Score Cloud Analysis for Rule-Aware Ranking Robustness Under Discrete Judgment Uncertainty

Abstract

Rule-defined rankings often transform continuous marks, discrete judgments, trimming rules, caps, truncation, and tie-breaking variables into a single official order. When ranking margins are small, a formally valid outcome may nevertheless be sensitive to marginal changes in the recorded decision state. This article presents Score Cloud Analysis as a rule-aware statistical sensitivity-reporting method for such systems. The method represents the official score as a deterministic function of recorded inputs and recomputes scores and ranks under finite perturbations, rather than relying on local linear approximations. It defines deterministic diagnostics, including directional Group A/Group C (A/C) decision exposures and their aggregate contested-point exposure, total sensitivity exposure, fragility-to-margin ratios, Score Cloud overlap, and single-call rank sensitivity, and separates these from scenario-conditional Monte Carlo Rank Cloud frequencies. The method is illustrated using a synthetic Wushu Taolu case study because that setting contains majority decisions, trimmed rater marks, discrete difficulty values, Head Judge adjustments, and tie-break rules. The synthetic experiment is an internal-consistency stress test, not an empirical validation and not an estimate of judging-error rates. A small sensitivity study varies perturbation scale and intra-athlete dependence to show which conclusions are scenario-specific. The method separates procedural validity from local rank robustness and is transferable to other reconstructable, rule-based, rater-mediated ranking systems.

1. Introduction

Rankings are often reported as if they were simple orderings of observed quantities. In many applied settings, however, the reported order is generated by a rule-defined scoring procedure. The score may combine continuous marks, binary or categorical judgments, trimming rules, caps, truncation conventions, and tie-breaking variables. In such systems, the official result can be perfectly valid under the rule while the local ordering remains sensitive to small changes in recorded decision states. The statistical problem is therefore not only how to compute the official score, but also how to report whether the induced ranking is stable under explicitly specified perturbations.

A minimal example makes the operational question concrete. Consider two athletes separated by only 0.003 points. If one recorded 2–1 decision in Group A or Group C (A/C) is reversed, the resulting score change may equal or exceed that official margin. Full score recomputation can then produce a score tie or inversion, after which the official tie-break chain must also be recomputed. The recorded result can remain procedurally valid while its local stability is limited; Score Cloud Analysis is designed to report that distinction.

This problem appears in league tables, educational accountability systems, clinical composite scores, grant panels, rater-mediated assessments, and judged sports. Previous statistical work has discussed uncertainty in institutional rankings and league tables [1,2], uncertainty and sensitivity analysis for composite-indicator rankings [3], global and local sensitivity analysis [4,5], uncertainty propagation [6,7], rater reliability and generalizability [8,9,10,11,12], judged-sport rater behavior [13,14], and robustness of estimators under contamination [15,16]. The present article does not replace these studies. It links them to a narrower reporting problem: how to diagnose local rank sensitivity when the ranking is generated by a known nonlinear rule containing discrete decisions and ordered statistics.

Modern Wushu Taolu is used as a synthetic case study because its rulebook offers a compact example of a nonlinear, rater-mediated scoring grammar. The 2024 International Wushu Federation rules define events with Degree of Difficulty using Quality of Movements, Overall Performance, and Degree of Difficulty components, and events without Degree of Difficulty, including Duilian and Jiti, using Quality of Movements and Overall Performance [17,18]. The rules include majority decisions, a five-mark trimmed Overall Performance component, Head Judge adjustments, and event-specific tie-breaks. These features make the setting useful for testing a rule-aware sensitivity-reporting method. The manuscript does not analyze any empirical competition result.

The method proposed here is called Score Cloud Analysis. It treats the official score as the output of a deterministic rule function, then studies finite perturbations of selected input states. The key distinction is between procedural validity and rank robustness. Procedural validity answers who wins under the rule. Rank robustness asks how stable that ordering is under declared perturbations of marginal calls, retained marks, adjustment terms, or tie-break variables.

This distinction is deliberately modest. A ranking can be locally robust under the recorded rule and still be biased relative to an unobserved true performance order if all recorded evaluations share a systematic error. Conversely, a ranking can be locally sensitive even when the recorded calls are substantively correct. Score Cloud Analysis estimates neither true athletic merit nor empirical rater error. It reports sensitivity of the official ranking rule under explicit deterministic or probabilistic perturbation assumptions.

Figure 1 summarizes the analytical sequence. The official record is transformed by a score function and a ranking function. Deterministic diagnostics describe score and rank sensitivity from recorded marginal states. Scenario-conditional perturbation models add weights to perturbation states and produce Monte Carlo Rank Cloud frequencies. Empirical calibration would require independent review or repeated panels.

Table 1. Positioning of the method relative to adjacent statistical approaches.

Related Approach	Usual Target	What Is Retained Here	Added Reporting Object
Uncertainty propagation	Distribution of a measured or model-derived quantity	Monte Carlo propagation through the complete rule-defined map	Rank frequencies after official tie-break recomputation
Finite-difference sensitivity	Response to a specified input perturbation	Exact recomputation under thresholds, caps, trimming, truncation, and tie-break rules	Directional score exposures and single-call rank sensitivity
Rank uncertainty	Uncertainty in an ordering or league position	Pairwise inversion and rank-persistence frequencies	Rule-aware Rank Cloud with official tie-breaks
Rater reliability	Agreement, severity, or variance components	Recognition that recorded marks are rater-mediated	No reliability, severity, or variance component is estimated without repeated empirical ratings
Robust statistics	Resistance to outliers or contamination	Trimmed and winsorized summaries motivate Overall Performance (Group B) sensitivity	Re-entry of discarded marks into tie-breaks can be diagnosed

positions the proposed reporting method relative to adjacent statistical approaches and makes explicit which elements are retained from existing methods and which reporting objects are added here.

