arXiv subjects

Ioannis Tsiokos

Publications and source records attributed to Ioannis Tsiokos.

3 recordsLinked to original sources

arXiv2026

A Strongly Aperiodic Monotile in Three Dimensions

Socolar and Taylor asked for a single, simply connected three-dimensional prototile that forces nonperiodicity by shape alone, admitting no weakly nonperiodic tiling; the Schmitt-Conway-Danzer biprism and the three-dimensional Socolar-Taylor tile admit screw motions or a periodic stacking direction. We exhibit a rational polyhedral $3$-ball $Q$, which we call Chair44 (R44): a seven-cube chair whose $24$ exposed unit panels carry tiny square-pyramid features, and prove, as a proof submission, that $Q$ admits tilings of $\mathbb{R}^3$ by congruent copies, reflections allowed, and that every such tiling has no translational period and a symmetry group of order at most $24$; every tiling is homochiral and carries a unique infinite hierarchy of nested supertiles. The solid was designed to a reading of the aperiodic-monotile phenomenon reached with the Six Birds emergence calculus (Section 3.3), and the construction turns on a single finite test, checked by machine: the tile's own contact rule survives coarsening, so that the decoded parent tiling obeys the tile's rule and no other. The proof combines a written geometric argument, that the features force every tiling onto a registered lattice, with exhaustive finite enumerations; the companion census is replayed by two independent implementations, every finite gate is kernel-checked in Lean 4 (modulo a named compiler hook per native_decide theorem), and the written geometric lemmas and the logical assembly are Lean theorems as well, so that the theorem is kernel-checked modulo the named compiler hooks; the written proofs remain as exposition.

math.CO↗

arXiv2026

To Throw a Stone with Six Birds: On Agents and Agenthood

Six Birds Theory (SBT) treats macroscopic objects as induced closures rather than primitives. Empirical discussions of agency often conflate persistence (being an object) with control (making a counterfactual difference), which makes agency claims difficult to test and easy to spoof. We give a type-correct account of agency within SBT: a theory induces a layer with an explicit interface and ledgered constraints; an agent is a maintained theory object whose feasible interface policies can steer outside futures while remaining viable. We operationalize this contract in finite controlled systems using four checkable components: ledger-gated feasibility, a robust viability kernel computed as a greatest fixed point under successor-support semantics, feasible empowerment (channel capacity) as a proxy for difference-making, and an empirical packaging map whose idempotence defect quantifies objecthood under coarse observation. In a minimal ring-world with toggles for repair, protocol holonomy, identity staging, and operator rewriting, matched-control ablations yield four separations: calibrated null regimes with single actions show zero empowerment and block model-misspecification false positives; enabling repair collapses the idempotence defect; protocols increase empowerment only at horizons of two or more steps; and learning to rewrite operators monotonically increases median empowerment (0.73 to 1.34 bits). These results provide hash-traceable tests that separate agenthood from agency without making claims about goals, consciousness, or biological organisms, and they are accompanied by reproducible, audited artifacts.

cs.AI↗

arXiv2026

Six Birds: Foundations of Emergence Calculus

We develop a discipline-agnostic emergence calculus that treats theories as fixed points of idempotent operators acting on descriptions. We show that, once processes are composable but access to the underlying system is mediated by a bounded observational interface, a canonical toolkit of six closure-changing primitives (P1--P6) is unavoidable. The framework unifies order-theoretic closure operators with dynamics-induced endomaps $E_{τ,f}$ built from a Markov kernel, a coarse-graining lens, and a time scale $τ$. We introduce a computable total-variation idempotence defect for $E_{τ,f}$; small retention error implies approximate idempotence and yields stable "objects" packaged at the chosen $τ$ within a fixed lens. For directionality, we define an arrow-of-time functional as the path-space KL divergence between forward and time-reversed trajectories and prove it is monotone under coarse-graining (data processing); we also formalize a protocol-trap audit showing that protocol holonomy alone cannot sustain asymmetry without a genuine affinity in the lifted dynamics. Finally, we prove a finite forcing-style counting lemma: relative to a partition-based theory, definable predicate extensions are exponentially rare, giving a clean anti-saturation mechanism for strict ladder climbing.

cs.LO↗