Pentarchic Theory
Everything in this project starts here: a proposal that the deep structure of physics rests on a simple five-cornered geometric shape — and that a surprising amount of what we observe follows from that one seed.
Hello.
I’m Tim — an independent researcher living north of Atlanta. I spent my career building data and decision systems in regulated industries: credit risk, identity, payments, the machinery that quietly decides things all day long. I hold prior U.S. patents from that work. Alongside it, I have seminary formation in Catholic theology.
The research collected here is the meeting of those two lives. It began with a question in geometry and grew — carefully, and with the seams showing on purpose — into a connected body of work spanning physics, mathematics, analytical method, economics, and theology. I think of it as an atlas rather than a stack of papers: each document is a chart of one territory, drawn so the charts join at their edges.
A few working principles, so you know what you’re getting. Conditional results are labeled conditional. Failures are reported as failures, because failures are data. And like any honest mapmaker, I revise the plates as the surveying improves — the newest edition supersedes, but the history stays on record.
Companion works in theology and the humanities appear under the pen name T. Petrarch.
The corpus, as an atlas
Think of what follows as plates in an atlas. Each paper charts one territory; the plates share a projection, so they join at their edges. Some are published — those carry a DOI you can follow to the full text, free. Some are still on the drafting table — those are marked pending. Each gets a plain-language introduction below, because a map you can’t read isn’t much of a map.
Plate I
Foundations — physics & mathematics
The Relational Structure of Fundamental Constants
Why do nature’s constants have the values they do? This paper looks at the relationships among them and finds they fit a small geometric pattern rather than a grab bag of coincidences.
Epipentarchic Theory: Biogenesis, Noogenesis, Christogenesis
If geometry underlies matter, what about life and mind? This one climbs the ladder — physics to biology to consciousness — asking whether each new level repeats the same structural pattern at a larger scale.
The Prime-Forced Rectification Tower
The Riemann Hypothesis is mathematics’ most famous open problem. An earlier version of this work sketched a conditional route toward it; that claim has been withdrawn. What survives is sharper and more honest: the geometry yields a natural candidate operator with a real spectrum — the kind of object a proof would need — together with an obstruction showing why this particular route cannot reach the hypothesis as stated. A candidate and an honest wall, reported as such.
Character-Twisted Rectification Towers
A companion to the previous plate, extending the same approach to the Generalized Riemann Hypothesis. The same withdrawal applies, and the same kind of finding: a symmetry built into the construction rules out the structure a proof along this line would require. Reported as an obstruction, not a result — which is its own kind of progress.
Forward Mathematical Genesis
Can you build the structure of physics from nothing but the prime numbers and one free choice? This paper tries, step by step — and keeps score of exactly which steps are forced and which aren’t.
Pending
Fractal Pentarchic
Coastlines look similar whether you view them from ten feet or ten thousand. This paper asks whether the framework’s core pattern repeats the same way — the same shape, echoed across scales.
Pending
Pentarchic Paradoxes
Every framework worth its salt runs into paradoxes. This one collects them, looks them in the eye, and sorts out which are real problems and which only look like problems.
Pending
Plate II
Method — how the surveying is done
The Rectified Pentachoron Framework
The instrument behind much of the work: a disciplined way to break a complex subject into exactly five essential aspects and study how they interact — including how they fail together.
Pending
Binary Bias
We love sorting the world into two boxes — on/off, us/them, right/wrong. This essay is about why two boxes are so often the wrong number, and what to reach for instead.
Pending
The Abstraction Pentachoron
An application of the five-aspect method to the concept of abstraction itself — and the method holds. Abstraction resolves cleanly into five essential aspects, confirmed against the same discipline every other chart in this atlas has to pass.
Pending
Anatomy of a Fault Class
When systems fail, the failures come in families — and each family has an anatomy. A close study of one, for anyone who has ever asked why the same kind of thing keeps breaking.
Pending
Decision-Making
What are the essential ingredients of any real decision? An application of the five-aspect method that proposes an answer — and locates where decisions most often quietly go wrong.
Pending
Plate III
Applications — systems, markets, machines
The Shadow Simplex: Failure Modes in Self-Evolving Multi-Agent RL Systems
When AI systems learn and modify themselves, they can fail in patterned ways. This paper catalogs those failure modes geometrically — a map of what can go wrong, drawn before it does.
ACEL — a small language for describing AI agents
A companion to the plate above, given back freely and open source. It lets you define an AI agent by its five essential aspects — what it wants, how it perceives the world, what it remembers, how it decides, and how it acts — and it ships a checker that flags the aspect people most often leave out: whether the agent is actually grounded in the world, rather than confidently making things up.
srel.net Open source
The Five Invisible Hands of Emergent Macroeconomic Coordination
Adam Smith gave us one invisible hand. This paper argues a real economy is coordinated by five distinct mechanisms that must work together — and shows the recognizable trouble when any one goes missing.
The Lambda Curve: Return Profiles in Private Credit and Lending Strategies
A look at private credit and lending through the shape of their returns — why these investments behave the way they do over time, and what their risk actually looks like.
Plate IV
Faith & philosophy — the oldest territory
Magnifica Humanitas for the Catholic Technologist
A plain-language walk through Pope Leo XIV’s encyclical on artificial intelligence and human dignity — written for Catholics who work in technology and want to know what the Church is actually asking of them.
Finitude Confers Weight
A philosophical essay on mortality — arguing that the fact our time is limited isn’t merely a sad constraint, but the very thing that gives our decisions their weight.
The Five Sequences
The old Latin liturgy kept exactly five great hymns called sequences. This paper asks whether those five form a complete set — whether, together, they cover the whole territory they were written to sing.
Pending
The Five Involutions
A meditation on the five wounds of the crucified Christ, read structurally: why five, why those five, and what the pattern holds together.
Pending
The Shadow Pentachoron
An old theological idea — that evil isn’t a thing in itself but a corruption of good — given a geometric treatment. The surprising conclusion: the parasitism of evil on good can be stated as a theorem, not just a metaphor.
Pending
Get in touch
Questions, corrections, and serious disagreement are all welcome — a map improves fastest when other people check it against the terrain.
Email
tim@poschel.us
ORCID
0009-0008-0265-9761