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Constraining the low-frequency (LF) behavior of general circulation models (GCMs) requires reliable observational estimates of LF variability. This two-part paper presents multiproxy reconstructions of Niño-3.4 sea surface temperature over the last millennium, applying two techniques [composite plus scale (CPS) and hybrid regularized expectation maximization (RegEM) truncated total least squares (TTLS)] to a network of tropical, high-resolution proxy records. This first part presents the data and methodology before evaluating their predictive skill using frozen network analysis (FNA) and pseudoproxy experiments. The FNA results suggest that about half of the Niño-3.4 variance can be reconstructed back to A.D. 1000, but they show little LF skill during certain intervals. More variance can be reconstructed in the interannual band where climate signals are strongest, but this band is affected by dating uncertainties (which are not formally addressed here). The CPS reliably estimates interannual variability, while LF fluctuations are more faithfully reconstructed with RegEM, albeit with inevitable variance loss. The RegEM approach is also tested on representative pseudoproxy networks derived from two millennium-long integrations of a coupled GCM. The pseudoproxy study confirms that reconstruction skill is significant in both the interannual and LF bands, provided that sufficient variance is exhibited in the target Niño-3.4 index. It also suggests that FNA severely underestimates LF skill, even when LF variability is strong, resulting in overly pessimistic performance assessments. The centennial-scale variance of the historical Niño-3.4 index falls somewhere between the two model simulations, suggesting that the network and methodology presented here would be able to capture the leading LF variations in Niño-3.4 for much of the past millennium, with the caveats noted above. [ABSTRACT FROM AUTHOR]