Part VI - COLLAPSE: a quantitative method for detecting systemic bias

Part VI: COLLAPSE A quantitative directional collapse test for detecting systemic bias James D. Atkinson 2025 Abstract This paper introduces a quantitative method for identifying systemic bias through the structured assessment of statistically implausible multi-domain outcome patterns. The approach formalises the collapse analysis, a reproducible statistical instrument designed with layered operational depth: a first-stage independence check that requires only domain classification and counting, a second-stage Monte Carlo dependency model for intermediate correlation stress-testing, and a final Bayesian update that formalises posterior convergence under conservative assumptions. Where the probability of the observed configuration arising under the innocent model falls beneath actuarial materiality, the method treats that event as a positive evidentiary signal of systemic bias. The framework is implemented through a five-stage protocol: domain extraction, temporal clustering, construction of a conservative stochastic model, correlation stress-testing, and calculation of an irreconcilability threshold. Particular attention is given to the distinction between directional coherence and random variation, and to the evidential significance of uniform multi-domain deviation following a triggering perturbation. Under this framework, statistical collapse is not presented as proof of motive; it is the quantitative rejection of the hypothesis that the observed pattern could reasonably emerge by chance. A fully synthetic case study is used to demonstrate the method in practice. The analysis shows that a fifteen-domain adverse pattern, emerging immediately after a triggering perturbation event, is actuarially incompatible with any plausible model of benign system behaviour. The resulting probability collapse provides a replicable foundation for evidential inference in legal, regulatory, and governance contexts. 1 The paper therefore contributes a generalisable statistical test for systemic bias that is intentionally transparent, computationally lightweight at its entry level, and progressively rigorous under deeper modelling layers. Unlike existing approaches — which typically rely on narrative reconstruction or comparator analysis — collapse analysis provides a formally quantifiable criterion for when an innocent organisational explanation becomes probabilistically untenable. The method requires no narrative reconstruction, no subjective interpretation, and no external corroboration beyond the observed outcome vector itself, operating solely on probabilistic structure under explicitly stated assumptions and remaining formally neutral with respect to motive. Keywords: systemic bias, directional collapse, multi-domain analysis, probability collapse, dependency modelling, Monte Carlo simulation, Bayesian convergence, forensic statistics, governance systems, evidential statistics 2 Contents 1 Introduction 6 2 Methodology 7 2.1 Contradiction matrices as temporal diagnostic instruments ...... 7 2.2 Domain representation and problem formalisation ........... 9 2.3 Baseline independence model ...................... 9 2.4 Directional Collapse Under Exchangeable Gaussian Dependence . . 10 2.5 Admissible dependence and copula models ............... 13 2.6 Directional Collapse Under Bounded Dependence (Distribution-Free Alternative) ................................. 13 2.7 Explicit Exponential Constants for Directional Collapse ........ 15 2.7.1 Gaussian-Copula Collapse Constant ............... 15 2.7.2 Distribution-Free Bounded-Dependence Constant ....... 16 2.7.3 Interpretive Summary ....................... 16 2.8 Decision-Theoretic Collapse Threshold ................. 17 2.9 Dependency structures and correlation stress-testing ......... 18 2.10 Bayesian Collapse of the Innocent Hypothesis ............. 18 2.11 Bayesian update ............................... 19 2.12 Decision-theoretic interpretation and governance thresholds . . . . 20 3 Mathematical formalism 20 3.1 Probability space and domain vector ................... 20 3.2 Competing hypotheses ........................... 21 3.3 Independent Bernoulli model ....................... 21 3.4 Correlated Gaussian-copula model .................... 22 3.5 Implausibility thresholds .......................... 23 3.6 Asymptotic interpretation ......................... 24 3.7 Bayesian update, Bayes factor, and posterior convergence ...... 24 3.8 Collapse analysis .............................. 25 4 Asymptotic properties of directional collapse 25 4.1 Exchangeable dependence and exponential decay ........... 26 4.2 Bayes factor divergence under directional collapse .......... 27 5 Limitations 28 5.1 Dependence on domain specification .................. 29 5.2 Binary representation of outcomes .................... 29 5.3 Absence of causal attribution ....................... 29 5.4 Model dependence and parameter selection .............. 29 5.5 Assumption of temporal clustering .................... 30 3 5.6 Dependence on available records ..................... 30 5.7 Interpretational boundary ......................... 30 5.8 Generalisability and recalibration ..................... 30 5.9 Summary of limitations ........................... 31 5.10 Comparator independence and misuse boundary ............ 31 6 Future Work 32 6.1 Model refinement and parameter sensitivity .............. 32 6.2 Integration with governance and behavioural data ........... 32 6.3 Expansion to multi-case and cross-sectional analysis ......... 32 6.4 Alignment with evidentiary standards .................. 33 6.5 Integration with contradiction-field modelling ............. 33 6.6 Tooling, automation, and audit integration ................ 33 6.7 Ethical and regulatory implications .................... 33 7 Strengths and Contribution 34 7.1 A formal test for statistical implausibility ................ 34 7.2 Directionality and systemic inference .................. 34 7.3 Reproducibility and evidential transparency ............... 35 7.4 Position within the wider literature .................... 35 8 Conclusion 35 A How to use this method without a maths background 37 A.1 Who this is for ................................ 37 A.2 What this test does ............................. 37 A.3 What you need ................................ 37 A.4 How to run it ................................. 38 A.5 What the result means ........................... 38 A.6 What to do with it .............................. 39 A.7 Important notes ............................... 39 B Standard operating procedure for union representatives (Equality Act as an example) 39 B.1 Purpose of this appendix .......................... 39 B.2 When to use a Collapse analysis ...................... 40 B.3 Suggested wording for correspondence ................. 