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From Stochastic to Stable: Rank Stability and Structural Sufficiency in AI Visibility Measurement

Ronald Sielinski

Year: 2026Venue: arXiv preprintArea: stat.APType: PreprintEmbeddings: 103

Intelligence

Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 96%

Last extracted: 7/18/2026, 11:59:20 AM

Summary

The paper introduces a sequential convergence framework for AI visibility measurement, addressing the lack of principled methods to determine when sufficient data has been collected for comparative analysis. It proposes two criteria: rank stability, which evaluates if the rank-correlation trajectory has reached a structural plateau, and structural sufficiency, which assesses if the spread of citation shares among established domains exceeds measurement uncertainty. Applied across 30 platform-topic combinations (Gemini, SearchGPT, Perplexity), the framework demonstrates that fixed collection budgets are unjustified and that convergence can be evaluated based on the structure of the observed citation distribution.

Entities (10)

Gemini · platform · 99%Perplexity · platform · 99%Ronald Sielinski · person · 99%SearchGPT · platform · 99%Sequential Convergence Framework · methodology · 98%Rank Stability · metric · 97%Structural Sufficiency · metric · 97%citation share · metric · 95%Bootstrap Confidence Intervals · statistical-method · 94%Power-Law Distribution · concept · 92%

Relation Signals (9)

Gemini → testedwith → Sequential Convergence Framework

confidence 99% · Applied across 30 platform-topic combinations spanning Gemini, SearchGPT, and Perplexity

SearchGPT → testedwith → Sequential Convergence Framework

confidence 99% · Applied across 30 platform-topic combinations spanning Gemini, SearchGPT, and Perplexity

Perplexity → testedwith → Sequential Convergence Framework

confidence 99% · Applied across 30 platform-topic combinations spanning Gemini, SearchGPT, and Perplexity

Sequential Convergence Framework → uses → Structural Sufficiency

confidence 98% · We introduce a sequential convergence framework based on two complementary criteria: rank stability... and structural sufficiency

Sequential Convergence Framework → uses → Rank Stability

confidence 98% · We introduce a sequential convergence framework based on two complementary criteria: rank stability... and structural sufficiency

Structural Sufficiency → evaluates → spread of citation shares

confidence 95% · structural sufficiency evaluates whether the spread of citation shares among established domains... exceeds the uncertainty

Rank Stability → evaluates → rank-correlation trajectory

confidence 95% · rank stability evaluates whether the rank-correlation trajectory has reached a structural plateau

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Abstract

Abstract:AI visibility measurement is comparative: practitioners want to know which domains generative search engines cite most often and whether observed differences are large enough to support decisions. Yet the industry lacks a principled way to determine whether enough data has been collected. Collection budgets vary widely across studies and platforms, and conclusions are often drawn from rankings whose stability and precision are unknown. We introduce a sequential convergence framework based on two complementary criteria: rank stability evaluates whether the rank-correlation trajectory has reached a structural plateau, while structural sufficiency evaluates whether the spread of citation shares among established domains -- those whose confidence intervals exclude zero -- exceeds the uncertainty of those estimates. Together, these criteria distinguish rankings that have merely stabilized from those sufficiently resolved to support inference. Both are derived from regularities in the observed citation distribution, including its rank structure, uncertainty profile, and the boundary between observed and established domains. The framework retains a small number of structural constants but requires no externally specified query count, correlation target, or confidence-interval width target; stopping is driven by observed measurement uncertainty and remains robust across a range of sufficiency thresholds. Applied across 30 platform-topic combinations spanning Gemini, SearchGPT, and Perplexity, the framework adapts to platform- and topic-specific citation distributions. Results show that no fixed collection budget can be justified across contexts and that convergence can instead be evaluated from the structure of the observed distribution. The framework provides a practical basis for determining when AI visibility measurements are ready to support comparative analysis.

