How Three Manifolds is Produced

Methodology, data sources, AI usage, and human review process for the Three Manifolds — Weekly Market Reading.

Looking for the mathematical foundations of the measurement? Read Your Portfolio Lives in a Space You Never See → For the same measures applied to a fund, see the Socrate II case study →

At a glance

The Three Manifolds weekly reading flows through six explicit gates. Gates 1–3 are deterministic (no AI). Gate 4 uses AI assistance for prose drafting only. Gates 5–6 are manual human steps.

DATA → COMPUTATION → READING → DRAFTING → REVIEW → PUBLICATION
 G1       G2          G3        G4          G5       G6
deterministic      deterministic   AI-assisted   human   human

Updated 27 September 2026 — the definitions of the Topological Survival Score, the Ricci Tomography index and the Flash Condition Index are restated in plain statistical terms; the description of the base no longer refers to scalar threads; the reproducibility section lists the working papers as background. Computations are unchanged. These rules supersede those in force before 27 September 2026; readings published earlier are left as published.

Why a manifold at all

Correlation matrices have a natural home. A correlation matrix is not an arbitrary table of numbers: it is symmetric and positive definite, so it lives on SPD(n), the space of symmetric positive-definite matrices. Working on that space has two practical consequences.

First, everything computed on it remains a valid correlation structure: an average of market states, a path between two states, or a stress scenario scaled up from an observed shock. A straight-line extrapolation between correlation matrices can produce a matrix that no market could generate; a computation on SPD(n) cannot, and the platform states how far a scenario can be amplified before it stops being a possible market state.

Second, changes are measured on the scale that matters for diversification. A move from a correlation of 0.10 to 0.30 and a move from 0.70 to 0.90 have the same size on a ruler that measures differences, but the second one destroys far more diversification; measurements on SPD(n) register that difference.

Every instrument described on this page is computed on SPD(n) with the same operations at every reading. The instrument describes the dependence structure of a universe and how it moves; it does not forecast.

Gate 1 — Data Ingestion

Inputs: end-of-day market data only, sourced from publicly available feeds:

  • US equity sector ETFs (SPDR family) — daily close prices
  • European sector indices (STOXX 600 super-sectors) — daily close
  • Spot FX rates (12 major pairs) — daily close
  • Short-term rates (3-month proxies, 10 instruments) — daily close
  • CPI series (9 country indices) — monthly, with daily forward-fill
  • Commodities and stress instruments (9 instruments) — daily close
  • US macro indicators (11 series, including industrial production) — official release cadence

The base. Beneath the four panels, the base is read from 13 macro-liquidity coordinates at their native frequency — US and euro real rates, breakevens, curve slopes, VIX, MOVE, VSTOXX, the dollar and the yuan, Fed net liquidity, the ECB deposit facility — sourced from OECD (composite leading indicator), FRED (Treasury yields, breakevens, high-yield spread, Fed balance sheet), the ECB Data Portal, the Bundesbank (Bund yields) and EODHD (FX, indices). Monthly series enter as slow modifiers, never as daily coordinates; nothing is interpolated.

Reading date. Some inputs are published one business day late — US Treasury yields and the high-yield spread on FRED, business days only. At the nightly run, the latest row of a panel that holds them carries a copy of the previous close for those instruments. A reading built on such a row is provisional: the honest output of the data as it stood, nothing patched, flagged when at least two instruments of a block, or five of a panel, were copied. Once the next run has seen the actual prints, the same date reads settled and its geometry no longer changes with new data. The calendar follows: the platform shows the last settled session (D−1) and the run day as preliminary; the Sunday weekly reads the last settled session, Thursday; Friday’s geometry is published on Tuesday. Every reading states its data date. One declared exception: the FCI is normalised on its whole history, so its past values can still move by about a hundredth between runs until its reference window is frozen.

No proprietary feeds. No labelled crisis dataset. No per-input tuning. Every reading is reproducible from the same public sources by any user with the same lookup window.

Universe: 77 nodes across 7 structural families on the macro panel. 11 S&P 500 sectors (13 nodes including joint commodity references). 19 STOXX 600 super-sectors. 99 names on the technology panel — extended from 52 on 26 August 2026, when its Block Registry was frozen (week-on-week panel-level comparisons across that recomposition are not published).

Gate 2 — Geometric Computation (deterministic, no AI)

The covariance trajectory of the universe is computed on a rolling window, then normalised to a correlation matrix before any measure is taken — R = D⁻¹ Σ D⁻¹, where D is the diagonal matrix of volatilities, with a small ridge ε·I enforcing strict positive definiteness. The result is a sequence of points on the manifold of symmetric positive-definite matrices, denoted SPD(n).

