============================================================================== OCEAN HEAT LAG: HOW MUCH WARMING IS ALREADY IN THE PIPELINE Science Journaling Club, Volume 2 Issue 3, Spring 2026 ============================================================================== This output is a computation. No physical measurement was made by us. Python : 3.12.3 numpy : 2.4.2 Seed : 20260315 (numpy default_rng / PCG64) Baseline parameters, all CMIP5 multimodel means from Geoffroy et al. (2013, J. Climate 26, 1841-1857), quoted, not fitted by us: C = 7.30 W yr m^-2 K^-1 upper box (equiv. 77 m mixed layer) C0 = 106.00 W yr m^-2 K^-1 deep ocean lambda= 1.13 W m^-2 K^-1 climate feedback (+- 0.31 intermodel) gamma = 0.70 W m^-2 K^-1 heat uptake efficiency (range 0.5-1.2) F_2x = 3.450 W m^-2 half the published F_4x = 6.9 ============================================================================== SECTION A. THE TWO TIMESCALES, DERIVED ============================================================================== Eigenvalues of A are -1/tau. Solving the characteristic polynomial: b = 0.2572887051 yr^-1 b* = 0.2440811579 yr^-1 delta = 0.0621085659 yr^-2 sqrt(d) = 0.2492159022 yr^-1 quantity closed form eigenvalue difference fast timescale tau_f [yr] 3.94863141 3.94863141 -1.33e-15 slow timescale tau_s [yr] 247.74542675 247.74542675 -8.53e-14 mode amplitudes: a_f = 0.60492977 a_s = 0.39507023 deep/surface phi_f = -0.02677402 phi_s = 2.57219181 product tau_f*tau_s = 978.255373 yr^2 identity C*C0/(l*g) = 978.255373 yr^2 difference -3.411e-13 Two limits the timescales have to respect. As C0 -> infinity the deep box becomes a sink that never warms, and the upper box relaxes at C/(lam+gam): C/(lam+gam) = 3.9890710383 yr tau_f at C0 = 1e+04 = 3.9886449329 yr difference -4.26e-04 tau_f at C0 = 1e+06 = 3.9890667775 yr difference -4.26e-06 tau_f at C0 = 1e+08 = 3.9890709937 yr difference -4.45e-08 tau_f at C0 = 1e+10 = 3.9890728380 yr difference 1.80e-06 The last row is worse than the one above it, not better. tau_f is computed as b - sqrt(delta), and at C0 = 1e10 those two numbers agree to eleven digits, so the subtraction throws most of them away. The limit is right; the arithmetic runs out first. Nothing else in this study sits near that regime: the realistic C0 range keeps b and sqrt(delta) well apart. As C -> 0 the slow mode approaches C0(1/lam + 1/gam) from above; Geoffroy et al. (2013 I) state the inequality tau_s > C0(1/lam + 1/gam): C0(1/lam + 1/gam) = 245.2339 yr ours = 247.7454 yr OK ============================================================================== VALIDATION V4. MODE IDENTITIES ============================================================================== Geoffroy et al. (2013 Part I) state a_f + a_s = 1 and phi_f*a_f + phi_s*a_s = 1 for any admissible parameter set. identity club value required difference a_f + a_s 1.000000000000000 1.000000000000000 -2.22e-16 phi_f a_f + phi_s a_s 1.000000000000000 1.000000000000000 -2.22e-16 sign checks (Part I): phi_f < 0 : True ; a_f, a_s, phi_s > 0 : True ============================================================================== VALIDATION V1. NUMERICAL INTEGRATION vs CLOSED FORM ============================================================================== Abrupt doubling of CO2: F jumps to 3.450 W m^-2 at t = 0 and stays there. RK4, fixed step dt = 0.005 yr, 3000 yr, 600,000 steps. integration done: 30001 stored points, 