Binary Structure and Stopping-Time Asymptotics in the Collatz Map: An Empirical Study to 10^7

Abstract

We present a large-scale empirical study of Collatz stopping times for all positive integers up to (N = 10^7). Exhaustive computation reveals that the binary representation of the seed integer — specifically, the number of trailing 1-bits — governs the mean stopping time with remarkable precision: each additional trailing 1-bit adds approximately 6.15 steps to the mean total stopping time, a relationship that holds with near-perfect linearity over five orders of magnitude of the seed count. We provide a mechanistic explanation rooted in the forced odd-step cascades induced by trailing 1-bits and connect the observed increment to the logarithmic constant (\log_2(3/2)). Additionally, we characterize the full distribution of stopping times, finding it approximately log-normal with detectable negative skewness ((-0.37)) and platykurtosis ((-0.19)) in log-space, with a Jarque–Bera statistic of 243,496 decisively rejecting exact log-normality. The maximum stopping time grows as (C \cdot (\ln N)^2) with an empirical constant (C \approx 2.64). Sequential autocorrelation at lag 1 is (r = 0.402), reflecting shared trajectory fragments among nearby integers. We report 53 record-breaking stopping times, whose spacing grows superlogarithmically, and confirm that stopping times violate Benford's Law, as expected from a distribution concentrated within a single order of magnitude. A complete one-file Python script using only the standard library accompanies the paper, enabling full reproduction of every reported statistic.

1. Introduction

The Collatz conjecture, formulated by Lothar Collatz in 1937, concerns the iteration of the map

[ T(n) = \begin{cases} n/2 & \text{if } n \equiv 0 \pmod{2}, \ 3n+1 & \text{if } n \equiv 1 \pmod{2}, \end{cases} ]

applied to any positive integer (n). The conjecture asserts that the orbit of every positive integer eventually reaches 1. Despite its elementary statement, the conjecture remains unresolved and is widely regarded as one of the most intractable problems in number theory. Erdős famously remarked that "mathematics may not be ready for such problems."

The total stopping time (\sigma(n)) of a positive integer (n) is the number of iterations required to first reach 1. Terras (1976) introduced the notion of stopping time and proved that almost all integers (in the sense of natural density) have finite stopping times. Lagarias (1985) provided an extensive survey of the (3x+1) problem and its connections to ergodic theory, Markov chains, and computability. More recently, Tao (2019) proved that almost all Collatz orbits attain almost bounded values, providing the strongest density result to date.

While the theoretical literature has established probabilistic heuristics — notably, the stochastic model treating the parity of iterates as independent fair coin flips — quantitative empirical studies at scale remain valuable for several reasons. First, they provide precise constants (e.g., the coefficient relating maximum stopping time to ((\ln N)^2)) that heuristic arguments predict only in order of magnitude. Second, they can reveal structural phenomena not visible in probabilistic models. Third, they serve as benchmarks for computational verification of the conjecture itself.

In this paper, we contribute an exhaustive computation of all stopping times for (1 \le n \le 10^7) and present a detailed statistical analysis. Our centerpiece finding is that the mean stopping time, when partitioned by the number of trailing 1-bits in the binary representation of the seed, follows a strikingly precise linear relationship: each additional trailing 1-bit adds approximately 6.15 steps. We provide a mechanistic explanation for this pattern and connect it to the arithmetic of the Collatz map. We also characterize the overall distribution shape, the growth of extremes, sequential correlations, and residue-class structure.

2. Data and Methods

2.1 Stopping-Time Computation

We computed the total stopping time (\sigma(n)) for every integer (n) from 1 to (N = 10{,}000{,}000) by direct iteration. The stopping time is defined as the smallest (k) such that (T^{(k)}(n) = 1). We used an optimized iterative procedure with a lookup table for previously computed values to avoid redundant computation of shared orbit tails.

2.2 Statistical Methods

All statistics were computed in Python using only the standard library (no external packages). This design choice ensures complete reproducibility without dependency management. The following analyses were performed:

• Descriptive statistics: mean, standard deviation, maximum, and the count of distinct stopping-time values.
• Distribution characterization: moments of (\ln(\sigma(n))), including skewness and excess kurtosis, tested against the normal distribution via the Jarque–Bera statistic.
• Growth of extremes: the maximum stopping time (M(N) = \max_{1 \le n \le N} \sigma(n)) evaluated at powers of 10, with the ratio (M(N)/(\ln N)^2) computed to estimate the asymptotic constant.
• Residue-class analysis: mean stopping time partitioned by residue class modulo (2^k) for (k = 2, 3, 4, 5, 6), with one-way ANOVA (F)-statistics.
• Trailing 1-bit analysis: mean stopping time partitioned by the number of trailing 1-bits in the binary representation.
• Sequential autocorrelation: Pearson correlation between (\sigma(n)) and (\sigma(n+\ell)) for lags (\ell = 1, 2, 3, 5, 10, 100).
• Benford's Law test: chi-squared goodness-of-fit test comparing the leading-digit distribution of stopping times against the Benford distribution.
• Record-breaking analysis: identification of all record-setting stopping times and the growth rate of their indices.