Table 2. Claims and non-claims.

Supported by the Method	Not Supported Without Empirical Calibration
Sensitivity of the official scoring and ranking rule under declared perturbations.	Probability that a judge, athlete, panel, or federation made an error.
Deterministic exposure and scenario-conditional Rank Cloud frequencies.	True underlying athletic ranking or true performance order.
Comparison of alternative perturbation scenarios in a synthetic proof-of-concept.	Prevalence of fragility in real Wushu competitions.
Identification of pairs whose ranking relation is active under the specified stress test.	Operational policy thresholds or official review triggers.

records the main claims and non-claims. These boundaries are central to the interpretation of the synthetic experiment.

2. Materials and Methods

2.1. Structural Representation of a Rule-Defined Ranking

Let competitors be indexed by $i=1,\dots ,n$. The observed record for competitor i is denoted

$${\mathbf{x}}_i=({\mathbf{z}}_i^{AC},{\mathbf{b}}_i,{\mathbf{h}}_i,{\mathbf{t}}_i,{\mathbf{q}}_i),$$

(1)

where ${\mathbf{z}}_i^{AC}$ denotes discrete A/C decision states, ${\mathbf{b}}_i$ denotes rater marks, ${\mathbf{h}}_i$ denotes rule-based adjustment states, ${\mathbf{t}}_i$ denotes tie-break variables, and ${\mathbf{q}}_i$ contains other rule-defined quantities. The official score is represented as

$$s_i=f_{\mathcal{R}}\left({\mathbf{x}}_i\right),$$

(2)

where $\mathcal{R}$ is the rulebook. The official ranking is then

$$\mathbf{r}={\rho }_{\mathcal{R}}(\mathbf{s},\mathbf{t}),$$

(3)

where $\mathbf{s}=(s_1,\dots ,s_n)$ is the score vector, $\mathbf{t}$ contains the rule-defined tie-break variables, $f_{\mathcal{R}}$ is the score map, and ${\rho }_{\mathcal{R}}$ is the ranking map. Equations (2) and (3) are deterministic. Randomness enters only when the analyst assigns a distribution to perturbation states.

This representation is compatible with structural counterfactual notation, but it does not identify causal effects. Structural causal models use deterministic assignments together with uncertainty over some determinants [19,20]; here, selected recorded components are replaced and the deterministic rule is recomputed. The perturbations therefore define rule counterfactuals rather than identified causal effects.

Let ${\omega }_i$ denote a perturbation state:

$${\omega }_i=({\mathit{z}}_i^{AC,\ast },{\mathit{b}}_i^{\ast },{\mathit{h}}_i^{\ast },{\mathit{t}}_i^{\ast }).$$

(4)

The perturbed input is ${\mathbf{x}}_i\left({\omega }_i\right)$ and the recomputed score is

$$s_i\left({\omega }_i\right)=f_{\mathcal{R}}\left({\mathbf{x}}_i\left({\omega }_i\right)\right).$$

(5)

For a pair $(i,j)$, the recomputed ranking is

$$\mathbf{r}\left(\mathbf{\omega }\right)={\rho }_{\mathcal{R}}(\mathbf{s}\left(\mathbf{\omega }\right),\mathbf{t}\left(\mathbf{\omega }\right)).$$

(6)

This formulation avoids a linear-additive assumption. Additive decompositions may be used as screening summaries, but the operative computation is the rule-aware recomputation in Equations (5) and (6).

2.2. Wushu Taolu as a Synthetic Case Study

In the synthetic case study, the official score has the schematic form

$$s_i={\tau }_3(A_i+B_i+C_i-H_i+G_i),$$

(7)

where $A_i$ is the quality or execution component, $B_i$ is the Overall Performance component, $C_i$ is the Degree of Difficulty component when applicable, $H_i$ is a signed Head Judge deduction term expressed as a subtraction, $G_i$ is a bonus or adjustment term when present, and ${\tau }_3$ denotes the three-decimal truncation operator used in the synthetic reporting layer. Real applications must follow the event-specific official reporting rule.

The Overall Performance component is calculated from five displayed marks. If 

$$b_{i\left(1\right)}\le b_{i\left(2\right)}\le b_{i\left(3\right)}\le b_{i\left(4\right)}\le b_{i\left(5\right)},$$

(8)

then

$$B_i={\tau }_3\left\{{\displaystyle \frac{b_{i\left(2\right)}+b_{i\left(3\right)}+b_{i\left(4\right)}}{3}}\right\}.$$

(9)

The lower discarded mark is

$$B_i^{LD}=b_{i\left(1\right)}.$$

(10)

This quantity is excluded from the B-score estimator in Equation (9) but may re-enter the official ranking function as a tie-break variable in events where the rulebook specifies it. Thus, full reconstruction of tied-score ranking requires the five-mark vector, not only the trimmed mean. Supplementary File S1 gives the event-specific rule details and the treatment of residual official ex aequo.

For the synthetic output files, a final athlete identifier may be used only to sort rows deterministically in comma-separated values (CSV) tables. It is not part of ${\rho }_{\mathcal{R}}$ and has no sporting interpretation.