1.6 s so far t [yr] RK4 T [K] closed form eigen-decomp RK4-exact eig-exact 0.00 0.000000000000 0.000000000000 0.000000000000 -8.47e-16 -8.47e-16 0.10 0.046672870385 0.046672870385 0.046672870385 -1.92e-15 -7.98e-16 0.20 0.092190557522 0.092190557522 0.092190557522 -2.71e-15 -8.60e-16 0.30 0.136581944519 0.136581944519 0.136581944519 -3.52e-15 -1.11e-15 0.50 0.222097757161 0.222097757161 0.222097757161 -5.33e-15 -5.55e-17 0.70 0.303437249451 0.303437249451 0.303437249451 -6.49e-15 -5.00e-16 1.00 0.418064209026 0.418064209026 0.418064209026 -8.16e-15 -1.11e-16 1.20 0.489842999839 0.489842999839 0.489842999839 -9.55e-15 -6.11e-16 1.30 0.524409647057 0.524409647057 0.524409647057 -1.04e-14 -7.77e-16 1.60 0.623079323378 0.623079323378 0.623079323378 -1.11e-14 -1.11e-16 1.80 0.684873775062 0.684873775062 0.684873775062 -1.19e-14 -6.66e-16 2.10 0.771977753295 0.771977753295 0.771977753295 -1.30e-14 -4.44e-16 2.40 0.852813362087 0.852813362087 0.852813362087 -1.35e-14 -5.55e-16 2.80 0.951630924601 0.951630924601 0.951630924601 -1.47e-14 -5.55e-16 3.30 1.062129469740 1.062129469741 1.062129469741 -1.62e-14 -6.66e-16 3.80 1.159766790179 1.159766790179 1.159766790179 -1.60e-14 -4.44e-16 4.40 1.262094687001 1.262094687001 1.262094687001 -1.49e-14 -4.44e-16 5.10 1.363891657561 1.363891657561 1.363891657561 -1.49e-14 -4.44e-16 5.90 1.460792157050 1.460792157050 1.460792157050 -1.53e-14 -8.88e-16 6.90 1.558272599838 1.558272599838 1.558272599838 -1.35e-14 -2.22e-16 8.00 1.641704072191 1.641704072191 1.641704072191 -1.24e-14 -2.22e-16 9.20 1.711169074254 1.711169074254 1.711169074254 -1.20e-14 -2.22e-16 10.70 1.774983097801 1.774983097801 1.774983097801 -8.88e-15 -2.22e-16 12.40 1.825881758277 1.825881758277 1.825881758277 -6.44e-15 -4.44e-16 14.40 1.866864047873 1.866864047873 1.866864047873 -4.66e-15 -2.22e-16 16.70 1.898640869438 1.898640869438 1.898640869438 -4.44e-15 -2.22e-16 19.40 1.924183791824 1.924183791824 1.924183791824 -3.55e-15 2.22e-16 22.50 1.945436180368 1.945436180368 1.945436180368 -2.44e-15 -2.22e-16 26.10 1.965029175754 1.965029175754 1.965029175754 2.22e-16 -2.22e-16 30.30 1.984906787910 1.984906787910 1.984906787910 -1.78e-15 -2.22e-16 35.10 2.005991117116 2.005991117116 2.005991117116 -4.44e-16 -4.44e-16 40.70 2.029581542251 2.029581542251 2.029581542251 -2.66e-15 0.00e+00 47.20 2.056134090495 2.056134090495 2.056134090495 -8.88e-16 0.00e+00 54.80 2.086263039054 2.086263039054 2.086263039054 -4.44e-16 -4.44e-16 63.50 2.119627321363 2.119627321363 2.119627321363 -8.88e-16 -4.44e-16 73.70 2.157279255912 2.157279255912 2.157279255912 1.33e-15 -4.44e-16 85.50 2.198946497709 2.198946497709 2.198946497709 -8.88e-16 0.00e+00 99.10 2.244571412126 2.244571412126 2.244571412126 0.00e+00 0.00e+00 115.00 2.294831554949 2.294831554949 2.294831554949 1.33e-15 0.00e+00 133.30 2.348823106156 2.348823106156 2.348823106156 8.88e-16 0.00e+00 154.60 2.406843436873 2.406843436873 2.406843436873 2.22e-15 0.00e+00 179.30 2.468166657483 2.468166657483 2.468166657483 5.77e-15 -4.44e-16 208.00 2.532150174894 2.532150174894 2.532150174894 2.66e-15 0.00e+00 241.30 2.597669877322 2.597669877322 2.597669877322 -6.22e-15 0.00e+00 279.80 2.663218899867 2.663218899867 2.663218899867 -6.22e-15 0.00e+00 324.50 2.727582565395 2.727582565395 2.727582565395 -4.88e-15 0.00e+00 376.40 2.789105389506 2.789105389506 2.789105389506 -2.22e-15 0.00e+00 436.60 2.846054187146 2.846054187146 2.846054187146 -6.22e-15 -4.44e-16 506.30 2.896826594619 2.896826594619 2.896826594619 -7.55e-15 0.00e+00 587.20 2.940361750582 2.940361750582 2.940361750582 -7.11e-15 -4.44e-16 681.10 2.975926085015 2.975926085015 2.975926085015 -8.88e-16 0.00e+00 790.00 3.003374574419 3.003374574419 3.003374574419 -9.33e-15 0.00e+00 916.20 3.023220937447 3.023220937447 3.023220937447 -6.66e-15 0.00e+00 1062.60 3.036551418346 3.036551418346 3.036551418346 -2.66e-15 0.00e+00 1232.50 3.044763246654 3.044763246654 3.044763246654 1.78e-15 0.00e+00 1429.40 3.049332947325 3.049332947325 3.049332947325 4.00e-15 0.00e+00 1657.90 3.051600633926 3.051600633926 3.051600633926 8.88e-16 -4.44e-16 1922.90 3.052583778868 3.052583778868 3.052583778868 5.33e-15 0.00e+00 2230.20 3.052948784541 3.052948784541 3.052948784541 9.77e-15 0.00e+00 2586.60 3.053062096758 3.053062096758 3.053062096758 5.33e-15 0.00e+00 3000.00 3.053090700755 3.053090700755 3.053090700755 2.22e-15 0.00e+00 max |RK4 - closed form|, surface : 1.621e-14 K over 61 points max |eigen - closed form|, surface: 1.110e-15 K max |RK4 - closed form|, deep : 2.665e-14 K largest surface anomaly in the run: 3.053091 K relative worst-case agreement : 5.31e-15 VERDICT: three independent solutions agree to better than 1e-9 K ============================================================================== VALIDATION V3. ENERGY CONSERVATION ============================================================================== Summing the two box equations gives, exactly, d/dt (C*T + C0*T0) = F - lambda*T so the running integral of the net top-of-atmosphere imbalance must equal the heat stored in the two boxes at every step. Two residuals: (a) carried as a third RK4 state variable, checked at all 600,000 steps. worst absolute residual : 2.467e-11 W yr m^-2 at t = 1804.390 yr heat stored at t = 3000 : 345.914069 W yr m^-2 worst residual / stored : 7.132e-14 (b) independent Simpson quadrature over the stored temperature series, which shares no arithmetic with the integrator's own accumulator. t [yr] int (F - lam T) dt C*T + C0*T0 difference relative 1.0 3.2039483061 3.2039483019 4.21e-09 1.31e-09 5.0 12.6645240288 12.6645240153 1.35e-08 1.07e-09 10.0 20.9445301112 20.9445300938 1.73e-08 8.28e-10 25.0 40.6381198284 40.6381198096 1.88e-08 4.63e-10 50.0 69.9536388221 69.9536388033 1.88e-08 2.69e-10 100.0 120.3879877433 120.3879877244 1.88e-08 1.56e-10 250.0 222.8175529714 222.8175529526 1.88e-08 8.45e-11 500.0 301.0408110637 301.0408110448 1.88e-08 6.26e-11 1000.0 339.9522789562 339.9522789373 1.88e-08 5.54e-11 2000.0 345.8106057207 345.8106057018 1.88e-08 5.45e-11 3000.0 345.9140691133 345.9140690945 1.88e-08 5.44e-11 worst Simpson residual : 1.884e-08 W yr m^-2 (1.31e-09 relative) VERDICT: energy is conserved to quadrature precision ============================================================================== VALIDATION V2. THE ZERO HEAT UPTAKE LIMIT ============================================================================== With gamma = 0 the deep box is disconnected. The upper box must then obey C dT/dt = F - lambda*T, T(t) = (F/lambda)(1 - exp(-t*lambda/C)) a single exponential with timescale C/lambda, and the deep ocean must never warm at all. expected single-box timescale C/lambda : 6.4601769912 yr gamma tau_f [yr] tau_s [yr] tau_f - C/lam 1 3.3751171256 202.89 -3.085e+00 0.3 5.0892918131 448.51 -1.371e+00 0.1 5.9322450079 1154.33 -5.279e-01 0.03 6.2928130845 