The Contrast That Survives

This is not a claim that the war-peace cycle exists. That claim (H1: a roughly 35-year rhythm in how many wars occur) is tracked separately within the Triptych framework. What follows is a different, narrower claim (H2), and the two must not be read as one finding standing or falling together.

The question

Correlates of War records, at the level of individual participants rather than individual wars, from 1816 to 2007. Fold the series into a recurring window of about 35 years — the same window H1 specifies, used here purely as a fixed partition. H2 doesn't require that window to mark a real periodic clock; it only asks whether the shape of engagement inside repeated copies of the window differs by epoch, which is a question about clustering, not about periodicity, and stands on its own regardless of how H1 is eventually resolved. Split the record at 1914, and ask: is the year-to-year structure within that window different before 1914 than after it? Not whether wars became more frequent — whether their footprint, once one occurred, began clustering into a shape it had not had before.

The contrast was statistically significant in the specified analysis. Before removing any years, the chi-square contrast between epochs (M2, defined as χ² after 1914 minus χ² before it) was 24.37, with p = 0.019. This was calculated on the detrended series; "before removal" does not mean unprocessed raw data.

Wearing it down

The robustness test asked whether the contrast would survive removing the five highest values in the detrended post-1914 series. That removal rule and the decision point at k = 5 were specified before the robustness test was run.

It did survive. After removing five years, the contrast remained statistically significant (M2 = 24.91, p = 0.0085). It also remained significant after removing ten (M2 = 30.23, p = 0.002). These results support a precise conclusion: the contrast survives the specified removal of extreme years.

The rise in the normalized statistic does not mean that the underlying phase structure became stronger. As the technical report explains, the unnormalized phase-bin discrepancy in the post-1914 epoch decreased as the highest values were removed, while the variance used in normalization fell faster. The distinction matters: a larger test statistic is not, by itself, evidence of a larger underlying effect.

The random-removal comparison supplies useful context. Its mean M2 falls from 24.37 at k = 0 to 21.05 at k = 10. However, removing random years does not reproduce the variance reduction caused by removing the highest values, so this band is only a partial comparator. It is not a model of "no structure," and the separation between the curves should not be read as a direct measure of increased structural strength. Winsorization, which caps extreme values instead of deleting years, also leaves a statistically significant contrast at both tested thresholds.

Two further checks clarify the scope. Excluding all twelve world-war years (1914–1918 and 1939–1945) leaves a positive contrast (M2 = 14.65), but it no longer meets the 0.05 significance threshold (p = 0.096). This does not establish that the contrast is absent; it does mean that this variant does not provide statistically significant support at the chosen threshold. The highest value in the participant-engagement series occurs in 1991, the year of the Gulf War coalition. That fact alone cannot settle the contribution of the world-war years.

The war-count series does not show a corresponding statistically significant contrast between epochs in the reported checks. Participant engagement and war counts measure different quantities. The participant-series result therefore cannot be transferred to the frequency of war, and the comparison does not identify a causal mechanism.

What it is, and is not

The supported conclusion is that, using the specified 35.1-year folding window, the participant-engagement series shows a contrast between the pre-1914 and post-1914 epochs that survives the declared five-year removal test and remains significant after ten removals. The window length is an assumption of the analysis, not an estimated cycle length.

This is a result about the distribution of participant engagement, examined under H2. It does not establish a periodic clock, identify the mechanism behind the contrast, or provide a forecast for 2027–2030 or any other future window. The positive result of the primary robustness test remains, alongside the limitation shown by excluding the world-war years. The Test 4 robustness protocol and its decision rule were specified before that test was run.

The two panels below show the phase-bin means of the detrended series for both epochs, followed by the robustness curve and the partial random-removal comparator.

Correction, September 21, 2026: the interpretation has been revised to distinguish survival of the contrast from strengthening of the underlying pattern, explain the limits of the random-removal comparison, and state explicitly that the world-war-exclusion variant does not meet the 0.05 significance threshold. The reported numerical results are unchanged.

Full protocol, code, and the decision log entries behind every number here are in the GitHub repository; questions and objections belong in the discussions.