2.3 Trailing 1-Bit Extraction

For each odd integer (n), the number of trailing 1-bits in its binary representation is (\nu_1(n) = \nu_2(n+1)), the 2-adic valuation of (n+1). For even integers, the count is zero by definition. We partitioned all (n \le 10^7) by their trailing 1-bit count and computed mean stopping times within each group.

3. Results

3.1 Overview

The mean stopping time over all integers from 1 to (10^7) is 155.27 with a standard deviation of 61.76. The maximum stopping time observed is 685, achieved at (n = 8{,}400{,}511). A total of 589 distinct stopping-time values occur within this range.

Table 1. Summary statistics for Collatz stopping times, (1 \le n \le 10^7).

3.2 Trailing 1-Bit Structure

Our principal empirical finding is that the mean stopping time, when partitioned by the number of trailing 1-bits in the binary representation of the seed, follows a remarkably precise linear relationship. This is summarized in Table 2.

Table 2. Mean stopping time by number of trailing 1-bits in binary representation.

For trailing 1-bit counts of 1 through 5, the increment is extraordinarily stable at approximately 6.15 steps per additional trailing 1-bit. The initial increment from 0 to 1 trailing bits is somewhat smaller (5.19), and the increment at (k=6) is anomalously large (12.85), representing a compounding effect at (n \equiv 63 \pmod{64}), where all six lowest-order bits are 1.

Mechanistic explanation. When a positive integer has (k) trailing 1-bits in its binary representation, the Collatz map is forced into a specific sequence of operations. Each trailing 1-bit guarantees that the current iterate is odd, triggering the (3n+1) rule. The result of (3n+1) applied to a number ending in a 1-bit produces an even number, which is then halved. This paired operation — one odd step ((3n+1)) followed by one even step ((\div 2)) — yields a net multiplication by approximately (3/2).

Each such forced odd–even pair contributes 2 iterations to the stopping time but also inflates the magnitude of the iterate by a factor of (3/2). The inflated iterate then requires additional iterations to "work off" this extra magnitude. In the standard heuristic model, the expected number of iterations to reduce a factor of (x) is (\log_2(x) / \log_2(4/3)), since each step has an expected shrinkage of ((1/2 + 3/4)/2 = 3/4)... but more directly, we can observe that the total additional stopping time per trailing 1-bit is the sum of the 2 forced iterations plus the expected additional iterations needed to compensate for the (3/2) inflation.

The quantity (1/\log_2(3/2) \approx 1.71) gives the number of "equivalent random iterations" per factor of (3/2), but the total stopping-time increment of approximately 6.15 reflects the full cascade: the forced pair of steps, plus the downstream cost of the accumulated magnitude increase propagating through the remainder of the orbit. The precise value of this constant (approximately 6.15) is an empirical finding that merits further theoretical investigation.

The one-way ANOVA (F)-statistics for residue classes modulo (2^k) confirm that the binary structure of the seed is overwhelmingly significant:

Table 3. ANOVA (F)-statistics for residue classes modulo (2^k).

All (p)-values are effectively zero. The decreasing (F)-statistic with increasing modulus reflects the finer partitioning into more groups with smaller sample sizes per group, not a weakening of the effect. The total between-group variance continues to increase with finer modulus.

3.3 Residue-Class Extremes

The extremes of the residue-class means at modulus 64 reveal the full range of binary structure effects:

Table 4. Extreme mean stopping times by residue class modulo 64.

The range between these extremes is 74.66 steps — nearly half the overall mean — demonstrating that the low-order bits of the seed exert a powerful deterministic influence on the stopping time, well beyond what a purely random model would suggest.

3.4 Growth of Maximum Stopping Time

The maximum stopping time (M(N)) over (1 \le n \le N) was evaluated at each power of 10. Heuristic arguments predict (M(N) \sim C \cdot (\ln N)^2) for some constant (C). Table 5 reports the empirical ratio.

Table 5. Maximum stopping time and the ratio (M(N)/(\ln N)^2) at powers of 10.

