Enhanced Size Uniformity for Near-resonant Planets

The relevant sample for the entirety of this work is 299 multiple-planet SE and SN systems within the California Kepler Survey (CKS) catalog (Weiss et al., 2018b), which provides a high-purity collection of uniformly derived planet and host parameters for a sizable subset of Kepler systems. The CKS observational program yielded substantial improvements on the stellar parameters for 1305 Kepler Objects of Interest (KOIs) via high-resolution spectroscopic follow-up (Petigura et al., 2017) with the High Resolution Echelle Spectrometer (HIRES; Vogt et al. 1994) at the W.M. Keck Observatory, allowing for the provision of 2025 planet candidates with precise radii and host parameters (Johnson et al., 2017). These measurements were further refined via the consideration of parallaxes from Gaia Data Release 2 (Fulton & Petigura 2018; Gaia Collaboration et al. 2018). Following the removal of false-positive transit signals, diluted hosts, grazing ($b>0.9$) transits, low signal-to-noise ratio (S/N; $\text{S/N}<10$) transit candidates, and potential eclipsing binaries ($R_{p}>22.4R_{\oplus}$) from the CKS-Gaia catalog, 892 high-fidelity planet candidates remained as part of 349 multiple-planet systems (Weiss et al., 2018b). Of these 349 systems, we excluded from our analysis 50 systems hosting at least one planet larger than Neptune ($R_{p}>4R_{\oplus}$), thus arriving at the 299 multiple-planet systems (containing 745 total planets) of SE and SN we consider in this work.

As a result of uniform provision in photometric and spectroscopic data for the CKS hosts provided respectively by Kepler and HIRES, the 349 CKS multiple-planet systems, and our subset of 299 systems by extension, comprise a homogeneous sample of planet parameters for which a high degree of statistical integrity may be maintained for a relevant population analysis (Weiss et al., 2018b). Such homogeneity eliminates the complex biases that may arise from differences in survey construction, instrumentation, heuristics of analysis, nonstandard treatment of error, and other such systematic discrepancies that may exist in a heterogeneous sample. The elimination of such biases within this sample, as well as the inherent level of measurement precision on planetary radii and orbital parameters, serves to greatly reduce the amount of statistical noise present in our evaluations of resonance and intra-system size uniformity, thus promoting more effective comparison within our intended population analyses.

We distinguish between the near-resonant and nonresonant subsets of our sample using the resonance parameter $\zeta_{n,j}$ from Fabrycky et al. (2014):

where $\mathcal{P}$ is the outer-to-inner period ratio of a neighboring planet pair (always larger than unity), $j$ is the order of resonance under consideration, and $(n+1)$ is the order of the adjacent resonances that set the period ratio tolerance for demarcation of the resonant "neighborhood".

Throughout this work, we will consider only proximity to first-order MMR, as these are the most dominant mode of resonance to which Kepler systems lie in proximity (Fabrycky et al., 2014). As such, we follow the prescription of Fabrycky et al. (2014) to assign period ratios to neighborhoods of first-order resonances using the $\zeta_{2,1}$ parameter in particular, for which