Covariance and correlation. The covariance matrix records co-movements in the assets’ own volatility units: its diagonal carries the variances, and doubling one asset’s volatility doubles its whole row and column. The correlation matrix is the same object with the volatilities normalised out — diagonal fixed at 1, off-diagonal entries in [−1, 1] — so only the co-movement structure remains. The instrument takes its measures on the correlation matrix for one reason: the reading must capture structure, not volatility level. On correlation matrices the trace is fixed at n, so any move away from the identity necessarily spreads the spectrum — Dp, the TSS and the anisotropy read the dispersion of the spectrum and nothing else. On raw covariances, a uniform rise in volatilities would inflate the same metrics without any change in co-movement; the normalisation removes that artefact.

Operations applied (same operations every reading, no per-crisis tuning):

  • The base — market conditions — the 13 coordinates are compared through a Mahalanobis metric frozen on 2015-2019 (covariance and thresholds estimated once, never re-fitted on the data they judge). Eight states name the day: four cycle states from the OECD leading indicator as it was known at the time, and four exception states (QE / central-bank support, China credit, Shock / fast move, Trade / FX) that override the cycle when their condition holds for five trading days. Velocity is the Mahalanobis norm of the five-day move; above its frozen Q95 it reads as a fast move. The base describes conditions; it never triggers timing and is not a forecast.
  • Papadopoulos Distance (Dp) — the distance between the current correlation structure and the identity matrix, the structure of fully independent assets, on SPD(n). Current implementation: the Log-Euclidean distance to the identity matrix, Dp(Σ) = ‖ ln λ(Σ) ‖₂ = √( Σᵢ ln²(λᵢ) ), where λ are the eigenvalues of Σ. Dp = 0 means independent assets; a rising Dp means the structure is moving away from independence. Dp measures position — where the structure stands, not how it is moving.
  • Geodesic Distance (D) — the geodesic length of the one-session step: how far the covariance structure moved on SPD(n) in one trading day, between two consecutive readings. Current implementation: D(t) = ‖ ln λ(Σ(t−1)⁻¹Σ(t)) ‖₂. Informally, D reads as the day’s pace of displacement of the covariance structure; the measured object itself is a length. Dp says where the structure stands; D says how large the latest one-session step was — the word ‘velocity’ belongs to the Geodesic Velocity ṁ, the rate of change of Dp.
  • The Atlas — blocks and System — large universes are read at two levels: each economically coherent block on its own recent window (its Topological Survival Score, its deformation, its lockstep alarms — causal thresholds, each asset compared only with its own past), and a System reading across blocks measuring whether they still move independently. Each block tile in the Atlas also carries the percentile of its current TSS* against the block’s own history, tagged record high above the 98th percentile and record low below the 2nd — the colour answers ‘how evenly spread now’, in absolute terms; the percentile answers ‘compared with the block’s own past’. An attribution module shares the alarms out across blocks over the trailing month. The Two-Prices framework was retired on 25 August 2026.
  • Topological Survival Score (TSS) — the ratio of the geometric mean to the arithmetic mean of the eigenvalues of the panel’s correlation matrix, expressed as a percentage: TSS = (GM / AM) × 100. For a full-rank correlation matrix this equals 100 · det(R)^(1/n), a classical measure of spectral sphericity. It reaches 100% when every direction of the structure carries the same variance and falls as the spectrum becomes uneven — whether because one common direction dominates or because some assets become near-duplicates of each other; the smallest eigenvalues weigh most in it. It describes the structure over the trailing window; it is not a forecast. TSS* divides the raw score by its expected value when the assets are independent, given the panel size and window (TSS* = TSS / TSS_ceiling); it can slightly exceed 1. TSS* removes the mechanical effect of the estimation window; across panels of very different size, compare each panel with its own history. The zone (Singularity below 0.50, Tension below 0.80, Isotropic above) is a descriptive band fixed in advance on TSS*; it is not a probability and not a signal. Reference: URF-2, SSRN 6140809.
  • Flash Condition Index (FCI) — a purely geometric composite of four equal-weight components: geodesic displacement, a spectral dispersion index, structural anisotropy and geodesic acceleration. It does not use price movements. A high FCI means the four components are high together; it is a descriptive reading of the structure, not a forecast of any market event. This is not the Financial Conditions Index of Goldman Sachs or the Chicago Fed, which measure different quantities by different means. Spectral Entropy — Shannon entropy of the normalised eigenvalue distribution.
  • Principal component decomposition on the contagion network — identifies the dominant variance direction
  • Contagion ring classification — sorts each component into CORE / INDUCED / PERIPHERY from the rank of its current reading (the z-score of its factor score against its own history) among the directions the panel can resolve — those above the resolution floor of its spectrum, the Marchenko-Pastur edge computed from the panel size and window and never tuned. Unresolved directions are periphery by rule and never name the epicenter; they stay in every computation (TSS and Dp are read on the full spectrum, nothing reduced). The epicenter is the resolved direction with the most abnormal reading; it is labelled by assignment, with the assets carrying it stated. It describes where the week’s largest deviation sits, not a cause. Links are shared membership between components — never a flow: components are orthogonal, and an epicenter orthogonal to the market factor reads as a rotation inside the panel, not as a systemic move. Its systemic norm is normalised within its own panel (the most concentrated component reads 1.00) and compares across dates on the same manifold only.
  • Ricci Tomography — a redundancy scan of the panel’s rolling correlation structure. For each principal direction i, a spectral index Rᵢᵢ = −½ Σⱼ≠ᵢ [ (ln λᵢ − ln λⱼ) / (λᵢ − λⱼ) ]² computed from the eigenvalues: the more negative, the thinner the direction, that is, the more the assets carrying it behave as near-duplicates of one another. The off-diagonal entries show how the directions share the same assets. Because estimation alone thins the smallest directions most, a raw Rᵢᵢ is not comparable across ranks: Ricci* restates it as a multiple of what a panel of the same size and history would show under a one-factor null (1× = no structure beyond the market factor). The scan then reads each direction against its own 60-session history and assigns one of four states, thresholds fixed in advance: liquefaction (z below −2.33 — the direction is thinning unusually fast, its assets merging into one line), tension (z below −1.64), resilient (in range), easing (z above +1.64 — assets that moved together dissociate). The assets carrying each direction are shown with it. Warm-up: 250 sessions before the first state is read. Never read a raw Rᵢᵢ across ranks: read Ricci* and the state. A panel with no structure beyond its market factor shows no liquefaction at all — that is the test the null enforces. The index is a function of the eigenvalues; it is not the Riemannian curvature of SPD(n).

