The Symmetric Convergence Engine: A Stochastic, Self-Correcting Method for Fair and Equitable Allocation

Part II – The Symmetric Convergence Engine: A Stochastic, Self-Correcting Method for Fair and Equitable Allocation James D. Atkinson 2025 Abstract The Symmetric Convergence Engine (SCE) provides the procedural core of the mathematics of integrity, extending the evidential logic of The Contradiction Trap into a dynamic, falsifiable method for fair allocation under symmetric stress. SCE models allocation as a stochastic governance process in which fairness is not presumed but empirically demonstrated: declared proportions must emerge through convergence across repeated, entropy-certified draws. In this framework, deviation becomes evidence, drift becomes a diagnostic signal, and convergence becomes an operational test of procedural honesty. The engine integrates three interacting components –entropy-certified randomness, symmetric weighting, and a convergence-regulating update rule – to form a stochastic dynamical system that is unpredictable at the micro level yet statistically self-correcting at the macro level. Unlike deterministic quotas or naive lotteries, SCE produces a transparent, reproducible, and adversarially robust audit trail: every draw is a symmetry test, every residual an evidential unit, every correction bounded by publicly declared commitments. SCE functions as the procedural bridge in a trilogy on evidential integrity. Where The Contradiction Trap converts contradiction into epistemic evidence, and the PRIME Sentinel Framework converts symmetry into institutional evidence, SCE converts proportional drift into procedural evidence. It establishes a unified evidential paradigm for allocation systems – one in which legitimacy is not a policy declaration but an empirically verifiable pattern of symmetric convergence. Keywords: mathematicsofintegrity; symmetricconvergence; evidentialintegrity; stochastic governance; entropy-certified randomness; weighted stochastic processes; convergencedynamics; proportionaldrift; proceduralfairness; fairness-by-construction; symmetric stress-testing; behavioural diagnostics; adversarial robustness; allocation integrity; auditability; governance mathematics. 1 Contents 1 Visual Abstract 7 2 Notation and Symbols 8 3 Introduction 10 3.1 Relation to The Contradiction Trap (Paper1) .............. 12 4 Definition & Core Structure 13 4.1 Constituent Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.2 Functional Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.3 ConceptualSummary............................ 16 5 Terminology Standardisation 16 5.1 Procedural Fairness (Ex Ante) . . . . . . . . . . . . . . . . . . . . . . . 17 5.2 Proportional Equitableness (Long-Run) . . . . . . . . . . . . . . . . . 17 5.3 VerifiableEntropy.............................. 18 5.4 Coherence Cost: KL vs Quadratic Approximation . . . . . . . . . . . . 19 5.5 Residuals and Diagnostic Quantities . . . . . . . . . . . . . . . . . . . 19 5.6 Terminology Summary Table . . . . . . . . . . . . . . . . . . . . . . . . 20 6 Origins and Distinction 20 6.1 HistoricalContext.............................. 20 6.2 Comparison with Classical Mechanisms . . . . . . . . . . . . . . . . . 21 6.3 Architectural Distinction . . . . . . . . . . . . . . . . . . . . . . . . . . 22 6.4 Theoretical Positioning . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 6.5 Summary................................... 23 7 Coherence Cost Estimation 23 7.1 From Logical to Statistical Consistency . . . . . . . . . . . . . . . . . . 24 7.2 Canonical Coherence Cost . . . . . . . . . . . . . . . . . . . . . . . . . 24 7.3 Transition from Logical to Statistical Cost . . . . . . . . . . . . . . . . 25 7.4 Quadratic Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 25 7.5 Expected Tranche-Level Cost . . . . . . . . . . . . . . . . . . . . . . . 26 7.6 Diagnostic Thresholds . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 7.7 Interpretation................................ 27 8 Methodology for SCE Implementation 27 8.1 EntropyAcquisition............................. 27 8.2 Dynamic Weighting & Data Integration . . . . . . . . . . . . . . . . . . 29 8.3 Tranche Definition and Credit Mechanics . . . . . . . . . . . . . . . . . 29 2 8.4 ParameterSelection ............................ 29 8.5 AuditLogOverview............................. 30 8.6 Simulation and Validation . . . . . . . . . . . . . . . . . . . . . . . . . 31 8.7 Operational Safeguards . . . . . . . . . . . . . . . . . . . . . . . . . . 31 8.8 Summary................................... 31 9 Analytical Function 32 9.1 Expectation and Variance . . . . . . . . . . . . . . . . . . . . . . . . . 32 9.2 MartingaleFairness............................. 33 9.3 Proportional Convergence . . . . . . . . . . . . . . . . . . . . . . . . . 33 9.4 Martingale Central Limit Approximation . . . . . . . . . . . . . . . . . 34 9.5 Stability of Credit Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 34 9.6 DynamicEquilibrium ............................ 35 9.7 Tracking Non-Stationary Weights . . . . . . . . . . . . . . . . . . . . . 35 9.8 Identifiability of Deviation Sources . . . . . . . . . . . . . . . . . . . . 35 9.9 Interpretation................................ 36 9.10 Summary................................... 36 10 Empirical Convergence & Scale Thresholds 37 10.1 Minimum Viable Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 10.2 Small-KRegimes .............................. 39 10.3 Interval Fairness Claims (IFC) . . . . . . . . . . . . . . . . . . . . . . . 40 11 Applications & Operational Envelope 42 11.1 Motivation&Scope............................. 42 11.2 Empirical Convergence Thresholds . . . . . . . . . . . . . . . . . . . . 42 11.3 Minimum Convergence Threshold . . . . . . . . . . . . . . . . . . . . . 42 11.4 Small-k Reporting Protocol (SKRP) . . . . . . . . . . . . . . . . . . . . 44 11.5 Operational Envelope . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 11.6 Cross-Domain Deployment . . . . . . . . . . . . . . . . . . . . . . . . 45 11.7 Audit Dashboard Metrics & Reporting Standards . . . . . . . . . . . . 46 11.8 Summary................................... 47 12 The Core Principle: Symmetry Without Intervention 47 12.1 Fairness as Falsifiability . . . . . . . . . . . . . . . . . . . . . . . . . . 47 12.2 From Logical Symmetry to Procedural Symmetry . . . . . . . . . . . . 47 12.3 The Meaning of Non-Intervention . . . . . . . . . . . . . . . . . . . . . 48 12.4 Symmetry as Epistemic Discipline . . . . . . . . . . . . . . . . . . . . 48 12.5 Entropy as the Moral Constant . . . . . . . . . . . . . . . . . . . . . . . 48 12.6 Convergence as Evidence of Integrity . . . . . . . . . . . . . . . . . . 49 12.7 Implications for Governance Philosophy . . . . . . . . . . . . . . . . . 49 12.8 Mapping Epistemic Necessity to Procedural Necessity . . . . . . . . . 49 3 12.9 Summary................................... 49 13 Game-Theoretic Formalisation 50 13.1 Motivation .................................. 50 13.2 GameStructure............................... 51 13.3 Payoff Function and Expected Utility . . . . . . . . . . . . . . . . . . . 51 13.4 Equilibrium Characterisation . . . . . . . . . . . . . . . . . . . . . . . 52 13.5 Equilibrium under Partial Transparency . . . . . . . . . . . . . . . . . 53 13.6 System as Meta-Player . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 13.7 Repeated Interaction and Learning Dynamics . . . . . . . . . . . . . . 54 13.8 Information Symmetry and Trust Collapse . . . . . . . . . . . . . . . . 54 13.9 Deviations and Detection . . . . . . . . . . . . . . . . . . . . . . . . . . 54 13.10 Interpretive Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 13.11 Summary................................... 55 14 Interdisciplinary Relevance & Contemporary Context 55 14.1 Algorithmic Fairness & Governance . . . . . . . . . . . . . . . . . . . . 55 14.2 Organisational Systems & Bureaucratic Neutrality . . . . . . . . . . . 55 14.3 Information Theory & Entropy Ethics . . . . . . . . . . . . . . . . . . . 56 14.4 Behavioural & Cognitive Dimensions . . . . . . . . . . . . . . . . . . . 56 14.5 Political & Ethical Philosophy . . . . . . . . . . . . . . . . . . . . . . . 56 14.6 Artificial Intelligence & Machine Learning . . . . . . . . . . . . . . . . 57 14.7 Sociotechnical & Epistemic Implications . . . . . . . . . . . . . . . . . 57 14.8 Summary................................... 57 15 Comparative Context & Accountability Frameworks 58 15.1 SCE in the Landscape of Algorithmic Accountability . . . . . . . . . . 58 15.2 Relation to Doshi-Velez & Kim (2017): Accountability via Simulatability 58 15.3 ComparisonwithFairness-Through-UnawarenessandFairness-Aware Learning ................................... 59 15.4 SCE as Procedural Alternative to Interpretability . . . . . . . . . . . . 59 16 Operational Envelope & Deployment Architecture 60 16.1 Operational Envelope: Minimum Requirements & Failure Modes . . . 60 16.2 Weights Legitimacy Protocol (WLP) . . . . . . . . . . . . . . . . . . . . 61 16.3 Parameter Governance: Declaring & Justifying Weights . . . . . . . . 62 16.4 Audit Logging & Forensic Reconstruction . . . . . . . . . . . . . . . . 63 16.5 Organisational Integration . . . . . . . . . . . . . . . . . . . . . . . . . 63 16.6 Failure Protocols: Entropy Loss, Drift Detection, & Anomaly Flags . . 64 16.7 AuditProtocol................................ 65 16.8 Entropy Service-Level Agreements (SLAs) . . . . . . . . . . . . . . . . 65 16.9 Deployment Tiers: Private, Public, & Adversarial . . . . . . . . . . . . 66 4 16.10 Minimum Volume & Threshold Heuristics . . . . . . . . . . . . . . . . 66 16.11 Convergence Dashboards & Real-Time Monitoring . . . . . . . . . . . 67 17 Empirical Validation Under Adversarial and Non-Stationary Conditions 68 17.1 Adversarial Entropy Model (AEM) . . . . . . . . . . . . . . . . . . . . . 68 17.2 Non-Stationary Weights: Step, Ramp, and Seasonal Drift . . . . . . . 69 17.3 CorrelatedArrivals ............................. 69 17.4 Entropy Degradation: Sticky and Biased RNG . . . . . . . . . . . . . . 69 17.5 Partial Observability of Declared Weights and Real-Data Proxy Experiments .................................... 70 17.6 Required Sample Size for Operational Guarantees . . . . . . . . . . . 70 18 Ethical Limits, Misuse Modes, & Governance Guardrails 70 18.1 The Domain of Ethical Use . . . . . . . . . . . . . . . . . . . . . . . . . 71 18.2 Legitimacy of Declared Proportions . . . . . . . . . . . . . . . . . . . . 71 18.3 MisuseModes................................ 71 18.4 Governance Guardrails . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 18.5 Epistemic Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 18.6 Absolute Exclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 18.7 Summary................................... 73 19 Future Research Programme 73 19.1 Purpose and Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . 73 19.2 Direction I: Theoretical Extensions . . . . . . . . . . . . . . . . . . . . 74 19.3 Direction II: Empirical Validation . . . . . . . . . . . . . . . . . . . . . 75 19.4 Direction III: Ethical and Regulatory Integration . . . . . . . . . . . . 75 19.5 Direction IV: Philosophical Development . . . . . . . . . . . . . . . . 76 19.6 Synthesis and Forward Trajectory . . . . . . . . . . . . . . . . . . . . . 76 19.7 SCE-Prime: Toward Adaptive Fairness Without Control Loss . . . . . 77 20 Summary of Contributions 79 21 Conclusion 79 21.1 Toward Evidential Governance . . . . . . . . . . . . . . . . . . . . . . 81 A Parallel Concepts Table 81 B Public Verifiability Protocol (PVP) 81 B.1 InputstotheAuditor............................ 82 B.2 Verification of Entropy Provenance . . . . . . . . . . . . . . . . . . . . 82 B.3 Reconstruction of Allocations . . . . . . . . . . . . . . . . . . . . . . . 83 B.4 Tranche-Level Consistency . . . . . . . . . . . . . . . . . . . . . . . . 83 5 B.5 Cumulative Behaviour Verification . . . . . . . . . . . . . . . . . . . . 83 B.6 Completeness Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . 84 B.7 Interpretation................................ 84 B.8 Entropy Acquisition Layer . . . . . . . . . . . . . . . . . . . . . . . . . 84 B.9 Dynamic Weighting Layer . . . . . . . . . . . . . . . . . . . . . . . . . 84 B.10 Tranche–Credit Correction Layer . . . . . . . . . . . . . . . . . . . . . 85 B.11 Audit and Logging Layer . . . . . . . . . . . . . . . . . . . . . . . . . . 85 B.12 OperationalNotes.............................. 86 C Python scripts for reproducability 86 C.1 Convergence: Deviation vs K . . . . . . . . . . . . . . . . . . . . . . . . 86 C.2 Drifttracking................................. 87 C.3 Small-Knoise ................................ 89 C.4 Alphasensitivity............................... 90 C.5 Creditfunction................................ 91 C.6 Quadratic approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 93 C.7 Ensemble Convergence: Geometric Mean ± Geometric SD (20 Runs) 95 C.8 Ensemble Convergence: 95% Geometric Confidence Interval . . . . 98 C.9 Convergence: Log–Binned Median Curve . . . . . . . . . . . . . . . . 100 C.10 SCEsimulations...............................102 C.11 Plot SCE simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6 1 Visual Abstract Certified Randomness Stochastic Draws Residuals Δ𝑖(𝑇) Credit Update 𝜅𝑖(𝑇) Convergence 𝑋𝑖/𝐾→𝑤𝑖 Figure 1: Visual abstract for SCE: a stochastic–corrective pipeline that achieves symmetry-preserving convergence under certified entropy. 