Is the Unibrow Quantile Model the New Hottest GF in Town?
You judge. Sina @Sina_21st and @TheRealPlanC are power law researchers who post most of their work on X. They recently introduced an alternative way to look at the power law model, presenting it as a superior and completely different way to interpret Bitcoin’s price dynamics. While it is still rooted in the same basic power law principles, they’re framing it as a fundamentally different model with greater explanatory power — an assertion that misrepresents its nature. In truth, although quantile regression can offer useful insights if implemented correctly, the specific implementation they propose is conceptually flawed and produces misleading results.
Introduction to the Unibrow Quantile Regression Approach
A popular way to analyze Bitcoin’s long-term growth is to regress the log(price) against log(days from Genesis Block). Traditionally, researchers have used an ordinary least squares (OLS) “power law” fit, where one seeks a single linear relationship of the form:
log(price) = m · log(days_from_GB) + c
to describe much of Bitcoin’s historical price dynamics. This simplified approach does an excellent job of capturing Bitcoin’s overall price dynamics. One measure of its accuracy is the coefficient of determination (R²). While R² is not the only metric for evaluating a model’s fit, the exceptionally high value here strongly suggests there is a genuine signal in Bitcoin’s behavior — namely, a power law.
The overall power-law trend is so strong that it is visible by visual inspection alone. More sophisticated methods reveal the true nature of Bitcoin: a scale-invariant system.
There are many ways to describe and characterize the deviations from the trend that seem to be a different type of additional signal on top of the power law behavior. Parallel, % deviations from the main trend or oscillatory patterns (to illustrate the apparent periodicity of the bubbles) are valid approaches.
Power laws differ from other curves because they possess a critical property: scale invariance. In simpler terms, the functional form remains valid across different magnitudes, which is why power laws can appear almost “magical.” We’ll delve deeper into this concept in the following sections. For now, let’s return to our examination of the so-called “alternative” approach and see how it stacks up against a true power-law framework. However, the Unibrow Quantile Regression approach attempts to capture the entire distribution of Bitcoin’s price behavior, rather than focusing on a single “best-fit” line. In principle, it’s a reasonable objective: although a single power law captures Bitcoin’s overall behavior quite well, there is value in examining the role of the speculative bubbles under one unified framework. One particular approach (proposed by @Sina_21st and @TheRealPlanC ) to achieve this goal is to implement a Quantile Regression.
In practice, it leverages the same inputs as a power-law OLS fit — namely, the time since Bitcoin’s Genesis Block (transformed by the logarithm) on the x-axis and the logarithm of Bitcoin’s price on the y-axis. The decision to use a logarithmic scale on both axes implicitly assumes a power-law relationship in the data — whether stated outright or not. Grasping this core premise is essential for making sense of the broader discussion that follows. This assumption already indicates that Quantile Regression is not a fundamentally new “Bitcoin model.” Rather, it’s a method of probing the underlying power-law dynamics, with an added layer designed to account for large deviations from that power law. The main difference from the simple power law fit is that the Quantile Regression focuses on different portions (quantiles) of the price distribution. Instead of looking for one line that minimizes the squares of residuals, the model:
- **Selects a particular quantile} (e.g., 30% or 70%) as a target.
- Assigns asymmetric weights to the data points, such that the chosen quantile has a strong influence on the resulting fit.
- Outputs a family of lines — one for each quantile — offering a broader statistical description of the dataset.
By doing so, this particular implementation of the Quantile Regression aims to characterize Bitcoin’s lower, median, and upper ranges, highlighting how each segment might behave (and scale) in relation to time. On the surface, this can look like a more comprehensive power-law model — one that describes the entire dataset via multiple slope–intercept pairs.
However, as we will discuss, this “one-size-fits-all” extension of a power-law approach can lead to confusing or contradictory results — especially when Bitcoin’s price data is known to exhibit two distinct regimes: a power-law “core” and bubble-driven outliers. Simply adding multiple quantile lines may obscure these different underlying behaviors rather than clarify them. And for this reason I like to call their quantile approach the “Unibrow Quantile Model” because it merges two distinct regions (the power-law core and the bubble regime) into a single, continuous framework — much like a unibrow is two brows fused into one.
Below, we take a closer look at how the Unibrow Quantile Regression works and why it struggles to provide a single, coherent power-law exponent for all Bitcoin price data. We also highlight an alternative method — separating the lower, power-law portion from the upper, bubble-like portion — that offers clearer, coherent and scientifically meaningful insights into Bitcoin’s long-term growth trajectory.