While the $n=1$, $j=1$ case of $\zeta_{1,1}$ may also be used to evaluate proximity to first-order resonance, the rationale for using $\zeta_{2,1}$ is that the boundaries of the resonant neighborhoods will be established by the nearest third-order resonances, and are therefore centered more closely around the first-order MMRs of interest, while second-order neighborhoods are also excluded from consideration by construction; $\zeta_{2,1}$ thus provides more effective constraints on proximity to first-order MMR while simultaneously avoiding the erroneous inclusion of systems in nearby second-order resonances, as may occur with $\zeta_{1,1}$ (Fabrycky et al., 2014). More specifically, for a given first-order MMR of index $i$, $\zeta_{2,1}$ will be equal to 0 at the exact MMR ratio $\mathcal{P}=(i+1)/i$ and will extend to values of $-1$ and 1 at the ends of its resonant neighborhood as defined by the two adjacent third-order resonances at $(3i+4)$:$(3i+1)$ and $(3i+2)$:$(3i-1)$ (i.e. the 3:2 MMR at $\mathcal{P}=1.50$ will be bounded by the 10:7 MMR, at $\mathcal{P}\approx 1.43$, and the 8:5 MMR at $\mathcal{P}=1.6$). We may also see here the general benefit of employing the $\zeta_{n,j}$ formalism instead of a simpler method, such as assessing fractional deviation from integer period ratio: while both methods appropriately scale the size of the resonant neighborhood to the specific resonance being considered, the use of $\zeta_{n,j}$ automatically eliminates contamination from higher-order resonances sufficiently close to the desired MMR, a bias for which simpler methods have no effective correction.

Given that $\zeta_{2,1}=0$ at the exact period ratios of first-order MMRs and $|\zeta_{2,1}|\leq 1$, close proximity to first-order resonance is best defined by low values of $|\zeta_{2,1}|$ such that, for some arbitrary value $\zeta_{lim}$, near-resonant systems may be effectively characterized by $|\zeta_{2,1}|\leq\zeta_{lim}$. We establish and motivate our choice of $\zeta_{lim}$ in Section 3, and discuss the impacts of altering this boundary in Section 4.

Following the methodology of Goyal & Wang (2022), we adopt the adjusted Gini index (Gini 1912; Deltas 2003), a statistic commonly used in economics to quantify income or wealth inequality in a given population, as our primary metric for the assessment of intra-system size uniformity across our sample. For a given data vector $x$ with size $N$, the standard Gini index can be calculated as follows:

which is equivalent to half of the average difference between all possible pairwise combinations of the data, normalized to the mean $\overline{x}$. The index is unit-normalized such that an ideally uniform dataset would yield $G=0$, and a maximally diverse dataset would yield $G=1$.

However, Deltas (2003) demonstrates the standard Gini index exhibits a substantial bias for which sufficiently small sample sizes ($N\lesssim 10$) will consistently overestimate ($\gtrsim 15\%$ deviation) the true degree of uniformity for a given population, an effect to which planetary systems are highly susceptible since the vast majority of Kepler architectures host a number of planets $N_{p}\lesssim 5$. To account for this intrinsic property of the metric, Deltas (2003) also defines an "adjusted" Gini index with an additional corrective prefactor:

We have demonstrated in Goyal & Wang (2022) that use of the adjusted Gini index provided by Deltas (2003) effectively neutralizes the aforementioned sample size bias within the same 299 CKS systems utilized in this work, as the mean standard Gini index of 1000 random draws of $N_{p}$ planets exhibits a $\sim 37\%$ variation over $N_{p}\in[2,5]$, while the mean adjusted Gini is successfully stabilized to within $\sim 2\%$ across the same range of planet multiplicities. We thus adopt for the entirety of this work the adjusted Gini index for all relevant calculations of planetary size uniformity.

Having established the heuristic basis for the primary metrics within our analysis, we utilize the resonance parameter $\zeta_{2,1}$ to partition our sample of 446 neighboring planetary pairs across 299 multiple-planet systems into near-resonant and nonresonant subgroups. In establishing an appropriate value of $\zeta_{lim}$ for which planet pairs with $|\zeta_{2,1}|\leq\zeta_{lim}$ are considered to be "near" resonance, we find for our sample that a boundary of $\zeta_{lim}=0.25$ corresponds to a maximum period ratio deviation from first-order MMR of 4.2% and a median deviation of 1.2%, thus providing consistency with the near-resonant tolerance level of $0.04\mathcal{P}$ from Fabrycky et al. (2014) while simultaneously imposing a more stringent evaluation of proximity to resonance.

