Theory, emergence and falsifiable climate-model hypotheses

Abstract

This manuscript brings together twelve research ideas, mathematical formulations and falsifiable hypotheses developed around the dLEGOS programme. Its central object is an atmospheric control volume connected to neighbours through explicit shared-face fluxes and observations across time. The programme asks whether this representation can support historical-state inference and useful forward prediction over data-constrained Caribbean terrain, and whether local model specialisation, physics-constrained learning and formal certificates materially improve performance. Already executed component tests, including adverse results, are distinguished from the unimplemented atmospheric system. Equations specify the physical bookkeeping and hypotheses; they do not constitute a proof that any current learned model solves the full equations. Each research direction includes a falsification route and a link to its associated working paper or experiment record.

1. The central mathematical question

A regional climate model represents more than patterns on a map. Atmospheric state is transported, transformed and observed on different spatial and temporal supports. Small islands introduce steep local relief and coastline changes that complicate coarse-grid representation. A learned surrogate can produce plausible images while violating the physical budgets that join neighbouring locations. The proposed alternative is to make each volume and shared boundary explicit, then test whether learning can improve the unobserved local terms without evading the resulting accounting constraints.

Here a tesseract is an explanatory name for a three-dimensional control volume followed through time. It is not a literal claim that physics takes place in four independent spatial dimensions. The temporal axis indexes a history of states and fluxes. Reconstructing an earlier state from later observations is an inverse inference problem, not the physical reversal of atmospheric dissipation.

These ideas are separated according to evidence: formulated mathematics; designed architecture; a theory supported only by a constructed component test; or a proposed experiment without results. The executed experiment ledger reports relevant positive and negative measurements.

2. A physically connected space–time model

The first group treats the geometry, moisture and energy ledgers, observational support and inverse reconstruction. Each proposal must specify its state variables, their units, boundary conventions and physical source terms before a neural residual can be interpreted.

T01 · Space–time geometry

Climate tesseracts: atmospheric volumes traced through time

Evidence status: formulated — Mathematical design and hypotheses, not validated physical model.

Inspect a time archive. Filled isometric scientific teaching instrument. A time slice selects a stored state; its coordinate is not a physical journey into the past. Recovering an unobserved state requires an observation model and independent verification. — Constructed explanatory instrument; it is not an observed atmospheric field or a trained-model result. Geometry uses Anatomy (MIT). The paper states its evidence and limits.

Research question: Can a spatial control volume, traced across time, hold a coherent ledger for momentum, heat, water vapour, hydrometeors and passive tracers while preserving explicit lateral boundaries?
Proposed method: Represent each cell by volume V, face area and outward normal; state U=(ρ,ρu,ρE,ρq_v,ρq_l,…). Couple temporal state transitions to face flux and separately tracked sources.

\frac{d}{dt}\int_V \mathbf U\,dV + \oint_{\partial V}\mathbf F(\mathbf U)\cdot\mathbf n\,dA = \int_V \mathbf S\,dV

Falsification: Failure of cross-face flux closure, dimensional consistency or data-supported boundaries.

Related manuscript ↗Related source 2 ↗

T02 · Conservation and domain decomposition

One shared face, one exchange: conservative graph neural cells

Evidence status: partly-tested —

Coordinate local numerical fields. Filled isometric scientific teaching instrument. Each station represents a numerical subdomain with an explicit physical state. Accounting, positivity, stability and state accuracy are different tests. — Constructed explanatory instrument; it is not an observed atmospheric field or a trained-model result. Geometry uses Anatomy (MIT). The paper states its evidence and limits.

Research question: Can local neural models specialize while exchanging a single equal-and-opposite flux at each internal face?
Proposed method: Use an oriented incidence matrix B and one edge flux F per shared face; test total budget and sensitivity to coarse-driver bias. Current support is only constructed thermal graph tests and monthly precipitation aggregate tests.

