AURORA · Davis Geometric

Why AURORA

Machine-precision conservation. Analytical regime classification. No tunable parameters.

Every climate simulation makes errors it doesn't report. AURORA makes conservation an algebraic identity — each timestep either proves itself or halts. The predictability horizon is a field you can map, not a scalar you fit.

25,000×

Casimir energy drift improvement over state-of-the-art finite-difference dycore, achieved with DEC on the cubed-sphere

8ε

Machine-precision receipt tolerance — closed-form bound on integrator roundoff, never a configurable threshold

Threshold parameters in regime classification — ENSO, NAO, MJO boundaries emerge from the causal-state geometry of the data

For dycore developers

Conservation laws are algebraic identities on the integrator, verified at machine epsilon. You get a receipt on every timestep — not a drift number in the appendix of a paper published two years later.

For climate centres

Regime boundaries — ENSO phase, NAO index, MJO propagation — emerge from the causal-state geometry of the reanalysis. No threshold calibration. No index revision cycle. No committee.

For forecasters

The predictability capacity C is a field on the sphere. You see exactly where the trajectory bundle decoheres — not a global e-folding time, but a local map of where tomorrow's forecast is trustworthy.

The Contract

Three clauses. No third state between pass and refuse.

Climate models have lived with energy drift, mass non-closure, and PV non-conservation because the alternative — failing the run — was institutionally unthinkable. AURORA makes the alternative cheap and routine.

Every observable is gauge-invariant.

Under coordinate choice on the sphere, frame choice between inertial and rotating, and resolution choice at mesh refinement. What you measure cannot depend on how you labeled it.

Every operation is local.

No global flux correctors. No post-hoc energy fixers. No operations outside the bundle structure. Mass conservation is the same primitive as divergence cleaning, inside the integration step, not bolted on.

Every claim has no tunable tolerance.

Conservation receipts pass or fail at the algebraic identities of the integrator — machine epsilon times a closed-form bound, never a configurable threshold. A run emits a valid receipt or it emits a refusal.

The Math

Two field equations on the rotating sphere.

The Davis Field Equations, instantiated for geophysical fluid dynamics. Capacity sets the predictability horizon. Closure decomposes every state into a regime assignment with no residual.

Capacity

C = τ ⁄ K

Predictability as a field, not a scalar.

τ is the local eddy decorrelation time on the configuration manifold of geostrophic flows. K is the sectional curvature of the slow manifold induced by the Poisson bracket structure. C is the predictability capacity — the local horizon beyond which the trajectory bundle decoheres. Lyapunov exponents are recovered as 1 / ⟨C⟩ over the attractor measure, derived rather than fitted.

Closure

S + d² = 1

Regime assignment with no residual.

S is the causal-state structural information of the local atmospheric regime — Shannon information of an ε-machine state distribution conditioned on the past-future predictive equivalence class. d² is the geodesic distance squared to the nearest regime centroid on the Fisher metric. ENSO, NAO, MJO become equivalence classes, not threshold classifications.

The Receipt

Every timestep, the substrate emits proof or it refuses.

In a standard dycore, conservation drift is measured post-hoc and reported as a number in a paper appendix. In AURORA, it is checked at every step as an algebraic identity on the Casimir invariants of the Poisson bracket.

Schema · per timestep ● receipt: pass

step                 :: uint64
wall_time            :: monotonic_ns
casimir_energy       :: f64      # H, bit-exact across compatible profiles
casimir_mass         :: f64      # continuity as semidirect-product Casimir
casimir_pv_l1        :: f64      # ∫ρ q dV — Ertel
casimir_pv_l2        :: f64      # ∫ρ q² dV — potential enstrophy
kelvin_holonomies    :: map<loop_id, f64>  # PLAQUETTE sums
c_field_summary      :: { min, max, mean, attractor_avg }
s_d2_residual        :: f64      # max |1 − S − d²| over mesh
refusal_reason       :: Option<{ casimir, expected, actual, eps_mult }>

The refusal threshold is not tunable. It is fixed at eps_mult ≤ 8 per Casimir — a closed-form bound on the integrator's worst-case roundoff amplification, derived once and pinned.

The Verbs

AURORA introduces no new primitives. It instantiates the GIGI verb set on the rotating sphere.

SYMPLECTIC_FLOWintegrator

Störmer-Verlet on the semidirect-product bracket. Preserves the Hamiltonian and the Casimirs by construction.

PROJECT_GAUSSprojection

Tikhonov-regularized CG projection onto the divergence-free manifold. Inside the integration step, not after it.

GAUGE_FIELDconnection

SO(3) principal bundle over the sphere. Coriolis is the connection coefficient — frame choice is a gauge transformation.

PLAQUETTEholonomy

Discrete loop integral of the connection. Kelvin's circulation theorem becomes a holonomy receipt, not a tendency equation.

TRANSPORTcovariant

Parallel transport of vorticity along the flow. The covariant derivative D = d + A acts on every section.

ATTENDassimilation

Geometric weighted average over observation points. Replaces hand-tuned 4D-Var and EnKF covariance matrices.

EPISODICmemory

Rolling reanalysis window with curvature-aware decay. Low-K regions get long memory; high-K regions get short.

DREAMensemble

Ensemble generation on the learned slow manifold. Cloud operators are trained under hard conservation constraints.

DHOOMserialization

Curvature-aware wire format. 66–84% token savings over JSON on reanalysis tiers, full round-trip.

The Test Plan

Each test must pass before the next begins.

Williamson 2, 5, and 6 are the dycore-equivalent of MNIST. Any working geometric integrator passes them — AURORA's claim is that it passes them with zero drift, not just with skill.

Williamson 2 — steady-state geostrophic

Williamson et al. 1992

L2 ≤ 10× analytic bound

Williamson 5 — zonal flow over mountain

ibid.

Casimir drift below 8ε / 15d

Williamson 6 — Rossby-Haurwitz wave

ibid.

Phase speed Hamiltonian-exact

Held-Suarez — forced-dissipative

Held & Suarez 1994

Within ensemble spread

Aquaplanet

APE protocol

ITCZ, MJO, Hadley

AMIP with prescribed SST

CMIP6 AMIP

First test vs. obs

The Non-Claims

What AURORA does not do.

Restraint is the credibility move. The reviewer will look for these. The substrate names them first.

Does not

Collapse the IPCC ECS range. The dominant climate-sensitivity uncertainty is low-cloud feedback — it requires data, not a better dycore.

Does not

Close the 10-to-14-day predictability gap by itself. That gap is dominated by initial-condition uncertainty and physics error. AURORA removes the dycore-attributable component only.

Does not

Conserve quantities to machine precision indefinitely. It conserves with bounded roundoff drift and no secular term, which is the honest version of the claim.

Does not

Replace the need for observations. Reanalysis still needs the obs. AURORA stops corrupting them with non-conserving assimilation.

Does not

Ship as a single binary. It is a substrate. Operational deployment requires a named dycore collaborator. See below.