44 B.4 Explaining the Collapse analysis ..................... 44 B.5 Linking to section 136 Equality Act 2010 ................ 45 B.6 Requesting disclosure and justification ................. 45 B.7 Practical tips for union representatives ................. 45 B.8 Conclusion .................................. 46 4 C Worked example of a collapse analysis 46 C.1 Defining the domains ............................ 46 C.2 Rating before and after the trigger .................... 47 C.3 Interpretation ................................ 48 D One–page quick reference for Union Reps 48 E References and DOI links 49 F Collapse Analysis Protocol (CAP) 49 F.1 Overview ................................... 50 F.2 Input schema ................................ 50 F.3 Validity conditions .............................. 51 F.4 Null model specification .......................... 51 F.5 Computation of evidential probability .................. 52 F.6 Bayesian evaluation ............................. 52 F.7 Interpretation thresholds ......................... 53 F.8 Conditions under which the protocol cannot be applied ........ 53 F.9 Summary ................................... 54 5 1 Introduction This paper introduces collapse analysis, a formal evidential criterion under which uniform high-dimensional adverse outcomes become statistically incompatible with stochastic innocence. It sets out a quantitative method for evaluating whether an observed pattern of system decision-making can be reconciled with ordinary, benign random fluctuation. It provides an actuarial modelling framework that identifies when the probability of such a benign explanation becomes statistically implausible. Where the null model — that adverse outcomes arise through random variation — falls below any credible evidential threshold, the resulting implausibility constitutes evidence of a directional process. The underlying problem is well established across legal, regulatory, and governance domains. Findings of systemic adverse treatment or structural bias typically depend on narrative reconstruction, subjective assessment of credibility, and comparators that are often incomplete or ill-specified. These approaches lack a formal method for quantifying whether a multi-domain adverse pattern is compatible with chance. As a result, biased systems may remain undetected where the governance record is sparse, fragmented, or inconsistent. This paper addresses that evidential gap. It introduces a statistical mechanism for assessing directional drift across multiple domains of system conduct following a triggering perturbation. The method applies standard actuarial techniques: domain extraction, temporal clustering, dependency modelling, and Monte Carlo simulation. These tools are used to test whether the observed configuration of outcomes can be accommodated by an innocent stochastic process. Where the observed pattern is materially at odds with that model, the divergence is treated as statistical implausibility. The contribution is twofold. First, it provides a reproducible analytic procedure for detecting systemic bias without requiring assumptions about motive or intent. Second, it demonstrates how statistical implausibility — quantified formally and subject to correlation stress-testing — may serve as evidence that system behaviour has departed from benign fluctuation and entered the domain of systemic adverse treatment. A fully synthetic case study is included to illustrate the method in practice. The observed pattern comprises a sequence of adverse changes across fifteen governancesystem domains following a triggering perturbation. The model presented here evaluates whether this pattern is reconcilable with ordinary stochastic variation. The resulting findings demonstrate a level of statistical implausibility that, on any reasonable assumption, materially undermines the innocent explanation. The framework is generalisable. Although the case study illustrates systemic adverse 6 treatment, the methodology can be applied to any multi-domain system in which adverse outcome patterns must be distinguished from random fluctuation. The model therefore offers a structured, evidence-based mechanism for identifying directional bias and for strengthening the evidential foundations of legal, governance, and regulatory decision-making. Throughout this paper, “systemic bias” is used in a strictly statistical sense: a directional process that produces outcome patterns whose probability under the innocent model lies below any credible evidential threshold. 2 Methodology This section sets out a quantitative methodology for assessing whether a multi-domain pattern of adverse outcomes observed in a governance system can be reconciled with ordinary stochastic fluctuation, or whether it constitutes statistical evidence of a directional process. As in actuarial and forensic statistical practice, the analysis does not make findings of fact or law. Its purpose is to quantify the degree to which an observed configuration of outcomes can plausibly arise under a null model of benign system behaviour. The methodological framework rests on three principles: 1. Binary domain modelling: each governance-system domain is represented as a binary variable indicating adverse or neutral/positive movement. 2. Competing hypotheses: •𝐻0: outcomes arise from ordinary stochastic system fluctuation; •𝐻1: a directional process affects system-level decision-making. 3. Quantitative reconciliation: the task is to evaluate the magnitude of ℙ(𝐷∣𝐻0), where 𝐷denotes the observed outcome pattern. 2.1 Contradiction matrices as temporal diagnostic instruments The diagnostic framework supporting the quantitative model relies on two structured analytic tools: the Contradiction Matrix and the Cross-Examination Matrix. These instruments do not assign motive or liability. Their function is evidential: to test whether the system’s stated rationale can be reconciled with its own preand post-perturbation record. 