TN-020 — Eigenvalue Stability of the Finnish Energy–Economy System
Technical Note · Stability Regions and Phase Transitions
Domain D-1 · D-3 · D-4 · Version 0.1 (Open Working Draft) · June 2026
Builds on: CN-024 · CN-025 · TN-019 · SM-013
Abstract
This technical note reformulates the PPA allocation dynamics from CN-024 and TN-019 as a linearised stability problem. The central result: Finland's energy–economy system is not a single-equilibrium market but a multi-regime eigenstructure system, where PPA penetration acts as a structural operator that shifts eigenvalues — determining which actor groups absorb instability — rather than simply redistributing prices. Three dominant eigenmodes are identified: PPA compression, liquidity collapse, and institutional lag. Four phase regimes are defined, with the current Finnish system estimated to occupy Phase B (distorted equilibrium).
Status note: This is an open working draft. Key parameters ($\alpha, \beta, \gamma_i, \theta_c$) are structural placeholders pending calibration from Fingrid hourly data and Statistics Finland input-output tables. The qualitative phase structure is robust; quantitative thresholds are indicative.
§ 01 — System (Minimal Form)
The CN-026 macro–energy state space reduces to a coupled dynamical system. Linearising around equilibrium $x^*$:
$$\dot{\delta x} = A^* \delta x$$
where $A^*$ is the Jacobian of the coupled energy–economy system. Stability depends entirely on the eigenvalues of $A^*$:
Mode 3 — Institutional Lag Mode
$$\lambda_3 \approx \frac{1}{\tau} - \delta_{\text{policy}}$$
When $\tau$ exceeds the system's own adjustment cycle. Effect: policy responses overshoot; corrections amplify rather than dampen volatility; structural drift accumulates. Corresponds to SM-013's drift regime and D-suppression.
§ 03 — Eigenvector Allocation: Who Absorbs Instability
The eigenvectors of $A^*$ determine which actors are loaded by each instability mode. The system does not eliminate volatility — it assigns it:
$v_1$ (industrial sector) — insulated via PPA; near-zero loading on Mode 1
$v_4$ (households) — residual sink; SM-013 SurplusCitizen condition is the social expression of $v_4$ dominance
PPA does not change mean prices — it shifts eigenvalues, determining which eigenvector components carry instability. The observable consequence is not average price but distributional volatility: stable for Layer 1, amplified for Layers 3–4.
§ 04 — Phase Transitions
A critical surface in $(K_p, L)$ space determines regime transitions:
$$K_p \cdot \frac{1}{L} \geq \theta_c$$
Empirical calibration of $\theta_c$: CN-024 estimates PPA share reaching 30–50% of zero-carbon generation by 2028 with 15–25% liquidity decline. This implies $\theta_c$ in the range of 0.6–0.8 (indicative). Fingrid quarterly data on contracted vs exchange-traded volumes would permit direct estimation.
Phase
Condition
Eigenvalue signature
Finnish analogue
A — Classical equilibrium
Low $K_p$, high $L$, fast $\tau$
All $\Re(\lambda_i) < 0$
Pre-2022 energy market
B — Distorted equilibrium
Moderate $K_p$, declining $L$
Mixed: some $\Re(\lambda_i) \approx 0$
Current estimated position
C — Critical transition
$K_p/L \geq \theta_c$
$\Re(\lambda_{\max}) \to 0$
2027–2029 projection
D — Structural instability
High $K_p$, low $L$, high $\tau$
$\Re(\lambda_{\max}) > 0$
Scenario A in SM-013 skenaariotyö
§ 05 — One-Line Summary
Finland is a regime-switching eigenvalue system where PPA penetration moves the economy across stability boundaries by reallocating volatility rather than eliminating it.
§ 06 — Calibration Requirements and Open Questions
Open calibration gap
Parameters $\alpha, \beta, \gamma_i$ (sector absorption coefficients) and $\theta_c$ (critical threshold) are structural placeholders. Calibration requires: (1) Fingrid hourly spot + contracted volume data 2019–2026; (2) Statistics Finland input-output tables for energy intensity by sector; (3) Farm and forestry income cycle data (MTT/Luke). Without calibration, this note provides qualitative regime structure, not quantitative forecasts. Claims about which phase Finland currently occupies are directional estimates, not measurements.
The equilibrium point $x^*$ for linearisation is sensitive to choice of base year. Pre-energy-crisis (2021) and post-stabilisation (2025) give different Jacobian structures. A robust result would hold across both.
§ 07 — Falsification Conditions
FC-1 — Eigenvalues of $A^*$ remain stable (all $\Re(\lambda_i) < 0$) across observed PPA penetration 2022–2026. If true, the regime-shift hypothesis fails.
FC-2 — Agriculture and forestry do not appear as dominant eigenvector loads in observed volatility events (2022–2024 energy crisis data). If Layer 3 exposure was not amplified relative to Layer 4 (households), the eigenvector assignment claim fails.
FC-3 — Institutional lag $\tau$ does not correlate with policy overshoot amplitude. If policy corrections were appropriately damped despite delay, Mode 3 is not active.
FC-4 — BESS growth rate $dK_b/dt$ is sufficient to offset liquidity decline $dL/dt$ over 2024–2028. If so, the liquidity collapse mode is stabilised by market response alone, reducing the urgency of CM-1/CM-2.
Builds on CN-024 — PPA as Invisible Allocation Mechanism CN-025 — The Unprotected Layer TN-019 — PPA Rigidity and Price Formation SM-013 — Social Contract Calibration Failure (drift regime) Pending
CN-026 — Unified Allocation System (full state-space model, open draft, parametrisation pending)
SM-014 — Phase A–D: Policy Implications Without Equations (in preparation)