No machine learning. No backtesting on labelled outcomes. The framework is descriptive: the same deterministic computations at every reading.

Three distinct measurements. Dispersion (σ), zone (position) and epicenter (location of the largest deviation) measure different things on the same manifold. The epicenter is a surprise of return, never a level of price; a direction can be unresolved (below the resolution floor) and still enter every measure of the panel — resolution decides who may name the epicenter, nothing else.

Computation cost: 15-minute daily batch on standard infrastructure. No GPU required for the production stack.

Named instruments and tests

EconoSysmographe™ is a production instrument that records the dependence structure of an asset universe continuously, the way a seismograph records ground motion — it does not forecast, it registers. At each reading it computes the correlation trajectory of the universe as a sequence of points on SPD(n), and reads it as an atlas: each block’s Topological Survival Score, deformation and lockstep alarms, the System across blocks, the attribution of alarms, and the contagion topology showing which directions and assets currently move together most strongly. The same operations are applied at every reading, with no per-crisis tuning. Operated as a continuous dated series since May 2026 and published weekly. Developed by Trident-AI and commercialised by SmartGreenInvest Ltd.

Trident-AI Engine — the computational engine powering EconoSysmographe™. Current version: v3.0. Operates on four live panels: Global Systemic Composite SPD(77), S&P 500 SPD(13), STOXX 600 SPD(19), Technology SPD(99) — above one base, the market’s state space of 13 macro-liquidity coordinates. Every panel is read on its full spectrum: as many components as assets, nothing reduced.

Gate 3 — Reading Synthesis

Each daily computation produces four dashboards:

  • Macro Sysmographe — gauge, timeline, contagion network, principal-component decomposition
  • S&P 500 Sysmographe — same structure, 11 sector ETFs
  • STOXX 600 Sysmographe — same structure, 19 European super-sectors
  • Technology Sysmographe — same structure, 99 names, read through the Atlas (nine blocks and a System reading)

Each dashboard exports both a visual (PNG) and a structured JSON with all numerical outputs. The JSON is the canonical record; the PNG is for human and downstream agent consumption.

Gate 4 — AI-Assisted Drafting

This is the only gate where AI is used.

