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THE FRACTALS OF FINANCE | Research Series Phase 2 | Episode 5 of 9

The Structure Persists

Forty years of algorithmic trading, quantitative finance, and massive capital deployment have not diminished the feedback structure. The adaptive markets hypothesis predicts convergence. The data shows none.

There are exactly three possible states for the feedback structure we have documented across Episodes 1 through 4. State 1: it does not exist, and what we measured is noise. State 2: it exists but is decaying, the residual hum of an inefficiency being arbitraged away. State 3: it exists and is persisting, a structural feature of how markets process information.

Episode 4 killed State 1. Three independent tests, the rolling autocorrelation, the regime-conditional variance ratio, and the conditional correlation, all reveal the same spectral structure: positive and negative feedback alternating in every market. The random walk is dead.

This episode tests State 2. The adaptive markets hypothesis, the most sophisticated version of the efficiency story, makes a specific prediction about what should happen to a market anomaly over time: it should decay. Capital should flow toward the signal. Models should crowd the trade. Each decade should show less structure than the last, as more money chases the same pattern.

This is a testable prediction. If the feedback structure is an exploitable anomaly, its oscillation amplitude should have shrunk over forty years. The autocorrelation should swing less far from zero in 2020 than it did in 1990. The variance ratio should sit closer to 1.0. The convergence should be measurable.

We tested it. The structure has not decayed. After four decades of adaptive pressure, its amplitude is statistically indistinguishable from where it started. State 2 is dead.

The Falsification

We divided the forty-year sample into four decades. For each of the sixty-eight contracts, we computed the standard deviation of the rolling autocorrelation within each decade. This measures the oscillation amplitude: how far the autocorrelation swings from zero in either direction. Under State 2, this number should decline monotonically.

Figure 5.1 ACF oscillation amplitude by decade. Blue bars are observed. The dashed red line is what efficiency predicts: monotonic decay. The data shows no decline. Paired t-test: p = 0.27.

The dashed red line is what efficiency predicts: a steady decline as adaptive agents exploit and erode the pattern. The blue bars are what we observe: no decline. The most recent decade shows the highest point estimate of the four, at 0.0558.

A paired t-test comparing the most recent decade to the first, using the fifty-five contracts with data in both periods, yields p = 0.27. A Wilcoxon signed-rank test yields p = 0.12. Neither is close to significant. The honest conclusion is not that amplitude is rising. It is that amplitude has not fallen. After four decades, the feedback structure oscillates with the same force it had when the data begins.

This is a stronger result than it may initially appear. State 2 does not merely predict slight decline. It predicts convergence. The amount of capital deploying quantitative strategies has grown by orders of magnitude since 1986. The number of CTAs, algorithmic funds, and systematic strategies has exploded. If these forces were eroding the structure, forty years should be more than enough to show it. The data shows nothing.

Where the Feedback Is Getting Louder

The aggregate statistic conceals an important asymmetry. When we compare amplitude by asset class between the first and most recent decades, a pattern emerges.

Figure 5.2 ACF amplitude by asset class: first decade vs most recent. Six of eight classes are louder now. The largest, most liquid markets show the strongest gains.

Six of eight asset classes show higher amplitude in the most recent decade than the first. Metals lead at plus fifty-two percent. Livestock is up sixty-five percent, FX up seventeen percent, grains and energy show single-digit to modest increases. Fixed income is up modestly. Only equity indices and softs show declines, both modest.

The pattern is notable. The asset classes showing the strongest amplitude gains include some of the most liquid, capital-intensive markets on earth. If adaptive pressure were eroding the structure, these are the markets where the erosion should be most visible. They show the opposite.

This does not prove acceleration. The sample sizes within each asset class are small, the uncertainty is large, and this is a descriptive comparison, not a formal test. But the direction is consistent: the markets where the most capital has been deployed against the pattern show no sign of decay.

The Honest Split

The aggregate and asset-class stories could be accused of selection bias. Let the individual contracts speak for themselves.

Figure 5.3 Amplitude change by contract: 62% show higher amplitude now, 38% lower. The mean tilts positive. Neither test is significant.

Thirty-four of fifty-five paired contracts, sixty-two percent, show higher amplitude in the most recent decade. Twenty-one show lower. The mean difference is positive but not statistically significant. This is not a landslide. It is a tilt.

