https://chatgpt.com/share/68f00731-5bc8-8010-9b69-ed5464c64256
https://osf.io/ne89a/files/osfstorage/68effd340c8fad784bc40616
Nested Uplifts Inevitability: A Sequential-Evidence and Small-Gain Theory of Regime Switching in Open Dissipative Systems
1. Introduction
Open, dissipative systems—biological populations, online platforms, supply chains, financial ecosystems—often undergo abrupt “regime switches” (uplifts) from slow, additive change to fast, multiplicative growth, followed by a new steady regime after a suitable re-scaling. We propose a general, testable theory explaining when such uplifts are not accidental but structurally inevitable under mild, observable conditions. The core idea is that (i) many observables evolve multiplicatively, (ii) closed-loop feedback creates a small but persistent positive drift in log-returns, and (iii) a cumulative, sequential-evidence process inevitably crosses a decision threshold, triggering a measurable regime change and stabilization into a new “additive” world under an appropriate transform.
Minimal working vocabulary. We will use three primitives. First, a multiplicative observable with log-returns:
Y_{t+1} = Y_t · r_t, u_t := log r_t. (1.1)
Second, a cumulative log-evidence (e.g., log-likelihood ratio or GLR-type statistic) with a stopping boundary Λ:
S_t = ∑_{k=1}^t s_k, τ := inf{ t : S_t ≥ Λ }. (1.2)
Third, a loop discriminant linking macro feedback, micro amplification, and damping:
Δ := gβ − γ. (1.3)
Intuitively, Δ > 0 induces a positive drift μ(Δ) := 𝔼[u_t] > 0, so that S_t grows linearly on average and, by standard hitting-time results, crosses Λ with probability 1 and finite expected time. At τ, a regime switch is defined to occur (e.g., a Markov-kernel jump in rule parameters or a stability-class bifurcation). After the switch, dissipative dynamics ensure convergence to a new attractor; if Δ remains nonnegative and cross-observer objectivity is certified, the same logic recurses under a further re-scaling (e.g., log–log), producing nested uplifts.
Why this matters. Existing accounts of tipping points are often model-specific (e.g., SIR in epidemics, S-shaped adoption in platforms) or descriptive (change-point detection without mechanism). Our contribution is a mechanism-agnostic, sequential, and closed-loop explanation that (a) identifies observable levers (g, β, γ), (b) provides a provable route from feedback to positive drift to threshold crossing, and (c) supplies an operational pipeline to turn subjective signals into objective evidence before declaring a new regime.
Contributions.
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Sequential-evidence inevitability. We show that under mild mixing/tail conditions and Δ > 0, the cumulative statistic S_t hits Λ with probability 1 and finite mean time, triggering a regime switch.
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Closed-loop small-gain link. We formalize the map Δ ↦ μ(Δ) with μ′(Δ) > 0 in a neighborhood of 0, establishing that positive loop discriminant implies positive drift in u_t.
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Operational Log-Gauge Fixing. We define a practical residual-whitening and standardization pipeline that yields cross-source consistency; objectivity is certified when an agreement index exceeds a threshold R*.
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Falsifiable predictions and a minimal case study. We derive testable predictions about hitting times, re-scaling (log → log–log) residual whitening, and policy levers that delay/cancel τ; we provide a small, reproducible study to illustrate each.
Scope and assumptions (at a glance). Our main theorem assumes: (i) multiplicative updates (1.1) with u_t that are i.i.d. or α-mixing and have finite variance or sub-exponential tails; (ii) a closed-loop linearization yielding Δ (1.3) and a regularity link to μ(Δ); (iii) a sequential statistic S_t (1.2) with an admissible stopping rule; (iv) dissipative post-switch dynamics admitting a Lyapunov function; and (v) an objectivity check via Log-Gauge Fixing. We also delineate failure modes (heavy tails with α < 2, long memory with H > 0.5, large delays, unwhitenable sources) where inevitability can break.
What is new. Methodologically, we combine sequential analysis (hitting-time inevitability) with small-gain reasoning (Δ-driven drift), then elevate “objectivity” from rhetoric to an operational, testable criterion. Conceptually, we show how additive → multiplicative → log-projected additive transitions can recur under re-scaling, generating a nested hierarchy of regimes observable in diverse domains.
Roadmap. Section 2 situates our work within sequential analysis, multiplicative processes, small-gain theory, dissipative stability, and consensus metrics. Section 3 formalizes the model, assumptions, and measurable definition of regime switching. Section 4 states the main theorem (INU). Section 5 details the proof architecture—five lemmas and the bridges connecting them. Section 6 specifies the Log-Gauge Fixing pipeline and the objectivity threshold. Section 7 derives falsifiable predictions and testing procedures. Section 8 presents a minimal, reproducible case study. Section 9 analyzes robustness and failure modes. Section 10 discusses applications and design levers. Section 11 concludes.
Reader guidance. Readers seeking the theorem statement can jump to Section 4; those wanting the logic flow should read Section 5. Practitioners can go directly to Sections 6–8 (pipeline, tests, and case study). Robustness and limitations are in Section 9. Appendices collect notation, full proofs, algorithms, and reproducibility materials.
2. Related Work
This section situates our results within five established strands: sequential analysis and stopping rules; multiplicative processes and large deviations; feedback and small-gain theory; dissipative systems and Lyapunov methods; and objectivity/consensus metrics linked to residual whitening. We close by explaining how INU unifies these lines and what is new.
