The global temperature started to rise after the Little Ice Age of the 19th century [1] . Compared with the Little Ice Age, the present day may be called a modern warm period. Additionally, atmospheric CO₂ has risen simultaneously, although direct measurements of CO₂ were limited before 1958 [2] . There is a relationship between the change rate of the CO₂ concentration (drco₂/dt, where rco₂ = CO₂ concentration, t = time) and the change in global temperature (ΔT), as shown in Equation (1), but with a temperature-leading time lag [3] .

drco₂/dt = γΔT (γ: a constant)(1)

The time lag is approximately 0.5 - 1 year.

The relationship between the global temperature and CO₂ concentration has been investigated [3] - [6] . On the basis of Equation (1), a cross-correlation between drco₂/dt and ΔT with a temperature-leading time lag was found [6] , where a correlation coefficient r can be defined as follows (x = drco₂/dt and y = ΔT):

$r=\frac{\frac{1}{n}{{\displaystyle \sum }}_{i=1}^n\left(x_i-\bar{x}\right)\left(y_i-\bar{y}\right)}{\sqrt{\frac{1}{n}{{\displaystyle \sum }}_{i=1}^n{\left(x_i-\bar{x}\right)}^2}\sqrt{\frac{1}{n}{{\displaystyle \sum }}_{i=1}^n{\left(y_i-\bar{y}\right)}^2}}$(2)

The satellite-based 13-month average of the temperature anomaly and the annual average of the rate of CO₂ increase corresponding to Equation (1) are correlated over 40 years, as reported in a previous paper [3] . Its correlation coefficient, r, was 0.73, and the correlation was relatively good [6] . A detailed analysis indicated that the temperature-leading time lag was 5 months [6] .

An increase in global temperature, an increase in soil respiration (Rs), and a subsequent increase in global CO₂ emissions were recognized in our papers [4] [5] . In other words, as the temperature increases, thermally induced CO₂ is emitted, and the CO₂ concentration increases [5] . This natural process can be clearly detected during periods of increasing temperature, specifically during El Niño events. Figure 1 summarizes the Rs control process for global warming. In the mechanism based on the Rs control process, the increase in temperature during the modern warm period increases CO₂ emissions because of an increase in Rs. The emitted CO₂ can be considered thermally induced CO₂ [5] . As a result, the CO₂ concentration in the atmosphere increases. Although there is a cross-correlation between temperature and CO₂ concentration, a temperature-leading time lag is observed because it is a process mediated by Rs. However, according to the Intergovernmental Panel on Climate Change (IPCC) under the United Nations, as anthropogenic CO₂ emissions increase, the CO₂ concentration in the atmosphere increases, and the atmospheric temperature increases due to the greenhouse effect of CO₂ [7] , even though anthropogenic CO₂ is a minor constituent of global CO₂. The IPCC model may predict a CO₂-leading time lag if there is a cross-correlation between the CO₂ concentration and temperature. Therefore, our recent results cast strong doubts that anthropogenic CO₂ is the cause of global warming.

Tropical rainforests account for 6% - 7% of the Earth’s land surface, whereas temperate rainforests account for only 0.2% - 0.3%. However, deciduous and coniferous forests each occupy more than 10% of the Earth’s land surface, and if forests in subtropical regions are included, these forests account for approximately 30% of the Earth’s land area [8] [9] . Since the temperature in midlatitude

forests changes seasonally in contrast to the temperature in tropical rainforests, soil respiration in midlatitude forests changes seasonally depending on changes in the annual temperature. Additionally, an increase in global temperature affects the degree of soil respiration in midlatitude forests rather than in tropical rainforests. Soil respiration emits a large amount of CO₂ that exceeds anthropogenic CO₂ emissions, and an increase in CO₂ emissions due to an increase in temperature during the modern warm period must be significant.