But the question is not whether the structure is getting louder. The question is whether it is getting quieter. State 2 makes a directional prediction: decline. For that prediction to hold, we should see a clear majority of contracts with falling amplitude, a negative mean difference, and statistical significance. We see none of these. The mean is positive. The majority tilts toward increase. The test is not significant in either direction.

The structure has not decayed. That is the finding. Not acceleration, but persistence. And persistence alone kills State 2, because State 2 requires convergence.

The Continuous View

Decade snapshots can mask trends within decades. The continuous rolling view provides finer resolution.

Figure 5.4 Continuous mean |ACF| from 1986 to 2026. The trend is slightly negative: 16% decline over 40 years. The structure retains 84% of its original magnitude.

The continuous mean absolute ACF shows a slight negative trend: a decline of approximately sixteen percent over forty years. This is real. The structure is marginally quieter in terms of mean absolute departure from zero than it was four decades ago.

Two points of context. First, the decline is small in economic terms. The feedback structure retains roughly eighty-four percent of its original magnitude. After four decades of exponential growth in systematic capital, the structure has barely budged. Second, this metric measures the average departure from zero, not the oscillation amplitude. The decade standard deviations, which measure how far the autocorrelation swings in each direction, show no decline. The feedback swings just as far; its centre of gravity has shifted marginally closer to zero.

These are different things. A system that swings from +0.10 to −0.10 has a larger mean absolute value than one that swings from +0.12 to −0.08, but the second system has a wider amplitude. The persistence of amplitude alongside a slight decline in mean level is consistent with a structure that is maintaining its force while becoming slightly more symmetric.

The S&P 500: An Extreme Case

Figure 5.5 Left: S&P 500 cumulative ACF drift to −19.5 (extreme negative-feedback bias). Right: universe splits roughly evenly between positive and negative drift. The S&P is an outlier.

The S&P 500’s cumulative ACF drifts to negative 19.5 over forty years, driven by its persistent negative-feedback bias. This is a genuine finding about the S&P 500 specifically. It tells us that the feedback structure in the world’s most important equity index has a strong directional character: negative feedback dominates positive, and the imbalance accumulates over time.

But it is not a universe-wide finding. Across all sixty-eight contracts, the cumulative drift is split roughly evenly: fifty-three percent positive, forty-seven percent negative. Some markets accumulate positive feedback. Others accumulate negative. The S&P 500 is at the extreme negative end.

This matters because it clarifies what the feedback structure is. It is not a single force pushing all markets in one direction. It is a spectral mechanism that runs in every market but with different tuning. Some markets spend more time at the trending end. Others spend more at the oscillating end. The S&P 500 is oscillation-dominant. The structure persists in all of them, but it persists differently.

Why the Structure Persists

The adaptive markets hypothesis assumes that capital chasing anomalies will erode them. More trend followers should mean less trend. More mean-reversion traders should mean less reversion. Competition should flatten the landscape.

Forty years of data reject this prediction. Three explanations are consistent with the evidence.

First, the structure may not be an anomaly at all. If positive and negative feedback are the mechanism by which prices discover value, not a flaw to be exploited but the process itself, then capital cannot arbitrage it away. You cannot arbitrage away price discovery. You can only participate in it.

Second, the population of adaptive agents may amplify rather than dampen the cycle. Trend followers extend the positive-feedback phase. Mean-reversion traders sharpen the negative-feedback reversal. More players do not flatten the landscape. They deepen the grooves. The structure feeds on the capital deployed against it.

Third, the structure may operate at a timescale that resists exploitation. A feedback cycle with median regime duration of four to five months and maximum duration measured in years is invisible to strategies operating at millisecond to daily horizons. The structure persists because most capital operates at the wrong frequency to erode it.

State 3

The evidence across five episodes now stands as follows.

The feedback structure exists. Three independent tests confirm it: rolling autocorrelation, regime-conditional variance ratio, and conditional correlation. State 1 is dead.

The feedback structure has not decayed. Oscillation amplitude is statistically indistinguishable from forty years ago. Six of eight asset classes show higher amplitude in the most recent decade. The continuous mean absolute ACF has declined marginally, by roughly sixteen percent over four decades, retaining eighty-four percent of its original magnitude despite exponential growth in systematic capital. State 2 is dead.

