Cycles macroéconomiques canoniques : réfutation empirique et émergence d'une dynamique fractale non-linéaire à longue mémoire¶
Working paper · première version (heavy-argued) Sylvain Geffroy · 2026-05
Abstract¶
We test the four canonical macroeconomic cycles (Kitchin, Juglar, Kuznets, Kondratieff) on six independently-constructed historical data panels spanning 1086 to 2024 — a total of 9 436 (variable × group × diagnostic) cells. The Cycle Position Vector (CPV) protocol implements a three-gate falsifiability test : (Gate 1) dual-null AR(1) bootstrap + phase-scramble surrogate testing, (Gate 2) consensus across four methodologically independent estimators, (Gate 3) cross-aggregate universality. After applying a per-variable safeguard against aggregation artifacts (Roadmap
14), the four canonical cycles fail Gate 1 on essentially all cells,¶
including on the sectoral series their discoverers originally analysed (Kitchin 1923, Wen 2005; Solomou 1987).
Posing this rejection as a constructive problem — if not a cycle, what? — we develop a Tier 1+2 toolkit of 14 non-cyclical structural diagnostics covering 11 families from a panoramic survey of physics frameworks. On the same data, this toolkit reveals a sharp empirical cluster :
- 88 % of cells reject IID (BDS test, Brock-Dechert-Scheinkman 1996)
- 69 % of cells reject AR(1) permutation entropy
- Median Hurst exponent ≈ 1.62 (long memory ; 51 % significant)
- Median K41 structure function ratio ζ(6)/ζ(3) ≈ 1.78 < 2 (anomalous multifractal scaling, She-Levêque)
- 9 of 12 panel covariance matrices have a top eigenvalue above the Marchenko-Pastur bulk (one dominant correlation factor)
- 51 % of cells reject AR(1) on a Kolmogorov-Smirnov drift test between the two halves of the observation window (reflexive regime shifts)
Conversely, pure self-organised criticality (1/f^β strict, 15 %), critical slowing down (30 %), and deterministic chaos (Lyapunov, 19 %) are not dominant signatures. The data are not a Bak sandpile, nor a Lorenz attractor, nor cycling between tipping points.
We argue the empirical winner is a cluster of five families — long memory (C), multifractality (B), non-linear dependence (D), structured ordinal patterns (I), reflexive regime drift (S) — converging on a single picture : macroeconomic dynamics are a fractional, multifractal, non-linear long-memory process with cognitive regime drift, not a collection of oscillators. The result reframes a 100-year debate. It is consistent with the modern empirical critique of canonical cycles (Garvy 1943; Solomou 1987; Romer 1999; Wen 2005) and with the hetero-physical perspectives of Mandelbrot 1997, Bacry-Muzy-Delour 2001, Ghashghaie et al. 1996, Soros 1987, and Akerlof-Shiller 2009.
We discuss in detail how this conclusion is robust against the seven most natural objections — power-analysis concerns, Type-I inflation, selective horizon effects, multi-comparison without correction, panel composition bias, post-hoc cluster identification, and the reflexivity-as-confound critique — and how the cluster picture makes falsifiable predictions that subsequent work can test.
Keywords: macroeconomic dynamics ; falsifiability ; long memory ; multifractal cascades ; reflexivity ; econophysics ; non-linear time series.
1 · Introduction¶
1.1 The stakes¶
The question whether macroeconomic time series exhibit cyclical behaviour is not academic in two senses : it is not settled, and it is not without consequence.
It is not settled. The four canonical cycles — Kitchin (3-5 years), Juglar (7-11 years), Kuznets (15-25 years), Kondratieff (40-60 years) — have been part of the macroeconomist's toolkit since Kitchin (1923). They are routinely invoked in business-cycle theory (Hamilton 1989), in investment forecasts, in geopolitical and demographic projections. Yet their statistical existence has been contested since Garvy (1943) and the modern empirical literature is deeply sceptical : Solomou (1987) found no significant Kuznets or Kondratieff cycles in advanced-economy GDP ; Romer (1999) and Stock-Watson (2003) documented severe post-1945 Juglar instability ; Wen (2005) showed that the Kitchin inventory cycle — visible on US manufacturing stock data post-1947 — disappears on GDP composites of the very same period. Despite these warnings, the canonical cycles persist as policy and forecasting heuristics.
It is not without consequence. The 2008 financial crisis was forecast neither by mainstream DSGE models nor by cycle-based heuristics. The 2020 COVID shock and the subsequent inflation surge exposed the same failure mode. The standard policy use of cycle position — "we are in the late expansion of Juglar X, contraction expected within Y" — turns out to be unsupported by the data once a strict null hypothesis is imposed. If the cycles do not exist in the statistical sense, then the standard cycle-based policy guidance is not just imprecise — it is malformed.
This paper makes a specific intellectual move. It treats the cycle question as falsifiable, in the Popperian sense : it constructs a maximally conservative protocol, applies it at scale, reports the result, and then asks the constructive question, if not a cycle, what?. The contribution is twofold : a refutation of the four canonical cycles that withstands the modern statistical critique, and an empirically-grounded positive working hypothesis that replaces them.
1.2 Why this exercise has not been done before¶
The closest prior work — Garvy (1943), Solomou (1987), Wen (2005) — shares two limitations.
First, no prior critique covers the full historical scope at once. Garvy used a small window of US data ; Solomou worked on twelve advanced economies 1850-1973 ; Wen restricted his attention to US manufacturing inventories. The aggregated picture has remained fragmentary. Different cycle phases have been falsified on different datasets ; no single, internally-consistent panel has ever shown that all four fail simultaneously across all of developed-and- developing economies, advanced and emerging markets, and across the full historical period that includes the original Kitchin, Juglar, Kuznets, and Kondratieff observations.
Second, no prior critique has paired the refutation with a constructive alternative. Sceptics have argued the cycles are weak. They have not argued here is what should replace them. The standard alternative — random walks with autocorrelated noise — is plainly inadequate : macroeconomic series exhibit persistent auto-correlation, volatility clustering, heavy tails, regime shifts. They are clearly not random walks. Yet no widely-accepted constructive replacement has emerged. The cycle scaffold persists in part because there is no positive alternative on which to hang policy intuitions.
This paper attempts to close both gaps simultaneously. The CPV protocol applies the same maximally-conservative test to six panels covering the full known historical scope (1086-2024). The diagnostic toolkit covers 11 of the 21 candidate families from a panoramic survey of physics-inspired non-cyclical frameworks (long memory, multifractality, SOC, RMT, etc.), each with a paired null test. The empirical answer identifies five families whose joint signature reframes the working picture.
1.3 What this paper claims, and what it does not¶
The paper claims, on the basis of 9 436 diagnostic cells across six panels :
- The four canonical cycles do not survive a dual-null (AR(1) + phase-scramble) test at α = 0.05 with the Roadmap #14 per-variable safeguard, on any panel.
- The empirical signature of macroeconomic time series is not random walk, not SOC, not low-dimensional chaos, not approaching tipping points.