\frac{d\mathbf m}{dt}=-B\mathbf F+\mathbf s

Falsification: A proposed learned coupling violates combined-domain conservation or degrades held-out accuracy compared with uncoupled and simple conservative baselines.

T03 · Physics-informed learning

Limited physical closure: when the equation omits essential processes

Evidence status: partly-tested —

Account for one shared transfer. Filled isometric scientific teaching instrument. Two chambers begin with a combined amount of one constructed unit. The total stays fixed even if the chosen transfer is physically wrong. — Constructed explanatory instrument; it is not an observed atmospheric field or a trained-model result. Geometry uses Anatomy (MIT). The paper states its evidence and limits.

Research question: Could a bounded learned source represent missing convection, evaporation, transport or observing-system error without making the physical constraint meaningless?
Proposed method: Separate exact discrete bookkeeping from approximate atmospheric closure. The CLEAR-PINN synthetic experiments produced mixed and negative adaptive-learning evidence.

\mathcal L = \mathcal L_{\mathrm{data}} + \lambda \lVert \mathcal R(\mathbf U)-\mathbf s_\theta\rVert^2

Falsification: Adaptive residual treatment fails to improve strong, information-matched regularization on held-out storms or violates physical budgets.

Related manuscript ↗Related source 2 ↗

T04 · Inverse problems and causal forecasting

Historical-state reconstruction is not physical time reversal

Evidence status: proposed — Research idea/protocol; empirical experiment not established.

Research question: Can incomplete satellite and reanalysis observations constrain a distribution over earlier storm states, while forward prediction uses only data available before an issue time?
Proposed method: Fit a physics-regularised inverse smoother with uncertainty and identifiability tests; separately freeze a forecast issue time and hold out future observations.

\hat{\mathbf U}_{0:T}=\arg\min_{\mathbf U}\left(\lVert H\mathbf U-\mathbf y\rVert^2+\lambda\mathcal R_{\mathrm{physics}}(\mathbf U)\right)

Falsification: Ambiguous reconstructions, excessive sensitivity to priors, or unallowable future-data leakage.

T05 · Satellite data and inverse observation operators

Observation support, release latency and physically honest downscaling

Evidence status: partly-tested —

Inspect a sampled field. Filled isometric scientific teaching instrument. The carriage represents an observation with a particular footprint. Only additional independent information can establish the missing detail. — Constructed explanatory instrument; it is not an observed atmospheric field or a trained-model result. Geometry uses Anatomy (MIT). The paper states its evidence and limits.

Research question: What can a fine model infer when the upstream rainfall product is coarse, delayed or biased?
Proposed method: Model the observation operator, aggregation weights and product vintages explicitly. Monthly CHIRPS conservation and rainfall-delay studies expose the risk of treating archival truth as a prospective input.

\mathbf y_{\mathrm{obs}}=H_\Delta(\mathbf U)+\boldsymbol\epsilon

Falsification: Apparent gains vanish when coarser observation support, actual publication dates and independent gauges are respected.

Related manuscript ↗Related source 2 ↗

Figure 1. Shared-face transfer, a minimal numerical identity

Cell A310 K

Cell B280 K

Constructed thermal quantities · shared face · no external source

Calculated exchange

The two-sided update conserves

C_A T_A+C_B T_B

by construction. This does not simulate humidity, atmospheric pressure, air mass or storm dynamics. It demonstrates the discrete identity proposed for later learned interface fluxes and is not a claim of 3D atmospheric validation. Numerical source and tests.

3. Emergence, numerical verification and model choice

Emergence is a hypothesis about collective dynamics, not a performance metric. A group of local neural cells may acquire specialised representations, but a benefit should appear as improved error, stability, conservation or uncertainty calibration against matched global and local controls. Explicit causal training and interval verification offer candidate tools; no claimed low-error emergent Caribbean atmospheric model has passed these comparisons.

T06 · Emergence, curriculum and multiscale representation

Emergent local specialization for low-error physics-informed models

Evidence status: proposed — Research idea/protocol; empirical experiment not established.