3627.31 -1.674e-01 0.01 6.4034748185 10693.86 -5.670e-02 0.001 6.4544647276 106093.81 -5.712e-03 0.0001 6.4596053412 1060093.81 -5.716e-04 1e-05 6.4601198217 10600093.81 -5.717e-05 1e-06 6.4601712702 106000093.81 -5.721e-06 0 6.4601769912 infinite 0.000e+00 direct integration with gamma = 0, against the single-box exact solution: t [yr] RK4 T [K] single-box exact difference deep T [K] 1.0 0.43784111421546 0.43784111421546 -1.50e-15 0.00e+00 2.0 0.81289194707843 0.81289194707843 -2.11e-15 0.00e+00 5.0 1.64507800211631 1.64507800211631 -2.22e-15 0.00e+00 10.0 2.40375071572737 2.40375071572738 -3.55e-15 0.00e+00 20.0 2.91499135064413 2.91499135064413 -1.33e-15 0.00e+00 50.0 3.05176866528992 3.05176866528992 4.00e-15 0.00e+00 100.0 3.05309676690351 3.05309676690351 -1.33e-15 0.00e+00 200.0 3.05309734513246 3.05309734513263 -1.77e-13 0.00e+00 max |RK4 - single box| : 1.772e-13 K max deep-ocean warming : 0.000e+00 K (must be identically zero) tau_f at gamma=1e-6 minus C/lambda : -5.721e-06 yr VERDICT: the model collapses to the single box as it must ============================================================================== SECTION B. TIMESCALES MEASURED FROM THE MODEL'S OWN IMPULSE RESPONSE ============================================================================== Nothing above measured anything; it solved algebra. So now we throw a delta function of heat at the model and read the timescales off the decay, the way you would if you did not know the answer. A pulse of 1 W yr m^-2 is delivered over the first 0.05 yr, then the forcing is switched off and the system is integrated for 4000 yr. peak surface response : 0.13443596 K at t = 0.100 yr timescale measured analytic difference rel err slow, log-linear fit over 1200-3600 yr 247.745427 247.745427 1.42e-13 0.0000% fast, after removing the slow mode 3.948631 3.948631 7.68e-14 0.0000% fit windows chosen before looking at the residuals; the fast window ends at 14 yr because beyond 3.5 fast e-foldings the residual is numerical dust. ratio of the two timescales tau_s/tau_f : 62.74 published order of magnitude (Geoffroy et al. 2013 I): ~4 yr and ~250 yr club tau_f 3.95 yr vs published ~4 yr : -1.3% club tau_s 247.7 yr vs published ~250 yr : -0.9% ============================================================================== SECTION C. EQUILIBRIUM vs TRANSIENT SENSITIVITY ============================================================================== ECS = F_2x / lambda = 3.0531 K TCR = surface anomaly at year 70 of a 1%-per-year CO2 ramp (linear forcing) = 1.8982 K TCR / ECS = 0.6217 warming still owed at the moment of doubling = 1.1549 K numeric ramp integration at year 70 = 1.8981536493 K closed-form ramp solution at year 70 = 1.8981536493 K difference = -1.11e-15 K Against the IPCC AR6 assessed best estimates (Forster et al. 2021, ch.7): quantity club AR6 difference ECS [K] 3.05 3.00 +0.05 TCR [K] 1.90 1.80 +0.10 We are not fitting to AR6. Our ECS and TCR fall out of parameters taken from CMIP5 model calibrations, so the agreement is a consistency check on the two-box reduction, not an independent confirmation of anything. ============================================================================== SECTION D. COMMITTED WARMING AFTER AN ABRUPT DOUBLING ============================================================================== Forcing steps to 3.450 W m^-2 at year 0 and is held there forever. The system has to get