Glossary

The canonical definitions of record for EconoSysmographe™ and the Universe Risk Framework. Any other formulation attributed to these terms is not canonical.

Finance does not live on a flat space. A correlation matrix is not a point in a vector space: it lives on SPD(n), which is curved. On a curved space the shortest path between two regimes is a geodesic, not a straight line, and the midpoint between two correlation structures is not their arithmetic average. Most standard risk tools assume a flatness that is not there, and the error they introduce grows precisely when the structure moves — which is when the measurement matters.

On this page

Foundation

SPD(n)

The manifold of symmetric positive-definite matrices — the space in which correlation and covariance matrices actually live. It is curved, not flat. Distances, midpoints and averages on it are geodesic quantities, not arithmetic ones. Every metric in this glossary is defined on SPD(n).

Core metrics

Topological Survival Score (TSS)

Formula. TSS = (GM / AM) × 100

The ratio of the geometric mean to the arithmetic mean of the panel’s covariance spectrum, expressed as a percentage. It reaches 100% when risk is spread evenly across every direction of the structure, and collapses toward zero as the spectrum concentrates onto a few directions. A high TSS means the panel survives stress well; a low TSS means risk has collapsed onto a single direction.

Reading. A low TSS means high systemic stress. A high TSS means a healthy topology.

The attainable maximum of the raw TSS depends on panel dimension: estimation noise mechanically depresses the score as the number of assets approaches the estimation window. Comparability across manifolds is restored by the dimension-normalised reading TSS* = TSS / TSS_ceiling, in [0, 1] — see TSS Dimensional Ceiling.

Comparability. Raw TSS levels are not comparable across manifolds, and neither are zones computed on the raw scale. TSS* is. Intra-manifold deltas compare on either scale.

Reference: URF-2, SSRN 6140809.

TSS Dimensional Ceiling

Formula. TSS_null(n, w) = 100 · exp( (1/n) · Σ ψ((w−i)/2) + ln(2/w) ) — ψ is the digamma function, n the number of assets, w the estimation window. Defined only for w > n.

The attainable maximum of the raw TSS for a given panel: the expected TSS when the true correlation is the identity, given the resolution of the estimator. The ceiling shrinks with panel dimension — a 13-asset panel on a 60-day window tops out near 88, a 52-asset panel near 49. Dividing the raw TSS by its ceiling yields the dimension-normalised reading TSS*, in [0, 1], which compares across manifolds. The ceiling is undefined when w ≤ n: the sample matrix is structurally singular and the panel must be reconfigured, not corrected.

Reading. TSS* close to 1 means the panel is as diversified as its estimator can resolve; close to 0 means the spectrum has collapsed. As of August 2026, no panel measured by the instrument has reached Isotropic levels (TSS* ≥ 0.80): a common market factor has always been present. The zone functions as an asymptotic reference.

Engine implementation validated against Monte-Carlo simulation.

Papadopoulos Distance (Dp)

Role. The distance between the current correlation structure and perfect diversification on SPD(n).

Current implementation. Dp(Σ) = ‖ ln λ(Σ) ‖₂ = √( Σᵢ ln²(λᵢ) )

The Log-Euclidean distance from a covariance structure to the identity matrix on SPD(n): λ are the eigenvalues of Σ. The identity matrix is the point of perfect diversification, so Dp measures how far the structure has moved away from it. Dp measures position — where the structure stands, not how it is moving.

Reading. Dp = 0 means perfect diversification. A rising Dp means the structure is moving away from equilibrium, approaching the Singularity Wall.

Geodesic Distance (D)

Role. 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)⁻¹ Σ(t)) ‖₂

λ are the generalised eigenvalues of the pencil (Σ(t), Σ(t−1)). D is unsigned: it sees every movement of the structure — including movement at constant Dp.

Reading. A low D means stability — the structure barely moves. A high D means a regime transition is in progress. Informally, D reads as the day’s pace of displacement of the covariance structure; the measured object itself is a length.

Distinction. Dp says where the structure stands; D says how large the latest one-session step was; the Geodesic Velocity ṁ says how fast Dp itself is drifting. Three measurements, three questions — the word ‘velocity’ belongs to ṁ, never to D.

Reference: URF-1, SSRN 6055375.

