Research
The mathematics behind the readings, published and citable — and the weekly analyses where the discipline is applied to live markets, in public.
The Three Manifolds — weekly analyses
Every week since April 2026, we publish our structural reading of the market: what the manifolds show, against which thresholds, and what the reading cannot tell us. Dates and values are on public record before events unfold.
- The Sell-Off Hit Cyclicals, Not Technology. Market Structure Did Not Move.
- NVIDIA Printed $96 Billion. Warsh Warned on 3.7%. The Stress Sits in the Chip-Making Machines.
- Covid-19: What Market Geometry Revealed While Stock Indices Hit Record Highs
- $500bn secured on GPUs. Your screens say nothing moved — every transmission root did.
- The VIX says calm. The geometry says re-armament.
Published papers
The framework is documented in five research papers available on SSRN. They cover the geometric representation of markets, the stress measures computed on it, and their statistical behaviour.
- The Geometry of Risk: Predicting Financial Crises 332 Days Early Across a 26-Year Sample — SSRN 6566658, April 2026.
- Why Martingales Cannot Predict Markets — And What Riemannian Geometry Can Do Instead — SSRN 6400438, March 2026.
- Beyond Portfolio Death: Geometric Rupture Detection and the Cone de Evangelos — A Riemannian Critique of Lifecycle Investing and the 100% Equity Illusion — SSRN 6140809, February 2026.
- The Minsky Singularity: A Geometric Discovery of the Economic Termination Signal — SSRN 6212120, January 2026.
- The Universe Risk Framework: Geometric Foundations of the EconoSysmographe — SSRN 6055375, January 2026.
The framework explained
Three short videos: what the instrument measures, how a single model produces every view, and how the reading behaves through a correlation crisis.
Methodology in brief
The universe of listed instruments is represented as a geometric structure — a manifold — computed deterministically from public price data. On this structure, the platform measures each block’s Topological Survival Score and its zone, curvature-based measures that locate concentration, and a System reading across blocks that measures whether they still move independently. The same inputs always produce the same reading; every historical reading can be re-run and verified.
The Eratosthenes test
Around 240 BC, Eratosthenes of Cyrene measured the circumference of the Earth without leaving Egypt. He compared the shadow cast at two known points on the same day, assumed an underlying geometry, and derived a figure that held. He did not travel the distance; he checked that two independent measurements of the same object agreed.
The Eratosthenes test applies that logic to a risk model. The same regime transition is measured under two different metrics on SPD(n), and the two measurements are required to agree within a threshold fixed in advance. If they diverge, the model’s geometry is not internally consistent, and the divergence localises where.
What it answers that a backtest cannot. A backtest asks whether a model matched history. That question needs out-of-sample data, and it can only be asked once the history exists. The Eratosthenes test asks a different question — whether the model’s own geometry holds together — and needs no out-of-sample data at all. A supervisor, an allocator or an internal validation team can pose it on the model as it stands today.
What it does not do. It does not establish predictive power, and it does not certify a model. A model can be perfectly self-consistent and still describe the wrong thing. The test rules out one specific failure — a geometry that contradicts itself — and reports the threshold it was measured against. It is a diagnostic, in the same register as everything else published here.
The formal definition of record is in the glossary. The test is run as a scoped engagement rather than as a product tier — contact us to discuss whether it applies to your model.
Glossary
Manifold — the geometric representation of a market universe, built from public price data, on which all measures are computed.
TSS (Topological Survival Score) — the headline measure of remaining diversification capacity on the manifold, expressed as a percentage with defined regime thresholds. A low TSS signals high structural stress; a high TSS, a healthy topology.
TSS zone — the qualitative state attached to a panel’s TSS reading, assigned against thresholds fixed in advance and identical on every manifold: Singularity below 50, Tension from 50 to 80, Isotropic at 80 and above.
Atlas — the market manifold assembled from its economically coherent blocks: each block measured as a structure of its own, a System reading across them, and an attribution module locating where alarms concentrate. Two-Prices was retired on 25 August 2026 and is superseded by Two Rates.
Epicentre — the node of the contagion network where structural stress concentrates; surrounding nodes are classified as core, induced or periphery.
FCI (Flash Condition Index) — the flash-risk indicator computed on the structure; a reading above its stated threshold triggers a Flash Warning on the dashboard.
Entropy — the dispersion measure of the structure, tracked alongside the FCI on the daily timeline.
Deterministic computation — no stochastic or machine-learning step: identical inputs always yield identical readings, making every reading independently verifiable.
See the research applied, live.
In a 30-minute discovery session, we demonstrate the platform on current market data — the same measures described in the papers, running on this week’s market.