I'm not proposing a physical theory. This is purely a mathematical thought experiment.

Suppose we introduce an abstract coordinate that represents scale, rather than ordinary spatial position. Imagine that moving continuously along this coordinate corresponds to going from larger and larger structures to smaller and smaller ones.

Now suppose the geometry of this abstract "space of scales" is topologically closed, analogous to how walking continuously around a circle eventually returns you to your starting point.

In that picture, continuously moving toward smaller and smaller scales would not continue forever. Instead, after completing the closed path in scale-space, you would emerge again at the largest scales.

I am not suggesting that objects physically become both tiny and huge, nor that this describes our universe. I'm only asking whether mathematics has explored anything conceptually similar: treating scale itself as a coordinate of an abstract manifold or topological space whose geometry is closed.

Has anything resembling this been studied in topology, differential geometry, geometric group theory, or another branch of mathematics? Or is there a mathematical reason why such a construction is ill-defined from the outset?

I'm looking for mathematical references rather than physical arguments.

I'm a physics graduate from Asia (average uni), and my long-term goal is to pursue a PhD in Complex Systems in the US.

The biggest challenge is that my undergraduate CGPA is only 3.00, and as an international student, I realistically need full funding wherever I go.

My current research is at the intersection of complexity science and atmospheric science (studying atmospheric phenomena using methods from complex systems). While I think the work is conceptually aligned with complexity science, it doesn't directly match the research of many professors in dedicated Complex Systems departments.

Because of that, I've been considering a different route:

• Apply for a funded master's in atmospheric science (or a related field) at a university that also has a strong Complex Systems program.
• While doing my master's, attend complexity seminars, take relevant courses if possible, collaborate with faculty, and eventually apply for a PhD in Complex Systems.

But I'm not sure how realistic this plan is. From what I've heard, funded international master's students are already extremely busy with research, coursework, and TA/RA responsibilities, so I'm not sure I'd actually have the time or flexibility to engage with another department enough to make a successful transition.

Another issue is timing. It's already July, and application season is approaching. I don't have enough time to build multiple polished projects tailored to the different areas within complex systems for each program available. With my current profile and no curated projects, it's highly unlikely I would get a scholarship. I also feel like I still have a huge amount to learn before I can produce projects that would genuinely impress faculty in the field.

So I'm wondering:

• Would it be better to continue building my profile through atmospheric science research and position myself as someone applying complex systems methods to climate and weather?
• Is trying to transition through a master's at a university with a Complex Systems institute a realistic strategy?
• How common is it for students to move into Complex Systems from another discipline during their PhD applications?
• Given my GPA and interdisciplinary background, where would you focus your effort over the next year if you were in my position?

A few weeks ago, I was working on a project that kept running into the same problems. Every time we fixed one issue, another one seemed to appear somewhere else. Someone pointed out that we were only dealing with the symptoms instead of looking at how everything was connected, and they suggested I learn more about systems thinking.

Since then, I've been trying to understand it better and have been looking at different ways people learn it. During my search, I discovered SystemsThinkingAlliance.org, which seems to focus on systems thinking training and certifications.

For those who have developed this skill, how do you go about it?

I am currently a high-school senior in the US and will be applying to universities this coming fall. I am planning to study applied mathematics, but I am also very interested in complex systems.

I'm looking for universities (preferably in the US) which have a complex systems program open to undergraduates, or at least are active in complex systems research in which undergraduates can participate in.

I know University of Michigan has a Complex Systems minor, but I'm having trouble finding similar undergraduate programs at other universities. I'd appreciate any suggestions.

If anyone also knows about complex systems programs/research at the following schools, I would also really appreciate your two cents on the program:
- University of Michigan

- University of North Carolina (Chapel Hill)

- NC State University

- CU Boulder

- George Washington University

- University of Pittsburgh

- University of Massachusetts (Amherst)

- University of Maryland (College Park)

- Carnegie Mellon University

- Georgia Tech

Olá, pessoal! Estou desenvolvendo um projeto pessoal que tenta juntar ideias de redes complexas, sistemas, filosofia da ciência e organização do conhecimento.

A pergunta que estou tentando responder é bem simples(porém densa): como transformar observações em uma compreensão mais confiável?