2.3. Deterministic Score-Cloud Diagnostics

A marginal A/C decision is a recorded decision state whose outcome would change if one member of the three-judge group changed vote. For example, under a two-out-of-three threshold, 2–1 and 1–2 states are one-vote reversible. Let ${\mathcal{M}}_i$ be the set of marginal A/C decisions for competitor i. Each decision $\mathsf{\ell }\in {\mathcal{M}}_i$ has a signed finite-difference effect

$${\delta }_{i\mathsf{\ell }}=f_{\mathcal{R}}\left({\mathbf{x}}_i^{\left(\mathsf{\ell }\right)}\right)-f_{\mathcal{R}}\left({\mathbf{x}}_i\right),$$

(11)

where ${\mathbf{x}}_i^{\left(\mathsf{\ell }\right)}$ differs from ${\mathbf{x}}_i$ only by the specified rule-counterfactual reversal of decision ℓ.

Directional A/C exposure is defined as

$$E_i^{-}=\sum _{\mathsf{\ell }\in {\mathcal{M}}_i}\max \{-{\delta }_{i\mathsf{\ell }},0\},E_i^{+}=\sum _{\mathsf{\ell }\in {\mathcal{M}}_i}\max \{{\delta }_{i\mathsf{\ell }},0\},$$

(12)

with aggregate contested-point exposure (CPE)

$${\mathrm{CPE}}_i^{AC}=E_i^{-}+E_i^{+}.$$

(13)

The exact Score Cloud is defined first as the image of the admissible perturbation set,

$${\mathcal{C}}_i^{\mathrm{exact}}=\{f_{\mathcal{R}}\left({\mathbf{x}}_i\left({\omega }_i\right)\right):{\omega }_i\in {\mathrm{\Omega }}_i\},$$

(14)

with exact min–max hull

$${\mathcal{H}}_i^{\mathrm{exact}}=\left[\underset{c\in {\mathcal{C}}_i^{\mathrm{exact}}}{\min }c,\underset{c\in {\mathcal{C}}_i^{\mathrm{exact}}}{\max }c\right].$$

(15)

A separate additive screening envelope is used when one-at-a-time finite differences are summarized without exhaustive joint enumeration. For A/C calls alone,

$${\mathcal{E}}_i^{AC,\mathrm{scr}}=[s_i-E_i^{-},s_i+E_i^{+}].$$

(16)

This envelope is not called exact unless the admissible perturbations are jointly additive and interaction-free under the rule. For $|{\mathcal{M}}_i|$ binary A/C decisions, exhaustive enumeration may scale as $2^{|{\mathcal{M}}_i|}$. Monte Carlo sampling is a computational approximation to weighted enumeration when the perturbation space is large or when scenario weights are assigned.

For adjacent pair $(i,j)$ with official leader i, define the official margin

$${\Delta }_{ij}=s_i-s_j.$$

(17)

The raw A/C fragility-to-margin ratio (FMR) is

$${\mathrm{FMR}}_{ij}^{AC}={\displaystyle \frac{E_i^{-}+E_j^{+}}{|{\Delta }_{ij}|}},{\Delta }_{ij}\ne 0.$$

(18)

When ${\Delta }_{ij}=0$, the raw ratio is undefined and the pair is tie-break active. For plotting or screening only, an epsilon-regularized value may be computed as

$${\mathrm{FMR}}_{ij}^{AC,\epsilon }={\displaystyle \frac{E_i^{-}+E_j^{+}}{\max \left(\right|{\Delta }_{ij}|,\epsilon )}}.$$

(19)

For the synthetic implementation, the deterministic B screening half-width is the declared quantity

$$U_{B,i}^{\mathrm{scr}}=u_{B,i}\ge 0.$$

(20)

The value $u_{B,i}$ is pre-specified by the synthetic B variation profile and reported as B_block_halfwidth in Supplementary File S2: 0.019 for low-variation profiles, 0.030 for moderate-variation profiles, and 0.055–0.070 for high-variation profiles. It is independent of the Gaussian Monte Carlo scale and is neither a distributional quantile nor a confidence limit. If $d_{H,i}$ is the signed total-score change when the Head Judge perturbation state is activated, define directional terms

$$U_{H,i}^{-}=\max (-d_{H,i},0),U_{H,i}^{+}=\max (d_{H,i},0).$$

(21)

The total athlete-level screening interval displayed in Figure 2 is therefore

$${\mathcal{E}}_i^{\mathrm{tot},\mathrm{scr}}=\left[s_i-E_i^{-}-U_{B,i}^{\mathrm{scr}}-U_{H,i}^{-},s_i+E_i^{+}+U_{B,i}^{\mathrm{scr}}+U_{H,i}^{+}\right].$$

(22)

For an adjacent official leader i and follower j, screening contact is declared when the leader’s lower endpoint is no greater than the follower’s upper endpoint; equality is recorded separately as tie activation. The corresponding total sensitivity exposure (TSE) is

$${\mathrm{TSE}}_{ij}=E_i^{-}+E_j^{+}+U_{B,i}^{\mathrm{scr}}+U_{B,j}^{\mathrm{scr}}+U_{H,i}^{-}+U_{H,j}^{+}.$$

(23)

The total epsilon-screened fragility-to-margin ratio is

$${\mathrm{FMR}}_{ij}^{\mathrm{tot},\epsilon }={\displaystyle \frac{{\mathrm{TSE}}_{ij}}{\max \left(\right|{\Delta }_{ij}|,\epsilon )}}.$$

(24)

The synthetic study uses $\epsilon =0.001$ score units, matching the public three-decimal reporting grid. Recomputing the synthetic labels with $\epsilon \in \{0.0005,0.001,0.002,0.005\}$ leaves the baseline red/amber/green pair counts unchanged (11/10/8). Total FMR is interpreted only as a screening ratio, not as a probability.

Figure 3 shows why FMR must be interpreted with caution near zero margins. The statistic is informative as a normalized screening value, but it is undefined at exact zero margin and can become very large for millipoint-scale separations.