Two Rates (ρ)

Formula. ρ = λ_real / λ_geo

The successor of the Two-Price theory — the Two-Prices name was retired on 25 August 2026. Two Rates compares two speeds measured on a block’s correlation trajectory: the rate the geometry would permit (curvature bound) and the rate the market actually realises. Regimes are defined on their ratio ρ — compression (ρ < 0, reconvergence), suture (0 ≤ ρ < 1, absorption dominates: the market’s normal state), rupture (ρ ≥ 1, structure rewriting faster than it absorbs). Full definitions and measured occupancies are published with the URF-3.3 revision; ρ is research-grade and is not a timing signal.

Naming. Two Rates supersedes Two-Prices, retired from the product on 25 August 2026. Full definitions are published with the URF-3.3 revision. Earlier papers that use the Two-Prices name remain valid as historical documents.

Flash Condition Index (FCI)

Purely geometric composite of four equal-weight components. It does NOT use price movements. A high FCI indicates the structure approaching a flash-crash regime.

Reading. A high FCI indicates the structure approaching a flash-crash regime.

Reference: URF-1, SSRN 6055375, §3.8.

Structural states

TSS zone

Thresholds. Singularity · Tension · Isotropic — canonical values 50 and 80

The qualitative state attached to a panel’s TSS reading, assigned against thresholds fixed in advance. The canonical threshold values 50 and 80 are conserved across methodology versions; methodology v2 (August 2026) applies them to the normalised scale — TSS* below 0.50 Singularity, 0.50–0.80 Tension, 0.80 and above Isotropic — instead of the raw scale, because the attainable range of the raw scale depends on panel dimension. Published readings state which scale they use.

Reading. Isotropic is a healthy panel; Singularity is isotropic collapse. The zone follows the TSS and inherits its direction — a high score is the healthy end.

Note. The console adds two display refinements that are not zones: an alert state below 15, and a borderline marker within two points of a boundary.

The base — market conditions

Base (Market State Space) (B_c)

Formula. d(v1, v2)^2 = (v1 - v2)^T Sigma_B^-1 (v1 - v2)

The market’s state space beneath the manifolds: 13 macro-liquidity coordinates (US and euro real rates, breakevens, curve slopes, VIX, MOVE, VSTOXX, FX, Fed net liquidity, ECB deposit facility), measured at native frequencies and compared through a frozen Mahalanobis metric. The manifolds (SPD panels) are the fibres above it. The base describes conditions; it never triggers timing. VIX and VSTOXX are the equity implied-volatility indices (US, euro); MOVE is the ICE BofA MOVE Index — the bond-market equivalent, implied volatility of US Treasury yields from one-month options; its 20-day jump is an s7 condition but its level is excluded from the STRESS flag (the MOVE level left its 2015-2019 regime for good in 2022).

Base states (s1-s8) (s1-s8)

Eight frozen states of the base. The cycle states s1 Recovery, s2 Expansion, s3 Downturn, s4 Contraction come from the published OECD CLI phase (level vs 100 x 3-month direction, read at its ~6-week availability, never earlier). Four exception states are computed from the base’s own daily dynamics and override the phase; when several are true the same day, the most acute one labels the day — Shock / fast move > Trade war / FX > China credit > QE (codes s7 > s8 > s6 > s5): s7 Shock / fast move (velocity or 20-day VIX/MOVE jump above frozen Q95), s8 Trade war / FX shock, s6 China credit engine, s5 QE / CB support. States persist 5 days; occupancy, dwell times and transitions are measured results, never inputs. QE stands for Quantitative Easing: central banks supporting markets by expanding their balance sheet; the s5 condition reads it directly from the plumbing — the Fed’s net liquidity (assets minus reverse repo minus Treasury account) or the ECB’s deposit facility growing over 13 weeks beyond its frozen Q85.

Reading. s7 reads ‘the base is moving beyond its Q95 speed’ – it is a speed state, not a crisis label; the separate STRESS flag (VIX above its frozen Q95 level) tells a Minsky-type episode from a fast move toward calm.

Slow layer / fast layer

The base is read on two time scales. The slow layer is the published OECD phase: confirmed context, known with a ~6-week publication lag, revised ex post. The fast layer is the daily base state, velocity and shock flags: the real-time signal. Between two CLI publications only the fast layer brings new information; that six-week window is the analysis object.

Base velocity

Formula. v(t) = || b_t - b_{t-5} ||_{Sigma_B}

How fast the base is moving through its state space, in units of its own frozen covariance. Velocity above the frozen Q95 (with the VIX/MOVE 20-day jumps) is the entry condition of the s7 Shock / fast move state. Velocity carries no sign: a pivot toward calm and a crisis can be equally fast.

