SciExam for ENSO · Model equations

Model equations

12 offline final models + 2 Web checkpoints · verified equations · back to the results

Hover over an original-model term for its ablation results. Colors identify term types, not importance, in both original and simplified models. Hover changes use each experiment’s stated comparison baseline; the tables compare simplified and original models. Tildes denote refitted coefficients. Throughout, \(\dot{\mathbf{W}}\) denotes Gaussian white noise and \(\Delta\mathbf{W}\) its increment over one numerical step. \(\dot{\boldsymbol{\xi}}\) denotes the non-Gaussian innovations of GPT-5.6-Sol and Claude Opus 5, described in their notes.

Structure labels describe the drift in the displayed state coordinates. Linear dynamics include offsets and seasonal coefficients. Additive noise is state-independent, but may vary with season or be correlated across equations; multiplicative noise has state-dependent amplitude. Numerical safeguards are retained separately.

Linear couplingNonlinear driftSeasonal driftStochastic forcingOffset

Claude Fable 5.1

round_56Original transcription verified

Original model

\[\begin{aligned} \frac{du}{dt} &= \class{term kind-linear tip-fable51-1}{a_{11}u} + \class{term kind-seasonal tip-fable51-2}{a_{12}(t)h_W} + \class{term kind-linear tip-fable51-3}{a_{13}T_C} + \class{term kind-linear tip-fable51-4}{a_{14}T_E} + \class{term kind-linear tip-fable51-5}{a_{15}\tau} + \class{term kind-noise tip-fable51-6}{\sigma_u(t)\dot W_u} \\ \frac{dh_W}{dt} &= \class{term kind-linear tip-fable51-7}{a_{21}u} + \class{term kind-seasonal tip-fable51-8}{a_{22}(t)h_W} + \class{term kind-linear tip-fable51-9}{a_{23}T_C} + \class{term kind-linear tip-fable51-10}{a_{24}T_E} + \class{term kind-linear tip-fable51-11}{a_{25}\tau} + \class{term kind-nonlinear tip-fable51-12}{a_{26}\frac{T_E^2}{1+(T_E/T_{s,h})^2}} + \class{term kind-nonlinear tip-fable51-13}{a_{27}h_W^3} + \class{term kind-offset tip-fable51-14}{a_{28}} + \class{term kind-noise tip-fable51-15}{\sigma_h(h_W,t)\dot W_h} \\ \frac{dT_C}{dt} &= \class{term kind-linear tip-fable51-16}{a_{31}u} + \class{term kind-seasonal tip-fable51-17}{a_{32}(t)h_W} + \class{term kind-seasonal tip-fable51-18}{a_{33}(t)T_C} + \class{term kind-linear tip-fable51-19}{a_{34}T_E} + \class{term kind-linear tip-fable51-20}{a_{35}\tau} + \class{term kind-nonlinear tip-fable51-21}{a_{36}T_C^2} + \class{term kind-nonlinear tip-fable51-22}{a_{37}T_C^2\,\mathbb{1}_{T_C>0}} \\ &\quad + \class{term kind-nonlinear tip-fable51-23}{a_{38}T_C^3} + \class{term kind-offset tip-fable51-24}{a_{39}} + \class{term kind-noise tip-fable51-25}{\sigma_C(T_C)\dot W_C} \\ \frac{dT_E}{dt} &= \class{term kind-linear tip-fable51-26}{a_{41}u} + \class{term kind-seasonal tip-fable51-27}{a_{42}(t)h_W} + \class{term kind-linear tip-fable51-28}{a_{43}T_C} + \class{term kind-seasonal tip-fable51-29}{a_{44}(t)T_E} + \class{term kind-linear tip-fable51-30}{a_{45}\tau} + \class{term kind-nonlinear tip-fable51-31}{a_{46}(t)\frac{T_E^2}{1+(T_E/T_{s,E})^2}} \\ &\quad + \class{term kind-nonlinear tip-fable51-32}{a_{47}T_E^3} + \class{term kind-offset tip-fable51-33}{a_{48}} + \class{term kind-noise tip-fable51-34}{\sigma_E(T_E,t)\dot W_E} \\ \frac{d\tau}{dt} &= \class{term kind-linear tip-fable51-35}{a_{51}u} + \class{term kind-linear tip-fable51-36}{a_{52}h_W} + \class{term kind-linear tip-fable51-37}{a_{53}T_C} + \class{term kind-linear tip-fable51-38}{a_{54}T_E} + \class{term kind-linear tip-fable51-39}{a_{55}\tau} + \class{term kind-noise tip-fable51-40}{\sigma_{\tau}(T_C,t)\dot W_{\tau}} \end{aligned}\]
\[\text{after each Euler--Maruyama sub-step: } x_i \leftarrow \min(\max(x_i,-x_{\max}),x_{\max}),\quad x_i\in\{u,h_W,T_C,T_E,\tau\}\]

Simplified model · linear dynamics + multiplicative noise

\[\begin{aligned} \frac{du}{dt} &= \class{kind-linear}{\widetilde a_{11}u} + \class{kind-linear}{\widetilde a_{12}h_W} + \class{kind-linear}{\widetilde a_{13}T_C} + \class{kind-linear}{\widetilde a_{14}T_E} + \class{kind-linear}{\widetilde a_{15}\tau} + \class{kind-noise}{\sigma_u(t)\dot W_u} \\ \frac{dh_W}{dt} &= \class{kind-linear}{\widetilde a_{21}u} + \class{kind-linear}{\widetilde a_{22}h_W} + \class{kind-linear}{\widetilde a_{23}T_C} + \class{kind-linear}{\widetilde a_{24}T_E} + \class{kind-linear}{\widetilde a_{25}\tau} + \class{kind-offset}{\widetilde a_{28}} + \class{kind-noise}{\sigma_h(h_W,t)\dot W_h} \\ \frac{dT_C}{dt} &= \class{kind-linear}{\widetilde a_{31}u} + \class{kind-linear}{\widetilde a_{32}h_W} + \class{kind-seasonal}{\widetilde a_{33}(t)T_C} + \class{kind-linear}{\widetilde a_{34}T_E} + \class{kind-linear}{\widetilde a_{35}\tau} + \class{kind-offset}{\widetilde a_{39}} + \class{kind-noise}{\sigma_C(T_C)\dot W_C} \\ \frac{dT_E}{dt} &= \class{kind-linear}{\widetilde a_{41}u} + \class{kind-linear}{\widetilde a_{42}h_W} + \class{kind-linear}{\widetilde a_{43}T_C} + \class{kind-seasonal}{\widetilde a_{44}(t)T_E} + \class{kind-linear}{\widetilde a_{45}\tau} + \class{kind-offset}{\widetilde a_{48}} + \class{kind-noise}{\sigma_E(T_E,t)\dot W_E} \\ \frac{d\tau}{dt} &= \class{kind-linear}{\widetilde a_{51}u} + \class{kind-linear}{\widetilde a_{52}h_W} + \class{kind-linear}{\widetilde a_{53}T_C} + \class{kind-linear}{\widetilde a_{54}T_E} + \class{kind-linear}{\widetilde a_{55}\tau} + \class{kind-noise}{\sigma_{\tau}(T_C,t)\dot W_{\tau}} \end{aligned}\]
\[\text{after each Euler--Maruyama sub-step: } x_i \leftarrow \min(\max(x_i,-x_{\max}),x_{\max}),\quad x_i\in\{u,h_W,T_C,T_E,\tau\}\]

34 drift coefficients, including offsets and seasonal SST growth. Independent noise drivers retain state-dependent amplitudes.

ModelStatisticsDAPredictionTotal
Original0.71450.63870.27380.5155
Simplified0.80990.59970.29730.5418
Change vs original+0.0953 (+13.34%)-0.0390 (-6.11%)+0.0235 (+8.57%)+0.0263 (+5.10%)

Kimi K3

round_59Original transcription verified

Original model

\[\begin{aligned} \frac{du}{dt} &= \class{term kind-linear tip-kimi-1}{a_{11}u} + \class{term kind-linear tip-kimi-2}{a_{12}h_W} + \class{term kind-linear tip-kimi-3}{a_{13}T_C} + \class{term kind-linear tip-kimi-4}{a_{14}T_E} + \class{term kind-linear tip-kimi-5}{a_{15}\tau} + \class{term kind-offset tip-kimi-6}{a_{16}} + \class{term kind-noise tip-kimi-7}{\big(\mathbf{B}(x,t)\,\dot{\mathbf{W}}\big)_{u}} \\ \frac{dh_W}{dt} &= \class{term kind-seasonal tip-kimi-16}{\mu_h(t)}\big[\class{term kind-linear tip-kimi-8}{a_{21}u} + \class{term kind-seasonal tip-kimi-9}{a_{22}(t)h_W} + \class{term kind-linear tip-kimi-10}{a_{23}T_C} + \class{term kind-linear tip-kimi-11}{a_{24}T_E} + \class{term kind-linear tip-kimi-12}{a_{25}\tau} + \class{term kind-offset tip-kimi-13}{a_{26}}\big] + \class{term kind-nonlinear tip-kimi-14}{a_{27}\max(T_E,0)^2} + \class{term kind-offset tip-kimi-15}{a_{28}} \\ &\quad + \class{term kind-noise tip-kimi-17}{\big(\mathbf{B}(x,t)\,\dot{\mathbf{W}}\big)_{h_W}} \\ \frac{dT_C}{dt} &= \class{term kind-linear tip-kimi-18}{a_{31}u} + \class{term kind-linear tip-kimi-19}{a_{32}h_W} + \class{term kind-seasonal tip-kimi-20}{a_{33}(t)T_C} + \class{term kind-linear tip-kimi-21}{a_{34}T_E} + \class{term kind-linear tip-kimi-22}{a_{35}\tau} + \class{term kind-offset tip-kimi-23}{a_{36}} + \class{term kind-noise tip-kimi-24}{\big(\mathbf{B}(x,t)\,\dot{\mathbf{W}}\big)_{T_C}} \\ \frac{dT_E}{dt} &= \class{term kind-linear tip-kimi-25}{a_{41}u} + \class{term kind-linear tip-kimi-26}{a_{42}h_W} + \class{term kind-linear tip-kimi-27}{a_{43}T_C} + \class{term kind-seasonal tip-kimi-28}{a_{44}(t)T_E} + \class{term kind-linear tip-kimi-29}{a_{45}\tau} + \class{term kind-nonlinear tip-kimi-30}{a_{46}\,\mathbb{1}[\bar T<0]\,T_E} + \class{term kind-offset tip-kimi-31}{a_{47}} \\ &\quad + \class{term kind-noise tip-kimi-32}{\big(\mathbf{B}(x,t)\,\dot{\mathbf{W}}\big)_{T_E}} \\ \frac{d\tau}{dt} &= \class{term kind-linear tip-kimi-33}{a_{51}u} + \class{term kind-linear tip-kimi-34}{a_{52}h_W} + \class{term kind-linear tip-kimi-35}{a_{53}T_C} + \class{term kind-linear tip-kimi-36}{a_{54}T_E} + \class{term kind-linear tip-kimi-37}{a_{55}\tau} + \class{term kind-nonlinear tip-kimi-38}{a_{56}\max(\bar T,0)} + \class{term kind-nonlinear tip-kimi-39}{a_{57}\min(\bar T,0)} + \class{term kind-offset tip-kimi-40}{a_{58}} \\ &\quad + \class{term kind-noise tip-kimi-41}{\big(\mathbf{B}(x,t)\,\dot{\mathbf{W}}\big)_{\tau}} \end{aligned}\]
\[\bar T=(T_C+T_E)/2,\qquad \mu_h(t)=1+h_{\mathrm{mod}}(m),\quad m=\lfloor t\rfloor\bmod12\]
\[\mathbf B(x,t)=\operatorname{diag}\!\left(n_u(m),n_h(m),n_C(m)(1+\epsilon_CT_C),n_E(m)(1+\epsilon_ET_E),n_\tau(m)\right)\mathbf B_0\]

Simplified model · linear dynamics + multiplicative noise

\[\begin{aligned} \frac{du}{dt} &= \class{kind-linear}{\widetilde a_{11}u} + \class{kind-linear}{\widetilde a_{12}h_W} + \class{kind-linear}{\widetilde a_{13}T_C} + \class{kind-linear}{\widetilde a_{14}T_E} + \class{kind-linear}{\widetilde a_{15}\tau} + \class{kind-offset}{\widetilde a_{16}} + \class{kind-noise}{\sigma_u n_u(m)\dot W_u} \\ \frac{dh_W}{dt} &= \class{kind-seasonal}{\mu_h(t)}\big[\class{kind-linear}{\widetilde a_{21}u} + \class{kind-seasonal}{\widetilde a_{22}(t)h_W} + \class{kind-linear}{\widetilde a_{23}T_C} + \class{kind-linear}{\widetilde a_{24}T_E} + \class{kind-linear}{\widetilde a_{25}\tau} + \class{kind-offset}{\widetilde a_{26}}\big] + \class{kind-noise}{\sigma_{h_W} n_h(m)\dot W_{h_W}} \\ \frac{dT_C}{dt} &= \class{kind-linear}{\widetilde a_{31}u} + \class{kind-linear}{\widetilde a_{32}h_W} + \class{kind-seasonal}{\widetilde a_{33}(t)T_C} + \class{kind-linear}{\widetilde a_{34}T_E} + \class{kind-linear}{\widetilde a_{35}\tau} + \class{kind-offset}{\widetilde a_{36}} + \class{kind-noise}{\sigma_C n_C(m)(1+\epsilon_C T_C)\dot W_C} \\ \frac{dT_E}{dt} &= \class{kind-linear}{\widetilde a_{41}u} + \class{kind-linear}{\widetilde a_{42}h_W} + \class{kind-linear}{\widetilde a_{43}T_C} + \class{kind-seasonal}{\widetilde a_{44}(t)T_E} + \class{kind-linear}{\widetilde a_{45}\tau} + \class{kind-offset}{\widetilde a_{47}} + \class{kind-noise}{\sigma_E n_E(m)(1+\epsilon_E T_E)\dot W_E} \\ \frac{d\tau}{dt} &= \class{kind-linear}{\widetilde a_{51}u} + \class{kind-linear}{\widetilde a_{52}h_W} + \class{kind-linear}{\widetilde a_{53}T_C} + \class{kind-linear}{\widetilde a_{54}T_E} + \class{kind-linear}{\widetilde a_{55}\tau} + \class{kind-offset}{\widetilde a_{58}} + \class{kind-noise}{\sigma_\tau n_\tau(m)\dot W_\tau} \end{aligned}\]

Independent noise drivers; seasonal and state-dependent amplitudes retained.

