Sexual dimorphism is thought to have evolved via selection on both sexes. Ostracodes display sexual shape dimorphism in adult valves; however, no previous studies have addressed temporal changes on evolutionary timescales or examined the relationships between sexual shape dimorphism and selection pressure and between sexual shape dimorphism and juvenile shape. Temporal changes in sexually dimorphic traits result from responses of these traits to selection pressure. Using the Gaussian mixture model for the height/length ratio, a valve-shape parameter, we identified sexual differences in the valve shape of Krithe dolichodeira s.l. from deep-sea sediments of the Paleocene (62.6–57.6 Ma) and estimated the proportion of females in the fossil populations at 11 time intervals. Because the proportion of females in a population is altered by the mortality rate of adult males, it is reflective of selection pressure on males. We attempted to correlate the height/length ratios between the sexes with the proportion of females, taking into consideration that the valve shape was not linked with the selection pressure on males. In time-series data of the height/length ratio, both sexes indicate no significant changes on evolutionary timescales, even though the sex ratio of the population changed from female skewed to male skewed during the late Paleocene. The sexual shape dimorphism was not driven by sexual selection. The static allometry between the height/length ratio and length indicates that the sexual shape dimorphism did not function for sexual display. The absence of change over time in the female allometric slope suggests that the evolution of valve shape was constrained by stasis.

#### Supplementary Figure 1

Bar plot of the means and standard deviations of the H/L ratios in (A) all of the adult specimens (Welch’s t-test: t = 2.8395, df = 167.56, p-value = 5.1 x 10−3), (B) in morphotype B (Welch’s t-test: t = 1.5646, df = 114.7, p-value = 0.12), (C) in morphotype S (Welch’s t-test: t = 2.9211, df = 52.548, p-value = 5.1 x 10−3), and (D) in juveniles (Welch’s t-test: t = 2.0779, df = 70.788, p-value = 0.041). n indicates the number of valves. The numbers in the bars indicate the mean ± standard deviation.

SuppleFig01rv_HLRatios_LV_RV_final.pdf

#### Supplementary Table 1

The datum events of calcareous nannofossil biostratigraphy and carbon isotope stratigraphy (Norris et al. 2014; Yamaguchi et al. submitted). Abbreviations: B = bottom and T = top.

SupplementaryTable01_1129_2016.xlsx

#### Supplementary Table 2

Measurements of Krithe dolichodeira s.l. Abbreviation: LV = left valve, RV = right valve.

SupplementaryTable02_1129_2016.xlsx

#### Supplementary Table 3

In each sex (Sex) of the time interval in mega-annum (From TimeInterval_start_Ma to TimeInterval_end_Ma), the significant allometry equation consists of the slopes (Slope_b), their standard errors at 1-sigma (Slope_SE), the intercept in common logarithm of micrometer (Intercept_log10(a)), and their standard errors at 1-sigma (Intercept_SE). TimeInterval_mean_Ma and n indicate the mean age of the sediment samples in mega-annum and the number of measurements, respectively.

Supplementary_Table03.xlsx

#### Supplementary Table 4

In the time interval in mega-annum (from TimeInterval_start_Ma to TimeInterval_end_Ma); depth in m CCSF of the top sample (Sample_top_m) and of the bottom sample (Sample_bottom_m), geological ages in mega-annum of the top sample (Sample_top_age_Ma) and of the bottom sample (Sample_bottom_age_Ma) and the mean age of the samples (Sample_mean_age_Ma); the number of females (n_females), the mean of female H/L ratio (Female_mean_H/L), the standard deviation of female H/L ratio at 1-sigma (Female_SD_H/L), the mean of female length in micrometer (Female_mean_L), the standard deviation of female length at 1-sigma in micrometer (Female_SD_L); the number of males (n_males), the mean of male H/L ratio (Male_mean_H/L), the standard deviation of female H/L ratio at 1-sigma in micrometer (Male_SD_H/L), the mean of male length in micrometer (Male_mean_L), the standard deviation of male length at 1-sigma in micrometer (Male_SD_L); the proportion of females (Proportion_females); the number of juveniles (n_juveniles), the mean of juvenile H/L ratio (Juvenile_mean_H/L), the standard deviation of juvenile H/L ratio at 1-sigma (Juvenile_SD_H/L).

Supplementary_Table04.xlsx