Math for emergence, open access
A math-only framework for how stable objects and effective "laws" form and evolve when you compress a complex system through a limited observational window.
Six Birds treats emergence as a three-step workflow. First, choose what you can observe (a "lens"). Second, compress what you see into stable objects (a "closure"): objects are whatever survives being compressed twice. Third, run independent audits to check what is real versus an artifact of your lens. The paper's headline idea is that this same workflow applies not just to living systems, but to theory-building itself: new objects and new "laws" appear whenever a new descriptive layer stabilizes, and they can change when the lens changes.
“Plato imagined reality written in timeless Forms. Six Birds treats reality as something that keeps building new dictionaries: new objects appear, new variables become meaningful, and the ‘laws’ update when a new layer stabilizes. That’s true for life—and just as true for physics and mathematics.”
— Ioannis Tsiokos
Theories build new dictionaries
Adding a new yes/no distinction to a theory almost never "does nothing." It typically splits existing categories into finer ones, expanding what the theory can describe.
Objects are stable compressions
A "thing" is what survives re-compression. If you compress a pattern into a coarse label and then compress again, a genuine object is one whose label does not change the second time.
No fake arrows of time
Zooming out can hide directionality, but it cannot create it. If a coarse summary shows a forward-vs-backward asymmetry, that asymmetry was already present in the detailed data.
Open-endedness needs new closures
Repeating a single rule eventually saturates: nothing new happens. Sustained novelty requires changing the rule itself, which means building a new descriptive layer, not just running more iterations.
Core trio
What you can see
Lens
A lens is what you can reliably tell apart. Two fine-grained states count as "the same" if the lens gives them the same label. Choosing a different lens changes which objects you can see.
How "things" form
Closure
A closure is how a description becomes stable. You compress the data, then compress again. If the second compression changes nothing, the description has closed: it is self-consistent at that level.
How you stay honest
Audits
Independent checks that prevent self-deception. Stability, novelty, and directionality are three separate questions. Each must be verified on its own; passing one does not guarantee the others.
Highlighted results
A stability score you can compute
Defines an "idempotence defect": compress your data into objects, then compress those objects again. The defect measures how much the second pass still changes. A small defect means the objects are genuinely stable.
Nothing stays constant under extension
Proves (in a precise finite setting) that adding a new yes/no distinction almost always splits at least one existing category. The theory strictly grows; the old vocabulary was not enough.
Arrow-of-time audit
The arrow of time is measured as the asymmetry between a trajectory played forward and the same trajectory played in reverse. The paper proves that zooming out can reduce this asymmetry, never inflate it.
The protocol trap
If a hidden schedule or clock is controlling the order of steps, the process can look irreversible even when it is not. Making the clock visible removes the false arrow, unless a genuine directional bias exists.
Methods & reproducibility
Sanity checks
Media-ready
The repository includes deterministic scripts that regenerate every figure from scratch. Useful for journalists and readers who want to independently verify the claims. For packaged images, captions, or a walkthrough, contact the author.
• When looking more closely helps vs. hurts (objects can appear or dissolve)
• Arrow-of-time audit: what zooming out can hide, and what it cannot fabricate
• Protocol trap: how a hidden clock can fake directionality
Limitations & scope
Resources
Read the paper (arXiv)
arXiv: 2602.00134
DOI on Zenodo
10.5281/zenodo.18365949
Code & reproducibility
Lean proof core + deterministic Python harness
Open access
CC BY 4.0 · No funding · No conflicts of interest.
Six Birds papers
To Wake a Stone with Six Birds: A Life is A Theory
Applies the framework to life: strict baselines, separable drive channels, and two concrete substrates (particle and neural).
Life & audits
To Become a Stone with Six Birds: A Physics is A Theory
Tests the same closure checks on four physics models: quantum dephasing, kinetic theory, turbulence filtering, and gravitational averaging.
Physics & closure
To Throw a Stone with Six Birds: On Agents and Agenthood
Separates "being a stable object" (agenthood) from "making a difference" (agency), measured by survival guarantees and causal impact.
Agency & controls
To Notch a Stone with Six Birds: Time as a Theory Artifact of Order, Measure, and Arrow
Treats time not as a background given, but as something assembled from stable ordering, repeatable ticks, and irreversible records.
Time & arrows
To Plot a Stone with Six Birds: A Geometry is A Theory
Builds points, distances, and curvature from scratch: points are indistinguishable groups, distance is transition cost, curvature is loop mismatch.
Geometry & holonomy
A Six-Birds' Eye View of Quantum Theory
Recasts quantum "collapse" as a packaging step: objects form when stable records do, not through a new physical law.
Quantum & records
A Six Birds' Eye View of Dark Energy
Asks whether dark energy could be a correction term forced by averaging an inhomogeneous universe, rather than a fundamental substance.
Cosmology & closure
Citation
Ioannis Tsiokos (2026). Six Birds: Foundations of Emergence Calculus. Zenodo. https://doi.org/10.5281/zenodo.18365949
BibTeX
@misc{tsiokos2026sixbirds,
title = {Six Birds: Foundations of Emergence Calculus},
author = {Tsiokos, Ioannis},
year = {2026},
publisher = {Zenodo},
doi = {10.5281/zenodo.18365949},
url = {https://doi.org/10.5281/zenodo.18365949}
}Press & contact
For media inquiries, figures, or walkthroughs of the reproducibility harness, reach out directly.
Ioannis Tsiokos
ioannis@automorph.io
Corresponding author · Press contact