How many iterations does the calibration need?

The four gates

A budget passes only if all four hold at once. They are deliberately different in kind — the first three are self-consistency checks that a broken sampler can still pass, and the fourth is the one that catches it.

What you are turning, and what you are watching

Two different vocabularies get mixed together in the output. These are the knobs:

And these are the calibration parameters being watched. R̂ and ESS are reported per parameter; the gates use the worst one.

Three models, three budgets

All 81 cells complete, N = 1513 coretop sites, 4 chains, one seed per cell. Each recommendation is the cheapest cell clearing all four gates, measured against the 1000/1000 incumbent.

Finding 1 — warmup binds; whether sampling helps depends on the model

Each cell holds the number of parameters left with R̂ > 1.01. The univariate panel is the one that produced the original headline: read across a row and sampling three times longer barely moves it, read down and crossing warmup 400 collapses it to zero.

The two EIV panels do not behave that way. Both axes matter there, because what is being counted is dominated by 3026 per-site latent variables — and those do respond to more draws.

The counts are not comparable across panels — the univariate model has about seven parameters and the EIV models about 3040, so a 7 and a 754 describe very different failures. Within a panel the gradient is the message.

The practical advice survives, with its scope narrowed: if a fit will not converge, raise warmup first. Adding draws to an unadapted chain buys ESS on a posterior that has not settled, which makes a bad answer look more precise. For the EIV models, raising both is what actually clears the latent variables.

Two univariate cells still break the pattern: 500 / 500 reports 7 while its neighbours report 0, and 300 stays bad where 200 / 500 is nearly clean. Each cell is a single seed, so these are almost certainly noise — see the caveats.

Finding 2 — the strict gate works, but it is gating on nuisance parameters

The EIV models add a latent true value per site for both G_2/3 and NO₃⁻ — 3026 extra parameters, each informed by roughly one observation. They dominate the maximum, and they are not the calibration.

How completely they dominate it is worth stating precisely: across the full grid, the worst-R̂ parameter is a per-site latent in 27 of 27 EIV cells and 27 of 27 bounded‑T cells. Not once, in 54 fits, is the slowest parameter one the calibration actually uses.

The same question, outside the grid

The proxy-convention comparison (Part 2) refits the whole calibration under three crenarchaeol ring conventions at a deliberately generous 1000/1000. Three of those posteriors are flagged as failing — and every one fails on a latent, exactly as the grid predicts.

The calibration behind the main-text figures carries a headline R̂ of 1.01023 — and a calibration-parameter R̂ of 1.00001. Every parameter that reaches a reconstruction is converged with room to spare.

Part 3 — the inverse model asks a different question

It is tempting to carry the forward budget across to the reconstructions. The Stan source says not to. The forward EIV model is hierarchical with ~3000 correlated latent variables, which is why its warmup binds. The inverse model declares one parameter block:

The calibration draws arrive as data, not parameters, and the target is a sum of per-sample terms with no coupling between samples. The posterior therefore factorises into N independent one-dimensional posteriors, where a diagonal metric is exact rather than approximate — the easiest geometry HMC ever sees.

So N is a cost knob, not a difficulty knob, and what binds instead is M and saturation: where Ring Index approaches the upper asymptote the likelihood goes flat in temperature and t_est is prior-dominated. Neither is fixed by more iterations. Part 3 sweeps 4 budgets × M ∈ {300, 500} against 200 coretop sites spread over the proxy range rather than sampled at random — the near-saturated warm end is 15.6% of the compilation and 30% of this subset, and it is where the paleo sites live.

Because coretop temperature is known, this half can be scored on accuracy as well as convergence: bias, RMSE, and 68% / 90% interval coverage against measured SST. The sites trained the calibration, so those numbers are a tuning diagnostic and never a validation statistic.

RMSE varies by 0.02 °C across a 11.5× spread in cost, and M = 300 is indistinguishable from M = 500. The defensible SI statement is therefore not “we picked this budget” but “the answer does not depend on it”, which is the stronger claim.

Coverage is the open question: 68% intervals contain the measured SST 62%–64% of the time and 90% intervals 86%–86%, stable across every cell, so it is not sampling noise. Some of that is by construction — a constant prior mean, an in-sample set, a subset deliberately weighted toward the hard end — but systematic under-coverage deserves an explanation before it reaches an SI.

Caveats worth stating in the SI

• One seed per cell in the forward grid. Every number there is a single draw from a noisy process, and the two univariate anomalies are the visible evidence. Part 3 now carries a seed replicate; the forward grid still does not, and a defensible published recommendation wants 3–5 seeds near the threshold.
• E‑BFMI is not being reported. This cmdstanpy build does not expose it, so the diagnostic returns -1 / UNKNOWN for every run. It is therefore not one of the four gates, and no claim about energy-transition efficiency should be made from this sweep.
• Iteration counts transfer between machines; seconds do not. Every speed-up here is specific to this 8-core box, running nothing else alongside.
• Thresholds are conventions, not laws. R̂ < 1.01 and ESS ≥ 400 follow Vehtari et al. (2021); the |z| ≤ 0.1 accuracy bound and the 0.1 °C drift bound are this study’s own choices and should be stated as such.
• Part 2 reused the SRI03 calibration rather than refitting it. The resume logic found it cached, so the SRI03 column of the proxy comparison comes from the earlier run, not from this sweep.

What happened next

Every item this page listed as pending has now been resolved:

• The remaining EIV cells finished, including the 1000/4000 reference the drift gate needs.
• Bounded‑T ran as a third model — and came back easier than the parent, not harder.
• The EIV gate question was settled empirically rather than by switching the gate: strict gating works, core gating is 2.2× cheaper, and the choice between them is now an SI argument with numbers behind it.
• A seed repeat was added — to Part 3, where it turned out to be load-bearing rather than a refinement.

Still open: the corrected Part 3 run, the under-coverage question, and a seed-repeat pass on the forward cells nearest the threshold.