7 The contradiction matrix. This matrix evaluates the relationship between recorded behaviour across multiple governance domains in two temporal windows: the pre-perturbation and post-perturbation periods. Each domain is assessed for: (i) consistency between behaviours; (ii) neutral or indeterminate alignment; or (iii) contradiction, where the asserted explanation cannot be reconciled with the observed conduct. By comparing the preand post-perturbation profiles, the matrix identifies where stable, coherent system practice gives way to contradiction or adverse divergence following a triggering perturbation. In doing so, it converts qualitative inconsistency into a structured pattern of directional deviation. Where contradictions cluster across domains that are independent or only weakly correlated, the plausibility of a benign explanation decreases sharply. The cross-examination matrix. This instrument subjects each major system decision to a three-stage analytic test: 1. Procedural compliance — whether the decision was taken in accordance with policy, statutory requirements, and recorded governance norms; 2. Substantive justification — whether the rationale is contemporaneous, documented, and internally coherent; 3. Outcome trajectory — whether the effect of the decision aligns with established system practice and precedent. A decision passes only if all three criteria are satisfied. Any failure is recorded and mapped separately for the preand post-perturbation windows. When aligned with the Contradiction Matrix, the combined output produces a multidimensional profile of systemic inconsistency and temporal drift. Diagnostic value. The matrices provide an evidential lens orthogonal to the probabilistic modelling. While the quantitative framework assesses whether the observed post-perturbation pattern could plausibly arise under an innocent stochastic process, the diagnostic matrices test whether the system itself provides a coherent non-stochastic explanation across its own record. Convergence between statistical implausibility and structural contradiction produces a strong multi-modal evidential signature of directional system behaviour. Taken together, these instruments define the admissible classification of domains for the stochastic model. Let 𝒞denote the set of domains in which narrative explanation is inconsistent with the documentary record. For all 𝑖∈𝒞, the model treats 𝑋𝑖=1as 8 an observed adverse outcome requiring probabilistic reconciliation under 𝐻0. In this way, the diagnostic instruments constrain the domain state space without imposing assumptions about motive or causation. 2.2 Domain representation and problem formalisation Let 𝑋1,𝑋2,…,𝑋𝑛(2.1) denote binary indicators of domain outcomes. For each domain 𝑖: 𝑋𝑖=⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩0, neutral or positive movement, 𝑏𝑒𝑔𝑖𝑛𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛3𝑝𝑡]1, adverse movement.(2.2) Let the observed configuration be 𝐷={𝑋1=1,𝑋2=1,…,𝑋𝑛=1}. (2.3) The competing hypotheses are: 𝐻0∶outcomes arise from ordinary stochastic system fluctuation, 𝑏𝑒𝑔𝑖𝑛𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛3𝑝𝑡]𝐻1∶a directional process induces correlated adverse outcomes. (2.4) The analysis proceeds by examining whether the likelihood of 𝐷under 𝐻0is so small that the hypothesis becomes actuarially untenable. 2.3 Baseline independence model To give the null hypothesis every benefit of the doubt, we adopt a conservative baseline model in which: • each domain has marginal adverse probability 𝑝; and • all domains behave independently. The resulting probability is: ℙ(𝐷∣𝐻0)=𝑝𝑛.(2.5) 9 Numerical Illustration. If 𝑝=0.20, then Φ−1(0.8)≈0.84, so 𝛼(0.20,𝜌)≈0.353(1−𝜌). (2.33) For 𝜌=0.5, this yields 𝛼≈0.176, implying a collapse probability on the order of 𝑒−0.176𝑛. 2.7.2 Distribution-Free Bounded-Dependence Constant Under the distribution-free framework of Theorem 2.5, assume: ℙ(𝑋𝑖=1)=𝑝, Corr(𝑋𝑖,𝑋𝑗)≤𝜌<1. (2.34) Let 𝑆𝑛=∑𝑛 𝑖=1𝑋𝑖. Then Var(𝑆𝑛)≤𝑛𝑝(1−𝑝)(1+(𝑛−1)𝜌). (2.35) Applying the Bernstein–Chernoff inequality yields: ℙ(𝑆𝑛=𝑛)≤exp ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝−𝑛2(1−𝑝)2 2𝑛𝑝(1−𝑝)(1+(𝑛−1)𝜌)+2 3𝑛(1−𝑝) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠.(2.36) Hence, the explicit exponential decay constant is: 𝛾(𝑝,𝜌,𝑛)= 𝑛(1−𝑝)2 2𝑝(1−𝑝)(1+(𝑛−1)𝜌)+2 3(1−𝑝).(2.37) For each fixed 𝑝∈(0,1)and 𝜌<1, this yields a strictly positive exponential decay rate in 𝑛, confirming that directional collapse remains exponentially implausible under any bounded-dependence innocent model. 2.7.3 Interpretive Summary Both model regimes yield explicit exponential decay: ℙ(𝐷∣𝐻0)≤⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩exp(−𝛼(𝑝,𝜌)𝑛), Gaussian copula, 𝑏𝑒𝑔𝑖𝑛𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛6𝑝𝑡]exp(−𝛾(𝑝,𝜌,𝑛)), distribution-free bounded dependence. (2.38) In both cases, the innocent hypothesis collapses exponentially in the number of aligned adverse domains unless the system is driven by a near-deterministic global synchronis16 ing mechanism. 2.8 Decision-Theoretic Collapse Threshold We now reframe the Bayesian collapse of 𝐻0as a formal decision-theoretic result. Under this framework, continued reliance on the innocent hypothesis becomes strictly suboptimal once its posterior support falls below a specified evidential threshold. Theorem 2.3 (Decision-Theoretic Collapse of 𝐻0).Let 𝐻0denote the hypothesis of stochastic innocence, and 𝐻1a directional alternative. Suppose the observed pattern 𝐷satisfies: ℙ(𝐷∣𝐻0)≤𝜀, ℙ(𝐷∣𝐻1)≈1, (2.39) for some small threshold 𝜀 ≪ 0.01representing actuarial materiality. Let ℙ(𝐻0),ℙ(𝐻1) be prior beliefs, and let 𝐿(⋅,⋅)denote a decision-theoretic loss function over hypotheses and actions. Then for any decision rule 𝛿satisfying: 𝛿(𝐷)=“retain 𝐻0”,(2.40) there exists a competing decision rule 𝛿′such that: 𝔼[𝐿(𝛿′(𝐷),𝐻)]<𝔼[𝐿(𝛿(𝐷),𝐻)], (2.41) under all loss functions in which: • false negatives under 𝐻1incur strictly greater loss than false positives under 𝐻0, • and posterior odds satisfy ℙ(𝐻1∣𝐷)≫ℙ(𝐻0∣𝐷). In particular, if 𝜀≪10−6, then 𝛿(𝐷)=“retain 𝐻0” becomes irrational under all admissible loss-sensitive models. Proof. From Bayes’ rule: ℙ(𝐻1∣𝐷) ℙ(𝐻0∣𝐷) =ℙ(𝐷∣𝐻1) ℙ(𝐷∣𝐻0)⋅ℙ(𝐻1) ℙ(𝐻0).(2.42) Under the assumptions, this ratio exceeds 1/𝜀, which is very large. Thus, posterior belief overwhelmingly favours 𝐻1. 17 Under any loss function 𝐿with asymmetric penalty (i.e., 𝐿(retain 𝐻0,𝐻1)≫𝐿(retain 𝐻1,𝐻0)), expected loss is minimised by choosing the action corresponding to 𝐻1. Therefore, retaining 𝐻0becomes strictly suboptimal. Corollary 3 (Governance Trigger Threshold).Let 𝜀denote the actuarial materiality threshold (e.g., 𝜀=10−6). Then for any configuration 𝐷satisfying: ℙ(𝐷∣𝐻0)≤𝜀, (2.43) the rational institutional response is to treat 𝐻1as evidentially dominant. Inaction becomes the higher-risk decision. Remark 4. This result provides a formal bridge between probabilistic inference and governance decision-making. It defines the evidential point at which benign explanation is no longer rationally defensible under any risk-aware institutional model. It suffices for this conclusion that the expected loss incurred by retaining 𝐻0when 𝐻1 is true strictly exceeds the loss incurred by retaining 𝐻1when 𝐻0is true. No further assumption regarding the magnitude of this asymmetry is required. 2.9 Dependency structures and correlation stress-testing Let 𝜌denote pairwise correlation between domains under 𝐻0. Correlated outcomes are generated via a Gaussian copula construction: 𝑋𝑖=𝕀(𝑍𝑖>𝜏), 𝑍∼𝒩(0,Σ𝜌). (2.44) Monte Carlo simulations are used to evaluate whether admissible correlation structures rescue 𝐻0. Results show that only near-perfect synchronisation (𝜌≈0.9) yields material probabilities of full-domain collapse. Such coupling is itself evidence of a unifying causal mechanism and therefore inconsistent with decentralised benign fluctuation. Thus, the only dependency structures capable of numerically sustaining 𝐻0simultaneously invalidate it conceptually. 