Editing Openai/6938382f-f9e4-8003-a158-6d812c1bce84 (section)

==== 3. Mathematical and logical consistency ====

===== 3.1 Dimensions and units =====

A few issues stand out:
* He writes the CMIP5 LWCF error as “±4 Wm⁻² year⁻¹ model⁻¹” and compares it to an annual GHG forcing increment of ~0.035 Wm⁻² yr⁻¹. Propagation_of_Error_and_the_Re… - In Lauer & Hamilton, LWCF RMSE is in W/m², not W/m² per year. It’s a bias/variance in mean cloud radiative effect. American Meteorological Society Journals<ref>{{cite web|title=American Meteorological Society Journals|url=https://journals.ametsoc.org/view/journals/clim/26/11/jcli-d-12-00451.1.xml|publisher=American Meteorological Society Journals|access-date=2025-12-10}}</ref> - Frank effectively inserts “per year” to match the GHG forcing trend, but that’s mixing a level error with a time derivative.
* He then turns this into a “lower limit of model resolution”: anything smaller than ±4 W/m² per year is claimed to be unresolved. Propagation_of_Error_and_the_Re… - That’s not how resolution works in time-stepping models constrained by energy conservation; a bias in mean LWCF does not imply a ±4 W/m² white-noise floor on annual changes.

===== 3.2 Algebraic steps & limiting behavior =====
* Given u ≈ constant per year, Eq. (6) makes uncertainty grow as σ_T ∝ √N. For N ≈ 100 years and u ≈ O(0.5°C), you get ±5°C–10+°C by year 2100, which is what he reports (±15°C). Propagation_of_Error_and_the_Re…
* If his interpretation were physically correct, GCM trajectories of global mean temperature would look like huge random walks with rapidly diverging spread, even for identical forcing scenarios. - In reality, ensembles show bounded spreads and robust means under identical forcings; global SAT does not execute an unbounded random walk in GCMs. IPCC<ref>{{cite web|title=IPCC|url=https://www.ipcc.ch/site/assets/uploads/2018/02/WG1AR5_Chapter09_FINAL.pdf|publisher=ipcc.ch|access-date=2025-12-10}}</ref> - So his uncertainty propagation predicts a behavior that actual GCMs demonstrably do not exhibit, which is a strong sign that the error model is mis-specified.

Limiting cases:
* Take the limit where the LWCF bias is constant in time: then, under common practice of working with temperature anomalies, most of that bias cancels between baseline and projection period. The residual effect is on climate sensitivity, not on a random walk of global mean T. Frank explicitly chooses not to treat it that way.

===== 3.3 “Magic steps” and circular logic =====

Some semi-magical steps:
* From cloud-fraction RMSE to annual independent flux errors. The jump from a 25-year climatological RMS bias in cloud fields Propagation_of_Error_and_the_Re…  to “annual average ±4 Wm⁻² year⁻¹ uncertainty injected at every simulation step” is asserted more than derived.
* From “models have large structural cloud errors” to “therefore no anthropogenic signal can be evidenced.” That conclusion assumes his particular error-propagation model is the correct representation of how these structural errors affect trend detection, which is precisely the point in dispute.

There is also some implicit circularity:
* He assumes the cloud-error-induced uncertainty is so large that model projections are meaningless, then uses that conclusion to dismiss all detection & attribution work that could have provided empirical checks on his premise.