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From Stochastic to Stable Rank Stability and Structural Sufficiency in AI Visibility Measurement Ronald Sielinski IQRush ron@iqrush.ai Abstract AI visibility measurement is fundamentally a comparative exercise: practitioners want to know which domains generative search engines cite most often and whether observed differences are large enough to support competitive decisions. Yet the industry lacks a principled way to determine whether enough data has been collected to support those comparisons. Collection budgets vary widely across studies and platforms, and conclusions are often drawn from rankings whose stability and precision are unknown. We introduce a sequential convergence framework based on two complementary criteria: rank stability evaluates whether the rank-correlation trajectory has reached a structural plateau, and structural sufficiency evaluates whether the spread of citation shares among established domains (those whose citation-share confidence intervals exclude zero) exceeds the uncertainty of those estimates. Together, these criteria distinguish rankings that have merely stabilized from rankings that are sufficiently resolved to support inference. Both criteria are derived from regularities already present in the observed citation distribution, including its rank structure, uncertainty profile, and the boundary between observed and established domains. The framework retains a small number of structural constants but requires no externally specified query count, correlation target, or CI-width target; the stopping decision is driven by directly observed measurement uncertainty and is robust across a range of sufficiency thresholds. Applied across 30 platform-topic combinations spanning Gemini, SearchGPT, and Perplexity, the framework adapts automatically to platform- and topic-specific citation distributions. The results demonstrate that no fixed collection budget can be justified across contexts and that convergence can instead be evaluated from the structure of the observed citation distribution itself. The framework provides a practical basis for determining when AI visibility measurements are ready to support comparative analysis. CCS Concepts: Information systems → Information retrieval → Evaluation of retrieval results; Information systems → World Wide Web → Web searching and information discovery. Additional Keywords and Phrases: generative search, generative search optimization, answer engine optimization, AI visibility. 1 Introduction The purpose of AI visibility measurement is comparative analysis. Practitioners want to know which domains a generative search engine cites most often, how that standing changes over time, and whether observed differences are large enough to matter for competitive strategy. The goal is not the measurement itself, but the decisions that it is meant to enable. The industry quickly converged on a multi-query process for sampling multiple aspects of a topic and measuring how often a brand’s web pages are cited (or its name is mentioned) in generative search responses. But the industry has paid far less attention to a critical question: whether the resulting measurements yield enough data to support the comparisons they are used to make. Collection budgets differ by nearly an order of magnitude across published studies and industry practice, with no principled basis for choosing one over another. Comparative conclusions are routinely drawn from measurements that may not be adequate to support them. The core difficulty is that citation share is a sample statistic, not a fixed quantity. Generative search engines are stochastic: the same query submitted on different occasions can produce different responses and cite different sources. A multi-query design is intended to average over prompt-level idiosyncrasies, but it also aggregates many variable outputs, so repeating the same measurement process will likely produce different results even when nothing meaningful has changed. Citation counts and shares are therefore random variables, and the rank ordering derived from them inherits the same variability. When adjacent domains have similar citation metrics, their relative ranks may remain statistically unresolved even after substantial data collection. Our prior work (Sielinski, 2026) established these properties empirically. The question this paper addresses is what follows from them: how much data must be collected before rank-based conclusions can be trusted? The answer requires two conditions to hold simultaneously. The first is rank stability: the rank ordering must have settled. During early collection, the ordering shifts substantially with each additional response, because the domains that will eventually become top-cited have not yet established consistent leads. Because generative engines cite only a handful of sources per response—SearchGPT, for example, cites between 4 and 6—the eventual leaders may not appear at all in the first several responses. Stability emerges gradually as cumulative citation counts begin to reflect the underlying distribution. The second condition is sufficiency: even after rank order has stabilized, the measurement must be precise enough to support the comparisons it is used to make. A stable rank ordering can still be too imprecise to act on—if the domain ranked first and the domain ranked second have nearly identical underlying shares, the ordering does not reliably distinguish winners from runners-up. Sufficiency requires that the spread of citation shares among the reliably present domains exceed the typical measurement uncertainty—that the signal be at least as large as the noise. Both conditions depend on the structure of the underlying citation distribution, which varies substantially across platforms and topics. Citation density differs by more than sixfold between platforms; the concentration of the distribution and the rate at which rank order stabilizes follow suit. No fixed collection budget can be justified: the amount of data required depends on the structure of the citation distribution, not on a schedule set in advance. A fixed rank-correlation threshold fails for the same reason: convergence response orders under a fixed ρ ≥ 0.95 criterion range from 40 responses to never across the 30 platform-topic combinations studied here, and no platform converges at a rate that is consistently fastest or slowest. The framework this paper introduces avoids external calibration by deriving both criteria directly from the structure already present in the data. Neither criterion requires the analyst to specify a target rank correlation, a desired confidence interval width, or a predetermined query count. Rank stability is detected as a structural plateau in the rank correlation trajectory, identified by model selection rather than level crossing. Sufficiency is established when the spread of established-domain shares exceeds the mean measurement uncertainty—a ratio whose threshold follows from the geometry of signal and noise rather than from an external specification. The same criterion automatically demands more data wherever citations accrue slowly or established shares are tightly packed. The key to this self-calibration is a consistent structural property of generative citation distributions. Citation distributions in generative search engines are broadly consistent with a power-law form: a small number of highly cited domains accumulate citations at an elevated rate while the great majority appear sporadically. This pattern holds across all platforms and topics studied here; what varies are the parameters—citation density, power-law exponent, noise boundary, convergence rate. Because both criteria are expressed in terms of observed distributional features—rank trajectory, citation density, share spread, and empirical uncertainty—the framework calibrates itself to the citation distribution it encounters. The questions this paper addresses fall within the scope of sequential analysis, the branch of statistics concerned with data collection protocols that adapt based on accumulated evidence (Wald, 1947). Sequential estimation treats sample size as a random outcome rather than a fixed input, stopping when a specified precision criterion has been met. The convergence framework developed here instantiates this logic for citation visibility measurement. The remainder of this paper is organized as follows. Section 2 provides background on citation visibility metrics as random variables, the power-law structure of citation distributions, and the two manifestations of rank uncertainty that motivate the convergence framework, and closes with a review of related work. Section 3 describes the dataset and methods, including bootstrap CI estimation, power-law segmentation, the rank stability criterion, and the structural signal-to-noise ratio that operationalizes sufficiency. Section 4 presents results across 30 platform-topic combinations: distributional structure, rank stability trajectories and convergence response orders, rank confidence interval widths at full sample size, the conjunctive convergence results, and a validation analysis comparing stopped orderings to end-of-window orderings. Section 5 discusses implications and limitations, and Section 6 concludes. 2 Background 2.1 Citation Visibility Metrics as Random Variables In our prior work, we established empirically that citation visibility metrics are random variables, not fixed values (Sielinski, 2026). The same queries submitted to the same platform on different occasions produce different cited source sets: platform-level Jaccard similarity across repeated runs of the same query is approximately 0.30 for Gemini, 0.40 for SearchGPT, and 0.50 for Perplexity, meaning that even the most consistent platform shares only half its cited domains across repeated samples. This is a structural property of generative systems that use stochastic sampling at generation time, not measurement error to be corrected. The practical consequence is that the citation share for a domain, defined as the domain’s citation count divided by the total citations across all responses, is a sample statistic with quantifiable uncertainty. Its variance under a multinomial sampling model is approximately θ(1−-θ)/M, where θ is the true underlying citation probability and M is the total citation count, so CI width scales as 1/ xM. Each doubling of data yields only approximately 29% improvement in CI width. The law of diminishing returns applies directly. Bootstrap confidence intervals, computed by resampling at the response level and recomputing share for each replicate, make this uncertainty explicit. Our prior work found that these intervals are wide relative to the differences between domains: for most frequently cited domains on SearchGPT, 95% CI spans range from 3 to 6 percentage points, and apparent differences between many adjacent domains are statistically indistinguishable from sampling noise. A domain reported at 9.5% citation share and another at 6.0% may have confidence intervals that overlap substantially, so the data cannot rule out that the second domain’s true share exceeds the first’s. This finding has a direct implication for rank that the prior paper did not explore. If the share estimates underlying a rank ordering carry uncertainty wide enough to make adjacent domains indistinguishable, then the rank ordering itself carries uncertainty that a single point estimate of rank cannot capture. Domain A may be ranked first and domain B second in a single sample, while the bootstrap distributions of their shares overlap substantially. The rank is real, but the uncertainty around it is also real. In a repeated sample, B could easily rank above A. Ignoring that uncertainty produces the same false precision as reporting share without a confidence interval. 2.2 Two Manifestations of Rank Uncertainty The tension between rank’s utility and its uncertainty manifests in two distinct ways, corresponding to two phases of measurement. The first is rank instability during collection. The domains that will ultimately be most frequently cited need time to accumulate enough citations to establish themselves above the competition. During early collection, the first 20 to 60 queries, depending on topic and platform, the rank ordering changes substantially with each additional response, and the confidence intervals of the top-cited domains overlap broadly enough that many apparent rank differences are statistically indistinguishable. This is a structural feature of how citations accumulate, not a transient artifact: top-cited domains establish themselves faster than mid-tier domains, but not instantaneously. Reporting ranks from an insufficient sample produces a misleading picture of competitive standing, because the ordering has not yet stabilized. The second is rank uncertainty in a stable distribution. Even after rank order has stabilized, adjacent domains in the dense middle of the distribution can exchange positions across successive samples without either showing a statistically significant change in share. The distinction matters: stability concerns whether the ordering is still changing over collection; uncertainty concerns whether it is precise enough at a fixed collection size to support inference. A domain that moves from rank 10 to rank 20 between two measurement periods may have undergone a meaningful competitive shift, or it may be oscillating within the band of sampling noise that is indistinguishable from stability at its share level. Distinguishing these cases requires tools that go beyond a point estimate of rank and beyond a share-based significance test. Figure 1 illustrates three patterns that appear consistently across topics and platforms. Rank positions among the top-five domains shift frequently during the first 20 to 40 responses on every platform—the early citation landscape is volatile. Even after rank order stabilizes, confidence intervals for individually ranked domains continue to overlap, so platform preference among similarly-ranked domains remains statistically ambiguous. Measurable uncertainty persists even in the most stable configurations: observed ranks are point estimates, not fixed readings. Figure 1: Citation share accumulation over response order for the five domains that finish with the highest citation share at end of collection. Each line tracks a domain’s cumulative citation share as responses accumulate; the shaded bands represent the domain’s share CI. Histories are shown only for the domains that eventually rank in the top five; earlier in collection these domains may have been ranked below their final position, which is not shown. 