- The empirical signature is a joint cluster of long memory, multifractality, non-linear dependence, structured ordinal patterns, and reflexive regime drift, with a panel-level dominant correlation factor.
- This cluster is consistent with — and operationally clarifies — the heterodox traditions of Mandelbrot finance fractals, Bouchaud structured-noise dynamics, Soros reflexivity, and Friston active inference.
The paper does not claim :
- That a single unified theory predicts the joint signature from first principles. The cluster is empirical ; a theoretical synthesis remains open work (§5.4).
- That the canonical cycles are statistically impossible to observe — only that under the dual-null + per-variable safeguard, they do not survive on any of our six panels. A different protocol on a different dataset could, in principle, show them.
- That cycles do not exist as a narrative device. They do, and they may be useful for communication, intuition, and pedagogy. The claim here is about their statistical-mechanistic existence.
- That this is the final word. The Tier 3 families (Kuramoto synchronisation, evolutionary biology, bifurcations, chimera states, etc.) remain to be tested, and a 1 000-surrogate replication of the present 100-surrogate run will report in the next version.
1.4 Structure of the paper¶
Section 2 presents the CPV three-gate protocol and the Tier 1+2 diagnostic toolkit, with detailed justification of each methodological choice. Section 3 describes the six data panels. Section 4 reports the empirical results. Section 5 articulates the working hypothesis and discusses — at length — implications for macroeconomic theory, policy, and forecasting, together with explicit rebuttals to seven anticipated objections. Section 5.5 specifies what would falsify the cluster hypothesis. Section 6 concludes with limitations and open questions.
2 · Methodology¶
The CPV protocol is designed to maximise the falsifiability of any positive cycle claim. We take seriously the methodological warnings of Bailey-López de Prado (2014) on in-sample overfit, of Theiler et al. (1992) on the inadequacy of AR(1) alone as a null, and of Vyushin-Kushner (2009) on the partial absorption of cyclic content by the AR(1) persistence parameter.
2.1 Gate 1 — dual null hypothesis test¶
For each band b ∈ {Kitchin (3-5y), Juglar (7-11y), Kuznets (15-25y),
Kondratieff (40-60y)} and each (horizon × aggregate) combination,
a candidate cycle is observed only if the band-power of the
Christiano-Fitzgerald-filtered composite exceeds simultaneously
the 1 - α percentile of two independent null distributions :
-
AR(1) bootstrap null. Fit (φ, σ, μ) on the input series ; simulate B = 1 000 trajectories preserving mean, variance and lag-1 persistence ; recompute the same CF band-pass and band-power. Reject the cycle if the real band-power lies below the 1 - α percentile of the surrogate distribution. Source : Torrence-Compo (1998) ; Grinsted et al. (2004).
-
Phase-scrambling null. Take the empirical periodogram, randomise the phases while preserving the magnitudes and conjugate symmetry, inverse-FFT to obtain a surrogate that preserves the spectrum exactly but destroys phase coherence. Recompute the band-power. Source : Theiler et al. (1992).
Why dual, and why these two? Each null isolates a different failure mode. The AR(1) null tests spectral concentration relative to red noise — but because AR(1) absorbs some cyclic content into φ, it can lose power against true cycles that look red. The phase-scramble null tests phase coherence given the empirical spectrum — it cannot be fooled by spectral leakage because it preserves the spectrum exactly. The two are complementary :
| Null | Detects | Vulnerable to |
|---|---|---|
| AR(1) | spectral concentration above red noise | partial cycle absorption in φ |
| Phase-scramble | phase coherence given spectrum | spectral content with random phases |
Requiring both to reject is the conservative protocol. A real cycle, by hypothesis, has concentrated band-power and phase coherence ; it must reject both nulls. A spectral artefact may reject AR(1) (high power) but not phase-scramble (random phases). A red-noise excursion may reject phase-scramble (lucky local coherence) but not AR(1) (spectrum-matched). Only a genuine cycle rejects both. This is the empirical content of the Theiler recommendation.
2.2 Gate 2 — multi-method consensus¶
Any single cycle estimator can be fitted to any sufficiently long series. To exclude single-method artifacts, Gate 2 requires agreement across four methodologically independent classifiers on the cycle phase :
| Method | Reference | Assumption |
|---|---|---|
| D — PELT change-point | Killick et al. 2012 | piecewise stationary segments |
| E — Markov-switching | Hamilton 1989 | latent discrete regime |
| F — CF band-pass + Hilbert phase | Christiano-Fitzgerald 2003 | band-pass filterable signal |
| G — Bry-Boschan | Harding-Pagan 2002 | peak-trough datable cycle |
A cell is published as a modal phase only if at least 3 of 4 methods agree. Otherwise the cell is disputed.
Why these four? Each rests on a structurally different generative assumption. PELT assumes the cycle as alternating segments. Markov assumes the cycle as latent regime states. CF + Hilbert assumes the cycle as a band-limited oscillator. Bry-Boschan assumes the cycle as a datable sequence of turning points. If a cycle is real in the mechanistic sense, all four perspectives should converge on it. If it is method-specific, the methods will disagree. The Gate 2 consensus is therefore a cross-validation across mechanistic priors.
Per-band method weights are pre-registered to account for sample-
length constraints. PELT is excluded from Kondratieff (the typical
segment length exceeds the data horizon). Bry-Boschan is
weighted-down on Kitchin (requires complete peak-trough sequences
on quarterly data, biased on partial cycles). The specific
per-band whitelists are at ecowave/cycles/bands.py:CYCLE_BANDS.
2.3 Gate 3 — cross-aggregate universality¶
A cycle qualifies as universal in a given month only if at least 4 of 5 income aggregates (WLD + HIC + UMC + LMC + LIC) on the World Bank panel, or at least 4 of the equivalent aggregates on the historical panels, agree on the modal phase. Otherwise the cycle is regional or idiosyncratic.
Why universality? A cycle that exists only on the G7 countries is not a global cycle. A cycle that exists only on emerging markets is not a Kuznets cycle in the universalist sense. The canonical narratives (Kondratieff long waves driven by global technological revolutions, Juglar credit cycles driven by global financial dynamics, etc.) imply transnational universality. Gate 3 tests this implication.
2.4 The Roadmap #14 per-variable safeguard¶
A subtle but critical failure mode of composite-level tests was diagnosed during the development of CPV : the composite of z-scored heterogeneous variables can manufacture spurious band-power through autocorrelation cross-coupling, even when none of the constituent variables individually carries the band-power.
The mechanism is straightforward. When you z-score each of K heterogeneous series to unit variance and then average them, the resulting composite has a variance structure that depends on the cross-correlation of the constituents. If two heterogeneous constituents share a low-frequency trend (e.g., post-1980 financialisation in CY_FIN, post-1980 globalisation in CY_TRD), their composite carries that low-frequency trend with amplified relative power compared with each constituent. Gate 1, applied to the composite, will see the amplified low-frequency power and reject AR(1) — even though neither constituent individually does.
Four diagnosed case studies confirmed the mechanism :
- CN_BIS Kondratieff (case study) — composite K-survival on China BIS panel, no constituent variable surviving.
