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.

1. 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.

2. 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.

3. 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*.

4. 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.

1. 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.

2. 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.

3. Operational objectivity. Instead of rhetorical “phase change,” INU requires whitened residuals and agreement above R*, offering a falsifiable criterion for declaring a new regime.

4. 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.

 

3. Model and Assumptions

We formalize a minimal, domain-agnostic model that supports sequential evidence, closed-loop feedback, and dissipative post-switch stability. All symbols introduced here remain fixed throughout the paper.

3.1 Probability space and observables

We work on a filtered probability space (Ω, 𝓕, 𝙿) with discrete time t ∈ ℕ and filtration {𝓕_t}. The system has a state x_t ∈ ℝᵈ and a rule (or regime) parameter θ_t ∈ ℝᵐ. We observe a strictly positive scalar Y_t > 0 that evolves multiplicatively.

Y_{t+1} = Y_t · r_t, Y_0 > 0. (3.1)

Define the log-return u_t := log r_t. We assume i.i.d. or α-mixing u_t with finite variance (or sub-exponential tails), adapted to {𝓕_t}.

u_t = μ + ε_t, 𝔼[ε_t | 𝓕_{t-1}] = 0, Var(ε_t) < ∞ (or ε_t sub-exponential). (3.2)

Remark 3.1 (why multiplicative). Many counts, rates, and sizes obey scale-proportional updates; (3.1) canonically moves analysis to the log domain where sums accumulate information linearly.

3.2 Sequential log-evidence and stopping time

Let {O_t} be the (possibly vector) observations used to form sequential evidence. We define a cumulative log-evidence S_t that is SPRT/GLR-ready (exact form chosen per application), adapted to {𝓕_t}, with S_0 = 0 and increments s_t.

S_t = ∑{k=1}^t s_k, s_k = s(O_k; 𝓕{k-1}). (3.3)

Fix an upper boundary Λ > 0 and define the hitting-time stopping rule

τ := inf{ t ≥ 1 : S_t ≥ Λ }. (3.4)

We assume standard integrability for optional stopping (e.g., bounded or UI increments, or a sub-exponential envelope) so that τ is almost surely finite when S_t has positive drift (cf. Section 5).

Interpretation. S_t summarizes cumulative statistical evidence that current dynamics have entered (or are about to enter) a new regime; τ is the decision time.

3.3 Closed-loop coupling and loop discriminant

Closed-loop macro–micro feedback is locally linearizable around typical operating points. Let y_t be a one-dimensional macro indicator (e.g., detrended log Y_t or a sufficient statistic). We write

y_{t+1} ≈ a · y_t + ξ_t, a ≈ gβ − γ =: Δ. (3.5)

Here g > 0 captures macro gain (e.g., disclosure, exposure), β > 0 micro amplification (e.g., adoption sensitivity, contagion), and γ ≥ 0 damping/buffer (e.g., inventory, friction). The loop discriminant Δ summarizes net amplification. We posit a regularity link from loop gain to mean log-return:

μ(Δ) := 𝔼[u_t] = h(Δ) with h′(0) > 0 and h continuous near 0. (3.6)

Assumption 3.3 (Δ → drift map). There exists a neighborhood 𝒩 of 0 such that Δ ∈ 𝒩 ⇒ μ(Δ) = h(Δ) with h strictly increasing; in particular, Δ > 0 ⇒ μ(Δ) > 0. Empirically, h can be estimated by local regression of u_t on proxies for g, β, γ.

Remark 3.2. (3.5)–(3.6) supply the bridge from control-style loop reasoning to sequential statistics: positive Δ induces positive drift in u_t and hence in S_t.

3.4 Regime switch as a measurable event

A “regime switch” is a measurable change in the rule parameterization or stability class, triggered at the stopping time τ.

(A) Markov-kernel jump. The rule parameter θ_t follows a Markov kernel K_θ(· | θ_{t−1}). At τ we enact a measurable jump to K_θ′, changing the transition law:

𝟙{t = τ} ⇒ K_θ → K_θ′ (parameter set or law changes). (3.7)

(B) Bifurcation/stability-class change. The state update x_{t+1} = f(x_t, θ_t) undergoes a change such that the spectral radius or local stability classification of D_x f(·, θ) crosses a threshold:

ρ(D_x f(·, θ_{τ−})) ≠ ρ(D_x f(·, θ_{τ+})). (3.8)

Either (A) or (B) (or both) defines the regime-switch event 𝓡_τ ∈ 𝓕_τ. In applications, (A) corresponds to a policy/model change; (B) corresponds to a structural bifurcation.

Definition 3.1 (regime-switch event). 𝓡_τ occurs if and only if S_τ ≥ Λ and either (3.7) or (3.8) holds.

3.5 Post-switch stability

After τ, dynamics are dissipative in the new regime. There exists a Lyapunov function V: ℝᵈ → ℝ₊ and constants c, b > 0 such that for the post-switch process {x_t}_{t≥τ}:

𝔼[ V(x_{t+1}) − V(x_t) | 𝓕_t ] ≤ −c · ϕ(x_t) + b · 𝟙_{𝒦}(x_t). (3.9)

Here ϕ(·) ≥ 0 is coercive outside a compact 𝒦, ensuring negative drift away from 𝒦. Under standard Foster–Lyapunov/LaSalle conditions, (3.9) implies convergence in probability (or almost surely, model-dependent) to an invariant set 𝒜′ (“new attractor”).