2.10 Bayesian Collapse of the Innocent Hypothesis We now formalise the posterior collapse of the innocent model under observed directional outcome patterns. 18 Theorem 2.4 (Bayesian Collapse).Let 𝑋1,…,𝑋𝑛be binary domain outcomes with observed configuration 𝐷={𝑋1=⋯=𝑋𝑛=1}. Let ℙ(𝐷∣𝐻0)and ℙ(𝐷∣𝐻1)denote the probabilities of this configuration under: •𝐻0: ordinary stochastic fluctuation (e.g., copula or bounded dependence model), •𝐻1: a directional mechanism inducing full-domain adverse movement. Suppose: ℙ(𝐷∣𝐻1)≈1, ℙ(𝐷∣𝐻0)≤e−𝛼𝑛.(2.45) Then for any prior odds ℙ(𝐻1) ℙ(𝐻0)=𝑟>0, the posterior odds satisfy: ℙ(𝐻1∣𝐷) ℙ(𝐻0∣𝐷) ≥𝑟⋅e𝛼𝑛.(2.46) In particular, when 𝑛is sufficiently large that ℙ(𝐷∣𝐻0)≪10−6, the posterior overwhelmingly supports 𝐻1under any reasonable prior. Proof. Bayes’ Rule gives: ℙ(𝐻1∣𝐷) ℙ(𝐻0∣𝐷) =ℙ(𝐷∣𝐻1) ℙ(𝐷∣𝐻0)⋅ℙ(𝐻1) ℙ(𝐻0).(2.47) By assumption, ℙ(𝐷∣𝐻1)≈1and ℙ(𝐷∣𝐻0)≤e−𝛼𝑛, so: ℙ(𝐻1∣𝐷) ℙ(𝐻0∣𝐷) ≥𝑟⋅e𝛼𝑛.(2.48) Remark 5. This result provides a quantitative evidential threshold: when the observed pattern 𝐷renders the null probability exponentially small, the directional hypothesis becomes not merely plausible but overwhelmingly probable in posterior belief. 2.11 Bayesian update Under 𝐻1, we have: ℙ(𝐷∣𝐻1)≈1. (2.49) Under 𝐻0, across admissible (𝑝,𝜌): ℙ(𝐷∣𝐻0)≪10−6.(2.50) 19 Thus, the Bayes factor satisfies: ℙ(𝐷∣𝐻1) ℙ(𝐷∣𝐻0)≫106,(2.51) implying overwhelming posterior support for 𝐻1under all reasonable priors. 2.12 Decision-theoretic interpretation and governance thresholds When ℙ(𝐷∣𝐻0)≤𝜀, 𝜀≪0.01, (2.52) continued reliance on 𝐻0becomes statistically irrational under any loss-sensitive system governance model. At this threshold, inaction itself becomes the higher-risk decision, and remedial review, regulatory scrutiny, or system intervention become the rationally required responses. This does not constitute a finding of liability. It formalises the point at which stochastic innocence ceases to be probabilistically defensible. 3 Mathematical formalism This section formalises the probabilistic framework underlying the methodology. The aim is to specify the null and alternative models, the structure of the random variables, and the evidential criterion under which statistical implausibility is treated as evidence of a directional process. 3.1 Probability space and domain vector Let (Ω,ℱ,ℙ) be a probability space representing all possible configurations of outcomes across a fixed set of system domains. Let 𝑛 ∈ ℕdenote the number of domains under consideration. For each 𝑖 ∈ {1,…,𝑛}, define a Bernoulli random variable: 𝑋𝑖∶Ω→{0,1}, 𝑋𝑖(𝜔)=⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩1, adverse movement in domain 𝑖, 𝑏𝑒𝑔𝑖𝑛𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛2𝑝𝑡]0, neutral or positive movement in domain 𝑖. (3.1) 20 The joint outcome vector is: 𝑋=(𝑋1,…,𝑋𝑛)∈{0,1}𝑛.(3.2) Let 𝐷⊆{0,1}𝑛denote an observed configuration. In the fully directional case, 𝐷={(1,1,…,1)}={𝑋1=1,…,𝑋𝑛=1}. (3.3) The total number of adverse domains is: 𝐾= 𝑛 􏾜 𝑖=1𝑋𝑖,(3.4) and the fully directional configuration satisfies 𝐾=𝑛. 3.2 Competing hypotheses Two competing explanations for the observed configuration are defined: 𝐻0∶outcomes arise from ordinary stochastic system fluctuation,(3.5) 𝐻1∶a directional process induces correlated adverse outcomes.(3.6) Under 𝐻0, adverse outcomes reflect benign stochastic variation; under 𝐻1, they arise from a persistent directional mechanism. The fundamental quantity of interest is: ℙ(𝐷∣𝐻0)=ℙ(𝑋∈𝐷∣𝐻0). (3.7) 3.3 Independent Bernoulli model As a deliberately conservative baseline, assume under 𝐻0: • each domain has the same marginal probability 𝑝∈(0,1)of an adverse outcome; and • all domains are independent. 21 Formally, 𝑋𝑖∣𝐻0∼Bernoulli(𝑝), 𝑖=1,…,𝑛. (3.8) Then: ℙ(𝐷∣𝐻0)=𝑝𝑛.(3.9) Under the maximally entropy-favourable choice 𝑝=0.5, ℙ(𝐷∣𝐻0)=2−𝑛.(3.10) For the synthetic illustrative value 𝑛=15, 2−15 =1 32,768≈3.05×10−5.(3.11) More realistic values yield: 𝑝=0.20∶ (0.2)15 =3.3×10−11, 𝑝=0.10∶ (0.1)15 =1.0×10−15, 𝑝=0.05∶ (0.05)15 =3.1×10−20.(3.12) 3.4 Correlated Gaussian-copula model To allow dependence between domains, consider a Gaussian-copula construction. Let 𝑍=(𝑍1,…,𝑍𝑛)∼𝒩(0,Σ𝜌), (3.13) where Σ𝜌is the equicorrelation matrix: (Σ𝜌)𝑖𝑖 =1, (Σ𝜌)𝑖𝑗 =𝜌, 𝑖≠𝑗, 𝜌∈[0,1). (3.14) Let 𝜏satisfy ℙ(𝑍𝑖>𝜏)=𝑝 ⟺ 𝜏=Φ−1(1−𝑝). (3.15) Define 𝑋𝑖=𝕀(𝑍𝑖>𝜏). (3.16) The distribution of 𝐾 = ∑𝑛 𝑖=1𝑋𝑖has no closed form and is estimated via Monte Carlo simulation. Across realistic (𝑝,𝜌)values, the observed probabilities of 𝐾 = 𝑛lie well below standard actuarial materiality thresholds. 22 Only correlations approaching the singular limit 𝜌→1yield non-negligible probabilities of full-domain collapse. Such coupling is itself inconsistent with benign decentralised fluctuation and implies a unifying causal mechanism. 3.5 Implausibility thresholds Let 𝜀>0be a pre-specified actuarial threshold. An event 𝐷is classified as statistically implausible under 𝐻0if ℙ(𝐷∣𝐻0)≤𝜀. (3.17) With 𝜀typically in the range 10−4–10−5, both the independent and correlated models satisfy ℙ(𝐷∣𝐻0)≪𝜀in the fully directional case. Exponential scaling in the number of domains. The evidential behaviour of the framework is governed by the exponential dependence on the number of contradictory domains 𝑛. Even under the least favourable null assumption 𝑝=0.5, the probability of a fully directional configuration satisfies: ℙ(𝐷∣𝐻0)=2−𝑛.(3.18) Each additional contradictory domain halves the probability mass available to the innocent explanation. For realistic values 𝑝≪0.5, the decay accelerates further as 𝑝𝑛. No admissible correlation structure alters this exponential dependence on 𝑛without collapsing the interpretation of 𝐻0itself. Correlation modifies constant factors but does not change the order of magnitude. Rejection of stochastic innocence is therefore asymptotically controlled by 𝑛, not by fine-grained parameter tuning. Dominant driver of evidential collapse In all admissible null models, the dominant driver of evidential implausibility is the number of contradictory domains 𝑛. The probability of a fully directional configuration decays exponentially in 𝑛as 𝑝𝑛. Correlation modifies the geometry of the distribution but does not change the exponential dependence on domain cardinality without itself implying a unifying causal mechanism. 23 3.6 Asymptotic interpretation The results above establish a sharp asymptotic separation between stochastic innocence and directional structure. Under any admissible null model, the probability of a fully directional adverse configuration decays exponentially in the number of domains: ℙ(𝐷∣𝐻0)≤e−𝛼𝑛.