What remains is State 3: the feedback structure is a persistent feature of how markets process information. It has survived four decades of adaptive pressure, and in the markets where the most capital has been deployed against it, it shows no sign of erosion.

We do not claim the structure is accelerating. The evidence is suggestive in some asset classes but not statistically conclusive. What we claim, and what the data firmly supports, is that the structure is not going away. The remaining episodes build on this foundation.

The feedback structure runs everywhere, but it does not run the same way everywhere. Fixed income autocorrelation looks nothing like equity autocorrelation. Energy regime durations bear no resemblance to FX regime durations. The structure has a common architecture but asset-specific signatures.

Episode 6 maps the fingerprint. Eight asset classes. Eight distinct spectral configurations. One underlying mechanism.

Endnotes

Methodology

Decade amplitude comparison: for each of the 68 contracts, the rolling 504-day ACF(1) series (21-day steps) was divided into four decades (1986–1995, 1996–2005, 2006–2015, 2016–2026). The standard deviation of ACF readings within each decade measures oscillation amplitude. Universe means: D1 = 0.0491 (n=55), D2 = 0.0537 (n=67), D3 = 0.0532 (n=68), D4 = 0.0558 (n=68). Bootstrap 95% CI computed from 2,000 resamples (seed=42): D1 [0.0429, 0.0557], D2 [0.0479, 0.0607], D3 [0.0495, 0.0569], D4 [0.0514, 0.0608]. All four CIs overlap substantially. Paired t-test on 55 contracts with data in both D1 and D4: t = 1.10, p = 0.27. Wilcoxon signed-rank: W = 586, p = 0.12.
Asset class D1 vs D4 amplitude: Livestock +65.3% (n=3), Metals +51.9% (n=6), FX +16.9% (n=7), Grains +9.0% (n=7), Energy +8.5% (n=6), Fixed Income +3.5% (n=8), Equity Index −1.0% (n=8), Softs −9.6% (n=10). Small n within each class limits statistical power; these comparisons are descriptive.
Continuous mean |ACF| trend: linear regression of monthly universe-mean |ACF(1)| on time index. Slope = −0.0000012, r = −0.08. Over the full sample, this represents a decline from approximately 0.063 to 0.053. The decline is modest: a 16% reduction in mean |ACF| over 40 years, retaining 84% of original magnitude.
Cumulative ACF drift: for each contract, the sum of all rolling ACF readings from inception to 2026. S&P 500: −19.5 (extreme negative, reflecting its strong negative-feedback dominance). Universe: 32 contracts negative (47%), 36 positive (53%). Universe mean final cumulative: +2.7. The S&P 500 is a multi-sigma outlier in the negative direction.
State 2 falsification framework: the adaptive markets hypothesis (Lo, 2004) predicts that exploitable patterns should decay as adaptive agents deploy capital against them. We test the specific prediction that oscillation amplitude should decline over time. The null hypothesis of the paired test is H0: D4 amplitude = D1 amplitude. We cannot reject this null (p = 0.27). The one-sided p-value for H1(State 2): D4 < D1 is approximately 0.86, meaning the data provides strong evidence against the State 2 prediction.

Data

Same dataset as Episodes 1 through 4: 68 CSI ratio-adjusted continuous futures, September 1984 to January 2026. 68 contracts had sufficient data for rolling ACF computation. 55 contracts had sufficient data in both the first and most recent decades for paired comparison.

Figures

Figure 5.1: Decade amplitude bars with 95% bootstrap CI and State 2 prediction overlay.
Figure 5.2: Paired horizontal bars comparing D1 and D4 amplitude by asset class.
Figure 5.3: Sorted bar chart of D4 minus D1 amplitude change for 55 paired contracts.
Figure 5.4: Continuous mean |ACF| over time with 5-year rolling mean and linear trend.
Figure 5.5: Left: S&P 500 cumulative ACF drift. Right: histogram of final cumulative ACF across 68 contracts.

This research series is drawn from The Fractals of Finance: Determinism, Adaptation and the Geometry of Markets

The book explores the full architecture of feedback, fat tails, and fractal structure in financial markets, and what it means for how we trade, invest, and understand risk.

Available now on Amazon in paperback, hardcover, and Kindle.

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