Applying our $|\zeta_{2,1}|\leq\zeta_{lim}=0.25$ selection limit to 446 neighboring planet pairs within the 299 systems in our sample, we find that 54 pairs in 48 systems lie sufficiently close to first-order MMR to be considered near-resonant while the remaining 392 pairs comprise the corresponding collection of nonresonant pairs. We illustrate this partitioning of the global $\zeta_{2,1}$ distribution in the left panel of Figure 1, where the pink and black (hatched) portions of each histogram respectively correspond to groupings of near-resonant and nonresonant pairs. We find a notable correspondence in the near-resonant portion of our sample as compared to the analogous distribution from Fabrycky et al. (2014), where we observe an identical spike from $-0.2\lesssim\zeta_{2,1}\lesssim-0.1$, which, according to the construction of the resonance parameter given by Equation 2, further confirms the preference for near-resonant Kepler systems to lie slightly wide of their corresponding perfect-integer MMR period ratios (Delisle et al. 2012; Fabrycky et al. 2014). We map both sets of planet pairs onto period ratio space in the right panel of Figure 1, where the dashed lines indicate the perfect MMR period ratios around which pairs were found in proximity (all first-order resonances up to 6:5). We observe perhaps more readily here the aforementioned preference for asymmetric distributions about the first-order MMR period ratios, as well as the reduction in relative size of the resonant neighborhoods with increasing resonant index (note that the histogram bin width is itself broader than the entire resonant neighborhood for the 4:3, 5:4, and 6:5 MMRs), thus demonstrating the efficacy of the $\zeta_{2,1}$ parameter in assessing close proximity to MMR even for tightly spaced, higher-index resonances.

We calculate the size Gini index for each of the 446 neighboring planet pairs in our sample and assess the behavior of pair size uniformity across the resonance partition illustrated in Figure 2. In performing a nonparametric comparison of the two Gini distributions, we employ the two-sample Anderson-Darling (AD) test due to its greater efficacy in evaluating discrepancies at the distribution tails than the similar two-sample Kolmogorov-Smirnov test. In the context of our sample, the AD test is thereby more sensitive in assessing whether the near- and nonresonant populations differ in their respective propensities for extremely high- or low-uniformity pairs. Applying the AD test to our two distributions, we obtain a probability of $p=0.009$ for emergence from a common parent distribution, indicating a statistically significant level of discrepancy. The near-resonant pairs are entirely contained within $\mathcal{G}_{R}\leq 0.38$, a value above which 10% of the nonresonant pairs reside. The near-resonant pairs also exhibit specifically strong clustering near the highest-uniformity values of $\mathcal{G}_{R}\leq 0.1$, with 57% of near-resonant pairs residing within this regime compared to 43% of nonresonant pairs. Additionally, we calculate the outer-to-inner planetary size ratios for either pair classification, recovering respective distributions of $R_{p,i+1}/R_{p,i}=1.08\pm 0.33$ and $1.14\pm 0.56$ for the near-resonant and nonresonant pairs. We see that, while both median size ratios are slightly larger than unity, illustrative of first-order consistency of either population with observed trends of intra-system size uniformity and size ordering (Weiss et al., 2018a), the dispersion of the nonresonant population is $\sim 70\%$ greater than that of its near-resonant analog, suggesting that the latter may harbor a greater occurrence of especially highly uniform planetary pairs.