2.3 Power-Law Structure and What It Means for Rank Citation count distributions follow an approximate power-law form, the rank-frequency counterpart of Zipf’s Law (Zipf, 1949), although the fitted parameters vary by platform and topic. On log-log axes, citation counts decline roughly linearly with rank: a small number of highly cited domains account for a disproportionate share of citations, followed by progressively less-cited domains down the ordering. Our prior work documented this structure across multiple platforms and found the overall shape to be stable across repeated samples, even when individual domain positions changed between draws. The existence of sampling variability therefore does not imply the absence of structure; rather, the rank fluctuations of the preceding section occur within an underlying distributional form that is itself comparatively consistent. What matters here is not the internal structure of any single fit but the variation across fits. The fitted exponent and the citation density differ substantially across platform-topic combinations, and those differences are what make the recurring pattern of the preceding section (early rank instability, then persistent confidence-interval overlap even among top-ranked domains) resolve at different rates. The power-law fit therefore serves as descriptive background that quantifies how citations concentrate across platforms and topics; it is not the basis for any stopping decision. The convergence criterion derives the established domain set directly from bootstrap confidence intervals rather than from a fitted boundary (§3.4), and the variation in fits is precisely why no single collection budget transfers across contexts. 2.4 Contributions of This Paper This paper develops a principled framework for determining when enough data has been collected for rank-based citation visibility to be trustworthy. We make three contributions. First, we establish rank stability during collection as a convergence criterion: using weighted Spearman rank correlation tracked across response order, we show when the citation rank ordering has stabilized sufficiently to support reporting and characterize how that threshold varies by platform and topic. Second, we show that even after rank order stabilizes, confidence intervals around rank positions remain wide relative to the differences between adjacent domains; because CI width shrinks only with the square root of sample size, continued collection closes that gap slowly and at diminishing returns. Third, we introduce structural sufficiency as a principled stopping criterion: a signal-to-noise ratio that compares the spread of citation shares across the established domain set (domains whose citation-share CI excludes zero) to their mean CI half-width, providing a data-driven answer to the cost-benefit tradeoff between collection effort and inferential precision. 2.5 Related Work Stochastic citation systems. Citation visibility in generative search has attracted growing empirical attention as answer engines have displaced traditional search results for many queries. Gao et al. (2023) established that LLM-generated answers can be evaluated for citation quality independently of generation quality, formalizing the grounded-generation task that underlies any systematic citation measurement. Empirical audits have documented that citation patterns are stochastic, platform-specific, and concentrated in ways traditional search metrics do not predict: Li and Sinnamon (2024) found geographic and commercial bias in citation patterns across ChatGPT, Bing Chat, and Perplexity across repeated sampling, and Zhang et al. (2025) showed that citation distributions in LLM-based search engines are substantially more concentrated than those of traditional search, with 37% of cited domains unique to LLM-based systems. That concentration pattern directly motivates the distributional analysis in §3.4: rank comparisons carry meaning only over the subset of domains that appear reliably, rather than those observed only incidentally. Mökander et al. (2023) argue that application-level LLM audits must be systematic and reproducible, a requirement the convergence criterion developed here is designed to satisfy. Rank stability and uncertainty. The use of rank as a visibility metric connects this work to a long tradition in information retrieval. Järvelin and Kekäläinen (2002) established that rank position carries graded importance, the foundational motivation for any rank-based visibility metric, and Moffat and Zobel (2008) showed that rank-based retrieval metrics must account for experimental uncertainty and the elevated importance of top positions. Webber et al. (2010) introduced rank-biased overlap (RBO), a probabilistic similarity measure for ranked lists that weights top-rank agreement and accommodates non-conjoint lists, which is relevant here because different platforms frequently cite different domains entirely. The rank stability criterion developed in this paper is methodologically adjacent: it uses weighted Spearman correlation rather than RBO, but similarly concentrates the stability signal at the top of the distribution. Vigna (2015) generalized weighted rank correlation to handle ties, a necessary extension when many domains share identical or near-identical citation counts, as is common in the citation tail. Sequential stopping rules. The problem of determining when enough data has been collected falls within sequential analysis, where sample size is treated as a random outcome determined by accumulated evidence rather than a fixed input (Wald, 1947). Chow and Robbins (1965) formalized the fixed-width sequential confidence interval, which stops when a CI has narrowed to a specified target width, the direct methodological antecedent of the sufficiency criterion developed here. Neither the classical sequential analysis literature nor the retrieval evaluation literature has addressed the convergence problem specific to citation-share distributions: the joint determination of rank stability and measurement precision in a stochastic, power-law distributed system with substantial between-platform variation in effective sample size. More broadly, both criteria are calibrated to the data’s own structure rather than to externally specified targets: the plateau detector requires no target correlation level, and the SNR criterion requires no target CI width. The framework discovers whether enough data has been collected from the properties of the data itself. 3 Methods 3.1 Data The analyses reported here use a new dataset collected under the same protocol as our prior paper (Sielinski, 2026). The dataset is based on a set of ten topics that span a range of market segments: consumer goods, travel, and medical devices, among others. For each topic, 125 queries were generated by prompting an LLM (ChatGPT) to produce queries conditioned on a randomly selected query type drawn from a fixed set of twenty categories: strengths and weaknesses, popular options, emerging trends, expert recommendations, feature comparisons, and others. The resulting query sets were submitted to Perplexity Search, OpenAI SearchGPT, and Google Gemini, and citation extraction followed platform-specific procedures. The resulting records link each cited URL to its registered domain, query identifier, response identifier, platform, and topic. All analyses operate at the domain level. The dataset structure is summarized in Table LABEL:tab:1. 3.2 Bootstrap Resampling and Citation Share Estimation Citation visibility metrics are sample statistics computed from a finite collection of responses and are therefore subject to sampling variability. We characterize this variability using multinomial bootstrap resampling at the query level rather than the citation level, which correctly propagates within-response citation clustering: citations within a single response are a joint draw, not independent observations, and treating them as exchangeable understates uncertainty by assuming independence that does not hold. The bootstrap procedure resamples queries with replacement at each iteration, recomputing citation share as the domain’s citation count divided by total citations across the resampled responses. Citation share for domain d within platform p at bootstrap iteration b is: s^d,p(b)=cd,p(b)∑d′cd′,p(b) s_d,p^(b)= c_d,p^(b) _d^ c_d^ ,p^(b) where both numerator and denominator are recomputed from each bootstrap draw, correctly propagating the dependence of share on total citation volume. Confidence intervals used in reporting are computed using the BCa (bias-corrected and accelerated) method throughout (Efron, 1987), which adjusts for the systematic downward bias in bootstrap distributions for sparse citation data. All bootstrap confidence intervals reported in this paper use B = 1,000 replicates. Doubling B from 500 to 1,000 moves roughly half the sufficiency firing orders later by one to two responses, and the rank-stability test involves no resampling at all, so the stopping decisions are insensitive to the replicate budget at this scale. BCa applies to the share-level intervals that define the established tier and gate the sufficiency criterion; derived statistics, including the rank confidence intervals of §4.3 and the share-trajectory bands, use bias-corrected percentile intervals, because the acceleration term does not transfer through post-transforms. All resampling draws whole queries within a platform-topic bucket, which conditions every interval on citation-bearing responses; the rank confidence intervals additionally share one query draw across platforms, a paired design stratified by topic. 3.3 Reliability of Bootstrap Confidence Intervals Bootstrap confidence intervals on citation share are not uniformly reliable across the citation count distribution, and rank confidence intervals inherit the same structure. The cause is within-query citation clustering: a domain cited five times within a single query behaves very differently under bootstrap resampling than one cited once across five queries, despite identical total counts. The former contributes its entire citation mass only when that query is resampled, producing a zero-inflated distribution that percentile intervals understate, and the BCa correction addresses this only partially in the sparse regime. The practical consequence is a danger zone at roughly k = 5–20 citations, where interval widths appear stable while the underlying distributions remain skewed (Gemini reaches distributional symmetry near k ≈ 20 citations; SearchGPT does not within the observed range). Because rank is computed from share, rank confidence intervals carry the same caveat, which we flag when interpreting them in §4.3. 3.4 Observed and Established Domains It is natural to expect that the domains a platform cites most often will be the first to surface, so that the leaders announce themselves early in collection. They do tend to appear sooner than tail domains, but only on average: because each response cites just a handful of sources, the eventual leaders can be missing from the first several responses, and the early ordering is correspondingly noisy. A second and more consequential expectation is that new domains will eventually stop appearing, which would hand the analyst a natural signal to stop collecting once the vocabulary has settled. They do not. Figure 2 shows previously unseen domains entering the result set throughout the window, at a decelerating but non-vanishing rate. Figure 2: The number of new (previously unseen) domains per response order for the “creamy milk candy” topic, all three platforms. This continued growth is what the distributional structure predicts. The vocabulary follows Heaps’ Law (Heaps, 1978): the number of distinct domains grows as a concave power of collection size, so new domains keep arriving and never stop within any feasible window. Its companion, Zipf’s Law (Zipf, 1949), the rank-frequency form of the same power law (§2.3), explains what those newcomers are: a few domains accumulate citations rapidly while the great majority are cited only a handful of times, many of them just once. Heaps’ Law guarantees the stream of new domains; Zipf’s Law guarantees that most of them never amount to anything. Waiting for the vocabulary to stop growing is therefore not a usable stopping rule. The problem this creates is one of distinction: most domains that appear are noise, seen once or twice and unlikely to recur, while a minority are reliably present, and a convergence criterion must operate on the latter. Within the observed citation vocabulary we therefore separate two nested tiers by statistical detectability. A domain is observed once it has been cited at least once. A domain is established when its citation-share confidence interval excludes zero, that is, when its observed presence is robust to response-level resampling of the query set. Observed domains include everything seen in the collection; established domains are the subset whose presence is reliable enough to persist across repeated runs of the same analysis. The established tier is the domain set over which