmowski metric) defines a distance between two random variables or vectors. LK-metric is not a metric as it does not satisfy the identity of indiscernibles axiom of the metric; for the same random variables, its value is greater than zero, providing they are not both degenerated.