The ratio stabilizes in the range 2.6–2.7 for (N \ge 10^5), consistent with the heuristic prediction of (O((\ln N)^2)) growth. Our empirical estimate of the asymptotic constant is (C \approx 2.64). The slight non-monotonicity at (N = 10^6) (ratio 2.745 vs. 2.641 at (10^5)) is consistent with the discreteness of record-breaking events.

3.5 Distribution Shape

The distribution of stopping times is unimodal and right-skewed in natural scale. Following standard practice, we examine the distribution in log-space, where a log-normal distribution would appear Gaussian.

Table 6. Moments of (\ln(\sigma(n))) for (1 \le n \le 10^7).

The exponential of the log-space mean is (e^{4.9623} = 142.92), which is lower than the arithmetic mean of 155.27, as expected for a right-skewed distribution. The negative skewness in log-space ((-0.37)) indicates a left-tail excess (the log-distribution is pulled leftward), while the negative excess kurtosis ((-0.19)) indicates platykurtosis — thinner tails than a Gaussian. The Jarque–Bera statistic of 243,496 decisively rejects exact log-normality ((p \approx 0)). However, the departures are modest in absolute terms, so the log-normal remains a useful first approximation for the bulk of the distribution.

3.6 Sequential Autocorrelation

The stopping times of consecutive integers are positively correlated, as shown in Table 7.

Table 7. Autocorrelation of stopping times at selected lags.

The lag-1 autocorrelation of 0.402 is substantial and reflects a well-understood mechanism: if (n) is even, then (T(n) = n/2) and (T(n+1) = 3(n+1)+1), but the orbits of (n) and (n+1) frequently merge within a few steps, after which they share a common trajectory to 1. This trajectory sharing means that nearby integers tend to have similar stopping times. The slow decay of autocorrelation — still 0.100 at lag 100 — indicates that this effect persists over moderate ranges.

3.7 Odd vs. Even Seeds

Odd seeds have a mean stopping time of 161.46 (standard deviation 62.61), compared to 149.09 (standard deviation 60.26) for even seeds. The difference of 12.37 steps is highly significant ((t = 318.29)). This is entirely expected: an odd seed immediately undergoes the (3n+1) operation, adding at least one step before any halving can occur, whereas an even seed begins with a division by 2, immediately reducing its magnitude.

3.8 Benford's Law

The first-digit distribution of stopping times was tested against Benford's Law (the expected leading-digit distribution for data spanning multiple orders of magnitude). The chi-squared statistic is 4,934,712 with 8 degrees of freedom, emphatically rejecting conformity.

This result is entirely expected and should be understood as a negative result that confirms the nature of the distribution rather than revealing an anomaly. Benford's Law applies to data spanning several orders of magnitude, but Collatz stopping times for (n \le 10^7) are concentrated in the range roughly 50–400, barely exceeding a single order of magnitude. The leading digit 1 is massively over-represented at 56.9% (versus the Benford expectation of 30.1%), reflecting the distribution's peak in the range 100–199. The digit 3 is correspondingly under-represented at 2.1% (versus 12.5% expected). This confirms that the stopping-time distribution is too concentrated for Benford's Law to be relevant.

3.9 Record-Breaking Stopping Times

We identified 53 record-breaking stopping times in the range (1 \le n \le 10^7), where a record is defined as an (n) whose stopping time exceeds that of all smaller integers. The last record occurs at (n = 8{,}400{,}511) with (\sigma = 685).

The ratio of the index of each record to the natural logarithm of that index increases from 2.39 (for early records) to 3.29 (for late records), suggesting superlogarithmic growth of record-setting integers. This is consistent with the heuristic expectation that records should become increasingly sparse as (N) grows, but the accelerating growth ratio indicates that the spacing of records may exceed what a simple logarithmic model predicts.

4. Discussion

4.1 The Trailing 1-Bit Effect

The most striking finding of this study is the precise linear dependence of mean stopping time on the trailing 1-bit count. While it is well known that the parity sequence of a Collatz orbit determines the orbit completely, and that the low-order bits of the seed determine the first several parity choices, the quantitative regularity of the increment — approximately 6.15 steps per trailing 1-bit, stable to within (\pm 0.1) for (k = 2) through (k = 5) — has not, to our knowledge, been previously documented at this precision.

The mechanistic explanation is clear in qualitative terms: each trailing 1-bit forces one odd step ((3n+1)) followed by at least one even step ((\div 2)), a pair of operations that multiplies the iterate by approximately (3/2). The total cost of each such inflation is not merely the 2 forced iterations but includes the downstream penalty of carrying a larger iterate through the remainder of the orbit. The per-bit increment of 6.15 thus encapsulates both the direct cost (2 iterations) and the indirect cost (approximately 4.15 additional iterations to compensate for the (3/2) inflation).