Figure 3. Fragility-to-margin ratio (FMR) as a function of official pairwise margin for several fixed exposure levels. Dashed horizontal reference lines at FMR = 1 and FMR = 5 are the synthetic screening thresholds used in Table 3; they are illustrative, not empirically validated cut-points. The figure also shows the instability of raw ratios near zero margins.

Figure 3. Fragility-to-margin ratio (FMR) as a function of official pairwise margin for several fixed exposure levels. Dashed horizontal reference lines at FMR = 1 and FMR = 5 are the synthetic screening thresholds used in Table 3; they are illustrative, not empirically validated cut-points. The figure also shows the instability of raw ratios near zero margins.

Table 3. Synthetic adjacent-pair screening labels used in the stress test. The rules are declared conventions for the artificial examples only. A/C denotes Group A/Group C decisions.

Label	Synthetic Rule
Red	At least one of the following holds: ${\widehat{\pi }}_{ij}^{R,\mathrm{inv}}\ge 0.20$; ${\mathrm{FMR}}_{ij}^{\mathrm{tot},\epsilon }\ge 5$; A/C single-call tie/rank activity is present; or the official score margin is zero.
Amber	The pair is not red, and at least one of the following holds: ${\widehat{\pi }}_{ij}^{R,\mathrm{inv}}\ge 0.05$; ${\mathrm{FMR}}_{ij}^{\mathrm{tot},\epsilon }\ge 1$; or deterministic screening envelope overlap/tie activity is present.
Green	Neither the red nor the amber condition holds under the declared synthetic screening convention.

Table 3. Synthetic adjacent-pair screening labels used in the stress test. The rules are declared conventions for the artificial examples only. A/C denotes Group A/Group C decisions.

Label	Synthetic Rule
Red	At least one of the following holds: ${\widehat{\pi }}_{ij}^{R,\mathrm{inv}}\ge 0.20$; ${\mathrm{FMR}}_{ij}^{\mathrm{tot},\epsilon }\ge 5$; A/C single-call tie/rank activity is present; or the official score margin is zero.
Amber	The pair is not red, and at least one of the following holds: ${\widehat{\pi }}_{ij}^{R,\mathrm{inv}}\ge 0.05$; ${\mathrm{FMR}}_{ij}^{\mathrm{tot},\epsilon }\ge 1$; or deterministic screening envelope overlap/tie activity is present.
Green	Neither the red nor the amber condition holds under the declared synthetic screening convention.

2.4. Single-Call Rank Sensitivity

A pair is A/C single-call active when changing one marginal A/C decision is sufficient to produce a score inversion, a score tie, or tie-break activation under the official ranking rule. Let

$$L_{ij}=\max \{\underset{\mathsf{\ell }\in {\mathcal{M}}_i}{\max }{(-{\delta }_{i\mathsf{\ell }})}_{+},\underset{\mathsf{\ell }\in {\mathcal{M}}_j}{\max }{\left({\delta }_{j\mathsf{\ell }}\right)}_{+}\}.$$

(25)

The pair is single-call rank active if

$$L_{ij}\ge {\Delta }_{ij}$$

(26)

on the official reporting grid. Equality is important because a score tie can activate official tie-breaks.

2.5. Group B Sensitivity and Discarded Mark Re-Entry

Before display quantization, a single retained-mark perturbation of $+0.01$ changes the retained three-mark mean by

$$\Delta {\overline{B}}_{\mathrm{ret}}={\displaystyle \frac{0.01}{3}}=0.003\overline{3},$$

(27)

provided that the retained set is unchanged. After applying ${\tau }_3$, the displayed change is instead

$$\Delta B_{\mathrm{disp}}={\tau }_3({\overline{B}}_{\mathrm{ret}}+0.01/3)-{\tau }_3\left({\overline{B}}_{\mathrm{ret}}\right),$$

(28)

and need not equal $0.003\overline{3}$ exactly. Likewise, a common-mode $+0.01$ shift in all three retained marks produces an unquantized retained-mean change of $0.01$, while the displayed change remains subject to the reporting operator. If retained-mark perturbations have common variance ${\sigma }_B^2$ and pairwise correlation $\rho$, the retained-block surrogate, before quantization, re-ordering, re-trimming, and truncation, has

$${\sigma }_{B,\mathrm{eff}}^2={\displaystyle \frac{{\sigma }_B^2}{3}}(1+2\rho ).$$

(29)

The mark-level synthetic baseline perturbs all five displayed B marks, returns them to the two-decimal grid, orders them, trims them, recomputes $B_i$, recomputes $B_i^{LD}$, and then recomputes ${\rho }_{\mathcal{R}}$. The retained-block model is retained only as a comparative surrogate.

The lower discarded B mark creates a rule-sensitivity feature. The trimming procedure excludes $b_{i\left(1\right)}$ from Equation (9), yet $B_i^{LD}=b_{i\left(1\right)}$ can be used to resolve a tied official score and tied B score. A mark excluded from score estimation can therefore re-enter the ranking function. This is not an error in the rule; it is a feature that must be retained in a rule-aware ranking analysis. Alternative summaries such as the lowest retained central mark, a winsorized mean, or a dispersion-based panel-consensus criterion could produce a different ordering under the same five marks. The implication is not that the official tie-break is wrong, but that equal-score cases can be sensitive to the chosen ranking statistic.

