How the numbers were chosen

Normalization

Every overlaid data series is rescaled to the interval [-1, 1] using its own minimum and maximum over the visible window. The cycle curves are sinusoids of unit amplitude (the amplitude_normalized field on every cycle is 1.0). The vertical axis is therefore dimensionless: visual peak heights do not represent real-world magnitudes, only relative shape over time. That is convenient for eyeballing shape against a normalized sinusoid, and it is lossy: it hides absolute magnitude and makes level differences invisible. Two points stand out:

• A series with one enormous spike (e.g. global conflict deaths in WWII) compresses every other variation toward a thin band. The visible shape near the peaks is real; the visible shape away from them is attenuated.
• Because normalization is per-series, you cannot compare amplitudes across series. Only across time within a single series.
• Because the min/max is taken over the visible window, brushing the time range re-normalizes the series: normalized heights are comparable only within the current window, and a narrow window forces even trivial local variation to span the full −1 to +1 height.

Why Pearson is the wrong tool

The calibration panel reports a Pearson correlation between the data series and the cycle curve. Pearson assumes two things this context violates:

• Phase sensitivity. For two sinusoids of the same period, Pearson r reduces to cos(Δφ), where Δφ is the phase offset between them (exactly so for continuous integration or evenly spaced samples over whole periods; approximately for the finite, partial-period records actually correlated here). A perfect cosine evaluated over one full period has r = 1 with itself, r = 0 with a quarter-period shift, and r = −1 with a half-period shift - even though all three are the same cycle in any structural sense. Pearson therefore measures phase alignment, not cyclic similarity, and the calibration slider primarily moves r by changing Δφ.
• Independence of observations. Time series are autocorrelated, so classical Pearson significance tests are anti-conservative on data like ours: the effective sample size is smaller than the row count, and naive p-values overstate significance. The calibration drawer therefore reports the r value but not a p-value.

The panel exposes Pearson anyway because the single most important question - "how much is the peak-year choice doing?" - is visible just from watching the correlation change as you move the slider. That diagnostic use is valid. Treating the number as a test statistic is not. Three further honesty notes: the readout is always computed over the series' full record, not the brushed window shown in the chart (equal weight per row, so densely sampled modern years dominate an irregular series); it is computed on the transformed values where a transform applies (log1p for Project Mars), so it is the correlation of the logged series; and because the sliders let you tune both phase and period against the same observations, a slider-maximized r is an in-sample search result, not evidence - do not hunt for the peak.

Better tools for cyclic data include cross-correlation at varying lags, the Fourier periodogram, Lomb-Scargle for unevenly sampled records (which one of the present series is: the WID wealth series' pre-1913 points arrive at 30-, 20-, 10-, and 3-year gaps before annual coverage begins), and wavelet decomposition for non-stationary signals. The periodogram leg is now implemented - see the spectral-testing section below. Cross-correlation remains future work, and wavelets are deliberately excluded (pointwise wavelet significance is a known false-positive trap).

Spectral testing

Every verdict, one row per pairing. A pairing is testable only when its record spans at least 3.0 full periods of the theory's stated cycle; 0 of the 9 do.

Record is the span of the series each verdict actually tests — for some pairings a different cut from the one drawn on the chart, named on that cycle's page. Years short is how much longer that record would need to be to reach three periods.

Since August 2026 every cycle–series pairing carries a pre-registered spectral verdict, computed by a committed script (scripts/spectral_verdict.py) from a frozen analysis manifest and published at /data/spectral/verdicts.json with one figure per pairing. The question is narrow: does the paired series contain significant power at the theory's exact stated period, above an autocorrelated (red-noise) null? In plain terms: does the data actually repeat at the rhythm the theory names, more strongly than slow-drifting noise would produce by chance? Frequencies are never fitted or scanned - the test is a harmonic regression at exactly 1/P (cosine + sine + linear trend) compared by likelihood ratio against the same model without the sinusoid, with the p-value calibrated by parametric bootstrap (99,999 draws) from a fitted AR(1) null and re-checked against an AR(2) null. Multiple tests are Holm-corrected within pre-registered families. The multitaper spectrum on each figure (NW = 2, K = 3) is the descriptive picture only; it is never the verdict. Inference always runs on unsmoothed, uninterpolated records: TFP on Fernald's raw annual dtfp_util (never the 5-year-averaged display series) and the wealth series only from its annual 1913+ span.

Before any spectrum, an eligibility gate: a pairing is testable only if its record spans at least 3.0 full target periods - a deliberately conservative site rule, not a theorem (period estimation conventionally wants ~5). Below the gate the verdict is INSUFFICIENT_DATA and no code path emits a p-value; the test suite enforces that, not just convention. There are exactly four verdict states: INSUFFICIENT_DATA, NO_SIGNIFICANT_TARGET_POWER, MODEL_SENSITIVE (the AR(1) and AR(2) nulls disagree at the Holm-adjusted threshold, so no verdict is claimed), and SIGNIFICANT_TARGET_POWER. The 2026 run's headline: 0 of the 9 paired constructions reach the gate - none of these records is long enough to clear it, which is itself the finding. INSUFFICIENT_DATA is an eligibility outcome - the site declines to run its test below three periods - not evidence that the data contain no information about the cycle. A secondary cross-grid panel re-pairs each period with every series long enough to clear the gate (19 cells, labelled as re-pairings, not the site's claims). A 54- and a 55-year period differ by 0.000337 cycles per year - separating them would take a ~3,000-year record under the Rayleigh resolution criterion, a spectral-resolution heuristic rather than a bound on every parametric method - so no verdict text distinguishes Kondratiev from Perez; every result in that band is one ~54–55-year statement.