to 3.0531 K eventually. The question is when. t [yr] T surf [K] T deep [K] realized committed imbal N heat [W yr] 1 0.4181 0.0014 13.69% 2.6350 2.9776 3.20 2 0.7437 0.0053 24.36% 2.3094 2.6097 5.99 5 1.3504 0.0265 44.23% 1.7027 1.9241 12.66 10 1.7479 0.0772 57.25% 1.3052 1.4749 20.94 20 1.9288 0.1915 63.18% 1.1243 1.2705 34.38 30 1.9835 0.3044 64.97% 1.0696 1.2086 46.75 50 2.0673 0.5176 67.71% 0.9858 1.1139 69.95 70 2.1438 0.7142 70.22% 0.9093 1.0275 91.36 100 2.2475 0.9810 73.61% 0.8056 0.9103 120.39 150 2.3947 1.3597 78.44% 0.6584 0.7440 161.61 200 2.5151 1.6691 82.38% 0.5380 0.6080 195.29 300 2.6937 2.1288 88.23% 0.3594 0.4061 245.31 500 2.8928 2.6408 94.75% 0.1603 0.1811 301.04 700 2.9816 2.8692 97.66% 0.0715 0.0808 325.90 1000 3.0318 2.9983 99.30% 0.0213 0.0241 339.95 1500 3.0503 3.0458 99.91% 0.0028 0.0032 345.12 2000 3.0527 3.0521 99.99% 0.0004 0.0004 345.81 3000 3.0531 3.0531 100.00% 0.0000 0.0000 345.91 time to reach 50.0% of equilibrium : 6.55 yr time to reach 63.2% of equilibrium : 20.10 yr time to reach 75.0% of equilibrium : 113.37 yr time to reach 90.0% of equilibrium : 340.38 yr time to reach 95.0% of equilibrium : 512.10 yr time to reach 99.0% of equilibrium : 910.83 yr Headline: 73.6% of the equilibrium response is realized 100 years after the step; 0.8056 K of the 3.0531 K total is still in the pipeline. Fraction of the remaining commitment carried by each mode at t = 100 yr: fast mode : 6.0687e-12 of ECS (2.300e-09 % of what is left) slow mode : 0.26386125 of ECS (100.00000000 % of what is left) After the first few decades the pipeline is the deep ocean and nothing else. ============================================================================== SECTION E. SWEEP OF THE OCEAN HEAT UPTAKE EFFICIENCY, gamma ============================================================================== lambda held at 1.13 W m^-2 K^-1, so ECS is fixed at 3.0531 K in every row. Only the route to it changes. The published CMIP5 range is 0.5 to 1.2. gamma tau_f[yr] tau_s[yr] a_s TCR[K] TCR/ECS f(100yr) lag[yr] 0.00 6.4602 inf 0.0000 2.7713 0.9077 100.00% 54.61 0.10 5.9322 1154.33 0.0821 2.5723 0.8425 92.47% 55.25 0.20 5.4801 624.79 0.1531 2.4086 0.7889 86.96% 55.63 0.35 4.9133 398.21 0.2424 2.2120 0.7245 81.14% 55.74 0.50 4.4493 307.82 0.3158 2.0578 0.6740 77.18% 55.26 0.60 4.1844 272.75 0.3578 1.9725 0.6461 75.21% 54.52 0.70 3.9486 247.75 0.3951 1.8982 0.6217 73.61% 53.34 0.80 3.7374 229.03 0.4285 1.8328 0.6003 72.31% 51.65 0.90 3.5472 214.50 0.4585 1.7750 0.5814 71.24% 49.61 1.00 3.3751 202.89 0.4856 1.7234 0.5645 70.33% 47.59 1.20 3.0759 185.52 0.5327 1.6356 0.5357 68.93% 44.45 1.50 2.7138 168.22 0.5894 1.5324 0.5019 67.47% 41.67 2.00 2.2675 151.00 0.6589 1.4096 0.4617 66.02% 39.48 gamma does not change where the system ends up. It changes how much of the response is handed to the slow mode: a_s runs from 0.000 to 0.659 across this sweep, and TCR/ECS falls from 0.908 to 0.462. ============================================================================== SECTION F. SWEEP OF THE FEEDBACK PARAMETER, lambda ============================================================================== gamma held at 0.70 W m^-2 K^-1. Now ECS moves, and so does the lag. lambda ECS[K] tau_f[yr] tau_s[yr] TCR[K] TCR/ECS f(100yr) 0.60 5.7500 5.5036 334.76 2.6606 0.4627 58.70% 0.70 4.9286 5.1245 308.16 2.4741 0.5020 62.61% 0.82 4.2073 4.7323 284.87 2.2815 0.5423 66.47% 0.90 3.8333 4.5021 272.82 