For these reasons, soil respiration in midlatitude forests is highly dependent on temperature changes during the modern warm period and may play a significant role in controlling atmospheric CO₂ concentrations. Compared with that in the tropics, the rate of change in the CO₂ concentration at midlatitudes (≒ 50 N) significantly responds to temperature changes [4] . The temperature difference between land and sea areas is greater in the north (20 N - 90 N) than in the south (20S - 90S) [4] . These results support the critical role of soil respiration in midlatitude forests. For example, the Pacific temperate rainforest is located between No. California and S. Alaska along the Pacific Ocean. The region of Olympic National Park has approximately 3500 mm/year of precipitation (see Figure 2 ). CO₂ emission by soil respiration from forests is significant.

The United Nations and each government have begun to take the initiative to reduce anthropogenic CO₂ emissions to prevent global warming. This raises the question of whether reducing anthropogenic CO₂ emissions actually decrease the global CO₂ concentration. This question is investigated in this paper.

Atmospheric CO₂ concentrations are reported by the National Oceanic and Atmospheric Administration (NOAA). Further details are available on their website [10] . The annual emission of anthropogenic carbon was determined according to Boden et al. [11] . The global carbon budget of the IPCC [12] was also used.

The temperature datasets were obtained from the University of Alabama in Huntsville (UAH), and the 13-month average of the lower troposphere anomaly values was used [13] .

The change rate of the CO₂ concentration has been the focus of our previous papers [3] - [6] . In this paper, temporal changes in global CO₂ concentrations are the focus. Figure 3 shows the changes in atmospheric CO₂ and satellite-based temperature anomalies between 1979 and 2023. The CO₂ concentrations in Figure 3 are de-seasonalized values representing the long-term trend reported by NOAA [10] . The atmospheric CO₂ concentrations and global temperature anomalies are correlated, with r = 0.81. Notably, the small inflection points, a-j, in the temporal change in CO₂ concentrations correspond to ENSO events [3] . The changes in CO₂ concentrations around the inflection points are small for La Niña events, whereas the changes in CO₂ concentrations are greater for El Niño events. Figure 4 shows examples of inflection points, c and d, which represent changes in CO₂ concentrations. The inflection point c corresponds to La Niña, and the CO₂ concentration shows a small change, whereas the inflection point d corresponds to El Niño, and the CO₂ concentration shows a great change.

Figure 4. Changes in atmospheric CO₂ around inflection points c and d in Figure 3 .

In the Rs control process summarized in Figure 1 , the increase in temperature due to the modern warm period increases CO₂ emissions because of an increase in Rs. The emitted CO₂ can be considered thermally induced CO₂. As a result, the CO₂ concentration in the atmosphere increases. For a short period, similar natural processes can be detected during El Niño and La Niña events. The inflection points of temporal changes in CO₂ concentrations demonstrate that ENSO events change temperature, followed by changes in CO₂ [3] [4] .

The same temporal changes in atmospheric CO₂ and satellite-based temperature anomalies are shown in Figure 5 to compare the rates of acceleration between the two. The red solid line represents a linear regression of the temperature anomaly, (temperature) = 0.01438 × (year) − 28.83. The regression line indicates that the rate of increase in temperature is 0.1438˚C/decade. Notably, the atmospheric CO₂ concentration increases with time, but the rate of increase is greater than the rate of increase in temperature. Next, the change rate of the CO₂ concentration over 10 years was analyzed to determine the trend of changes in CO₂ over a long time range. Figure 6 shows the averages of increased annual CO₂ over ten years between 1970 and 2020, for example, (a value of CO₂ concentration in 1970 − a value of CO₂ concentration in 1960)/10. The increase in annual CO₂ over ten years is not constant but rather has increased over time or accelerated. If atmospheric CO₂ concentrations increase, followed by an increase in the global temperature, the global temperature may show a similar change to the accelerated increase in CO₂. However, the global temperature increases as a linear function, and no acceleration is observed.

The acceleration of the increase in CO₂ concentration is critical in predicting future CO₂ during the modern warm period. As described later, anthropogenic CO₂ is a small proportion of the total carbon cycle budget, so even if a small portion of anthropogenic CO₂ is partially reduced, the decrease in CO₂ concentration may be difficult because of this acceleration of the CO₂ concentration.