(3.19) This behaviour is invariant to weak, moderate, or copula-induced dependence. By contrast, under a directional alternative 𝐻1, such configurations are typical rather than exceptional. The resulting Bayes factor therefore diverges at least exponentially fast in 𝑛, and posterior mass concentrates on 𝐻1for any fixed prior 𝜋1>0. The evidential force of the framework is therefore controlled primarily by the cardinality of the directional pattern, not by parameter tuning, distributional detail, or calibration finesse. Each additional contradictory domain induces a multiplicative contraction of the innocent hypothesis. This asymptotic geometry explains why the method is both simple and powerful: highdimensional directional coherence is statistically lethal to stochastic innocence even under conservatively favourable assumptions. The collapse is not the product of modelling aggressiveness but a structural consequence of high-dimensional tail behaviour. 3.7 Bayesian update, Bayes factor, and posterior convergence Let 𝜋0=ℙ(𝐻0)and 𝜋1=ℙ(𝐻1)denote priors with 𝜋0+𝜋1=1. The Bayes factor is: 𝐵10 =ℙ(𝐷∣𝐻1) ℙ(𝐷∣𝐻0).(3.20) We assume that under a directional mechanism 𝐻1, the fully adverse configuration occurs with non-negligible probability; that is, there exists a constant 𝑐∈(0,1]such that ℙ(𝐷∣𝐻1)≥𝑐. (3.21) Under a directional process, 𝐷is typical, so ℙ(𝐷 ∣ 𝐻1) ≈ 1. Under 𝐻0, the modelling above yields: ℙ(𝐷∣𝐻0)≪10−6.(3.22) 24 Thus: 𝐵10 ≫106.(3.23) The posterior probability of 𝐻1is: ℙ(𝐻1∣𝐷)= 𝐵10𝜋1 𝐵10𝜋1+𝜋0.(3.24) Posterior convergence. As 𝐵10 →∞, ℙ(𝐻1∣𝐷)→1 for every fixed 𝜋1>0. (3.25) Hence unless one assigns an extraordinarily small prior 𝜋1, posterior mass converges overwhelmingly on 𝐻1whenever ℙ(𝐷∣𝐻0)collapses into the extreme tail. 3.8 Collapse analysis Let 𝐷be an observed configuration. If: 1. ℙ(𝐷∣𝐻0)is computed under a conservative null model, 2. ℙ(𝐷∣𝐻0)≤𝜀for a pre-specified actuarial threshold, and 3. ℙ(𝐷∣𝐻1)≈1, then: •𝐷is statistically implausible under 𝐻0; and • the resulting Bayes factor 𝐵10 provides quantitative evidence favouring 𝐻1. 4 Asymptotic properties of directional collapse In this section we formalise the asymptotic behaviour of fully directional configurations under admissible dependence structures. The results justify the claim that, within a wide class of stochastic models, evidential collapse is driven primarily by the number of domains 𝑛rather than by fine-grained parameter calibration. 25 Outside these conditions, the framework loses diagnostic validity and must not be used to support evidential inference. These constraints define the epistemic boundary of the method. When they are not satisfied, the protocol is inapplicable by definition. This boundary ensures that the framework functions as a protective diagnostic instrument rather than as a general-purpose rhetorical weapon. 6 Future Work The quantitative method set out in this paper establishes a replicable framework for identifying statistical implausibility as an evidentiary indicator of directional system behaviour. Several avenues for future development follow directly from this initial formulation. 6.1 Model refinement and parameter sensitivity The present Monte Carlo approach relies on conservative and respondent-favourable parameter choices. Further work should examine the behaviour of the model under alternative dependency structures, non-exchangeable correlations, and heterogeneous marginal probabilities. Systematic sensitivity analyses across these dimensions would strengthen the evidential robustness of the method and enable its application to systems with more complex internal dynamics. 6.2 Integration with governance and behavioural data The model treats each domain as a measurable outcome of system behaviour. Extending the framework to incorporate temporal covariates, governance inputs, escalation pathways, and behavioural markers would permit a more granular assessment of directional drift. Such integration would also allow regulators and audit bodies to evaluate whether certain governance configurations exhibit elevated susceptibility to statistically implausible outcome patterns. 6.3 Expansion to multi-case and cross-sectional analysis While the method is demonstrated here through a single synthetic illustration, the underlying framework is generalisable. Future work could apply the methodology to co32 horts of cases, identifying whether directional system behaviour manifests through consistent statistical signatures across organisations, sectors, or decision-making environments. This would support the construction of population-level baselines and facilitate comparative risk benchmarking. 6.4 Alignment with evidentiary standards A further line of inquiry is the formal mapping of statistical implausibility to legal and regulatory evidentiary thresholds. Although this paper does not purport to make legal findings, the relationship between statistical rejection of the innocent model and evidential inference warrants systematic examination. Establishing principled criteria for when implausibility becomes probative could improve consistency across tribunals, regulators, and enforcement bodies. 6.5 Integration with contradiction-field modelling The observed directional drift across multiple domains indicates that system behaviour may be represented as a structured contradiction field in which pressure, omission, and procedural decay exert measurable influence. Future work should explore formal unification of the statistical framework with contradiction-field dynamics, allowing directional behaviour to be modelled not only as an outcome pattern but as a structural failure mode with quantifiable properties. 6.6 Tooling, automation, and audit integration The method lends itself naturally to automation. Development of software tools capable of ingesting system records, detecting directional divergence, and quantifying statistical implausibility would provide regulators, auditors, and practitioners with a practical mechanism for early detection. Such tools could support proactive governance, reduce litigation exposure, and identify systemic risk before harm crystallises. 