2.1687 0.5658 68.64% 1.00 3.4500 4.2437 260.48 2.0423 0.5920 71.01% 1.13 3.0531 3.9486 247.75 1.8982 0.6217 73.61% 1.25 2.7600 3.7101 238.36 1.7819 0.6456 75.65% 1.44 2.3958 3.3859 226.72 1.6241 0.6779 78.31% 1.60 2.1562 3.1535 219.09 1.5112 0.7009 80.15% 1.80 1.9167 2.9041 211.47 1.3903 0.7254 82.06% 2.00 1.7250 2.6911 205.39 1.2872 0.7462 83.64% A less sensitive climate (large lambda) is also a faster one. tau_s falls from 335 yr at lambda = 0.60 to 205 yr at lambda = 2.00, because the same feedback that limits the destination also shortens the journey. ECS varies by a factor of 3.33 across this sweep; TCR only by 2.07. ============================================================================== SECTION G. THE (gamma, lambda) GRID ============================================================================== Realized fraction T(100 yr)/ECS after an abrupt doubling, in per cent. gamma \ lam 0.82 1.00 1.13 1.44 1.80 0.35 75.54% 79.13% 81.14% 84.67% 87.41% 0.50 70.73% 74.85% 77.18% 81.33% 84.60% 0.70 66.47% 71.01% 73.61% 78.31% 82.06% 0.90 63.62% 68.44% 71.24% 76.31% 80.40% 1.20 60.82% 65.93% 68.93% 74.42% 78.89% TCR/ECS on the same grid. gamma \ lam 0.82 1.00 1.13 1.44 1.80 0.35 0.654 0.699 0.724 0.771 0.809 0.50 0.598 0.646 0.674 0.726 0.769 0.70 0.542 0.592 0.622 0.678 0.725 0.90 0.500 0.551 0.581 0.640 0.691 1.20 0.454 0.505 0.536 0.597 0.650 range of realized fraction across the grid : 60.8% to 87.4% range of TCR/ECS across the grid : 0.454 to 0.809 ============================================================================== SECTION H. AN IDEALISED HISTORICAL RUN ============================================================================== Everything above is a thought experiment about step functions. This section is the closest the study comes to the real world, and it is still an idealisation. Forcing grows exponentially from 1750 to 2019, F(t) = F_2019 * exp((t - 2019)/tau_g) with F_2019 = 2.72 W m^-2, the total anthropogenic effective radiative forcing assessed in IPCC AR6 chapter 7 (Forster et al. 2021). That endpoint is a published number. The exponential shape is our choice, made because it has one parameter and we can set that parameter from one observation: the growth time tau_g is tuned so the model's top-of-atmosphere imbalance in 2019 matches the 0.87 W m^-2 reported by von Schuckmann et al. (2020). The AR6 forcing time series is not used. Section 9 of the article says what that costs us. bracketing: imbalance at tau_g = 2 yr is 2.1582 W m^-2, at tau_g = 5000 yr it is 0.3796 W m^-2, so the 0.87 W m^-2 target is bracketed. fitted forcing growth time tau_g : 87.3456 yr model imbalance in 2019 : 0.870000 W m^-2 target (von Schuckmann et al. 2020) : 0.8700 W m^-2 forcing in 1850 under this fit : 0.3929 W m^-2 quantity value modelled warming 2019 relative to 1850 1.4077 K observed 2011-2020 vs 1850-1900 (AR6) 1.0900 K club minus observed +0.3177 K club / observed 129.2% equilibrium for the 2019 forcing 2.4071 K realized by 2019 (rel. 1750) 1.6372 K still committed at 2019 forcing 0.7699 K realized fraction 68.0% If the composition were frozen at its 2019 value, the model says: 10 yr later (2029) : T = 1.7231 K, 71.6% of equilibrium, 0.6840 K left 20 yr later (2039) : T = 1.7546 K, 72.9% of equilibrium, 0.6525 K left 50 yr later (2069) : T = 1.8293 K, 76.0% of equilibrium, 0.5777 K left 100 yr later (2119) : T = 1.9349 K, 80.4% of equilibrium, 0.4721 