7 2 Notation and Symbols Symbol Meaning Core Allocation Structure 𝑅={𝑟𝑖}Recipient set (groups, agents, categories). 𝑤𝑖Declared proportional weight for recipient 𝑖,∑𝑖𝑤𝑖=1. 𝑤𝑖(𝜏) Declared weight at tranche 𝜏(stationary or drifting). 𝑈𝑡Entropy variate for allocation event 𝑡,𝑈𝑡∼Unif[0,1). 𝑟𝑡Recipient selected at event 𝑡. 𝑋𝑖(𝐾) Cumulative allocations received by group 𝑖after 𝐾events. 𝑤𝑖(𝐾) Empirical frequency: 𝑤𝑖(𝐾)=𝑋𝑖(𝐾)/𝐾. Tranches, Deviations, and Credits 𝑇Tranche size: number of allocation events per tranche. 𝜏Tranche index. 𝑋𝑖(𝜏) Allocations to group 𝑖within tranche 𝜏. Δ𝑖(𝜏) Tranche residual: 𝑇𝑤𝑖(𝜏)−−𝑋𝑖(𝜏). 𝑔(Δ𝑖)Bounded correction function producing credit 𝜅𝑖. 𝜅𝑖(𝜏) Credit for group 𝑖after tranche 𝜏(adjusts probability). 𝛼Responsiveness parameter blending credits with declared weights. 𝜋𝑖(𝑡) Effective participation probability at event 𝑡: 𝜋𝑖(𝑡)=𝛼𝑤𝑖(𝜏(𝑡))+(1−𝛼)𝜅𝑖(𝜏(𝑡)−1). Entropy and Provenance 𝑒𝑡Entropy sample for event 𝑡(raw output from beacon/QRNG). ℎ𝑡Hash-chain commitment: ℎ𝑡=𝐻(𝑒𝑡‖ℎ𝑡−1). ℰEntropy provenance ledger. BEACON Public randomness source (e.g. NIST, drand). Deviation and Fairness Metrics 𝛿𝑖(𝐾) Deviation from target: 𝑤𝑖(𝐾)−−𝑤𝑖. 𝑅𝑖(𝐾) Signed residual: 𝑋𝑖(𝐾)−−𝐾𝑤𝑖. 𝐶(𝐾) Canonical coherence cost: 𝐷KL( 𝑤(𝐾)‖𝑤). 𝐶𝜏Tranche-level KL divergence. 𝐶quad Quadratic surrogate: ∑𝑖𝛿2 𝑖/(2𝑤𝑖ln2). 𝐶abs Non-canonical 𝐿1deviation measure. 𝐻norm Normalised entropy of empirical distribution. 8 Symbol Meaning Stochastic Limits and Noise 𝐸[𝐶𝜏]Expected KL noise floor: (𝑛−1)/(2𝑇ln 2). 𝜎stoch Stochastic KL deviation width: √(𝑛−1)/(2𝑇ln 2). 𝐾min Minimum viable scale for convergence (typically 1000𝑛). 𝑧𝑝Statistical significance multiplier for anomaly detection. Convergence and Martingale Structure 𝑀𝑖(𝐾) Martingale difference sum: ∑𝑡≤𝐾(𝑋𝑖(𝑡)−𝜋𝑖(𝑡)). ℱ𝑡Natural filtration generated by entropy draws. 𝐴1–𝐴6 Structural assumptions for convergence (entropy independence, bounded feedback, stationary or slow-drift weights, etc.). 𝑤𝑖(𝐾) Converges in probability to 𝑤𝑖under the SCE update rule. Credit Function and Stability 𝑔′(0) Local slope of correction function; must satisfy 0<𝑇𝑔′(0)<2. 𝑔max Max correction amplitude (clipping bound). 𝜌Stability parameter linking credit dynamics to variance decay. 𝜅∗ 𝑖Equilibrium credit when 𝐸[Δ𝑖]≈0. Non-Stationary and Drift-Tracking Quantities 𝑤𝑖(𝜏+1)−𝑤𝑖(𝜏) Drift step; bounded by 𝑐/𝑇under stability assumptions. Θdrift Drift intensity (magnitude of weight change per tranche). IFC Interval Fairness Claim: uncertainty band for small-𝐾systems. Game-Theoretic Structure 𝑆𝐶𝐸-game The behavioural game ⟨𝐴,𝐵,𝑆𝐵,𝑈𝐴,𝑈𝐵⟩. 𝑈𝐴Auditor utility: positive coherence cost (+𝐶). 𝑈𝐵System utility: negative coherence cost (−𝐶). BR𝐵Best-response set; empty when deviation is unavoidable. Δ(𝑟) Epistemic update produced by observed response 𝑟. Diagnostic Events 𝐼𝜏Tranche-level incident flag (KL anomaly, entropy failure, credit oscillation). WNLogged incident: undeclared or unjustified weight drift. Entropy (fail) Allocations suspended due to entropy provenance failure. Simulation and Experimental Parameters 𝐾Total number of simulated events. 𝑚Number of simulation runs in ensemble analysis. 9 quencies align with declared weights within statistical tolerance (Devroye, 1986; Doob, 1953; Hall & Heyde, 1980). 4.3 Conceptual Summary SCE formalises fairness as a dynamic equilibrium between randomness and proportionality. It is neither purely probabilistic nor centrally deterministic; it occupies the epistemic midpoint where stochastic processes self-regulate through bounded correction (Kushner & Yin, 2003; Robbins & Monro, 1951). Fairness is measurable but not absolute: it strengthens with scale, stabilises through feedback, and remains auditable at every step (Self, Wang, & Kehl, 2023). SCE thus transforms fairness from an ethical aspiration into a falsifiable statistical proposition whose precision improves with repetition. Entropy Input Weighted Draw Proportional DeviationCredit Update Figure 5: Feedback loop of SCE. Note on simulations All simulations in this paper use NumPy’s PCG64 pseudo-random generator. SCE’s convergence guarantees hold for any unbiased entropy source. PRNGs suffice for simulation because they reproduce the distributional properties relevant to SCE’s behaviour. In operational deployments, however, TRNGs are required to maintain auditability and prevent entropy manipulation (National Institute of Standards and Technology, 2022; Vinod, 2013b). 5 Terminology Standardisation SCE inherits vocabulary from epistemic governance (Paper 1) and classical allocation theory. To support auditability, regulatory usage, and cross-disciplinary clarity, we 16 standardisethreecoreterms: procedural fairness,proportional equitableness, and verifiable entropy. Eachcapturesadistinctproperty ofthe mechanism and preventstheterminological collapse common in the broader fairness literature (Binns, 2018; Mitchell et al., 2021). 5.1 Procedural Fairness (Ex Ante) Definition. Procedural fairness requires that each allocation event is governed by an unbiased random process matching the declared weights 𝑤(𝜏)at the moment of selection. If ℱ𝜏,𝑘denotes the filtration up to event 𝑘of tranche 𝜏, then ℙ(𝐴𝜏,𝑘=𝑖|ℱ𝜏,𝑘)=𝑤𝑖(𝜏) (5.1) modulo the bounded adjustments induced by the tranche–credit mechanism. This formalises the per-event symmetry requirement introduced in TCT Section 5 and aligns with ex ante fairness formulations in computational social choice (Caragiannis et al., 2019; Procaccia & Tennenholtz, 2013). Purpose. Proceduralfairnessgovernsthemicro-scale: itcertifiesthateachopportunity is allocated impartially, independent oflong-run outcomesortemporaldrift indeclared proportions. Misinterpretation avoided. Proceduralfairnessisnotaconvergencepropertyanddoes not guarantee proportional outcomes over any finite horizon. It prohibits per-event manipulation and defines the baseline trust condition for the allocation system (Barocas et al., 2019b; Binns, 2020). 5.2 Proportional Equitableness (Long-Run) Definition. Proportional equitableness requires that cumulative allocations converge to the declared distribution: 1 𝐾𝐾 ∑ 𝑡=11{𝐴𝑡=𝑖} 𝑝 −→𝑤⋆ 𝑖(5.2) where 𝑤⋆ 𝑖is either the stationary weight or the time-average of a drift process 𝑤𝑖(𝜏). This is the procedural analogue of TCT’s evidential resolution (Section 8) and follows from martingale convergence (Doob, 1953; Hall & Heyde, 1980). Purpose. Thispropertyensuresthat, atscale, eachrecipientreceivesallocationsinproportion to their declared share. It is a macroscopic guarantee supported by stochastic 17 approximation and the tranche–credit feedback loop (Kushner & Yin, 2003; Robbins & Monro, 1951). Misinterpretation avoided. Proportional equitableness is not a moral intuition nor a quota system. It is a statistical law of large numbers operating over allocations, contingent on the sample size 𝐾and the stability region 0<𝑇𝑔′(0)<2(Devroye, 1986). Clarification: SCE Is Not a Quota System. Proportional equitableness must not be interpreted as enforcing deterministic quotas. SCE does not constrain tranche-level totals and imposes no fixed recipient counts. Instead, it ensures that the residual trajectory converges in probability under unbiased entropy and bounded feedback. SCE enforces statistical symmetry, not administrative control (Aziz et al., 2019). 5.3 Verifiable Entropy Definition. Verifiableentropyisrandomnesswhoseprovenanceispublic,unpredictable prior to release, and tamper-evident. Suitable examples include: • NIST/FIPS-certified DRBGs with published seeds, • public randomness beacons (e.g., NIST Beacon, drand), • multi-party commit–reveal schemes with hash chaining. Purpose. SCE does not require metaphysical “true randomness”; it requires auditable randomness (Self, Wang, & Kehl, 2023). Verifiable entropy ensures operators cannot manipulate or reseed entropy streams without detection (Drand Consortium, 2021; National Institute of Standards and Technology, 2022). Misinterpretation avoided. Verifiable entropy is not a distributional requirement but aprovenance requirement. PRNGs suffice for simulation because provenance is irrelevant in the experimental setting. Deployment requires entropy whose history can be publicly reconstructed (Brunton & Nissenbaum, 2015). Simulation vs Deployment. Simulation studies in this paper use high-quality PRNGs (e.g. PCG64) because distributional correctness is sufficient for empirical analysis. In deployment, SCE prohibits PRNGs for allocation decisions: fairness depends on public verifiability, not on statistical similarity. Formally, simulation randomness ≠deployment entropy. 18 5.4 Coherence Cost: KL vs Quadratic Approximation SCE defines coherence cost using the Kullback–Leibler divergence 𝐶KL =𝐷KL(𝑃‖𝑊) (5.3) which measures informational distortion between realised and target distributions and extends the evidential logic formalised in TCT (Kullback & Leibler, 1951). For analytic purposes, we use the quadratic surrogate 𝐶quad =(𝑃−𝑊)⊤𝑊−1(𝑃−𝑊) (5.4) which is valid when ‖𝑃−𝑊‖is small. The quadratic form is symmetric, lightweight, and robust for local analysis; KL remains the canonical diagnostic measure (Vinod, 2013b). Standardisation. 𝐶KL is the primary audit metric; 𝐶quad is reserved for derivations and robustness checks. 5.5 Residuals and Diagnostic Quantities Residuals. For group 𝑖after 𝐾cumulative events: 𝑅𝑖(𝐾)=𝑋𝑖(𝐾)−−𝐾𝑤𝑖(5.5) measures the signed deviation from proportional expectation. Noise Band. The expected tranche-level noise floor is 𝔼[𝐶𝜏]≈ 𝑛−1 2𝑇ln 2(5.6) providing a baseline for anomaly detection. KL excursions exceeding a 𝑧𝑝-scaled multiple of this value are statistically significant (Hoeffding, 1963). Incident. AnytrancheproducingaKL-boundviolation, entropyanomaly, credit-oscillation event, or stagnant cumulative KL is flagged as an incident 𝐼𝜏. Incidents represent departures from expected stochastic behaviour and require audit investigation (DoshiVelez & Kim, 2017; Self, Wang, & Kehl, 2023). 19 Diagnostic vs Convergent Regimes. SCE operates in the diagnostic regime when the cumulative event count is below the convergence threshold 𝐾min (Section 11.4). Beyond this threshold, the system enters the convergent regime, where proportional equitableness becomes statistically enforceable. This prevents misinterpretation of earlystage deviations (Binns, 2018; Rahwan et al., 2019). 5.6 Terminology Summary Table Term Standardised Meaning Procedural fairness Per-event symmetry: unbiased sampling from 𝑤(𝜏). Proportional equitableness Convergence of cumulative allocations to declared proportions. Verifiable entropy Randomness whose provenance is auditable and tamper-evident. Coherence cost Canonical: KL divergence. Local analysis: quadratic surrogate. Tranche–credit feedback Boundedcorrectionensuringstatisticalproportionality. Stability region Parameter regime 0<𝑇𝑔′(0)<2preventing oscillatory dynamics. Table 2: Standardised terminology used throughout SCE. 6 Origins and Distinction 6.1 Historical Context SCE emerges from a lineage of algorithmic fairness research grounded in procedural transparency and epistemic accountability. Following Floridi (2019), who frames transparency as the precondition for external audit, SCE treats fairness as a falsifiable empirical property rather than a normative aspiration. This positions it against outcomesbasedapproaches-paritymetrics,reweighting,orpost-hoccorrection-whichassertfairness through statistical alignment alone (Barocas et al., 2019b; Corbett-Davies & Goel, 2023; Mitchell et al., 2021). By contrast, SCE inherits the evidential posture of The Contradiction Trap: fairness is not declared but demonstrated through observable behaviour under symmetric stochastic pressure. 20 This procedural focus echoes Binns (2018), who warns that distributional definitions of fairness can obscure process-based harms. Where traditional algorithmic systems often hide trade-offs within opaque model logic, SCE foregrounds deviation as evidence: systematic residue becomes the statistical fingerprint of structural asymmetry. This reflects the argument of Rahwan et al. (2019), who proposes that algorithmic systems should be studied as behavioural entities whose actions are observable, testable, and open to scrutiny over time (Doshi-Velez & Kim, 2017; Selbst et al., 2019). The underlying governance problem is ancient. Sortition in Athens, medieval lotteries, nineteenth-century rotation systems, and modern welfare algorithms all wrestle with the same dilemma: impartiality in selection versus equity in outcome (Headlam, 1906; Stone, 2011). Computational administration reframed the dilemma in mechanisticterms-howtoguaranteefairnesswithoutsmugglingbiasintothemechanism(Moulin, 2019). Deterministic schedulers offered repeatability but not transparency. Early stochastic methods-shuffles, seeded PRNGs, weighted lotteries-introduced the appearance of chance while retaining algorithmic determinism. Their randomness was reproducible theatre: predictable on audit and capable of encoding structural bias when examined in aggregate (National Institute of Standards and Technology, 2022; Vinod, 2013b). SCE departs from this lineage by introducing a probabilistic governance mechanism: an allocationprotocolgroundedinverifiableentropyandlarge-sampleconvergence(Drand Consortium, 2021; Kushner&Yin, 2003). At small scales, SCEfunctions asa diagnostic of institutional behaviour by exposing deviations; at large scales, it becomes a stability process that converges visibly toward declared proportions. 