6.7 Ethical and regulatory implications Finally, deployment of this framework across operational and regulatory contexts raises important questions regarding the ethical use of statistical evidence, the governance of 33 algorithmic audit tools, and the potential for misuse. Future work should examine safeguards, transparency requirements, and interpretative standards to ensure that statistical implausibility functions as a protective diagnostic instrument rather than a punitive or rhetorical one. 7 Strengths and Contribution Existing quantitative approaches to adverse outcomes typically address individual events, external comparators, or population-level disparities. They do not provide a formal statistical test for evaluating the plausibility of a single high-dimensional directional outcome pattern. This paper therefore fills a methodological gap: it operationalises evidential implausibility as a measurable quantity within a stochastic framework, providing an instrument not currently present in systemic adverse outcome analysis, organisational modelling, or actuarial practice. This paper makes three core contributions to the quantitative analysis of directional system behaviour. Each contribution is deliberately confined to statistical inference and evidential reasoning, and does not purport to make findings of fact, motive, or law. The strengths of the method arise from its formal structure, its evidential clarity, and its reproducibility across contexts. 7.1 A formal test for statistical implausibility The first contribution is a structured framework for assessing whether an observed pattern of outcomes is reconcilable with innocent stochastic fluctuation. By modelling marginal probabilities, dependency structures, and correlation effects explicitly, the method provides a quantitative means of determining when the null model collapses. This transforms what would otherwise remain a qualitative concern into a formally testable evidential question: is the observed pattern statistically compatible with random variation? 7.2 Directionality and systemic inference The second contribution lies in the treatment of directionality. Traditional approaches often assess adverse events in isolation. By contrast, the present framework treats a sequence of uniformly adverse outcomes as a single directional vector. This captures systemic drift rather than episodic discrepancy, and permits inference at the level of 34 system behaviour rather than individual incidents. The framework therefore identifies when a pattern, taken as a whole, exhibits structural properties that cannot reasonably be attributed to stochastic variation. 7.3 Reproducibility and evidential transparency The third contribution is the model’s reproducibility. All assumptions, parameters, and computational steps are explicit, capable of independent replication, and amenable to external scrutiny. This ensures transparency in both academic analysis and applied evidential settings. The framework can be deployed across diverse system contexts, provided that input probabilities are specified conservatively and the dependency structure is articulated clearly. The result is a generalisable and methodologically coherent instrument for detecting evidential implausibility. 7.4 Position within the wider literature Finally, the paper addresses a persistent gap in current practice. While qualitative analyses of systemic adverse treatment and structural bias are well established, quantitative methods for evaluating the plausibility of innocent stochastic explanations remain limited. This framework provides a structured alternative grounded in actuarial modelling and statistical inference. Its contribution is not to assert motive, but to demonstrate when stochastic explanations cannot be sustained on probabilistic grounds. In evidential contexts, this distinction is decisive. 8 Conclusion This paper has introduced a quantitative method for identifying patterns of system behaviour that are statistically incompatible with ordinary stochastic fluctuation. By modelling the expected distribution of outcomes under conservative assumptions and comparing those distributions to observed directional configurations, the framework provides a structured mechanism for determining when a benign stochastic explanation can no longer be sustained on probabilistic grounds. The concept of statistical implausibility is used not to assert motive, but to establish that certain outcome configurations cannot be reconciled with randomness without invoking assumptions that are themselves unrealistic, untestable, or internally inconsistent. 35 The contribution of the paper is twofold. First, it formalises a replicable, evidencebased framework that integrates actuarial reasoning, dependency modelling, and structured contradiction analysis into a single evaluative instrument. Second, it demonstrates how this framework can function as an evidentiary diagnostic in contexts where explanatory records are fragmented, incomplete, or structurally absent. In such settings, quantitative analysis of outcome patterns provides a defensible means of assessing whether prevailing explanations remain probabilistically viable. The findings do not replace legal, regulatory, or organisational judgment. Rather, they delineate the probabilistic boundaries within which such judgment can be exercised without departing from statistical coherence. When observed behaviour occupies regions of the outcome space that actuarial models treat as effectively zero-probability events, continued reliance on stochastic innocence becomes analytically indefensible. At that point, the evidential burden shifts from randomness to structure. Statistical implausibility therefore functions not as proof but as a threshold. Once crossed, the domain of credible innocent explanations collapses, and analytical focus necessarily reorients toward directional system behaviour. The framework presented here provides a disciplined mechanism for identifying that threshold and for quantifying the strength of the resulting evidential signal. Accordingly, the methodological contribution of this paper is a formally defined, replicable statistical test for evidential implausibility in system outcomes: a diagnostic criterion specifying when the null model of stochastic fluctuation is no longer a coherent probabilistic hypothesis under observed directional alignment. This criterion unifies stochastic modelling, dependency analysis, and Bayesian updating into a single framework suitable for academic, regulatory, and applied governance contexts. In this sense, directional collapse functions as a phase transition in evidential geometry: beyond the threshold, stochastic innocence ceases to exist as a viable statistical state. 36 Appendix note Throughout this appendix, ‘effectively zero’ is used in the actuarial sense: probabilities below material decision thresholds, not literal impossibility. Where domain ratings are materially contested, results should be presented as indicative rather than determinative. A How to use this method without a maths background A.1 Who this is for This guide is for people who are facing serious problems with decisions made about them — especially in employment, legal, regulatory, or institutional settings — and who believe the system’s decisions may not be fair. You do not need to know mathematics to use this guide. A.2 What this test does The collapse test checks whether a set of negative outcomes can be explained by random chance, or whether the pattern is so unlikely that something directional — like bias — must be causing it. It gives you a number: if that number is extremely small, then the system’s explanation (”it’s just how things happened”) is statistically not believable. A.3 What you need To run the collapse test, you need three things: 1. A list of the different domains where decisions were made about you — for example: training, funding, access, promotion, email response, complaints handling, supervision, etc. 2. For each domain, decide if the change after a key event (e.g. a complaint, a disclosure, or conflict) was: 37 •Adverse (clearly worse), •Neutral or better (no change or improved), or •Not applicable (no record). 