Introducing Capital Tensor: From Money-as-a-Tensor to a Live Module

Capital Tensor is the second module in the Vortex Research Suite. Where Diagnostics scores a single asset's structural health, Capital Tensor steps up one level of abstraction: it asks how capital is distributed across a panel of (group, cohort) pairs, where the equilibrium of that distribution sits, how the cells couple, and whether the system is mean-reverting.

The framework grounding the module is the Money-as-a-Tensor paper. For DeFi the panel is 5 protocols × 4 wallet classes; for crypto it is 6 categories × 5 cap tiers; for equity it is 4 sectors × 3 cap tiers. Each weekly observation is a matrix M(t), and the module fits six interrelated objects on top of it: M* (the time-mean equilibrium), K (cross-pair coupling on z-scored deltas), A (the Jacobian linking ΔM(t+1) to M(t)−M*), λ(A) (eigenvalues with α(A) = max real part), Γ (per-pair mean-reversion friction), and V(t) (the Lyapunov stress curve under counterfactual policies).

Each object answers a different question. M* asks 'where is capital tending to sit, on average, over this window.' K asks 'when capital leaves cell i, where does it go — and which cells share a common driver.' A and α(A) ask 'is the panel mean-reverting on this window — does the residual decay in expectation.' Γ asks 'how fast does the residual decay per pair.' V(t) asks 'how much energy does the residual carry, and how does it dissipate under three counterfactual policy regimes.' The Elite-only policy simulator lets the user re-run the V(t) integration with their own damping coefficients.

The most opinionated step in the pipeline is subtracting M* before any further analysis. A weekly capital panel is non-stationary almost by construction — protocols grow, sectors rotate, fashion cycles. Subtracting an empirical mean removes the largest non-stationary component, so the residual M(t) − M* is roughly mean-zero on the observed window. By construction M* is the time-mean over the fit window, not the long-run equilibrium. Re-fitting on a different window gives a different M*. The module surfaces this caveat directly: every snapshot carries data_through and computed_at fields so the reader can see exactly which window the equilibrium was measured against.

What Capital Tensor doesn't do is forecast the panel. α(A) < 0 says the residual decays in expectation under the fitted dynamics on this window — it does not say the next window will be stable. α(A) ≥ 0 is a near-critical or unstable fit on this window and should be treated as a diagnostic of the window, not a prediction. The module's value is in making the structure of the panel legible — coupling clusters that consensus doesn't see, friction asymmetries between protocols, the gap between no-policy and targeted-damping recovery times — not in extrapolating it forward.

The full mathematical treatment lives in the Money-as-a-Tensor paper (linked from /research) and the operational reference is on the methodology page at /methodology.