6.2 Comparison with Classical Mechanisms Traditionalweightedlotteriesenforceproportionalityonlythroughstatic,exanteweights. Over repeated use, stochastic variance accumulates and produces long-run distortions that vanish only with extremely large sample sizes. Attempts to mitigate this-deficit recovery, rotating schedules, heuristic rebalancing-correct imbalances only through external calibration (Procaccia & Tennenholtz, 2013; Walsh, 2011). Game-theoretic constructs such as the Shapley value, envy-free divisions, or Nash bargaining mechanisms provide rigorous fairness guarantees, but only under assumptions of complete information, rational agents, and stable environments (Brams & Taylor, 2005; Moulin, 2019). Such conditions rarely hold in operational governance. These mechanisms are theoretically elegant but practically brittle. SCE diverges from both families. It treats allocation as a dynamic stochastic process 21 and seeks asymptotic proportionality through runtime behaviour rather than designtime perfection (Devroye, 1986; Robbins & Monro, 1951). Fairness does not depend on agent rationality or administrative tuning; it emerges statistically within observable error bands through the combination of verifiable entropy and bounded feedback. 6.3 Architectural Distinction SCE’s architecture departs from prior art in three fundamental respects: 1. Entropy Verification. SCE replaces algorithmic pseudorandomness with certified entropy-randomness whose provenance is public and tamper-evident-removing theepistemicopacityinherentin PRNG-basedfairnessclaims(DrandConsortium, 2021; National Institute of Standards and Technology, 2022). 2. Tranche–Credit Dynamics. The bounded, memory-light feedback loop compensatesforshort-rundisproportionality. Convergenceisstatisticalratherthanexact: deviations shrink in expectation as event volume increases (Hall & Heyde, 1980; Kushner & Yin, 2003). 3. Auditability as a Design Primitive. Each allocation is paired with a verifiable record of entropy input, probability vector, and tranche state. Fairness becomes empirically inspectable and falsifiable, not assumed via authorial or institutional authority (Doshi-Velez & Kim, 2017; Self, Wang, & Kehl, 2023). 6.4 Theoretical Positioning SCE sits at the intersection of three conceptual traditions: •Probability theory: its convergence derives from the Law of Large Numbers and theCentralLimitTheorem, bothrequiringsubstantialsamplesizesforvalidity(Devroye, 1986; Kolmogorov, 1933); •Game theory: it emphasises behaviour under symmetric rules over outcome optimisation, echoing the adversarial clarity structures in TCT (Brams & Taylor, 2005; Caragiannis et al., 2019); •Ethical governance: fairness is treated as an auditable property of system behaviour rather than a moral narrative or design-time declaration (Floridi, 2019; Mitchell et al., 2021; Selbst et al., 2019). 22 Whereearlier systemssought legitimacythroughopacity-trust inexpertise, complexity, or institutional authority-SCE grounds its legitimacy in transparency and reproducible statistical evidence (Rahwan et al., 2019; Self, Wang, & Kehl, 2023). Fairness is demonstrated, not proclaimed. 6.5 Summary SCE occupies a distinctive conceptual space. It is not an algorithm that simulates fairness, but a mechanism that tests it. Deterministic schedulers enforce equality through control; naive randomness abandons equality to chance. SCE shows that control and chancecancoexistwithinasinglestochasticdisciplinegovernedbylaw-likeconvergenceprovidedthescaleofeventsissufficientforregularitiestoemerge(Kushner&Yin,2003; Robbins & Monro, 1951). In this formulation, fairness becomes neither a political aspiration nor an engineering artefact, but a measurable property of entropy constrained by proportion and volume (Floridi, 2019; Vinod, 2013b). Table 3: Fairness paradigms contrasted with SCE’s procedural model. Method Fairness Interpretation Demographic Parity Statistical equality across group outcomes (Mitchell et al., 2021). Individual Fairness Similar individuals receive similar treatment (Dwork et al., 2012). Counterfactual Fairness Outcomes invariant under perturbations of sensitive attributes (Kusner et al., 2017). Post-hoc Correction Output adjusted after allocation, often opaquely (Hardt et al., 2016). SCE (proposed) Fairness as stochastic convergence: transparent, self-correcting, auditable via deviation trace. 7 Coherence Cost Estimation SCEextendsthecoherence-costframeworkintroducedinThe Contradiction Trap (TCT), translating epistemic inconsistency into a statistical domain. In TCT, inconsistency carries an information cost 𝐶(𝑟), representing the epistemic strain required to maintain 23 contradictory positions under symmetric interrogation. SCE generalises this idea: deviation from proportional allocation becomes a statistical analogue of inconsistency, and the coherence cost measures the informational tension between observed behaviour and declared proportions (Kullback & Leibler, 1951; Vinod, 2013b). Unlike paraconsistent logics, which tolerate contradiction by redefining consistency thresholds (Priest, 2006), SCE treats deviation as a falsifiable claim about institutional symmetry. Its posture is diagnostic, not permissive: sustained deviation is empirical evidence of procedural imbalance, mitigated only through bounded tranche–credit correction. Coherence is not logical survivability, but statistical falsifiability (Binns, 2018; Rahwan et al., 2019). 7.1 From Logical to Statistical Consistency In TCT, coherence cost quantifies the epistemic price of preserving inconsistent propositions. SCE reinterprets this as the cost of preserving fairness under stochastic pressure. The central question becomes: how much information is lost when the realised allocation diverges from the declared proportions? Let 𝑤𝑖denote the declared weight for recipient 𝑟𝑖, and let the empirical frequency after 𝐾draws be 𝑤𝑖(𝐾)= 1 𝐾𝐾 ∑ 𝑡=11{𝐴𝑡=𝑖} (7.1) Define the deviation 𝛿(𝐾)=(𝛿1,…,𝛿𝑛), 𝛿𝑖(𝐾)= 𝑤𝑖(𝐾)−−𝑤𝑖(7.2) SCE’s feedback mechanism operates to reduce 𝛿(𝐾)in expectation as 𝐾grows, without compromising procedural fairness (Kushner & Yin, 2003; Robbins & Monro, 1951). 7.2 Canonical Coherence Cost Heuristic Absolute-Deviation Cost (Non-canonical) For quick diagnostic inspection, one may use the 𝐿1deviation 𝐶abs(𝐾)= 𝑛 ∑ 𝑖=1|𝑤𝑖(𝐾)−−𝑤𝑖|(7.3) 24 This is not thecanonical coherencecost usedbySCE andcarries noinformation-theoretic interpretation. It is useful only as a coarse visual indicator or early-stage sanity check. Convergence guarantees apply exclusively to the KL-based metric. SCE adopts the Kullback-Leibler divergence as its canonical coherence metric: 𝐶(𝐾)=𝐷KL( w(𝐾)‖w)=𝑛 ∑ 𝑖=1 𝑤𝑖(𝐾)log2𝑤𝑖(𝐾) 𝑤𝑖(7.4) When 𝐶(𝐾)=0, empirical proportions match the declared distribution. For small deviations, a second-order Taylor expansion yields the quadratic approximation 𝐶(𝐾)≈ 1 2ln2𝑛 ∑ 𝑖=1𝛿𝑖(𝐾)2 𝑤𝑖(7.5) This form is symmetric, computationally convenient, and accurate whenever ‖𝛿(𝐾)‖∞is small (Hall & Heyde, 1980; Kullback & Leibler, 1951). SCE’s dynamics are designed so that 𝔼[𝐶(𝐾)]→0 as 𝐾→∞ (7.6) butonlyonce𝐾exceedstheregimeinwhichLLNandCLTeffectsdominatefinite-sample noise (Devroye, 1986; Kolmogorov, 1933). 7.3 Transition from Logical to Statistical Cost 7.4 Quadratic Approximation Lemma 7.1 (Quadratic Approximation of KL Divergence).Let 𝑃be empirical frequencies and 𝐹the declared distribution with 𝑃𝑖,𝐹𝑖>0, and suppose 𝑃𝑖=𝐹𝑖+𝜀𝑖with ∑𝑖𝜀𝑖=0and ‖𝜀‖∞small. Then 𝐷KL(𝑃‖𝐹)=12𝑛 ∑ 𝑖=1𝜀2 𝑖 𝐹𝑖+𝑂(‖𝜀‖3∞)(7.7) Proof. Immediate from the Taylor expansion of log(1+𝑥)applied to (𝐹𝑖+𝜀𝑖)log((𝐹𝑖+𝜀𝑖)/𝐹𝑖) and cancellation of linear terms (Kullback & Leibler, 1951). 25 9 Analytical Function This section formalises SCE as a stochastic process with bounded feedback, establishing its expectation, variance, convergence, stability, and drift-tracking properties. The analysisparallelstheevidentiallogic ofThe Contradiction Trap: insteadofevaluatinginference under symmetric pressure, we evaluate allocation behaviour under stochastic symmetry. Fairness is not assumed but emerges from exposure to verifiable randomness and bounded correction (Borkar, 2008; Kushner & Yin, 2003; Ljung, 1977). Assumptions for Proportional Convergence SCE’s convergence guarantees hold under the following structural conditions: A1. Entropy independence. Entropy draws (𝑈𝑡)are i.i.d. Unif[0,1)or satisfy a strong-mixing condition with 𝛼(𝑘)→0(Hoeffding, 1963). A2. Measurability. Allocation indicators 𝑋𝑖(𝑡)are adapted to the filtration ℱ𝑡= 𝜎(𝑈1,…,𝑈𝑡)and satisfy 𝔼[𝑋𝑖(𝑡)∣ℱ𝑡−1]=𝜋𝑖(𝑡). A3. Bounded credit update. 𝑔∶ℝ→ℝis bounded, Lipschitz near 0, and satisfies 𝑔(0)=0. A4. Finite tranche size. Each tranche has fixed finite size 𝑇. A5. Stationary or bounded-drift weights. Either 𝑤𝑖(𝜏)is constant, or satisfies |𝑤𝑖(𝜏+1)−𝑤𝑖(𝜏)|≤𝑐/𝑇(Robbins & Monro, 1951). A6. Responsiveness. 0<𝛼≤1ensures every update contains a stationary component. 9.1 Expectation and Variance Let 𝑤𝑖(𝐾)be the empirical allocation proportion after 𝐾events. Under the assumptions above and bounded 𝑔,𝔼[𝑤𝑖(𝐾)]=𝑤𝑖+𝑂(𝐾−1)(9.1) Var[𝑤𝑖(𝐾)]=𝑂(𝐾−1)(9.2) withthevarianceterminflatedonlybyasmallconstantproducedbythecredit-feedback mechanism. Thus SCE exhibits the same 1/𝐾decay rate as an unbiased Bernoulli process, modulo mild correction noise (Devroye, 1986; Hall & Heyde, 1980). 32 9.2 Martingale Fairness Define the effective participation probability 𝜋𝑖(𝑡)=𝛼𝑤𝑖(𝜏(𝑡))+(1−𝛼)𝜅𝑖(𝜏(𝑡)−1) (9.3) where 𝜅𝑖denotes the tranche-level credit. Lemma 9.1 (Martingale Fairness).Under Assumptions 9, 𝑀𝑖(𝐾)= 𝐾 ∑ 𝑡=1(𝑋𝑖(𝑡)−−𝜋𝑖(𝑡))(9.4) is a martingale with bounded increments |𝑋𝑖(𝑡)−𝜋𝑖(𝑡)|≤1. Proof. Since 𝑋𝑖(𝑡)is produced by an independent entropy draw with success probability 𝜋𝑖(𝑡),𝔼[𝑋𝑖(𝑡)∣ℱ𝑡−1]=𝜋𝑖(𝑡). The bounded-increment property follows from 𝑋𝑖(𝑡)∈{0,1}. This recasts SCE as a bounded martingale system. Deviations reflect stochastic fluctuations, not structural bias, unless they persist after variance decay. 9.3 Proportional Convergence Theorem 9.2 (ProportionalConvergence).Let 𝑤𝑖(𝐾)=1 𝐾∑𝐾 𝑡=1𝑋𝑖(𝑡)be the empirical share of allocations to group 𝑖. Under Assumptions 9, 𝑤𝑖(𝐾) 𝑝 ⟶𝑤𝑖(9.5) as 𝐾→∞. Proof. Write𝑋𝑖(𝑡)=𝜋𝑖(𝑡)+𝐷𝑖(𝑡), with𝐷𝑖(𝑡)themartingale-differencetermfromLemma9.1. Summing, 𝑤𝑖(𝐾)−𝑤𝑖=1 𝐾𝐾 ∑ 𝑡=1𝐷𝑖(𝑡)+1 𝐾𝐾 ∑ 𝑡=1(𝜋𝑖(𝑡)−𝑤𝑖)(9.6) ThemartingaletermvanishesinprobabilitybytheWeakLawforboundedincrements(Hoeffding, 1963). Bounded credits imply the second term is 𝑂(𝐾−1). 33 9.4 Martingale Central Limit Approximation Theorem 9.3 (Central Limit Approximation).Under Assumptions 9 and the martingale Lindeberg condition, √𝐾(𝑤(𝐾)−𝑤)𝑑 −→ 𝒩(0,Σ) (9.7) where Σ𝑖𝑗={𝑤𝑖(1−𝑤𝑖)(1+𝜀𝜅), 𝑖=𝑗, −𝑤𝑖𝑤𝑗(1+𝜀𝜅), 𝑖≠𝑗, (9.8) and 𝜀𝜅captures bounded credit-induced variance inflation (Hall & Heyde, 1980; Kushner & Yin, 2003). This provides the asymptotic sampling distribution needed for uncertainty estimation, interval construction, and anomaly testing. 9.5 Stability of Credit Dynamics Let Δ𝑖(𝜏)denote the tranche residual, and 𝜅𝑖(𝜏)=𝑔(Δ𝑖(𝜏)) (9.9) the credit update. Lemma 9.4 (Local Stability).If 𝑔is differentiable at 0with derivative 𝑔′(0), then the linearised dynamics are stable iff 0<𝑇𝑔′(0)<2 (9.10) Proof. Linearising gives Δ𝑖(𝜏+1)=(1−−𝑇𝑔′(0))Δ𝑖(𝜏)+𝜀𝜏(9.11) Stability of the deterministic recurrence requires |1−−𝑇𝑔′(0)|<1, giving the claimed condition (Borkar, 2008; Ljung, 1977). Corollary 9.5 (Global Stability).Let 𝐿=sup 𝑥≠0|𝑔(𝑥)|/|𝑥| (9.12) be the global slope bound. If 0<𝑇𝐿<2, then Δ𝑖(𝜏) 𝑝 ⟶0 (9.13) 34 even when 𝑔is clipped or nonlinear (Kushner & Yin, 2003). Thus the same stability criterion that governs the linear case extends to the full nonlinear system. 9.6 Dynamic Equilibrium At equilibrium, 𝔼[Δ𝑖]=0, 𝔼[𝜅𝑖]=0 (9.14) Perturbations decay geometrically: 𝔼[Δ𝑖(𝜏+𝑘)]≈(1−−𝑇𝑔′(0))𝑘Δ𝑖(𝜏) (9.15) The convergence rate is therefore a function of 𝑔′(0)and tranche size 𝑇, but becomes empiricallyobservableonlyoncesamplingnoisefallsbelowthedynamicssignal(Borkar, 2008). 9.7 Tracking Non-Stationary Weights SCE can track slowly varying weight sequences: ‖𝑤(𝜏+1)−𝑤(𝜏)‖∞≤𝛿 (9.16) Lemma 9.6 (Drift Tracking).Under Assumptions 9, if declared weights evolve under bounded drift 𝛿, then sup 𝜏𝔼[|𝑤𝑖(𝜏𝑇)−𝑤𝑖(𝜏)|] ≤ 𝐶1𝛿+𝐶2 𝜏(9.17) Thus SCE tracks dynamic proportions up to an error floor proportional to the drift rate 𝛿. Fairness can follow real changes only as fast as statistical stability allows (Kushner & Yin, 2003; Robbins & Monro, 1951). 9.8 Identifiability of Deviation Sources Lemma 9.7 (Identifiability).Under Assumptions 9, the following deviation sources are distinguishable from tranche-level statistics: I1. Stochastic noise: 𝐶𝜏=𝑂(√(𝑛−1)/(2𝑇ln2))with no serial correlation. 35 I2. Mis-tuned 𝑔: persistent oscillation with autoregression coefficient | 𝜌|≈|1−𝑇𝑔′(0)|> 0.6. I3. Entropy degradation: variance systematically below 𝑤𝑖(1−𝑤𝑖). I4. Concept drift: transient rises in 𝐶𝜏that decay within 𝑂(1/𝛼). I5. Systemic bias: 𝐶𝜏>𝐸[𝐶𝜏]+𝑧𝑝𝜎stoch for multiple tranches after ruling out I1–I4. This defines the evidential logic of SCE: unfairness is that which cannot be explained by noise, mis-tuning, entropy faults, or drift. 9.9 Interpretation Analytically, SCE is a lightly damped stochastic control system: entropy provides symmetric exploration; credits supply bounded correction; and proportionality emerges as the unique fixed point. Fairness is therefore not a declarative property of the algorithm but a statistical outcome verified through replication (Floridi, 2019; Jaynes, 2003). 