ModelStatisticsDAPredictionTotal
Original0.61980.57030.30530.4792
Simplified0.57530.57270.30220.4653
Change vs original-0.0445 (-7.18%)+0.0024 (+0.42%)-0.0031 (-1.03%)-0.0139 (-2.90%)

GPT-5.6-Sol

round_73Original transcription verified

Original model

\[\begin{aligned} \frac{du}{dt} &= \class{term kind-linear tip-gpt56-1}{a_{11}u} + \class{term kind-linear tip-gpt56-2}{a_{12}h_W} + \class{term kind-linear tip-gpt56-3}{a_{13}T_C} + \class{term kind-linear tip-gpt56-4}{a_{14}T_E} + \class{term kind-linear tip-gpt56-5}{a_{15}\tau} + \class{term kind-nonlinear tip-gpt56-6}{a_{16}\,\mathcal{C}(\hat u)} + \class{term kind-offset tip-gpt56-7}{a_{17}} \\ &\quad + \class{term kind-noise tip-gpt56-8}{\big(\mathbf{B}\,\dot{\boldsymbol{\xi}}\big)_{u}} \\ \frac{dh_W}{dt} &= \class{term kind-linear tip-gpt56-9}{a_{21}u} + \class{term kind-linear tip-gpt56-10}{a_{22}h_W} + \class{term kind-linear tip-gpt56-11}{a_{23}T_C} + \class{term kind-linear tip-gpt56-12}{a_{24}T_E} + \class{term kind-linear tip-gpt56-13}{a_{25}\tau} + \class{term kind-nonlinear tip-gpt56-14}{a_{26}\,\mathcal{C}(\hat h_W)} + \class{term kind-offset tip-gpt56-15}{a_{27}} \\ &\quad + \class{term kind-noise tip-gpt56-16}{\big(\mathbf{B}\,\dot{\boldsymbol{\xi}}\big)_{h_W}} \\ \frac{dT_C}{dt} &= \class{term kind-linear tip-gpt56-17}{a_{31}u} + \class{term kind-linear tip-gpt56-18}{a_{32}h_W} + \class{term kind-linear tip-gpt56-19}{a_{33}T_C} + \class{term kind-linear tip-gpt56-20}{a_{34}T_E} + \class{term kind-linear tip-gpt56-21}{a_{35}\tau} + \class{term kind-nonlinear tip-gpt56-22}{a_{36}\,\mathcal{C}(\hat T_C)} + \class{term kind-nonlinear tip-gpt56-23}{a_{37}\,\mathcal{C}(\max(\hat T_C,0))} + \class{term kind-offset tip-gpt56-24}{a_{38}} \\ &\quad + \class{term kind-noise tip-gpt56-25}{\big(\mathbf{B}\,\dot{\boldsymbol{\xi}}\big)_{T_C}} \\ \frac{dT_E}{dt} &= \class{term kind-linear tip-gpt56-26}{a_{41}u} + \class{term kind-linear tip-gpt56-27}{a_{42}h_W} + \class{term kind-seasonal tip-gpt56-28}{a_{43}(t)T_C} + \class{term kind-linear tip-gpt56-29}{a_{44}T_E} + \class{term kind-linear tip-gpt56-30}{a_{45}\tau} + \class{term kind-nonlinear tip-gpt56-31}{a_{46}\,\mathcal{C}(\hat T_E)} + \class{term kind-nonlinear tip-gpt56-32}{a_{47}\,\mathcal{C}(\min(\hat T_E,0))} \\ &\quad + \class{term kind-nonlinear tip-gpt56-33}{a_{48}\,\mathcal{Q}(\max(\hat T_C,0))} + \class{term kind-seasonal tip-gpt56-34}{a_{49}(t)} + \class{term kind-noise tip-gpt56-35}{\big(\mathbf{B}\,\dot{\boldsymbol{\xi}}\big)_{T_E}} \\ \frac{d\tau}{dt} &= \class{term kind-linear tip-gpt56-36}{a_{51}u} + \class{term kind-linear tip-gpt56-37}{a_{52}h_W} + \class{term kind-seasonal tip-gpt56-38}{a_{53}(t)T_C} + \class{term kind-linear tip-gpt56-39}{a_{54}T_E} + \class{term kind-linear tip-gpt56-40}{a_{55}\tau} + \class{term kind-nonlinear tip-gpt56-41}{a_{56}\,\mathcal{C}(\hat\tau)} + \class{term kind-seasonal tip-gpt56-42}{a_{57}(t)} \\ &\quad + \class{term kind-noise tip-gpt56-43}{\big(\mathbf{B}\,\dot{\boldsymbol{\xi}}\big)_{\tau}} \end{aligned}\]
\[\mathcal C(v)=\frac{v^3}{1+(v/a)^2},\qquad\mathcal Q(v)=\frac{v^2}{1+(v/a)^2},\qquad \hat x=x-m_x\]

Simplified model · nonlinear dynamics + additive noise

\[\begin{aligned} \frac{du}{dt} &= \class{kind-linear}{\widetilde a_{11}u} + \class{kind-linear}{\widetilde a_{12}h_W} + \class{kind-linear}{\widetilde a_{13}T_C} + \class{kind-linear}{\widetilde a_{14}T_E} + \class{kind-linear}{\widetilde a_{15}\tau} + \class{kind-nonlinear}{\widetilde a_{16}\,\big(\hat u\big)^3} + \class{kind-offset}{\widetilde a_{17}} \\ &\quad + \class{kind-noise}{\sigma_{u}\dot\xi_{u}} \\ \frac{dh_W}{dt} &= \class{kind-linear}{\widetilde a_{21}u} + \class{kind-linear}{\widetilde a_{22}h_W} + \class{kind-linear}{\widetilde a_{23}T_C} + \class{kind-linear}{\widetilde a_{24}T_E} + \class{kind-linear}{\widetilde a_{25}\tau} + \class{kind-nonlinear}{\widetilde a_{26}\,\big(\hat h_W\big)^3} + \class{kind-offset}{\widetilde a_{27}} \\ &\quad + \class{kind-noise}{\sigma_{h_W}\dot\xi_{h_W}} \\ \frac{dT_C}{dt} &= \class{kind-linear}{\widetilde a_{31}u} + \class{kind-linear}{\widetilde a_{32}h_W} + \class{kind-linear}{\widetilde a_{33}T_C} + \class{kind-linear}{\widetilde a_{34}T_E} + \class{kind-linear}{\widetilde a_{35}\tau} + \class{kind-nonlinear}{\widetilde a_{36}\,\big(\hat T_C\big)^3} + \class{kind-nonlinear}{\widetilde a_{37}\,\big(\max(\hat T_C,0)\big)^3} + \class{kind-offset}{\widetilde a_{38}} \\ &\quad + \class{kind-noise}{\sigma_{C}\dot\xi_{C}} \\ \frac{dT_E}{dt} &= \class{kind-linear}{\widetilde a_{41}u} + \class{kind-linear}{\widetilde a_{42}h_W} + \class{kind-seasonal}{\widetilde a_{43}(t)T_C} + \class{kind-linear}{\widetilde a_{44}T_E} + \class{kind-linear}{\widetilde a_{45}\tau} + \class{kind-nonlinear}{\widetilde a_{46}\,\big(\hat T_E\big)^3} + \class{kind-nonlinear}{\widetilde a_{47}\,\big(\min(\hat T_E,0)\big)^3} \\ &\quad + \class{kind-nonlinear}{\widetilde a_{48}\,\mathcal{Q}(\max(\hat T_C,0))} + \class{kind-seasonal}{\widetilde a_{49}(t)} + \class{kind-noise}{\sigma_{E}\dot\xi_{E}} \\ \frac{d\tau}{dt} &= \class{kind-linear}{\widetilde a_{51}u} + \class{kind-linear}{\widetilde a_{52}h_W} + \class{kind-seasonal}{\widetilde a_{53}(t)T_C} + \class{kind-linear}{\widetilde a_{54}T_E} + \class{kind-linear}{\widetilde a_{55}\tau} + \class{kind-nonlinear}{\widetilde a_{56}\,\big(\hat\tau\big)^3} + \class{kind-seasonal}{\widetilde a_{57}(t)} \\ &\quad + \class{kind-noise}{\sigma_{\tau}\dot\xi_{\tau}} \end{aligned}\]

Independent non-Gaussian drivers; no cross-equation noise mixing.

ModelStatisticsDAPredictionTotal
Original0.55440.51500.30630.4433
Simplified0.61310.52920.30380.4642
Change vs original+0.0588 (+10.60%)+0.0142 (+2.76%)-0.0025 (-0.82%)+0.0209 (+4.71%)

GPT-6 Astra

round_46Original transcription verified

Original model

\[\begin{aligned} \frac{du}{ds} &= \class{term kind-seasonal tip-gpt6astra-1}{a_{11}(s)\,u} + \class{term kind-seasonal tip-gpt6astra-2}{a_{12}(s)\,H} + \class{term kind-seasonal tip-gpt6astra-3}{a_{13}(s)\,T_C} + \class{term kind-seasonal tip-gpt6astra-4}{a_{14}(s)\,T_E} + \class{term kind-seasonal tip-gpt6astra-5}{a_{15}(s)\,\tau} + \class{term kind-offset tip-gpt6astra-6}{a_{16}} + \class{term kind-noise tip-gpt6astra-7}{\big(\mathbf{B}(x,s)\,\dot{\mathbf{W}}\big)_{u}} \\ \frac{dH}{ds} &= \class{term kind-seasonal tip-gpt6astra-8}{a_{21}(s)\,u} + \class{term kind-seasonal tip-gpt6astra-9}{a_{22}(s)\,H} + \class{term kind-seasonal tip-gpt6astra-10}{a_{23}(s)\,T_C} + \class{term kind-seasonal tip-gpt6astra-11}{a_{24}(s)\,T_E} + \class{term kind-seasonal tip-gpt6astra-12}{a_{25}(s)\,\tau} + \class{term kind-nonlinear tip-gpt6astra-13}{a_{26}\,T_E^2} + \class{term kind-offset tip-gpt6astra-14}{a_{27}} + \class{term kind-noise tip-gpt6astra-15}{\big(\mathbf{B}(x,s)\,\dot{\mathbf{W}}\big)_{H}} \\ \frac{dT_C}{ds} &= \class{term kind-seasonal tip-gpt6astra-16}{a_{31}(s)\,u} + \class{term kind-seasonal tip-gpt6astra-17}{a_{32}(s)\,H} + \class{term kind-seasonal tip-gpt6astra-18}{a_{33}(s)\,T_C} + \class{term kind-seasonal tip-gpt6astra-19}{a_{34}(s)\,T_E} + \class{term kind-seasonal tip-gpt6astra-20}{a_{35}(s)\,\tau} + \class{term kind-nonlinear tip-gpt6astra-21}{a_{36}\,T_C^3} + \class{term kind-nonlinear tip-gpt6astra-22}{a_{37}\tanh\!\left(a_C T_C + z_C\right)} + \class{term kind-offset tip-gpt6astra-23}{a_{38}} \\ &\quad + \class{term kind-noise tip-gpt6astra-24}{\big(\mathbf{B}(x,s)\,\dot{\mathbf{W}}\big)_{T_C}} \\ \frac{dT_E}{ds} &= \class{term kind-seasonal tip-gpt6astra-25}{a_{41}(s)\,u} + \class{term kind-linear tip-gpt6astra-26}{a_{42}\,H} + \class{term kind-seasonal tip-gpt6astra-27}{a_{43}(s)\,T_C} + \class{term kind-seasonal tip-gpt6astra-28}{a_{44}(s)\,T_E} + \class{term kind-seasonal tip-gpt6astra-29}{a_{45}(s)\,\tau} + \class{term kind-nonlinear tip-gpt6astra-30}{a_{46}\,T_E^3} + \class{term kind-nonlinear tip-gpt6astra-31}{a_{47}\tanh\!\left(a_E T_E + a_H H + z_E\right)} \\ &\quad + \class{term kind-nonlinear tip-gpt6astra-32}{a_{48}(s)\,\tau\tanh\!\left(\frac{T_E - T_C}{w_g}\right)} + \class{term kind-offset tip-gpt6astra-33}{a_{49}} + \class{term kind-noise tip-gpt6astra-34}{\big(\mathbf{B}(x,s)\,\dot{\mathbf{W}}\big)_{T_E}} \\ \frac{d\tau}{ds} &= \class{term kind-linear tip-gpt6astra-35}{a_{51}\,u} + \class{term kind-linear tip-gpt6astra-36}{a_{52}\,H} + \class{term kind-seasonal tip-gpt6astra-37}{a_{53}(s)\,T_C} + \class{term kind-seasonal tip-gpt6astra-38}{a_{54}(s)\,T_E} + \class{term kind-seasonal tip-gpt6astra-39}{a_{55}(s)\,\tau} + \class{term kind-nonlinear tip-gpt6astra-40}{a_{56}\,T_C^2} + \class{term kind-nonlinear tip-gpt6astra-41}{a_{57}(s)\log\cosh\!\left(\frac{\tau}{w_{\tau}}\right)} + \class{term kind-seasonal tip-gpt6astra-42}{a_{58}(s)} \\ &\quad + \class{term kind-noise tip-gpt6astra-43}{\big(\mathbf{B}(x,s)\,\dot{\mathbf{W}}\big)_{\tau}} \\ h_W &= H - \eta\left(\sqrt{H^2 + L_H^2} - L_H\right) \end{aligned}\]

