Ioannis Tsiokos

Six Birds for Navier-Stokes: A Mechanized No-Zeno Scaffold

Energy estimates for the three-dimensional incompressible Navier–Stokes equations bound the kinetic energy, but they do not stop velocity gradients from concentrating in ever smaller regions. This is why the classical regularity criteria, such as Beale–Kato–Majda, ask for control...

When a Copy Is Not a Continuation

Thought experiments about copying, teleporting and uploading ask whether the result is the same subject. This paper asks a prior question that can be answered exactly. When the organization described by a closure architecture for consciousness is moved to a different finite syste...

Why Mathematics Even Works

Many objects that mathematics uses to solve a problem do not belong to the setting in which the problem is posed: imaginary numbers are not real, infinitesimals are not real numbers, roots of an irreducible polynomial need not lie in the base field, and a cohomology class is not...

To Create a Stone with Six Birds: Emergent Geometric and Thermodynamic Regimes from a Minimal Stochastic Substrate

How much macroscopic regularity can a random Markov chain acquire when it is allowed to rewrite itself using only a small set of closure operations? We study this question in a minimal stochastic substrate: a random transition kernel on n states that is repeatedly rewritten, gate...

To Count a Stone with Six Birds: A Mathematics is A Theory

Limits, completions, and analytic continuations are usually introduced as definitions. This paper asks an operational question instead: when does a large discrete protocol — a difference quotient, a cumulative sum, a truncated Euler product — admit a stable continuous closure, an...

To Count a Stone with Six Birds: A Mathematics is A Theory

Limits, completions, and analytic continuations are usually introduced as definitions. This paper asks an operational question instead: when does a large discrete protocol — a difference quotient, a cumulative sum, a truncated Euler product — admit a stable continuous closure, an...

To Plot a Stone with Six Birds: Constructing and Auditing Emergent Geometry from Markov Dynamics

Can points, distances and curvature be built from nothing more than a Markov chain and a decision about what to ignore? This paper studies that question for finite Markov dynamics, within the Six Birds emergence calculus. A lens groups microstates into macro states, each represen...

Six Birds for Incompleteness: Fixed Packages, Package Change, and Conditional Arithmetic Lift

Gödel's theorems tell us that no consistent, effectively axiomatized theory containing enough arithmetic can settle every arithmetic truth, so mathematics grows by adding axioms. They do not, by themselves, say how to describe that growth or how to compare different ways of growi...

To Chart a Stone with Six Birds: Emergence Phase Diagrams for Effective Theories in Control Space

Scientists routinely draw maps of macroscopic behavior: phase diagrams, stability charts, bifurcation diagrams. Emergence is rarely mapped in the same way. Usually one asks for the best coarse-graining or the best score. We propose to ask instead where a good macroscopic descript...

Strict Theory Extension on a Lawful Continuous Cantor Shell

We study a twenty-state stochastic substrate in which six coupled mechanisms (operator rewrite, admissibility gating, timescale adaptation, lens selection, packaging, and a budget ledger) rewrite a Markov kernel at every step. We describe it at two levels: a growth theory, which...

A Six-Birds' Eye View of Quantum Theory: Defining Objects by What We Can Distinguish, Not What We Imagine

Summary Quantum theory uses a single object, the quantum state, for two different jobs: describing how a system evolves and keeping track of what has been recorded about it. This paper keeps the two jobs apart. Using the packaging language of Six Birds Theory (SBT), a closure-bas...

To Cast a Stone with Six Birds: A Closure-Deficit Account of Randomness under Packaging and Budget

Why does a process look random? This paper develops one precise answer within the Six Birds framework: much of what a description calls randomness is information the description has chosen not to keep. We model a description by a packaging map Π that groups microstates X_t into c...

Six Birds Protocol Trap: Holonomy Without Entropy Production

When a system is driven through a fixed sequence of operations, two different things can happen. The outcome can depend on the order in which the operations are applied (route dependence, or holonomy), and the motion can acquire a preferred direction in time (an arrow, measured b...

To Classify a Stone with Six Birds: Primitive Activation and Observable Separation of Layer-Birth Classes

A layer birth is the point at which a coarse-grained description of a stochastic system starts to behave like a theory in its own right, with stable macrostates that it reproduces under its own dynamics. This paper asks whether such births come in different kinds, and how one wou...

To Kill Three Stones with Six Birds: A Common Grammar for the SM, QM, and GR

Can one small mathematical vocabulary organize two very different open problems in fundamental physics? We test this with the Six Birds emergence calculus, whose working core is simple: a coarse description is a quotient of a finer one, a quantity either passes through the quotie...

Glass Is an Unclosed Layer: Kovacs Memory as Predictive-Quotient Residue

Glass Is an Unclosed Layer: Kovacs Memory as Predictive-Quotient Residue asks, on finite Markov models small enough to settle it by exact computation, whether a declared readout forms a closed description: whether its present value fixes its own future statistics. Kovacs's 1963 c...

Glass Is an Unclosed Layer: Kovacs Memory as Predictive-Quotient Residue

Glass Is an Unclosed Layer: Kovacs Memory as Predictive-Quotient Residue asks, on finite Markov models small enough to settle it by exact computation, whether a declared readout forms a closed description: whether its present value fixes its own future statistics. Kovacs's 1963 c...

To Become a Stone with Six Birds: A Physics is A Theory

Physics routinely replaces a detailed description with a coarser one — density matrices with probabilities, kinetic distributions with fluid moments, resolved fields with filtered fields, heterogeneous ensembles with averages — and then treats the coarse description as a theory i...

To Notch a Stone with Six Birds: Time as a Theory Artifact of Order, Measure, and Arrow

To Notch a Stone with Six Birds asks an operational question: when does a system have time, rather than merely being described with a time parameter? Within Six Birds Theory (SBT), a layer of description has time when it can sustain three things: a stable notion of “next” (order)...

To XOR a Stone with Six Birds: Closure Diagnostics for Emergent Bits, Gates, and Booleanity

When does a physical or stochastic system deserve to be described in terms of bits and logic gates? We treat logic not as a starting point but as something a coarse-grained description has to earn. Working in the Six Birds emergence calculus, we call a description a logic layer w...