9.10 Summary SCE satisfies three core analytical properties: 1. Expectation: Unbiased entropy ensures 𝔼[𝑤𝑖(𝐾)]=𝑤𝑖+𝑂(𝐾−1)(9.18) 2. Convergence: Bounded feedback yields 𝑤𝑖(𝐾) 𝑝 →𝑤𝑖(9.19) under scale. 3. Stability: Residual dynamics are stable whenever 0<𝑇𝑔′(0)<2(or globally 0< 𝑇𝐿<2). SCE therefore formalises fairness as a probabilistic equilibrium: not exact, not moral, but empirically demonstrable. It guarantees proportionality only relative to the declared weights; the legitimacy of those weights is a separate question of governance and procedural justice (Binns, 2018; Rahwan et al., 2019). 36 10 Empirical Convergence & Scale Thresholds SCE’s analytical guarantees apply in the limit, but deployment requires an understanding of how quickly proportional accuracy emerges at finite scales. This section summarises the empirically observed decay of deviation under stochastic symmetry and provides operational thresholds for minimum safe allocation volumes (Borkar, 2008; Devroye, 1986; Hoeffding, 1963). 10.1 Minimum Viable Scale For target absolute accuracy 𝜀for all 𝑛recipient groups simultaneously at overall confidence 𝑝, a conservative worst-case bound is obtained by applying the Šidák correction to the multinomial variance: 𝐾min ≈𝑧2 𝑝1/𝑛 4𝜀2,(10.1) where 𝑧𝑝1/𝑛 is the corresponding normal quantile (Lehmann & Romano, 2006). This boundassumesmaximalvariance𝑤𝑖(1−𝑤𝑖)=1/4, stationary weights, and noadversarial reporting; SCE’s tranche–credit loop typically achieves faster empirical variance decay once past the burn-in (Ljung, 1977). Illustrative values (worst case): 𝜀=±2%, 𝑝=0.95, 𝑛=10 ⇒ 𝐾min ≈4,900, (10.2) 𝜀=±1%, 𝑝=0.99, 𝑛=10 ⇒ 𝐾min ≈27,000. (10.3) ThesethresholdsalignwithSCE’sempirical𝑂(1/𝐾)variancedecay: below𝐾min, stochastic noise dominates and SCE functions primarily as a diagnostic monitor rather than a guaranteed proportional allocator. Table 6: Conservative 𝐾min thresholds for achieving ±𝜀accuracy for all 𝑛groups simultaneously at overall confidence 𝑝, using the Šidák-adjusted normal bound 𝐾min ≈ 𝑧2 𝑝1/𝑛/(4𝜀2). Rounded to the nearest hundred. 𝑛Target accuracy Confidence 𝑝Recommended 𝐾min 5±5% 90% 500 95% 700 99% 1000 (continued) 37 (continued) 𝑛Target accuracy Confidence 𝑝Recommended 𝐾min 5±2% 90% 3300 95% 4100 99% 6000 5±1% 90% 13300 95% 16500 99% 23900 10 ±5% 90% 700 95% 800 99% 1100 10 ±2% 90% 4100 95% 4900 99% 6800 10 ±1% 90% 15000 95% 19000 99% 27000 20 ±5% 90% 900 95% 1000 99% 1200 20 ±2% 90% 4900 95% 5700 99% 7600 20 ±1% 90% 19500 95% 22700 99% 30300 50 ±5% 90% 900 95% 1100 99% 1400 50 ±2% 90% 5900 95% 6700 99% 8600 50 ±1% 90% 23600 95% 27000 99% 34600 38 These values should be treated as deployment heuristics rather than rigid constraints. When 𝐾 <𝐾min, SCE still functions, but the variance floor overwhelms the feedback signal and proportionality cannot be expected to manifest reliably. 10.2 Small-K Regimes Figures7–9showtypicalbehaviourat𝐾=100,500, and 1000. Theprogressionillustrates three characteristic regimes (Borkar, 2008; Kushner & Yin, 2003): •Volatility-dominated (𝐾≲300): noise masks all structure. •Transitional (300≲𝐾≲1500): variancebegins to contractbutthecorrectivesignal remains weak. •Early convergence (𝐾≳1000): empirical proportions begin to shadow declared weights. These regimes reflect the martingale variance bound discussed in Lemma 9.1: until 1/√𝐾drops below the residual tranche–credit signal, convergence cannot manifest visually or statistically. Figure 7: SCE Small-𝐾Regime (K = 100). Raw 𝐿1deviation ‖ 𝑤(𝐾)−−𝑤‖1as a function of allocation count. Behaviour is volatility-dominated: noise masks weight structure for 𝐾≲300. 39 Figure 8: SCE Small-𝐾Regime (K = 500). Deviation begins transitioning from volatilitydominated to contraction: the variance floor remains high, but the tranche–credit signal becomes visible. Figure 9: SCE Small-𝐾Regime (K = 1000). Early convergence emerges: empirical proportions begin shadowing declared weights, marking the point where 1/√𝐾drops below the residual credit signal. 10.3 Interval Fairness Claims (IFC) SCE’s proportional guarantees require sufficiently large allocation volumes. When 𝐾is small, the stochastic component dominates the tranche-credit correction, and proportional error is governed primarily by binomial variance rather than convergence dynam40 ics. To provide evidentially meaningful fairness reporting in these small-𝐾settings, we introduce Interval Fairness Claims (IFC) (Blaker, 2000; Clopper & Pearson, 1934). Definition (IFC). Let 𝑋𝑖denote the realised number of allocations to group 𝑖out of 𝐾 total draws, under target proportion 𝑤𝑖. For a desired simultaneous coverage level 1−𝛼 across all 𝑛groups, define the Šidák-adjusted per-group level 𝛼′=1−−(1−𝛼)1/𝑛 (10.4) For each group 𝑖, compute the exact Clopper–Pearson interval [𝐿𝑖,𝑈𝑖]for 𝑋𝑖/𝐾at confidence level 1−𝛼′. Interval Fairness Claim (IFC): with probability at least 1−𝛼, the following holds simultaneously: 𝐿𝑖≤𝑋𝑖 𝐾≤ 𝑈𝑖, 𝑖=1,…,𝑛 (10.5) Interpretation. IFC supplies a finite-sample fairness statement that remains valid even when 𝐾is far below the convergence threshold. The IFC bands quantify the range of deviations attributable to sampling variance alone. Observed proportions outside these bands constitute evidential asymmetry in the small-𝐾regime and may indicate directional drift or structural misalignment. Reporting Rule. In all SCE deployments, fairness reporting shall include: 1. realised proportions 𝑋𝑖/𝐾for each group; 2. the corresponding IFC bands [𝐿𝑖,𝑈𝑖]at the chosen confidence level; 3. tranche–credit residuals Δ𝑖(𝑇)and credit balances 𝜅𝑖(𝑇); 4. a statement of whether all groups fall within IFC simultaneously. This yields a unified evidential framework: IFC for small-𝐾settings and convergence diagnostics for large-𝐾deployments. Example. With 𝑛=4groups, 𝐾=80allocations, and a 95% simultaneous coverage target, the adjusted per-group level is 𝛼′=1−−0.951/4 ≈0.0127. If a group receives 𝑋𝑖=14allocations,theexactintervalfor𝑋𝑖/𝐾is[0.110,0.255]. Anobservedshareoutside this range breaches the IFC and signals small-𝐾fairness drift. 41 eters are declared, they must remain untouched. Any midstream adjustment corrupts theevidentialenvironment. Symmetrybecomesanepistemicconstraint: fairnessmust emerge under even exposure or fail publicly. 12.3 The Meaning of Non-Intervention Non-interventionisbothmethodologicalnecessityandethicalboundary. Deterministic governance enforces balance through control; SCE reveals balance through statistical correction(Binns, 2018; Mitchelletal., 2021). The systempermitsshort-runimbalance precisely because it refuses to override randomness with discretion. Interfering with the stochastic process-adjusting a draw, re-weighting a result, retroactively selecting an outcome-equates to falsifying experimental conditions. SCE’s integrity therefore lies in its indifference. As with a randomised controlled trial, fairness arises from how exposure is allocated, not from administrator intent (Lehmann & Romano, 2006). 12.4 Symmetry as Epistemic Discipline SCE encodes two linked forms of symmetry: 1. Input symmetry: all recipients experience the same procedure, governed by common entropy and tranche–credit rules. 2. Outcome symmetry: across sufficiently many events, empirical frequencies converge to declared proportions. These mirror the distinction between hypothesis and observation. Declared weights represent an institutional claim; the empirical trace provides its test. Deviation becomes evidential when it exceeds what stochastic symmetry predicts. In this sense, symmetry functions not merely as a fairness constraint but as a mode of challengeability: integrity can only be demonstrated when it can be meaningfully challenged (Floridi, 2019). 12.5 Entropy as the Moral Constant Entropy is SCE’s moral constant: an indifferent arbiter that equalises exposure. The entropy term here functions as an epistemic temperature — regulating the balance be48 tweenorderanduncertaintymuchasthermal systemstradeenergyforstructure. Byreplacing subjective discretion withcertifiedrandomness, SCEconverts ethicaljudgment into procedural risk-sharing (Rahwan et al., 2019). Fairness is therefore a property of process, not outcome. Symmetric ignorance renders deviation legible and prevents bias from being concealed behind explanation. 12.6 Convergence as Evidence of Integrity SCE’s evidential claim is cumulative. Convergence toward declared weights is not a performance target but a form of empirical testimony: the system has survived sustained entropic pressure. Failure to converge within tolerance is not a moral indictment but a procedural signal. SCE does not assert “I am fair”; it offers instead: “Observe me over time.” 12.7 Implications for Governance Philosophy Symmetry without interventionimpliesadifferentgovernanceethic. Traditionalinstitutions govern through interpretive discretion; every correction embeds an unseen preference. SCE proposes the opposite: commit upfront, then abstain. Correction arises only through the fixed feedback mechanism, not through human revision. This model asks institutions to relinquish the illusion of control and accept that fairness must be exposed, not curated (Mitchell et al., 2021; Mittelstadt et al., 2016). 12.8 Mapping Epistemic Necessity to Procedural Necessity SCEinherits the structurallogicofTCT. The TCTprinciple thatasymmetry without necessity implies intent becomes, in SCE, deviation without stochastic cause implies process failure. The mapping is direct: The two frameworks thus share a common evidential structure. TCT exposes inconsistency across commitments; SCE exposes inconsistency across behaviour. In both, deviation is not noise but information. 12.9 Summary SCE’s core principle is symmetry without intervention. It replaces discretion with exposure, intent with evidence, and reassurance with reproducibility. Fairness becomes a 49 TCT Concept SCE Procedural Analogue Asymmetry without necessity Deviation exceeding tranche-level KL noise Contradiction under symmetric framing Stagnant or divergent cumulative KL Coherence cost Tranche or cumulative KL divergence Narrative–rationale conflict Declared weights vs realised allocations Evasion (delay, reframing, suppression) Entropy anomalies, stale credits, drift patterns Information gain Δ𝐼 Audit utility from incident flags and diagnostics No equilibrium possible No stationary distribution when 𝑇𝑔′(0)∉(0,2) Evidential conclusion: intent likely Procedural conclusion: fairness failure likely Table 8: Mapping epistemic necessity (TCT) to procedural necessity (SCE). property that emerges under challenge, not a claim insulated from it. Where TCT operationalises epistemic honesty in argument, SCE operationalises it in allocation. Neither guarantees fairness; both require the system to demonstrate it. 13 Game-Theoretic Formalisation 13.1 Motivation Any mechanism that allocates opportunity creates incentives for strategic behaviour. Participantsmaymisreport inputs; administrators mayattempt subtleintervention; observers may misinterpret stochastic variation as bias. SCE neutralises these distortions not by policing agents, but by embedding symmetry directly into the mechanism. Strategic manipulation yields no advantage because the procedure itself is indifferent. The resulting architecture behaves as a repeated stochastic game in which the dominant strategy is honesty-not by moral appeal, but by strategic futility. This situates SCE within the incentive-compatible tradition of mechanism design, but replaces de50 terrence with evidential neutrality (Maskin, 1985; Myerson, 1979; Roth, 2002). 13.2 Game Structure Let the allocation procedure define a stochastic game 𝐺=⟨𝑅,𝐴,𝑆,𝑓,𝑃⟩ (13.1) with components: •𝑅={𝑟1,…,𝑟𝑛}– recipients (players); •𝐴𝑖={𝑡𝑖,𝑚𝑖}– each may transmit truthful weights 𝑡𝑖=𝑤𝑖or manipulate them 𝑚𝑖=𝑤′𝑖; •𝑆– system state (declared weights, credits, tranche history); •𝑓∶𝑆×𝐴→𝑆′– deterministic update under SCE’s tranche–credit rule; •𝑃𝑖– public stochastic payoff function (allocation probability). Each event 𝑡proceeds as follows: (i) players choose actions from 𝐴𝑖; (ii) the system updates via 𝑓; (iii) allocations occur via verifiable entropy. Fairness is defined not per draw, but in expectation: the relevant object is the long-run empirical distribution 𝑤𝑖(𝐾). 13.3 Payoff Function and Expected Utility For recipient 𝑟𝑖, define the expected utility 𝑈𝑖(𝑡)=𝔼[1{𝐴𝑡=𝑖}]−−𝑐𝑖(𝑎𝑖)(13.2) where 𝑐𝑖represents reputational, procedural, or audit exposure cost. Since entropy is non-manipulable and the tranche–credit mechanism corrects deviations over time, 𝔼[1{𝐴𝑡=𝑖}∣𝑎𝑖=𝑚𝑖]≈𝔼[1{𝐴𝑡=𝑖}∣𝑎𝑖=𝑡𝑖]=𝜋𝑖(𝑡), (13.3) 51 and thus 𝑈𝑖(𝑡)=𝜋𝑖(𝑡)−−𝑐𝑖(𝑎𝑖), (13.4) with 𝑐𝑖(𝑚𝑖)>0. There is no expected gain from misreporting, making truth-telling a weakly dominant strategy. 13.4 Equilibrium Characterisation Figure 13: Game-theoretic dynamics in SCE. Left: Strategic choice diagram showing weak dominance of honesty. Centre: Expected payoff structure under repeated exposure and audit risk. Right: Stability region in which manipulation becomes irrational as evidence volume increases. Together, these illustrate a probabilistic Nash equilibrium driven by statistical transparency rather than punitive threat. Theorem 13.1 (Probabilistic Honest Equilibrium).If entropy is verifiable and audit logs are public, then the honest strategy profile (𝑎𝑖=𝑡𝑖)𝑛𝑖=1constitutes a Nash equilibrium of 𝐺 in expectation. 