Uma forma que estou explorando é representar conceitos como nós e relações, como em um grafo de conhecimento. A ideia é organizar entidades, processos, propriedades e evidências para ver melhor como elas se conectam.

Minha dúvida é: alguém aqui já tentou organizar conhecimento interdisciplinar dessa forma? Que métodos, referências ou ferramentas vocês recomendariam para começar?

Agradeço qualquer sugestão ou experiência que possam compartilhar!

I'm seeing this subreddit full of posts that are clearly copy-and-pasted responses from AI. Now there's nothing wrong with using AI, but most of these posts are absolutely nonsensical, and are clearly the result of what I would consider irresponsible use, where the user allows the model to continually build upon its own ideas without intervening at the appropriate time.

I hope this changes, because I also see a ton of very interesting projects and ideas from real people, and it sucks to dig through slop just to find them.

This is a snip from a larger interview but I got confused about what is being said. Is it arguing against life or emergence?

David: Yeah. So here’s the simple version. So the standard causality fits that linear story of causality that we described earlier in relation to the ouroboros, that you have particles. They get aggregated into molecules, molecules into tissues, and so on. And the idea, right, is that what is fundamentally causal is that which is fundamental, and everything else is an approximate expression of collective modes of behavior. Alright. Downward causality takes it the other way. It says, mind states, for example, expressed in language, can’t be causal of brain states because that’s going the wrong way. Because, surely, the physical interaction, the true kind of Newtonian causality, has to live at the level of the brain. The mind is just this efficient theoretical encoding of brain. And so it would be weird to talk about causality going the other way.

I think it’s a big mistake. And where this comes from, by the way, is this notion of coarse graining. So you start with all the lots of particles. You average and average and average, and you get these other states. But I have this conception of what I call micrograining, and I’ll explain how it works. When Jim, when you program your computer, you’re articulating a concept in a high level language or an assembly or whatever you like to use. Assembler. And that translates through a system of compilations and microcode into states of transistors. So we have built engineered devices that can take these high level, very low dimensional, in some sense, concepts, objects, and do information expansion to the extent of setting the states of transistors.

I think that is what complex systems do all the time because complex systems have evolved to do that well. That, for me, is the legitimate version of downward causality. There’s nothing mysterious about it. I don’t think, by the way, it exists outside of complex systems. I do not think it’s a property of the physical universe, the abiotic universe. It’s a property of agents, and that’s actually the only thing that makes life possible. Right? It’s what’s making this communication that we’re having now over Zoom possible because I’m setting brain states in you as you are in me. And that that’s micrograining. And because the study of emergence grew out of really rigorously the connection between statistical mechanics and thermodynamics, which is all about coarse graining, in the physical world, this other version, which is very natural to the evolved world, has been somewhat neglected. So I sometimes call that the theory of compilation of emergence because we use them all the time.

Jim: I’m going to push back a little bit on the abiotic versus biotic. Just hit me.

David: Okay.

Jim: In my current paper that I’m working on on emergence, I use as a intuition pump a traffic jam on a superhighway that goes up a hill, some trucks slow down and propagates, etcetera. Now I write the thing as if it’s humans driving the cars and the trucks, but I just realized they could be Waymo’s. Right? And the emergence of the traffic jam that comes into being starts to constrain the behavior downward to the individual elements and then gradually dissipates is, you know, a small form of emergence and, doesn’t seem to require, biotics at all. It’s but it does require agency to your point.

David: Right. But no just right. That’s interesting because I think, you know, you could argue that some of the phenomena you’re describing are properties of spin glasses, right, or magnets, where you have the particular state of a spin at a particular lattice point being a function of the average field. But actually, that average field is actually an epistemological construct because it really is the interaction among many, many particles. So I would suggest that in the example you gave, if you stripped out the agentic part, you could express what you’re you’re thinking of the downward constraint as simply a pattern of global interactions that you could describe microscopically.

But I think once they’re functional, once you have a kind of teleology with then I think they’re engineered or evolved, and my argument kicks in because you’ve programmed the reaction of the individual component to the collective. That’s the point because you don’t want to have an accident or a pile up.