Simplified model · nonlinear dynamics + additive noise

\[\begin{aligned} \frac{du}{ds} &= \class{kind-seasonal}{\widetilde a_{11}(s)\,u} + \class{kind-seasonal}{\widetilde a_{12}(s)\,H} + \class{kind-seasonal}{\widetilde a_{13}(s)\,T_C} + \class{kind-seasonal}{\widetilde a_{14}(s)\,T_E} + \class{kind-seasonal}{\widetilde a_{15}(s)\,\tau} + \class{kind-offset}{\widetilde a_{16}} + \class{kind-noise}{\big(\mathbf{B}(s)\,\dot{\mathbf{W}}\big)_{u}} \\ \frac{dH}{ds} &= \class{kind-seasonal}{\widetilde a_{21}(s)\,u} + \class{kind-seasonal}{\widetilde a_{22}(s)\,H} + \class{kind-seasonal}{\widetilde a_{23}(s)\,T_C} + \class{kind-seasonal}{\widetilde a_{24}(s)\,T_E} + \class{kind-seasonal}{\widetilde a_{25}(s)\,\tau} + \class{kind-offset}{\widetilde a_{27}} + \class{kind-noise}{\big(\mathbf{B}(s)\,\dot{\mathbf{W}}\big)_{H}} \\ \frac{dT_C}{ds} &= \class{kind-seasonal}{\widetilde a_{31}(s)\,u} + \class{kind-seasonal}{\widetilde a_{32}(s)\,H} + \class{kind-seasonal}{\widetilde a_{33}(s)\,T_C} + \class{kind-seasonal}{\widetilde a_{34}(s)\,T_E} + \class{kind-seasonal}{\widetilde a_{35}(s)\,\tau} + \class{kind-offset}{\widetilde a_{38}} \\ &\quad + \class{kind-noise}{\big(\mathbf{B}(s)\,\dot{\mathbf{W}}\big)_{T_C}} \\ \frac{dT_E}{ds} &= \class{kind-seasonal}{\widetilde a_{41}(s)\,u} + \class{kind-linear}{\widetilde a_{42}\,H} + \class{kind-seasonal}{\widetilde a_{43}(s)\,T_C} + \class{kind-seasonal}{\widetilde a_{44}(s)\,T_E} + \class{kind-seasonal}{\widetilde a_{45}(s)\,\tau} + \class{kind-nonlinear}{\widetilde a_{47}\tanh\!\left(a_E T_E + a_H H + z_E\right)} \\ &\quad + \class{kind-offset}{\widetilde a_{49}} + \class{kind-noise}{\big(\mathbf{B}(s)\,\dot{\mathbf{W}}\big)_{T_E}} \\ \frac{d\tau}{ds} &= \class{kind-linear}{\widetilde a_{51}\,u} + \class{kind-linear}{\widetilde a_{52}\,H} + \class{kind-seasonal}{\widetilde a_{53}(s)\,T_C} + \class{kind-seasonal}{\widetilde a_{54}(s)\,T_E} + \class{kind-seasonal}{\widetilde a_{55}(s)\,\tau} + \class{kind-seasonal}{\widetilde a_{58}(s)} \\ &\quad + \class{kind-noise}{\big(\mathbf{B}(s)\,\dot{\mathbf{W}}\big)_{\tau}} \\ h_W &= H - \eta\left(\sqrt{H^2 + L_H^2} - L_H\right) \end{aligned}\]

The bounded thermocline feedback of T_E is the retained nonlinearity. Seasonal coefficients keep their annual cycle only. The noise is additive with an annual modulation (state dependence is inactive in both models). The gated wind term of the original has a zero coefficient and is not shown.

ModelStatisticsDAPredictionTotal
Original0.52980.62880.23720.4425
Simplified0.59840.61090.24810.4620
Change vs original+0.0686 (+12.96%)-0.0179 (-2.85%)+0.0109 (+4.61%)+0.0196 (+4.43%)

Claude Opus 5

round_72Original transcription verified

Original model

\[\begin{aligned} \frac{du}{d\tilde t} &= \class{term kind-linear tip-opus5-1}{a_{11}u} + \class{term kind-linear tip-opus5-2}{a_{12}H} + \class{term kind-linear tip-opus5-3}{a_{13}T_C} + \class{term kind-linear tip-opus5-4}{a_{14}T_E} + \class{term kind-linear tip-opus5-5}{a_{15}\tau} + \class{term kind-noise tip-opus5-6}{\big(\mathbf{B}(x,t)\,\dot{\boldsymbol{\xi}}\big)_{u}} \\ \frac{dH}{d\tilde t} &= \class{term kind-linear tip-opus5-7}{a_{21}H} + \class{term kind-seasonal tip-opus5-8}{a_{22}(t)T_E} + \class{term kind-linear tip-opus5-9}{a_{23}T_C} + \class{term kind-linear tip-opus5-10}{a_{24}u} + \class{term kind-linear tip-opus5-11}{a_{25}\tau} + \class{term kind-nonlinear tip-opus5-12}{a_{26}T_E^2} + \class{term kind-nonlinear tip-opus5-13}{a_{27}\frac{T_E^3}{D_H(T_E)}} \\ &\quad + \class{term kind-nonlinear tip-opus5-14}{a_{28}F_E\frac{T_E^2}{T_d^2+T_E^2}} + \class{term kind-offset tip-opus5-15}{a_{29}} + \class{term kind-noise tip-opus5-16}{\big(\mathbf{B}(x,t)\,\dot{\boldsymbol{\xi}}\big)_{H}} \\ \frac{dT_C}{d\tilde t} &= \class{term kind-linear tip-opus5-17}{a_{31}H} + \class{term kind-seasonal tip-opus5-18}{a_{32}(t)T_C} + \class{term kind-linear tip-opus5-19}{a_{33}T_E} + \class{term kind-linear tip-opus5-20}{a_{34}\tau} + \class{term kind-nonlinear tip-opus5-21}{a_{35}T_C^2} + \class{term kind-linear tip-opus5-22}{a_{36}u} + \class{term kind-nonlinear tip-opus5-23}{a_{37}T_E\tanh(T_E/T_a)} \\ &\quad + \class{term kind-nonlinear tip-opus5-24}{a_{38}T_C^3} + \class{term kind-nonlinear tip-opus5-25}{a_{39}T_C^5} + \class{term kind-offset tip-opus5-26}{a_{3,10}} + \class{term kind-noise tip-opus5-27}{\big(\mathbf{B}(x,t)\,\dot{\boldsymbol{\xi}}\big)_{T_C}} \\ \frac{dT_E}{d\tilde t} &= \class{term kind-linear tip-opus5-28}{a_{41}H} + \class{term kind-seasonal tip-opus5-29}{a_{42}(t)T_E} + \class{term kind-linear tip-opus5-30}{a_{43}T_C} + \class{term kind-linear tip-opus5-31}{a_{44}\tau} + \class{term kind-nonlinear tip-opus5-32}{a_{45}T_E^2} + \class{term kind-linear tip-opus5-33}{a_{46}u} + \class{term kind-nonlinear tip-opus5-34}{a_{47}T_E^3} + \class{term kind-nonlinear tip-opus5-35}{a_{48}T_E^3\tanh(T_E/T_a)} \\ &\quad + \class{term kind-nonlinear tip-opus5-36}{a_{49}T_E^5} + \class{term kind-nonlinear tip-opus5-37}{a_{4,10}\tanh(H/H_s)\,(a_{41}H + a_{46}u + a_{44}\tau)} + \class{term kind-offset tip-opus5-38}{a_{4,11}} + \class{term kind-noise tip-opus5-39}{\big(\mathbf{B}(x,t)\,\dot{\boldsymbol{\xi}}\big)_{T_E}} \\ \frac{d\tau}{d\tilde t} &= \class{term kind-linear tip-opus5-40}{a_{51}\tau} + \class{term kind-linear tip-opus5-41}{a_{52}T_C} + \class{term kind-linear tip-opus5-42}{a_{53}T_E} + \class{term kind-linear tip-opus5-43}{a_{54}H} + \class{term kind-noise tip-opus5-44}{\big(\mathbf{B}(x,t)\,\dot{\boldsymbol{\xi}}\big)_{\tau}} \end{aligned}\]
\[H=h_W+bT_E,\qquad D_H(T_E)=1+(T_E/T_s)^2\left[1+k_s\tanh(T_E/T_a)\right]\]
\[F_E\text{ denotes the deterministic }T_E\text{ tendency.}\]

Simplified model · nonlinear dynamics + additive noise

\[\begin{aligned} \frac{du}{d\tilde t} &= \class{kind-linear}{\widetilde a_{11}u} + \class{kind-linear}{\widetilde a_{12}H} + \class{kind-linear}{\widetilde a_{13}T_C} + \class{kind-linear}{\widetilde a_{14}T_E} + \class{kind-linear}{\widetilde a_{15}\tau} + \class{kind-noise}{\big(\mathbf{B}(t)\,\dot{\boldsymbol{\xi}}\big)_{u}} \\ \frac{dH}{d\tilde t} &= \class{kind-linear}{\widetilde a_{21}H} + \class{kind-seasonal}{\widetilde a_{22}(t)T_E} + \class{kind-linear}{\widetilde a_{24}u} + \class{kind-linear}{\widetilde a_{25}\tau} + \class{kind-noise}{\big(\mathbf{B}(t)\,\dot{\boldsymbol{\xi}}\big)_{H}} \\ \frac{dT_C}{d\tilde t} &= \class{kind-linear}{\widetilde a_{31}H} + \class{kind-seasonal}{\widetilde a_{32}(t)T_C} + \class{kind-linear}{\widetilde a_{33}T_E} + \class{kind-linear}{\widetilde a_{34}\tau} + \class{kind-linear}{\widetilde a_{36}u} + \class{kind-nonlinear}{\widetilde a_{39}T_C^5} + \class{kind-noise}{\big(\mathbf{B}(t)\,\dot{\boldsymbol{\xi}}\big)_{T_C}} \\ \frac{dT_E}{d\tilde t} &= \class{kind-linear}{\widetilde a_{41}H} + \class{kind-seasonal}{\widetilde a_{42}(t)T_E} + \class{kind-linear}{\widetilde a_{44}\tau} + \class{kind-nonlinear}{\widetilde a_{45}T_E^2} + \class{kind-linear}{\widetilde a_{46}u} + \class{kind-nonlinear}{\widetilde a_{49}T_E^5} + \class{kind-offset}{\widetilde a_{4,11}} + \class{kind-noise}{\big(\mathbf{B}(t)\,\dot{\boldsymbol{\xi}}\big)_{T_E}} \\ \frac{d\tau}{d\tilde t} &= \class{kind-linear}{\widetilde a_{51}\tau} + \class{kind-linear}{\widetilde a_{52}T_C} + \class{kind-linear}{\widetilde a_{53}T_E} + \class{kind-linear}{\widetilde a_{54}H} + \class{kind-noise}{\big(\mathbf{B}(t)\,\dot{\boldsymbol{\xi}}\big)_{\tau}} \end{aligned}\]

State-independent, seasonally varying noise. Skewed innovations and shared forcing are retained.