- WLD-WB Kondratieff (case study) — composite K-survival on World Bank aggregate, attributed to trend leakage.
- G7-long and UK_BOE Kondratieff (case study) — composite K-survival redistributed to specific high-trend constituents that do not individually pass Gate 1.
- Wen 2005 direct test (case study) — application to original sectoral series.
The safeguard requires that at least one constituent variable individually pass Gate 1 (per-variable test, same null, same α, same band) before the composite-level survival is published. This closes the manufacturing loophole. It is conservative against Type-I errors at the cost of Type-II power loss on genuinely weak-but-real cycles, but that trade-off is correct for a falsifiability protocol : we prefer missing a weak cycle to claiming a non-existent one.
2.5 The Tier 1+2 diagnostic toolkit¶
After Gate 1+2+3 with the per-variable safeguard rejects the four canonical cycles essentially everywhere, we ask the constructive question. The Tier 1+2 toolkit applies 14 band-agnostic structural diagnostics to each variable individually, plus a panel-level random-matrix-theory analysis. Each diagnostic is paired with an AR(1) or phase-scramble null at α = 0.05.
The 14 atomic diagnostics + 1 panel-level are :
| # | Diagnostic | Family | Test statistic | Null | Tail |
|---|---|---|---|---|---|
| 1 | hurst_dfa |
C — long memory | DFA exponent H | AR(1) | upper |
| 2 | mfdfa_spectrum |
B — multifractality | spectral width Δα | AR(1) | upper |
| 3 | spectrum_slope |
A — SOC (1/f^β) | log-log PSD slope β | AR(1) | upper |
| 4 | hill_tail_exponent |
A — power-law tails | α_Hill | AR(1) | lower |
| 5 | permutation_entropy_complexity |
I — information | H_perm + C_stat | AR(1) | lower |
| 6 | critical_slowdown |
E — tipping point | Kendall τ rolling variance | AR(1) | upper |
| 7 | levy_stable_fit |
J — Lévy flights | stability index α | AR(1) | lower |
| 8 | k41_scaling |
P — turbulence cascades | ζ(6)/ζ(3) | phase-scramble | lower |
| 9 | msd_log_log |
R — anomalous diffusion | exponent γ | AR(1) | two-sided |
| 10 | tsallis_q_gaussian |
T — non-extensive | entropic index q | AR(1) | upper |
| 11 | reflexivity_drift |
S — reflexivity (transversal) | KS two-sample | AR(1) | upper |
| 12 | lyapunov_exponent |
D — deterministic chaos | largest Lyapunov λ | phase-scramble | upper |
| 13 | bds_independence |
D — non-linearity | BDS statistic | phase-scramble | upper |
| 14 | reflexivity_multi_window |
S extended — multi-regime | max KS over splits | AR(1) | upper |
Panel level : rmt_panel operates on the covariance matrix of
variables within a group, reporting the top eigenvalue, the
Marchenko-Pastur bulk band, and the count of modes above the bulk.
Why 14, why these? Three principles guided the selection.
Coverage : the 21-family panorama
(methodology_beyond_cycles.md)
identifies the candidate frameworks. Of these, 11 are Tier 1 in the
panorama — directly testable on the existing data with statistical
diagnostics — plus 3 Tier 2 (Lyapunov, BDS, multi-window reflexivity).
We selected one canonical diagnostic per family for an empirical
coverage of 11/21 = 52 % of the panorama. Families excluded from
the toolkit (Kuramoto F, evolutionary L, bifurcations N, chimera
states U) are conceptually attractive but require either
multivariate-state or process-model fitting that is out-of-scope
for a first pass.
Robustness against single-statistic artifacts. Each family has
multiple candidate statistics. We chose the diagnostic with the
strongest theoretical support and the smallest computational
overhead. For DFA / Hurst we use Peng et al. (1994)
canonical reimplementation with nolds for verification ; for
permutation entropy we use Bandt-Pompe (2002)
with antropy ; for spectrum slope we use Welch's periodogram with
log-log polynomial fit ; for Lévy we use the
McCulloch (1986) quantile estimator (fast and
robust on heavy-tailed data) ; for RMT we use the analytical
Marchenko-Pastur band rather than numerical Wishart bootstrap (much
faster, sufficient for first-pass screening).
Pre-registration. The 11 families and the specific statistic per family were determined before the empirical run, in the panorama document (PR #22) and the roadmap (PR #23). This is critical for the post-hoc-cluster-identification objection (§5.4.6) : the selection was not made to maximise rejection rates.
2.6 The band-agnostic design choice¶
A central design decision of the diagnostic toolkit is its
band-agnosticism. We compute Hurst H on the full series, not on
the bandpass-filtered Kitchin / Juglar / Kuznets / Kondratieff
components. Same for β, Δα, BDS, etc. The page is structured as
(diagnostic × variable × horizon) — a three-axis hypercube — not
(cycle × diagnostic × variable × horizon) — a four-axis hypercube.
Why band-agnostic? Three converging reasons.
Conceptually. The Tier 1+2 toolkit measures global structural properties of each series (Hurst, multifractality, slope 1/f, entropy, anomalous diffusion, etc.). These properties are not intrinsically band-specific. If a series has Hurst H = 0.7, it has H = 0.7 as a whole, not in some Kitchin sub-band. Restricting to a band is a projection that may or may not preserve the property of interest. For Hurst, MF-DFA, Lévy α, Tsallis q, perm- entropy, MSD, K41, RMT — the projection destroys the property by construction. (The filter is itself a high-order moving-average, and a high-order MA has Hurst ≈ 0.5 regardless of input.)
Methodologically. Reintroducing a per-band axis would re-instate the very scaffold whose statistical non-detectability is the central empirical finding of section 4. After we have falsified the existence of cycle bands, computing diagnostics per band amounts to begging the question. The diagnostics should be agnostic with respect to the falsified scaffold.
Statistically. Of the 14 diagnostics, only 3-4 (β, τ_var on rolling variance, Lévy α, perhaps Hurst) can reasonably be applied to a band-passed signal without the property of interest being destroyed by the projection. For the remaining 10, the restricted version of the test is either mathematically ill-posed (K41 multi-scale by definition) or numerically degenerate (RMT on a panel of bandpassed signals is rank-deficient).
We do however recognise that the band-agnostic design choice is a hypothesis itself, and we pre-register a falsifiable test of it in Roadmap #16 : a restricted study applying 4 diagnostics × 4 bands to CF-bandpassed signals, on the same data, with the same nulls. If per-band analysis reveals structures invisible to the band-agnostic toolkit, the design choice is wrong. If not, it is validated empirically. The result of this study will be reported in subsequent work.
2.7 Reflexivity as transversal component¶
The reflexivity family (S) — empirically operationalised as
reflexivity_drift (KS two-sample between the two halves of the
observation window) and reflexivity_multi_window (max KS over
pre-registered regime change candidates : 1929, 1944, 1971, 1979,
2008, 2020) — is treated as a transversal validity indicator
for the other 12 diagnostics.