structural SNR is computed and the primary reporting-relevant rank uncertainty is evaluated. The boundary is set directly by the data: a domain becomes established when its accumulated citations are enough for a bootstrap confidence interval to clear zero. Because BCa intervals correct for the zero-inflation and asymmetry of sparse bootstrap distributions, the boundary adapts automatically to within-query clustering and citation density. No minimum domain count is required, and no power-law fit is needed to define membership; the framework does not depend on fitting the distribution at all, and the fits reported in §4.1 are descriptive. Which observed domains will ever become established cannot be known in advance, and the established tier grows more slowly than the full vocabulary as that status emerges from the data. 3.5 Rank Stability During Collection Rank order is not immediately interpretable from the first query. During early collection, the domains that will eventually become the most frequently cited have not yet accumulated enough citations to establish themselves: their confidence intervals overlap broadly with mid-tier and even lower-tier domains, and the rank ordering changes substantially as each additional query is processed. This is not a transient artifact but a structural property of Zipfian processes: the top-cited domains need time to separate from the field. We assess whether rank order has stabilized using the weighted Spearman rank correlation between successive collection windows. Successive windows overlap—each contains the observations of the previous window plus the new increment—so the correlation trajectory is strongly autocorrelated and, by construction, slow-moving: ρt = ρSpearman(θ^t-w, θ^t), wd= θ^d,t-w+θ^d,t2 _t\ =\ _Spearman ( θ_t-w,\ θ_t ),\ \ w_d=\ θ_d,t-w+ θ_d,t2 Weights are the average shares across the two windows, so rank swaps among high-share domains contribute more to the correlation than swaps in the low-share tail, where single-citation fluctuations routinely reorder positions without competitive meaning. The statistic is deliberately computed over the full observed vocabulary rather than the established tier alone: the share weights already place roughly three quarters of its mass on established domains at end of window (0.42 to 0.92 across the 30 combinations), and leaving the universe unrestricted keeps rank stability free of the bootstrap machinery that defines establishment, so the two convergence conditions supply independent evidence. The residual gap between full-ordering stability and established-set precision is precisely what the sufficiency condition and the validation analysis of §4.6 measure. The two collections compared at response order n are the cumulative samples through response n and through response n −- w, where w = max(3, ⌈ 0.10 ⋅· n⌉ ). The proportional lag counteracts a mechanical drift in ρ that would otherwise appear as n grows. At n = 10, a single new response shifts cumulative citation shares by roughly 10%; at n = 100 the same response moves them by 1%. Consecutive snapshots therefore grow more similar as n increases, pushing ρ toward 1 even when ranks are still settling. Stretching the lag in proportion to n holds the share-movement gap between the two snapshots roughly constant, so the correlation trajectory remains informative at every sample size. This construction gives the statistic a natural form of smoothing that requires no explicit filter. Because the windows are cumulative, the increment from one response order to the next is a fraction on the order of 1/n of the accumulated counts, so each additional response perturbs the ordering less than the one before, and the trajectory is mechanically damped as collection proceeds—once an initial warm-up period has passed and the established set has accumulated sufficient citations for the 1/n dampening to dominate window-to-window noise. The share weighting compounds this: the top-cited domains both anchor the correlation and extend their lead fastest, so once a top domain has accumulated a substantial count it becomes increasingly difficult to displace, and the weighted correlation it dominates settles even while the long tail keeps churning. The trajectory therefore rises and plateaus rather than oscillating freely, which is what makes a plateau—rather than a particular level—the natural object to detect, and what lets the detector below operate on the raw correlation without a separate smoothing step. The weighted correlation tends to increase as collection proceeds, although local declines remain possible. A natural first approach might be to declare rank order stable when the correlation crosses a fixed threshold (e.g., ρ ≥ 0.95). The results of this paper, however, expose a structural problem with any fixed level: citation density determines the ceiling ρ can reach, regardless of whether the rank order has stopped changing. A sparse platform’s correlation may plateau well below 0.95 even after the citation distribution has stabilized, because sampling noise between successive collection windows suppresses the observed rank correlation even when the true distribution is no longer changing. The criterion this paper uses detects that plateau directly with a segmented model fit to the raw correlation trajectory (Figure 7, in the Results section, shows the full collection of trajectories to which this test is applied). The rising phase is fit by isotonic (monotone non-decreasing) regression via the pool-adjacent-violators algorithm (PAVA; Barlow et al., 1972). Because the trajectory rises toward a ceiling, the isotonic fit absorbs the window-to-window noise without a bandwidth or smoothing constant and, unlike a straight line, does not misrepresent the concave, Heaps- and Zipf-shaped approach to that ceiling. A flat (constant) tail is then appended at a changepoint; both the changepoint location and whether a changepoint is warranted at all are selected by the Bayesian information criterion against a pure-rise model, and the tail must in addition be better described as flat than as linear under an effective sample size neff = n(1 −- ρ 1)/(1 + ρ 1), where ρ 1 is the lag-1 autocorrelation of the linear-fit residuals (Bartlett, 1946). Rank stability (rank_corr_ok) is declared once such a flat tail is established and spans at least a minimum length—15 responses throughout this paper—the single structural constant of the rank-stability test. This minimum plays the role of the earlier floor on the number of observations required before evaluation and serves as a monotone conservatism dial: a longer required tail fires later but is more robust to a transient early plateau. The test introduces no slope floor, no significance level, and no smoothing half-life; it is model selection together with one length floor. Concretely, the detector at response order n proceeds in five steps. (1) Compute the raw weighted Spearman correlation between the cumulative share vectors at n and n −- w, over the union of domains observed in either window; domains missing from the earlier window enter at share zero, ties receive average ranks, and weights are the mean of the two share vectors, normalized to sum to one. (2) For each candidate changepoint τ from 15 to n −- 15, fit the rising segment on the first τ observations by pool-adjacent-violators and a constant to the remainder, and score the model by BIC = n ⋅· log(RSS/n) + k ⋅· log(n), where k counts the isotonic fit’s distinct levels plus two, one for the tail level and one for the changepoint location. (3) Retain the best τ only if its BIC improves on a pure-rise model scored the same way. (4) Certify the tail as flat only if a constant model beats a linear model on the tail by BIC evaluated at the autocorrelation-corrected effective sample size defined above. (5) Declare rank stability at the first n with a certified flat tail of at least 15 responses; the verdict latches. Because both segments must span at least 15 observations, the earliest possible firing is response 30. Figure 3: BIC-iso rank stability detection for the “coffee mugs” topic on SearchGPT. The series shows the weighted Spearman ρ computed at each response order between successive cumulative citation-share windows. Isotonic regression (PAVA) is fit to the full ρ history; a two-segment model—a monotone-rising phase followed by a flat tail—is preferred over a pure-rise model when it achieves a lower BIC. τ denotes the BIC-selected plateau-onset index: the earliest response order at which the correlation series transitions from its rising phase to a sustained flat tail. Rank stability is declared at τ provided the flat tail spans at least 15 consecutive responses and passes a flatness test corrected for window autocorrelation. Once declared, the verdict latches. Because the test locates a plateau rather than a level crossing, it certifies stability at whatever correlation the platform’s citation density admits: a plateau at 0.88 is certified the same as one at 0.94. This is the property a fixed threshold lacks. Once certified, the verdict latches—rank stability is a terminal, one-way decision in the sequential-analysis sense (Wald, 1947), and the slow residual creep of the cumulative correlation toward its ceiling is asymptotic approach, not renewed re-ordering. Throughout this paper, we use a threshold-based criterion (ρ ≥ 0.95) as a reference benchmark; the contrast between the two is itself an empirical demonstration of where fixed thresholds fail. For period-over-period drift detection, a collection whose rank trajectory has plateaued provides a principled baseline for future checkpoint comparisons, regardless of the absolute correlation level achieved. 3.6 Rank Uncertainty Quantification Even after rank order has stabilized, individual domain ranks carry uncertainty that a point estimate does not capture. Rank confidence intervals give the plausible range of rank positions for a domain across repeated samples. Dense rank within platform is computed at each bootstrap iteration over the domain set, and the resulting distribution yields a CI on rank position. Because rank is discrete, these intervals are interpreted as a range of positions rather than a continuous estimate. Rank CIs inherit the reliability limitations described in §3.3: in the k ≈ 5–20 citation danger zone, the underlying share bootstrap distributions remain skewed and rank CIs derived from them may misrepresent coverage. 3.7 Structural Signal-to-Noise Ratio Structural SNR operationalizes the question of whether the spread of citation shares across the established set (domains whose share CI excludes zero) is large relative to the precision of individual share estimates. SNRest=σ(sd:d∈Dest)h¯CISNR_est= σ(s_d:d∈ D_est) h_CI where Dest is the established domain set, sd the citation share of established domain d, and Ud and Ld the upper and lower bounds of its 95% bootstrap confidence interval. The per-domain half-width and its mean across the established set are hCI,d=Ud−Ld2,h¯CI=1|Dest|∑d∈DesthCI,dh_CI,d= U_d-L_d2, h_CI= 1|D_est| _d∈ D_esth_CI,d The numerator is the standard deviation of established-domain citation shares (the signal); the denominator is the mean half-width of their citation-share confidence intervals (the noise). When SNR < 1, the typical uncertainty around established-domain shares is larger than the cross-domain spread, so the observed ordering is dominated by estimation uncertainty. When SNR ≥ 1, the share distribution carries enough structure, at the set level, to support rank-based reporting at the top of the distribution, though it does not imply that every adjacent rank is resolved. We treat structural SNR as a practical inferential sufficiency heuristic rather than a theoretically derived criterion, and the SNR = 1 threshold as an operational default rather than a statistically canonical one. Its motivation is geometric: the question is whether the spread of shares across established domains is large relative to how precisely those shares are measured. If the shares are tightly clustered, even narrow CIs may not resolve the rank ordering; if the shares are spread out, even moderate CI widths are sufficient. The criterion adapts to this automatically because both of its terms are estimated from the same accumulating sample: the spread of established shares and the per-domain measurement uncertainty it is weighed against both come from the data in hand, so the criterion sets its own bar rather than testing against a target fixed in advance. The validation analysis in §4.6 confirms empirically that at this threshold the criterion accepts precisely the orderings that agree with the full-window result and refuses precisely those that agree least. In the sequential analysis framing, the criterion belongs to the family of relative-precision stopping rules: rather than requiring CI width to fall below a fixed target (as in fixed-width procedures), it requires CI width to fall below a fraction of the signal standard deviation, making the stopping target adaptive to the observed share spread. A compact scaling argument explains why structural SNR rises predictably with collection and why it is self-calibrating. The established set is defined by a citation-share confidence interval that excludes zero, and that test requires the response-level bootstrap: the analytic Wilson lower bound is strictly positive for any cited domain and so cannot distinguish a share from zero, whereas the bias-corrected and accelerated (BCa) bootstrap lower bound can reach zero. Membership therefore rests on the bootstrap, and so does the CI half-width that enters the SNR denominator. The scaling itself does not depend on the interval’s closed form. A citation share is a ratio statistic over clustered citations; under independent citations its variance would be θ(1−-θ)/M, where M is the total citation count; within-response citation clustering inflates this by the design