ModelStatisticsDAPredictionTotal
Original0.54760.58930.24730.4400
Simplified0.62930.56710.28570.4732
Change vs original+0.0817 (+14.92%)-0.0223 (-3.78%)+0.0384 (+15.53%)+0.0332 (+7.54%)

GPT-5.5

round_71Original transcription verified

Original model

\[\begin{aligned} \frac{du}{dt} &= \class{term kind-linear tip-gpt55-1}{a_{11}u} + \class{term kind-linear tip-gpt55-2}{a_{12}h_W} + \class{term kind-linear tip-gpt55-3}{a_{13}T_C} + \class{term kind-linear tip-gpt55-4}{a_{14}T_E} + \class{term kind-linear tip-gpt55-5}{a_{15}\tau} + \class{term kind-nonlinear tip-gpt55-6}{a_{16}u^3} + \class{term kind-offset tip-gpt55-7}{a_{17}} \\ &\quad + \class{term kind-noise tip-gpt55-8}{\big(\mathbf{B}(\mathbf{x},t)\,\dot{\mathbf{W}}\big)_{u}} \\ \frac{dh_W}{dt} &= \class{term kind-linear tip-gpt55-9}{a_{21}u} + \class{term kind-linear tip-gpt55-10}{a_{22}h_W} + \class{term kind-linear tip-gpt55-11}{a_{23}T_C} + \class{term kind-linear tip-gpt55-12}{a_{24}T_E} + \class{term kind-linear tip-gpt55-13}{a_{25}\tau} + \class{term kind-nonlinear tip-gpt55-14}{a_{26}h_W^3} + \class{term kind-nonlinear tip-gpt55-15}{a_{27}[h_W]_-^2} + \class{term kind-nonlinear tip-gpt55-16}{a_{28}T_E[T_E-\theta_R]_+} + \class{term kind-offset tip-gpt55-17}{a_{29}} \\ &\quad + \class{term kind-noise tip-gpt55-18}{\big(\mathbf{B}(\mathbf{x},t)\,\dot{\mathbf{W}}\big)_{h_W}} \\ \frac{dT_C}{dt} &= \class{term kind-linear tip-gpt55-19}{a_{31}u} + \class{term kind-linear tip-gpt55-20}{a_{32}h_W} + \class{term kind-linear tip-gpt55-21}{a_{33}T_C} + \class{term kind-linear tip-gpt55-22}{a_{34}T_E} + \class{term kind-linear tip-gpt55-23}{a_{35}\tau} + \class{term kind-nonlinear tip-gpt55-24}{a_{36}T_C^3} + \class{term kind-nonlinear tip-gpt55-25}{a_{37}[T_C]_-^2} + \class{term kind-nonlinear tip-gpt55-26}{a_{38}[T_C+\theta_C]_-^2} + \class{term kind-offset tip-gpt55-27}{a_{39}} \\ &\quad + \class{term kind-noise tip-gpt55-28}{\big(\mathbf{B}(\mathbf{x},t)\,\dot{\mathbf{W}}\big)_{T_C}} \\ \frac{dT_E}{dt} &= \class{term kind-linear tip-gpt55-29}{a_{41}u} + \class{term kind-linear tip-gpt55-30}{a_{42}h_W} + \class{term kind-linear tip-gpt55-31}{a_{43}T_C} + \class{term kind-seasonal tip-gpt55-32}{a_{44}(t)T_E} + \class{term kind-linear tip-gpt55-33}{a_{45}\tau} + \class{term kind-nonlinear tip-gpt55-34}{a_{46}T_E^3} + \class{term kind-nonlinear tip-gpt55-35}{a_{47}[T_E]_+^2} + \class{term kind-nonlinear tip-gpt55-36}{a_{48}[T_E]_-^2} + \class{term kind-offset tip-gpt55-37}{a_{49}} \\ &\quad + \class{term kind-noise tip-gpt55-38}{\big(\mathbf{B}(\mathbf{x},t)\,\dot{\mathbf{W}}\big)_{T_E}} \\ \frac{d\tau}{dt} &= \class{term kind-linear tip-gpt55-39}{a_{51}u} + \class{term kind-linear tip-gpt55-40}{a_{52}h_W} + \class{term kind-linear tip-gpt55-41}{a_{53}T_C} + \class{term kind-linear tip-gpt55-42}{a_{54}T_E} + \class{term kind-linear tip-gpt55-43}{a_{55}\tau} + \class{term kind-nonlinear tip-gpt55-44}{a_{56}\tau^3} + \class{term kind-offset tip-gpt55-45}{a_{57}} \\ &\quad + \class{term kind-noise tip-gpt55-46}{\big(\mathbf{B}(\mathbf{x},t)\,\dot{\mathbf{W}}\big)_{\tau}} \\ \mathbf{x}_{n+1} &= \min\!\Big(\max\big(\mathbf{x}_n + \mathbf{f}(\mathbf{x}_n,t_n)\,\Delta t + \mathbf{B}(\mathbf{x}_n,t_n)\,\Delta\mathbf{W}_n,\ \mathbf{x}^{\min}\big),\ \mathbf{x}^{\max}\Big) \end{aligned}\]

Simplified model · nonlinear dynamics + additive noise

\[\begin{aligned} \frac{du}{dt} &= \class{kind-linear}{a_{11}u} + \class{kind-linear}{a_{12}h_W} + \class{kind-linear}{a_{13}T_C} + \class{kind-linear}{a_{14}T_E} + \class{kind-linear}{a_{15}\tau} + \class{kind-nonlinear}{a_{16}u^3} + \class{kind-offset}{a_{17}} \\ &\quad + \class{kind-noise}{\widetilde\sigma_{u}(t)\dot W_{u}} \\ \frac{dh_W}{dt} &= \class{kind-linear}{a_{21}u} + \class{kind-linear}{a_{22}h_W} + \class{kind-linear}{a_{23}T_C} + \class{kind-linear}{a_{24}T_E} + \class{kind-linear}{a_{25}\tau} + \class{kind-nonlinear}{a_{26}h_W^3} + \class{kind-nonlinear}{a_{27}[h_W]_-^2} + \class{kind-nonlinear}{a_{28}T_E[T_E-\theta_R]_+} + \class{kind-offset}{a_{29}} \\ &\quad + \class{kind-noise}{\widetilde\sigma_{h}(t)\dot W_{h_W}} \\ \frac{dT_C}{dt} &= \class{kind-linear}{a_{31}u} + \class{kind-linear}{a_{32}h_W} + \class{kind-linear}{a_{33}T_C} + \class{kind-linear}{a_{34}T_E} + \class{kind-linear}{a_{35}\tau} + \class{kind-nonlinear}{a_{36}T_C^3} + \class{kind-nonlinear}{a_{37}[T_C]_-^2} + \class{kind-nonlinear}{a_{38}[T_C+\theta_C]_-^2} + \class{kind-offset}{a_{39}} \\ &\quad + \class{kind-noise}{\widetilde\sigma_{C}(t)\dot W_{T_C}} \\ \frac{dT_E}{dt} &= \class{kind-linear}{a_{41}u} + \class{kind-linear}{a_{42}h_W} + \class{kind-linear}{a_{43}T_C} + \class{kind-seasonal}{a_{44}(t)T_E} + \class{kind-linear}{a_{45}\tau} + \class{kind-nonlinear}{a_{46}T_E^3} + \class{kind-nonlinear}{a_{47}[T_E]_+^2} + \class{kind-nonlinear}{a_{48}[T_E]_-^2} + \class{kind-offset}{a_{49}} \\ &\quad + \class{kind-noise}{\widetilde\sigma_{E}(t)\dot W_{T_E}} \\ \frac{d\tau}{dt} &= \class{kind-linear}{a_{51}u} + \class{kind-linear}{a_{52}h_W} + \class{kind-linear}{a_{53}T_C} + \class{kind-linear}{a_{54}T_E} + \class{kind-linear}{a_{55}\tau} + \class{kind-nonlinear}{a_{56}\tau^3} + \class{kind-offset}{a_{57}} \\ &\quad + \class{kind-noise}{\widetilde\sigma_{\tau}(t)\dot W_{\tau}} \\ \mathbf{x}_{n+1} &= \min\!\Big(\max\big(\mathbf{x}_n + \mathbf{f}(\mathbf{x}_n,t_n)\,\Delta t + \widetilde{\boldsymbol{\Sigma}}(t_n)\,\Delta\mathbf{W}_n,\ \mathbf{x}^{\min}\big),\ \mathbf{x}^{\max}\Big) \end{aligned}\]

Nonlinear drift unchanged. The noise is additive, independent across variables and seasonally modulated.

ModelStatisticsDAPredictionTotal
Original0.41180.59530.25150.4027
Simplified0.39030.60010.25120.3976
Change vs original-0.0216 (-5.23%)+0.0048 (+0.80%)-0.0003 (-0.12%)-0.0052 (-1.28%)
Also qualifies: linear drift + multiplicative noise0.38700.59010.25500.3951
Change vs original-0.0248 (-6.03%)-0.0052 (-0.88%)+0.0035 (+1.39%)-0.0076 (-1.89%)

A second reduction of the other family also qualifies: removing every nonlinear drift term while keeping the state-dependent noise (linear drift + multiplicative noise). The vote keeps the higher-scoring additive-noise reduction shown above.

Gemini 3.8 Flash (medium)

round_69Original transcription verified

Original model

\[\begin{aligned} \frac{du}{dt} &= \class{term kind-linear tip-gemini38-1}{a_{11}u} + \class{term kind-linear tip-gemini38-2}{a_{12}h_W} + \class{term kind-linear tip-gemini38-3}{a_{13}\tau} + \class{term kind-noise tip-gemini38-4}{\sigma_u \dot W_u} \\ \frac{dh_W}{dt} &= \class{term kind-linear tip-gemini38-5}{a_{21}h_W} + \class{term kind-linear tip-gemini38-6}{a_{22}T_E} + \class{term kind-linear tip-gemini38-7}{a_{23}\tau} + \class{term kind-linear tip-gemini38-8}{a_{24}u} + \class{term kind-noise tip-gemini38-9}{\sigma_h \dot W_h} \\ \frac{dT_C}{dt} &= \class{term kind-linear tip-gemini38-10}{a_{31}h_W} + \class{term kind-linear tip-gemini38-11}{a_{32}T_C} + \class{term kind-linear tip-gemini38-12}{a_{33}T_E} + \class{term kind-linear tip-gemini38-13}{a_{34}u} + \class{term kind-nonlinear tip-gemini38-14}{a_{35}[\max(0,T_C)]^2} + \class{term kind-nonlinear tip-gemini38-15}{a_{36}T_C^3} + \class{term kind-offset tip-gemini38-16}{a_{37}} + \class{term kind-noise tip-gemini38-17}{\sigma_C \dot W_C} \\ \frac{dT_E}{dt} &= \class{term kind-seasonal tip-gemini38-18}{a_{41}(t)h_W} + \class{term kind-linear tip-gemini38-19}{a_{42}T_C} + \class{term kind-linear tip-gemini38-20}{a_{43}T_E} + \class{term kind-seasonal tip-gemini38-21}{a_{44}(t)\tau} + \class{term kind-linear tip-gemini38-22}{a_{45}u} \\ &\quad + \class{term kind-nonlinear tip-gemini38-23}{a_{46}(t)[\max(0,h_W + b_h\tau)]^2} + \class{term kind-nonlinear tip-gemini38-24}{a_{47}T_E^3} + \class{term kind-seasonal tip-gemini38-25}{a_{48}(t)} + \class{term kind-noise tip-gemini38-26}{\sigma_E \dot W_E} \\ \frac{d\tau}{dt} &= \class{term kind-linear tip-gemini38-27}{a_{51}\tau} + \class{term kind-seasonal tip-gemini38-28}{a_{52}(t)T_C} + \class{term kind-seasonal tip-gemini38-29}{a_{53}(t)T_E} + \class{term kind-noise tip-gemini38-30}{\sigma_\tau(T_C) \dot W_\tau} \end{aligned}\]

Simplified model · nonlinear dynamics + additive noise

\[\begin{aligned} \frac{du}{dt} &= \class{kind-linear}{a_{11}u} + \class{kind-linear}{a_{12}h_W} + \class{kind-linear}{a_{13}\tau} + \class{kind-noise}{\sigma_u \dot W_u} \\ \frac{dh_W}{dt} &= \class{kind-linear}{a_{21}h_W} + \class{kind-linear}{a_{22}T_E} + \class{kind-linear}{a_{23}\tau} + \class{kind-linear}{a_{24}u} + \class{kind-noise}{\sigma_h \dot W_h} \\ \frac{dT_C}{dt} &= \class{kind-linear}{a_{31}h_W} + \class{kind-linear}{a_{32}T_C} + \class{kind-linear}{a_{33}T_E} + \class{kind-linear}{a_{34}u} + \class{kind-nonlinear}{a_{35}[\max(0,T_C)]^2} + \class{kind-nonlinear}{a_{36}T_C^3} + \class{kind-offset}{a_{37}} + \class{kind-noise}{\sigma_C \dot W_C} \\ \frac{dT_E}{dt} &= \class{kind-seasonal}{a_{41}(t)h_W} + \class{kind-linear}{a_{42}T_C} + \class{kind-linear}{a_{43}T_E} + \class{kind-seasonal}{a_{44}(t)\tau} + \class{kind-linear}{a_{45}u} \\ &\quad + \class{kind-nonlinear}{a_{46}(t)[\max(0,h_W + b_h\tau)]^2} + \class{kind-nonlinear}{a_{47}T_E^3} + \class{kind-seasonal}{a_{48}(t)} + \class{kind-noise}{\sigma_E \dot W_E} \\ \frac{d\tau}{dt} &= \class{kind-linear}{a_{51}\tau} + \class{kind-seasonal}{a_{52}(t)T_C} + \class{kind-seasonal}{a_{53}(t)T_E} + \class{kind-noise}{\widetilde\sigma_\tau \dot W_\tau} \end{aligned}\]

Nonlinear drift unchanged. The wind noise becomes state-independent.