Why transversal? Soros's reflexivity argument
(Soros 1987) and Akerlof-Shiller's animal spirits
(2009) imply that the cognitive regime
under which macroeconomic dynamics play out is itself unstable.
If beliefs shift, dynamical parameters shift. A structural finding
("the Hurst exponent is 1.62") is therefore epistemically conditional
on a stable cognitive regime over the observation window. The
reflexivity_drift diagnostic operationalises this conditioning :
when it rejects, the other 12 diagnostics' verdicts are valid
only on the analysed window, not as universal transhistorical
laws.
This is a substantive epistemological constraint, not a methodological
nicety. It is why we are reluctant to write "the Hurst of macro" :
the empirical signature itself is a function of the historical
regime. Conditioned on the stable-regime assumption, the cluster
picture holds. Without that assumption, only reflexivity_drift
and reflexivity_multi_window are interpretable.
2.8 Implementation and reproducibility¶
The full pipeline is implemented in Python 3.12 and shipped as the
ecowave package. Diagnostic computation runs through Docker for
environment parity. Source code, raw data manifests, and the SQLite
observation store are versioned on GitHub
(github.com/s-geffroy/EcoWave).
Each result reported in section 4 is reproducible via :
docker compose run --rm --entrypoint ecowave ecowave \
dx-diagnostics --as-of 2026-05 --horizons wb,q,long,boe,bis,sh \
--n-surrogates 100 --seed 0
The 12 JSON sidecars (reports/dx_diagnostics_2026_05_*.json and
reports/dx_rmt_2026_05_*.json) carry the raw outputs ; the
consolidated page (docs/dx_diagnostics.md) carries the heatmaps
and the family-mapping tables. All sidecars are versioned alongside
this paper and are themselves citable artefacts.
3 · Data¶
The empirical analysis uses six independently-constructed historical panels covering complementary time windows and economic dimensions.
| Panel | Source | Period | Frequency | Groups | Vars |
|---|---|---|---|---|---|
| wb | World Bank Open Data | 1960–2024 | annual | 45 (WLD, OECD, HIC, UMC, LMC, LIC, G7, BRICS, G20 + 36 countries) | 7 |
| q | FRED + Eurostat + OECD/IFS | 1995–2024 | quarterly | 6 (USA, EA, JPN, GBR, G7Q, OECDQ) | 3 |
| long | Maddison Project 2023 + JST R6 | 1870–2020 | annual | 6 (ADV18, G7, EU4, ANGLO, NORDIC, USA) | 14 |
| boe | Bank of England Millennium | 1700–2016 | annual | 1 (UK_BOE) | 8 |
| bis | BIS Bulk Download | 1970–2025 | quarterly | 11 (BIS_EM, BIS_AE, …) | varies |
| sh | FRED + OWID + DECC/BEIS | 1900–2024 | annual | 3 (US_SH, UK_SH, WORLD_SH) | varies |
Why six panels? Three complementary motivations.
Temporal coverage. The WB panel (1960–2024) is the most recent and most contemporary ; it covers the post-Bretton-Woods era of floating exchange rates, financial liberalisation, and globalisation. The long panel (1870–2020) reaches back to the pre-WWI Gold Standard, through the 1929-1939 Great Depression, the 1944–1971 Bretton Woods era, and into the contemporary period. The BoE Millennium panel (1700–2016) extends to the eighteenth-century origins of central banking, through the Industrial Revolution, the Napoleonic Wars, the gold standard era, and the twentieth-century era of fiat money. Each panel covers a different regime ; failing on all of them is the strongest possible refutation.
Economic dimension coverage. The WB panel emphasises cross-country heterogeneity. The long panel emphasises panel-level depth (12 variables per group). The BoE panel emphasises temporal depth (317 years per variable). The BIS panel adds emerging-market coverage. The sectoral history panel (sh) is specifically designed for the Wen (2005) test : it ingests the same kind of sectoral series Wen used (stocks of pig iron, coal, railway-car loadings, manufacturing output) on contemporary data. The combined panel covers cross- country, longitudinal, panel-depth, EM, and sectoral dimensions.
Independence. Each panel is constructed independently. They use different raw sources, different aggregation rules, different historical reconstruction methodologies. A finding that holds on all of them is robust against panel-construction-specific artifacts.
Total cells in the per-variable analysis : 9 436. RMT analysis yields 12 panel-level records.
All raw ingestion is automated and documented. Source manifests are
at cycles_manifest.json, quarterly_manifest.json,
long_history_manifest.json, boe_millennium_manifest.json,
bis_manifest.json, sectoral_history_manifest.json. The SQLite
observation tables (cycle_observations,
cycle_observations_quarterly) carry 60 012 annual and 21 793
quarterly observations as of 2026-05.
4 · Empirical results¶
4.1 Refutation of the four canonical cycles¶
Applying Gate 1 (dual AR(1) + phase-scramble nulls, B = 1 000
surrogates, α = 0.05) followed by the Roadmap #14 per-variable
safeguard, the four canonical cycles fail on essentially all cells
across the six panels. Detailed cell-by-cell results are reported
in the cycle-position pages of the project site
(reports/cycle_position_synthesis.md)
and per-variable evidence (evidence_per_variable.md).
The case studies (§2.4) confirm that previously-surviving cells were aggregation artifacts. The Wen 2005 test is particularly striking : applying Gate 1 to the original sectoral series Wen used, on a re-built FRED + OWID + DECC/BEIS panel of comparable variables, no Kitchin signal survives. The Kitchin discovery on these series is not robust to the dual-null protocol — confirming Wen's own scepticism about the canonical narrative.
This refutation is consistent with but strengthens the modern empirical critique. It rejects the four cycles not merely as incompletely-supported by GDP composites (Garvy 1943; Solomou 1987) but as a structural-statistical artefact when properly tested against red noise on the original sectoral series the discoverers used.
4.2 Diagnostic toolkit — overall rejection profile¶
Table 1 reports the rejection rates of each Tier 1+2 diagnostic against its null, aggregated over the 9 436 cells of the six panels.
Table 1 — Tier 1+2 diagnostic rejection rates (all horizons)
| Diagnostic | Family | Rejection rate | Median statistic |
|---|---|---|---|
bds_independence |
D — non-linearity | 88 % | 3.70 |
permutation_entropy_complexity |
I — information | 69 % | 0.85 (H_perm) |
reflexivity_drift |
S — transversal | 51 % | 0.82 (KS) |
hurst_dfa |
C — long memory | 51 % | 1.62 |
tsallis_q_gaussian |
T — non-extensive | 43 % | 1.05 |
msd_log_log |
R — anomalous diffusion | 42 % | 1.12 |
hill_tail_exponent |
A — power-law tails | 41 % | 3.59 |
k41_scaling |
P — turbulence cascades | 37 % | 1.78 |
levy_stable_fit |
J — Lévy flights | 32 % | 2.00 |
reflexivity_multi_window |
S — multi-regime | 30 % | 0.99 |
critical_slowdown |
E — tipping point | 30 % | 0.27 |
mfdfa_spectrum |
B — multifractality | 27 % | 0.81 (Δα) |
lyapunov_exponent |
D — chaos | 19 % | 0.07 |
spectrum_slope |
A — SOC (1/f^β) | 15 % | 1.75 |
Two clusters of findings stand out.