To perform a more robust and objective assessment of a possible relationship between pairwise size uniformity and distance from MMR, we briefly disregard the imposition of a discrete $\zeta_{lim}$ partition and consider instead a means of fully bivariate evaluation of the $\zeta_{2,1}-\mathcal{G}_{R}$ space across the total pair sample. To remove potential biases that may result from assumptions of Gaussianity in the pairwise Gini distribution or linearity between the two variables under consideration, we conduct a nonparametric analysis of our sample using a constrained B-spline (COBS) smoothing algorithm (Ng & Maechler 2007, 2022), from which we construct smoothed 10th, 25th, 50th, 75th, and 90th percentile regression curves (Fig. 2, gray lines) for the total $\zeta_{2,1}-\mathcal{G}_{R}$ distribution. While the COBS contours are less suggestive of strong variation in the high-uniformity regime, with the bottom half of Gini distribution generally contained within $\mathcal{G}_{R}\lesssim 0.1$ across the full range of $|\zeta_{2,1}|$, they maintain strong support for discrepancies in the high-uniformity region of the distribution, with the 75th percentile curve for $|\zeta_{2,1}|\gtrsim 0.25$ lying near or above the 90th percentile curve for $|\zeta_{2,1}|\lesssim 0.25$ for a large section of the relevant domain ($0.3\lesssim|\zeta_{2,1}|\lesssim 0.8$), while the median value of the 90th percentile curve for $|\zeta_{2,1}|>0.25$ $(\mathcal{G}_{R}=0.35)$ is nearly identical to the maximum Gini value achieved for $|\zeta_{2,1}|<0.25$ $(\mathcal{G}_{R}=0.38)$. Most prominently, we observe that, even in the absence of a defined near-resonant partition within the sample, there still exists a strong qualitative augmentation in the Gini values of the 75th and 90th percentile curves with increasing $|\zeta_{2,1}|$, indicating that, regardless of any prescribed classification scheme, the most nonuniform planetary pairs tend to favor configurations far from MMR.

We hereafter classify an entire planetary system as near-resonant if it contains at least one of the 54 planetary pairs with $|\zeta_{2,1}|\leq 0.25$, and analogously deem a whole system as nonresonant if all of its constituent planet pairs have $|\zeta_{2,1}|>0.25$. We thus sort our 299 CKS systems into 48 near-resonant and 251 nonresonant systems, and plot the corresponding size Gini index distributions (unit-normalized histogram and cumulative distribution function, hereafter CDF) for either population in Figure 3. As performed for the distributions of the individual planetary pairs, we apply the two-sample AD test to the Gini distributions for our near- and nonresonant systems, obtaining a probability of $p=0.079$ for incidence from a common parent distribution. While this value itself falls slightly above the conventional threshold for statistical significance, it is still representative of a level of discrepancy that manifests as readily identifiable qualitative differences between the distributions in question. In similar fashion to the aforementioned dichotomy between near- and nonresonant planetary pairs, we observe immediately that the distribution of near-resonant systems is far more pronounced at the low-Gini end, with 38% and 81% of near-resonant systems contained respectively within $\mathcal{G}_{R}=0.1$ and $\mathcal{G}_{R}=0.2$, compared to 28% and 65% of nonresonant systems. We note also for the near-resonant distribution the suppression of a high-Gini tail that appears to be present in the nonresonant sample, evidenced by the 75th ($\mathcal{G}_{R}=0.17$) and 90th ($\mathcal{G}_{R}=0.24$) percentile Gini indices of the near-resonant systems lying far below the analogous values ($\mathcal{G}_{R}=0.24$ and $\mathcal{G}_{R}=0.34$) for the nonresonant systems. We thus recover a tendency for near-resonant planetary systems to provide greater occupation of the low-Gini, high-uniformity regime while exhibiting a markedly lower propensity for highly heterogeneous planet sizes.

We note that the metrics and statistical results presented thus are intended only as intuitive and heuristically simple suggestive evidence for the heightened propensity of near-resonant planetary pairs and systems to be more uniform in size than their nonresonant counterparts. A more detailed and statistically rigorous investigation of the same trends will serve as the subject of the remainder of this section.