ModelStatisticsDAPredictionTotal
Original0.40610.50330.24140.3694
Simplified0.38850.49900.24060.3625
Change vs original-0.0176 (-4.33%)-0.0043 (-0.86%)-0.0008 (-0.34%)-0.0069 (-1.87%)

DeepSeek V4.1 Flash

round_12Original transcription verified

Original model

\[\begin{aligned} \frac{du}{dt} &= \class{term kind-seasonal tip-dsflash-1}{a_{11}(t)u} + \class{term kind-linear tip-dsflash-2}{a_{12}h_W} + \class{term kind-linear tip-dsflash-3}{a_{13}T_C} + \class{term kind-linear tip-dsflash-4}{a_{14}T_E} + \class{term kind-linear tip-dsflash-5}{a_{15}\tau} + \class{term kind-noise tip-dsflash-6}{\big(\mathbf{B}(x)\,\dot{\mathbf{W}}\big)_{u}} \\ \frac{dh_W}{dt} &= \class{term kind-linear tip-dsflash-7}{a_{21}u} + \class{term kind-seasonal tip-dsflash-8}{a_{22}(t)h_W} + \class{term kind-linear tip-dsflash-9}{a_{23}T_C} + \class{term kind-linear tip-dsflash-10}{a_{24}T_E} + \class{term kind-linear tip-dsflash-11}{a_{25}\tau} + \class{term kind-noise tip-dsflash-12}{\big(\mathbf{B}(x)\,\dot{\mathbf{W}}\big)_{h_W}} \\ \frac{dT_C}{dt} &= \class{term kind-linear tip-dsflash-13}{a_{31}u} + \class{term kind-linear tip-dsflash-14}{a_{32}h_W} + \class{term kind-seasonal tip-dsflash-15}{a_{33}(t)T_C} + \class{term kind-linear tip-dsflash-16}{a_{34}T_E} + \class{term kind-linear tip-dsflash-17}{a_{35}\tau} + \class{term kind-noise tip-dsflash-18}{\big(\mathbf{B}(x)\,\dot{\mathbf{W}}\big)_{T_C}} \\ \frac{dT_E}{dt} &= \class{term kind-linear tip-dsflash-19}{a_{41}u} + \class{term kind-linear tip-dsflash-20}{a_{42}h_W} + \class{term kind-linear tip-dsflash-21}{a_{43}T_C} + \class{term kind-seasonal tip-dsflash-22}{a_{44}(t)T_E} + \class{term kind-linear tip-dsflash-23}{a_{45}\tau} + \class{term kind-nonlinear tip-dsflash-24}{a_{46}\,\frac{T_E^2}{1+(T_E/s_q)^2}} + \class{term kind-noise tip-dsflash-25}{\big(\mathbf{B}(x)\,\dot{\mathbf{W}}\big)_{T_E}} \\ \frac{d\tau}{dt} &= \class{term kind-linear tip-dsflash-26}{a_{51}u} + \class{term kind-linear tip-dsflash-27}{a_{52}h_W} + \class{term kind-linear tip-dsflash-28}{a_{53}T_C} + \class{term kind-linear tip-dsflash-29}{a_{54}T_E} + \class{term kind-seasonal tip-dsflash-30}{a_{55}(t)\tau} + \class{term kind-noise tip-dsflash-31}{\big(\mathbf{B}(x)\,\dot{\mathbf{W}}\big)_{\tau}} \end{aligned}\]

Simplified model · linear dynamics + multiplicative noise

\[\begin{aligned} \frac{du}{dt} &= \class{kind-seasonal}{\widetilde a_{11}(t)u} + \class{kind-linear}{\widetilde a_{12}h_W} + \class{kind-linear}{\widetilde a_{13}T_C} + \class{kind-linear}{\widetilde a_{14}T_E} + \class{kind-linear}{\widetilde a_{15}\tau} + \class{kind-noise}{\big(\mathbf{B}(x)\,\dot{\mathbf{W}}\big)_{u}} \\ \frac{dh_W}{dt} &= \class{kind-linear}{\widetilde a_{21}u} + \class{kind-seasonal}{\widetilde a_{22}(t)h_W} + \class{kind-linear}{\widetilde a_{23}T_C} + \class{kind-linear}{\widetilde a_{24}T_E} + \class{kind-linear}{\widetilde a_{25}\tau} + \class{kind-noise}{\big(\mathbf{B}(x)\,\dot{\mathbf{W}}\big)_{h_W}} \\ \frac{dT_C}{dt} &= \class{kind-linear}{\widetilde a_{31}u} + \class{kind-linear}{\widetilde a_{32}h_W} + \class{kind-seasonal}{\widetilde a_{33}(t)T_C} + \class{kind-linear}{\widetilde a_{34}T_E} + \class{kind-linear}{\widetilde a_{35}\tau} + \class{kind-noise}{\big(\mathbf{B}(x)\,\dot{\mathbf{W}}\big)_{T_C}} \\ \frac{dT_E}{dt} &= \class{kind-linear}{\widetilde a_{41}u} + \class{kind-linear}{\widetilde a_{42}h_W} + \class{kind-linear}{\widetilde a_{43}T_C} + \class{kind-seasonal}{\widetilde a_{44}(t)T_E} + \class{kind-linear}{\widetilde a_{45}\tau} + \class{kind-noise}{\big(\mathbf{B}(x)\,\dot{\mathbf{W}}\big)_{T_E}} \\ \frac{d\tau}{dt} &= \class{kind-linear}{\widetilde a_{51}u} + \class{kind-linear}{\widetilde a_{52}h_W} + \class{kind-linear}{\widetilde a_{53}T_C} + \class{kind-linear}{\widetilde a_{54}T_E} + \class{kind-seasonal}{\widetilde a_{55}(t)\tau} + \class{kind-noise}{\big(\mathbf{B}(x)\,\dot{\mathbf{W}}\big)_{\tau}} \end{aligned}\]
\[\mathbf B(x)=\operatorname{diag}\!\left(1,a_h(h_W),1,a_E(T_E),1\right)\mathbf B_0,\quad \text{row order }(u,h_W,T_C,T_E,\tau)\]
\[a_h(h_W)=\max\!\left(1-\mu_h\tanh(h_W/s_h),0.1\right),\qquad a_E(T_E)=\max\!\left(1+\mu_E\tanh(T_E/s_E),0.1\right)\]

Seasonal structure retained. Multiplicative noise retained for statistical fidelity; the additive alternative scores 0.3979. Original row below uses the reproducible local baseline; archived leaderboard scores differ. Original split-step integration retained.

ModelStatisticsDAPredictionTotal
Original0.41790.50770.18250.3507
Simplified0.41910.55930.24370.3910
Change vs original+0.0011 (+0.27%)+0.0516 (+10.17%)+0.0611 (+33.48%)+0.0403 (+11.48%)

DeepSeek V4 Pro

round_29Original transcription verified

Original model

\[\begin{aligned} \frac{du}{dt} &= \class{term kind-linear tip-deepseek-1}{a_{11}u} + \class{term kind-linear tip-deepseek-2}{a_{12}\tau} + \class{term kind-linear tip-deepseek-3}{a_{13}h_W} + \class{term kind-linear tip-deepseek-4}{a_{14}T_C} + \class{term kind-linear tip-deepseek-5}{a_{15}T_E} + \class{term kind-noise tip-deepseek-6}{\big(\mathbf{B}(x,t)\,\dot{\mathbf{W}}\big)_{u}} \\ \frac{dh_W}{dt} &= \class{term kind-linear tip-deepseek-7}{a_{21}h_W} + \class{term kind-linear tip-deepseek-8}{a_{22}T_E} + \class{term kind-linear tip-deepseek-9}{a_{23}T_C} + \class{term kind-linear tip-deepseek-10}{a_{24}u} + \class{term kind-noise tip-deepseek-11}{\big(\mathbf{B}(x,t)\,\dot{\mathbf{W}}\big)_{h_W}} \\ \frac{dT_C}{dt} &= \class{term kind-linear tip-deepseek-12}{a_{31}h_W} + \class{term kind-seasonal tip-deepseek-13}{a_{32}(t)T_C} + \class{term kind-linear tip-deepseek-14}{a_{33}T_E} + \class{term kind-linear tip-deepseek-15}{a_{34}\tau} + \class{term kind-linear tip-deepseek-16}{a_{35}u} + \class{term kind-nonlinear tip-deepseek-17}{a_{36}\frac{T_C}{1+e^{kT_C}}} + \class{term kind-nonlinear tip-deepseek-18}{a_{37}T_C^3} \\ &\quad + \class{term kind-noise tip-deepseek-19}{\big(\mathbf{B}(x,t)\,\dot{\mathbf{W}}\big)_{T_C}} \\ \frac{dT_E}{dt} &= \class{term kind-linear tip-deepseek-20}{a_{41}h_W} + \class{term kind-seasonal tip-deepseek-21}{a_{42}(t)T_E} + \class{term kind-linear tip-deepseek-22}{a_{43}T_C} + \class{term kind-linear tip-deepseek-23}{a_{44}\tau} + \class{term kind-linear tip-deepseek-24}{a_{45}u} + \class{term kind-nonlinear tip-deepseek-25}{a_{46}\frac{T_E}{1+e^{-kT_E}}} + \class{term kind-nonlinear tip-deepseek-26}{a_{47}T_E^3} + \class{term kind-nonlinear tip-deepseek-27}{a_{48}\max(T_E,0)^3} \\ &\quad + \class{term kind-noise tip-deepseek-28}{\big(\mathbf{B}(x,t)\,\dot{\mathbf{W}}\big)_{T_E}} \\ \frac{d\tau}{dt} &= \class{term kind-linear tip-deepseek-29}{a_{51}\tau} + \class{term kind-seasonal tip-deepseek-30}{a_{52}(t)T_C} + \class{term kind-linear tip-deepseek-31}{a_{53}T_E} + \class{term kind-linear tip-deepseek-32}{a_{54}h_W} + \class{term kind-linear tip-deepseek-33}{a_{55}u} + \class{term kind-nonlinear tip-deepseek-34}{a_{56}\tau^3} + \class{term kind-noise tip-deepseek-35}{\big(\mathbf{B}(x,t)\,\dot{\mathbf{W}}\big)_{\tau}} \end{aligned}\]
\[S(t)=\cos\!\left(2\pi(\lfloor2t\rfloor-11)/12\right),\qquad n(t)=1+0.2S(t)\]
\[\mathbf B(x,t)\dot{\mathbf W}=(\eta_u,\eta_h,\eta_C,\eta_E,\eta_\tau)^{\mathsf T}\]
\[\eta_u=\sigma_u n(t)\dot W_u,\quad \eta_h=\sigma_h n(t)\dot W_h,\quad \eta_\tau=\sigma_\tau n(t)\dot W_\tau\]
\[\eta_C=\sigma_C n(t)\dot W_C+\sigma_{C,m}|T_C|\dot W_6+\sigma_{C,a}[-T_C]_+\dot W_9\]
\[\eta_E=\sigma_E n(t)(\rho\dot W_C+\sqrt{1-\rho^2}\dot W_E)+\sigma_{E,m}|T_E|\dot W_7+\sigma_{E,a}[T_E]_+\dot W_8\]
\[\rho=0.75,\quad [z]_+=\max(z,0),\quad W_C,W_E,W_h,W_u,W_\tau,W_6,W_7,W_8,W_9\text{ independent.}\]
\[\text{after each Euler--Maruyama step: } x_i \leftarrow \min(\max(x_i,-b_i),b_i),\quad \mathbf b=20\mathbf 1,\quad \mathbf x=(u,h_W,T_C,T_E,\tau)^{\mathsf T}\]

Simplified model · linear dynamics + multiplicative noise

\[\begin{aligned} \frac{du}{dt} &= \class{kind-linear}{\widetilde a_{11}u} + \class{kind-linear}{\widetilde a_{12}\tau} + \class{kind-linear}{\widetilde a_{13}h_W} + \class{kind-linear}{\widetilde a_{14}T_C} + \class{kind-linear}{\widetilde a_{15}T_E} + \class{kind-noise}{\big(\mathbf{B}(x,t)\,\dot{\mathbf{W}}\big)_{u}} \\ \frac{dh_W}{dt} &= \class{kind-linear}{\widetilde a_{21}h_W} + \class{kind-linear}{\widetilde a_{22}T_E} + \class{kind-linear}{\widetilde a_{23}T_C} + \class{kind-linear}{\widetilde a_{24}u} + \class{kind-noise}{\big(\mathbf{B}(x,t)\,\dot{\mathbf{W}}\big)_{h_W}} \\ \frac{dT_C}{dt} &= \class{kind-linear}{\widetilde a_{31}h_W} + \class{kind-seasonal}{\widetilde a_{32}(t)T_C} + \class{kind-linear}{\widetilde a_{33}T_E} + \class{kind-linear}{\widetilde a_{34}\tau} + \class{kind-linear}{\widetilde a_{35}u} \\ &\quad + \class{kind-noise}{\big(\mathbf{B}(x,t)\,\dot{\mathbf{W}}\big)_{T_C}} \\ \frac{dT_E}{dt} &= \class{kind-linear}{\widetilde a_{41}h_W} + \class{kind-seasonal}{\widetilde a_{42}(t)T_E} + \class{kind-linear}{\widetilde a_{43}T_C} + \class{kind-linear}{\widetilde a_{44}\tau} + \class{kind-linear}{\widetilde a_{45}u} \\ &\quad + \class{kind-noise}{\big(\mathbf{B}(x,t)\,\dot{\mathbf{W}}\big)_{T_E}} \\ \frac{d\tau}{dt} &= \class{kind-linear}{\widetilde a_{51}\tau} + \class{kind-seasonal}{\widetilde a_{52}(t)T_C} + \class{kind-linear}{\widetilde a_{53}T_E} + \class{kind-linear}{\widetilde a_{54}h_W} + \class{kind-linear}{\widetilde a_{55}u} + \class{kind-noise}{\big(\mathbf{B}(x,t)\,\dot{\mathbf{W}}\big)_{\tau}} \end{aligned}\]

Linear drift with the original state-dependent noise.