52 Proof. For any player 𝑖,𝔼[𝑈𝑖(𝑎𝑖=𝑚𝑖)]≤𝔼[𝑈𝑖(𝑎𝑖=𝑡𝑖)], (13.5) becauseexpectedallocationprobabilityisunchanged,whilemanipulationincursnonzero auditcost. Under symmetric exposure andfullyvisible logs, no player benefitsinexpectation from deviation. Hence truthful reporting forms a Nash equilibrium (Fudenberg & Tirole, 1991). 13.5 Equilibrium under Partial Transparency Perfect transparency is not always available. Even when only summary statistics or delayed logs are released, SCE can preserve incentive alignment through probabilistic auditing. Lemma 13.2 (ApproximateEquilibriumwithAudit Probability).Let 𝜌∈(0,1]be the probability that a tranche is audited, and Πthe penalty if manipulation is detected. If 𝐺denotes the one-tranche gain from deviation, then 𝜌Π≥𝐺 ⇒ Honest play is a Bayesian Nash equilibrium. (13.6) Proof. A deviator’s expected utility is 𝑈dev =𝐺−−𝜌Π, (13.7) while honest play yields 𝑈hon =0. If 𝜌Π≥𝐺, deviation is unprofitable. Increasing 𝜌or Πpreserves equilibrium under uncertainty. Policy implication. The minimal audit rate maintaining equilibrium is 𝜌min =𝐺 Π.(13.8) Randomisedauditswithpublicdisclosureenforcestrategyneutralityevenunderpartial transparency (Becker, 1968; Fudenberg & Levine, 1998). 13.6 System as Meta-Player SCE behaves as a meta-agent whose “actions” are entropy sampling, bounded correction, and log publication. If the system itself deviates-biased reseeding, selective adjustment, withheld logs-the deviation produces a measurable coherence cost 𝐶(𝐾), ex53 posing SCE to its own evidential test. Integrity thus becomes self-confirming: anomalies indict the mechanism automatically through violation of its statistical profile. 13.7 Repeated Interaction and Learning Dynamics Under repeated exposure, rational agents update beliefs about the mechanism. When convergence consistently matches declared weights, manipulation is empirically discredited. Equilibrium becomes self-enforcing: not through deterrence, but through accumulated evidence that deviation yields no advantage. Belief revision drives behavioural stability (Young, 2004). 13.8 Information Symmetry and Trust Collapse Conventional allocation systems rely on asymmetric information: administrators possess privileged knowledge. SCE collapses this asymmetry. All players observe identical logs, draws, and credit updates. Trust becomes unnecessary; verification replaces reputation. Authority is grounded in reproducibility rather than secrecy (Floridi, 2019; Rahwan et al., 2019). 13.9 Deviations and Detection Any attempt to manipulate inputs or internal parameters leaves statistical traces: •variance inflation in allocation frequencies, •stagnant or rising coherence cost, •persistent residual patterns across tranches. Detection does not rely on whistleblowers or normative assumptions; deviation reveals itself through empirical inconsistency. 13.10 Interpretive Summary SCE does not enforce virtue; it disables strategy. Manipulation becomes unprofitable by design and unviable in expectation. Where traditional governance appeals to ethics or enforcement, SCE appeals to evidence. Its equilibrium is not ideal behaviour but 54 the absence of a better alternative. Fairness emerges not from intent, but from the strategic futility of deviation. 13.11 Summary SCE defines a stochastic governance game in which honesty is the only stable strategy undertransparency. Nosecretmoves, noprivilegedinformation, nomanipulablelevers. The system does not participate in the game-it exposes it. In this design, fairness is not engineered; it is the equilibrium that remains. 14 Interdisciplinary Relevance & Contemporary Context 14.1 Algorithmic Fairness & Governance SCE occupies an unusual position in the algorithmic fairness landscape. Most frameworks address predictive bias-modifying model outputs so groups receive statistically similar treatment (Barocas & Selbst, 2016; Mitchell et al., 2021). SCE focuses instead on the fairness of access to decision-making itself. It does not evaluate model accuracy or correct for downstream disparities; it scrutinises the procedural distribution of opportunity. Governance systems typically define fairness through outcome parity or representational balance-metrics that are descriptive rather than procedural. SCE reverses this logic. It tests neutrality before decisions occur, treating fairness as the absence of predictable advantage under entropy rather than as equality of end states. Fairness becomes a probabilistic property of symmetry in process, not a retrospective moral claim. 14.2 Organisational Systems & Bureaucratic Neutrality Traditional bureaucracy enforces impartiality through authority: oversight, hierarchy, and supervision stand in for neutrality. SCE replaces this apparatus with statistical feedback. It enforces impartiality not through discretion but through verifiable entropy and bounded correction (Goodsell, 2015; Weber, 1947). This yields a form of administrative neutrality that is empirical rather than performative-fairness demonstrated by convergence, not inferred from ritual. 55 The result is a kind of bureaucracy without bureaucracy: a system in which procedural integrity arises from reproducible randomness rather than managerial intervention. 14.3 Information Theory & Entropy Ethics SCE’s use of entropy extends beyond mechanics into ethics. The requirement for unpredictability becomes a moral principle: distributed ignorance that prevents foresight frombecomingadvantage. Entropyenforces proceduralblindness, makinginformation symmetry a precondition for justice (Floridi, 2019; Shannon, 1948). In a culture dominated by prediction and optimisation, SCE’s commitment to randomness is countercultural. It is an ethic of epistemic restraint-fairness achieved not by insight, but by controlled ignorance. 14.4 Behavioural & Cognitive Dimensions SCE alters how agents perceive fairness. Instead of trusting intentions or hierarchy, participants infer fairness from the visible behaviour of the system. Each allocation becomes a micro-experiment; over time, trust emerges not from authority but from observed regularity (Kahneman, 2011). This induces a cognitive shift from deterministic expectation to probabilistic reasoning. Participants cease expecting specific outcomes and instead learn to trust convergence. Fairness becomes a pattern, not a promise. 14.5 Political & Ethical Philosophy Political theory often frames fairness as a tension between libertarian chance and egalitarian correction. SCE synthesises these through structured randomness: impartiality generated by entropy, not empathy. It operationalises the Rawlsian veil of ignorance as a procedural mechanism rather than a thought-experiment (Rawls, 1971). Italsoexposesthefragilityofinstitutionalneutrality. Manyinstitutionsdeclarefairness while retaining discretionary authority. SCE challenges this structure: deviations are not moral lapses but statistical anomalies-signals that can be measured, audited, and falsified. 56 14.6 Artificial Intelligence & Machine Learning Where machine learning extracts and amplifies patterns, SCE suppresses predictable structuretoenforcefairness. Itis,ineffect,acounter-algorithm: astochasticsafeguard against historical overfitting. Integrated into AI systems, SCE creates a two-layer architecture: • an optimisation layer that predicts outcomes, • and a fairness layer that allocates opportunity impartially. Accuracy is tempered by ignorance; prediction is bounded by entropy. This yields a model of accountable automation that is transparent by design and epistemically selfauditing (Cath, 2018; Rahwan et al., 2019). 14.7 Sociotechnical & Epistemic Implications SCE challenges the assumption that fairness requires human judgement. It demonstratesthatconstrainedautomation,whenexposedtotransparencyandstatisticalpressure, can be more accountable than discretionary governance. It does not replace human values; it embeds them in reproducible procedure. Epistemically, SCE reframes trust as verification. In an era of black-box systems and invisible algorithmic asymmetry, this may be the only durable form of institutional trust available: not trust in intention, but trust in evidence. 14.8 Summary SCE’s contribution is synthetic. It unifies probability theory, bureaucratic design, information ethics, and machine learning under a single operational thesis: fairness is integrity under symmetric exposure. It extends the epistemic logic of The Contradiction Trap: systems demonstrate honesty not by avoiding scrutiny, but by surviving it. SCE makes that scrutiny statistical and continuous. 57 4. Irreversibility. SCE outputs are final decision tokens. 5. Audit hooks. Dashboards, logs, and metrics must be externally inspectable. Summary. IntegrationdisciplinepreventsSCEfrombecomingadecorativerandomiser. 16.6 Failure Protocols: Entropy Loss, Drift Detection, & Anomaly Flags SCE must fail loudly, not silently. 1. Entropy checks. Freshness, uniqueness, and variance must be continuously monitored. 2. Drift detection. Unexpected deviation patterns indicate upstream manipulation or misconfiguration. 3. Automatic suspension. Allocation halts when diagnostic thresholds are exceeded. 4. Forensic reconstruction packages. Auditors receive complete logs up to the failure point. 5. Clear recovery paths. Restoration must follow documented, precommitted procedures. Summary. Silent degradation produces fake fairness. Failure protocols prevent it. 64 16.7 Audit Protocol SCE Audit Protocol: Minimal Requirements A verifier needs only five public fields: A1. Entropy commitments. A2. Declared weights 𝑤(𝜏). A3. Credit state Δ(𝜏). A4. Ordered allocation sequence. A5. Hash-chain anchor 𝐻𝜏. These suffice for full reconstruction and tamper-evident verification. 16.8 Entropy Service-Level Agreements (SLAs) Entropy is SCE’s epistemic constant. Deployments require explicit SLAs for availability, freshness, failover, and recovery. Availability. Beacon delay must satisfy Pr(delay >𝜏𝑑)<𝜀𝑑. Freshness. Entropy older than 𝜏𝑓is invalid. Failover. Failover must occur within 𝜏failover ≤2s. Hard failures. SCE suspends, issues warnings, and switches beacons. Recovery time objective (RTO). Recovery must occur within RTO ≤30s. Degraded mode. Fallback commit–reveal mode is permitted for at most 𝑇max =2𝑇 events. 65 Summary. Entropy cannot be assumed; it must be governed. 16.9 Deployment Tiers: Private, Public, & Adversarial Different environments impose different integrity requirements. Private. Medium integrity, low adversarial risk. Public / regulated. High integrity, medium adversarial risk, mandatory transparency. Adversarial. Maximum integrity, cryptographic safeguards required. Summary. SCE scales from discipline to protocol depending on its threat model. 16.10 Minimum Volume & Threshold Heuristics SCE’s guarantees are asymptotic; small-𝐾fairness is diagnostic, not normative. 1. Minimum volume. Stable proportionality requires 𝐾≥1000𝑛. 2. Tranche size. Recommended: 50≤𝑇≤200. 3. Smoothing. Suggested: 0.4≤𝛼≤0.7. 4. Credit function. Stability requires |𝑇𝑔′(0)|<2. 5. Low-volume interpretation. Small-𝐾deviations reflect noise, not malfeasance. Summary. Convergence needs statistical room to breathe. Below threshold, SCE diagnoses; it does not guarantee. 66 16.11 Convergence Dashboards & Real-Time Monitoring Fairness is a trajectory, not a snapshot. 1. Live deviation tracking. Monitor 𝐷𝑡=‖𝑤(𝑡)−𝑤‖1with confidence envelopes. 2. Tranche summaries. Publish deviation, credit state, allocation frequencies, and commitments. 3. Entropy health. Expose freshness, uniqueness, and statistical quality metrics. 4. Baseline envelopes. Compare observed behaviour to Monte Carlo predictions. 5. Threshold alerts. Trigger suspension and alerting on sustained divergence. 6. Public dashboards. High-stakesdeploymentsrequirepublicvisibilityandauditability. Summary. Monitoring transforms SCE from a claim into continuous evidence. Together, these requirements define SCE as an institutional discipline. When its constraints are honoured, SCE produces fairness as a reproducible regularity. When they are not, it produces randomness without integrity. The next section addresses the ethical obligations that follow from a system capable of revealing injustice as reliably as it enforces proportionality. Reference frameworks. Theseoperationalprinciplesalignwithestablishedauditability standards (Kent, Souppaya, et al., 2012; Kokoris-Kogias et al., 2020), verifiable randomness guidance (Turán et al., 2023), and accountability-by-design frameworks in machine learning (Brundage et al., 2020; Carlini et al., 2020). They collectively situate SCE within the broader movement toward verifiable governance infrastructure. 67 17 Empirical Validation Under Adversarial and Non-Stationary Conditions SCE is designed for governance contexts where data-generating processes are rarely IID and often strategically pressured. The purpose of empirical validation is not to demonstrate idealised convergence, but to test operational resilience: whether the algorithm preserves proportional fairness under degraded entropy, evolving weights, correlated arrivals, or incomplete observability. These experimental regimes extend the evidential testing logic established in Atkinson (2025): deviation under symmetric pressure constitutes empirical evidence of procedural failure. 