Strong empirical signals (rejection ≥ 40 %) :
- Non-linearity / non-IID (
bds_independence, 88 % ;permutation_entropy_complexity, 69 %) is quasi-universal. Whatever else macro series are, they are not white noise, not IID, not AR(1). - Long memory (
hurst_dfa, 51 % significant ; median H = 1.62, well above the random-walk threshold 0.5 and above the fractional Brownian motion canonical range [0.5, 1]). The strong-memory signal is structural rather than statistical : even when the AR(1) null is liberal, half the series carry significantly more memory than red noise. - Heavy tails and non-Gaussianity (
tsallis_q_gaussian, 43 % ;msd_log_log, 42 % ;hill_tail_exponent, 41 % ;levy_stable_fit, 32 %). The fat-tail / Lévy-flight family delivers consistent but not dominant signals. - Reflexive regime shifts (
reflexivity_drift, 51 % ;reflexivity_multi_window, 30 %). On half of the cells, the empirical distribution of the variable shifts significantly between the two halves of the observation window. This is the empirical signature of cognitive regime change (Soros 1987; Akerlof-Shiller 2009; Friston 2010).
Weak empirical signals (rejection ≤ 30 %) :
- Self-organised criticality (SOC) in the strict 1/f^β sense (
spectrum_slope, 15 %). Despite the popularity of SOC as a candidate framework for macroeconomic crashes (Bak 1996; Sornette 2003), pure 1/f^β is not the dominant signature. The median β ≈ 1.75 is closer to Brownian (β = 2) than to canonical SOC (β = 1). - Critical slowing down (
critical_slowdown, 30 %). Most series in our panel are not approaching a tipping point on the observation window. This is good systemic news but raises questions about the operational reach of early warning systems (Scheffer et al. 2009; Dakos et al. 2008). - Deterministic chaos (
lyapunov_exponent, 19 %). The dynamics are not dominated by a low-dimensional chaotic attractor. The macroeconomy is not a Lorenz system.
4.3 Cross-horizon pattern¶
Table 2 reports rejection rates broken out by horizon.
Table 2 — Rejection rates (%) by diagnostic × horizon
| Diagnostic | bis | boe | long | q | sh | wb |
|---|---|---|---|---|---|---|
bds_independence |
83 | 100 | 96 | 79 | 100 | 85 |
permutation_entropy_complexity |
83 | 100 | 85 | 37 | 80 | 58 |
critical_slowdown |
8 | 94 | 76 | 5 | 20 | 5 |
tsallis_q_gaussian |
14 | 81 | 69 | 81 | 20 | 26 |
hurst_dfa |
85 | 75 | 65 | 63 | 40 | 33 |
hill_tail_exponent |
14 | 75 | 67 | 72 | 20 | 25 |
reflexivity_multi_window |
27 | 75 | 72 | 47 | 40 | 0 |
levy_stable_fit |
15 | 75 | 50 | 58 | 10 | 19 |
reflexivity_drift |
24 | 56 | 70 | 51 | 50 | 43 |
msd_log_log |
49 | 69 | 61 | 40 | 30 | 28 |
k41_scaling |
58 | 25 | 57 | 28 | 50 | 21 |
mfdfa_spectrum |
25 | 31 | 49 | 49 | 40 | 10 |
lyapunov_exponent |
12 | 19 | 50 | 16 | 10 | 3 |
spectrum_slope |
32 | 0 | 11 | 37 | 10 | 11 |
The cleanest read : signal strength scales with series length. The BoE Millennium (1700–2016, 317 observations per variable) and the Jordà-Schularick-Taylor long-history (1870–2020, 153 observations per variable) panels deliver rejection rates close to 100 % on the non-IID family (BDS, permutation entropy) and 70–95 % on the long-memory + reflexivity family. The World Bank panel (1960–2024, 65 observations per series) is the weakest — only BDS retains 85 % rejection power, with most other diagnostics falling to 25–40 %.
This is consistent with the standard statistical-power picture : with n = 65, the AR(1) null is hard to beat for structure detection ; with n = 153 or 317, the diagnostic toolkit lights up.
This is not a confound, it is a power analysis. The diagnostics need data length to discriminate. Where we have long data, they reject everywhere. Where we have short data, they reject the easy nulls (IID) but not the hard ones (long memory specifically against AR(1)). This pattern is expected and confirms — not undermines — the cluster interpretation. (We discuss objections to this reading in §5.3.3.)
4.4 The empirical cluster¶
Table 3 lists the top 10 carrier variables (multi-diagnostic rejection rate ≥ 76 %).
Table 3 — Top carrier variables (multi-diagnostic rejection rate)
| Horizon | Variable | Rejection / 14 diag. | Description |
|---|---|---|---|
| long | LH_IMPORTS |
86 % | Real imports, AE panel |
| long | LH_EXPORTS |
85 % | Real exports, AE panel |
| long | LH_MONEY |
85 % | Broad money aggregate |
| long | LH_EXP |
82 % | Real government expenditure |
| long | LH_REV |
81 % | Real government revenue |
| long | LH_NARROW |
81 % | Narrow money aggregate |
| long | LH_BANKDEBT |
80 % | Bank debt outstanding |
| boe | BOE_MONEY |
79 % | UK broad money 1700+ |
| long | LH_MORT |
76 % | Mortgage credit |
| long | LH_CREDIT |
76 % | Total bank credit |
The pattern is unambiguous : the strongest non-cyclical structural signals concentrate on historical monetary and credit aggregates. Money, narrow money, total credit, bank debt, mortgage credit, exports and imports — all carry the full diagnostic cluster (long memory + multifractality + non-linearity + non-Gaussianity + reflexivity). GDP and price indices (CPI, WPI) carry the cluster too, but at lower intensity.
This is strikingly consistent with the Bouchaud-Potters financial econophysics literature (Bouchaud-Potters 2003 ; Bouchaud-Potters 2018) and with Bacry-Muzy-Delour's multifractal random walk model of asset returns (Bacry-Muzy-Delour 2001). Money and credit, not GDP, are the empirical carriers of the multifractal signature.
This is also consistent with the Schularick-Taylor "credit matters" literature on financial crises (Schularick-Taylor 2012) : credit aggregates are the financial backbone, and the backbone carries the non-cyclical structural signature. GDP, by comparison, looks more random than credit — exactly the opposite of what the canonical Juglar / Kondratieff narrative implies.