We shall first verify that the populations of near-resonant and nonresonant systems individually exhibit intra-system size uniformity as compared to their respective random expectation, for which we subject either group to the null hypothesis procedure featured in Goyal & Wang (2022): For a given sample of systems, we compute the adjusted size Gini $\mathcal{G_{R}}$ of each system and compare the sum of these values across our real sample to a distribution of analogous summed Gini values for $10^{4}$ mock samples wherein all individual planetary radii were randomly shuffled (without replacement) across the sample itself. For our 48 near-resonant and 251 nonresonant systems, we find that intra-system planetary size uniformity is present against a null hypothesis of purely random assortment with respective significance values of $6.4\sigma$ and $4.8\sigma$, thereby indicating that system-level peas-in-a-pod size uniformity is indeed present within either population.

To ascertain the existence of a possible discrepancy in the strength of this size uniformity between the two populations, we employ a null hypothesis test similar in construction to the analyses presented by Goyal & Wang (2022) and Millholland et al. (2017). The null hypothesis in question is the assumption that size uniformity is wholly independent of proximity to first-order resonance, and may consequently be assessed via direct comparison of the aggregate uniformity across the near-resonant population to that of random, equally sized subsamples of the nonresonant population. As such, we measure the total size Gini index ($\sum\mathcal{G}_{R}$) for our 48 near-resonant systems against a control distribution of $\sum\mathcal{G}_{R}$ values generated for $10^{5}$ subsamples of 48 systems drawn randomly (without replacement) from the 251 systems in the nonresonant population.

Illustrating our results in Figure 4, we see that the total size Gini index of the near-resonant population ($\sum\mathcal{G}_{R}=6.28$, black line) falls well below the vast majority of the values which comprise the nonresonant control distribution ($\sum\mathcal{G}_{R}=8.03\pm 0.74$, pink histogram). Consequently, the null hypothesis is rejected at the level of $\sim 2.4\sigma$ confidence, thus consistent with enhanced intra-system size uniformity for near-resonant systems that holds a $<1\%$ probability of incidence from the nonresonant population.

Given that all near-resonant systems must, by definition, contain at least one planetary pair with period ratio $\mathcal{P}\lesssim 2$, while nonresonant systems have no such restriction, we shall attempt to verify that the dichotomy characterized by our result is not significantly confounded by physical differences between closely-spaced and more distant pairs, perhaps owing to varied degrees of similarity in their formation environment. To this end, we repeat our experiment for only the 110 systems wherein all constituent planetary pairs are at least as closely spaced as the widest near-resonant pair in our sample ($\mathcal{P}<2.1$). We find that the 35 near-resonant systems therein still display enhanced size uniformity with $2.2\sigma$ significance ($<1.5\%$ probability of chance occurrence) as compared to the 75 nonresonant systems of similar compactness, thereby affirming that our result is not driven by differences in pair spacing between the two sub-populations.

Given this level of statistical significance that may be attributed to the enhanced size uniformity of near-resonant systems within the CKS sample, we briefly consider here the possible means by which the observed strength of this trend itself may be bolstered for a larger overall sample. Operating under the assumptions that the intrinsic fraction of near-resonant multiple-planet systems ($f_{res}\approx 0.17$), as well as the true size Gini distributions for both near- and nonresonant systems, may be modeled adequately by the analogous quantities within the CKS sample thus considered, we attempt to determine the characteristic sample size $N_{3\sigma}$ at which the experimental procedure described in Figure 4 would reject its null hypothesis with $3\sigma$ significance. For various trial sample sizes $N_{sys}$, we respectively draw, at random, $N_{sys}f_{res}$ and $N_{sys}(1-f_{res})$ values from the near- and nonresonant CDFs in Figure 3 to serve as mock samples for either group, for which we then apply our null hypothesis test to obtain a corresponding significance value. We then perform 1000 bootstrap iterations of this process at each value of $N_{sys}$ and assign to $N_{3\sigma}$ the sample size for which the median significance level is equal to $3\sigma$, obtaining a nominal sample size of $N_{3\sigma}\approx 440$. While such a $\sim 50\%$ increase in sample size may be obtained, in principle, by the inclusion of other candidate systems to the sample considered here, we shall remark once again that the primary motivation for limiting our analysis to the CKS dataset alone is the especially high degree of statistical purity for the data therein, emergent from an extensive spectroscopic follow-up campaign, refinement with supplemental astrometry, and homogeneity in observational systematics and parameter estimation techniques (Johnson et al. 2017; Petigura et al. 2017; Fulton & Petigura 2018). As such, the CKS dataset remains at present the most suitable sample for the analyses considered thus, though the provision of a larger sample of multiplanet systems, subject to similarly detailed and homogeneous follow-up observations, may indeed be achieved with the Transiting Exoplanet Survey Satellite, or with the European Space Agency’s PLAnetary Transits and Oscillations of stars mission.