effect DEFF (Kish, 1965), the ratio of the jackknife variance of the share to its simple-random-sampling variance, so the effective citation count is M/DEFF and the standard error scales as 1/ x(M/DEFF). Because M grows in proportion to the citation-bearing response count R, the mean CI half-width across the established set falls as 1/ xR, while the signal σ, the standard deviation of established shares, is a property of the distribution’s shape and changes slowly once rank order has stabilized. Structural SNR, the ratio of the two, therefore grows as xR, and setting it to its target gives the minimum-response projection R* = R ⋅· (SNR_target / SNR)2. A flatter distribution (smaller σ) requires more data than a steep one, and a platform that accrues effective observations slowly, whether through sparse citation or heavy within-response clustering, requires more responses to drive the half-width down. Because the BCa interval has no closed-form half-width, the honest object of the derivation is this rate, not a plug-in sample-size equation. That distinction governs how the projection can be used. R* resembles a classical fixed-width sample-size formula, but it cannot be evaluated at the outset to set a collection budget. The signal σ is not stationary: as R grows and intervals tighten, more domains clear the zero boundary and enter the established set, reshaping σ and pulling the projected target downward, so R* settles only as σ settles, near convergence rather than before it. DEFF is itself estimated from the responses already collected, and the projection is denominated in citation-bearing responses, which a variable query-to-citation yield does not convert into a query budget in advance. R/SNR2 is therefore a confirmation signal that becomes trustworthy only in the final stretch before convergence, not a forecast: the criterion reports when enough data has accumulated, not how much will ultimately be required. When a decision fixes an absolute precision target (distinguishing a domain at 12% share from one at 10%, say, implies a target CI half-width near one percentage point), the analyst can simply collect until the half-width on the domains of interest falls below it; structural SNR is the self-calibrating default for the common case in which no such target can be stated in advance. The structural SNR, by contrast, is computed from a snapshot of the established set at each response order: its numerator, σ(established shares), and its denominator, mean CI half-width, both shift discretely whenever the established-set boundary moves or a new response accrues. Without smoothing, a single-response fluctuation could temporarily push SNR above the threshold and falsely declare sufficiency, or a transient dip could cancel a declaration that should persist. An exponentially weighted moving average (α, with a half-life that adapts to collection size: max(3, 0.10 ⋅· n) responses, equivalent to a smoothing constant near 0.2 early in collection and near 0.06 at n = 125) is therefore applied to the raw SNR series before the sufficiency_ok flag is evaluated. The smoothed series suppresses threshold crossings caused by discrete jumps in the established set while remaining responsive to genuine directional change. This smoothed series is what governs the sufficiency_ok flag and what is plotted in the Results section. The visual test is direct: compare the length of a typical error bar (the mean CI half-width, drawn as the red arrow) to the half-height of the blue band (the signal σ). When SNR < 1 the arrow extends beyond the half-band, when SNR = 1 the two match exactly, and when SNR > 1 the band extends beyond the arrow. Figure 4: Structural SNR = σ(established shares) / mean CI half-width, illustrated conceptually at three collection sizes using synthetic domains (D1–D12). The blue band marks ± 1 SD of established shares (signal σ = 0.062 in all panels, held constant). The red arrow marks the mean CI half-width (noise). Left panel (n = 21, SNR = 0.45): noise exceeds signal; the error bar extend beyond the blue band. Center panel (n = 43, SNR = 0.71): below the SNR = 1 threshold; signal does not yet match noise. Right panel (n = 116, SNR = 1.30): signal clearly exceeds noise. SNR grows slightly faster than n n at small n because the interval’s finite-sample correction relaxes as n grows; pure 1/n1/ n scaling from SNR = 0.45 at n = 21 would give 1.06 at n = 116, while the exact value is 1.30. Figure 5 illustrates the test with real data. At n = 34 (SNR = 0.61, EWMA-smoothed, the series the sufficiency gate thresholds), when rank_corr_ok is first determined, most CI half-widths extend well beyond the ± 1 SD band. Very little statistical separation exists among the established domains. At n = 81 (SNR = 1.00), when sufficiency_ok is determined, the number of established domains has increased to 44, and the CI bands of the top-ranked domains are increasingly differentiated from the dense middle. At n = 125 (SNR = 1.39), the end of the data gathering process, the CI bands are perceptibly narrower. In the dense middle, the signal is often greater than the noise, but adjacent domains still have overlapping CIs. σ barely changes across panels (0.016 → 0.016 → 0.015) because the true share distribution is stable; SNR improves almost entirely because hw falls as n grows, which is precisely what the metric is designed to capture. Between n = 81 and n = 125 the mean half-width falls faster than the 1/n1/ n per-domain rate alone would predict, because newly established domains enter with narrow absolute intervals and pull the set average down. Figure 5: Structural SNR using actual bootstrap CIs for the Perplexity / “coffee mugs” platform-topic combination at three collection sizes. Each point is a domain citation share with 95% bootstrap CI; the blue band marks ± 1 SD of the established domain shares. SNR annotations show the ratio and its components (σ, hw). Both criteria share a common design principle. Rank stability requires no target correlation level: the BIC-iso test detects a plateau regardless of whether it occurs at ρ = 0.88 or ρ = 0.98. Structural SNR requires no target CI width or precision level: the threshold SNR ≥ 1 emerges from the relationship between the observed signal and noise. Both stopping decisions are derived from the data’s own structure: the share spread of the established set, the empirical CI widths, and the plateau geometry of the rank-correlation trajectory. The framework specifies no target query count, correlation level, or CI width. What remains externally set are two structural constants: the 15-response minimum plateau length, and the α smoothing applied to the SNR signal, whose half-life scales with collection size, max(3, 0.10 ⋅· n) responses. Neither encodes a target the data is asked to reach; each governs how much confirmation the test demands before it fires. The 15-response minimum also serves as a data-maturity floor for the sufficiency test: structural SNR is not evaluated before 15 responses, because near-degenerate bootstrap intervals at very small samples can spuriously inflate the ratio. Sufficiency firing orders of 15 in the Results are therefore censored at this floor. 4 Results 4.1 Dataset and Distributional Structure The analysis covers ten topics spanning diverse market structures and information environments: “marathon training shoes”, “identity protection and fraud prevention”, “smoke detectors”, “healthy meal delivery for busy professionals”, “collagen protein”, “creamy milk candy”, “elevation-based corneal tomography”, “hand-thrown coffee mugs”, “unusual travel destinations”, and “ingredients for homemade chili”. All analyses cover three platforms: Gemini, SearchGPT, and Perplexity. Table LABEL:tab:1 summarizes citation-bearing response counts, citation counts, domain counts, and mean citations per citation-bearing response across all platform-topic combinations. For Gemini and Perplexity, virtually every submitted query yields a citation-bearing response; for SearchGPT, a topic-dependent fraction of queries return no citations, so the response count reflects only queries that produced at least one cited source. Citation density differs substantially across platforms and, to a lesser extent, across topics. Gemini produces between 19.9 and 50.1 citations per response depending on topic; Perplexity produces between 14.5 and 42.2; and SearchGPT produces between 4.2 and 6.3. Within-response clustering compounds the density gap: the design effect (DEFF, the ratio of jackknife to simple-random-sampling variance for a domain’s share), averaged across established domains at end of window, runs 2.9–5.3 on Gemini (median 3.9), 3.7–7.8 on Perplexity (median 4.3), and 1.3–4.0 on SearchGPT (median 2.1). The effective response count R/DEFF, the number of independent-citation-equivalent responses, therefore ends the window near a quarter of the nominal response count on Gemini and near half on SearchGPT. These differences drive every downstream result about rank stability timelines, rank CI width, and the sensitivity of period-over-period tests. Table 1: Dataset summary by topic and platform. Cit/Resp = mean citations per response. Topic Platform Responses Citations Domains Cit/Resp Mean DEFF Coffee mugs Gemini 125 3811 371 30.5 3.8 Perplexity 125 2117 137 16.9 3.7 SearchGPT 121 693 153 5.7 2.1 Collagen protein Gemini 125 4985 385 39.9 4.0 Perplexity 125 2627 213 21.0 4.0 SearchGPT 121 725 170 6.0 2.1 Corneal tomography Gemini 125 3785 254 30.3 3.6 Perplexity 122 2224 91 18.2 3.8 SearchGPT 117 494 50 4.2 1.7 Creamy milk candy Gemini 125 2488 435 19.9 2.9 Perplexity 125 1812 217 14.5 3.8 SearchGPT 119 556 161 4.7 1.3 Healthy meal delivery Gemini 124 4290 291 34.6 3.9 Perplexity 125 2184 154 17.5 4.3 SearchGPT 124 640 123 5.2 1.5 Homemade chili Gemini 113 5656 356 50.1 4.1 Perplexity 121 3956 180 32.7 4.8 SearchGPT 104 594 156 5.7 2.1 Identity protection Gemini 125 4516 544 36.1 5.3 Perplexity 125 2820 320 22.6 7.8 SearchGPT 104 660 230 6.3 2.9 Marathon training shoes Gemini 125 3763 276 30.1 4.0 Perplexity 124 2347 114 18.9 4.8 SearchGPT 123 673 139 5.5 1.9 Smoke detectors Gemini 122 4922 439 40.3 4.0 Perplexity 125 5270 184 42.2 7.8 SearchGPT 124 705 48 5.7 4.0 Travel destinations Gemini 122 3597 496 29.5 3.5 Perplexity 125 1907 190 15.3 4.4 SearchGPT 113 704 190 6.2 2.7 One structural feature of SearchGPT deserves mention. Unlike Gemini and Perplexity, SearchGPT produces a non-negligible fraction of responses containing no citations. In the “identity protection” topic, for example, SearchGPT returned 104 citation-bearing responses from a query batch in which Gemini and Perplexity each returned 125, a zero-citation rate of approximately 17%. The practical consequence is that the response counts in Table LABEL:tab:1 for SearchGPT understate the collection effort required to accumulate a given number of usable observations: a collection budget of 125 queries may yield substantially fewer than 125 citation-bearing responses. Convergence response orders for SearchGPT should therefore be read as sufficient citation-bearing responses, not sufficient queries submitted. Practitioners can account for this by inflating the query budget: if R citation-bearing responses are required and the platform’s observed yield rate is Y (the fraction of submitted queries returning at least one citation), the required query budget is Nsubmit = ⌈ / Y⌉ . For a target of 100 citation-bearing responses at Y ≈ 0.83, this implies submitting approximately 121 queries rather than 100. Citation count distributions are broadly consistent with a power-law model across the 29 platform-topic combinations for which fitting is feasible. Figure 6 shows one topic, “creamy milk candy,” on log-log axes. Citation count declines approximately linearly with rank—the characteristic signature of a power law—and the dotted line marks xminx_ , the fitted boundary between the structured body of the citation distribution and its sparse lower-ranked tail. Domains above xminx_ (blue) participate in the stable structure that defines the ranking, whereas those below it (tan) occur too infrequently to follow the same regular pattern. Figure 6: Citation count distributions in log-log scale for the “creamy milk candy” topic at full sample size, all three platforms. The dotted horizontal line marks xmin; blue points are domains that belong to the structured body of the citation distribution (count ≥ xmin); tan points fall below that boundary. Table LABEL:tab:2 reports the power-law fit parameters and the Heaps’ Law exponent for every platform-topic combination in the dataset. We fit each distribution with the KS-minimized discrete maximum-likelihood procedure of Clauset et al. (2009), which also supplies the semiparametric bootstrap goodness-of-fit test reported there. Table 2: Distributional structure by topic and platform. Domains = total distinct domains observed; xmin and α are the lower bound and exponent of the KS-minimized discrete power-law fit (Clauset et al., 2009); GOF p is the semiparametric bootstrap goodness-of-fit p-value, with values above 0.10 consistent with a power-law model; β is the fitted Heaps’ Law exponent for domain accumulation. Dashes (—) in the xmin, α, and GOF p columns indicate fewer than 50 domains, below the floor required for a stable fit; β does not require this floor. Topic Platform Domains xmin α GOF p β Coffee mugs Gemini 371 4 1.95 0.51 0.61 Perplexity 137 6 1.89 0.21 0.52 SearchGPT 153 2 2.02 0.20 0.66 Collagen protein Gemini 385 6 2.03 0.30 0.60 Perplexity 213 5 2.03 0.41 0.60 SearchGPT 170 1 1.84 0.089 0.69 Corneal tomography Gemini 254 4 1.82 0.69 0.59 Perplexity 91 2 1.56 0.26 0.49 SearchGPT 50 1 1.84 0.021 0.54 Creamy milk candy Gemini 435 4 2.26 0.99 0.69 Perplexity 217 5 2.18 0.49 0.66 SearchGPT 161 2 2.18 0.51 0.67 Healthy meal delivery Gemini 291 4 1.82 0.044 0.57 Perplexity 154 8 2.05 0.76 0.59 SearchGPT 123 1 1.92 0.37 0.64 Homemade chili