The connection — base to fibre

Connection (base to fibre)

The rule linking moves of the base to moves of the fibre geometry. Measured finding (2026-08): the full linear map A(b) = d vech(log Sigma)/db is not identifiable at accessible frequencies; the architecture is therefore asymmetric, with four implementations: (1) discrete state conditioning (Frechet mean and dispersion per state), (2) strict engine separation (risk timing belongs to the fibre’s curvature alarm; the base is a regime filter), (3) the divergence module, (4) accepted scalar threads. Never a price or an execution signal.

Macro-implied path

Formula. Σ β·Δx + k·t

The path an asset’s log price would have followed if only its macro conditions had moved it: the frozen base-connection coefficients applied to the base coordinates (real rates, dollar), plus the in-sample drift. The reference line the Macro Premium is measured against.

Reading. Not a fair value and not a target: a measured reference built from frozen coefficients, against which the premium is read.

Macro Premium

Formula. Macro Premium = ln(spot/P0) − Σ β·Δx − k·t

The measured gap between an asset’s price and its macro-implied path — the path that macro conditions (real rates, the dollar) justify under the frozen base connection. Positive = premium above what conditions carry; negative = discount. Read against the percentile of its own history (normal / elevated / widest since anchor). The premium is the product, not a model failure: like a credit spread, it is the measured object. Diagnostic — never a forecast.

Reading. A premium in the top decile of its own history means the market is paying well above what macro conditions justify — a measured dislocation, with no timing claim attached.

Scalar thread (invariant connection)

Formula. delta m_t = k + beta . delta b_t

A frozen, versioned scalar relation between daily increments of the base and one invariant of the fibre. Validated by the pre-registered R-2c protocol (out-of-sample vs a circular-shift null, anti-tautology control without volatility coordinates): TSS* accepted on all three test fibres, Dp accepted where its own out-of-sample record is positive, FCI rejected. Out-of-sample R-squared is 1-4% daily and 5-9% at 5 days: real, replicated, and small. Label everywhere: measured conditioning – not a forecast.

Divergence (sigma) (sigma)

Formula. sigma_t = m_t^observed - m_t^base-implied

The part of the geometry’s move the base does not carry. Computed from an accepted scalar thread: the observed invariant path against the base-implied path, cumulated from a common anchor. A structural anomaly reads: fibre compressed (low TSS*, curvature alarm) while the base is relaxed (low velocity, no flags). Divergence contextualises; it does not time. Timing stays with the curvature alarm, which never uses these threads.

Parallel transport (AIRM)

Formula. Gamma_{A->B}(X) = E X E^T, E = A^{1/2} (A^{-1/2} B A^{-1/2})^{1/2} A^{-1/2}

The tool that moves a direction (a deviation) from the fibre over one point to the fibre over another without deforming it, so geometries above different base states become comparable. Measured findings (Phase 3): no jump at state boundaries (transitions are transport, not resets), no measurable holonomy (memory), and a transported deviation is forgotten within about 20 trading days.

The Atlas — blocks and System

Block

An economically coherent group of 8 to 19 assets (one family × region: US equity sectors, EU equity sectors, global indices, rates & credit, FX, commodities), derived deterministically from the canonical asset dictionary and frozen in a versioned Block Registry — never edited by hand. Geometrically, each block is a chart of the atlas: a sub-manifold small enough for every measurement to work (the full panel is not). Each block carries its own TSS*, Papadopoulos Distance and curvature alarms.

Reading. A block is where measurement lives. Block-level readings are comparable over time because the topology is frozen and version-stamped into every snapshot.

Atlas

The market manifold assembled from its blocks — the mosaic view. Each block is a chart of the atlas in the geometric sense: a local map where measurement works. The Atlas view shows every block (tile area proportional to block size, colour = TSS* zone, pulsing border = curvature alarm with its attribution share) inside the System ring. Each tile also carries the percentile of the block’s current TSS* against its own history, tagged record high above the 98th percentile and record low below the 2nd. The two scales answer two different questions: the colour answers ‘how concentrated now’, in absolute terms; the percentile answers ‘compared with the block’s own past’. Both are printed, deliberately — a block at a record high of its own range can still be concentrated in absolute terms.

Reading. The Atlas answers ‘where?’ at a glance: which blocks are structurally fragile, which are alarming, and whether the System holds. On each tile, the colour answers ‘how concentrated now’; the percentile answers ‘compared with its own past’.