ModelStatisticsDAPredictionTotal
Original0.28460.53050.25100.3449
Simplified0.26910.53210.24950.3402
Change vs original-0.0155 (-5.45%)+0.0016 (+0.30%)-0.0014 (-0.57%)-0.0048 (-1.38%)

MiniMax-M3

round_36Original transcription verified

Original model

\[\begin{aligned} \frac{du}{dt} &= \class{term kind-linear tip-minimax-1}{a_{11}u} + \class{term kind-linear tip-minimax-2}{a_{12}\tau} + \class{term kind-linear tip-minimax-3}{a_{13}h_W} + \class{term kind-noise tip-minimax-4}{\sigma_u \dot W_u} \\ \frac{dh_W}{dt} &= \class{term kind-linear tip-minimax-5}{a_{21}h_W} + \class{term kind-linear tip-minimax-6}{a_{22}T_E} + \class{term kind-linear tip-minimax-7}{a_{23}T_C} + \class{term kind-linear tip-minimax-8}{a_{24}u} + \class{term kind-linear tip-minimax-9}{a_{25}\tau} \\ \frac{dT_C}{dt} &= \class{term kind-linear tip-minimax-10}{a_{31}h_W} + \class{term kind-linear tip-minimax-11}{a_{32}T_C} + \class{term kind-linear tip-minimax-12}{a_{33}T_E} + \class{term kind-linear tip-minimax-13}{a_{34}\tau} + \class{term kind-nonlinear tip-minimax-14}{a_{35}T_C\tau} + \class{term kind-nonlinear tip-minimax-15}{a_{36}[T_C]_+^3} + \class{term kind-nonlinear tip-minimax-16}{a_{37}[T_C]_-^3} \\ &\quad + \class{term kind-nonlinear tip-minimax-17}{a_{38}[T_C]_+^2} + \class{term kind-nonlinear tip-minimax-18}{a_{39}[T_C]_-^2} + \class{term kind-noise tip-minimax-19}{\sigma_C \dot W_C} \\ \frac{dT_E}{dt} &= \class{term kind-linear tip-minimax-20}{a_{41}h_W} + \class{term kind-linear tip-minimax-21}{a_{42}T_E} + \class{term kind-linear tip-minimax-22}{a_{43}T_C} + \class{term kind-linear tip-minimax-23}{a_{44}\tau} + \class{term kind-nonlinear tip-minimax-24}{a_{45}T_E\tau} + \class{term kind-nonlinear tip-minimax-25}{a_{46}[T_E]_+^3} + \class{term kind-nonlinear tip-minimax-26}{a_{47}[T_E]_-^3} \\ &\quad + \class{term kind-nonlinear tip-minimax-27}{a_{48}[T_E]_+^2} + \class{term kind-nonlinear tip-minimax-28}{a_{49}[T_E]_-^2} + \class{term kind-noise tip-minimax-29}{\sigma_E(T_E) \dot W_E} \\ \frac{d\tau}{dt} &= \class{term kind-linear tip-minimax-30}{a_{51}\tau} + \class{term kind-linear tip-minimax-31}{a_{52}T_E} + \class{term kind-linear tip-minimax-32}{a_{53}T_C} + \class{term kind-noise tip-minimax-33}{\sigma_\tau(\tau) \dot W_\tau} \\ \mathbf{x}_{n+1} &= \Pi\big(\mathbf{x}_n + \mathbf{f}(\mathbf{x}_n)\Delta t + \mathbf{G}(\mathbf{x}_n)\Delta\mathbf{W}_n\big),\ \Pi:\ |T_C|,|T_E| \le \bar T,\ |h_W|,|u|,|\tau| \le 1 \end{aligned}\]
\[[z]_+=\max(z,0),\qquad [z]_- =\min(z,0)\]
\[\sigma_\tau(\tau)=\sigma_{\tau,0}+\sigma_{\tau,1}[\tau]_+,\qquad W_u,W_C,W_\tau\text{ independent.}\]
\[\sigma_E(T_E)=0,\qquad \bar T=0.5,\qquad \text{seasonal multipliers equal }1.\]

Simplified model · nonlinear dynamics + multiplicative noise

\[\begin{aligned} \frac{du}{dt} &= \class{kind-linear}{\widetilde a_{11}u} + \class{kind-noise}{\sigma_u \dot W_u} \\ \frac{dh_W}{dt} &= \class{kind-linear}{\widetilde a_{21}h_W} + \class{kind-linear}{\widetilde a_{22}T_E} + \class{kind-linear}{\widetilde a_{24}u} \\ \frac{dT_C}{dt} &= \class{kind-linear}{\widetilde a_{31}h_W} + \class{kind-linear}{\widetilde a_{32}T_C} + \class{kind-linear}{\widetilde a_{33}T_E} + \class{kind-linear}{\widetilde a_{34}\tau} + \class{kind-nonlinear}{\widetilde a_{36}[T_C]_+^3} + \class{kind-nonlinear}{\widetilde a_{37}[T_C]_-^3} + \class{kind-noise}{\sigma_C \dot W_C} \\ \frac{dT_E}{dt} &= \class{kind-linear}{\widetilde a_{41}h_W} + \class{kind-linear}{\widetilde a_{42}T_E} + \class{kind-linear}{\widetilde a_{43}T_C} + \class{kind-linear}{\widetilde a_{44}\tau} + \class{kind-nonlinear}{\widetilde a_{45}T_E\tau} + \class{kind-nonlinear}{\widetilde a_{46}[T_E]_+^3} + \class{kind-nonlinear}{\widetilde a_{47}[T_E]_-^3} \\ \frac{d\tau}{dt} &= \class{kind-linear}{\widetilde a_{51}\tau} + \class{kind-linear}{\widetilde a_{52}T_E} + \class{kind-noise}{\sigma_\tau(\tau) \dot W_\tau} \end{aligned}\]
\[\text{after each Euler--Maruyama step: } x_i \leftarrow \min(\max(x_i,-b_i),b_i),\quad \mathbf b=(1,1,0.5,0.5,1)^{\mathsf T},\quad \mathbf x=(u,h_W,T_C,T_E,\tau)^{\mathsf T}\]
\[[z]_+=\max(z,0),\qquad [z]_- =\min(z,0)\]
\[\sigma_\tau(\tau)=\sigma_{\tau,0}+\sigma_{\tau,1}[\tau]_+,\qquad W_u,W_C,W_\tau\text{ independent.}\]

Equations use nondimensional states: SST / 7.5 °C, h_W / 150 m, u / 1.5 m/s, and wind / 5 m/s; one model-time unit is 2 months. Scores use the reproducible local original and the current v2 statistics/total; the baseline audit reproduces the archived leaderboard under its older scoring definition. Only the cold-side T_E quadratic group is removed; three nonlinear drift groups and state-dependent wind noise remain. Drift and noise already have no seasonal modulation. Exactly-zero original pathways are omitted from the compact equations. Original post-step clipping retained; maximum boundary-hit fraction 2.086% in the archived 50-year, 16-trajectory probe. One grading seed and post-hoc selection do not establish scientific equivalence or a globally minimal model.

ModelStatisticsDAPredictionTotal
Original0.23110.32640.13590.2216
Simplified0.22950.32260.13630.2202
Change vs original-0.0016 (-0.69%)-0.0038 (-1.15%)+0.0004 (+0.28%)-0.0015 (-0.66%)

GLM-5.2

round_33Original transcription verified

Original model

\[\begin{aligned} \frac{du}{dt} &= \class{term kind-linear tip-glm52-1}{a_{11}u} + \class{term kind-linear tip-glm52-2}{a_{12}\tau} + \class{term kind-linear tip-glm52-3}{a_{13}h_W} + \class{term kind-noise tip-glm52-4}{\sigma_u(t)\,\dot W_u} \\ \frac{dh_W}{dt} &= \class{term kind-linear tip-glm52-5}{a_{21}h_W} + \class{term kind-linear tip-glm52-6}{a_{22}\tau} + \class{term kind-seasonal tip-glm52-7}{a_{23}(t)T_C} + \class{term kind-seasonal tip-glm52-8}{a_{24}(t)T_E} + \class{term kind-nonlinear tip-glm52-9}{a_{25}(t)T_C^2} + \class{term kind-nonlinear tip-glm52-10}{a_{26}(t)T_E^2} \\ \frac{dT_C}{dt} &= \class{term kind-linear tip-glm52-11}{a_{31}T_C} + \class{term kind-nonlinear tip-glm52-12}{a_{32}(t)F_C(h_C)} + \class{term kind-linear tip-glm52-13}{a_{33}T_E} + \class{term kind-linear tip-glm52-14}{a_{34}u} + \class{term kind-nonlinear tip-glm52-15}{a_{35}[T_C]_+^3} + \class{term kind-noise tip-glm52-16}{\sigma_C\,\dot W_C} \\ \frac{dT_E}{dt} &= \class{term kind-linear tip-glm52-17}{a_{41}T_E} + \class{term kind-nonlinear tip-glm52-18}{a_{42}(t)F_E(h_E)} + \class{term kind-nonlinear tip-glm52-19}{a_{43}[T_E]_+^3} + \class{term kind-nonlinear tip-glm52-20}{a_{44}[T_E]_-^3} + \class{term kind-nonlinear tip-glm52-21}{a_{45}T_E[h_E]_+} \\ \frac{d\tau}{dt} &= \class{term kind-linear tip-glm52-22}{a_{51}\tau} + \class{term kind-seasonal tip-glm52-23}{a_{52}(t)T_C} + \class{term kind-seasonal tip-glm52-24}{a_{53}(t)T_E} + \class{term kind-nonlinear tip-glm52-25}{a_{54}(t)T_C^2} + \class{term kind-nonlinear tip-glm52-26}{a_{55}(t)T_E^2} \\ &\quad + \class{term kind-nonlinear tip-glm52-27}{a_{56}[\tau]_+^3} + \class{term kind-nonlinear tip-glm52-28}{a_{57}[\tau]_-^3} + \class{term kind-noise tip-glm52-29}{\sigma_\tau(T_E,t)\,\dot W_\tau} \end{aligned}\]
\[\tau_{\rm eq}=0.32S(t)\left(1.2T_C+0.6T_E+1.5T_C^2+1.4T_E^2\right),\qquad \tau_{\rm eff}=2.2\left(0.72\tau_{\rm eq}+0.28\tau\right)\]
\[h_E=h_W+\tau_{\rm eff},\qquad h_C=h_W+0.35\tau_{\rm eff}\]
\[F_C(h)=0.4500\tanh(h/0.4500),\qquad F_E(h)=\begin{cases}1.000\tanh(h/1.000) & h>0\\ 0.1000\tanh(h/0.1000) & h\le 0\end{cases}\]
\[S(t)=1+0.5\cos(2\pi t/6-3),\qquad S_n(t)=1+0.5\cos(2\pi t/6+1)\]
\[\sigma_u(t)=\sigma_{u,0}S_n(t),\qquad \sigma_\tau(T_E,t)=\sigma_{\tau,0}(1+3[T_E]_+)S_n(t)\]
\[[z]_+=\max(z,0),\qquad [z]_-=\min(z,0),\qquad W_u,W_C,W_\tau\text{ independent.}\]
\[\text{after each Euler--Maruyama step: } x_i \leftarrow \min(\max(x_i,-b_i),b_i),\quad \mathbf b=10\mathbf 1,\quad \mathbf x=(u,h_W,T_C,T_E,\tau)^{\mathsf T}\]

Simplified model · linear dynamics + additive noise

\[\begin{aligned} \frac{du}{dt} &= \class{kind-linear}{\widetilde a_{11}u} + \class{kind-linear}{\widetilde a_{12}\tau} + \class{kind-linear}{\widetilde a_{13}h_W} + \class{kind-noise}{\sigma_u\,\dot W_u} \\ \frac{dh_W}{dt} &= \class{kind-linear}{\widetilde a_{21}h_W} + \class{kind-linear}{\widetilde a_{22}\tau} + \class{kind-linear}{\widetilde a_{23}T_C} + \class{kind-linear}{\widetilde a_{24}T_E} \\ \frac{dT_C}{dt} &= \class{kind-linear}{\widetilde a_{31}T_C} + \class{kind-linear}{\widetilde a_{32}h_C} + \class{kind-linear}{\widetilde a_{33}T_E} + \class{kind-linear}{\widetilde a_{34}u} + \class{kind-noise}{\sigma_C\,\dot W_C} \\ \frac{dT_E}{dt} &= \class{kind-linear}{\widetilde a_{41}T_E} + \class{kind-linear}{\widetilde a_{42}h_E} \\ \frac{d\tau}{dt} &= \class{kind-linear}{\widetilde a_{51}\tau} + \class{kind-linear}{\widetilde a_{52}T_C} + \class{kind-linear}{\widetilde a_{53}T_E} + \class{kind-noise}{\sigma_\tau\,\dot W_\tau} \end{aligned}\]
\[\text{after each Euler--Maruyama step: } x_i \leftarrow \min(\max(x_i,-b_i),b_i),\quad \mathbf b=10\mathbf 1,\quad \mathbf x=(u,h_W,T_C,T_E,\tau)^{\mathsf T}\]
\[\tau_{\rm eq}=0.32\left(1.2T_C+0.6T_E\right),\qquad \tau_{\rm eff}=2.2\left(0.72\tau_{\rm eq}+0.28\tau\right)\]
\[h_E=h_W+\tau_{\rm eff},\qquad h_C=h_W+0.35\tau_{\rm eff}\]
\[F_C(h_C)=h_C,\quad F_E(h_E)=h_E,\qquad W_u,W_C,W_\tau\text{ independent.}\]

Equations use nondimensional states: SST / 7.5 °C, h_W / 150 m, u / 1.5 m/s, and wind / 5 m/s; one model-time unit is 2 months. Scores use the reproducible local original and the current v2 statistics/total; the baseline audit reproduces the archived leaderboard under its older scoring definition. All registered nonlinear drift groups and seasonal modulation are removed; the remaining noise is constant and additive. Original post-step clipping retained; maximum boundary-hit fraction 0.000% in the archived 50-year, 16-trajectory probe. Linear describes the interior drift; the clipped numerical model is not a globally linear SDE. One grading seed and post-hoc selection do not establish scientific equivalence or a globally minimal model.