K left 200 yr later (2219) : T = 2.0917 K, 86.9% of equilibrium, 0.3153 K left 500 yr later (2519) : T = 2.3131 K, 96.1% of equilibrium, 0.0939 K left 1000 yr later (3019) : T = 2.3946 K, 99.5% of equilibrium, 0.0125 K left THE COMPARISON, STATED PLAINLY. Our modelled 2019 warming is 1.4077 K. The observational assessment gives 1.09 K. We are +29.2% off, a gap of +0.318 K. Our own numerical error is of order 1e-13 K, so none of that gap is arithmetic. It is the model. The same three published numbers, rearranged, give the effective feedback the real record implies: lambda_eff = (ERF - EEI)/dT = (2.72 - 0.87)/1.09 = 1.6972 W m^-2 K^-1 implied energy-budget ECS = F_2x/lambda_eff = 2.0327 K our CMIP5-mean lambda = 1.1300 W m^-2 K^-1, ECS = 3.0531 K lambda_eff is 1.83 intermodel SD above the CMIP5 mean. The gap between a feedback diagnosed from a century of history and one diagnosed from a long model run is the pattern effect. Armour (2017) and Zhou et al. (2021) are the references. We do not model it. Rerunning the same forcing history with lambda = 1.697 instead: modelled warming 2019 rel. 1850 : 1.0482 K (observed 1.09 K) ECS : 2.0327 K realized fraction at 2019 : 76.2% tau_s : 215.2 yr The lag is not what is wrong. The feedback is. ============================================================================== SECTION I. MONTE CARLO OVER THE PARAMETER SPREAD ============================================================================== Trials : 400000 Seed : 20260315, stream spawned from the master SeedSequence Sampling distributions, and where each one comes from: lambda ~ Normal(1.13, 0.31), truncated to (0.30, 2.60) mean and SD are the published CMIP5 intermodel values. gamma ~ Uniform(0.50, 1.20) the published range across the 16 models. Uniform is OUR choice: we have the range but not the shape of the distribution. C0 ~ Normal(91, 27), truncated to (30, 220) published mean and SD with the outlying INM-CM4 excluded. C held fixed at 7.3. We do not have a published intermodel SD for it, and Section E shows C only sets the fast mode, which contributes 6.07e-12 of ECS to the pipeline after a century. These are not a posterior. They are a spread of calibrated models, sampled independently, which ignores the real correlations between the parameters. draws rejected by truncation : 6242 of 400000 (1.560%) trials retained : 393758 Results. Standard errors are the sample SD divided by sqrt(n) from these very trials; nothing is assumed about the shape of the output distribution. fast timescale tau_f [yr] mean 3.75330 SD 0.75562 SE 0.001204 median 3.62178 5-95% 2.77957 to 5.18059 2.5-97.5% 2.66735 to 5.60568 slow timescale tau_s [yr] mean 206.77712 SD 73.26998 SE 0.116765 median 198.21407 5-95% 102.57731 to 339.01715 2.5-97.5% 89.05423 to 374.50376 ECS [K] mean 3.32365 SD 1.14574 SE 0.001826 median 3.05086 5-95% 2.10409 to 5.46779 2.5-97.5% 1.98577 to 6.37298 TCR [K] mean 1.89811 SD 0.35622 SE 0.000568 median 1.84073 5-95% 1.42710 to 2.57034 2.5-97.5% 1.36818 to 2.76058 TCR / ECS mean 0.59523 SD 0.08242 SE 0.000131 median 0.60373 5-95% 0.44583 to 0.71474 2.5-97.5% 0.40691 to 0.73145 realized fraction at 100 yr mean 0.73441 SD 0.07911 SE 0.000126 median 0.74382 5-95% 0.59033 to 0.84618 2.5-97.5% 0.54847 to 0.86516 committed warming left at 100 yr [K] mean 0.95659 SD 0.67882 SE 0.001082 median 0.77200 5-95% 0.34725 to 2.17448 2.5-97.5% 0.29824 to 2.79215 baseline (no