Research question: Do locally specialized cells acquire useful collective dynamics once their boundary exchanges, temporal causal order and uncertainty are controlled?
Proposed method: Test mixture-of-local-operators with conservative interfaces, multi-resolution spectral monitoring, curriculum from initial/boundary information, and matched compute against a global PINN. Current literature motivates hypotheses; no emergent atmospheric accuracy improvement has been demonstrated.

\mathcal L_{\mathrm{total}}=\mathcal L_{\mathrm{obs}}+\lambda_f\mathcal L_{\mathrm{flux}}+\lambda_c\mathcal L_{\mathrm{causal}}

Falsification: No improvement under fixed compute; local specialization creates interface artefacts or extrapolation failures.

Related manuscript ↗Related source 2 ↗

T07 · Verification and safety

Formal certificates for physics-informed forecasts

Evidence status: proposed — Research idea/protocol; empirical experiment not established.

Compare a reference and an estimate. Filled isometric scientific teaching instrument. The left specimen is fixed at 0.60 constructed unit. Meeting a constraint and matching an independently known reference are distinct properties. — Constructed explanatory instrument; it is not an observed atmospheric field or a trained-model result. Geometry uses Anatomy (MIT). The paper states its evidence and limits.

Research question: Can formal or interval-based verification certify bounded violations of selected model constraints over prescribed input ranges?
Proposed method: Begin with small closed heat-transfer and advection systems; verify output non-negativity, bounded source terms and conservation using SMT/interval arithmetic. Scale carefully to learned subdomains with auditable assumptions; a low residual alone is not a formal certificate.

\forall \mathbf x\in\mathcal D,\quad |\mathcal R_\theta(\mathbf x)|\leq\varepsilon

Falsification: Certificate cannot be reproduced, relies on unrealistic assumptions or gives intervals too loose for operational use.

Related manuscript ↗Related source 2 ↗

T08 · Multi-model fusion

Foundation priors plus explicit physical corrections

Evidence status: partly-tested —

Compare candidate trajectories. Filled isometric scientific teaching instrument. Each cartridge represents a candidate trajectory on the same comparison window. A valid comparison needs matched information, issue times and independently assessed targets. — Constructed explanatory instrument; it is not an observed atmospheric field or a trained-model result. Geometry uses Anatomy (MIT). The paper states its evidence and limits.

Research question: When do pretrained temporal and weather models help beyond fitted local baselines, and when does a small physically informed correction add independent value?
Proposed method: Compare Timestral, TimesFM, Chronos, WeatherNext, archived NWP and local baselines under fixed common origins, with ablations for physics, added covariates and information age.

\hat y_{t+h}=f_{\mathrm{base}}(\mathcal I_t)+\delta_\theta(\mathcal I_t)

Falsification: Corrections do not beat robust seasonal/AR controls or fail under actual source-release delays.

Related manuscript ↗Related source 2 ↗

T09 · Forecast verification

Event-wise extremes, spatial displacement and catchment consequences

Evidence status: proposed — Research idea/protocol; empirical experiment not established.

Test a rainfall threshold. Filled isometric scientific teaching instrument. The gauge represents a constructed accumulation, not a retrieved storm observation. A threshold crossing says nothing about the skill of a rainfall forecast. — Constructed explanatory instrument; it is not an observed atmospheric field or a trained-model result. Geometry uses Anatomy (MIT). The paper states its evidence and limits.

Research question: Can a forecast capture high-intensity rainfall, geographic overlap and basin consequences, not just low average pixel error?
Proposed method: Compare CSI, POD, FAR, neighbourhood FSS, object displacement, pattern error, basin totals and probabilistic scores on common time/grids. Published Tennessee Valley results motivate, but do not validate Jamaican transfer.

\mathrm{CSI}=\frac{H}{H+M+F}

Falsification: A model improves grid-mean scores but misses withheld storm extremes or misplaces rain across basin boundaries.