Having established the statistical prevalence of enhanced size uniformity for systems containing at least one near-resonant planetary pair compared to those devoid of such pairs, we now consider potential differences in the degrees of size uniformity for near-resonant and nonresonant substructures within these systems themselves. The question remains of whether the trend elicited by the system-level comparison in Figure 4 is the result of enhanced size uniformity for all planets in a near-resonant systems, including its constituent nonresonant pairs, or enhanced size uniformity that is exclusive to near-resonant pairs and chains alone. In order to ascertain which of these modes is dominant, we perform here three modified versions of the null hypothesis testing scheme described in Section 3.3.

We first probe the possibility of inherently discrepant degrees of size uniformity between near-resonant and nonresonant planetary pairs within the same system (Figure 5, left panel). Considering from our sample the 27 near-resonant systems with an equal or greater number of nonresonant pairs as near-resonant pairs, we sum the size Gini indices of the individual resonant pairs within each system, then summing across all 27 systems to obtain a total Gini index for the near-resonant pairs (black line). We then randomly select from each system nonresonant pairs equal in number to the near-resonant pairs of the same system, repeating the summation process to obtain nonresonant control values. This random selection process allows for 2304 possible combinations of nonresonant pairs between the 27 systems considered, the corresponding total Gini values of which serve to comprise our nonresonant control distribution (pink histogram). We see that the total size Gini of the near-resonant pairs ($\sum\mathcal{G}_{R}=3.22$) lies below the nonresonant control distribution ($\sum\mathcal{G}_{R}=3.60\pm 0.17$) at the $\sim 2.2\sigma$ level ($<1.4\%$ probability of chance occurrence), providing evidence for the enhanced size uniformity of near-resonant planetary pairs compared to nonresonant pairs within the same system.

We then compare the respective degrees of intra-system uniformity for the nonresonant components of near-resonant systems and systems that are entirely nonresonant (Figure 5, central panel). We determine the nonresonant portions of our near-resonant systems via the removal of individual planets belonging to resonant pairs, and in order to preserve the greatest integrity of extant architectures and prevent the creation of nonphysical gaps in orbital structure, we exclusively consider the systems that lose their resonant signature from the removal of only their innermost or outermost planet(s). This removal process results in 23 systems rendered nonresonant from this truncation, which are then compared to the 251 naturally nonresonant systems in our total sample via the same null hypothesis process described in Section 3.3. We see that the total size Gini of the 23 truncated systems ($\sum\mathcal{G}_{R}=3.20$) displays consistency with the nonresonant control distribution ($\sum\mathcal{G}_{R}=3.84\pm 0.54$) within $\sim 1.2\sigma$, suggesting that the nonresonant components of near-resonant systems do not exhibit any statistically significant enhancement in their size uniformity.

Finally, we perform an experiment complementary to the previous test by assessing intra-system size uniformity for only the near-resonant components of near-resonant systems (Figure 5, right panel). For all 48 near-resonant systems, we remove all nonresonant neighboring pairs and compare the collection of remaining near-resonant pairs or chains in each system to the 251 nonresonant systems via the same null hypothesis test from Section 3.3. We find that the total Gini index for these resonant-only configurations ($\sum\mathcal{G}_{R}=5.46$) lies beneath the nonresonant control distribution ($\sum\mathcal{G}_{R}=8.03\pm 0.74$) with $\sim 3.5\sigma$ confidence ($<0.03\%$ probability of chance occurrence), implying that the near-resonant pairs or chains within a given system exhibit a greater enhancement in their size uniformity than the same system considered in its entirety.

