PHAETHON — A Curvature-of-Section Formulation of Multi-Machine Stability

Paper 1: Equivalence to Classical Small-Signal at the Conservative Skeleton, with a Saddle-Energy Mass-Gap Extension

What this deposit contains

• phaethon_paper.pdf — 33-page paper (amsart), 11 sections + 3 appendices, full proofs
• phaethon_paper.tex + phaethon_paper.bib — LaTeX source and BibTeX bibliography
• prereg_predictions.json — pre-registered falsifiers (AEMO 2016, 2003 Northeast blackout, ERCOT 2023 West Texas), timestamped audit trail before event-data analysis
• reproducibility_kit_v1.zip — self-contained verification kit (~70 KB): IEEE 9-/39-bus case files, precomputed PF / K_eff / contingency-sweep / K_bus / gap_k JSONs, and a standalone numpy+scipy verifier (verify_identity.py) that reproduces all four headline numerical claims of the paper in one command. See Reproducibility kit below.

The full PHAETHON implementation (Rust workspace + React frontend, validated on IEEE 9-/39-/118-bus and ACTIVSg 10,000-bus) is not released with this deposit. See Reproducibility kit below.

The one-paragraph claim

Multi-machine power-system stability admits a single operator-theoretic reading — the curvature of a section in the sense of the Davis Duality — that exactly reproduces the conservative-skeleton small-signal A-matrix on its classical domain, and extends past it (a) topologically, via an augmented state (Φ_t, r_t) with a winding-number vector tracking pole-slip sector transitions, and (b) energetically, via a finite-graph saddle-energy mass-gap theorem bounding the inter-sector barrier. Equivalence is verified to machine precision on the IEEE 39-bus benchmark; the novel content is the topological/energetic extension, with quantitative falsifiers pre-registered.

The math, in five frames

1. The master identity

C = τ / K

read in two forms:

• static: $C(x_{\mathrm{op}}) = \tau / K(x_{\mathrm{op}})$ — evaluates curvature at the operating point.
• dynamic: $C_{\mathrm{dyn}}(t) = \tau / (K(t) + \eta \cdot T(t))$ — adds a trajectory-excursion term $\eta T(t)$ where $\eta$ is a dynamic correction rate (sign data-dependent, not constrained ≥ 0).

K is the curvature of a section in the Davis Duality sandwich; on grids it is the inertia-weighted Hessian of the swing-Lyapunov function at the equilibrium.

2. Static K for multi-machine grids

The generalized eigenproblem on the post-Kron-reduction internal-node network:

$$ K_{\mathrm{red}} \cdot v = \omega^2 \cdot \widetilde{M} \cdot v $$

with $\widetilde{M} = \operatorname{diag}(2 H_{k} / \omega_{0})$. Equivalently, in symmetric form,

$$ K_{\mathrm{eff}} \cdot w = \omega^2 \cdot w, \qquad K_{\mathrm{eff}} \equiv \widetilde{M}^{-1/2} \cdot K_{\mathrm{red}} \cdot \widetilde{M}^{-1/2}. $$

The Davis coherence reading is $C_{\mathrm{Dav}} = \tau_{\mathrm{freq}} / f_{\mathrm{min}}$ with $f_{\mathrm{min}} = \sqrt{\lambda_{\mathrm{min}}(K_{\mathrm{eff}})} / (2\pi)$ — both in Hz, dimensionless ratio.

3. Augmented state (Φ_t, r_t)

The configuration manifold is the N_gen-torus of generator angles, with universal cover the real space of lifted angles. The augmented state lifts the dynamic state to a pair (Φ_t, r_t), where Φ_t lives in the continuous bulk-state space (bus voltages and angles modulo 2π) and r_t is an integer vector in dimension N_gen with one entry per generator. The discrete component is given by

$$ r_{t,k} = \operatorname{round}\left( \frac{\tilde\delta_{k}(t) - \tilde\delta_{k}(0)}{2\pi} \right). $$

Here r_t is the signed branch-crossing count — the deck-group label on the universal cover. The static form is valid while r_t is constant; transitions r_{t,k} → r_{t,k} ± 1 are pole-slip events of generator k.

4. Finite-graph saddle-energy mass gap

On the universal cover, under standard controlling-UEP nondegeneracy and conditional on the existence of an e_k-direction controlling-UEP (generic but not universal), define the inter-sector energy barrier as the mountain-pass infimum: the lowest peak energy attained by any continuous path that takes the system from the equilibrium δ_0 (sector 0) to its translate in sector e_k. The result is

$$ \mathrm{gap}{k} = E(\tilde\delta{k}^{*}) - E(\tilde\delta_{0}) > 0, $$

where δ*_k is the controlling unstable equilibrium point on the stability boundary in the e_k direction.