2.6. Scenario-Conditional Rank Clouds

A deterministic Score Cloud enumerates or selects admissible perturbation states. A Rank Cloud assigns a scenario distribution to those states and propagates them through the official score and ranking functions. Let ${\mathbf{\omega }}^{\left(q\right)}$ denote a perturbation state sampled at Monte Carlo iteration $q=1,\dots ,M$. The recomputed rank vector is

$${\mathbf{r}}^{\left(q\right)}={\rho }_{\mathcal{R}}(\mathbf{s}\left({\mathbf{\omega }}^{\left(q\right)}\right),\mathbf{t}\left({\mathbf{\omega }}^{\left(q\right)}\right)).$$

(30)

The tie-aware pairwise rank inversion frequency is

$${\widehat{\pi }}_{ij}^{R,\mathrm{inv}}={\displaystyle \frac{1}{M}}{\displaystyle \sum _{q=1}^M}\mathbb{I}\{r_j^{\left(q\right)}<r_i^{\left(q\right)}\}.$$

(31)

Separate quantities are reported for score inversion and score tie:

$${\widehat{\pi }}_{ij}^{S,\mathrm{inv}}={\displaystyle \frac{1}{M}}{\displaystyle \sum _{q=1}^M}\mathbb{I}\{s_j^{\left(q\right)}>s_i^{\left(q\right)}\},{\widehat{\pi }}_{ij}^{S,\mathrm{tie}}={\displaystyle \frac{1}{M}}{\displaystyle \sum _{q=1}^M}\mathbb{I}\{s_j^{\left(q\right)}=s_i^{\left(q\right)}\}.$$

(32)

The empirical Monte Carlo standard error (MCSE) of a frequency estimate $\widehat{p}$ is

$$\widehat{\mathrm{se}}\left(\widehat{p}\right)=\sqrt{{\displaystyle \frac{\widehat{p}(1-\widehat{p})}{M}}}.$$

(33)

If $x=M\widehat{p}$ events are observed, the two-sided 95% exact binomial Monte Carlo uncertainty interval is

$$\begin{array}{cc}\hfill I_{0.95}^{\mathrm{MC}}(x;M)& =[L_x,U_x],\hfill \\ \hfill L_x& =\left\{\begin{array}{cc}0,\hfill & x=0,\hfill \\ Q_{\mathrm{Beta}}(0.025;x,M-x+1),\hfill & x>0,\hfill \end{array}\right.\hfill \\ \hfill U_x& =\left\{\begin{array}{cc}1,\hfill & x=M,\hfill \\ Q_{\mathrm{Beta}}(0.975;x+1,M-x),\hfill & x<M.\hfill \end{array}\right.\hfill \end{array}$$

(34)

where $Q_{\mathrm{Beta}}(a;\alpha ,\beta )$ is the a-quantile of a Beta distribution with parameters $(\alpha ,\beta )$. This Clopper–Pearson interval is applied to the principal pairwise inversion frequencies and to every athlete–rank cell of the Rank Cloud. It quantifies only finite-simulation uncertainty conditional on the declared perturbation model; it is not an empirical confidence interval for judging reliability, latent performance, or ranking correctness. For athlete i, the individual Rank Cloud frequency at rank r is

$${\widehat{p}}_{i,r}={\displaystyle \frac{1}{M}}{\displaystyle \sum _{q=1}^M}\mathbb{I}\{r_i^{\left(q\right)}=r\}.$$

(35)

In the synthetic implementation, the declared tie-break chain is recomputed at every iteration. No residual sporting tie occurs in the generated baseline; if a real rulebook permits residual ex aequo, the state space must include the corresponding tied-rank category rather than resolving it with a numerical tolerance or arbitrary identifier. All Monte Carlo values in this article are scenario-conditional sensitivity frequencies. Under a Bayesian interpretation, a genuinely prior predictive construction requires a perturbation model specified before conditioning on the realized official record. If probabilities are elicited after observing that record, they are more accurately described as a subjective predictive scenario conditional on the observed record. Under empirical calibration, those probabilities would be estimated from independent review, repeated panels, or historical raw judging logs.

2.7. Choosing a Perturbation Model

The perturbation model should be selected before analysis.

Table 4. Practical workflow for perturbation model selection.

Model Choice	Use When	Required Information	Main Risk
Independent calls	No evidence of shared ambiguity or panel-level dependence is available.	Declared perturbation probabilities for each decision class.	May understate uncertainty if calls are correlated.
Intra-athlete random effect	Several marginal calls may share routine-level ambiguity.	Repeated review or elicited intraclass dependence.	Sensitivity depends on assumed dependence.
Panel-level effect	Rater-scale or panel interpretation may affect several marks.	Repeated panels, judge logs, or calibration exercises.	Confounding with athlete quality.
Bayesian predictive model	Expert beliefs or empirical data can be encoded as priors/posteriors.	Prior distributions or calibration data.	Results inherit prior and model assumptions.

gives a practical workflow. The independent model is a transparent baseline, but it can be non-conservative when calls share routine-level ambiguity, common rater interpretation, or panel-level conditions. A random effect model can represent shared uncertainty among calls of the same athlete. A panel-level structure is appropriate when the same rater or panel tendency affects multiple items. Empirical calibration is required before scenario-conditional frequencies can be interpreted as empirical predictive probabilities.

2.8. Diagnostic Quantities

Table 5. Principal quantities reported by Score Cloud Analysis. Definition/status distinguishes proposed diagnostics, reporting constructs, and adapted displays. CPE denotes contested-point exposure; TSE denotes total sensitivity exposure; FMR denotes fragility-to-margin ratio.