ModelStatisticsDAPredictionTotal
Original0.22240.26150.10810.1884
Simplified0.24120.38070.26510.2926
Change vs original+0.0188 (+8.47%)+0.1192 (+45.61%)+0.1570 (+145.19%)+0.1042 (+55.32%)

Qwen3-Max

round_78Original transcription verified

Original model

\[\begin{aligned} \frac{du}{dt} &= \class{term kind-linear tip-qwen3max-1}{a_{11}u} + \class{term kind-linear tip-qwen3max-2}{a_{12}\tau} \\ \frac{dh_W}{dt} &= \class{term kind-linear tip-qwen3max-3}{a_{21}h_W} + \class{term kind-linear tip-qwen3max-4}{a_{22}T_C} + \class{term kind-linear tip-qwen3max-5}{a_{23}T_E} + \class{term kind-linear tip-qwen3max-6}{a_{24}u} + \class{term kind-nonlinear tip-qwen3max-7}{a_{25}h_W^3} \\ \frac{dT_C}{dt} &= \class{term kind-seasonal tip-qwen3max-8}{a_{31}(t)h_W} + \class{term kind-seasonal tip-qwen3max-9}{a_{32}(t)T_C} + \class{term kind-nonlinear tip-qwen3max-10}{a_{33}(t)T_C^3} \\ \frac{dT_E}{dt} &= \class{term kind-seasonal tip-qwen3max-11}{a_{41}(t)h_W} + \class{term kind-seasonal tip-qwen3max-12}{a_{42}(t)T_E} + \class{term kind-nonlinear tip-qwen3max-13}{a_{43}(t)T_E^2} + \class{term kind-nonlinear tip-qwen3max-14}{a_{44}(t)T_E^3} \\ \frac{d\tau}{dt} &= \class{term kind-linear tip-qwen3max-15}{a_{51}\tau} + \class{term kind-linear tip-qwen3max-16}{a_{52}T_C} + \class{term kind-linear tip-qwen3max-17}{a_{53}T_E} + \class{term kind-noise tip-qwen3max-18}{\sigma_\tau \dot W_\tau} \end{aligned}\]
\[a_{3j}(t)=\overline a_{3j}[1+0.25\cos(2\pi t/6-1.05\pi/6)],\qquad a_{4j}(t)=\overline a_{4j}[1+0.39\cos(2\pi t/6-1.05\pi/6)]\]

Simplified model · linear dynamics + additive noise

\[\begin{aligned} \frac{du}{dt} &= \class{kind-linear}{\widetilde a_{11}u} + \class{kind-linear}{\widetilde a_{12}\tau} \\ \frac{dh_W}{dt} &= \class{kind-linear}{\widetilde a_{21}h_W} + \class{kind-linear}{\widetilde a_{22}T_C} + \class{kind-linear}{\widetilde a_{23}T_E} + \class{kind-linear}{\widetilde a_{24}u} \\ \frac{dT_C}{dt} &= \class{kind-linear}{\widetilde a_{31}h_W} + \class{kind-linear}{\widetilde a_{32}T_C} \\ \frac{dT_E}{dt} &= \class{kind-linear}{\widetilde a_{41}h_W} + \class{kind-linear}{\widetilde a_{42}T_E} \\ \frac{d\tau}{dt} &= \class{kind-linear}{\widetilde a_{51}\tau} + \class{kind-linear}{\widetilde a_{52}T_C} + \class{kind-linear}{\widetilde a_{53}T_E} + \class{kind-noise}{\sigma_\tau \dot W_\tau} \end{aligned}\]

Equations use nondimensional states: SST / 7.5 °C, h_W / 150 m, u / 1.5 m/s, and wind / 5 m/s; one model-time unit is 2 months. Scores use the reproducible local original and the current v2 statistics/total; the baseline audit reproduces the archived leaderboard under its older scoring definition. All registered nonlinear drift groups are removed. Seasonal modulation is retained, and the remaining noise is constant and additive. Only wind receives direct stochastic forcing. Prediction is zero for both models; its relative change is undefined. The original model has no state clipping. One grading seed and post-hoc selection do not establish scientific equivalence or a globally minimal model.

ModelStatisticsDAPredictionTotal
Original0.11830.06160.00000.0540
Simplified0.11830.06160.00000.0540
Change vs original-0.0000 (-0.00%)-0.0000 (-0.00%)+0.0000 (+0.00%)-0.0000 (-0.00%)

Fable 5.1 · Web checkpoint comparison

Original agent models from formal6X-20260920. Best means the highest archived composite score among 45 checkpoints; final means round 45. The existing offline Fable final remains above.

VALID WITH EXCEPTION · A search listing exposed the prohibited reference’s metadata and abstract. These supplementary checkpoints are not added to the offline leaderboard. Scores below retain this campaign’s own grading manifest.

Claude Fable 5.1 · Web best

formal6X-20260920 · round_24Source hashes and discrete update verified

Original model · nonlinear dynamics + diagonal multiplicative noise

\[\begin{aligned} \frac{d\hat u}{ds} &= \class{term kind-linear tip-fable51-web-best-1}{a_{11}\hat u} + \class{term kind-linear tip-fable51-web-best-2}{a_{12}\hat h_W} + \class{term kind-linear tip-fable51-web-best-3}{a_{15}\hat\tau} + \class{term kind-noise tip-fable51-web-best-4}{\sigma_u(t)\dot W_u} \\ \frac{d\hat h_W}{ds} &= \class{term kind-linear tip-fable51-web-best-5}{a_{21}\hat u} + \class{term kind-linear tip-fable51-web-best-6}{a_{22}\hat h_W} + \class{term kind-seasonal tip-fable51-web-best-7}{a_{23}(t)\hat T_C} + \class{term kind-seasonal tip-fable51-web-best-8}{a_{24}(t)\hat T_E} + \class{term kind-linear tip-fable51-web-best-9}{a_{25}\hat\tau} + \class{term kind-noise tip-fable51-web-best-10}{\sigma_h(t)\dot W_h} \\ \frac{d\hat T_C}{ds} &= \class{term kind-linear tip-fable51-web-best-11}{a_{31}\hat u} + \class{term kind-linear tip-fable51-web-best-12}{a_{32}\hat h_W} + \class{term kind-seasonal tip-fable51-web-best-13}{a_{33}(t)\hat T_C} + \class{term kind-linear tip-fable51-web-best-14}{a_{34}\hat T_E} + \class{term kind-linear tip-fable51-web-best-15}{a_{35}\hat\tau} \\ &\quad + \class{term kind-nonlinear tip-fable51-web-best-16}{a_{36}\hat T_C\mathcal S(\hat T_C)} + \class{term kind-nonlinear tip-fable51-web-best-17}{a_{37}\hat T_C^3} + \class{term kind-noise tip-fable51-web-best-18}{\sigma_C(\hat T_C)\dot W_C} \\ \frac{d\hat T_E}{ds} &= \class{term kind-linear tip-fable51-web-best-19}{a_{41}\hat u} + \class{term kind-linear tip-fable51-web-best-20}{a_{42}\hat h_W} + \class{term kind-linear tip-fable51-web-best-21}{a_{43}\hat T_C} + \class{term kind-seasonal tip-fable51-web-best-22}{a_{44}(t)\hat T_E} + \class{term kind-linear tip-fable51-web-best-23}{a_{45}\hat\tau} \\ &\quad + \class{term kind-nonlinear tip-fable51-web-best-24}{a_{46}\hat T_E\tanh\!\left(\frac{\mathcal S(\hat T_E-T_0)}{W_E}\right)} + \class{term kind-nonlinear tip-fable51-web-best-25}{a_{47}\hat T_E\mathcal S(\hat T_E)} + \class{term kind-nonlinear tip-fable51-web-best-26}{a_{48}\hat T_E^3} + \class{term kind-noise tip-fable51-web-best-27}{\sigma_E(\hat T_E,t)\dot W_E} \\ \frac{d\hat\tau}{ds} &= \class{term kind-linear tip-fable51-web-best-28}{a_{52}\hat h_W} + \class{term kind-seasonal tip-fable51-web-best-29}{a_{53}(t)\hat T_C} + \class{term kind-seasonal tip-fable51-web-best-30}{a_{54}(t)\hat T_E} + \class{term kind-linear tip-fable51-web-best-31}{a_{55}\hat\tau} + \class{term kind-noise tip-fable51-web-best-32}{\sigma_\tau(\hat T_C,t)\dot W_\tau} \end{aligned}\]

Temperature in °C, thermocline depth in m, currents and winds in m/s; t is in months and s is nondimensional time. All five noise drivers are independent standard Wiener processes in s. Colors identify term types, not evidence of physical mechanisms.

\[\begin{aligned} \hat T_C=T_C/7.5,\quad\hat T_E=T_E/7.5,\quad\hat h_W=h_W/150,\quad\hat u=u/1.5,\quad\hat\tau=\tau/5,\quad s=t/2,\quad\omega=2\pi/12 \end{aligned}\]
\[\begin{aligned} \mathcal S(z)=k^{-1}\log\!\left[1+\exp\!\left(\operatorname{clip}(kz,-50,50)\right)\right],\qquad k=25,\quad T_0=T_{0,E} \end{aligned}\]
\[\begin{aligned} a_{33}(t)=R_{C0}\left[1+R_{Ca}\cos\omega(t-\phi_C)+R_{Cb}\cos2\omega(t-\phi_{C2})\right] \end{aligned}\]
\[\begin{aligned} a_{44}(t)=R_{E0}\left[1+R_{Ea}\cos\omega(t-\phi_E)+R_{Eb}\cos2\omega(t-\phi_{E2})\right] \end{aligned}\]
\[\begin{aligned} (a_{23},a_{24})(t)=-(a_C,a_E)\left[1+a_s\cos\omega(t-\phi_{as})\right] \end{aligned}\]
\[\begin{aligned} (a_{53},a_{54})(t)=(m_C,m_E)\left[1+\mu_a\cos\omega(t-\phi_a)\right] \end{aligned}\]
\[\begin{aligned} \sigma_C=\sigma_{TC}\left[1+\mu_{Cn}\mathcal S(-\hat T_C)\right],\qquad\sigma_E=\sigma_{TE}\left[1+\mu_E\mathcal S(\hat T_E)\right] \end{aligned}\]
\[\begin{aligned} \sigma_\tau=\sigma_t\left[1+s_t\cos\omega(t-\phi_t)\right]\left[1+\mu_t\mathcal S(\hat T_C)\right] \end{aligned}\]
\[\begin{aligned} \sigma_h=\sigma_{h0}\left[1+s_h\cos\omega(t-\phi_h)\right],\qquad\sigma_u=\sigma_{u0}\left[1+s_u\cos\omega(t-\phi_u)\right] \end{aligned}\]

Implemented time-step rule

The cubic terms above describe the small-step drift limit. In the executable model, the non-cubic drift and Gaussian noise are advanced first, then temperature damping is applied using the rational map below. This is not ordinary Euler–Maruyama with an explicit cubic term.

\[\begin{aligned} \Delta s=\Delta t/2,\quad Y_j=\hat X_j+b_j\Delta s+\sigma_j\,\Delta W_j,\quad\Delta W_j\sim\mathcal N(0,\Delta s) \\ Z_C=\frac{Y_C}{1+c_C\Delta s\,Y_C^2},\quad Z_E=\frac{Y_E}{1+c_E\Delta s\,Y_E^2},\quad Z_j=Y_j\;(j=h,u,\tau),\quad\hat X_j^{+}=\operatorname{clip}(Z_j,-5,5) \end{aligned}\]

Here b is the displayed drift excluding the two cubic terms; c_C = −L₃₇ and c_E = −L₄₈. The source’s inactive quadratic T_C, T_CT_E, nonlinear wind, offset and seasonal thermocline-feedback terms are omitted because their coefficients are zero, not because they were ablated.

Eastern-Pacific noise is state-dependent but has no seasonal modulation in this checkpoint (s_E = 0).

Simplified model · cubic nonlinear dynamics + multiplicative noise

\[\begin{aligned} \frac{d\hat u}{ds} &= \class{kind-linear}{\widetilde a_{11}\hat u} + \class{kind-linear}{\widetilde a_{12}\hat h_W} + \class{kind-linear}{\widetilde a_{15}\hat\tau} + \class{kind-noise}{\sigma_u\dot W_u} \\ \frac{d\hat h_W}{ds} &= \class{kind-linear}{\widetilde a_{21}\hat u} + \class{kind-linear}{\widetilde a_{22}\hat h_W} + \class{kind-seasonal}{\widetilde a_{23}(t)\hat T_C} + \class{kind-seasonal}{\widetilde a_{24}(t)\hat T_E} + \class{kind-linear}{\widetilde a_{25}\hat\tau} + \class{kind-noise}{\sigma_h\dot W_h} \\ \frac{d\hat T_C}{ds} &= \class{kind-linear}{\widetilde a_{31}\hat u} + \class{kind-linear}{\widetilde a_{32}\hat h_W} + \class{kind-seasonal}{\widetilde a_{33}(t)\hat T_C} + \class{kind-linear}{\widetilde a_{34}\hat T_E} + \class{kind-linear}{\widetilde a_{35}\hat\tau} + \class{kind-nonlinear}{\widetilde a_{37}\hat T_C^3} + \class{kind-noise}{\sigma_C(\hat T_C)\dot W_C} \\ \frac{d\hat T_E}{ds} &= \class{kind-linear}{\widetilde a_{41}\hat u} + \class{kind-linear}{\widetilde a_{42}\hat h_W} + \class{kind-linear}{\widetilde a_{43}\hat T_C} + \class{kind-seasonal}{\widetilde a_{44}(t)\hat T_E} + \class{kind-linear}{\widetilde a_{45}\hat\tau} + \class{kind-nonlinear}{\widetilde a_{48}\hat T_E^3} + \class{kind-noise}{\sigma_E(\hat T_E)\dot W_E} \\ \frac{d\hat\tau}{ds} &= \class{kind-linear}{\widetilde a_{52}\hat h_W} + \class{kind-seasonal}{\widetilde a_{53}(t)\hat T_C} + \class{kind-seasonal}{\widetilde a_{54}(t)\hat T_E} + \class{kind-linear}{\widetilde a_{55}\hat\tau} + \class{kind-noise}{\sigma_\tau(\hat T_C)\dot W_\tau} \end{aligned}\]

Remove both warm-side dampers, T_E threshold growth, semiannual T_C growth modulation, and seasonal h_W/u/wind noise. Retain and refit both cubic dampers; all state-dependent noise factors remain. The original post-update cubic rule and safety bounds are retained.