The bound is characterized by the controlling-UEP energy plus a local quadratic approximation in K_eff; in the two-machine equivalent reduction it collapses to a closed form (Appendix B.4) coinciding with the textbook equal-area criterion in the lossless SMIB limit.

This is structurally analogous to the Yang–Mills mass gap of the Davis Yang–Mills paper (Davis 2026, v5.0) — the proof here is for a finite graph base manifold and does not invoke the open Clay-Millennium statement on continuum 4-space.

5. Trichotomy parameter Γ

$$ \Gamma = \frac{m \cdot \tau^2}{K_{\mathrm{max}} \cdot \log |S|}. $$

• Γ > 1 → determined regime (classical small-signal sufficient)
• Γ = 1 → critical (saddle-energy bifurcation locus, IBG inertia floor)
• Γ < 1 → underdetermined (augmented state required, sector transitions accessible)

The proposition: Γ → 1 along an inertia-loss family implies gap_{k*} → 0 for the leading mode (hypotheses: uniform existence of controlling-UEP, non-degenerate leading mode).

Numerical features

Limit recovery: identity to machine precision

Honest framing: the equivalence at the conservative skeleton is an algebraic identity, not an independent measurement — it earns the right to make predictions outside the linearization, and that is what the rest of the paper does.

Saddle-energy gap on IEEE 39-bus (base case)

The two-machine-equivalent gap_k ranks all 10 generators on the IEEE 39-bus benchmark. Gens 5 and 9 (most heavily loaded, smallest synchronizing margin) rank as most vulnerable to single-event pole slipping; gen 10 (New York equivalent) ranks safest. Slip angles deviate from the textbook SMIB value π − δ_{0,k} by the multi-machine power-angle correction, computed via Hiskens–Pai trajectory-sensitivity continuation.

Dynamic-form η regression

The pipeline takes PMU samples of δ(t_n) and ω(t_n) and an event log, regresses

$$ \Delta K_{\mathrm{op}}(t) = \alpha \cdot T(t) + \beta \cdot \int_{0}^{t} (\omega^{\top} \omega) \mathrm{d}t' + \sum_{e \in \mathrm{events}} \gamma_{e} \cdot \mathbf{1}[t \ge t_{e}]. $$

returning the framework's three η-components: η_exc (geometric excursion), η_cfg (configuration jumps), η_dis (dissipative). On synthetic IEEE 39-bus cascades the pipeline recovers sign and order of magnitude in both inertia-loss and line-outage directions — a sanity check, not a quantitative validation.

Per-bus dashboard layer K_bus

$$ K_{\mathrm{bus},i} = w_{V} \cdot K_{V,i} + w_{M} \cdot K_{M,i} + w_{Q} \cdot K_{Q,i} + w_{L} \cdot K_{L,i} $$

— voltage sensitivity, local inertia deficit, reactive gap, line thermal stress. Three operational tiers (stable / drift / critical). Threshold defaults are operator policy heuristically anchored to the modal spectrum on IEEE 39-bus; the 39-bus sanity check is not a validation of the thresholds, which require recalibration per network class.

Pre-registered falsifiers (deposit-locked)

Three historical events, three distinct mechanisms, falsifiers fixed before event-data analysis:

Predictions and methodology deposited to Zenodo before event-data loading. The DOI timestamp provides the audit trail.

Reproducibility kit

To let independent readers verify the headline numerical claims, this deposit includes reproducibility_kit_v1.zip (~70 KB). Unpack and run:

unzip reproducibility_kit_v1.zip
cd reproducibility_kit_v1
pip install numpy scipy
python verify_identity.py

Expected output: 4 / 4 checks passed.

The kit contains:

• cases/ — IEEE 9-bus and IEEE 39-bus case files (TOML and JSON, with machine H and X_d′, bus topology, line impedances, scheduled generation)
• precomputed/:
  • ieee39_pf_base.json — converged power-flow solution at the base operating point (V and θ at all 39 buses)
  • ieee39_keff_base.json — the 10 K_eff eigenvalues and the 9 classical conservative-skeleton ω_n² for direct identity comparison
  • ieee39_contingency_sweep.json — 33 N−1 line outages with λ_min(K_eff) per contingency
  • ieee39_kbus_base.json and ieee39_kbus_stressed.json — per-bus K_bus four-way decomposition for base and stressed operating points
  • ieee39_gap_k_base.json — per-generator gap_k table (10 gens, two-machine-equivalent with confidence flags)
• prereg_predictions.json — full pre-registered falsifier set for AEMO 2016, 2003 NE blackout, and ERCOT 2023 West Texas
• verify_identity.py — standalone verifier (numpy + scipy only) that runs four independent checks:
  1. K_eff vs classical ω_n² identity at machine precision (paper §4.2)
  2. N−1 contingency sweep ranking consistency (Spearman ρ = 1.000)
  3. K_bus stress-direction under contingency + IBG ride-through (§8.3)
  4. gap_k ranking — gens 5 and 9 most vulnerable (§5.5)