3. An estimate of how likely a negative change would be in each domain if everything were fair (e.g. 20% means that normally, 1 in 5 people might have an adverse outcome in that domain). A.4 How to run it You can use the test in this way: •Simple version: If you had 𝑛domains with adverse outcomes, and each has a fair-chance adverse rate of 𝑝, then: Collapse probability =𝑝𝑛(A.1) If this number is smaller than 1 in 10,000, the chance explanation becomes statistically very unlikely. Plain English. Think of this number like getting a sequence of heads in a row. Getting 5 heads might just be luck. But getting 15 heads in a row isn’t luck – it’s a weighted coin. Every additional ‘head’ (adverse outcome) makes the ‘luck’ explanation exponentially harder to believe, until the only rational conclusion is that the game is rigged. Worked example. Let 𝑝=0.5and 𝑛=15, the the collapse probability is (0.5)15 =0.5×0.5×0.5×0.5...×0.5(15 times) (A.2) =0.000030517578125 (A.3) =0.00305% (A.4) ≈1 in 32,000 (A.5) A.5 What the result means If your collapse probability is: •Greater than 1% — the result is not unusual enough to conclude systemic bias. •Less than 1 in 1,000 — the explanation of “just chance” is starting to look weak. 38 •Less than 1 in 100,000 — this is a strong signal that something directional is happening. A.6 What to do with it If the test shows collapse: • Include the result in your legal or regulatory documents. • Present it to investigators or mediators. • Submit it as part of a complaint or appeal. This test does not prove motive. But it does prove that the system’s pattern is increasingly unlikely, level by level, to be explained by random variation. A.7 Important notes • You do not need to prove intent or conspiracy — only that the pattern is statistically incompatible with fairness. • This test does not require external comparators, or internal emails, or personal statements. It works only on the outcomes. • The more domains you include — even small ones — the stronger the test becomes. This method is free to use for anyone and was developed to help people facing injustice in systems that often leave no clear trail. B Standard operating procedure for union representatives (Equality Act as an example) B.1 Purpose of this appendix This appendix is written for union representatives. It explains how to use a Collapse analysis to show that a member’s pattern of adverse treatment is not compatible with innocent organisational fluctuation. 39 The aim is simple: • to provide a clear method any rep can apply; • to help identify cases where discrimination, victimisation, or structural detriment is likely; and • to give unions language that fits the Equality Act 2010 burden of proof under section 136. The Collapse analysis does not replace legal advice. It provides a structured, quantitative way to describe patterns that many members experience but struggle to evidence. B.2 When to use a Collapse analysis A Collapse analysis is useful when a member reports multiple changes to their working conditions, especially where: • the changes occur after a trigger event, for example disability disclosure, a grievance, whistleblowing, or requesting adjustments; • the changes are spread across different aspects of the job, not just one decision; and • the overall pattern feels like a sudden deterioration rather than normal organisational noise. Typical trigger events include: • disclosure of a disability or long term health condition; • bringing a grievance or raising a protected concern; • taking or requesting maternity, parental, or carers leave; • raising health and safety issues; • union activity or acting as a representative. If the member can say, in plain language, “Everything went downhill at once after X”, the case is a candidate for Collapse analysis. 40 Step 1: Define the operational domains An operational domain is any distinct aspect of the working environment that can become better or worse over time. Examples include: 1. Line management and supervision 2. Workload and task allocation 3. Deadlines and time pressure 4. Access to information and systems 5. Inclusion in meetings and decisions 6. Access to training and development 7. Quality of communication from management 8. Support from colleagues and team 9. Autonomy and control over work 10. Clarity of role and expectations 11. Career progression and opportunities 12. Fairness of performance management 13. Health and safety impact 14. Access to reasonable adjustments 15. Governance and oversight of decisions 16. Any other domain relevant to the member’s role There is no fixed list. The key points are: • domains should be practical and recognisable; • each domain should describe one aspect of working conditions; and • if it matters to the member and can get better or worse, it can be included as a domain. In many cases, sixteen domains provide a good balance between detail and practicality, but fewer or more can be used if necessary. 41 C.3 Interpretation Even under generous assumptions (high correlation, skewed priors, or clustered risk), the probability of fifteen adverse shifts arising through normal organisational fluctuation is effectively zero. This is a classical Collapse event: • tightly time–locked to a trigger, • spanning many independent domains, • with no offsetting improvements. Under s. 136 EqA, this establishes facts from which the Tribunal may conclude unlawful treatment unless the employer provides a coherent, contemporaneous explanation. The burden therefore shifts. D One–page quick reference for Union Reps What is Collapse? A structured diagnostic for identifying whether a worker’s pattern of detriment can realistically be explained by chance. How to apply it: 1. Define 12–20 operational domains. 2. Rate each domain before and after the trigger (+1, 0, -1). 3. Count the number of adverse shifts (the Collapse score). 4. If 𝐾≥12, especially after a trigger event, the pattern is statistically incompatible with innocent fluctuation. Legal link (s.136 EqA): If the Collapse score is high, and no contemporaneous justification exists, the tribunal may draw an inference of unlawful treatment. The burden shifts to the employer. 