It is all power laws! Nah…
Let’s see what the results are when we apply the Vanilla Quantile Regression to Bitcoin data. It basically draws several power laws.
The Unibrow Quantile Model in a log-log chart. This is the exact model used by Sina and PlanC but in a log of price and log of time space. It describes the behavior of Bitcoin as if it was made by an infinite number of power law from the bottom all the way to the tops. If you don’t like the power law you should like this even less given it is an infinite number of them.
In the log-linear form (log of price and linear time) it look like what usually PlanC and Sina show in their posts.
The log-linear form of the Unibrow Quantile Model. It seems innocuous enough but it has several conceptual and practical problems. This is the form that is used by PlanB and Sina. It seems to me this is done on purpose to hide the fact the lines are actually straight lines in a log-log graph and therefore power laws.
When we obtain a result in our data analyses we need always to take a pause and interpret and understand the results. What is the meaning of all the powers laws? What do they really represent? One revealing clue comes from a chart originally created by PlanC and proudly showcased by Sina. Ironically, Sina does not recognize that this very graphic illustrates why the “Unibrow Quantile Model” fundamentally mischaracterizes Bitcoin’s behavior.
The graph illustrates how the slopes change between the real power laws (found at the bottom) and the fictitious power laws (introduced at the top). This visualization clearly distinguishes between self-consistent power laws, which reflect Bitcoin’s true scale-invariant behavior, and misleading power laws that at best provide a rough approximation of bubble decay, and at worst, obscure a deeper understanding of Bitcoin’s dynamics.
The lower slopes remain stable and coherent, reinforcing the idea that Bitcoin follows a consistent power law over time — a hallmark of scale invariance. Meanwhile, the upper slopes fluctuate significantly, revealing that these so-called “power laws” are merely statistical artifacts rather than genuine, fundamental relationships. By conflating the two, the Unibrow model dilutes the core insight: Bitcoin’s long-term trajectory is governed by a real power law, but bubbles and their decay follow an entirely different process that cannot be accurately described using the same framework.
This graph, created by PlanC, is indeed accurate. It illustrates how the slopes of the straight-line fits evolve across different quantiles. Notably, the Unibrow Quantile Model remains nearly flat up to about the 45% quantile. This means all the power laws in this range have similar slopes, indicating a consistent, scale-invariant pattern for all the data in that lower range. This is illustrated in the graph below, where we apply a 45% quantile cutoff to the Bitcoin data. By filtering out bubble-driven outliers, we uncover the fundamental signal: a remarkably consistent power law that holds across more than eight orders of magnitude (including early transactions). The precision of this relationship is extraordinary — not just for financial or social sciences, but even by the rigorous standards of physical sciences.
The lower slopes remain stable and coherent, reinforcing the idea that Bitcoin follows a consistent power law over time — a hallmark of scale invariance. Meanwhile, the upper slopes fluctuate significantly, revealing that these so-called “power laws” are merely statistical artifacts rather than genuine, fundamental relationships. By conflating the two, the Unibrow model dilutes the core insight: Bitcoin’s long-term trajectory is governed by a real power law, but bubbles and their decay follow an entirely different process that cannot be accurately described using the same framework.
This is what makes Bitcoin unique. No matter how extreme the speculative cycles become, the price always finds its way back to the power-law trajectory — as if it inherently “knows” where its fundamental growth path lies. This resilience suggests that Bitcoin’s price is not just driven by chaotic speculation but is anchored to a deeper, structured process — one governed by the principles of scale invariance and network effects
When the bubbles are removed from the dataset, a true and powerful signal emerges — Bitcoin follows a remarkably consistent power law, one that has remained stable and coherent despite repeated, long-lasting exponential deviations from it.
This is what makes Bitcoin unique. No matter how extreme the speculative cycles become, the price always finds its way back to the power-law trajectory — as if it inherently “knows” where its fundamental growth path lies. This resilience suggests that Bitcoin’s price is not just driven by chaotic speculation but is anchored to a deeper, structured process — one governed by the principles of scale invariance and network effects.
