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Math for emergence, open access

Six Birds: Foundations of Emergence Calculus

A math-only ‘calculus of emergence’ for how stable objects and effective “laws” form—and evolve—when you compress reality through a limited lens.

Preprint · v1 Not peer reviewed Published Jan 25, 2026 Open access · CC BY 4.0
Read the paper (PDF) Code & reproducibility

Plain-language overview

Six Birds treats emergence as a practical workflow: choose what you can observe (a lens), compress what you see into stable objects (a closure), and run independent audits to check what is real versus an artifact. The paper’s headline idea is that this same emergence mechanics applies not only to living systems, but to theory-building itself: new objects and new “laws” appear when a new layer becomes stable—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

At a glance

Theories build new dictionaries

Adding a new yes/no distinction almost never “does nothing”: it typically creates new stable categories and changes what the theory can say.

Objects are stable compressions

A “thing” is what survives re-packaging: once you compress a pattern into an object, re-applying the same compression shouldn’t change it.

No fake arrows of time

Summaries can hide directionality, but they can’t manufacture it. If your coarse data shows an arrow of time, the underlying system had one.

Open-endedness needs new closures

Iterating one rule saturates. Sustained novelty requires changing the closure itself—new layers, not just more iterations.

Core trio

Lens, completion, and audits

Open-access preprint with formal proofs + reproducible code.

What you can see

Lens

A lens is what you can reliably tell apart. Two microstates are “the same” at this level if the lens labels them the same.

How “things” form

Closure

A closure is how a description becomes stable: you compress, then check that re-compressing doesn’t keep changing the result.

How you stay honest

Audits

Independent checks that prevent self-deception: stability, novelty, and directionality are different questions—and must be certified separately.

Highlighted results

What the paper proves

Three separate checks: stability, novelty, and directionality (don’t confuse them).

A stability score you can compute

Defines an “idempotence defect”: how much your description still changes if you package it again. Small defect signals a genuinely stable closure.

Nothing stays constant under extension

Shows (in a precise finite setting) that adding a new distinction is overwhelmingly likely to split at least one existing category—strictly extending the theory.

Arrow-of-time audit

Time’s arrow is defined as forward-vs-reversed asymmetry of whole trajectories. The paper proves coarse-graining can reduce this signal, not invent it.

The protocol trap

Order effects can look like irreversibility if you hide a clock/schedule. Restoring the hidden phase removes the spurious arrow unless a real bias exists.

Methods & reproducibility

How the results are supported

  • Math-first framework (preprint): definitions, theorems, and worked toy examples.
  • Formal proof artifacts (Lean) for core closure/packaging algebra.
  • Reproducible code that regenerates figures and sanity checks end-to-end.
  • Small Markov-chain “laboratory” examples that make the ideas visible: objects appear, saturate, and sometimes dissolve under refinement.

Sanity checks

  • • Coarse summaries never increase the arrow-of-time score (they can hide it, not create it).
  • • “Protocol trap” demo: a hidden schedule can fake directionality; making the clock explicit removes the illusion unless a true bias is present.

Media-ready

Figures & demos

The repo includes a deterministic script bundle that regenerates the key visuals from scratch—useful for journalists and readers who want to sanity-check the claims. If you need packaged images, captions, or a walkthrough, contact the author.

• When more detail helps vs. hurts (objects can appear—or dissolve)

• Arrow-of-time audit: what coarse summaries can hide (and can’t fabricate)

• Protocol trap: how a hidden clock can fake directionality

Regenerate figures from code

Limitations & scope

Read-this-first caveats

  • Status: this is a research preprint (not yet peer reviewed).
  • Math-only: the paper provides a general calculus and certificates, not domain-specific empirical claims.
  • The three audits are independent: stability, novelty, and directionality do not imply one another without an explicit operator.
  • Results depend on the chosen lens and timescale: refinement can help or hurt depending on dynamics.

Resources

Downloads & links

Read the paper (PDF)

Open access, 521 KB

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.

Citation

How to cite

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

Talk to the author

For media inquiries, figures, or walkthroughs of the reproducibility harness, reach out directly.

Ioannis Tsiokos

ioannis@automorph.io

Corresponding author · Press contact

Questions welcome about the lens/completion/audit trio, the idempotence-defect criterion, and the protocol-trap examples.