48 Use this wording in letters: The pattern of detriment is statistically incompatible with normal organisational noise. Under section 136 Equality Act 2010, the burden now shifts to the employer to provide a coherent explanation consistent with the evidence. When Collapse is strongest: • sudden deterioration across many domains; • tightly bound to a protected act or disclosure; • no documented rationale or governance trail; • organisation cannot identify comparable patterns in others. E References and DOI links This toolkit is based on publicly archived, openly licensed research on structural integrity, contradiction dynamics, and pattern–collapse analysis. Core Paper •Collapse: A Quantitative Test for Systemic Bias • DOI: 10.5281/zenodo.17856052 The DOI link allow union representatives, legal advisers, and tribunals to verify the methodology, to cite it formally, and to review the underlying analytical framework. Mathematical proofs are provided in earlier papers in the series. F Collapse Analysis Protocol (CAP) This appendix sets out the full protocol for applying the statistical directional collapse framework developed in the main text. It formalises the evidential structure, input 49 constraints, modelling requirements, and interpretative rules governing the use of the method. The protocol is general: it does not rely on any specific case and may be applied to any system in which directional patterns of outcomes require probabilistic evaluation. F.1 Overview The protocol defines a five-stage evidential process: 1. domain definition and classification; 2. construction of the adverse outcome vector; 3. specification and evaluation of the null model 𝐻0; 4. Bayesian evaluation of the alternative 𝐻1; 5. interpretation of results relative to actuarial thresholds. The protocol is intended for researchers, auditors, regulators, and applied statisticians who require a replicable and formally defined diagnostic tool. F.2 Input schema The protocol requires the following inputs: •Domain set 𝒟 ={1,…,𝑛}: the procedural, governance, or behavioural domains to be assessed. •Temporal segmentation: a pre-perturbation and post-perturbation window identifiable from observable records. •Observable record: contemporaneous materials sufficient to classify each domain outcome. •Classification functions: 𝐶1∶𝒟 →{consistent,indeterminate,contradiction}, 𝐶2∶𝒟 →{procedural,substantive,trajectory}. 50 •Outcome mapping: construction of a Bernoulli vector 𝑋=(𝑋1,…,𝑋𝑛), where 𝑋𝑖= 1denotes an adverse domain. Domain definitions must remain stable across the temporal boundary. F.3 Validity conditions The protocol is admissible only when the following structural conditions are satisfied: 1. Domain stability: identical domains are observable in both temporal windows. 2. Record sufficiency: the classification functions 𝐶1and 𝐶2can be applied without speculative inference. 3. Directional coherence: the observed configuration exhibits a unidirectional pattern. 4. Event localisation: the shift occurs within a bounded temporal window consistent with a perturbation. 5. Model regularity: marginal probabilities and correlation structures can be meaningfully specified for 𝐻0. Failure to satisfy any condition renders the diagnostic test invalid. F.4 Null model specification The null hypothesis 𝐻0represents innocent stochastic fluctuation. Its components are: 1. Marginal probability: each domain follows a Bernoulli law with parameter 𝑝𝑖∈ (0,1). 2. Least favourable assumption: the baseline adopts 𝑝𝑖= 0.5for all 𝑖, maximising ℙ(𝐷∣𝐻0). 3. Independence model: ℙ(𝐷∣𝐻0)= 𝑛 􏾟 𝑖=1 𝑝𝑋𝑖 𝑖(1−𝑝𝑖)1−𝑋𝑖.(F.1) For a fully adverse configuration with 𝑘adverse domains, ℙ(𝐷∣𝐻0)=𝑝𝑘. 51 4. Dependency model: correlation may be introduced via a multivariate normal copula with pairwise correlation 𝜌. Monte Carlo simulation is used to evaluate the resulting tail probability. Correlation values approaching 𝜌 ≈ 0.9are treated as indicative of a single unifying influence rather than benign decentralised fluctuation. F.5 Computation of evidential probability Given the model specification: 1. compute ℙ(𝐷∣𝐻0)under independence; 2. simulate 𝑁realisations of the correlated model where applicable; 3. evaluate the empirical survival probability 𝑆(𝑘)=ℙ(𝐾≥𝑘∣𝐻0), (F.2) where 𝐾is the total number of adverse domains. Fully directional configurations constitute low-entropy events and carry additional inferential weight even under moderate correlation. F.6 Bayesian evaluation The Bayesian update integrates stochastic evidence with prior uncertainty: 1. assign a prior 𝜋1to the directional alternative 𝐻1; 2. compute the Bayes factor 𝐵10 =ℙ(𝐷∣𝐻1) ℙ(𝐷∣𝐻0);(F.3) 3. evaluate the posterior ℙ(𝐻1∣𝐷)= 𝐵10𝜋1 𝐵10𝜋1+(1−𝜋1);(F.4) 4. assess posterior convergence. 52 Posterior dominance is declared when ℙ(𝐻1∣𝐷)→1under all reasonable priors. Remark 6 (Prior Robustness).The result holds regardless of the specific prior beliefs assigned, so long as ℙ(𝐻1) ℙ(𝐻0)>0. (F.5) That is, the directional hypothesis 𝐻1need only be assigned nonzero prior mass. This is particularly important in legal and regulatory settings, where starting assumptions are often contested: no special weighting in favour of 𝐻1is required for posterior collapse to occur. F.7 Interpretation thresholds Evidential regimes are classified as follows: •ℙ(𝐷∣𝐻0)>0.10: ordinary stochastic fluctuation; •0.01<ℙ(𝐷∣𝐻0)≤0.10: unusual; explanatory review required; •ℙ(𝐷∣𝐻0)≤0.01: actuarially implausible; •ℙ(𝐷∣𝐻0)<10−4: deep-tail regime incompatible with innocent stochastic fluctuation. Fully directional collapse (uniform adverse movement across all domains) is treated as strong quantitative evidence of a single unifying directional influence. F.8 Conditions under which the protocol cannot be applied The protocol is inapplicable when: • domain boundaries cannot be identified in either temporal window; • no observable outcomes exist in the post-perturbation window; • adverse and neutral outcomes cannot be meaningfully distinguished; • no triggering boundary can be established; • the domain set changes so radically that pre–post comparison is not conceptually meaningful. 53 Documentary incompleteness does not preclude application. Missing records, procedural gaps, and absence of rationale may themselves constitute adverse outcomes and are incorporated directly into the domain vector. F.9 Summary The Collapse Analysis Protocol provides a replicable evidential structure for evaluating whether a multi-domain directional pattern can plausibly arise under innocent stochastic fluctuation. When the observed configuration resides far below actuarial materiality thresholds and Bayesian convergence overwhelmingly favours the alternative, the null model collapses under quantitative scrutiny. Beyond this threshold, continued reliance on stochastic innocence becomes probabilistically indefensible under collapse analysis. 54