Similar to the heuristic argument presented at the beginning of Section 3.3, we subject our 23 systems rendered nonresonant via truncation (5, center) and our 48 near-resonant systems with nonresonant planets removed (5, right) to the null hypothesis procedure featured in Goyal & Wang (2022) to ensure that either sample maintains intra-system size uniformity as compared to random expectation. We find that the two samples respectively demonstrate system-level peas-in-a-pod size uniformity with $3.8\sigma$ and $6.0\sigma$ significance, thereby indicating that intra-system size uniformity is indeed maintained regardless of the presence of near-resonant pairs, while the enhancement of this uniformity in a given sample is modulated by such pairs.

We note that recent work by Millholland & Winn (2021) has demonstrated the existence of intra-system architectural substructure amongst planets on either side of the radius valley, where isolated treatment of either SE or SN within a given system yields size uniformity twice as strong as that displayed by the system considered globally. Accordingly, we wish to ascertain if the differences in pairwise size uniformity thus presented are themselves subject to confounding effects from these split peas-in-a-pod architectures, the most prominent evidence for which would present as a higher intrinsic rate of homogeneity in classification (SE-SE or SN-SN) for near-resonant pairs as compared to nonresonant pairs. As performed in Millholland & Winn (2021), we classify each planet in our sample as a SE or SN based on its residence below or above the radius valley boundary parameterized by Van Eylen et al. (2018) as $\log_{10}(R_{p}/R_{\oplus})=m\log_{10}(P\text{days})+a$, with $m=-0.09^{+0.02}_{-0.04}$ and $a=0.37^{+0.04}_{-0.02}$. We perform 1000 bootstrap iterations of this classification, where the values of $m$ and $a$ are resampled uniformly within their $1\sigma$ errors. Across this process, we find that $68.5^{+5.6}_{-2.1}\%$ of the near-resonant pairs and $62.5^{+2.1}_{-1.5}\%$ of the nonresonant pairs are of homogeneous planetary type. Since these rates are consistent within the $2\sigma$ level, even when uncertainties are limited to the empirical scatter of the classification scheme itself, we affirm that the enhancement in near-resonant pairwise size uniformity presented in this work is not a byproduct of an augmented rate of homogeneity in planetary type, and is most likely driven primarily by proximity to first-order MMR.

Taken together, the results of the tests presented in Figures 4 and 5 may be best summarized as the following: although multiplanet systems demonstrate intra-system size uniformity regardless of their proximity to resonance, near-resonant systems display an enhancement in their size uniformity that is primarily driven by enhanced similarity of planetary pairs and chains close to first-order MMR. We expound upon the astrophysical underpinnings of these findings in Section 4.2, specifically in terms of their implications for the dynamical evolution of multiple-planet systems.

Our primary results thus strongly support the notion that near-resonant and nonresonant configurations are distinct in their provision of planetary size uniformity, but given that these results are inexorably linked to certain assumptions and prescriptions inherent to the construction of our analysis, it remains to be seen whether the emergence of this dichotomy is truly indicative of physical differences between the two populations, and is not itself a statistical artifact or product of bias. As such, we shall perform here a series of additional tests to demonstrate that our methodology is statistically and heuristically robust, and to further validate the notion that the emergence of enhanced size uniformity is primarily dependent upon physical proximity to first-order MMR.

Having thus validated the statistical prevalence of enhanced intra-system size uniformity for near-resonant pairs and systems as compared to their nonresonant counterparts, we shall now attempt to provide insight towards the dynamical and evolutionary mechanisms out of which the observed architectural phenomena may emerge for either configuration.