Deriving the exact value of this constant from first principles would require a precise model of how the inflated iterate interacts with the subsequent pseudo-random parity sequence. The heuristic model, which treats parity choices as independent coin flips after the forced steps, predicts a cost per factor of (3/2) of approximately (2/\log_2(4/3) \approx 5.15) total iterations (counting the 2 forced ones), which is in the right ballpark but does not precisely match the observed 6.15. Closing this gap is a natural direction for future theoretical work.

The anomalous increment at (k = 6) (12.85 steps rather than the expected ~6.15) is notable. Numbers with 6 trailing 1-bits satisfy (n \equiv 63 \pmod{64}). After six forced odd–even pairs, the iterate has been multiplied by approximately ((3/2)^6 \approx 11.39), and the sample size for this group is approximately (10^7/64 \approx 156{,}000), so the deviation is unlikely to be a sampling artifact. We suspect a compounding interaction that emerges when many consecutive forced steps are chained, though a full explanation remains open.

4.2 Connection to Prior Work

The residue-class dependence of stopping times has been explored in a general sense by Lagarias (1985), who noted that the first several iterates of the Collatz map are determined by the residue class of the seed modulo a sufficiently high power of 2. Our trailing 1-bit analysis can be seen as extracting the most explanatorily powerful single feature from this residue-class structure. The ANOVA results in Table 3 confirm that the full residue-class decomposition is significant at every level, but the trailing 1-bit count alone captures the dominant axis of variation.

The approximate log-normality of stopping times is consistent with the stochastic heuristic model of the Collatz map, in which the orbit length is a sum of approximately independent steps, each contributing a random multiplicative factor. By the central limit theorem applied to the logarithm, this produces an approximately log-normal distribution. The departures we observe — negative skewness and platykurtosis — indicate that the steps are not fully independent; the forced parity structure of low-order bits introduces systematic correlations that skew and thin the distribution.

Tao's (2019) result that almost all Collatz orbits attain almost bounded values is consistent with our observation that the maximum stopping time grows only polylogarithmically (as ((\ln N)^2)) with the bound (N), rather than as a power of (N). Our empirical constant of (C \approx 2.64) refines the heuristic prediction to a specific quantitative value that may be useful for computational planning and benchmarking.

4.3 Limitations

This study is entirely empirical. No result constitutes a proof. The linearity of the trailing 1-bit effect could, in principle, break down at higher bit counts or larger (N), and the asymptotic constant for maximum stopping time growth could drift as (N) increases beyond (10^7). The range of (N) considered here, while large, covers only a small fraction of the integers for which the Collatz conjecture has been computationally verified (well beyond (10^{20}) by other investigators).

Moreover, the trailing 1-bit analysis groups integers by a single binary feature. More refined analyses — partitioning by longer bit patterns, or by the full parity vector of the first (k) iterates — could reveal additional structure. We have opted for the single most explanatory feature in the interest of clarity and interpretability.

The reproducible script accompanying this paper uses only the Python standard library and is limited by the computational resources available in a single-threaded environment. Extending the computation to (10^8) or beyond would require parallelization or compiled-language implementations.

5. Conclusion

We have presented an exhaustive empirical study of Collatz stopping times for all integers up to (10^7), yielding several quantitative findings:

1. The number of trailing 1-bits in the binary representation of the seed determines the mean stopping time with a per-bit increment of approximately 6.15 steps, stable to within (\pm 0.1) for 2–5 trailing bits. This is the most precise quantification of the binary-structural effect on Collatz dynamics that we are aware of.

2. The mean stopping time ranges from 118.13 (for multiples of 64, which are "maximally even") to 192.79 (for (n \equiv 63 \pmod{64}), which have the maximum trailing 1-bits at this modulus), a range of 74.66 steps.

3. The maximum stopping time grows as approximately (2.64 \cdot (\ln N)^2), consistent with the heuristic prediction of (O((\ln N)^2)) growth.

4. The distribution of stopping times is approximately log-normal with detectable negative skewness ((-0.37)) and platykurtosis ((-0.19)) in log-space, decisively rejecting exact log-normality (Jarque–Bera = 243,496).

5. Sequential autocorrelation is substantial ((r(1) = 0.402)) and decays slowly, reflecting shared trajectory fragments among nearby integers.

6. There are 53 record-breaking stopping times up to (10^7), with a growth ratio suggesting superlogarithmic spacing.

These findings provide quantitative benchmarks for theoretical models of the Collatz map and highlight the trailing 1-bit structure as a powerful organizing principle for understanding stopping-time variation. The accompanying reproducible script enables independent verification of all reported results.

References

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