Quantity	Definition/Status	Required Inputs	Interpretation	Not Interpreted as
${\mathrm{CPE}}_i^{AC}$	$E_i^{-}+E_i^{+}$; proposed aggregate diagnostic	Marginal A/C calls and signed finite differences ${\delta }_{i\mathsf{\ell }}$	Aggregate contested-point exposure; $E_i^{-}$ and $E_i^{+}$ are the directional quantities	Error probability
${\mathrm{TSE}}_{ij}$	Equation (23); reporting construct	$E_i^{-}$, $E_j^{+}$, $U_{B,i}^{\mathrm{scr}}$, $U_{B,j}^{\mathrm{scr}}$, $U_{H,i}^{-}$, and $U_{H,j}^{+}$	Ranking-active pairwise screening exposure	Calibrated risk
FMR	Pairwise exposure divided by official margin; proposed diagnostic	Relevant pairwise exposure and ${\Delta }_{ij}$; $\epsilon$ regularization is used for synthetic screening and plotting	Exposure relative to the official margin	Probability
Single-call activity	$L_{ij}\ge {\Delta }_{ij}$, including tie-active equality; proposed diagnostic	Largest ranking-relevant marginal A/C call and the tie-aware ranking rule	Whether one A/C call can activate a tie or rank change	Proof of a wrong call
Score Cloud	Image of $f_{\mathcal{R}}$ under admissible perturbation states; adapted uncertainty-set display	Scoring function, admissible states, and score inputs	Rule-aware score support under declared states	True-score interval
Rank Cloud	Distribution of ${\rho }_{\mathcal{R}}$ under sampled perturbations; adapted rank uncertainty display	Ranking function, perturbation distribution, and tie-break variables	Scenario-conditional rank frequency; finite-Monte Carlo uncertainty is reported separately	True-rank probability
${\widehat{\pi }}_{ij}^{R,\mathrm{inv}}$	Equation (31); adapted Monte Carlo summary	Simulated ranks after recomputing ${\rho }_{\mathcal{R}}$	Scenario-conditional pairwise rank inversion frequency, reported with Equation (34)	Empirical judging-error rate

consolidates the principal quantities. The table separates deterministic screening quantities from model-dependent frequencies to reduce the risk of interpreting screening diagnostics as probabilities.

Table 6. Compact mathematical properties of the principal diagnostics.

Quantity	Boundedness	Monotonicity/Sensitivity	Zero Margin Behavior	Model Dependence
${\mathrm{CPE}}^{AC}$	Bounded by the finite sum of declared marginal A/C units and rule caps	Nondecreasing when same-direction marginal units are added	Not a ratio; defined independently of pairwise  margin	No
$\mathrm{TSE}$	Bounded by the included A/C, B screening, and directional Head Judge components	Nondecreasing in every included exposure component	Defined at zero margin; its raw ratio to margin is not	No
Raw FMR	Unbounded as $|{\Delta }_{ij}|\to 0$	Increases with exposure and decreases with nonzero margin	Undefined when ${\Delta }_{ij}=0$	No
Epsilon-screened FMR	Bounded above by maximum admissible exposure divided by the chosen $\epsilon$ floor	Used for declared synthetic screening and plotting at zero or near-zero margins	Uses $\max \left(\right|{\Delta }_{ij}|,\epsilon )$ as a screening/display convention; the raw ratio remains undefined at zero	No
Score Cloud	Bounded by the image of admissible perturbation states under score caps	Under fixed $f_{\mathcal{R}}$, nested admissible sets ${\mathrm{\Omega }}_{i,1}\subseteq {\mathrm{\Omega }}_{i,2}$ imply nested exact Score Clouds; non-nested changes imply no monotonicity	Defined at athlete level	Optional weights only
Rank Cloud	Frequencies sum to one over represented rank or rank/tie states	Changes with perturbation probabilities, dependence, and rule changes	May allocate mass to tie-break or residual-tie states when represented	Yes

summarizes boundedness, monotonicity, zero margin behavior, and model dependence for the principal diagnostics.

3. Results: Synthetic Stress Test

3.1. Design

The numerical experiment is an internal-consistency stress test, not empirical validation. It uses $n=30$ synthetic competitors and $M=50,000$ Monte Carlo replications. The design deliberately includes stable, borderline, and sensitivity-active adjacent pairs.

Table 7. Baseline reconstruction and deterministic screening design. All values are illustrative and are not calibrated to empirical competition data.

Design Element	Baseline Value or Rule	Purpose
Competitors	$n=30$ synthetic competitors	Creates adjacent-pair relations across several official margins.
Monte Carlo replications	$M=50,000$	Gives maximum plug-in binomial Monte Carlo standard error (MCSE) near 0.0022.
A/C marginal calls	Baseline outcome-reversal probability 0.28 for marginal calls; robust 3–0 calls have zero outcome-reversal probability	Controls one-vote decision perturbation activity.
Group B baseline model	Five two-decimal marks; centered Gaussian mark perturbations; mapping to the centesimal grid; clipping to the synthetic legal range $[0.00,3.00]$; sorting, trimming, ${\tau }_3$, and $B^{LD}$ recomputation	Preserves the rule-aware mark-level B state and tie-break input.
Deterministic B screening width	Pre-specified $U_{B,i}^{\mathrm{scr}}$: 0.019 (low variation), 0.030 (moderate), and 0.055–0.070 (high); independent of the Monte Carlo Gaussian scale	Supplies a transparent deterministic screening width for TSE and Figure 2.
Head Judge perturbation	Signed total-score changes of $\pm 0.10$ where declared; baseline activation probability 0.12	Tests directional Head Judge exposure in pairwise ranking relations.
Synthetic labels	Green/amber/red labels defined by declared FMR, single-call, envelope-contact, zero margin, and rank frequency rules	Provides descriptive stress test classes, not operational thresholds.

and

Table 8. Scenario-scale and dependence specification for the synthetic stress test.

