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.
One of those charts is the : five faculties of the knower meeting five aspects of the knowable. It is the method plate made visible — twenty-five couplings, a veridical spine, and an honest empty corner for what the grid cannot name.
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
The Intersection of Knower and Knowable
Five faculties meet five aspects of the object. The resulting twenty-five couplings are scored by κ; the diagonal is the veridical spine, where knowing is direct. Drawn as a plate you can read without opening a paper.
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.
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
A Geometric Framework for Neurodivergence
Autism, ADHD, dyspraxia and the rest are usually treated as separate diagnoses that happen to overlap. This paper proposes something simpler: they’re neighbouring regions of a single space, built from five essential capacities — how you attend, how you organize time, how you feel and sense, how you read other minds, and how you move. Seen this way, the frequent overlap stops being a puzzle and starts being the point — and every claim it makes is one a study could prove wrong.
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
Multi-AI Router
The piece that decides which model answers. A transparent, rule-driven router that sends each request to the right model by task and by cost, logs every decision so “why did it pick that?” always has an answer, and keeps a running tally of spend — built for regulated settings. Open source.
github.com/tposchel/multi-ai-router 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.
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
Epistemic Coupling Matrix · Pentarchic, cardinality-conforming
The Intersection of the Knower and the Knowable
Five faculties of the subject meet five aspects of the object. Each of the twenty-five cells is a coupling — the knower’s act engaging a face of the real — scored by the truth-coupling κ. The diagonal is the veridical spine: where each faculty couples the aspect that is its own twin, and knowing is direct. Everything off the spine is mediation.
knower ↓
knowable →
Metaphysicswhat is
Empiricalthe given
Axiologywhat weighs
Conceptualthe form
Epistemologythe access
Intellectnous
K
Direct acquaintance
Nous grasping being itself — insight into what is.
T
Abstraction
Universals drawn from instances; the inductive lift.
B
Moral intuition
Grasping the good by intellect; the is–ought seam.
T
A priori insight
Rational self-evidence; seeing that a form must hold.
B
Reflexive knowing
Knowing that one knows; the KK principle, contested.
Senseaisthesis
B
Veil of perception
Reaching being through appearance; phenomena vs. noumena.
K
Perception
Aisthesis receiving the given — the sensory boundary.
B
Felt value
The pleasant and the beautiful as sensed; aesthetic good.
T
Seeing-as
Perceptual concepts; the theory-ladenness of observation.
T
The given
Sensory warrant; foundationalism and the Myth of the Given.
Will · Affectorexis
B
Existential assent
Willing what is; belief in reality, the leap of faith.
B
Motivated attention
What one chooses to observe; evidence sought or avoided.
K
Valuation
Orexis weighing the good — desire meeting worth.
B
Conceptual commitment
Deciding to adopt a framework; the stable middle zone.
B
Ethics of belief
The will-to-believe; doxastic voluntarism, Clifford vs. James.
Reasonlogos
T
Metaphysical argument
Inferring being; cosmological and ontological reasoning.
T
Hypothesis
Theorizing the data; underdetermination, inference to best.
T
Practical reason
Deliberation; means to ends, the reasoning of action.
K
Inference
Logos structuring form — entailment and logical order.
T
Giving warrant
The logic of justification; how reasons become knowledge.
Judgmentkrisis
B
Ontological positing
The verdict on what exists; commitment to an ontology.
T
Weighing evidence
Assessment; setting credence to what the data bear.
B
Value-verdict
Evaluation; the judgment of worth, the ranking act.
T
Coherence check
Judging consistency; is the whole well-formed and sound.
K
Assent
Krisis coupling — the verdict itself, κ made act.
The score κ
◆
Veridical spine. Each faculty couples its own twin aspect. Knowing is direct; no mediation stands between.
K
Knowledge. Coupling realized and accessed — dom(κ) with warrant. The spine cells.
T
Truth-apt. A being-correlate is reachable here; the coupling can land, warrant pending.
B
Belief. The faculty is engaged but the coupling is unconfirmed, voluntary, or contested.
The reading
Knowledge is a coupling of two pentachora. The knowable presents five aspects; the knower brings five faculties to meet them. The matrix is the map κ : knower → knowable made explicit — twenty-five ways a subject engages the real.
The diagonal is knowledge proper. Intellect to being, sense to the given, will to the good, reason to form, judgment to access — each faculty at home in its own aspect. Direct acquaintance runs corner to corner.
The off-diagonal is where epistemology lives. The twenty mediated cells are the named problems: the veil of perception, theory-ladenness, the ethics of belief, underdetermination. Each is a faculty reaching an aspect not its own — oblique coupling, where truth must be earned and belief is the honest default until κ is confirmed.
Off the grid. The zero-vector lies outside all twenty-five cells — what needs a sixth aspect, or neither faculty nor object-face to couple. Per the framework’s self-bound, it is addressed as the null: the apophatic corner, well-formed as a position and empty as a coupling. The matrix specifies what it can specify and assigns the void to what it cannot.
κ-scores are recommendations — [I]; the two true-up merges (Language+Conceptual, Perception+Empirical) and the knower/knowable mirror are [D], reserved.
A.M.D.G.
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