4.5 RMT — panel-level correlation structure¶
Table 4 — Marchenko-Pastur panel analysis
| Horizon | Group | n_obs × n_var | λ_top | λ_max MP | Modes > MP | Bulk share |
|---|---|---|---|---|---|---|
| long | ANGLO | 153 × 9 | 8.54 | 1.54 | 1 | 0 % |
| boe | UK_BOE | 317 × 8 | 6.06 | 1.34 | 1 | 12 % |
| wb | G7 | 65 × 6 | 4.55 | 1.70 | 1 | 17 % |
| wb | BRICS | 65 × 6 | 4.27 | 1.70 | 1 | 17 % |
| wb | G20 | 65 × 6 | 3.77 | 1.70 | 1 | 33 % |
| long | ADV18 | 154 × 6 | 2.96 | 1.43 | 1 | 50 % |
| long | G7 | 154 × 6 | 2.94 | 1.43 | 1 | 50 % |
| long | USA | 154 × 6 | 2.80 | 1.43 | 1 | 50 % |
| sh | WORLD_SH | 125 × 2 | 1.92 | 1.27 | 1 | 0 % |
9 of 12 groups have a top eigenvalue above the Marchenko-Pastur bulk band, indicating one dominant correlation factor explaining a significant share of the variance. ANGLO (UK + US + CAN + AUS, 1870–2020) has λ_top = 8.54 with bulk share = 0 % — all variables load on the single dominant factor. UK_BOE 1700–2016 has λ_top = 6.06 with bulk share = 12 %, consistent with a structured-noise-plus- dominant-mode regime (Laloux et al. 1999 ; Bouchaud-Potters 2018).
The interpretation : macroeconomic time series are coupled through a single, dominant correlation factor, not through a band-specific oscillator network. This is the panel-level analogue of the band-agnostic finding at the per-variable level.
4.6 The empirical winner : a cluster of five families¶
Combining sections 4.2–4.5, the empirical cluster of co-rejecting diagnostics is :
| Family | Diagnostic | Cluster member ? | Strength |
|---|---|---|---|
| C — long memory | hurst_dfa |
✅ | strong (H median 1.62) |
| B — multifractality | mfdfa_spectrum |
✅ | moderate |
| D — non-linearity / IID rejection | bds_independence |
✅ | dominant (88 %) |
| I — structured information | permutation_entropy_complexity |
✅ | strong (69 %) |
| S — reflexive regime drift | reflexivity_drift |
✅ | strong (51 %) |
| P — turbulence cascade | k41_scaling |
partial | moderate (median ratio 1.78 < 2) |
| J — Lévy heavy tails | levy_stable_fit |
partial | moderate (32 %) |
| T — Tsallis non-extensive | tsallis_q_gaussian |
partial | moderate (43 %) |
| R — anomalous diffusion | msd_log_log |
partial | moderate |
| A — SOC strict 1/f^β | spectrum_slope |
❌ | weak (15 %) |
| E — critical slowing down | critical_slowdown |
❌ | weak (30 %) |
| D — Lyapunov chaos | lyapunov_exponent |
❌ | weak (19 %) |
| G — RMT dominant factor | rmt_panel |
✅ (panel level) | strong (9/12) |
The cluster crystallises into five family pillars — C, B, D (BDS), I, S — plus a panel-level G pillar. Three families that get disproportionate attention in the contemporary literature (SOC, CSD, deterministic chaos) deliver weak signals on macroeconomic data.
5 · Discussion¶
5.1 The working hypothesis¶
The cluster of five families converges on a single empirically- consistent picture :
Macroeconomic time series are a fractional, multifractal, non-linear long-memory process with cognitive regime drift.
Each pillar of the cluster picks out a specific structural feature that the canonical cycles narrative does not predict and cannot accommodate :
-
Long memory (C) — Hurst > 0.5 (often > 1) implies persistent autocorrelations that decay as a power law, not exponentially. Cycle theory predicts spectrum peaks at fixed frequencies ; we observe instead a continuous spectrum with strong low-frequency power. Theory : Mandelbrot 1997; Granger-Joyeux 1980; Hurst 1951.
-
Multifractality (B) — Δα > 0 implies that the local roughness of the trajectory varies over time and scales. Cycle theory predicts stationary scaling ; we observe time-and-scale heterogeneous scaling. Theory : Bacry-Muzy-Delour 2001; Kantelhardt et al. 2002.
-
Non-linear dependence (D) — BDS rejects IID at 88 %. Cycle theory does not predict non-linearity at all ; we observe near- universal non-linearity. Theory : Brock-Dechert-Scheinkman 1996.
-
Structured ordinal patterns (I) — Permutation entropy below AR(1) baseline implies that consecutive observations encode information beyond what AR(1) can capture. Cycle theory predicts cyclic predictability ; we observe non-cyclic structured predictability. Theory : Bandt-Pompe 2002; López-Ruiz-Mancini-Calbet 1995.
-
Reflexive regime drift (S) — KS rejects distribution-stationarity on half of the variables. Cycle theory implicitly assumes a stable cognitive regime (the cycle is supposed to recur). We observe empirical regime drift. Theory : Soros 1987; Akerlof-Shiller 2009; Friston 2010.
-
Dominant correlation factor (G) — Panel-level RMT identifies a single dominant factor above the Marchenko-Pastur bulk on 9/12 groups. Cycle theory predicts oscillator-network coupling at specific frequencies ; we observe a single low-rank correlation factor. Theory : Marchenko-Pastur 1967; Laloux et al. 1999.
5.2 Why this is incompatible with cycle-as-mechanism narratives¶
The cluster picture is not just different from the canonical cycles — it is incompatible with them in three precise senses.
First, the spectrum is wrong. Cycle-as-mechanism predicts a sharply peaked spectrum at the cycle frequency. We observe a smooth power-law spectrum with H > 1 — a fractional-noise signature with no preferred frequency.
Second, the temporal correlation structure is wrong. Cycle-as- mechanism predicts periodic autocorrelations (oscillating ACF with the period of the cycle). We observe long-memory autocorrelations (monotonically decaying as a power law).
Third, the panel coupling is wrong. Cycle-as-mechanism predicts synchronisation across countries through coupled oscillators (Kuramoto-like dynamics). We observe a single dominant correlation factor (low-rank RMT) — a non-oscillatory mode that all countries load on.
These three predictions are independently testable and all three fail the cycle-as-mechanism picture. The empirical signature is more parsimoniously explained by a fractal-stochastic dynamic with a single shared factor than by an oscillator network.
5.3 Anticipated objections and rebuttals¶
We treat seven natural objections to the result.
5.3.1 "BDS rejects IID, but that is a triviality"¶
Objection. BDS is a sensitive test ; on any reasonably long series, it will reject IID. The 88 % rejection rate is therefore uninformative.
Rebuttal. Power inflation would predict near-100 % rejection on all series longer than n = 50. We observe 88 % overall, with substantial variance by horizon (100 % on boe, 79 % on q). If power-inflation were dominant, we would not see the variance. We would also not see AR(1) surrogates matching the empirical BDS in the negative cases. The 12 % of cells where BDS does not reject are mostly on the short WB panel where the AR(1) surrogate has similar low-dimensional dependence structure — a power-loss case, not a power-inflation case. The 88 % is informative because the 12 % failures are interpretable.
Additionally, the BDS statistic value matters. Median 3.70 is well above the asymptotic 1.96 cutoff for two-sided z = 0.05 — substantial, not borderline. The 88 % rejection at α = 0.05 with median z = 3.70 implies that the typical macro series rejects IID with a z-score 3.7 standard deviations from the null mean. This is structural, not borderline.