The kit deliberately does not include the full PHAETHON Rust/React implementation, the GIGI fiber-bundle adapter, or the sparse 10k-bus refactor. The shipped artifacts are sufficient for any reviewer to audit the paper's central numerical claims; reproducing them from raw network data (i.e., re-deriving K_red and the Kron reduction) requires the implementation, which is held for Paper 2.

Implementation provenance (not released)

The identity numbers in Sections 4–5 of the paper were produced by a working Rust/React platform that has been independently validated on:

• IEEE 9-bus (WSCC, Anderson–Fouad)
• IEEE 39-bus (New England, Athay–Podmore–Virmani / Pai 1989) — including the all-mode identity table, the N−1 contingency sweep, and the gap_k ranking
• IEEE 118-bus
• ACTIVSg 10,000-bus synthetic case (Texas A&M), with sparse linear algebra throughout: PF solve ≈ 0.47 s, Kron reduction ≈ 3.75 s
• gap_k vs. Yang et al. 2021 published critical-clearing times (IEEE 39-bus)
• GESL real-PMU spectrum identification

Implementation release decisions (open source, dual-license, or commercial) are deferred to the follow-up paper introducing the inverter-mode operator K_inv.

Implementation gotchas (Appendix A, paper)

Eight common traps documented from the implementation experience:

1. ω₀ convention (rad/s vs per-unit speed deviation) — 19.4× error if confused
2. Per-unit conversion on Lyapunov function — energies in wrong base by factor of system MVA
3. Symplectic integrator required for mode-shape integration — leap-frog drifts, RK4 inflates amplitude
4. Sparse vs dense Kron at 10k buses — 543 s → 3.75 s
5. LAPACK ?SYGV vs ?GGEV solver branch on lossy networks — O(loss-fraction) eigenvalue error if you symmetrize K_lossy
6. Rigid-rotation null mode handling
7. Inertia-loss raises λ(K_eff) (counterintuitive — frequency operator goes up, energy-barrier object K_red separate)
8. K_BUS_DENSE_LIMIT topology gate at large N

What this paper does not claim

• The Yang–Mills mass-gap conjecture on continuum 4-space. Our statement is for a finite graph base.
• A continuum-quantum-field-theory derivation of the swing equations. We use the Davis framework structurally, not as an embedding of the standard model.
• A first-principles derivation of the K_bus thresholds. Those are operator policy.
• An IBG-mode predictive operator K_inv. The 8-state inverter-mode Jacobian and its full SCR-parametric bifurcation curve are reserved for the follow-up paper.
• Event-data validation. The pre-registered falsifiers are evaluable; the evaluation belongs to the second paper.

Bibliography highlights

• Davis framework: Davis Duality (Zenodo 10.5281/zenodo.19428406), Geometry of Fuel (Davis Lab 2026, ISBN 9798245040004), Yang–Mills v5.0, Davis–Landau Sonic Onset Law
• Classical power systems: Anderson–Fouad, Kundur, Sauer–Pai, Pai (Energy Function Analysis), Chiang (controlling-UEP / BCU), Athay–Podmore–Virmani, DeMarco–Bergen, Hiskens–Pai
• Inverter modes: Yazdani–Iravani, Harnefors et al., Wang–Blaabjerg, Wen et al.
• Numerical methods: Kato (perturbation theory), Sanz-Serna–Calvo (symplectic integrators), Hauer–Demeure–Scharf (Prony)
• Events: AEMO 2017 (Black System SA), US–Canada 2004 (NE 2003), NERC 2023 (Odessa / West Texas)

License and citation

The paper, LaTeX/BibTeX source, pre-registered predictions, and reproducibility kit are released for academic use. The PHAETHON implementation is not part of this deposit and remains under reserved rights pending the follow-up paper. Cite as:

Davis, B. R. (2026). PHAETHON — A Curvature-of-Section Formulation of Multi-Machine Stability. Zenodo. https://doi.org/10.5281/zenodo.20273563

Davis Geometric / Davis Lab — 2026.