As we enter bubble territory, the lines become artificial — they no longer reflect a true underlying pattern in the data but instead serve as rough and misleading approximations. There are no genuine power laws governing the bubbles; rather, the algorithm “discovers” them simply because its design and input constraints leave it with only one possible response: *”everything is a power law.” This is a contrived, circular outcome, not an insightful characterization of Bitcoin’s behavior. When all you have is a hammer, every problem starts to look like a nail. By misrepresenting this distorted output as a completely new model — when in reality, it is an arbitrary collection of infinite power laws — the Unibrow Quantile Model only adds confusion. It misleads the X audience, especially those who may not fully grasp the details of its construction, underlying methods, and assumptions. Instead of clarifying Bitcoin’s behavior, it obscures the real power-law structure with a flawed, overfitted narrative. Ironically, this approach undermines the very purpose for which it was created — to provide a more general framework that captures not just Bitcoin’s fundamental power-law behavior, but also the large deviations caused by its several years bubble cycles. Instead of offering clarity, it distorts both aspects, failing to meaningfully separate the underlying trend from speculative excesses. However, what the Unibrow Quantile Regression unintentionally reveals is the remarkable consistency and robustness of Bitcoin’s power-law behavior at the lower quantiles. When isolated from outliers, this fundamental trend stands out with striking clarity, reinforcing the idea that Bitcoin follows a true scale-invariant growth pattern. With all this said, is there a general way to describe bubble behavior without committing to a specific cycle length or placing too much emphasis on the apparent periodicity of these features? The Bimodal Nature of Bitcoin Bitcoin is a unique phenomenon, resembling a physical system more than conventional financial data. It exhibits a clear scale-invariant behavior — something straight out of a physics textbook. Many physical systems display scale invariance and are naturally described by power laws. Physicists are so enamored with power laws that you can even find a bit of self-deprecating novelty shirts online that read: “I went to a physics conference, and all I got was a lousy power law.”
Physicists may joke about power laws, but in Bitcoin, they’re no joke — unveiling a deep, scale-invariant signal that persists even through the chaos of speculative bubbles.
Discovering such a clear power law in Bitcoin’s data is a significant breakthrough. It has far-reaching implications, particularly for uncovering the mechanisms and causal relationships that drive Bitcoin’s behavior. This finding remains at the core of our scientific understanding of Bitcoin — proven time and again to be real, consistent, and robust.
Ironically, Sina himself has contributed to validating this data. In the past, he conducted multiple cross-validation tests, demonstrating how the power law reliably predicts future behavior when the data is divided into a training set and a forward-testing set. By attempting to describe Bitcoin’s large, long-lasting deviations from the fundamental power law using additional power laws, the core message — that Bitcoin is a scale-invariant system — becomes muddled. Instead of clarifying the distinction between its stable growth dynamics and speculative excesses, this approach dilutes and distorts the key insight: that Bitcoin follows a singular, consistent power law over time.
The introduction of fictitious, bubble-driven power laws creates an illusion of continuity where there is none, ultimately weakening our understanding of Bitcoin’s true underlying structure. A far more logical, rigorous, and scientifically sound approach is to separate the distinct components within Bitcoin’s data rather than forcing them into a single, oversimplified framework. For those unfamiliar with extracting real physical signals — perhaps including business professors more accustomed to statistical summaries — this idea might seem counterintuitive. However, in the fields of physics, engineering, and signal processing, decomposition is standard practice. Identifying and isolating different behaviors within complex systems is not just useful; it is fundamental to understanding how those systems actually function. To illustrate this in practical terms, let’s systematically isolate the deviations from the fundamental power-law trend. This is done by simply subtracting the trend from the actual price, leaving us with what are known as residuals — the variations that diverge from the core power-law behavior.
These deviations can also be expressed as percent changes relative to the trend, providing a clearer view of how Bitcoin’s price fluctuates around its fundamental trajectory. The trend itself can be identified in multiple ways, but for now, we use the power law as the natural boundary between true and fictitious power-law behavior.
From our analysis of the slope vs. quantile graph, we determined that the 45% quantile serves as a reasonable threshold for distinguishing the core power law from speculative deviations. With this in mind, we proceed by subtracting this 45% quantile power-law fit from the actual price data to isolate the excesses — both in magnitude and structure — above the fundamental trend. The residuals clearly appear as isolated peaks that decay over time. This naturally raises the question: is this decay purely random, or does it follow a specific mathematical pattern?
Sina argued that their Unibrow Method is superior, claiming it solves the issue of having only a limited number of bubbles to analyze. But does it really? No.
The reality is simple: there are only four bubbles, regardless of the method used to study them. If you attempt to use the upper quantiles to predict future tops, you are essentially doing the same thing — extrapolating from a small set of points to establish a trend. The difference is that in Sina’s case, this fact is obscured by statistical jargon rather than openly acknowledged. This reveals a fundamental lack of intuitive understanding of the data itself.