17.1 Adversarial Entropy Model (AEM) SCE’sguaranteesdependonverifiable, unpredictableentropy. Toevaluatefailuremodes, we simulate an adversarial entropy model representing the manipulation strategies available to a compromised operator. This structure mirrors the adversarial configurations of Atkinson (2025), substituting randomness for the logical symmetry constraint in The Contradiction Trap. Threat model. An adversary may bias allocations by manipulating entropy via: 1. Suppression: withholding beacon values that would select an unfavourable recipient. 2. Reseeding: replacing authenticated randomness with correlated local values. 3. Replay: reusing past entropy tokens 𝑒𝑡that favour preferred outcomes. 4. Bias injection: modifying 𝑈𝑡to deviate from uniformity. 5. Temporal drift: introducing persistent correlations in {𝑈𝑡}. Eachtacticproducessystematicdeviationsbetweenempiricalallocationsanddeclared weights 𝑤. Observable signatures include elevated or stagnant KL divergence, inflated residual variance, serial correlation, or breaks in the hash-chain (Doshi-Velez & Kim, 2017; Drand Consortium, 2021; Self, Moekander, & Floridi, 2023; Turán et al., 2023; Vinod, 2013a). 68 Response protocol. Upon detection of an entropy anomaly, SCE suspends allocation, switches to the next verified entropy source, logs the incident, and publishes a provenance notice. Because SCE’s fairness guarantees rest on externally verifiable evidence, adversarial manipulation manifests immediately in public metrics such as KL drift or variance distortion. 17.2 Non-Stationary Weights: Step, Ramp, and Seasonal Drift Many governance settings require evolving weight vectors 𝑤(𝜏). We evaluate three drift regimes: (D1) step, (D2) ramp, and (D3) seasonal modulation. Residuals converge at 𝑂(1/√𝜏), and cost spikes decay within 𝑂(1/𝛼)tranches, consistent with Lemma 9.6. Thesetrajectoriesalignwiththebounded-driftconvergencepropertiesofstochasticapproximation processes (Kushner & Yin, 2003; Ljung, 1977) and SCE’s analytical model in Section 9. 17.3 Correlated Arrivals To evaluate temporal dependence, we simulate AR(1) and MA(1) arrival structures: 𝑈𝑡=𝜌𝑈𝑡−1+√1−𝜌2𝜀𝑡, 𝜀𝑡∼Unif[0,1) (17.1) For |𝜌|≤0.5, results are indistinguishable from IID draws. For 𝜌≥0.8, variance bands widenandcorrectionslows, butconvergenceremainsunbiased. Thisbehaviourmatches the martingale central limit theorem for dependent increments (Hall & Heyde, 1980; Hoeffding, 1963) and the proportional-convergence analysis of Section 9.4. 17.4 Entropy Degradation: Sticky and Biased RNG Partial entropy loss is modelled as a “sticky” RNG: 𝑈𝑡={𝑈𝑡−1,with probability 𝜂, Unif[0,1), otherwise. (17.2) For small 𝜂the system behaves normally, but for 𝜂 ∈ [0.2,0.4], variance suppression emerges and 𝐶𝜏increases monotonically, reproducing the expected diagnostic pattern (Turán et al., 2023; Vinod, 2013a). For 𝜂>0.5, allocations fail the diagnostic threshold 𝐶𝜏>𝐸[𝐶𝜏]+𝑧𝑝𝜎stoch, consistent with SCE’s failure-protocol design in Section 8.1. 69 17.5 Partial Observability of Declared Weights and Real-Data Proxy Experiments In practical deployments, auditors may observe 𝑤(𝜏)only in batches (e.g. every 5–20 tranches). SCE reconstructs the missing intervals using the last published weights, producing bounded error that decays with batch size. Synthetic and historical routing data confirm that for 𝐾 ≥1000𝑛, allocations reproduce the underlying categorical distribution without measurable bias. This validates auditability-by-design principles (DoshiVelez & Kim, 2017; Self, Moekander, & Floridi, 2023) and procedural fairness under partial transparency (Binns, 2018; Rahwan et al., 2019). The reconstruction process thus realises the documentary principle of Atkinson (2025) within a stochastic evidential regime. 17.6 Required Sample Size for Operational Guarantees Using the martingale CLT (Theorem 9.3), the minimum sample size for auditability with accuracy 𝜀and power 1−𝑝is: 𝐾min =𝑧2 𝑝𝜎2 infl 𝜀2, 𝜎2 infl =(𝑛−1)(1+𝜀𝜅)(17.3) For 𝜀=0.02and 𝑝=0.95, this yields 𝐾min ≈1000𝑛, consistent with empirical observations and the Law of Large Numbers (Devroye, 1986; Hall & Heyde, 1980; Kolmogorov, 1933). Synthetic trials suffice because the objective is not domain performance, but resilience across adversarial, non-stationary, and partially observable regimes. All replication seeds, logs, and scripts are publicly released for independent verification. 18 Ethical Limits, Misuse Modes, & Governance Guardrails SCE guarantees fairness of procedure, not fairness of purpose. It reproduces whatever proportions it is given, with high statistical fidelity. Its ethical load therefore sits not in the algorithm but in the institutional conditions surrounding it. Without those constraints, SCE becomes a precision instrument for amplifying structural bias rather than mitigating it (Barocas et al., 2019a; Binns, 2018; Floridi, 2019; Mökander, 2023). 70 18.1 The Domain of Ethical Use SCE is appropriate only when: • the goods at stake are substitutable or reversible; • proportionality is a defensible normative standard; • participants can meaningfully consent to stochastic exposure; • declared weights represent legitimate claims to representation. Outsidethisdomain-whereallocationstouchrights,safety, dignity,orirreversibleoutcomesrandomisation is not fairness but abdication (Buhmann et al., 2020; Rahwan et al., 2019). Boundary principle. Procedural symmetry can support justice only when the domain is itself just. (18.1) This echoes the evidential caution expressed in Atkinson (2025): procedural integrity cannot redeem an unjust prior. 18.2 Legitimacy of Declared Proportions SCE assumes the weight vector 𝑤is normatively justified; it tests adherence, not moral legitimacy. Institutions must therefore provide an independent, inspectable, and challengeable justification for 𝑤(Brundage et al., 2020; Zhou, 2022). Legitimate declaration. 𝑤𝑖must encode a defensible, reviewable, and contestable proportional claim. (18.2) A fair process cannot redeem an unjust prior; SCE is not a moral cleanser. 18.3 Misuse Modes 1. Bias Laundering. Using SCE to reproduce exclusionary or prejudicial proportions under the veneer of “neutral” randomness-an instance of algorithmic laundering dis71 cussed in Barocas et al. (2019a) and Binns (2018). The result is injustice with an audit trail. 2. Transparency Theatre. Publishing partial logs, forged entropy receipts, or selectively releasing favourable tranches. This replaces verification with performance (Carlini et al., 2020; Kokoris-Kogias et al., 2020). 3. Coercive Randomisation. Applying SCE in contexts involving vulnerability or irreversible outcomes. Randomness does not absolve harm; consent to stochastic exposure is mandatory (Buhmann et al., 2020; Floridi, 2019). 4. Output Manipulation. Overriding or discarding SCE allocations without transparent justification breaks reconstructability and nullifies fairness guarantees (Drand Consortium, 2021; Kent, Souppaya, et al., 2012; Turán et al., 2023). 18.4 Governance Guardrails To prevent misuse, SCE requires governance constraints as stringent as its mathematical ones: •G1. Legitimate Declaration: Proportions must be set through independent, reviewable, and contestable procedures (Brundage et al., 2020; Zhou, 2022). •G2. Verifiable Entropy: Randomness must be externally auditable and tamperresistant (Drand Consortium, 2021; Turán et al., 2023). •G3. Immutable Audit Trails: Logs must be complete, append-only, and hashcommitted across tranches (Kent, Souppaya, et al., 2012; Kokoris-Kogias et al., 2020). •G4. Reversibility of Harm: SCE must not be used where outcomes cannot be undone, appealed, or compensated (Buhmann et al., 2020; Floridi, 2019). •G5. Public Oversight in Public Deployments: Reconstruction must be possible without insider privileges (Doshi-Velez & Kim, 2017; Self, Moekander, & Floridi, 2023). These guardrails are structural, not decorative. Without them, SCE ceases to be a fairness mechanism at all. 72 18.5 Epistemic Implications SCE reframes fairness as an evidential claim-a statement about observable behaviour under symmetric pressure. Legitimacy thus becomes a matter of institutional conduct, not moral intent (Floridi, 2019; Rahwan et al., 2019). A system capable of revealing fairness can equally expose injustice. The ethical burden lies not on the algorithm but on the architecture that constrains it (Atkinson, 2025; Mökander, 2023). 18.6 Absolute Exclusions This framework is designed for evidential and diagnostic use in institutional auditing, algorithmic governance, and procedural validation. It must not be applied to humansubjectallocationoradjudicationcontexts(e.g., criminaljustice, socialwelfare,orhealthcaretriage),wherefairnessisalegalorethicalrightratherthanaproceduralhypothesis. The Symmetric Convergence Engine provides a falsifiable method for evidential testing, not an automated decision policy. 18.7 Summary SCE’s ethical limits are load-bearing. Within them, the algorithm enforces procedural symmetry and reveals deviation. Outside them, it becomes a high-resolution amplifier of whatever prior it is given. The distinction between fairness and harm does not lie in entropy but in the intentions encoded in 𝑤– the one input SCE cannot interrogate. 19 Future Research Programme 19.1 Purpose and Orientation SCE marks a baseline: fairness expressed as measurable equilibrium under statistical constraint. But equilibria matter only insofar as they survive stress. The next phase of work concerns those stresses-quantifying SCE’s tolerance, extending its analytical scope, and embedding it within broader architectures of evidential governance. Where The Contradiction Trap subjected beliefs to symmetric pressure, SCE subjects procedures to stochasticpressure. Both treatintegrityas a propertyrevealedbyfailure, and both demand falsification (Atkinson, 2025; Floridi, 2019). 73 workwhereentropyactsasarbiterandmathematicsaswitness. Byformalisingfairness as convergence within measurable bounds, SCE translates justice from moral aspiration into statistical experiment (Floridi, 2019; Rahwan et al., 2019). It neither moralises nor excuses. It operates precisely as instructed and invites the world to test whether institutions do the same. This is not performative impartiality but evidential governance: fairness as traceable, replayable behaviour (Brundage et al., 2020; Zhou, 2022). Across this paper, SCE has been extended along multiple dimensions: • As a simulation framework, it demonstratesempiricalconvergence-showingthat fairnessstabiliseswithvolumeanddeviationdecayswithrepetition(Devroye,1986; Hall & Heyde, 1980). • As a game-theoretic construct, it makes honesty rational through transparency rather than sanction (R. Aumann, 1999; R. J. Aumann & Brandenburger, 1995; Rahwan et al., 2019). • As an ethical model, itenforcesguardrails-boundingrandomnesswithlegitimacy, auditability, and reversibility (Binns, 2018; Buhmann et al., 2020; Floridi, 2019; Mökander, 2023). • As an interdisciplinary critique,itchallengespredictiveoptimisationandfairness theatre-replacing intention with evidence and rhetoric with reproducibility (Barocas et al., 2019a; Brundage et al., 2020; Carlini et al., 2020). • As an audit mechanism,itrendersinstitutionalbehaviourreconstructible-turning opaque processes into empirical experiments (Kent, Souppaya, et al., 2012; Self, Moekander, & Floridi, 2023). SCE’s deeper implication is cautionary: a system that measures fairness also measures its failure. Once fairness becomes empirical, injustice becomes empirical too. The cost of transparency is exposure. This aligns with the epistemic theme linking the trilogy. Where The Contradiction Trap showed that truth survives only pressures capable of disproving it, SCE shows that fairness survives only pressures capable of randomising it. Both articulate a single law: integrity-logical, procedural, or institutional-is whatever remains coherent after symmetry hasremovedevery opportunityfor self-deception(Atkinson, 2025; Floridi, 2019; Priest, 2006). SCE also reveals a structural possibility beyond allocation: procedural fairness itself can be treated as an evidential property, independent of domain. Verifiable entropy, 80 symmetric pressure, and empirical falsifiability form a minimal scaffold for decision systems that are accountable not in principle but in behaviour. 21.1 Toward Evidential Governance SCEcompletesthe procedurallayerofthe evidential programme introducedin The Contradiction Trap. Paper 1 established contradiction as a diagnostic of belief integrity; Paper 2 operationalises that logic through stochastic allocation. The next step is institutional: to treat governance itself as a hypothesis subjected to continuous symmetric scrutiny (Brundage et al., 2020; Mökander, 2023; Zhou, 2022). SCE’s design principles-auditable entropy, documentary trace, convergent behaviourextend directly to institutional processes that claim neutrality, compliance, or due process. If fairness can be made falsifiable through allocation behaviour, then legitimacy can be evaluated by the same evidential standard (Doshi-Velez & Kim, 2017; Floridi, 2019; Rahwan et al., 2019; Self, Moekander, & Floridi, 2023). The third paper in this sequence develops this generalisation as evidential governance: a paradigm in which procedures are judged not by intent or narrative but by reproducible behaviour under symmetric challenge. SCE therefore stands not merely as a distribution mechanism but as a canonical example of how evidential principles can be embedded into real organisational systems. Fairness becomes meaningful when it can be wrong. Institutions become trustworthy when they can be tested. SCE represents one pathway toward that standard; evidential governance aims to extend it. Integrity must not be assumed-it must be proven, publicly, repeatedly, and under symmetry. A Parallel Concepts Table B Public Verifiability Protocol (PVP) SCE derives its normative strength from a simple demand: fairness must be demonstrable to any observer. This appendix sets out a protocol by which any independent auditor – institutional, academic, or public – can verify the entire allocation process from the published audit log alone. No trust in the operator is required; correctness is established purely through reconstruction. 