Gemini 356 3 1.78 0.37 0.62 Perplexity 180 8 1.90 0.74 0.64 SearchGPT 156 1 1.85 0.25 0.73 Identity protection Gemini 544 4 2.04 0.68 0.67 Perplexity 320 4 2.08 0.12 0.68 SearchGPT 230 3 2.32 0.29 0.75 Marathon training shoes Gemini 276 3 1.81 0.10 0.59 Perplexity 114 6 1.89 0.88 0.54 SearchGPT 139 2 1.94 0.12 0.63 Smoke detectors Gemini 439 6 2.06 0.019 0.62 Perplexity 184 17 2.02 0.53 0.58 SearchGPT 48 — — — 0.57 Travel destinations Gemini 496 4 2.08 0.42 0.68 Perplexity 190 4 2.01 0.66 0.63 SearchGPT 190 4 2.42 0.12 0.75 Fitted α values range from 1.56 (Perplexity / “corneal tomography”) to 2.42 (SearchGPT / “travel destinations”), with most combinations between 1.78 and 2.26. The goodness-of-fit test of Clauset et al. (2009) fails to reject the power-law model (p > 0.10) for 25 of the 29 valid fits. The four exceptions are Gemini / “smoke detectors” (p = 0.019), SearchGPT / “corneal tomography” (p = 0.021), Gemini / “healthy meal delivery” (p = 0.044), and SearchGPT / “collagen protein” (p = 0.089); these are treated as approximate rather than ruled out as power-law distributed. One combination warrants individual note: SearchGPT / “corneal tomography” has only 50 total domains. With xmin = 1, the fitted noise tier is empty: all 50 domains fall within the structured body of the fit. The established set is far smaller (12 domains at full sample, Table LABEL:tab:7), because establishment is defined by a share CI that excludes zero, not by the fit. The power-law fit is technically valid but does not become so until response 114, when the domain count finally reaches the 50-domain floor the KS fit requires. The established-set criterion does not depend on that fit: structural sufficiency fires early (response 15, the first order at which sufficiency is evaluated; the few established shares being well separated), so rank stability is the binding condition and the combination converges at n* = 61, gated by rank stability rather than precision. The Heaps’ Law analysis confirms that new domains enter the observed distribution throughout collection at a decelerating rate for every platform-topic combination, with fitted exponents (β, Table LABEL:tab:2) consistent with the theoretical prediction from the corresponding Zipf fits. This confirms that the continued appearance of new tail domains is the structural consequence of the power-law distribution rather than evidence of non-stationarity, and it explains the first-appearance latency asymmetry: domains that ultimately rank highest appear earliest in collection because they have the highest citation probabilities under the Zipfian process, while tail domains continue appearing throughout the 125-response window. 4.2 Rank Stability During Collection Figure 7 shows weighted Spearman rank correlation trajectories for the “travel destinations” topic as a representative example, separately for Gemini, Perplexity, and SearchGPT. Every platform-topic combination begins with low correlation and rises as citations accumulate, confirming that rank order is genuinely unstable during early collection and only becomes interpretable as established domains separate from one another. Table LABEL:tab:3 reports plateau-test results for all 30 combinations. Figure 7: Weighted Spearman rank correlation versus response order for the “travel destinations” topic, shown separately for Gemini, SearchGPT, and Perplexity. Each point represents the raw weighted ρ . The points are red prior to rank_corr_ok and green thereafter. The BIC-iso plateau fit is overlaid on each trajectory; the solid blue line illustrates the isotonic fit, and the green vertical band identifies the BIC-chosen flat tail. The dashed horizontal line marks the fixed stability threshold (ρ = 0.95). Table LABEL:tab:3 reports convergence orders for all 30 platform-topic combinations. Table LABEL:tab:3 reports three stability benchmarks for each platform-topic combination: the first response order at which raw ρ ≥ 0.95 is reached (nr), the first response order at which raw ρ ≥ 0.95 was sustained for 15 consecutive responses (ns), and the response order at which the BIC-iso plateau test first declares rank stability (n*). The BIC-iso test evaluates each possible changepoint as the start of a candidate terminal window: the trajectory before the changepoint is fit as a monotone rise, and the trajectory after the changepoint is tested as a flat tail. BIC selects whether this two-part model improves on a pure monotone-rise model, and stability is declared only when the selected flat tail spans at least 15 responses and passes the flatness check described in §3.5. In all 30 cases, the rank correlation reaches ρ ≥ 0.95, often early in the process (mean response order = 19.9). However, rank correlation is jittery (as seen in Figure 7) and can fall below the threshold within a few responses. To avoid spurious results, we can require that the test’s condition hold for 15 consecutive observations, the same duration used in the BIC-iso plateau test. However, only 19 platform-topic combinations sustain ρ ≥ 0.95 for 15 consecutive observations, and they take much longer to do so (mean response order = 69.2). In contrast, the BIC-iso test declares stability in all 30 combinations. This contrast—19 combinations reaching sustained stability versus all 30 reaching the BIC-iso plateau—demonstrates that ceiling effects are the primary limitation of any fixed-threshold approach: 11 combinations never sustain 15 consecutive observations above ρ = 0.95 not because their ordering is still changing, but because their citation density does not support reaching that level. When citation density is low, each response cites only a handful of distinct domains, and the set of cited domains changes substantially from query to query. Rank correlation measures agreement in relative ordering across responses; when few domains are shared between any two responses, the ordering is inherently noisy, and rank correlation will not consistently reach 0.95 regardless of whether an underlying stable ordering exists. The plateau test addresses this directly by detecting flatness rather than level. Moreover, the BIC-iso test fires much earlier (mean response order = 40.0); in 18 of the 19 cases where the BIC-iso and sustained threshold tests were both satisfied, BIC-iso fires at an earlier response order. (The only exception was SearchGPT / “corneal tomography”.) Table 3: Response order at which raw ρ ≥ 0.95 first reached (nr), response order at which raw ρ ≥ 0.95 was sustained for 15 periods (ns), response order at which the rank stability criterion (rank_corr_ok) first fired (n*), rank correlation at that point (ρ*), and rank correlation at end of collection (ρ ). Shaded cells identify ρ ≥ 0.95. Topic Platform nr ns n* ρ* ρ Coffee mugs Gemini 13 — 75 0.983 0.960 Perplexity 20 51 34 0.937 0.979 SearchGPT 17 80 47 0.977 0.955 Collagen protein Gemini 21 60 33 0.954 0.972 Perplexity 31 56 38 0.902 0.944 SearchGPT 24 — 37 0.921 0.963 Corneal tomography Gemini 15 76 43 0.968 0.946 Perplexity 10 50 34 0.928 0.972 SearchGPT 8 43 61 0.996 0.981 Creamy milk candy Gemini 20 — 40 0.947 0.970 Perplexity 29 — 48 0.989 0.908 SearchGPT 13 50 37 0.973 0.966 Healthy meal delivery Gemini 19 72 33 0.976 0.942 Perplexity 16 95 35 0.899 0.928 SearchGPT 15 48 37 0.989 0.968 Homemade chili Gemini 22 95 35 0.981 0.943 Perplexity 17 — 37 0.996 0.964 SearchGPT 18 — 33 0.991 0.954 Identity protection Gemini 26 — 34 0.876 0.976 Perplexity 36 — 42 0.861 0.920 SearchGPT 32 — 35 0.913 0.911 Marathon training shoes Gemini 23 66 34 0.954 0.972 Perplexity 27 76 43 0.948 0.982 SearchGPT 19 50 39 0.980 0.934 Smoke detectors Gemini 23 100 35 0.884 0.970 Perplexity 15 109 33 0.963 0.928 SearchGPT 4 40 33 0.997 0.969 Travel destinations Gemini 29 98 41 0.928 0.952 Perplexity 20 — 49 0.970 0.928 SearchGPT 26 — 46 0.906 0.956 The plateau test declares rank stability for all 30 platform-topic combinations within the 125-response collection window. Joint rank stability (all three platforms simultaneously stable) is achieved for all ten topics, from “smoke detectors” at response 35 to “coffee mugs” at response 75. Stability firing orders fall within a moderate band: every n* falls between 33 and 75 responses. As §4.5 shows, the wide variation in total data requirements comes from the sufficiency condition, not from stability. Two patterns stand out. First, the platform ordering of convergence speed varies across topics. No platform is consistently fastest or slowest: each of the three platforms has at least one topic on which it plateaus earliest, with n* = 33, and at least one on which it plateaus latest, sometimes by a substantial margin, as in Gemini / “coffee mugs” (n* = 75) and SearchGPT / “corneal tomography” (n* = 61). The same platforms that converge earliest for some topics are latest for others. This cross-topic variation indicates that convergence speed is jointly determined by topic-level market structure and platform citation behavior, rather than by platform identity alone. Second, the stability orders occupy a narrower band (33 to 75) than the conjunctive orders in §4.5, which range from 33 to 94 and include three combinations that never converge. Although stability fires last in most converged combinations (16 of 27, §4.5), it never fires later than response 75; the long delays and all three convergence failures come from sufficiency. These results establish the practical implication directly: no fixed query count can be justified across all topics and platforms. The BIC-iso plateau test resolves this by detecting a plateau rather than a level—all 30 combinations reach stability between n* = 33 and n* = 75, while the sustained ρ ≥ 0.95 benchmark leaves 11 of the 30 unsatisfied within the collection window. Stability, however, is only half the criterion. §4.5 shows that structural sufficiency is the binding constraint in 11 of the 27 converging combinations and is never achieved in three. 4.3 Rank Uncertainty at Full Sample Size Even where the ordering is stable, the precision of individual ranks varies enormously across the distribution, and this pattern holds across all 30 platform-topic combinations. Table LABEL:tab:4 reports median CI width across the full domain vocabulary for each combination. At the low end, SearchGPT / “smoke detectors” (48 domains) has a median CI of 25.5 rank positions; at the high end, Gemini / “identity protection” (544 domains) reaches 329.9 positions. Expressed as a fraction of total vocabulary, median CI widths span 38–66% with a median of 58%. Table 5 pools the same widths by reporting scope: among each combination’s ten most-cited domains the median CI spans 5.0 rank positions (90th percentile 14.1, with 18.7% exceeding ten positions); across the established tier it spans 69.1; across the full vocabulary, 131.0. Rank uncertainty persists well inside the reporting-relevant portion of the distribution. These large medians reflect the many tail domains whose rank assignments are essentially arbitrary, but even within the established set, CI widths widen substantially toward the noise boundary. A rank ordering can be fully stable and still be too imprecise at the lower portion of the established set to support meaningful competitive analysis. This is the gap the structural SNR criterion in §3.7 closes: it identifies whether the top of the established set is measured precisely enough to trust the ordering. Table 4: Rank CI width statistics at full sample size (n = 125 responses, 95% CI). Median CI width and percentage of domains with CI width exceeding 10 rank positions are shown by topic and platform. Topic Platform Domains Median rank CI width % with width > 10 Coffee mugs Gemini 371 217.0 96 Perplexity 137 65.0 92 SearchGPT 153 95.0 92 Collagen protein Gemini 385 199.0 95 Perplexity 213 106.1 91 SearchGPT 170 102.6 94 Corneal tomography Gemini 254 134.6 93 Perplexity 91 35.0 84 SearchGPT 50 27.5 74 Creamy milk candy Gemini 435 262.1 96 Perplexity 217 132.0 93 SearchGPT 161 104.1 93 Healthy meal delivery Gemini 291 156.7 94 Perplexity 154 80.0 92 SearchGPT 123 76.0 89 Homemade chili Gemini 356 213.0 94 Perplexity 180 102.0 89 SearchGPT 156 98.5 93 Identity protection Gemini 544 329.9 97 Perplexity 320 186.2 95 SearchGPT 230 151.0 96 Marathon training shoes Gemini 276 153.0 92 Perplexity 114 54.3 86 SearchGPT 139 82.0 94 Smoke detectors Gemini 439 227.0 94 Perplexity 184 98.4 87 SearchGPT 48 25.5 71 Travel destinations Gemini 496 303.6 96 Perplexity 190 110.1 91 SearchGPT 190 116.5 96 Figure 8 illustrates this gradient directly. At the top of the established set, CIs are narrow: the first eight or nine domains are clearly distinguishable from one another, and rank-based reporting is reliable. Deeper in the distribution, where established domains share similar citation frequencies, CIs widen substantially, and adjacent ranks become statistically indistinguishable. By ranks #26–#30, several CI widths span as many as 80 rank positions, making the apparent ordering unreliable even though all these domains are established. Rank matters; the question is whether you have collected enough data to know which ranks can be trusted. Figure 8: Rank CI forest plot for Gemini / “healthy meal delivery” at full sample size (n = 125). Each horizontal bar is the 95% bootstrap rank CI for one domain. The top-ranked domains have narrow CIs and are distinguishable, but adjacent domains in the middle of the distribution have overlapping intervals spanning 30–80+ rank positions. Table 5: Rank CI width by reporting scope, pooled across all 30 platform-topic combinations at full sample size (95% CIs). Top 10 = the ten most-cited domains per combination; established = domains whose citation-share CI excludes zero; widths in rank positions. Scope Domains Median rank CI width 90th percentile width % with width > 10 Top 10 domains 300 5.0 14.1 18.7 Established domains 1,693 69.1 217.2 84.2 Full vocabulary 7,067 131.0 327.0 93.3 4.4 Structural