Figure 7 shows the annual emissions of anthropogenic carbon [11] . The emission increases approximately linearly and shows no acceleration. Additionally, the global CO₂ in Figure 3 has no effect on the CO₂ corresponding to inflection points a, b, and c in Figure 7 . The annual emission of anthropogenic carbon from 1960 and 2020 was 2 - 10 PgC (or GtC). The unit conversion from PgC (or GtC) to ppm can be calculated via Equation (3) [14] :

(x₁ GtC × 3.67/44)/(5135 Eg/28.9) = x₂ ppm (3)

x₁ and x₂: variables, 3.67: a conversion factor from carbon to CO₂, 44: molecular mass of CO₂, 28.9: molecular mass of air, 5135 Eg: air mass on earth.

Therefore, 2 - 10 PgC/year is equivalent to 1 - 5 ppm/year. Additionally, according to the IPCC carbon cycle, CO₂ emissions from fossil fuels are 7.8 GtC (or 3.9 ppm) [12] . These CO₂ amounts are too small compared with the change in atmospheric CO₂ concentrations (330 - 430 ppm) shown in Figure 5 . Since the residence time of atmospheric CO₂ can be estimated to be 3 - 4 years [4] [14] , anthropogenic CO₂ does not accumulate in the atmosphere for a long period of time. For these reasons, anthropogenic CO₂ does not significantly affect global CO₂ concentrations.

Since Rs increases exponentially with temperature [15] , it is assumed that CO₂ thermally induced via Rs follows a first-order reaction rate, as shown in Equation (4).

k = α·exp(−β/T)(4)

k: reaction rate constant, T: temperature, α and β: constants.

Figure 8 shows a hypothetical chart showing the temperature (or time) dependence of atmospheric CO₂. As shown in Figure 5 , the temperature changes linearly with time, so the x-axis can be replaced with time. The hypothetical chart is similar to the change in CO₂ with time in Figure 5 . This assumption may explain the accelerated increase in CO₂ concentrations during the modern warm period.

As shown in Equation (1), there is a relationship between the change rate of the CO₂ concentration (drco₂/dt) and the change in global temperature (ΔT). To examine the response of the change rate of the CO₂ concentration to temperature changes at various latitudes, two variables were compared between 1979 and 2022 in the tropics, at northern latitudes, and at southern latitudes, as reported in a previous paper [3] . Although ΔT in the tropics strongly responds to El Niño, drco₂/dt at northern latitudes responds more strongly to ΔT than does that in the tropics. Therefore, the temperature dependence of the temporal change in atmospheric CO₂ may be greater at northern latitudes than in the tropics. In other words, changes in Rs with temperature at northern latitudes are more important than those in the tropics in controlling atmospheric CO₂.

Anthropogenic CO₂ emissions constitute only ~4% of the global CO₂ cycles, as discussed above. The anthropogenic CO₂ emissions and other CO₂ origins of the global CO₂ cycle budget are shown in Figure 9 . Notably, anthropogenic CO₂ emissions contribute too little to affecting the global CO₂ concentration. Furthermore, no sign of a reduction in atmospheric CO₂ concentrations is observed, regardless of an effort to reduce anthropogenic CO₂ emissions, as shown in Figure 5 . Additionally, this means that an increase in anthropogenic CO₂ since the Industrial Revolution has contributed too little to affect the global CO₂ concentration.

The plants that abound on Earth are produced through photosynthesis. The amount on Earth is enormous, and according to IPCC assessments, Rs is estimated to be involved in approximately half of the carbon cycle on Earth, as shown in Figure 9 . When the global CO₂ or carbon cycle is in balance, the same number of plants synthesized are decomposed through “soil respiration” and returned to CO₂. Therefore, the quantitative role of “soil respiration” in the carbon cycle on Earth far exceeds that of anthropogenic CO₂ emissions [4] .