Scenario Element	Declared Values	Purpose
A/C scale multipliers	Low 0.35, baseline 1.00, high 1.40; scaled probabilities clipped to $[0,1]$	Tests sensitivity to the declared call-perturbation scale.
Group B mark scales	Athlete-level standard deviations: 0.0076 (low), 0.025 (moderate), and approximately 0.048–0.060 (high); scenario multipliers 0.50, 1.00, and 1.50	Creates low-, moderate-, and high-variation mark profiles.
Group B dependence	Within-athlete mark correlations 0.10 (low), 0.20 (moderate), and 0.65 (high)	Tests dependence among marks entering the trimmed component.
Head Judge scale	Activation probabilities 0.05 (low), 0.12 (baseline), and 0.18 (high)	Varies the occurrence of declared signed Head Judge states.
Intra-athlete random effect	Independent baseline; correlated scenario uses a Beta random effect with the scaled marginal-call mean and concentration $\alpha +\beta =1/\rho -1$ at $\rho =0.30$	Represents shared ambiguity among calls for the same athlete.
Retained-block comparison	Gaussian perturbation of the retained B component, clipped to $[0.00,3.00]$; the lower discarded mark is held at baseline	Provides a compact surrogate comparison, not the baseline model.

separate baseline reconstruction and deterministic screening from scenario-scale and dependence assumptions. Full CSV files, scripts, the data dictionary, and computational consistency tests are included in Supplementary File S2. The design follows the general reporting logic of simulation studies: state the aims, data-generating mechanism, estimands, methods, and performance summaries [21].

The generator uses random seed 20260530. Gaussian Group B perturbations are centered at zero, scaled by the athlete-specific values summarized in Table 8, mapped to the two-decimal mark grid, and clipped to the synthetic legal interval $[0.00,3.00]$ before re-sorting, re-trimming, and truncation. Scores and all tie-break variables are then recomputed. The deterministic $U_{B,i}^{\mathrm{scr}}$ values in Table 7 are pre-specified independently of this Monte Carlo distribution. A/C probabilities are clipped to $[0,1]$ after scenario scaling. Equality and truncation are handled on the declared decimal grids; the implementation avoids tolerance-based sporting decisions. The exact numeric scales, Head Judge signed changes, seed, and software versions are included in Supplementary File S2. Computations were performed in Python 3.13.5 (Python Software Foundation, Beaverton, OR, USA) using the open-source packages NumPy 2.3.5, pandas 2.2.3, Matplotlib 3.10.8, and SciPy 1.17.0.

Table 3 gives the reproducible screening conventions used only in the synthetic stress test. These labels are not operational thresholds for real competitions.

3.2. Baseline Synthetic Case

Table 9. Selected adjacent-pair deterministic diagnostics in the baseline synthetic stress test. $U_B$ is the sum of the two athlete-level B screening half-widths and $U_H$ is the ranking-active directional Head Judge contribution.

Pair	Margin	${\mathit{E}}_{\mathit{i}}^{-}$	${\mathit{E}}_{\mathit{j}}^{+}$	${\mathit{U}}_{\mathit{B}}$	${\mathit{U}}_{\mathit{H}}$	TSE	FMRtot,ε	Single-Call	Label
A01–A02	0.003	1.300	0.300	0.116	0.000	1.716	572.00	yes	red
A03–A04	0.170	0.000	0.000	0.038	0.000	0.038	0.22	no	green
A06–A07	0.000	1.300	0.300	0.139	0.000	1.739	1739.00	tie-break active	red
A10–A11	0.130	0.100	0.000	0.049	0.100	0.249	1.92	no	amber

summarizes selected adjacent-pair relations. The examples show why a margin alone is insufficient. A large exposure with a small margin is sensitivity-active; a small exposure with a large margin is stable; and a zero score margin requires the tie-break rule rather than an FMR ratio.

Table 10. Scenario-conditional Monte Carlo frequencies for the same selected adjacent pairs. The 95% intervals are exact binomial Monte Carlo uncertainty intervals conditional on the baseline synthetic perturbation model; they are not empirical confidence intervals for judging reliability or ranking correctness.

Pair	Score Inversion	Score Tie	Rank Inversion	MCSE	95% Monte Carlo Interval
A01–A02	0.4837	0.0036	0.4853	0.0022	[0.4809, 0.4897]
A03–A04	0.0000	0.0000	0.0000	0.0000	$[0,7.38\times {10}^{-5}]$
A06–A07	0.4877	0.0032	0.4890	0.0022	[0.4846, 0.4934]
A10–A11	0.0377	0.0068	0.0428	0.0009	[0.0411, 0.0446]

reports the corresponding scenario-conditional Monte Carlo frequencies and finite-simulation uncertainty intervals.

For A03–A04, zero observed rank inversions and a zero plug-in MCSE do not establish that the underlying scenario probability is identically zero. With $M=50,000$ and no observed events, the two-sided 95% exact binomial Monte Carlo uncertainty interval is $[0,7.38\times {10}^{-5}]$; the corresponding one-sided 95% exact upper bound is $1-0.{05}^{1/M}\approx 5.99\times {10}^{-5}$.

The complete adjacent-pair values are provided in adjacent_pair_flags.csv. The numerical tables are included in the main text so that the synthetic demonstration can be checked without opening the Supplementary Files.

3.3. Rank Cloud and Score Cloud Behavior

Figure 2 illustrates total screening intervals for the top ten synthetic athletes. The endpoints are exactly those in Equation (22); they combine A/C directional exposure, the deterministic B screening half-width, and directional Head Judge score effects. Marker color does not encode a Group B variation profile. Instead, it records the most severe analyst-defined synthetic adjacent-pair screening class (red/amber/green) involving that athlete and either immediate ranking neighbor in the baseline result. The figure should be read as a diagnostic display of declared perturbation states, not as a confidence interval for true performance. Figure 4 shows the corresponding Rank Cloud frequencies for selected athletes under the baseline synthetic perturbation model.