81 TCT Section SCE Instantiation §3 Coherence KL-based coherence cost 𝐶𝜏 §4 Adversarial Configurations Tranche structure and event-level stochastic tests §5 Symmetry Verifiable entropy and unbiased sampling §6 Documentary Value Append-only audit log with provenance §7 Necessity Small-𝑘diagnostic regime, incident flags §8 Evidential Resolution Convergence in probability under bounded feedback Table 9: Section-level correspondence between Paper 1 (TCT) and Paper 2 (SCE). B.1 Inputs to the Auditor The auditor receives: 1. the append-only audit log AL; 2. public randomness streams from all entropy sources; 3. declared weights 𝑊𝜏for each tranche; 4. tranche size 𝑇and credit function 𝑔; 5. the hash-chain seed ℎ0. B.2 Verification of Entropy Provenance For each event 𝑡, the auditor: 1. retrieves 𝑒𝑡from the published entropy stream; 2. verifies its authenticity using the beacon signature or commitment chain; 3. recomputes the hash: ℎ(audit) 𝑡=𝐻(𝑒𝑡‖ℎ(audit) 𝑡−1 )(B.1) and checks equality with the logged ℎ𝑡. Any mismatch indicates tampering, replay, omission, or entropy substitution. 82 B.3 Reconstruction of Allocations For each event 𝑡: 1. compute the effective probabilities 𝜋(𝑡)from 𝑊𝜏(𝑡)and 𝜅𝜏(𝑡)−1; 2. map the raw variate 𝑈𝑡to the simplex partition induced by 𝜋(𝑡); 3. confirm that the reconstructed recipient equals the logged 𝑟𝑡. Any discrepancy is evidence of manipulation or an incorrect correction path. B.4 Tranche-Level Consistency For each tranche 𝜏, the auditor reconstructs: 𝑋𝜏,𝑃𝜏,𝐶𝜏,Δ𝜏,𝜅𝜏(B.2) and checks that: 1. reconstructed values match the published ledger; 2. 𝐶𝜏lies inside the expected noise band unless flagged; 3. incident flags 𝐼𝜏correctly reflect the reconstructed anomalies. B.5 Cumulative Behaviour Verification The auditor evaluates: 𝐶cum(𝐾),𝑅𝑖(𝐾),Var(𝑅𝑖(𝐾)) (B.3) and verifies that: • cumulative KL decays approximately as 𝒪(1/𝐾); • residual variances match multinomial baselines; • no unexplained serial correlations appear in {𝑈𝑡}or {𝑟𝑡}. 83 B.6 Completeness Criterion An SCE deployment is publicly verified if: All event-level checks ∧All tranche-level checks ∧All entropy checks (B.4) are satisfied. Any violation provides evidence of: 1. operator error, 2. adversarial interference, or 3. deviation from symmetric stochastic governance. B.7 Interpretation The PVP removes trust as an epistemic requirement. An auditor with no privileged access can reconstruct the entire trace and diagnose every irregularity. This closes the evidential loop: SCE’s fairness claims are not accepted but tested, and thus meaningful in the sense of Section 12.1. B.8 Entropy Acquisition Layer Entropyissourcedfromcertifiedquantumoratmosphericgenerators(ANU QRNG,RANDOM.ORG) using authenticated APIs. Each entropy packet is hashed (SHA-3-512) and signed with the institution’s public key to enable provenance checks. If external entropy is unavailable, SCE enters a pseudorandom fallback mode with degraded-trust status; such tranches are highlighted in the audit log. B.9 Dynamic Weighting Layer Declared proportional weights 𝑤𝑖are stored immutably. For event 𝑡, SCE computes instantaneous probabilities: 𝜋𝑖(𝑡)=(1−𝜆)𝑤𝑖+𝜆𝜅𝑖(𝑡−1) (B.5) where 𝜅𝑖is the correction credit and 𝜆∈[0,1]controls responsiveness. 84 B.10 Tranche–Credit Correction Layer After each tranche of 𝑇events, residuals Δ𝑖(𝑇)= 𝑤𝑖(𝑇)−𝑤𝑖update credits via: 𝜅𝑖(𝑇+1)=clip(𝜅𝑖(𝑇)−−𝑔(Δ𝑖(𝑇)),−𝑔max,𝑔max), 𝑔(𝑥)=𝛾𝑥 (B.6) Typical stable ranges are 𝛾∈[0.1,0.3]and 𝑔max ∈[0.05,0.1]. Conservative tuning slows correction; aggressive tuning risks oscillation. Practitioners should calibrate (𝛾,𝑔max) with Monte-Carlo trials on historical workload data. Δ𝑖(𝑇) Residual 𝑔(Δ𝑖) Credit Update 𝜅𝑖(𝑇+1) Credit Balance 𝜋𝑖(𝑡+1) Adjusted Probability Allocation Outcome Observed result 𝑔(𝑥)=𝛾𝑥 credit register next-event weighting Figure 16: Tranche–credit feedback loop. Residuals update credit balances, which modulate future probabilities until convergence. B.11 Audit and Logging Layer Eachallocationrecordcaptures: entropysourceID,trancheindex, pre-andpost-correction probabilities, residual Δ𝑖, and timestamp. Tranche-level coherence cost 𝐶𝜏and cumulative cost 𝐶(𝐾)are appended to a public ledger. Dashboards visualise convergence and highlight any tranche where 𝐶𝜏exceeds the expected stochastic range. 85 B.12 Operational Notes •Scale: Recommended minimum 𝐾≥10,000allocations for 95% accuracy within ±2%. •Latency: Credit updates require 𝑂(𝑛)per tranche. •Resilience: Entropy and audit subsystems are modular; pseudorandom fallback preserves continuity with integrity flags. •Ethics: Legitimacy of the declared proportions must be externally certified; SCE guarantees coherence to declared weights, not their moral validity. C Python scripts for reproducability C.1 Convergence: Deviation vs K import numpy as np import matplotlib.pyplot as plt np.random.seed(42) # Parameters n=5 w = np.array([0.20]*n) alpha = 0.5 T = 100 def credit(delta): return 0.5 * delta # simple stable linear credit: |g’(0)| = 0.5 < 1 def run_SCE(K): delta = np.zeros(n) kappa = np.zeros(n) counts = np.zeros(n) deviations = [] for t in range(1, K+1): # tranche index tau = (t-1) // T # effective participation pi = alpha * w + (1 -- alpha) * kappa pi = pi / pi.sum() 86 # allocation i = np.random.choice(np.arange(n), p=pi) counts[i] += 1 # update tranche deviation + credit ift%T==0: observed = counts / t delta = w -- observed kappa = credit(delta) # record deviation norm deviations.append(np.linalg.norm((counts / t) -- w, ord=1)) return np.array(deviations) # Run simulation K = 100000 dev = run_SCE(K) # Burn-in to remove early-stage instability burn_in = 200 dev_burned = dev[burn_in:] K_range = np.arange(burn_in+1, len(dev)+1) # Plot plt.figure(figsize=(8,5)) plt.plot(K_range, dev_burned, linewidth=1.5) plt.xscale(”log”) plt.yscale(”log”) plt.xlabel(”K␣(log␣scale)”) plt.ylabel(r”$\|\widehat{w}(K)␣--␣w\|_1$␣(log␣scale)”) plt.title(”SCE␣Convergence:␣Deviation␣vs␣K␣(–postburn-in)”) plt.grid(True, which=’both’, linestyle=’--’, linewidth=0.5) plt.tight_layout() plt.savefig(”SCE_convergence_burnin.png”, dpi=300) plt.show() C.2 Drift tracking import numpy as np import matplotlib.pyplot as plt np.random.seed(42) # Parameters 87 n=5 alpha = 0.5 T = 100 K = 10000 # Base weights w0 = np.array([0.20]*n) def credit(delta): return 0.5 * delta def drifting_weights(tau): # gentle sinusoidal drift drift = 0.02 * np.sin(0.0005 * tau) * np.array([+1, -1, +1, -1, 0]) w = w0 + drift w = np.maximum(w, 0.001) return w / w.sum() def run_SCE_drift(K): delta = np.zeros(n) kappa = np.zeros(n) counts = np.zeros(n) tracked = [] true_w = [] for t in range(1, K+1): tau = (t-1) // T w = drifting_weights(tau) true_w.append(w.copy()) pi = alpha * w + (1 -- alpha) * kappa pi = pi / pi.sum() i = np.random.choice(np.arange(n), p=pi) counts[i] += 1 ift%T==0: observed = counts / t delta = w -- observed kappa = credit(delta) tracked.append((counts / t).copy()) return np.array(tracked), np.array(true_w) tracked, true_w = run_SCE_drift(K) # Track one component for clarity (component 0) emp = tracked[:,0] 88 true = true_w[:,0] # Plot tracking behaviour plt.figure(figsize=(10,6)) plt.plot(emp, label=”Empirical␣frequency␣$w_0(t)$”, linewidth=2, alpha=0.8) plt.xlabel(”Allocation␣Events␣(t)”) plt.ylabel(”Proportion”) plt.title(”SCE␣Drift␣Tracking␣(Component␣0)”) plt.legend() plt.grid(True, linestyle=’--’, alpha=0.5) plt.tight_layout() plt.savefig(”SCE_drift_tracking.png”, dpi=300) plt.show() C.3 Small-K noise import numpy as np import matplotlib.pyplot as plt np.random.seed(42) # Parameters n=5 w = np.array([0.20]*n) alpha = 0.5 T = 1000 def credit(delta): return 0.5 * delta def run_SCE(K): delta = np.zeros(n) kappa = np.zeros(n) counts = np.zeros(n) deviations = [] for t in range(1, K+1): pi = alpha * w + (1 -- alpha) * kappa pi = pi / pi.sum() i = np.random.choice(np.arange(n), p=pi) counts[i] += 1 ift%T==0: observed = counts / t delta = w -- observed 89 ␣ ␣ ␣ ␣Execute␣one␣SCE␣trajectory␣and␣return␣deviation␣array ␣ ␣ ␣ ␣||␣w_hat(t)␣--␣w␣||_1␣for␣t␣=␣1␣...␣K. ␣ ␣ ␣ ␣””” delta = np.zeros(n) kappa = np.zeros(n) counts = np.zeros(n) deviations = [] for t in range(1, K + 1): # Effective participation probabilities pi = alpha * w + (1 -- alpha) * kappa pi /= pi.sum() # Stochastic allocation i = np.random.choice(np.arange(n), p=pi) counts[i] += 1 # Tranche correction ift%T==0: observed = counts / t delta = w -- observed kappa = credit(delta) # L1 deviation deviations.append(np.linalg.norm((counts / t) -- w, ord=1)) return np.array(deviations) # -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- # RUN ENSEMBLE OF 20 TRAJECTORIES # -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- all_runs = np.zeros((RUNS, K)) for r in range(RUNS): print(f”Run␣{r␣+␣1}/{RUNS}”) all_runs[r] = run_SCE(K) # -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- # GEOMETRIC MEAN AND GEOMETRIC SD # -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- eps = 1e-12 # safety for log log_runs = np.log(all_runs + eps) 96 # geometric mean geo_mean = np.exp(np.mean(log_runs, axis=0)) # geometric standard deviation factor geo_sd_factor = np.exp(np.std(log_runs, axis=0)) # lower and upper bands lower = geo_mean / geo_sd_factor upper = geo_mean * geo_sd_factor # -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- # DOWNSAMPLE FOR CLARITY # -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- step = 20 xs = np.arange(K)[::step] gm = geo_mean[::step] lower_ds = lower[::step] upper_ds = upper[::step] # -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- # PLOT # -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- plt.figure(figsize=(10, 6)) plt.plot(xs, gm, linewidth=2, label=”Geometric␣Mean”) plt.fill_between(xs, lower_ds, upper_ds, alpha=0.3, label=”Geometric␣±␣SD”) plt.xscale(”log”) plt.yscale(”log”) plt.xlabel(”K␣(log␣scale)”) plt.ylabel(r”$\|\widehat{w}(K)␣--␣w\|_1$␣(log␣scale)”) plt.title(”SCE␣Ensemble␣Convergence:␣Geometric␣Mean␣±␣Geometric␣SD␣(20␣Runs)”) plt.grid(True, linestyle=’--’, linewidth=0.5) plt.legend() plt.tight_layout() plt.savefig(”SCE_geo_mean_sd.png”, dpi=300) plt.show() 97 C.8 Ensemble Convergence: 95% Geometric Confidence Interval import numpy as np import matplotlib.pyplot as plt # -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- # SCE PARAMETERS # -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- np.random.seed(42) n=5 w = np.array([0.20] * n) alpha = 0.5 T = 1000 K = 100000 RUNS = 20 # -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- # CREDIT FUNCTION # -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- def credit(delta): ””” ␣ ␣ ␣ ␣Stable␣linear␣credit␣function. ␣ ␣ ␣ ␣Satisfies␣|g’(0)|␣=␣0.5␣<␣1␣for␣contraction. ␣ ␣ ␣ ␣””” return 0.5 * delta # -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- # SCE CORE LOOP # -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- def run_SCE(K): ””” ␣ ␣ ␣ ␣Execute␣one␣SCE␣trajectory␣and␣return␣deviation␣array ␣ ␣ ␣ ␣||␣w_hat(t)␣--␣w␣||_1␣for␣t␣=␣1␣...