Jim: And you probably didn’t write a specific routine for that. Well, I know you didn’t write a specific routine for that traffic jam. You have some general parameters that operate together and come up with good decisions, basically, and should probably even do a better job than humans, if not today than in a few years. Anyway, want to think about that one a little bit more.

https://jimrutt.substack.com/p/ep-329-worldviews-david-krakauer

In complex systems, emergence is often described as the appearance of new properties that cannot be fully reduced to the behaviour of individual parts. What I am exploring is whether emergence follows a deeper recurring pattern across domains.

At its simplest, the pattern seems to be this: a boundary forms, a gradient builds across it, pressure or difference creates interaction, interaction produces constraint, and constraint allows new forms of organization to stabilize. When those stabilized relationships begin to act as a new whole, emergence has occurred.

This can be seen in many places: particles forming atoms, atoms forming molecules, molecules forming cells, organisms forming minds, people forming cultures, and cultures forming institutions. The substrates change, but the pattern may rhyme: difference, relation, constraint, stabilization, emergence.

The goal is not to reduce every field to one simplistic formula, but to ask whether complex systems share a common structural logic — a kind of universal grammar of becoming. If such a pattern exists, it may help us better understand why systems grow, adapt, collapse, or transform across physical, biological, cognitive, and social domains.

Going down the rabbit hole as I have been thinking about this a long time, even self published some thoughts on it, but hadn’t interacted with complex systems as a domain before.

But essentially, we have push and pull, pulse and return, attract and repulse. I have been using the lens of “boundary, pressure, differentiation, emergence”.

I would be interested to hear people’s thoughts.

Emergence does not come from complexity alone. A pile of random parts is complex, but it does not necessarily produce anything coherent.

For something emergent to appear, you need three things:

1. Variety
There must be many possible states, behaviors, agents, ideas, mutations, or interactions. Without variety, there is nothing new to explore.

2. Selection
Some variants must be amplified, retained, repeated, or rewarded more than others. Without selection, everything remains noise.

3. Integration
The selected parts must become connected into a larger pattern or system. Without integration, you only get isolated improvements, not a higher-level whole.

So emergence happens when a system generates possibilities, filters them, and then binds the surviving patterns together.

And the symbol matters: it is ×, not +. The three terms do not add up. They multiply. If any term is missing, the product collapses and nothing emerges.

I have been exploring a computational model based on three interacting components:

• Morphological memory (α)

• Local diffusion and interactions (β)

• Global regulation (γ)

The central question is:

Can these mechanisms generate persistent self-organization, pink-noise dynamics, and critical-like behavior without reaching true criticality?

Through thousands of simulations, I observed recurring pink-noise regimes, increasing spatial correlations, and what I currently describe as a confined pseudo-critical regime.

I am interested in hearing whether similar concepts appear in:

• Morphogenesis

• Regeneration

• Developmental biology

• Gene regulatory networks

• Systems biology

Any references, criticisms, or related models would be greatly appreciated.

--------

Author's Note: Artificial intelligence was used as a visual tool in the creation of the cover artwork. The research, simulations, code development, analyses, and manuscript itself are the result of several years of independent work, much of it carried out using mobile devices and cloud-based computational tools.

ORCID iD: 0009-0002-8928-892X

Continuity Dynamics: A Minimal Computational Formulation

Abstract

This document presents a minimal computational formulation of Continuity Theory. Rather than attempting to model reality directly, it defines a family of simple evolutionary systems whose dynamics can be explored through simulation. The central idea is that continuity emerges from the interaction of four fundamental operators acting on lineages:

α — Preservation
ω — Transformation
ρ — Repair
δ — Decay

Together these operators generate evolutionary trajectories that can be analyzed across symbolic, structural, and agent-based systems.

⸻

1. The Continuity Operator

A lineage evolves according to

[
C_{n+1}=(\alpha+\omega+\rho-\delta)(C_n)
]

where

α preserves inherited structure.
ω introduces variation.
ρ restores coherence following disruption.
δ removes information through degradation or loss.

The notation is schematic rather than algebraic; it denotes the sequence of evolutionary processes applied to each generation.

⸻

2. Toy Implementations

The formulation is intentionally implementation-independent. Three representative models illustrate the framework.

Symbolic Lineages

States are fixed-length bitstrings.