Sufficiency Rank stability certifies that the ordering of citation shares has stopped changing; it does not certify that the shares are measured precisely enough to be informative. Structural sufficiency addresses that second condition. It asks whether the structure that remains, the spread of citation shares across the established set, is large enough relative to the uncertainty in those shares to support inference. When structural SNR ≥ 1, the spread of citation shares across the established set is large relative to a typical confidence interval, meaning the ordering reflects a real signal rather than estimation noise. Where structural SNR never reaches 1, additional collection would be required—or, in some cases, the platform-topic combination may not support reliable share estimation within any feasible collection window. Figure 9 shows the structural SNR trajectories for the “travel destinations” topic across all three platforms. The trajectories illustrate the range of outcomes. Gemini crosses the SNR ≥ 1 threshold at n = 29 responses, before its rank stability fires at n* = 41; sufficiency is not the binding constraint here. Perplexity crosses at n = 94, after rank stability at n* = 49, so sufficiency is the binding constraint in this case. SearchGPT achieves rank stability at n* = 46, but the structural SNR never crosses the threshold within the collection window, and the combination does not converge. Figure 9: Structural SNR trajectories for the “travel destinations” topic, shown separately for each platform. The Structural SNR increases slowly as more data are gathered and the mean CI of established domains decreases (yellow dotted line). The horizontal dashed line marks the SNR = 1 sufficiency threshold, and the dotted vertical line indicates the response order at which the EWMA-smoothed SNR crosses the threshold. The points are colored red prior to sufficiency_ok and green thereafter. Table LABEL:tab:6 reports the structural SNR for all 30 platform-topic combinations. Across all 30 platform-topic combinations, structural sufficiency is the binding constraint in 11 of the 27 that converge: rank stability fires first in these combinations, and the conjunction must wait for structural SNR ≥ 1. Three combinations do not converge within the 125-response collection window, all on SearchGPT. For “identity protection” and “travel destinations”, the end-of-window structural SNR is 0.74 and 0.90 respectively. For “homemade chili”, structural SNR ends the window at 0.89. In each signal-limited case the established citation shares are distributed too narrowly—relative to the width of the confidence intervals—for the criterion to fire within the 125-response collection window. 4.5 Conjunctive Convergence Results The preceding sections treat rank stability (§4.2), rank uncertainty (§4.3), and structural sufficiency (§4.4) as separate analyses. Their conjunction defines the paper’s central convergence criterion: a platform-topic combination is declared sufficient when both rank_corr_ok (the BIC-iso plateau test on the rank correlation trajectory) and sufficiency_ok (structural SNR ≥ 1) hold simultaneously. Table LABEL:tab:6 reports, for each combination, the first response order at which each condition fired and the first response order at which the conjunction held; the conjunctive order is in every case the later of the two component orders. The criterion converges for 27 of 30 combinations within the 125-response collection window. Median conjunctive convergence orders are 42.0 for Gemini, 37.0 for SearchGPT (across its seven converging combinations), and 51.0 for Perplexity. The two conditions share the binding role: structural SNR fires last in 11 of the 27 converged combinations, and rank stability fires last in the other 16. Sufficiency is not a formality. Where sufficiency binds, it delays the stop by anywhere from a few responses to 47 responses (e.g., in Perplexity / “coffee mugs”, rank stability fires at response 34 but structural sufficiency does not fire until response 81). In three combinations, all on SearchGPT, structural sufficiency never fires within the window (§4.4). Table 6: Conjunctive convergence by platform-topic combination: first firing of rank stability (n*), first firing of structural sufficiency (nSNR), first firing of the conjunction (nconj), and the EWMA-smoothed structural SNR at end of window (SNRend). The conjunctive order equals the later of n* and nSNR in every converged case. Dashes (—) indicate that the test criterion was not met within the 125-response collection window. Values of nSNR = 15 coincide with the first response order at which sufficiency is evaluated (§3.7) and are censored at that floor. Topic Platform n* nSNR nconj SNRend Coffee mugs Gemini 75 60 75 1.59 Perplexity 34 81 81 1.39 SearchGPT 47 34 47 1.64 Collagen protein Gemini 33 32 33 2.12 Perplexity 38 58 58 1.61 SearchGPT 37 24 37 2.22 Corneal tomography Gemini 43 15 43 4.71 Perplexity 34 15 34 3.62 SearchGPT 61 15 61 5.77 Creamy milk candy Gemini 40 27 40 2.09 Perplexity 48 55 55 1.69 SearchGPT 37 21 37 2.57 Healthy meal delivery Gemini 33 15 33 3.06 Perplexity 35 22 35 2.07 SearchGPT 37 20 37 3.53 Homemade chili Gemini 35 33 35 2.50 Perplexity 37 30 37 2.35 SearchGPT 33 — — 0.89 Identity protection Gemini 34 63 63 1.39 Perplexity 42 74 74 1.32 SearchGPT 35 — — 0.74 Marathon training shoes Gemini 34 45 45 2.40 Perplexity 43 47 47 2.32 SearchGPT 39 45 45 1.74 Smoke detectors Gemini 35 55 55 1.66 Perplexity 33 43 43 1.94 SearchGPT 33 22 33 2.31 Travel destinations Gemini 41 29 41 1.84 Perplexity 49 94 94 1.30 SearchGPT 46 — — 0.90 Shaded rows: not converged. Figure 10 illustrates the conjunctive convergence process for the “travel destinations” topic. The four-row panel shows domain accumulation, rank correlation, structural SNR, and the conjunctive convergence flag together across all three platforms in a single view. Gemini achieves conjunction at n = 41: structural SNR clears 1 at response 29, well ahead of rank stability at n* = 41, so stability is the binding constraint. Perplexity achieves rank stability at n* = 49, but structural SNR does not clear the threshold until response 94, making sufficiency the binding constraint and setting the conjunctive order to n = 94. SearchGPT achieves rank stability at n* = 46 but does not cross the SNR threshold within the collection window, and the combination does not converge. Figure 10: Conjunctive convergence for the “travel destinations” topic across Gemini, Perplexity, and SearchGPT. Top row: unique domain accumulation. Second row: weighted Spearman rank correlation with BIC-iso plateau fit. Third row: structural SNR (EWMA-smoothed) with the sufficiency threshold at SNR = 1. Bottom row: conjunctive convergence flag, True when both rank_corr_ok and sufficiency_ok hold simultaneously. Conjunctive convergence orders: Gemini nconj= 41, Perplexity nconj= 94. SearchGPT does not converge within the 125-response collection window. The mechanism is the same in all three non-converging combinations: at roughly six citations per response, SearchGPT’s confidence intervals cannot narrow enough within 125 responses to resolve a signal σ that is already small. More collection would eventually push structural SNR past 1; the criterion’s verdict is that 125 responses is not enough here, which is precisely the platform-topic dependence that makes a fixed universal budget untenable. The zero-citation response rate noted in §4.1 affects the interpretation but not the validity of the convergence orders. The criterion asks whether the observed citation data (however many citation-bearing responses have been accumulated) is sufficient to support rank-based reporting. That question is correctly answered in terms of citation-bearing responses. The operational implication for collection planning is separate: because SearchGPT returns no citations in a topic-dependent fraction of queries, achieving a target number of citation-bearing responses requires submitting more total queries than the convergence order implies. The convergence criterion is not misspecified; the query budget must account for platform yield. 4.6 Validation: Do Stopped Orderings Persist? The validation supports the criterion on both sides of its decision, and its value is concentrated at the tail. The orderings it refuses include the weakest agreements in the dataset, and the orderings it certifies avoid the worst failures that stopping at rank stability alone would have admitted. The convergence criterion certifies that an ordering has stabilized and is measured precisely enough to act on. The central question is whether stopping at the firing order actually yields the ordering one would have obtained by collecting the full budget. The check is direct. For each combination, we compare the cumulative established-set ordering at the stopping point against the ordering at the end of the 125-response window, using the same weighted Spearman correlation as the stability criterion (weights proportional to average shares) over the established domains at full sample (those whose citation-share CI excludes zero), along with the overlap between the top-10 lists. Because the two orderings share the data collected up to the stopping point, this is an internal consistency check against the best available estimate, not a comparison with external ground truth. Fixing the established set at its full-sample membership likewise uses information unavailable at the stopping point, so the agreement figures are retrospective audits of the certified orderings, not quantities an analyst could have computed at the stop. Stronger validation would compare stopped orderings against independent repeated collections or held-out responses; we leave that design to future work. Three results stand out. First, stopping at rank stability alone is not safe: agreement at n* ranges from 0.364 to 0.986 (median 0.823) across all 30 combinations, and the NC refusal with the weakest ordering—SearchGPT / “identity protection” (0.364, the single worst)—demonstrates exactly why the sufficiency condition is needed. Perplexity / “identity protection” (0.713) and Gemini / “identity protection” (0.722) also show weak agreement at n*. This is exactly why sufficiency is required as a second condition: the criterion’s refusals are vindicated by the data. Second, where structural sufficiency delays the stop it usually helps and rarely hurts, though the average gain is modest: agreement improved in all 11 combinations where structural SNR delayed the stop beyond n*, with most of the benefit concentrated in a few cases (Perplexity / “identity protection” from 0.713 to 0.866; Perplexity / “travel destinations” from 0.842 to 0.978) and small elsewhere. Third, the certified stops are sound rather than flawless: across the 27 converged combinations, agreement at nconj spans 0.717 to 0.986 with a median of 0.844, and the median top-10 overlap is 80%, so on average two of the top ten domains still differ from the full-window ranking. The criterion’s contribution shows up at the floor: it lifts the worst certified agreement to 0.717, well above the 0.364 that rank stability alone would have admitted. Table 7: Agreement between stopped and end-of-window established-set orderings: weighted Spearman correlation evaluated at the rank-stability firing order (n*) and at the conjunctive firing order (nconj). A dash (—) indicates that the test criterion was not met within the 125-response collection window. The established set is fixed at the full-sample established tier (domains whose citation-share CI excludes zero). dest = number of domains with citation-share CI excluding zero at full sample. Topic Platform dest n* ρ at n* nconj ρ at nconj Coffee mugs Gemini 88 75 0.947 75 0.947 Perplexity 57 34 0.861 81 0.954 SearchGPT 37 47 0.781 47 0.781 Collagen protein Gemini 109 33 0.746 33 0.746 Perplexity 66 38 0.791 58 0.821 SearchGPT 45 37 0.837 37 0.837 Corneal tomography Gemini 57 43 0.915 43 0.915 Perplexity 31 34 0.909 34 0.909 SearchGPT 12 61 0.986 61 0.986 Creamy milk candy Gemini 81 40 0.802 40 0.802 Perplexity 41 48 0.876 55 0.877 SearchGPT 30 37 0.890 37 0.890 Healthy meal delivery Gemini 98 33 0.833 33 0.833 Perplexity 47 35 0.825 35 0.825 SearchGPT 23 37 0.885 37 0.885 Homemade chili Gemini 80 35 0.877 35 0.877 Perplexity 34 37 0.885 37 0.885 SearchGPT 25 33 0.818 — — Identity protection Gemini 122 34 0.722 63 0.838 Perplexity 64 42 0.713 74 0.866 SearchGPT 30 35 0.364 — — Marathon training shoes Gemini 71 34 0.822 45 0.839 Perplexity 36 43 0.841 47 0.862 SearchGPT 37 39 0.703 45 0.717 Smoke detectors Gemini 142 35 0.722 55 0.817 Perplexity 49 33 0.768 43 0.844 SearchGPT 12 33 0.799 33 0.799 Travel destinations Gemini 97 41 0.758 41 0.758 Perplexity 42 49 0.842 94 0.978 SearchGPT 30 46 0.716 — — Shaded rows: not converged. This analysis also answers the post-firing decline objection raised by Table LABEL:tab:3, where several combinations fire with ρ* above 0.95 and end the window below it (Gemini / “homemade chili” declines from 0.981 to 0.943, Perplexity / “smoke detectors” from 0.963 to 0.928). The declining quantity is the window-to-window correlation, which oscillates as marginal citations shift positions near the threshold. The ordering itself does not degrade: the certified orderings for those two combinations agree with the end-of-window orderings at 0.88 and 0.84 by the validation measure, with top-10 overlaps of 90% and 80% respectively. Threshold jitter in ρ is a property of the monitoring statistic, not of the ranking being certified. 