␣K. ␣ ␣ ␣ ␣””” delta = np.zeros(n) kappa = np.zeros(n) counts = np.zeros(n) deviations = [] 98 for t in range(1, K + 1): pi = alpha * w + (1 -- alpha) * kappa pi /= pi.sum() i = np.random.choice(np.arange(n), p=pi) counts[i] += 1 ift%T==0: observed = counts / t delta = w -- observed kappa = credit(delta) deviations.append(np.linalg.norm((counts / t) -- w, ord=1)) return np.array(deviations) # -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- # ENSEMBLE OF TRAJECTORIES # -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- all_runs = np.zeros((RUNS, K)) for r in range(RUNS): all_runs[r] = run_SCE(K) # -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- # GEOMETRIC MEAN AND 95% GEOMETRIC CONFIDENCE INTERVAL # -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- eps = 1e-12 log_runs = np.log(all_runs + eps) log_mean = np.mean(log_runs, axis=0) log_sd = np.std(log_runs, axis=0) geo_mean = np.exp(log_mean) lower_CI = np.exp(log_mean -- 1.96 * log_sd) upper_CI = np.exp(log_mean + 1.96 * log_sd) # -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- # DOWNSAMPLE FOR VISUAL CLARITY # -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- step = 20 99 xs = np.arange(K)[::step] gm = geo_mean[::step] lower = lower_CI[::step] upper = upper_CI[::step] # -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- # PLOT # -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- plt.figure(figsize=(10,6)) plt.plot(xs, gm, linewidth=2, color=”black”, label=”Geometric␣Mean”) plt.fill_between(xs, lower, upper, color=”gray”, alpha=0.25, label=”95%␣Geometric␣CI”) plt.xscale(”log”) plt.yscale(”log”) plt.xlabel(”K␣(log␣scale)”) plt.ylabel(r”$\|\widehat{w}(K)␣--␣w\|_1$␣(log␣scale)”) plt.title(”SCE␣Ensemble␣Convergence:␣95%␣Geometric␣Confidence␣Interval”) plt.grid(True, linestyle=’--’, linewidth=0.5) plt.legend() plt.tight_layout() plt.savefig(”geo_95ci.png”, dpi=300) plt.show() C.9 Convergence: Log–Binned Median Curve import numpy as np import matplotlib.pyplot as plt np.random.seed(42) # Parameters n=5 w = np.array([0.20]*n) alpha = 0.5 T = 1000 K = 100000 def credit(delta): return 0.5 * delta def run_SCE(K): delta = np.zeros(n) 100 kappa = np.zeros(n) counts = np.zeros(n) deviations = [] for t in range(1, K+1): pi = alpha*w + (1-alpha)*kappa pi /= pi.sum() i = np.random.choice(n, p=pi) counts[i] += 1 ift%T==0: observed = counts/t delta = w -- observed kappa = credit(delta) deviations.append(np.linalg.norm((counts/t)-w, ord=1)) return np.array(deviations) # Generate a long-run trajectory dev = run_SCE(K) burn_in = 10 dev_burn = dev[burn_in:] K_burn = len(dev_burn) # Logarithmic bins bins = np.unique( np.logspace( np.log10(1), np.log10(K_burn), 150 ).astype(int) ) binned_x = [] binned_y = [] for i in range(len(bins)-1): start, end = bins[i], bins[i+1] window = dev_burn[start:end] if len(window) > 0: binned_x.append((start + end)/2 + burn_in) binned_y.append(np.median(window)) binned_x = np.array(binned_x) binned_y = np.array(binned_y) # Plot plt.figure(figsize=(10,6)) # Raw noise (faint) plt.plot( 101 np.arange(burn_in+1, burn_in + K_burn + 1), dev_burn, alpha=0.15, linewidth=1, label=”Raw␣deviation” ) # Binned curve (bold) plt.plot(binned_x, binned_y, linewidth=2.5, color=”black”, label=”Log-binned␣median”) plt.xscale(”log”) plt.yscale(”log”) plt.xlabel(”K␣(log␣scale)”) plt.ylabel(r”$\|\widehat{w}(K)␣--␣w\|_1$␣(log␣scale)”) plt.title(”SCE␣Convergence:␣–LogBinned␣Median␣Curve”) plt.grid(True, linestyle=’--’, linewidth=0.5) plt.legend() plt.tight_layout() plt.savefig(”logbinned_curve.png”, dpi=300) plt.show() C.10 SCE simulations import numpy as np import math # ============================================================ # VECTORISED GAUSSIAN-COPULA AR(1) →UNIFORM ENTROPY # ============================================================ def ar1_uniform(T, rho=0.8, seed=123): ”””Generate␣fresh␣AR(1)␣Gaussian␣noise␣and␣map␣to␣Uniform(0,1)␣via␣�. ␣␣␣␣␣␣␣Uses␣math.erf␣under␣np.vectorize␣for␣safe␣cross-platform␣behaviour.””” rng = np.random.RandomState(seed) z = np.empty(T) z[0] = rng.randn() for t in range(1, T): z[t] = rho * z[t-1] + np.sqrt(1 -- rho**2) * rng.randn() phi = np.vectorize(lambda x: 0.5 * (1.0 + math.erf(x / math.sqrt(2)))) U = phi(z) return np.clip(U, 0.0, 1.0) def sticky_entropy(T, eta=0.3, seed=123): rng = np.random.RandomState(seed) 102 U = np.empty(T) U[0] = rng.rand() for t in range(1, T): if rng.rand() < eta: U[t] = U[t-1] else: U[t] = rng.rand() return U # ============================================================ # CORE TRANSFORMS # ============================================================ def soft_h(delta): return delta / (1.0 + np.abs(delta)) def compute_pi(w, delta, alpha): pi = (1 -- alpha) * w + alpha * soft_h(delta) pi = np.clip(pi, 0.0, None) s = pi.sum() if s <= 0: return w.copy() return pi / s def sample_allocation(pi, U): cum = np.cumsum(pi) idx = np.searchsorted(cum, U, side=”right”) return min(idx, len(pi)-1) def update_delta_avg(delta, allocation, w, alpha_resid): target = -w.copy() target[allocation] += 1.0 return (1 -- alpha_resid) * delta + alpha_resid * target def apply_correction(delta, g_slope=0.03, g_max=0.20): g = np.clip(g_slope * delta, -g_max, g_max) return delta + g # ============================================================ # DRIFT REGIMES # ============================================================ def drift_weights_step(tau, n, tau_star=50): 103 if tau < tau_star: return np.ones(n) / n w = np.linspace(1, n, n) return w / w.sum() def drift_weights_ramp(tau, n, tau_start=20, tau_end=120): if tau < tau_start: return np.ones(n) / n if tau > tau_end: w = np.arange(1, n+1) return w / w.sum() frac = (tau -- tau_start) / (tau_end -- tau_start) w0 = np.ones(n) / n w1 = np.arange(1, n+1) w1 = w1 / w1.sum() return (1-frac)*w0 + frac*w1 def drift_weights_seasonal(tau, n, period=40): base = np.ones(n) / n mod = 0.1 * np.sin(2*np.pi*tau/period) w = base + mod * np.linspace(-1, 1, n) w = np.maximum(w, 1e-6) return w / w.sum() # ============================================================ # SIMULATIONS (IID, CORRELATED, STICKY, DRIFT, BATCHED, REAL PROXY) # ============================================================ def simulate_SCE_iid(n=3, K=20000, T=200, alpha=0.25, g_slope=0.03, g_max=0.20, seed=123): rng = np.random.RandomState(seed) w = np.ones(n)/n delta = np.zeros(n) deltas = [] alpha_resid = 1.0/T num_tranches = K//T for _ in range(num_tranches): deltas.append(delta.copy()) for _ in range(T): pi = compute_pi(w, delta, alpha) a = sample_allocation(pi, rng.rand()) delta = update_delta_avg(delta, a, w, alpha_resid) delta = apply_correction(delta, g_slope, g_max) return np.array(deltas) 104 def simulate_SCE_correlated(n=3, K=20000, T=200, rho=0.8, alpha=0.25, g_slope=0.03, g_max=0.20, seed=123): delta = np.zeros(n) deltas = [] w = np.ones(n)/n alpha_resid = 1.0/T num_tranches = K//T for tau in range(num_tranches): deltas.append(delta.copy()) U = ar1_uniform(T, rho=rho, seed=seed+tau) for u in U: pi = compute_pi(w, delta, alpha) a = sample_allocation(pi, u) delta = update_delta_avg(delta, a, w, alpha_resid) delta = apply_correction(delta, g_slope, g_max) return np.array(deltas) def simulate_SCE_sticky(n=3, K=20000, T=200, eta=0.3, alpha=0.25, g_slope=0.03, g_max=0.20, seed=123): delta = np.zeros(n) deltas = [] w = np.ones(n)/n alpha_resid = 1.0/T num_tranches = K//T for tau in range(num_tranches): deltas.append(delta.copy()) U = sticky_entropy(T, eta=eta, seed=seed+tau) for u in U: pi = compute_pi(w, delta, alpha) a = sample_allocation(pi, u) delta = update_delta_avg(delta, a, w, alpha_resid) delta = apply_correction(delta, g_slope, g_max) return np.array(deltas) def simulate_SCE_drift(n=3, num_tranches=200, T=200, alpha=0.25, g_slope=0.03, g_max=0.20, seed=123, drift_type=”step”): rng = np.random.RandomState(seed) delta = np.zeros(n) 105 Cath, C. (2018). Governing artificial intelligence: Upholding human rights & dignity. Nature Machine Intelligence,1, 8–9. Clopper,C. J.,& Pearson, E. S. (1934). The use ofconfidenceorfiducial limitsillustrated in the case of the binomial. Biometrika,26(4), 404–413. Corbett-Davies, S., & Goel, S. (2023). What to measure, how to measure, and how to act: Practical guidance for fairness in machine learning. Communications of the ACM,66(5), 80–88. Devroye, L. (1986). Non-uniform random variate generation. Springer-Verlag. Doob, J. L. (1953). Stochastic processes. Wiley. Doshi-Velez, F., & Kim, B. (2017). Towards a rigorous science of interpretable machine learning. arXiv preprint arXiv:1702.08608. Drand Consortium. (2021). Drand public randomness beacon [Public randomness network]. Dwork, C., Hardt, M., Pitassi, T., Reingold, O., & Zemel, R. (2012). Fairness through awareness. Proceedings of the 3rd Innovations in Theoretical Computer Science Conference, 214–226. Floridi, L. (2019). The logic of information: A theory of philosophy as conceptual design. Oxford University Press. Fudenberg, D., & Levine, D. K. (1998). The theory of learning in games. MIT Press. Fudenberg, D., & Tirole, J. (1991). Game theory. MIT Press. Goodsell, C. T. (2015). The mission of public administration: Public service, governance, and democracy. Jossey-Bass. Hall, P., & Heyde, C. C. (1980). Martingale limit theory and its application. Academic Press. Hardt, M., Price, E., & Srebro, N. (2016). Equality of opportunity in supervised learning. Advances in Neural Information Processing Systems,29. Headlam, J. W. (1906). Election by lot at athens. Cambridge University Press. Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables. Journal of the American Statistical Association,58(301), 13–30. Jaynes,E.T.(2003). Probability theory: The logic of science.CambridgeUniversityPress. Kahneman, D. (2011). Thinking, fast and slow. Farrar, Straus; Giroux. Kent, K., Souppaya, M., et al. (2012). Guide to computer security log management (nist sp 800-92). National Institute of Standards and Technology. Kokoris-Kogias, E., Jovanovic, P., Gailly, N., Syta, E., & Ford, B. (2020). Trust in decentralized systems: A systematic review. Proceedings of the IEEE,108(7), 1266– 1301. https://doi.org/10.1109/JPROC.2020.2992412 Kolmogorov, A. N. (1933). Foundations of the theory of probability. Chelsea Publishing Company. Kullback, S., & Leibler, R. A. (1951). On information and sufficiency. Annals of Mathematical Statistics,22(1), 79–86. 112 Kushner, H. J., & Yin, G. G. (2003). Stochastic approximation and recursive algorithms and applications. Springer. Kusner, M. J., Loftus, J., Russell, C., & Silva, R. (2017). Counterfactual fairness. Advances in Neural Information Processing Systems,30. Lehmann, E. L., & Romano, J. P. (2006). Testing statistical hypotheses (3rd). Springer. Ljung, L. (1977). Analysis of recursive stochastic algorithms. IEEE Transactions on Automatic Control,22(4), 551–575. Maskin, E. (1985). The implementation of social choice rules: Some general results about nash equilibrium. Review of Economic Studies,52(2), 301–309. Metropolis, N., & Ulam, S. (1949). The monte carlo method. Journal of the American Statistical Association,44(247), 335–341. Mitchell, M., Wu, S., Zaldivar, A., Barnes, P., Vasserman, L., Hutchinson, B., Spitzer, E., Raji, I., & Gebru, T. (2021). Model cards for model reporting. Communications of the ACM,64(12), 56–65. Mittelstadt, B. D., Allo, P., Taddeo, M., Wachter, S., & Floridi, L. (2016). The ethics of algorithms: Mapping the debate. Big Data & Society,3(2). Mökander, J. (2023). Ethics-based auditing of automated decision-making systems. https://ora.ox.ac.uk/objects/uuid:93b3e15c-5c12-4a73-884d-7a6b3f1c4d7c Moulin, H. (2019). Fair division and collective welfare. MIT Press. Myerson, R. B. (1979). Incentive compatibility and the bargaining problem. Econometrica,47(1), 61–73. National Institute of Standards and Technology. (2022). Nist randomness beacon v2.0 [Accessed 2025]. Popper, K. R. (1959). The logic of scientific discovery. Routledge. Priest, G. (2006). In contradiction: A study of the transconsistent (2nd). Oxford University Press. Procaccia, A. D., & Tennenholtz, M. (2013). Approximate mechanism design without money. Proceedings of the 10th International Conference on Autonomous Agents and Multiagent Systems (AAMAS), 927–934. Rahwan, I., Cebrian, M., Obradovich, N., Bongard, J., Bonnefon, J.-F., Breazeal, C., Crandall, J. W., Christakis, N. A., Couzin, I. D., Jackson, M. O., et al. (2019). Machine behaviour. Nature,568(7753), 477–486. https://doi.org/10.1038/s41586-0191138-y Rawls, J. (1971). A theory of justice. Harvard University Press. Ribeiro, M. T., Singh, S., & Guestrin, C. (2016). ”why should i trust you?”: Explaining the predictions of any classifier. Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 1135–1144. Robbins, H., & Monro, S. (1951). A stochastic approximation method. Annals of Mathematical Statistics,22(3), 400–407. 113 Roth,A.E.(2002). Theeconomistasengineer:Gametheory,experimentation,andcomputation as tools for design economics. Econometrica,70(4), 1341–1378. https: //doi.org/10.1111/1468-0262.00335 Selbst, A. D., Boyd, D., Friedler, S., Venkatasubramanian, S., & Vertesi, J. (2019). Fairness and abstraction in sociotechnical systems. Proceedings of the Conference on Fairness, Accountability, and Transparency (FAT*), 59–68. Self, J., Wang, L., & Kehl, D. (2023). Auditability as a foundation for algorithmic accountability. AI & Society. Self, J., Moekander, J., & Floridi, L. (2023). Auditable ai: Designing for forensic transparency and procedural accountability. AI and Ethics.https://doi.org/10.1007/ s43681-023-00309-9 Shannon, C. E. (1948). A mathematical theory of communication. Bell System Technical Journal,27(3), 379–423. Shapley, L. S. (1953). A value for n-person games (Vol. 2). Skyrms,B.(2010).Signals: Evolution, learning, and information.OxfordUniversityPress. Stone, P. (2011). Lottery voting: A new proposal. The Good Society,20(1), 2–12. Turán, M., Barker, E., Kelsey, J., McKay, K., Bassham, L., Boyle, M., & Chen, J. (2023). Recommendation for the entropy sources used for random bit generation (nist sp 800-90b). https://doi.org/10.6028/NIST.SP.800-90B Vinod, H. D. (2013a). Hands-on intermediate econometrics using r: Randomness, bias, and simulation integrity. World Scientific. Vinod, H. D. (2013b). Entropy-based statistical fairness measures. Communications in Statistics—Simulation and Computation,42(5), 1093–1113. Wachter, S., Mittelstadt, B., & Russell, C. (2017). Counterfactual explanations without opening the black box: Automated decisions and the gdpr. Harvard Journal of Law & Technology,31(2), 841–887. Walsh, T. (2011). Uncertainty in mechanism design. Proceedings of the Twenty-Fifth AAAI Conference on Artificial Intelligence, 1323–1328. Weber, M. (1947). The theory of social and economic organization. Free Press. Young, H. P. (1995). Equity: In theory and practice. Princeton University Press. Young, H. P. (2004). Strategic learning and its limits. Oxford University Press. Zemel, R., Wu, Y., Swersky, K., Pitassi, T., & Dwork, C. (2013). Learning fair representations. International Conference on Machine Learning, 325–333. Zhou, L. (2022). Algorithmic accountability: Bridging human rationality and machine logic. AI and Ethics,2(4), 541–554. https://doi.org/10.1007/s4368102100107-w 114