2.1 Sequential analysis and stopping rules
Classical sequential analysis studies cumulative evidence processes that stop the experiment once a boundary is crossed. Wald’s Sequential Probability Ratio Test (SPRT) shows that a log-likelihood ratio with suitable thresholds achieves optimality in terms of expected sample size under Type I/II constraints. More broadly, generalized likelihood ratio (GLR) statistics and mixture-based tests extend the idea to composite hypotheses and drifting parameters. For stochastic processes adapted to a filtration, optional stopping theorems give conditions under which stopped martingales remain integrable and expectations are conserved. Hitting-time results for random walks and diffusions—both in discrete and continuous time—provide sharp control of probabilities and moments of the first-passage time. INU leverages this body of work by (i) modeling log-evidence as a cumulative sum S_t with a boundary Λ, and (ii) invoking positive drift conditions to guarantee 𝙿(τ < ∞) = 1 and 𝔼[τ] < ∞ under mild regularity.
2.2 Multiplicative processes and large deviations
Multiplicative dynamics are ubiquitous: Y_{t+1} = Y_t · r_t with log-returns u_t := log r_t. (2.1)
Under i.i.d. or mixing assumptions with finite variance or sub-exponential tails, the law of large numbers (LLN) implies t^{-1} ∑_{k=1}^t u_k → μ, while large deviation principles (LDP) quantify the exponential rarity of deviations from μ. These tools control both typical growth (geometric mean) and the tail of first-passage events for cumulative sums. In INU, the sequential statistic S_t inherits the drift of u_t, so that positive μ yields almost-sure boundary crossing and finite hitting times. This connects the probabilistic skeleton of “inevitability” to standard limit and deviation theory rather than bespoke assumptions.
2.3 Feedback and small-gain theory
Closed-loop systems often admit a local linearization of the macro-micro feedback chain, yielding an effective loop gain. We encode this by a loop discriminant Δ := gβ − γ, where g is macro gain, β is micro amplification, and γ aggregates damping/buffer terms. (2.2)
Small-gain theorems provide stability windows; root-locus and Nyquist-type analyses show how gains shift poles and alter transient/steady-state behavior. Queueing and congestion models likewise map feedback to throughput and delay. Our contribution is to link Δ—not merely to stability—but to statistical drift in log-returns: we formalize a local map Δ ↦ μ(Δ) with μ′(Δ) > 0 near Δ = 0, hence Δ > 0 ⇒ μ(Δ) > 0. (2.3)
This bridge converts control-style loop reasoning into sequential-statistical inevitability of crossing, a link that is rarely made explicit in prior literature.
2.4 Dissipative systems and Lyapunov methods
Dissipative dynamics are characterized by energy-like functions that decrease along trajectories. Foster–Lyapunov criteria (for Markov chains/processes) and LaSalle’s invariance principle (for deterministic ODEs) provide convergence to invariant sets or attractors when drift inequalities hold. In regime-switching contexts, such criteria can certify post-switch stability provided the new regime admits an appropriate Lyapunov function with negative drift outside a compact set. INU relies on this methodology to guarantee that, once the sequential boundary is hit and a rule change is enacted, trajectories settle into a new attractor—thereby turning a statistical stopping event into a dynamical phase with predictable long-run behavior.
2.5 Objectivity and consensus
Declaring a “new regime” requires more than a boundary crossing; it also requires objectivity—independence from observer-specific artifacts. Two strands are relevant. First, inter-rater agreement metrics (e.g., Fleiss κ, Krippendorff α) quantify consensus across observers. Second, residual-whitening practices in econometrics/signal processing ensure that transformed series have minimal autocorrelation and cross-source bias. INU operationalizes objectivity by a Log-Gauge Fixing pipeline: source-wise standardization (often via log-link GLMs or variance-stabilizing transforms), residual whiteness tests (ACF, Ljung–Box, Durbin–Watson, ADF), and a consensus threshold R* on agreement indices. The combination offers a practical certification that the post-switch “new additive regime” is not an observer artifact.
Positioning: what INU unifies and what is new
Unified view. INU composes five mature threads into a single pipeline: (i) multiplicative growth supplies a natural log-domain; (ii) small-gain feedback turns Δ into a positive drift μ(Δ); (iii) sequential analysis elevates positive drift to almost-sure boundary crossing in finite mean time; (iv) dissipative Lyapunov methods stabilize the post-switch phase; (v) Log-Gauge Fixing provides an operational test for objectivity and cross-observer reproducibility.
What is new.
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Control→Statistics bridge. Prior work typically treats loop gain as a stability notion; INU formalizes Δ ↦ μ(Δ), making loop gain a statistical driver of sequential evidence accumulation.
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Inevitability with recursion. Standard stopping-time analyses yield first-passage properties once drift is assumed; INU shows how closed-loop structure induces that drift and how admissible re-scalings (e.g., log→log–log) can recur, producing nested uplifts.
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Operational objectivity. Instead of rhetorical “phase change,” INU requires whitened residuals and agreement above R*, offering a falsifiable criterion for declaring a new regime.
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Model-agnostic applicability. The framework does not hinge on a specific domain model (SIR, GBM, Bass, etc.); it specifies observable levers (g, β, γ), measurable statistics (S_t, τ), and reproducible diagnostics (whiteness, κ/α) that transfer across domains.