As we demonstrate below, when one examines the higher quantiles, only a small cluster of points actually shapes the trend, meaning the method does not add any real statistical rigor. While it is true that the quantile regression formally includes all the data, in practice, only a limited weight is assigned to anything below the selected quantile. This means that even though the full dataset is processed, the actual fit is still heavily determined by just a few key points — exactly the issue it claims to solve.
No amount of statistical complexity can escape this fundamental limitation. Instead, the method is framed as a more precise and statistically valid approach, when in reality, the facts and the data show otherwise. See the graph below for an illustration of this issue.
This figure illustrates the limitations of focusing on top quantiles to describe Bitcoin’s price tops or to construct a predictive model. While the quantile regression lines (e.g., 70%, 80%, 99%) might seem to provide a robust framework for modeling bubble behavior, they inherently suffer from the same issue: they rely on a few isolated clusters of points at the bubble peaks. This clustering limits their predictive power, as it cannot overcome the fundamental problem of having only a small number of data points in these regions. Claiming that quantile regression resolves this limitation is both misleading and incorrect — the problem lies in the dataset itself, not the method. The figure reinforces that the lower quantiles capture the fundamental power-law trend, but the upper quantiles suffer strong limitations in describing the dynamics of the tops. If in addition we use the wrong functional form the description is nonsensical and useless.
Furthermore, from the deviation graph, it becomes immediately clear that a power law is not the appropriate fit for the decay phase — because the decay follows an exponential pattern, not a power law. In a log-linear chart, the decline appears as a straight line, which is the hallmark of exponential decay. This directly contradicts the assumptions of the Unibrow Method.
Not only does this approach fails to add any real rigor to the statistical analysis of Bitcoin’s tops, but it also introduces the wrong functional form to describe their decay. Instead of properly capturing the exponential nature of the bubble corrections, it forces a power-law framework onto a process that fundamentally does not follow one.
In other words, the Unibrow Method is not just ineffective — it is actively misleading, creating a distorted view of how Bitcoin’s bubbles form and resolve over time. Can we still find an approach that while recognizing the limitation in the data available is more meaningful and coherent in describing the reality of the data in front of us? A better approach is to plot the deviations from the power law on a log-log scale, similar to how we analyzed the tops, but this time applying quantile regression to the entire bubble structure rather than just the peaks. This shift in focus acknowledges the exponential nature of the decay while allowing us to analyze the full bubble dynamics, rather than isolating only the highest points.
Instead of forcing an artificial power-law fit on individual tops, this method highlights how the entire bubble structure exhibits a systematic decay, reinforcing the idea that Bitcoin’s corrections follow an exponential reversion rather than a self-similar power-law pattern. This represents a secondary signal that exists on top of and in addition to Bitcoin’s fundamental power-law behavior. Rather than distorting the primary trend, this approach properly separates the two dynamics — recognizing that Bitcoin’s long-term growth follows a power law, while its bubbles and corrections exhibit a distinct, exponential decay.
This is the correct way to perform signal analysis on complex data: identify the fundamental trend first, then analyze deviations as a separate process, rather than conflating them into a single misleading model. Below, we present the results of the quantile regression using the correct functional forms — exponentials, not power laws — applied to the deviations. This approach provides an intuitive and accurate representation of how Bitcoin’s bubble corrections unfold over time.
While this method effectively models the decay of the tops, we still need to connect this representation back to the original price action and visualize how these quantile-based structures manifest in a price vs. time graph.
This can be achieved by reconstructing the bubble component of Bitcoin’s price action. Since we expressed deviations as a percentile difference from the power-law trend:
Δ = (Price — Trend) / Trend
and since the quantile regression captures these deviations, we can use it to compute a quantile-based deviation price:
Price = Trend × (1 + Δ)
The reconstructed price dynamics, both in log-log and log-linear space, are shown in the figures below. This visualization makes it clear how Bitcoin’s price oscillates around the power-law trend, with bubbles rising and decaying in a structured, predictable manner.
This is a meaningful resolution to the ongoing debate. It improves on the single power-law description of Bitcoin by explicitly acknowledging the distinct dynamics of bubbles and their reversion to the baseline. Additionally, it avoids committing to a rigid periodicity for the bubbles, allowing for a more flexible and accurate representation of their behavior. This approach captures both the scale-invariant growth that defines Bitcoin’s long-term trajectory and the structured, exponential decay of its speculative cycles.