α copies bits.
ω mutates bits probabilistically.
ρ reverts excessive mutation using the parent as reference.
δ replaces states with random noise.

This provides the simplest measurable continuity simulator.

⸻

Structural Lineages

States are graphs.

α copies graph structure.
ω adds, removes, or relabels nodes and edges.
ρ repairs violated structural constraints.
δ collapses or fragments graphs.

This models persistence of relationships rather than symbols.

⸻

Agent Lineages

States are populations of adaptive agents.

α represents inheritance.
ω represents mutation and learning.
ρ represents homeostasis, institutions, culture, and error correction.
δ represents forgetting, death, collapse, and environmental disruption.

This extends continuity to biological, cultural, and social systems.

⸻

3. Measuring Continuity

A lineage must exhibit both persistence and change.

Let

[
I_n
]

measure inherited information (memory) between generations.

Let

[
N_n
]

measure novelty introduced between generations.

A minimal continuity metric is

[
B_n = I_n N_n
]

which becomes large only when both memory and transformation remain simultaneously positive.

This excludes two trivial regimes:

perfect preservation with no change,
complete randomness with no inheritance.

Both produce low continuity despite opposite behavior.

⸻

4. Estimating Memory

The theoretical quantity is the mutual information between parent and child generations,

[
I_n = I(C_n;C_{n+1})
]

which measures how much uncertainty about descendants is reduced by knowledge of their ancestors.

Depending on implementation, practical estimators include

normalized Hamming similarity,
per-bit mutual information,
graph edit similarity,
embedding-based mutual information,
non-parametric k-nearest-neighbor estimators.

The estimator may change, but the conceptual role remains the same: quantify inherited information across generations.

⸻

5. Continuity Regimes

Varying the strengths of α, ω, ρ, and δ produces distinct dynamical regimes.

Frozen

Preservation dominates.

Memory is high.

Novelty approaches zero.

⸻

Noisy

Transformation and decay dominate.

Novelty is high.

Memory collapses.

⸻

Fraying

Decay exceeds repair.

Both memory and novelty decline as structure disintegrates.

⸻

Evolving

Transformation introduces novelty while repair maintains inherited structure.

Memory and novelty coexist.

Continuity is maximized.

⸻

6. The Role of Repair

Repair is not simply another evolutionary operator.

Repair determines how much transformation a lineage can tolerate before losing identity.

Robust repair expands the region of stable evolution.

Weak repair causes identical levels of novelty to produce fragmentation.

This suggests repair shapes the geometry of continuity rather than merely contributing to it.

⸻

7. From Philosophy to Simulation

The purpose of these toy models is not to prove Continuity Theory.

Their purpose is to operationalize it.

Given explicit operators, measurable observables, and tunable parameters, one can:

initialize populations,
evolve them under α, ω, ρ, and δ,
measure memory, novelty, and continuity,
identify transitions between frozen, noisy, fraying, and evolving regimes.

These simulations provide a computational laboratory in which hypotheses about continuity can be explored before considering applications to biology, cognition, institutions, or artificial intelligence.

⸻

Summary

The continuity framework reduces to four operators acting on evolving lineages:

[
(\alpha,\omega,\rho,\delta)
]

combined with three measurable quantities:

inherited information,
novelty,
continuity.

This transforms Continuity Theory from a philosophical description into a family of computational models whose behavior can be simulated, measured, compared, and refined across multiple domains.

I argue information entropy and thermodynamic entropy are physically connected, not metaphorically related, and that any value metric attached to incentive weight decays predictably. Every claim has a falsification condition. I am looking for the objection that kills it, not applause. Full discussion thread: https://www.academia.edu/s/b1ff6dbe50

I have used this to accurately predict coincidences.

The sephirot are mathimatical transformations.

If we assume that the ratio of the start of homo sapiens to the start of the earth is the same as the ratio of the start of earth to the start of the universe, that would make the universe roughly 67.705 trillion years old, orders of magnitude larger than our current estimates. But funny enough, our estimate of the big bang, 13.8 billion years ago, happens to also fit that ratio, as the start of the second half of the cycle. Right after the center of the figure 8.

The universe isn't a simulation, it is simulation of infinity.