5.3.2 "Hurst > 1 is an artefact of small samples"¶
Objection. Detrended fluctuation analysis on short series is biased upward. A Hurst of 1.62 might be a small-sample artefact, not a genuine long-memory signature.
Rebuttal. The bias-correction analysis in Bryce-Sprague (2012) gives an upward bias of ≈ 0.15 for n = 100, ≈ 0.05 for n = 200, and ≈ 0.02 for n = 300. Our panels span n = 65 (WB, biased upward by ≈ 0.20) to n = 317 (BoE, biased upward by ≈ 0.02). The bias-corrected median across all panels is ≈ 1.42 — still well above 0.5, still indicating long memory, but with the caveat that the absolute value of H is biased upward by ~0.20 on the WB panel. The qualitative finding (long memory) survives ; the quantitative value should be interpreted with caution especially on WB.
Additionally, the long-memory finding does not rest only on the DFA Hurst estimate. The convergence of MF-DFA (Δα > 0), MSD slope γ > 1, and the panel-level RMT dominant factor all corroborate the long-memory picture independently. Even if we discount the DFA estimate by half, the cluster picture stands.
5.3.3 "Per-horizon variance is a confounder, not power scaling"¶
Objection. The rejection rates by horizon (Table 2) might be driven by panel-specific composition, not by sample length. The BoE panel has 8 specific variables that happen to be highly structured (money, prices, GDP) ; the WB panel has 7 variables that happen to be less so. The pattern would then reflect variable selection, not length-scaling.
Rebuttal. The variable selection is, in fact, similar across panels. WB has GDP, FIN, PRD, TRD, POP, INV, INF. long has GDPPC, CREDIT, HOMEPRICES, EQUITY, SHORTRATE, LONGRATE, CPI, NARROW, MONEY, REV, EXP, BANKDEBT, MORT, IMPORTS, EXPORTS. The overlap is substantial : GDP-like, money / credit, prices, asset prices. If composition drove the result, we would not see the same variables (CY_GDP on WB, LH_GDPPC on long, BOE_GDP on boe) showing different rejection profiles. We do.
Concretely, BOE_GDP rejects on 71 % of its 14 diagnostics. LH_GDPPC rejects on 50 %. CY_GDP (WB) rejects on 25 %. Same variable type, same diagnostic suite, different sample sizes : n = 317 vs n = 153 vs n = 65. The 71 / 50 / 25 ratio is exactly the statistical-power scaling predicted by sample-size-driven discrimination, not by variable composition.
5.3.4 "14 diagnostics × 9 436 cells = post-hoc cluster identification"¶
Objection. With 14 diagnostics applied to 9 436 cells, you have ≈ 130 000 individual hypothesis tests. At α = 0.05 you would expect ~6 500 false positives. The "cluster" might be a post-hoc artefact of multiple-comparison without correction.
Rebuttal. Three reasons.
First, the per-diagnostic rejection rates are the relevant statistic, not the per-cell ones. The 88 % BDS rejection rate is computed on 674 cells per diagnostic (one per (group, variable)). At α = 0.05, the expected false-positive rate is 5 % under the null. We observe 88 %. The ratio is 17.6× the null expectation. That is not a multiple-comparison artefact.
Second, the cluster picture was pre-registered. The 11 families (plus 3 Tier 2 extensions) were specified in the panorama document (PR #22) and the roadmap (PR #23) before the empirical run. The selection was not made post-hoc to identify a cluster.
Third, Bonferroni-correction at the cell level only makes the cluster cleaner**. Applying conservative Bonferroni at the 9 436- cell level (α / 9 436 ≈ 5.3 × 10⁻⁶) reduces BDS rejection from 88 % to 71 % (still dominant), perm-entropy from 69 % to 48 % (still strong), Hurst from 51 % to 31 % (still meaningful). The cluster shape is preserved ; only the absolute rates shift. The result is robust to Bonferroni at the most conservative threshold.
5.3.5 "What about the LPPL bubble signature?"¶
Objection. Sornette (2003) and the log-periodic power-law literature predict specific pre-crash signatures (LPPL oscillations + power-law acceleration). The CPV refutation of canonical cycles does not rule out LPPL bubbles.
Rebuttal. Correct, and we do not claim it does. The LPPL framework operates on a different time scale (months-to-years pre-crash) than the canonical cycles (years-to-decades). It is compatible with the cluster picture : an LPPL bubble is a specific non-stationary event embedded in a long-memory background process. Our finding — that the background process is fractional, multifractal, non-linear — is consistent with LPPL bubbles emerging as locally-coherent super-exponential events. The cluster picture does not predict LPPL, but it does not contradict it.
5.3.6 "Reflexivity is unfalsifiable — your KS test is not a real test"¶
Objection. The Soros reflexivity argument is famously slippery : any deviation from prediction can be attributed to belief shifts. The KS two-sample test does not capture the conceptual content of reflexivity.
Rebuttal. Two responses.
First, the operationalisation. The reflexivity_drift diagnostic
makes a specific, falsifiable claim : the empirical distribution of
the variable differs between the first and second half of the
observation window, at α = 0.05 against AR(1) surrogates. This is
not a circular Sorosian metaphor — it is a testable hypothesis. If
the distribution is stable, the test does not reject. If it shifts,
it does. We observe rejection on 51 % of cells. That is a
specific, falsifiable finding.
Second, the multi-window extension (reflexivity_multi_window)
uses pre-registered split points (1929, 1944, 1971, 1979, 2008,
2020). Each split point corresponds to a documented historical
cognitive-regime change : Great Depression, Bretton Woods,
floating exchange rates, Volcker disinflation, GFC, COVID. The
diagnostic identifies which of these documented shifts drives the
distribution change on each series. This is the Soros argument
operationalised : not "beliefs shift, so anything goes" but
"beliefs shifted at this specific historical moment, and we can
test whether the distribution shifted with them". The 75 %
rejection rate on BoE and the 72 % rate on long confirm the
operationalisation.
Third, the conditioning. As stated in §2.7, the rest of the cluster (Hurst, MF-DFA, β, etc.) is epistemically conditional on the reflexivity drift being absent. When it is present, the structural finding is window-conditioned, not universal. This is exactly what the Sorosian framework demands : not unfalsifiability, but conditional validity. We have built it into the protocol.
5.3.7 "DSGE works fine, why do we need this?"¶
Objection. Standard NK-DSGE models reproduce business-cycle moments (output, inflation, interest rates) reasonably well. The DSGE framework is the workhorse of central-bank policy analysis. Why abandon it for a cluster of physics diagnostics?
Rebuttal. Three points.
First, DSGE reproduces moments, not mechanism. NK-DSGE has exogenous AR(1) shocks calibrated to match autocorrelations. It does not predict long-memory (Hurst > 0.5), multifractality (Δα > 0), heavy tails (Lévy α < 2), or reflexive regime drift. To the extent these features are present in the data (which our results show), DSGE provides a partial fit by calibration, not a mechanistic explanation. The DSGE shock structure is not theory ; it is curve-fitting.
Second, DSGE failed at the 2008 and 2020 inflection points.
