**Publications data.** Our data is derived from the MAG data, which archives publications from 1800 to 2021. The publications cover 292 secondary subjects in 19 major disciplines, including but not limited to Economics, Biology, Computer science, and Physics. We excluded patents, datasets, and repositories, utilizing the doctype field in the MAG data. We limited our focus to publications up to 2020 because recent literature was probably not sufficiently collected. Although we used literature from as early as 1800, the KQI was only calculated from 1920 because the citations were too sparse to be interconnected in the early years. We removed possible errors in the data, including self-citations, duplicate citations, and citations violating time order. After eliminating potentially incorrect publications and closing the data up to 2020, the analytical sample consisted of 213,715,816 publications and 1,762,008,545 citations. The subject data is split from the MAG data. While processing data from a particular subject, we only preserved citation relationships that both article and reference are on the same subject, thus guaranteeing that all nodes within the network are from the same subject.

**Patents data.** The Patents View data collect 8.1 million patents granted between 1976 and 2022 and their corresponding 126 million citations. We limited our focus to citations made to U.S. granted patents by U.S. patents up to 2020 because recent patents were probably not yet sufficiently collected. Although we used patents from 1976, the KQI was only calculated from 2000 because the citations were too sparse to be interconnected in the early years. We removed possible errors in the data, including self-citations, duplicate citations, and citations violating time order. After eliminating potentially incorrect patents and closing the data up to 2020, the analytical sample consisted of 7,627,229 patents and 101,148,606 citations.

**KQI calculation.** KQI 30 is a metric that quantifies knowledge from the perspective of information structurization. As described by the proposers of KQI, we constructed year-by-year publication citation graphs or patent citation graphs, which are directed acyclic graphs. We calculated the KQI of each node in a citation graph and added them up to obtain the KQI of the citation graph for each year, exploiting the additivity of the KQI as pointed out by the proposers.

**Search results in Google Scholar.** The two terms, "knowledge explosion" and "information explosion," are searched as phrases enclosed with quotation marks in Google Scholar (https://scholar.google.com) and filtered by year ranges. We recorded the number of results returned manually.

**Analysis of mathematical conjecture.** We collected 61 mathematical conjectures proven to be correct since 1960 (Supplementary Table 1). Conjectures proved wrong and those not yet proven are excluded. We chose a possible intermediate year for conjectures without a specific formulation year or proof year. Due to certain mathematical conjectures proved in the same year but with different durations and difficulty in determining their temporal order, we took the average duration of proofs within the same year. We used such a time series when conducting correlation analyses and hypothesis testing. Spearman and Kendall 45 rank correlation coefficients are non-parametric measures of the strength of monotonic association between two variables and are calculated by measuring the rank correlation between two variables. The range of these two coefficients is from − 1 to 1, with values closer to 0 indicating a weaker relationship between the two variables. Cox-Stuart 46 and Mann-Kendall 47 hypothesis tests assess whether there is a monotonic increasing or decreasing trend over time in a time series data. The null hypothesis for both hypothesis tests is the absence of a monotonic trend, so a p-value greater than 0.05 indicates the lack of a significant trend.

**Random graph generated by Barabási-Albert model.** The BA model 34 uses a preferential attachment process to generate random graphs. We used the BA model under undirected graphs. After generating the graph, we oriented each edge chronologically, from the node joined earlier to the node joined later. This method naturally resulted in directed acyclic graphs available for KQI calculation directly.

**Random graph generated by Erdős-Rényi model.** The ER model 33 generates a random graph with the same probability of existence for each edge between two arbitrary nodes. We used the ER model under undirected graphs and uniquely numbered each node. After generating the graph, we oriented each edge in numbered order, from the smaller numbered node to the larger numbered node. This method naturally resulted in directed acyclic graphs available for KQI calculation directly.

**Random graph generated by Watts-Strogatz model.** The WS model 35 generates graphs with small-world properties and is adjustable between regular and random graphs. Following the WS model, we constructed a regular ring lattice, rewired edges, and numbered each node incrementally and uniquely along the ring. After generating the graph, we oriented each edge in numbered order, from the smaller numbered node to the larger numbered node. This method naturally involved interpolating between a regular ring lattice and a random graph.

**Limitation of knowledge growth.** We prove that the growth rate of KQI has an upper bound regarding the graph size through a theoretical derivation from the KQI formula. We rewrite the formula of KQI:\({K}_{\alpha }=\sum _{\beta \to \alpha }\frac{{d}_{\alpha }^{i}{V}_{\beta }-{V}_{\alpha }}{{d}_{\alpha }^{i}W}{\text{log}\left(1+\frac{1}{\frac{{V}_{\alpha }}{{d}_{\alpha }^{i}{V}_{\beta }-{V}_{\alpha }}}\right)}^{\frac{{V}_{\alpha }}{{d}_{\alpha }^{i}{V}_{\beta }-{V}_{\alpha }}}.\)Applying the Euler limit formula, it is simplified as follows:\({K}_{\alpha }=\frac{a}{W}\left(\sum _{\beta \to \alpha }{V}_{\beta }-{V}_{\alpha }\right),\left(0<a<\text{log}e\right).\)We sum KQI over all nodes and note that W is the sum of the out-degrees of all nodes. The relation between K and W is thus derived:\(K\triangleq \sum _{\alpha }{K}_{\alpha }=\frac{a}{W}\sum _{\alpha }\left(\sum _{\beta \to \alpha }{V}_{\beta }-{V}_{\alpha }\right)=a\sum _{\alpha }\frac{\left({d}_{\alpha }^{o}-1\right)}{W}{V}_{\alpha }<a\mathbb{E}\left({V}_{\alpha }\right)<W\text{log}e.\)

**Discovery of inflection points.** We discover the inflection points using segmented regression models developed by Vito M. R. Muggeo 48. The segmented regression was performed on the curve of KQI over time, with the regression line breakpoint considered the inflection point. We started with the null hypothesis of no breakpoint and performed a score test to determine if there was an additional breakpoint 49. This process repeated until no additional breakpoint. The significance level was 0.01. To counteract the multiple comparisons problem, we employed Bonferroni correction, requiring that the p-values for each of the first k tests be smaller than 0.01/k. Once the number of breakpoints was determined, we used the segmented method (33) to estimate their positions.

**Estimation of inflection density.** The inflection density is a quantity that characterizes the distribution of the network state at its transition between two different regimes. The area under an inflection density curve represents the average times finding the system in the inflection state. The main text investigates the inflection density per unit mean coreness. Due to the estimation error of inflection points, we map the probability densities of normal distributions centered around the inflection points to the mean coreness using linear interpolation and summing in cases with multiple mapping values. The estimated standard deviation of the inflection point determines the standard deviation of the normal distribution. We estimated the inflection density of a discipline by summing the probability densities with respect to mean coreness during network evolution. The total inflection density is the mean of densities for all disciplines. To estimate confidence intervals, 1000 bootstrap resamplings are employed.

**Calculation of marginal KQI.** The marginal KQI is calculated by subtracting the KQI for a given graph of the previous year from the current year and dividing it by the number of nodes added in the current year. We applied the LOWESS (locally weighted scatterplot smoothing) nonparametric regression method to perform local regression of marginal KQI. To estimate the 95% confidence interval of the LOWESS fit, we performed 1000 bootstrap resamplings. The fraction of data used when estimating was 2/3.