sampling) values for comparison: tau_f 3.9486 tau_s 247.75 ECS 3.0531 TCR 1.8982 TCR/ECS 0.6217 f100 0.7361 The Monte Carlo mean of tau_s (206.8 yr) sits below the baseline (247.7 yr). That is not skew, it is a different centre. The sampling distributions are centred on gamma = 0.85 and C0 = 91, against the baseline 0.70 and 106. One run at those central values gives tau_s = 190.4 yr. That overshoots the sampled mean of 206.8 yr, so the spread pulls back the other way: tau_s is convex in both gamma and C0, and averaging a convex function over a spread raises the mean above the value at the centre. Every output here is right-skewed, because ECS goes as 1/lambda while lambda is sampled symmetrically. ECS mean 3.324 against median 3.051; committed warming mean 0.957 against median 0.772. The article quotes the medians and reports the mean with its standard error beside them. Convergence of the mean committed warming as trials accumulate: n running mean running SE |mean - final| 10 1.087037 0.293545 0.130448 30 0.963811 0.127602 0.007222 100 0.947222 0.057542 0.009366 300 0.883929 0.030003 0.072660 1000 0.935593 0.019903 0.020996 3000 0.960101 0.012434 0.003513 10000 0.965431 0.006985 0.008842 30000 0.958456 0.003995 0.001867 100000 0.958178 0.002151 0.001589 200000 0.958970 0.001526 0.002381 393758 0.956589 0.001082 0.000000 final SE / final mean : 0.1131% the running mean first falls inside 1 SE of its final value at n = 52 ============================================================================== SUMMARY OF VALIDATIONS ============================================================================== [PASS] V1 RK4 vs closed form vs eigendecomposition max deviation 1.62e-14 K over 61 points [PASS] V2 gamma -> 0 collapses to the single box max deviation 1.77e-13 K, deep box exactly 0 [PASS] V3 energy conservation, every step worst residual 2.47e-11 W yr m^-2 (7.1e-14 relative) [PASS] V3b independent Simpson quadrature worst residual 1.88e-08 W yr m^-2 [PASS] V4 Geoffroy mode identities a_f+a_s-1 = -2.2e-16, phi.a-1 = -2.2e-16 [PASS] V5 tau_f, tau_s against published statements club 3.95 / 247.7 yr vs published ~4 / ~250 yr [PASS] V6 ECS and TCR against IPCC AR6 club 3.05 / 1.90 K vs AR6 3.0 / 1.8 K ONE THING DOES NOT AGREE, and it is not a numerical problem: the idealised historical run of Section H gives 1.41 K of warming by 2019 against an observational assessment of 1.09 K, a gap of +0.32 K (+29%). There is no standard error to quote on it, because it is not a sampling question: with the parameters fixed the model is deterministic and always gives 1.4077 K. The nearest thing to a sigma is the intermodel spread. The observed warming needs lambda near 1.70, which is 1.83 intermodel standard deviations above the CMIP5 mean of 1.13. In the Monte Carlo of Section I, 3.34% of the 393758 retained trials drew a lambda at least that large, so the observed record is not impossible under this model. It is just not where the multimodel mean sits. ============================================================================== HEADLINE NUMBERS ============================================================================== fast timescale tau_f = 3.95 yr slow timescale tau_s = 247.7 yr equilibrium climate sensitivity ECS = 3.05 K transient climate response TCR = 1.90 K TCR / ECS = 0.622 realized 100 yr after abrupt doubling = 73.6% still committed at that moment = 0.81 K wall clock : 7.7 s