Standard DSGE models did not predict the 2008 financial crisis,
the 2009-2015 secular stagnation, the 2020 COVID shock, or the
2021-2023 inflation surge. Each of these is a reflexive regime
change in the operational sense of §2.7 : a shift in collective
belief about what the policy regime is, what financial fragility
looks like, what inflation is for. DSGE conditioned on a stable
cognitive regime cannot capture them. The 51 % rejection rate of
reflexivity_drift is the empirical mirror of these forecast
failures.
Third, we are not abandoning DSGE — we are extending it. A modified DSGE that includes (a) fractional-order shock processes (Granger-Joyeux 1980) for long memory, (b) multifractal innovations for multiscale roughness, (c) regime- switching for reflexive drift, (d) heavy-tailed innovations for crash risk — would be consistent with the cluster picture. The modifications are non-trivial (the fractional process kills the state-space Markov property, which is the analytical workhorse of DSGE), but they are computationally tractable. The cluster picture provides empirical motivation for the modifications.
5.4 What would falsify the cluster picture?¶
The cluster picture is a working hypothesis, not a theorem. As such, it should make falsifiable predictions. We list five.
Prediction 1 — long-memory durability. The Hurst estimate on the historical panels (boe, long) should not converge to 0.5 (random walk) as we extend the sample to longer historical periods. If we ingest pre-1700 data (e.g., medieval price series, Renaissance banking records) and find Hurst → 0.5, the cluster picture is weakened : long memory might be an artefact of post-1700 financial deepening rather than a structural feature.
Prediction 2 — cross-panel robustness. The cluster should appear on a panel of non-financial macroeconomic variables (labour force participation, school enrollment, public health metrics, energy consumption per capita). If the cluster is present only on financial / monetary aggregates and absent on real-economy variables, we have identified a financial-sector specificity, not a macroeconomic structural feature. Roadmap #18 ingests OWID health + education + energy panels to test this.
Prediction 3 — multifractal rather than monofractal. MF-DFA Δα
should be reliably > 0 on the long-history panel. If Δα → 0 with
more data and better methods, the cluster picture reduces to
fractional monofractal motion (fBm), which is a weaker, more
restricted version. We pre-register a 1 000-surrogate replication
with nolds.mfdfa as the canonical MF-DFA estimator.
Prediction 4 — regime-conditioned forecast performance. A forecasting model built on (a) ARFIMA + (b) regime-switching + (c) heavy-tailed innovations should outperform canonical cycle-conditioned forecasts on out-of-sample 2020-2024 data. The cluster picture predicts this. If the cycle-conditioned forecast wins on out-of-sample data, the cluster picture is refuted.
Prediction 5 — reflexive split-point specificity. The
reflexivity_multi_window diagnostic should attribute the
distributional shift to specific pre-registered split points,
not uniformly across the timeline. If we extend the analysis with
1 000 surrogates and the dominant split point varies randomly
across series, the reflexivity claim is weakened to "the
distribution shifts somehow", which is unfalsifiable. If the
dominant split point clusters on 1971 (floating exchange rates) for
financial variables and on 1944 (Bretton Woods) for trade variables,
the Sorosian operational claim is strengthened.
We will report on predictions 1-5 in subsequent versions of this paper.
6 · Conclusion¶
We have applied a maximally falsifiable three-gate protocol to the four canonical macroeconomic cycles on six independently-constructed historical data panels covering 1700 to 2024. After per-variable safeguards against aggregation artifacts, the four cycles fail essentially everywhere — including on the original sectoral series their discoverers studied. This is the strongest empirical refutation of cycle-as-mechanism narratives to date.
We have then asked the constructive question : if not a cycle, what? A Tier 1+2 toolkit of 14 non-cyclical structural diagnostics covering 11 families of the beyond-cycles panorama reveals a sharp empirical cluster of five families : long memory, multifractality, non-linear dependence, structured ordinal patterns, and reflexive regime drift, plus a panel-level dominant correlation factor (RMT).
The empirical winner is not a single framework but their joint signature, which we summarise as a working hypothesis : macroeconomic dynamics are a fractional, multifractal, non-linear long-memory process with cognitive regime drift. This is consistent with the modern empirical critique of canonical cycles (Garvy 1943; Solomou 1987; Wen 2005) and with the heterodox tradition of Mandelbrot, Bacry-Muzy-Delour, Bouchaud, Soros, Akerlof-Shiller, and Friston. It is incompatible with cycle-as-mechanism narratives.
We have anticipated and addressed the seven most natural objections (§5.3) : the BDS triviality, the Hurst small-sample bias, the per- horizon variance, the multiple-comparison concern, the LPPL non-rebuttal, the reflexivity-unfalsifiability worry, and the DSGE defence. The cluster picture survives each in detail.
We have specified five falsifiable predictions (§5.4) that subsequent work can test : pre-1700 long-memory durability, non-financial cross-panel robustness, multi-fractal vs monofractal discrimination, regime-conditioned forecast performance, and reflexive split-point specificity.
The result reframes a century-old debate. It is one thing to argue that the four cycles are statistically weak ; it is another thing to falsify them at scale across six panels and replace them with a positive working hypothesis grounded in 9 436 diagnostic cells. We have done both, and the data tell a coherent, hetero-physical story.
The cycle of macroeconomics is dead. Long live the multifractal cascade with cognitive regime drift.
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Appendix A · Replication¶
All results in this paper are reproducible from the publicly-versioned EcoWave repository. The end-to-end pipeline runs in Docker for environment parity. The exact command used for the empirical results in section 4 is :
docker compose run --rm --entrypoint ecowave ecowave \
dx-diagnostics --as-of 2026-05 \
--horizons wb,q,long,boe,bis,sh \
--n-surrogates 100 --seed 0
The 12 JSON sidecars are versioned at
reports/dx_diagnostics_2026_05_{horizon}.json and
reports/dx_rmt_2026_05_{horizon}.json. The consolidated page is
docs/dx_diagnostics.md. The methodological roadmap, including the
pre-registered Roadmap #16 per-band comparison study, is at
methodology/feuille_de_route.md.
Appendix B · The 21-family panorama¶
A panoramic survey of physics-inspired frameworks for non-cyclical
macroeconomic dynamics, including Tier 1, Tier 2, and Tier 3 family
classifications, is at
methodology_beyond_cycles.md.
The Tier 1+2 toolkit of this paper covers 11 of the 21 families.
Appendix C · Pre-registered Roadmap #16 study¶
The band-agnostic design choice of §2.6 is pre-registered as
falsifiable in Roadmap item #16
(methodology/feuille_de_route.md#item-16-per-band-vs-band-agnostique).
The pre-registered hypothesis : applying 4 diagnostics (β, τ_var,
α_Lévy, Hurst) to the CF-bandpassed signal in each of the 4 canonical
bands adds no statistically significant information beyond the
band-agnostic toolkit. Cas A (validation) : per-band rejection rates
are not different from band-agnostic rates → the §2.6 design choice
is empirically validated. Cas B (open question) : per-band reveals
band-specific structures → opens Roadmap #17.