The log-log version of the “Unicorn Power Law Quantile Regression” model. This graph represents the “Unicorn Price Action”, reconstructed using a proper modeling approach for the deviations with exponentially decaying quantiles. Unlike previous methods that attempt to merge fundamentally different behaviors into a single framework, this approach separates the core power-law growth at the bottom from the decaying channels associated with speculative bubbles at the top.
This is a meaningful resolution to the ongoing debate. It improves on the single power-law description of Bitcoin by explicitly acknowledging the distinct dynamics of bubbles and their reversion to the baseline. Additionally, it avoids committing to a rigid periodicity for the bubbles, allowing for a more flexible and accurate representation of their behavior. This approach captures both the scale-invariant growth that defines Bitcoin’s long-term trajectory and the structured, exponential decay of its speculative cycles.
An important feature of this model is that the lines at the top never cross (while this is one of the nonsensical limitation of the Unibrow model); instead, they gradually converge towards the power-law threshold, reflecting the exponential decay of bubbles without violating the underlying power-law growth at the core. This behavior offers a consistent and physically meaningful description of the available data, separating the fundamental power-law behavior from the speculative deviations in a way that remains faithful to Bitcoin’s scale-invariant properties.
This is how the Unicorn Quantile Model appears in log-linear space. Notably, the predictions of this model align perfectly with earlier estimates based on modeling only the tops, providing a conservative projection of 200K for the next cycle top. This stands in stark contrast to the overly optimistic estimates produced by the Unibrow method, which lack consistency and coherence.
An important feature of this model is that the lines at the top never cross (while this is one of the nonsensical limitation of the Unibrow model); instead, they gradually converge towards the power-law threshold, reflecting the exponential decay of bubbles without violating the underlying power-law growth at the core. This behavior offers a consistent and physically meaningful description of the available data, separating the fundamental power-law behavior from the speculative deviations in a way that remains faithful to Bitcoin’s scale-invariant properties.
This method represents the best of both worlds — a synthesis of rigor and intuition that resolves the fundamental flaws of the Unibrow Quantile Model while preserving the key insights from both the power-law structure and the bubble dynamics.
At its foundation, this approach retains the power-law quantile regression for the lower portion of the data, where Bitcoin’s scale-invariant behavior is strongest. This ensures that we properly capture Bitcoin’s long-term growth trajectory without distortion. But unlike the Unibrow method, it does not force an artificial power-law structure onto the bubble regime, where the data behaves fundamentally differently.
Instead, we apply a physically meaningful, exponential decay function to model how bubbles revert back to the power-law trend. This method achieves two critical improvements:
- It avoids the mistake of treating bubbles as extensions of the fundamental power law — a misstep that introduces fictitious trends and obscures the true nature of Bitcoin’s price dynamics.
- It does not impose an arbitrary cycle length on the bubbles, allowing them to emerge and decay naturally while following a well-defined mathematical process.
By using quantile regression correctly — applying power laws where they belong and exponentials where they are justified — this approach offers a coherent, empirically grounded resolution to the debate. It incorporates the predictive power of the quantile regression at the lower quantiles while faithfully representing the behavior of speculative excesses and their decay.
This framework is not just a compromise — it is the scientifically correct solution. It respects Bitcoin’s fundamental power-law structure while acknowledging that bubbles require a distinct treatment. By separating these two phenomena properly, we achieve a clearer, more accurate understanding of Bitcoin’s price evolution — one that aligns with both theoretical principles and empirical evidence. We can call this model the “Unicorn Quantile Power Law Regression Model” because, unlike the flawed Unibrow approach, it elegantly separates Bitcoin’s fundamental power-law growth from its speculative bubbles while preserving their true mathematical structure. The name is fitting — not only because it corrects the Unibrow mistake, but because its future projection forms a sleek, powerful, and coherent curve, resembling the majestic horn of a unicorn.
Just as a unicorn’s horn is both beautiful and formidable, this model combines mathematical elegance with scientific rigor, offering a framework that is both visually compelling and analytically sound. Unlike naive extrapolations that blur together fundamentally different behaviors, the Unicorn Model recognizes that Bitcoin’s price evolution is not a single, continuous process — it is a structured interplay between long-term scale invariance and short-term speculative cycles.
I don’t know you I prefer a Unicorn to a Unibrow.
PS
This article will be expanded in future iterations where I will add more insights into good modelling vs fake ones.