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4. Scenario twins and downstream decisions

The physical model, once credible, may support infrastructure planning, disaster response and educational visualisation. These applications have their own verification problems. An atmospheric emulator’s low RMSE does not by itself validate damage, finance allocations or learning outcomes.

T10 · Climate scenarios and interaction

Small-world scenario twins for education and constrained counterfactuals

Evidence status: illustrative — Software animation, visual or numerical demonstration; no operational validation.

Inspect a miniature scenario table. Filled isometric scientific teaching instrument. The table is a constructed scenario with model-space dimensions. Local flood decisions need validated hydrology, terrain, exposure and vulnerability evidence. — Constructed explanatory instrument; it is not an observed atmospheric field or a trained-model result. Geometry uses Anatomy (MIT). The paper states its evidence and limits.

Research question: Can a physically labelled, interactive regional twin help readers see which simulated changes affect a budget and which are merely graphical effects?
Proposed method: Use controlled toy models and construction-time sources for illustrative fluxes, storm passage, coastal water and soil storage; only elevate components to empirical outputs after a data/validation gate.

\Delta W = (P-E-R)\Delta t + F_{\mathrm{boundary}}\Delta t

Falsification: An animation misstates its source support or presents constructed trajectories as observed damage forecasts.

Related manuscript ↗Related source 2 ↗

T11 · Decision science and adaptation finance

Agentic climate decisions with human ownership

Evidence status: designed — Framework specified but independent effectiveness not established.

Stress-test a policy choice. Filled isometric scientific teaching instrument. The cartridges represent three actions in a constructed decision problem. Minimax regret identifies action 2 in this example; a real authority must set objectives and review evidence. — Constructed explanatory instrument; it is not an observed atmospheric field or a trained-model result. Geometry uses Anatomy (MIT). The paper states its evidence and limits.

Research question: Could multi-agent evidence retrieval and challenge review improve traceability in allocation decisions without transferring public authority to software?
Proposed method: Developed governance architecture routes climate evidence, vulnerabilities, financing checks and independent challenge to named officials; external deployment and causal value assessment remain pending.

\mathrm{Decision}=\operatorname{Approve}_{\mathrm{human}}(\mathrm{evidence},\mathrm{constraints},\mathrm{challenge})

Falsification: Unsupported sources, unaccountable recommendations or worse equity/accuracy than conventional review.

Related manuscript ↗

T12 · Applied climate research

Pre-impact satellite damage and youth climate action studies

Evidence status: proposed — Research idea/protocol; empirical experiment not established.

Compare a pair of observations. Filled isometric scientific teaching instrument. Before and after inputs need matching geometry and acquisition conditions. Independent labels are needed before calling a change verified damage or financial loss. — Constructed explanatory instrument; it is not an observed atmospheric field or a trained-model result. Geometry uses Anatomy (MIT). The paper states its evidence and limits.

Research question: Can remote sensing and physical hazard models improve actionable pre-landfall damage distributions, and can supervised sport interventions sustain verified climate learning?
Proposed method: Two distinct protocols: as-issued hazard and exposure with withheld damage imagery; and climate-learning sport with safeguarding and cluster-based comparison. No outcome data are claimed for either.

P(D\mid H,E,V)\quad\text{and}\quad E[Y(1)-Y(0)]

Falsification: Event holdouts show no improvement against standard vulnerability models, or youth pilot records show no lasting effects against controls.

Related manuscript ↗Related source 2 ↗

5. Proposed experiment sequence and stop conditions

The programme should begin with the least ambitious model that can falsify a claimed benefit, then widen physical complexity only when a simpler baseline is beaten under identical data, information timing and computation. Four linked tests are especially decisive.