System

The cross-block read. Each block is reduced to a single daily summary series; the System is the geometry of the correlations BETWEEN those series — it does not look inside the blocks (each block does that itself). It answers the one truly systemic question: are the blocks fusing into a single trade? It speaks with three voices: System TSS* (cross-block diversification), System Dp (cross-block deformation) and the System curvature alarm. The word ‘META’ is banned from every public surface.

Reading. High System TSS* = the blocks live their own lives — cross-block diversification intact. A collapsing System TSS* = the market is becoming one trade: the systemic signal no single block can see.

Attribution & HHI (HHI)

Formula. HHI = Σ s_b²

The attribution gauge says WHERE the alarms concentrate: each block’s trailing 20-day share of all curvature alarms, normalised to sum to one, plus their Herfindahl-Hirschman Index (the standard concentration measure used by antitrust regulators). HHI = 1: one block carries every alarm — a localised shock with a clear address. HHI toward 1/K: alarms spread across all blocks — a broad systemic event, and that dilution is itself the information. Quiet windows carry no attribution claim at all.

Reading. The HHI says whether the episode has an address or is everywhere. Attribution is strongest for localised shocks; broad shocks dilute concentration by nature.

Contagion structure

Contagion topology, epicenter and systemic norm

Contagion zones describe a component’s position in the stress structure, 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): CORE for the most abnormal readings, induced next, periphery for the rest and for every unresolved direction. The epicenter is the emitting resolved direction whose current reading is the most abnormal relative to its own history. It is labelled by assignment — each asset names one component and one only — so the label is not necessarily the largest weight; the assets carrying the direction are stated with it. Its systemic norm scores how concentrated that role is, on a 0–1 scale. Links on the map are shared membership: two components are linked when the same assets carry both, and the strongest links read as “who moves with the epicenter”. A link is never a flow: the components of one decomposition are orthogonal, and an epicenter can sit orthogonal to the market factor — that configuration reads as a rotation inside the panel, not as a systemic move.

Reading. Dispersion (σ), zone (position) and epicenter (source) are three distinct measurements on the same manifold. The epicenter is a surprise of return, never a level of price.

Comparability. The systemic norm is normalised within its own panel: the most concentrated component reads 1.00 by construction. It therefore compares across dates on the same manifold, and not across manifolds.

Reference: URF-3, SSRN 6400438. Rule in force since 5 September 2026.

Unresolved direction (resolution floor)

Formula. λ+ = σ² (1 + √(n/T))² — Marchenko-Pastur edge for n assets observed over T sessions; σ² re-estimated once below the first edge so the market factor does not inflate it

Any panel of n assets observed over T sessions shows small directions of co-movement even if the assets were independent. That is sampling error, and its largest possible size is the Marchenko-Pastur edge, computed from n and T alone and never tuned. A direction below that floor is unresolved: it cannot be told from chance, so it is periphery by rule and never names the epicenter. Nothing is added and nothing is removed: the direction stays in every computation — TSS and Dp are measured on the full spectrum, and the small eigenvalues are precisely what the TSS reads — and the floor moves with the spectrum every day, so a direction unresolved today can cross it tomorrow. Reducing the panel to its resolved directions would break the measures and introduce a tuned parameter; keeping every direction is a rule.

Reading. Unresolved is a statement about the instrument’s resolution, not about the data: no noise is added, and no signal is claimed below the floor. The console prints how many of a panel’s directions are unresolved today.

Comparability. The floor depends on n and T, so the count of unresolved directions is not comparable across panels of different size; the rule is.

Curvature and spectrum

Ricci Curvature (Ric(t)) and Ricci Tomography

On the panel’s rolling correlation structure, the diagonal Rii measures how much structure each principal direction i concentrates: the more negative, the more the assets carrying that direction move as one line. The off-diagonal Rij measures the geometric coupling between two directions. Because estimation alone bends 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 Ricci Tomography 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 collapsing now, its assets merging into one line), tension (z below −1.64), resilient (in range), easing (z above +1.64 — the direction regains diversification, assets that moved together dissociate). The assets carrying each direction are shown with it. Warm-up: 250 sessions before the first state is read.

Reading. Negative and falling means concentration along that direction; positive and rising means it is spreading out again. 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.

Reference: URF-1, SSRN 6055375. States and the one-factor null: methodology, Gate 2.

Scalar Curvature (R(t))

Formula. trace of the Ricci tensor

R close to zero indicates a liquid, healthy market. R falling sharply indicates systemic fatigue and accumulating structural fragility. A local minimum of R has historically preceded market dislocations.