3.4. Sensitivity to Perturbation Scale and Dependence

Sensitivity to perturbation assumptions is assessed by varying the principal scenario parameters. The stress test varies A/C perturbation scale, B perturbation scale, Head Judge activation probability, and intra-athlete random effect dependence.

Table 11. Aggregate scenario summaries. Counts are synthetic screening outputs; they are not empirical prevalence estimates.

Scenario	Red	Amber	Green	Pairs Changing Class	Mean Rank Inversion Frequency	Maximum Rank Inversion Frequency
Low perturbation	11	10	8	0	0.096	0.490
Baseline stress	11	10	8	0	0.162	0.640
High perturbation	12	9	8	1	0.197	0.758
Intra-athlete random effect	11	10	8	0	0.141	0.520
Retained-block surrogate	11	10	8	0	0.158	0.625

summarizes the aggregate scenario outputs. The scenarios do not define policy thresholds; they show how conclusions depend on the declared model.

Figure 5 reports a small two-dimensional scenario grid varying the A/C perturbation scale and intra-athlete random effect correlation under the mark-level B baseline. The aggregate counts conceal one class transition in the high-perturbation scenario: one pair assigned the baseline synthetic amber class moves to the synthetic red class; all other listed scenarios preserve their baseline synthetic labels. Thus, stable counts should not be used as a substitute for pairwise transition checks, while inversion frequencies can still vary substantially. This is why empirical use should pre-specify the perturbation model or calibrate it from independent review data.

3.5. Transferability Beyond the Case Study

The method is not specific to Wushu notation. Consider a generic judged scoring rule

$$s_i=\tau \left(w_1{y}_i+w_2\mathbb{I}\{z_{i1}+z_{i2}\ge 1\}+w_3\mathrm{trim}\left({\mathbf{b}}_i\right)-h_i\right),$$

(36)

where $y_i$ is continuous, $z_{i1},z_{i2}$ are discrete decisions, $\mathrm{trim}\left({\mathbf{b}}_i\right)$ is an ordered-statistic estimator, $h_i$ is an adjustment, and $\tau$ is a reporting operator. Figure 6 shows that a threshold term alone can make a small input change produce a nonlocal score jump. Any domain with a reconstructable scoring grammar and an explicit ranking rule can use the same logic.

4. Discussion

4.1. What Is New and What Is Not

The novelty lies in rule-aware reporting, not in the invention of new Monte Carlo machinery. The method combines known statistical operations in a domain where the official rule itself is nonlinear and partially discrete. The contribution is the explicit alignment of perturbation units with the official scoring grammar, the distinction between deterministic exposure and scenario-conditional Rank Clouds, and the reporting of pairwise rank sensitivity after recomputing the official ranking rule. Table 5 is therefore as important as the Monte Carlo algorithm: it prevents screening quantities from being read as probabilities.

4.2. Synthetic Demonstration Versus Empirical Validation

The stress test checks the computational behavior of the diagnostic quantities across deliberately stable, borderline, and sensitivity-active configurations. External calibration would require raw judging logs, five B marks, official tie-break variables, and independent review or repeated panels. A useful empirical endpoint would be whether pairs flagged as sensitivity-active under pre-specified rules show higher disagreement or more rank changes under independent review than pairs assigned the synthetic green class.

4.3. Bayesian Interpretation

Without empirical calibration, perturbation probabilities remain scenario inputs. A genuinely prior predictive construction is specified before conditioning on the official record; probabilities elicited after inspecting that record define a subjective predictive distribution conditional on it. A posterior predictive construction requires calibration data and an explicit likelihood. The Rank Cloud propagates whichever distribution is declared through the scoring and ranking rule; none is estimated from empirical data here.

4.4. Robustness Is Not Correctness

Rank robustness is a stability property of the official ranking rule under declared perturbations. It is not a claim that the official order equals a latent true order. A ranking may be highly robust because all perturbations preserve the order, yet still be far from a true performance order if the recorded evaluations are collectively biased. Conversely, a ranking may be sensitivity-active even when all official calls are substantively defensible. This distinction addresses the difference between rule sensitivity, rater reliability, and empirical correctness.

4.5. Applicability Beyond Wushu Taolu

The method applies to reconstructable ranking systems with four ingredients: a computable score function, observable or documentable decision units, a tie-aware ranking rule, and a declared perturbation model. Potential domains include judged sports, examination panels, grant-review panels, clinical composite scores, league tables, and other rule-based rater-mediated rankings. The substantive interpretation in each domain depends on its scoring grammar and available calibration data.

5. Limitations

The numerical study is synthetic and the perturbation model is analyst-specified; accordingly, the reported frequencies describe scenario behavior rather than empirical prevalence. Different probability scales, dependence structures, or calibration data can change Rank Cloud outputs. Empirical Wushu deployment would require raw judging logs, event-specific tie-break variables, and independent validation data. Score Cloud Analysis alone cannot identify a latent true ranking.

6. Conclusions

Score Cloud Analysis provides a rule-aware statistical method for reporting ranking robustness under discrete judgment uncertainty. It recomputes deterministic score and rank functions under finite perturbation states, separates deterministic screening diagnostics from scenario-conditional Rank Cloud frequencies, and distinguishes score ties from rank inversions. The numerical findings characterize the behavior of the method under the specified synthetic perturbation scenarios; they do not empirically confirm judging reliability, ranking correctness, or a latent true performance order. Future work should calibrate perturbation distributions using independent review or repeated panels and test whether sensitivity-active pairs show greater disagreement or rank instability than pairs assigned the synthetic green class. The method is applicable to rule-based, rater-mediated ranking systems whose scoring grammar and tie-break rule can be reconstructed.