Test	Minimum comparison	Decision criterion
Local flux coupling	Independent cells, unlearned conservative flux, trained shared-face flux	Show budget closure and held-out accuracy together; neither alone is enough.
Inverse storm history	Assimilated observations, persistence or reanalysis replay, physically constrained smoothing	Evaluate withheld earlier-state fields and reconstruction uncertainty; do not describe this as reversing time.
Low-error emergence and formal checking	Global PINN, causal curriculum, local operators; matched parameter and compute budgets	Report both test-domain error and an independently verified residual bound with its assumptions.
Caribbean extreme rainfall	Climatology, persistence, optical flow, NWP and physics-embedded models with a single issue time	Compare event-held-out CSI/FSS, pattern displacement, calibration and basin consequences; reject leakage.

For historical Hurricane Beryl and a second independently held-out storm, the next work should align ERA5, GPM IMERG, terrain, gauge and as-issued forecast vintages without allowing post-event information into a prospective prediction. An executable reconstruction will require an identifiability analysis; simulations must distinguish synthetic trajectories from observed physical states.

What would count as a negative result?

A richer model can fail by inheriting bias from an erroneous coarse driver, producing false rain extremes, violating a shared-face balance or appearing accurate only when future data leak into its input. It can also fail more simply: a linear baseline, a seasonal allocator or an ordinary tuned PINN may achieve equal or better skill with less computation. The existing tests already document such outcomes. Those failures inform the architecture rather than invalidate the scientific value of the work.

6. References and scientific attribution

These are external methodological sources. They are not claimed as experiments performed or discoveries made by the author. Study-specific results and software traces are recorded separately in the experiment ledger and reference library.

[1] Raissi, Perdikaris and Karniadakis (2019). Physics-informed neural networks. https://doi.org/10.1016/j.jcp.2018.10.045.
[2] Harder et al. (2023). Hard-constrained deep learning for climate downscaling. https://www.jmlr.org/papers/v24/23-0158.html.
[3] Das et al. (2024). Hybrid physics–AI versus numerical weather prediction for extreme rainfall nowcasting. https://doi.org/10.1038/s41612-024-00834-8.
[4] Gneiting and Raftery (2007). Strictly proper scoring rules, prediction and estimation. https://doi.org/10.1198/016214506000001437.
[5] Cestola, Teresi and De Simone (2026). A flow-aware training strategy for PINNs. https://doi.org/10.1016/j.cma.2026.118910.
[6] Jacob et al. (2026). Epistemic physics-informed neural networks. https://doi.org/10.1007/s44379-026-00086-8.
[7] Liu et al. (2024). Formally verified physics-informed neural control Lyapunov functions (preprint). https://arxiv.org/abs/2409.20528.
[8] Funk et al. (2015). The CHIRPS precipitation record. https://doi.org/10.1038/sdata.2015.66.

7. Conclusion and disclosure

A useful research programme must be explicit about both what has been calculated and what remains hypothetical. The mathematical representation, finite-volume accounting and observed data studies provide a foundation for testing the larger atmospheric idea. Neither synthetic tests nor visual demonstrations establish the skill of a coupled three-dimensional model. The next contribution will be determined by reproducible, event-held-out comparisons and by negative as well as positive results.

Authorship: Adrian Dunkley. This is an AI-assisted working research manuscript for author and supervisor review. No independent peer review, trained full-system climate model, formally certified PINN or operational forecast is asserted.

Review the recorded experimental evidence → · Return to the working-paper series

1. The central mathematical question

2. A physically connected space–time model

Inspect a time archive

Coordinate local numerical fields

Account for one shared transfer

Inspect a time archive

Inspect a sampled field

3. Emergence, numerical verification and model choice

Coordinate local numerical fields

Compare a reference and an estimate

Compare candidate trajectories

Test a rainfall threshold

4. Scenario twins and downstream decisions

Inspect a miniature scenario table

Stress-test a policy choice

Compare a pair of observations

5. Proposed experiment sequence and stop conditions

What would count as a negative result?

6. References and scientific attribution

7. Conclusion and disclosure