4.7 Sensitivity to the Sufficiency Threshold The structural-SNR threshold of 1.0 is an operational default, so we test how the criterion behaves as it moves. Holding the rank-stability test fixed, we sweep the threshold T from 0.5 to 1.5 and recompute, at each value, how many of the 30 combinations converge within the window, how long they take, and how closely the certified ordering agrees with the full-window result (Figure 11). The result is a smooth cost-fidelity tradeoff, not a fragile operating point. Below T ≈ 0.75 the certified orderings do not improve: every combination converges, but the worst certified agreement stays pinned near 0.70 and the median near 0.84, so a lower threshold only stops sooner and admits the weakest orderings the data produce. From there the floor rises steadily rather than plateauing (0.71 at 0.85, 0.72 at the default, 0.76 by 1.2, and 0.80 by 1.4), as the weakest combinations either accumulate enough data to sharpen or fall below the rising bar, while the median agreement climbs (0.84 at T = 1.0 to 0.89 by 1.2) and the median conjunctive order rises from 43 to 53 responses. Beyond T ≈ 1.3 combinations begin falling outside the 125-response budget, with 27 converging at 1.3, 25 at 1.35, and 23 by 1.4. T = 1.0 certifies 27 of the 30 combinations at a median agreement of 0.84 and a worst case of 0.717, and it carries the geometric meaning that the spread of established shares equals their measurement noise. The criterion is not fragile at T = 1.0: agreement and collection cost vary smoothly as the threshold changes, so the threshold can be treated as an operating choice rather than a brittle constant. A practitioner who can afford more collection, or who needs a tighter floor, can read a stricter operating point directly off the curve. Figure 11: Sensitivity of the conjunctive criterion to the structural-SNR threshold T, with the BIC-iso rank-stability test held fixed. Left axis: weighted Spearman agreement between the certified ordering and the full-window ordering, shown as the median across certified combinations (squares) and the worst case, or floor (circles). Right axis: median conjunctive convergence order (triangles). The dotted line marks the default T = 1.0. Agreement and collection cost both rise smoothly with the threshold; below T ≈ 0.8 agreement is flat, so lower thresholds add no fidelity while admitting the weakest orderings. 5 Discussion The conjunctive convergence criterion identifies a richer condition than either rank stability or structural sufficiency alone, and the results show both conditions doing real work. Rank stability can hold while confidence intervals remain wide enough to make adjacent domains statistically indistinguishable: structural SNR fired after stability in 11 of the 27 converged combinations and never fired in three more. Structural SNR can clear its threshold while the rank ordering is still settling: SNR fired before stability in the other 16, by margins as large as 46 responses (SearchGPT / “corneal tomography”: SNR at response 15, stability at 61). The conjunction protects against both failure modes: stability ensures the ordering is not changing, and sufficiency ensures the ordering is precise enough to be meaningful. The finding that citation density and within-query clustering—not platform identity—govern convergence speed has a direct practical implication. When it converges, SearchGPT does so at response counts comparable to Gemini and Perplexity. The key is effective sample size, not total citation volume: SearchGPT emits roughly one-sixth as many citations per response, but its low within-query citation clustering (median end-of-window DEFF 2.1) means each response contributes roughly half an effective response, while Gemini’s high clustering (median DEFF 3.9) means each response contributes roughly one-quarter (the effective response count R/DEFF of §4.1). Equal-n collection budgets are therefore not equally informative across platforms, and practitioners comparing platforms on the same nominal query count may be comparing unequal amounts of effective data. The ceiling-effect finding has a direct implication for collection protocol design. Any platform with low citation density will appear unstable under a fixed correlation threshold regardless of whether its ordering has settled, because the mechanics of sparse citation make the threshold unreachable rather than unmet. Practitioners relying on fixed thresholds for platforms like SearchGPT—or any future platform with similarly sparse citation behavior—will systematically over-collect. The plateau detector resolves this by asking whether correlation has stopped changing rather than whether it has reached a target level. The practical upshot is that collection protocols should not be calibrated to a fixed ρ target; they should be calibrated to whether the trajectory has flattened. The SNR sufficiency criterion belongs to the family of relative-precision stopping rules described by Chow and Robbins (1965). Unlike fixed-width procedures that require an absolute precision target, the SNR criterion sets the stopping target relative to the observed signal spread σ, making it self-calibrating to the citation distribution at hand. When the practitioner has a specific precision requirement—for instance, a target CI half-width of one percentage point for a domain with a ten percent share—that target can be stated directly. The SNR threshold is the appropriate default when no such requirement can be specified in advance. A threshold-sensitivity analysis (§4.7) confirms the choice is not fragile: agreement and collection cost vary smoothly across a wide range of the threshold, with no value at which the criterion’s behavior changes abruptly. The validation analysis of §4.6 provides an internal check on the framework’s central claim. A stopping rule is only as good as the orderings it certifies, and the certified orderings agree reasonably with the end-of-window orderings (median 0.844); more to the point, the refusals include the weakest orderings in the dataset, which is where the sufficiency condition earns its keep. The same analysis quantifies the cost of dropping the sufficiency condition: rank stability alone would have certified an ordering agreeing only 0.364 with the final ordering on SearchGPT / “identity protection”. Where structural sufficiency delayed the stop, the certified ordering improved in all 11 cases. The convergence criterion can serve as active stopping logic: once both conditions hold simultaneously, the collection can be concluded and reporting begun. In practice, the criterion is most naturally reviewed at intervals consistent with balanced query design—for instance, after each complete set of query types has been run—to preserve equal representation across prompt types, user profiles, and consumer intent categories. When the criterion is not satisfied at end of collection, the appropriate response is to continue gathering data, guided by the constraints of a balanced dataset. 5.1 Limitations The dataset covers ten topics across three platforms during a single collection period. Generalizability of the specific convergence thresholds to other platforms, other topic categories, or different time periods is not established empirically here. The qualitative findings—that citation density and clustering govern convergence speed, and that no fixed query count is universally sufficient—are structural consequences of the citation-generation process and expected to generalize. Convergence is defined relative to the query population that is sampled, not to a query-independent notion of visibility. The citation distribution a platform produces depends on which queries are posed, so a different query design (real user queries, a different generating model, or a different mix of query types) would sample a different distribution and could converge at a different point. The criterion certifies that a given query design has been sampled sufficiently; it does not certify that the design is itself representative of the questions a brand cares about. That representativeness is an input the analyst must supply. The framework assumes the citation distribution is stationary over the collection window. It asks whether enough data has been drawn from a fixed distribution, not whether that distribution has itself changed. Generative platforms are updated over time (models, indexes, and ranking logic all change), and a collection that spans such a change would see genuine movement in the citation distribution. The convergence criterion cannot separate that movement from ordinary sampling variability; within a single collection it would read real drift as continued instability or as a failure to converge. Detecting change between collection periods is a distinct problem, one this paper does not explore. The analysis treats all queries within a platform-topic combination as a single set. In practice, the queries used for measurement may contain multiple facets, query types with options such as ‘expert recommendations’ or ‘product comparisons,’ various geographies, and more. An analyst focusing attention on a single facet option would necessarily have a proportionally smaller effective response count C to work with, which widens bootstrap confidence intervals and raises the CI half-width entering the SNR denominator. Rank stability may likewise require more responses to certify on the reduced series. The sufficiency criterion captures this shrinkage automatically: a lower SNR on the filtered subset signals that the narrower dataset cannot yet resolve citation-share differences at the required precision, and additional queries of the target type are needed before conclusions can be drawn. Analysts performing query-type-stratified comparisons should treat convergence as a per-stratum property and report the sufficiency ratio separately for each stratum. Similarly, the analysis treats each platform-topic combination on its own. Combining several platforms into a single, all-up measure introduces a further complication, because platforms differ in citation density by more than sixfold: Gemini and Perplexity emit tens of citations per response while SearchGPT emits four to six. Pooling raw citation counts across platforms weights each platform by its citation volume, so a dense platform overshadows sparse ones and the aggregate comes to reflect mostly the densest. Share-based metrics avoid this by normalizing within each platform before the platforms are combined, so that each contributes equally regardless of volume. The same discipline must extend to the uncertainty estimates: confidence intervals, and therefore structural SNR and rank CIs, should be computed per platform and then averaged across platforms, not pooled from raw counts, since a pooled bootstrap would once again let the high-volume platform dominate the variance. Convergence, likewise, is best treated as a per-platform property; an all-up result is ready only once every constituent platform has individually converged. Bootstrap BCa confidence intervals for rank CIs and rank displacement CIs are computationally intensive at scale. For studies where bootstrap cost is prohibitive, a Wilson score interval (Wilson, 1927) with per-domain design effect adjustment (Wilson + DEFF) provides a fast analytical alternative for the convergence criterion. However, an important subtlety governs its use: a Wilson interval’s lower bound is strictly positive for any domain with at least one citation, so it cannot reproduce the boundary between observed and established domains used throughout this paper, which is defined by a confidence interval that excludes zero. A Wilson-based pipeline must therefore identify the established set against a positive reference, such as the noise floor implied by the power-law fit, rather than against zero, and the resulting set will differ slightly from the bootstrap’s. All results reported here use the bootstrap, whose lower bound can reach zero and so identifies the established set directly. 6 Conclusion We introduced a two-stage convergence framework for citation visibility measurement in generative search engines. The first stage, rank stability, determines when the rank ordering of observed domains, dominated by high-share domains through weighting, has stopped changing. The second stage, structural sufficiency, determines when citation share estimates are precise enough relative to the distributional spread to support meaningful rank-based inference. Both conditions are required; neither alone is sufficient. Applied to 30 platform-topic combinations across three platforms, the conjunctive criterion converges for 27 within a 125-response collection window. The three non-converging combinations are all on SearchGPT and all are signal-limited, with citation shares too tightly packed relative to measurement precision for structural SNR to clear its threshold within 125 responses. The validation analysis supports the criterion on both sides of the decision: the certified stops agree reasonably with the end-of-window orderings, and, more tellingly, the three signal-limited refusals include the worst-agreeing ordering in the dataset. The framework is fully data-adaptive: it conditions the stopping criterion on the observed citation density, within-query clustering, and share spread at each evaluation point, rather than on assumed parameter values. The stopping rule sets no target query count, correlation level, or CI width, and the structural constants it does retain govern how much confirmation it requires, not where it must end up; the framework tells you when you have arrived, not how far away you are. No fixed query count can be justified a priori across contexts; in this dataset no budget below 94 responses would have certified every converging combination, and three combinations were not certifiable within 125. A collection that passes both conditions supports rank-based reporting at the top of the distribution, share comparisons among established domains, and a principled baseline for period-over-period change detection. 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From Stochastic to Stable: Rank Stability and Structural Sufficiency in AI Visibility Measurement

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