What the AI does:

  • Reads the JSON exports + the dashboard screenshots from the current week
  • Reads the previous issue’s article (for continuity)
  • Fills the canonical article template with this week’s data
  • Writes the interpretation paragraphs in the “we observe” register

What the AI does NOT do:

  • It does not compute the signal — Gates 1–3 are deterministic and run before the AI is invoked
  • It does not decide which sectors to highlight — the dashboard JSON pre-determines that
  • It does not produce directional recommendations — the system prompt forbids “buy / sell / underweight / overweight / target price / outperform” vocabulary
  • It does not publish — Gate 6 is manual

Bounded by an explicit system prompt: the AI agent operates under a versioned instruction set that defines the canonical structure, the language register, the slug convention, the disclaimer rules, and the data-integrity rules. Any deviation by the agent is caught at Gate 5 and the system prompt is updated.

Gate 5 — Human Editorial Review

Every draft is reviewed by the author (Evangelos Papadopoulos, Independent Researcher) before publication.

What the review covers:

  • Cross-check every numerical value in the article against the JSON exports from Gate 3
  • Verify the register stays “we observe” — no predictive or directional language
  • Verify all factual claims requiring external sources against publicly available references
  • Verify continuity sentences vs the previous issue (no contradictions, no stale framing)
  • Verify that any figure carried over from a previous issue is recomputed, not reused
  • Approve or revise

Time budget: approximately one full day per week (typically Saturday afternoon and Sunday morning). The human time is the quality bottleneck — and that bottleneck is intentional.

Corrections: where a published reading is later found to be overstated, the correction is published in the following issue with the mechanism explained, not silently amended.

Gate 6 — Manual Publication

Publication is never automated.

  • Article uploaded to WordPress by hand
  • Distribution posts to LinkedIn published by hand
  • Inbound institutional inquiries handled via the contact form at econosysmographe.com/contact

Publication window: Sunday 12:00–14:00 Paris time, weekly.

The manual step is deliberate — the friction forces a final read-through, prevents auto-publication of a flawed draft, and creates an audit trail at the publication moment.

What this means in practice

  • The signal you see in the dashboards is 100% deterministic. No AI, no labelled training, no per-crisis tuning. Reproducible by any user with the same data window.
  • The prose you read in the article is AI-drafted, human-reviewed. The writing is assisted; the analytics are not. The author retains editorial accountability.
  • Publication is human-gated. Nothing reaches the public surface without explicit human approval.

On AI disclosure

We use AI in Gate 4 because that is what allows a one-person research operation to maintain a weekly cadence at institutional quality. We disclose this here, on a public methodology page, rather than on every article footer — because the AI is bounded to drafting and the signal itself does not depend on it. Hiding the AI usage would be dishonest. Mentioning it on every article would be misleading the other way (it would suggest the analytics are AI-generated, which they are not).

The positioning of Econosysmographe™ rests on three pillars: a deterministic and reproducible methodology, AI-assisted production discipline, and human editorial accountability. We treat the three as inseparable.

Beyond the weekly reading — the URF business application

The deterministic stack described above (Gates 1–3) also powers a separate institutional application currently in private staging. This application allows authorised users to interact with their own portfolio readings through a conversational analytical layer, ask the dashboards specific questions, and test stress scenarios on their allocations.

The application is not publicly accessible at this stage. Institutional evaluation access is available on request to risk officers and asset managers who want to assess the system on their own portfolio. Contact: contact@econosysmographe.eu.

Reproducibility

Every reading is reproducible from end-of-day public data using the definitions on this page. The research background of the framework is documented in working papers (not peer-reviewed); where a definition on this page differs from a paper, this page is the reference:

  • URF-1 — SSRN 6055375. Geometric foundations: the Papadopoulos Distance, the Flash Condition Index, spectral entropy, geodesic velocity, structural anisotropy.
  • URF-2 — SSRN 6140809. The Topological Survival Score and the zone classification.
  • URF-3 — SSRN 6400438. Two-Prices (historical, retired on 25 August 2026), residual stress and the contagion topology.
  • URF-4 — SSRN 6566658. The geometry of risk across a 26-year sample.
  • The Minsky Singularity — SSRN 6212120. A standalone working paper, outside the URF numbering.

Educational positioning

The Three Manifolds Weekly Market Reading is educational material. SmartGreenInvest Ltd (Reg. England & Wales No. 14636473) is not an FCA-authorised firm and does not provide investment advice, recommendations, or solicitation to invest. The structural readings are research outputs intended to inform institutional risk-management discussions, not to substitute for them.

Readings describe the dependence structure of historical price data. They are not forecasts. The framework describes how concentrated or spread out a structure is and how it has moved; it does not state which way it will move next, when, or whether it will.

For institutional evaluation discussions or investor conversations, contact: contact@econosysmographe.eu.

Page maintained by Evangelos Papadopoulos. Last reviewed: 27 September 2026. Next scheduled review: 1 November 2026 (Q4 update) or earlier if the pipeline changes.