Reference: URF-1, SSRN 6055375.

Spectral Entropy (H(t))

Formula. Shannon entropy of the normalised eigenvalue distribution

High H means risk is spread across many independent factors. Low H means risk is concentrated in few dominant factors — structural fragility, diversification collapse.

Reference: URF-1, SSRN 6055375.

Geodesic Velocity (ṁ(t))

Formula. rate of change of the Papadopoulos Distance

The rate of change of the Papadopoulos Distance: how fast Dp itself is drifting. Sudden increases in geodesic velocity signal instability and the onset of structural reconfiguration. Used in the FCI composite via the acceleration component.

Note. Not the Geodesic Distance D, which is the unsigned length of the one-session step — a displacement. The word ‘velocity’ belongs to ṁ.

Reference: URF-1, SSRN 6055375, Table 2.

Structural Anisotropy Index (Anis(t))

Formula. percentile rank of |Ricci_max − Ricci_min|

High anisotropy means directional distortion and localised stress — some factors collapse while others remain stable. Low anisotropy means a homogeneous, isotropic structure: true diversification.

Reference: URF-1, SSRN 6055375.

Geometric Inertia (IG)

High inertia identifies a structurally invariant asset; low inertia identifies curvature-breaking behaviour. Used to rank assets by their role as geometric anchors versus amplifiers.

Reference: URF-3, SSRN 6400438.

The instrument

EconoSysmographe

A production instrument that records the covariance geometry of an asset universe continuously, the way a seismograph records ground motion. It does not forecast; it registers. At each reading it computes the covariance 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 curvature alarms, the System across blocks, the attribution of alarms, and the contagion topology identifying which node transmits stress. The same operations are applied at every reading, with no per-crisis tuning. Operated as a continuous dated series since May 2026. Developed by Trident-AI and commercialised by SmartGreenInvest Ltd.

Trident-AI Engine

The computational engine powering Econosysmographe. 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.

Reading date — provisional and settled readings

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.

Reading. A number is never wrong for being provisional; it is not settled. Cite the data date, and hold the macro-context and FCI readings of a provisional day lightly — prices and equity blocks are unaffected.

Deterministic computation

No stochastic step and no machine-learning step enters the computation. Identical inputs always yield identical readings, so any historical reading can be re-run and verified independently. There is no per-crisis tuning: the same operations are applied at every reading, on every panel.

Eratosthenes test

A self-consistency test for stress-testing methodology. Around 240 BC, Eratosthenes of Cyrene measured the circumference of the Earth without leaving Egypt, by comparing the shadow cast at two known points and assuming an underlying geometry. The 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.

Reading. Where a backtest asks whether a model matched history, the Eratosthenes test asks whether the model’s own geometry is internally consistent — a question a supervisor can pose without any out-of-sample data.

Notes on naming

Two terms were published under other names in earlier issues of the weekly reading, and are corrected here.

  • The Topological Survival Score appeared as Topological Stress Score in issues 1 to 15. The published name inverted the reading: a high TSS is a healthy panel, not a stressed one.
  • Two-Prices was retired from the product on 25 August 2026 and is superseded by Two Rates — see the entry above. Earlier issues and URF-3 (SSRN 6400438) keep the historical name.
  • The Flash Condition Index appeared as Financial Conditions Index. It is not the Financial Conditions Index of Goldman Sachs or the Chicago Fed, which measure different quantities by different means.

Earlier issues are left as published. The definitions on this page are the ones of record.

References

  • URF-1 — SSRN 6055375. Geometric foundations: the Papadopoulos Distance, the Flash Condition Index, Ricci and scalar curvature, spectral entropy, geodesic velocity, structural anisotropy.
  • URF-2 — SSRN 6140809. The Topological Survival Score and the geometric zone classification.
  • URF-3 — SSRN 6400438. Why martingales cannot predict markets; Two-Prices (historical name, superseded by Two Rates), residual stress, geometric inertia and the contagion topology.
  • URF-4 — SSRN 6566658. The geometry of risk: geometric crisis detection across a 26-year sample.
  • The Minsky Singularity — SSRN 6212120. A geometric discovery of the economic termination signal (standalone paper, outside the URF numbering).

Full production methodology: econosysmographe.com/methodology

EconoSysmographe™ and Universe Risk Framework (URF)™ are proprietary technologies. Developed by Trident-AI, commercialised by SmartGreenInvest Ltd (Reg. England & Wales No. 14636473). Educational research — not investment advice. Last reviewed: 6 September 2026.