As described in the introduction, midlatitude forests cover approximately 30% of the Earth’s land area [8] [9] . Because the temperature of midlatitude forests varies seasonally, in contrast to that of tropical rainforests, soil respiration in midlatitude forests varies seasonally in response to changes in annual temperature. Thus, increasing global temperatures affect the extent of soil respiration in midlatitude forests more than they do in tropical rainforests. Therefore, the global Rs control process is highly dependent on temperature, and during the modern warm period, atmospheric CO₂ concentrations continue to increase and are difficult to reduce. Even if anthropogenic CO₂ is changed by human activities, atmospheric CO₂ will not change significantly.

The Little Ice Age lasted until the mid-18th century [1] . The Mendenhall Glacier near Juneau, AK, is an example of a receding glacier after the Little Ice Age. The Glacier started retreating in the mid-1700s before the rapid development of the Industrial Revolution [16] . The amount of thermally induced CO₂ may have started to increase before the Industrial Revolution. It is doubtful that anthropogenic CO₂ increased at the beginning of the Industrial Revolution, after which the global temperature simultaneously increased.

Hermann Harde [17] reported that the increase in CO₂ over recent years can be explained well by a single balance equation, which considers the total atmospheric CO₂ cycle. It comprises temperature-dependent natural emissions and uptake processes and human activities. This uptake is characterized by a single time scale, with a residence time of approximately 3 years. For a conservative assessment, he reported that the anthropogenic contribution to the observed CO₂ increase over the Industrial Era was significantly less than the natural influence. On average, between 2007 and 2016, anthropogenic emissions contributed no more than 4.3% to the total concentration. He noted that not anthropogenic emissions but rather natural processes, particularly temperature, have to be considered the dominant impacts for the observed CO₂ increase over the last 270 years. His analysis correlates well with our results in this paper.

To consider how to control the concentration of CO₂ in the atmosphere, our recent results [3] - [6] and the results from this work are summarized here.

1) Changes in temperature and CO₂ are correlated, but temperature leads to CO₂.

2) Thermally induced CO₂ is overwhelmingly larger than anthropogenic CO₂.

3) Thermally induced CO₂ plays a critical role in the atmospheric CO₂ balance during the modern warm period.

4) The temperature linearly increased, whereas atmospheric CO₂ did not linearly change but accelerated with time.

5) Atmospheric CO₂ significantly changes during El Niño events but does not change much during La Niña events.

6) There is no correlation between the atmospheric CO₂ concentration and anthropogenic CO₂ emissions.

As a result, anthropogenic CO₂ emissions constitute a small part of the overall CO₂ balance, and the total amount of CO₂ is controlled by thermally induced CO₂ and not by anthropogenic CO₂. After the Little Ice Age ended, the modern warm period began in the mid-18th century, and the amount of thermally induced CO₂ increased. Therefore, even if anthropogenic CO₂ emissions are reduced, the total CO₂ concentration will continue to increase during the modern warm period.

Figure 1 summarizes the Rs control process derived from our work on global warming. According to the IPCC, as anthropogenic CO₂ emissions increase, the CO₂ concentration in the atmosphere increases, and the atmospheric temperature increases due to the greenhouse effect of CO₂. In the Rs control process, the increase in temperature due to the modern warm period increases CO₂ emissions due to increased Rs. One of the main factors is that the Rs control process in the midlatitude forest zone changes significantly due to temperature changes. The emitted CO₂ can be considered thermally induced CO₂. As a result, the CO₂ concentration in the atmosphere increases. Therefore, although there is a cross-correlation between temperature and CO₂ concentration, a temperature-leading time lag is observed because it is a process mediated by Rs. Even though anthropogenic CO₂ has decreased, reducing total atmospheric CO₂ concentrations during the modern warm period is difficult. Additionally, this means that an increase in anthropogenic CO₂ since the Industrial Revolution has contributed too little to affect the global CO₂ concentration.