42 active scalar parameters (original: 55), including seasonal phases and noise parameters; not a drift-only coefficient count. Tildes identify candidate coefficients. The state scales and softplus function above are unchanged.

\[\begin{aligned} \class{kind-seasonal}{\widetilde a_{33}(t)=\widetilde R_{C0}[1+R_{Ca}\cos\omega(t-\phi_C)]} \end{aligned}\]
\[\begin{aligned} \class{kind-seasonal}{\widetilde a_{44}(t)=\widetilde R_{E0}[1+R_{Ea}\cos\omega(t-\phi_E)+R_{Eb}\cos2\omega(t-\phi_{E2})]} \end{aligned}\]
\[\begin{aligned} \class{kind-seasonal}{(\widetilde a_{23},\widetilde a_{24})(t)=-(\widetilde a_C,\widetilde a_E)[1+a_s\cos\omega(t-\phi_{as})]} \end{aligned}\]
\[\begin{aligned} \class{kind-seasonal}{(\widetilde a_{53},\widetilde a_{54})(t)=(\widetilde m_C,\widetilde m_E)[1+\mu_a\cos\omega(t-\phi_a)]} \end{aligned}\]
\[\begin{aligned} \class{kind-noise}{\sigma_C=\sigma_{TC}[1+\mu_{Cn}\mathcal S(-\hat T_C)],\qquad \sigma_E=\sigma_{TE}[1+\mu_E\mathcal S(\hat T_E)]} \end{aligned}\]
\[\begin{aligned} \class{kind-noise}{\sigma_\tau=\widetilde\sigma_t[1+\mu_t\mathcal S(\hat T_C)],\qquad \sigma_h=\widetilde\sigma_{h0},\quad \sigma_u=\widetilde\sigma_{u0}} \end{aligned}\]
Alternative · linear dynamics + multiplicative noise · total 0.5072

Alternative · linear dynamics + multiplicative noise

\[\begin{aligned} \frac{d\hat u}{ds} &= \class{kind-linear}{\widetilde a_{11}\hat u} + \class{kind-linear}{\widetilde a_{12}\hat h_W} + \class{kind-linear}{\widetilde a_{15}\hat\tau} + \class{kind-noise}{\sigma_u\dot W_u} \\ \frac{d\hat h_W}{ds} &= \class{kind-linear}{\widetilde a_{21}\hat u} + \class{kind-linear}{\widetilde a_{22}\hat h_W} + \class{kind-seasonal}{\widetilde a_{23}(t)\hat T_C} + \class{kind-seasonal}{\widetilde a_{24}(t)\hat T_E} + \class{kind-linear}{\widetilde a_{25}\hat\tau} + \class{kind-noise}{\sigma_h\dot W_h} \\ \frac{d\hat T_C}{ds} &= \class{kind-linear}{\widetilde a_{31}\hat u} + \class{kind-linear}{\widetilde a_{32}\hat h_W} + \class{kind-seasonal}{\widetilde a_{33}(t)\hat T_C} + \class{kind-linear}{\widetilde a_{34}\hat T_E} + \class{kind-linear}{\widetilde a_{35}\hat\tau} + \class{kind-noise}{\sigma_C(\hat T_C)\dot W_C} \\ \frac{d\hat T_E}{ds} &= \class{kind-linear}{\widetilde a_{41}\hat u} + \class{kind-linear}{\widetilde a_{42}\hat h_W} + \class{kind-linear}{\widetilde a_{43}\hat T_C} + \class{kind-seasonal}{\widetilde a_{44}(t)\hat T_E} + \class{kind-linear}{\widetilde a_{45}\hat\tau} + \class{kind-noise}{\sigma_E(\hat T_E)\dot W_E} \\ \frac{d\hat\tau}{ds} &= \class{kind-linear}{\widetilde a_{52}\hat h_W} + \class{kind-seasonal}{\widetilde a_{53}(t)\hat T_C} + \class{kind-seasonal}{\widetilde a_{54}(t)\hat T_E} + \class{kind-linear}{\widetilde a_{55}\hat\tau} + \class{kind-noise}{\sigma_\tau(\hat T_C)\dot W_\tau} \end{aligned}\]

Remove both remaining cubic dampers and refit. All linear pathways and their signs remain. Total −3.78%; Statistics −9.07%; DA −1.12%; Prediction +2.92% under v2. The T_E → T_C coefficient reaches its 0.1%-of-original lower bound.

40 active scalar parameters (original: 55), including seasonal phases and noise parameters; not a drift-only coefficient count. Tildes identify candidate coefficients. The state scales and softplus function above are unchanged.

The seasonal and noise forms shown for the cubic candidate also apply here, with separately refitted linear coefficients. No cubic update is applied; the original safety clip remains.

Scientific evaluation · v2
ModelStatisticsDAPredictionTotal
Original · round_240.79440.55390.30660.5271
Simplified · cubic drift0.80690.55460.30620.5309
Alternative · linear drift0.72230.54770.31550.5072
Cubic change vs original+0.0125 (+1.57%)+0.0007 (+0.13%)-0.0004 (-0.12%)+0.0038 (+0.72%)
Linear change vs original-0.0721 (-9.07%)-0.0062 (-1.12%)+0.0089 (+2.92%)-0.0199 (-3.78%)

24 post-hoc interventions plus original/full-refit controls; one fixed evaluation seed. Parameter fitting uses original-model trajectories. Candidate selection reuses the scientific grader; this is neither an autonomous submission nor evidence of physical or statistical equivalence.

Claude Fable 5.1 · Web final

formal6X-20260920 · round_45Source hashes and discrete update verified

Original model · nonlinear dynamics + diagonal multiplicative noise

\[\begin{aligned} \frac{d\hat u}{ds} &= \class{term kind-linear tip-fable51-web-final-1}{a_{11}\hat u} + \class{term kind-linear tip-fable51-web-final-2}{a_{12}\hat h_W} + \class{term kind-linear tip-fable51-web-final-3}{a_{15}\hat\tau} + \class{term kind-noise tip-fable51-web-final-4}{\sigma_u(t)\dot W_u} \\ \frac{d\hat h_W}{ds} &= \class{term kind-linear tip-fable51-web-final-5}{a_{21}\hat u} + \class{term kind-linear tip-fable51-web-final-6}{a_{22}\hat h_W} + \class{term kind-seasonal tip-fable51-web-final-7}{a_{23}(t)\hat T_C} + \class{term kind-seasonal tip-fable51-web-final-8}{a_{24}(t)\hat T_E} + \class{term kind-linear tip-fable51-web-final-9}{a_{25}\hat\tau} + \class{term kind-noise tip-fable51-web-final-10}{\sigma_h(t)\dot W_h} \\ \frac{d\hat T_C}{ds} &= \class{term kind-linear tip-fable51-web-final-11}{a_{31}\hat u} + \class{term kind-linear tip-fable51-web-final-12}{a_{32}\hat h_W} + \class{term kind-seasonal tip-fable51-web-final-13}{a_{33}(t)\hat T_C} + \class{term kind-linear tip-fable51-web-final-14}{a_{34}\hat T_E} + \class{term kind-linear tip-fable51-web-final-15}{a_{35}\hat\tau} \\ &\quad + \class{term kind-nonlinear tip-fable51-web-final-16}{a_{36}\hat T_C\mathcal S(\hat T_C)} + \class{term kind-nonlinear tip-fable51-web-final-17}{a_{37}\hat T_C^3} + \class{term kind-noise tip-fable51-web-final-18}{\sigma_C(\hat T_C)\dot W_C} \\ \frac{d\hat T_E}{ds} &= \class{term kind-linear tip-fable51-web-final-19}{a_{41}\hat u} + \class{term kind-linear tip-fable51-web-final-20}{a_{42}\hat h_W} + \class{term kind-linear tip-fable51-web-final-21}{a_{43}\hat T_C} + \class{term kind-seasonal tip-fable51-web-final-22}{a_{44}(t)\hat T_E} + \class{term kind-linear tip-fable51-web-final-23}{a_{45}\hat\tau} \\ &\quad + \class{term kind-nonlinear tip-fable51-web-final-24}{a_{46}\hat T_E\tanh\!\left(\frac{\mathcal S(\hat T_E-T_0)}{W_E}\right)} + \class{term kind-nonlinear tip-fable51-web-final-25}{a_{47}\hat T_E\mathcal S(\hat T_E)} + \class{term kind-nonlinear tip-fable51-web-final-26}{a_{48}\hat T_E^3} + \class{term kind-noise tip-fable51-web-final-27}{\sigma_E(\hat T_E,t)\dot W_E} \\ \frac{d\hat\tau}{ds} &= \class{term kind-linear tip-fable51-web-final-28}{a_{52}\hat h_W} + \class{term kind-seasonal tip-fable51-web-final-29}{a_{53}(t)\hat T_C} + \class{term kind-seasonal tip-fable51-web-final-30}{a_{54}(t)\hat T_E} + \class{term kind-linear tip-fable51-web-final-31}{a_{55}\hat\tau} + \class{term kind-noise tip-fable51-web-final-32}{\sigma_\tau(\hat T_C,t)\dot W_\tau} \end{aligned}\]

Temperature in °C, thermocline depth in m, currents and winds in m/s; t is in months and s is nondimensional time. All five noise drivers are independent standard Wiener processes in s. Colors identify term types, not evidence of physical mechanisms.

\[\begin{aligned} \hat T_C=T_C/7.5,\quad\hat T_E=T_E/7.5,\quad\hat h_W=h_W/150,\quad\hat u=u/1.5,\quad\hat\tau=\tau/5,\quad s=t/2,\quad\omega=2\pi/12 \end{aligned}\]
\[\begin{aligned} \mathcal S(z)=k^{-1}\log\!\left[1+\exp\!\left(\operatorname{clip}(kz,-50,50)\right)\right],\qquad k=25,\quad T_0=T_{0,E} \end{aligned}\]
\[\begin{aligned} a_{33}(t)=R_{C0}\left[1+R_{Ca}\cos\omega(t-\phi_C)+R_{Cb}\cos2\omega(t-\phi_{C2})\right] \end{aligned}\]
\[\begin{aligned} a_{44}(t)=R_{E0}\left[1+R_{Ea}\cos\omega(t-\phi_E)+R_{Eb}\cos2\omega(t-\phi_{E2})\right] \end{aligned}\]
\[\begin{aligned} (a_{23},a_{24})(t)=-(a_C,a_E)\left[1+a_s\cos\omega(t-\phi_{as})\right] \end{aligned}\]
\[\begin{aligned} (a_{53},a_{54})(t)=(m_C,m_E)\left[1+\mu_a\cos\omega(t-\phi_a)\right] \end{aligned}\]
\[\begin{aligned} \sigma_C=\sigma_{TC}\left[1+\mu_{Cn}\mathcal S(-\hat T_C)\right],\qquad\sigma_E=\sigma_{TE}\left[1+s_E\cos\omega(t-\phi_{En})\right]\left[1+\mu_E\mathcal S(\hat T_E)\right] \end{aligned}\]
\[\begin{aligned} \sigma_\tau=\sigma_t\left[1+s_t\cos\omega(t-\phi_t)\right]\left[1+\mu_t\mathcal S(\hat T_C)\right] \end{aligned}\]
\[\begin{aligned} \sigma_h=\sigma_{h0}\left[1+s_h\cos\omega(t-\phi_h)\right],\qquad\sigma_u=\sigma_{u0}\left[1+s_u\cos\omega(t-\phi_u)\right] \end{aligned}\]

Implemented time-step rule

The cubic terms above describe the small-step drift limit. In the executable model, the non-cubic drift and Gaussian noise are advanced first, then temperature damping is applied using the rational map below. This is not ordinary Euler–Maruyama with an explicit cubic term.

\[\begin{aligned} \Delta s=\Delta t/2,\quad Y_j=\hat X_j+b_j\Delta s+\sigma_j\,\Delta W_j,\quad\Delta W_j\sim\mathcal N(0,\Delta s) \\ Z_C=\frac{Y_C}{1+c_C\Delta s\,Y_C^2},\quad Z_E=\frac{Y_E}{1+c_E\Delta s\,Y_E^2},\quad Z_j=Y_j\;(j=h,u,\tau),\quad\hat X_j^{+}=\operatorname{clip}(Z_j,-5,5) \end{aligned}\]

Here b is the displayed drift excluding the two cubic terms; c_C = −L₃₇ and c_E = −L₄₈. The source’s inactive quadratic T_C, T_CT_E, nonlinear wind, offset and seasonal thermocline-feedback terms are omitted because their coefficients are zero, not because they were ablated.

Eastern-Pacific noise is both state-dependent and seasonally modulated in this checkpoint (s_E ≠ 0).

No ablation or scored simplification has been performed for this Web final checkpoint. Web best and offline Fable ablation effects do not apply here.

Post-hoc studies: seed 0 · Statistics use the v2 grader · DA denotes state reconstruction · Total weights 0.3 / 0.3 / 0.4. Simplified models are post-hoc candidates, not agent submissions or updated leaderboard entries. Small score changes do not establish physical or statistical equivalence. Page generation does not run the grader.