The failed-detection precedents that shaped this design: Korotayev & Tsirel (2010) report a significant Kondratiev wave only after replacing the World War years with geometric means, on a record of at most three cycles - contested, never retracted. Kuznets's ~20-year swings were killed as a moving-average artifact (Adelman 1965; Howrey 1968), the direct ancestor of this site's unsmoothed-TFP rule. Turchin himself reports the ~150-year cycle as one realized oscillation and offers no formal spectral test - candor this page treats as the standard. Method references: Mann & Lees (1996); Torrence & Compo (1998); Hamilton (2018); Meyers (2012).

Missing and sparse data

The curves cover 1600–2050. Every data series has shorter coverage. DW-NOMINATE: 46th–118th Congress (1879–2023); Fernald TFP annual series 1948–present, displayed 1948–present (the 1948 and 1949 values use a clipped, asymmetric window because a true 5-year centered window only becomes available at 1950 - treat the first two displayed points as edge artifacts); Project Mars conflict data 1800–2011; WID top-1% wealth modern coverage 1913–most-recent (with five earlier decadal points spliced from secondary sources, see below); Maddison US/world GDP share trimmed to 1870+ (earlier years have unstable country coverage); V-Dem 1789–present; Stimson Policy Mood 1952–2024. Gaps are simply absent from the chart - nothing is interpolated. If a series fails to load, its legend entry shows "data unavailable" and the rest of the viz keeps working.

Two finer caveats. The modern WID/Saez–Zucman US top-1% wealth series begins in 1913; the five pre-1913 points (1820, 1850, 1880, 1900, 1910) come from earlier historical sources spliced via OWID/ WID and have wider standard errors. The TFP 5-year centered rolling average is this project's derivation from Fernald's annual dtfp_util column (utilization-adjusted TFP growth), not Fernald's own published series; the build script keeps clipped (asymmetric) windows at the boundaries rather than dropping rows, so 1948 = mean of {1948, 1949, 1950} and 1949 = mean of {1948, 1949, 1950, 1951} - read those endpoints with appropriate skepticism.

A third caveat. The Maddison rebuild forward-fills each country's GDP between sparse benchmark observations but does not back-fill before each country's first observation. Many non-Western countries enter Maddison only at 1950, so the world denominator is systematically smaller pre-1950 than post-1950 - biasing US share of world GDP upward for early years. The 1870 value (~10.6%) and the magnitude of the 1870→1945 climb should both be read as "US share of countries Maddison covers in that year," not "US share of world GDP" literally.

Notes on individual pairings

V-Dem with Strauss-Howe, not Huntington. An earlier draft of this project paired V-Dem with Huntington as a secondary signal alongside DW-NOMINATE. We moved it to Strauss-Howe so no cycle would carry two series - nine of the ten cycles have exactly one paired series; Turchin's fathers-and-sons cycle has none. Both are arguments. The Strauss-Howe pairing reads V-Dem's recent decline as a Fourth-Turning institutional-stress signal; the Huntington pairing would have read it as the trough side of a creedal-passion cycle. The data is the same; the framing differs.

Project Mars covers conventional wars only. The conflict-deaths series registers years like 2010 as zero because no qualifying conventional war (interstate or civil war between states with differentiated militaries causing ≥500 deaths) was active that year - even though other conflict datasets (UCDP, COW, PRIO) record substantial casualties in 2010 (Afghanistan, Iraq, Mexican drug war). The series therefore measures conventional-war intensity, not all conflict deaths; read drops to zero accordingly.

Perez pairs with technology diffusion, not asset prices. Through Phase 13 Perez had no paired series; the HATCH diffusion-intensity composite closed that gap in Phase 14. Shiller's CAPE was the runner-up candidate - Perez's frenzy/turning-point mechanism is financial, and the CAPE valuation extreme came at the turn of 1999–2000, right at the anchor - but CAPE carries no explicit reuse license and measures paper values, not the economy-wide diffusion that is Perez's actual object. We cite CAPE here as anchor validation without redistributing it.

No paired series for Turchin's fathers-and-sons cycle. The natural series - Turchin's US Political Violence Database - is posted on his site (USPVD2010.xlsx) but carries no explicit reuse license, so we do not redistribute it. The cycle ships unpaired rather than paired to a construct-mismatched proxy.

Stimson Policy Mood with Schlesinger Jr. Of the ten cycles, Schlesinger's pairing is the closest the site gets to a direct measurement: Stimson's index is, by construction, an estimate of US public preference for liberal vs. conservative domestic policy - one component of what Schlesinger's cycle claims to track (mass preferences over the scope of government, not the whole political-historical rhythm), and Stimson's own reading stresses shorter, thermostatic swings rather than a fixed 30-year cycle. The catch is coverage: the series only starts in 1952. Inside the empirical window this construction (period 30, peak 1970) plots troughs at 1955, 1985, and 2015 and peaks at 1970 and 2000 - two complete trough-to-trough swings inside a record spanning 2.4 periods. The pre-1952 shape of the Schlesinger curve cannot be stress-tested against the paired data; treat the calibration drawer's Pearson r accordingly.

See each series' per-source provenance file for full retrieval and processing notes.