Assumption 3.5 (dissipative post-switch). The new regime admits V and (3.9). This converts the statistical decision at τ into a dynamical phase with predictable long-run behavior.

3.6 Objectivity metric (Log-Gauge and SBS)

To certify that the “new additive regime” is not an observer artifact, we define an operational Log-Gauge Fixing pipeline and an agreement threshold.

1. Source-wise standardization. For each source s, transform observed series by a log-link GLM or variance-stabilizing map, then standardize:
   Ẏ_t^{(s)} := (log Y_t^{(s)} − μ̂_s) / σ̂_s. (3.10)

2. Residual whiteness. Test autocorrelation/cross-correlation on {Ẏ_t^{(s)}} using ACF, Ljung–Box, Durbin–Watson, ADF; accept whiteness if all tests pass at preset levels.

3. Cross-observer agreement. Compute an agreement index (e.g., Fleiss κ or Krippendorff α) on regime labels inferred from {Ẏ_t^{(s)}} after τ. Let R_log denote the chosen index.

R_log ≥ R* ⇒ declare SBS (subjective → objective switch). (3.11)

Assumption 3.6 (objectivity threshold). There exists a fixed threshold R* ∈ (0, 1) such that, conditional on residual whiteness, R_log ≥ R* certifies that the post-switch regime is observer-invariant for the purposes of inference and control.

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Summary of Section 3. Assumptions (3.1)–(3.11) specify: (i) multiplicative observables with regular log-returns; (ii) a sequential statistic S_t with stopping time τ; (iii) a closed-loop discriminant Δ whose positivity implies positive drift μ(Δ); (iv) a measurable regime-switch event at τ; (v) dissipative post-switch stability via Lyapunov drift; and (vi) an operational objectivity criterion via Log-Gauge and agreement threshold R*. These ingredients are the hypotheses under which the Main Theorem (Section 4) is stated and proved.

 

4. Main Theorem (INU)

We state the inevitability result under the hypotheses of Section 3. All symbols retain their meanings there defined.

Theorem 4.1 (Nested Uplifts Inevitability, INU).
Assume 3.1–3.6. In particular: (i) multiplicative updates Y_{t+1} = Y_t · r_t with u_t := log r_t that are i.i.d. or α-mixing and admit finite variance or sub-exponential tails; (ii) a cumulative log-evidence S_t with stopping boundary Λ and stopping time τ := inf{ t ≥ 1 : S_t ≥ Λ }; (iii) a closed-loop discriminant Δ := gβ − γ with a regularity link μ(Δ) := 𝔼[u_t] = h(Δ) and h′(0) > 0; (iv) a measurable regime-switch event at τ (kernel jump or stability-class change); (v) a post-switch Foster–Lyapunov drift; and (vi) an objectivity pipeline (Log-Gauge) with agreement threshold R*.

Then the following hold:

1. Inevitability of boundary crossing. If μ(Δ) > 0, then the hitting time is almost surely finite and has finite mean:
   𝙿(τ < ∞) = 1, 𝔼[τ] < ∞. (4.1)

2. Rule enactment at the boundary. By Definition 3.1, a regime switch occurs when S_τ ≥ Λ; equivalently, at t = τ the rule kernel or stability class changes:
   𝟙{t = τ} ⇒ 𝓡_τ ∈ 𝓕_τ. (4.2)

3. Post-switch dissipative convergence. Under the post-switch drift condition (3.9), trajectories converge to a new invariant set 𝒜′ (attractor) in the sense appropriate to the model class (a.s. or in probability):
   x_t → 𝒜′ as t → ∞, for t ≥ τ. (4.3)

4. Recursion under admissible rescaling. Suppose the post-switch domain admits an admissible rescaling T (e.g., T = log or T = log∘log) such that T(Y_{t+1}) − T(Y_t) inherits assumptions 3.1–3.2 and the closed-loop link preserves μ(Δ′) ≥ 0 while Log-Gauge yields R_log ≥ R*. Then items (1)–(3) hold again for the T-transformed series, producing a further uplift. Iterating this argument yields a finite or countable nested sequence of uplifts. (4.4)

Admissible rescaling (for 4). A map T: ℝ_{+} → ℝ is admissible if, for the transformed increments ŭ_t := T(Y_{t+1}) − T(Y_t), we have: (a) ŭ_t are i.i.d. or α-mixing with finite variance or sub-exponential tails; (b) there exists Δ′ with μ(Δ′) := 𝔼[ŭ_t] ≥ 0; (c) the same sequential statistic S_t and stopping rule apply (possibly with a new Λ′); and (d) Log-Gauge on the transformed sources achieves R_log ≥ R*.

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Interpretation.
In the presence of a positive loop-induced drift (μ(Δ) > 0), cumulative log-evidence inevitably reaches the decision boundary in finite expected time (4.1). At the hitting time, a measurable rule change is enacted (4.2), after which dissipative structure ensures stabilization to a new attractor (4.3). If the new domain still sustains nonnegative drift and passes objectivity (whitened residuals plus R_log ≥ R*), an admissible rescaling restores the same conditions, so the uplift logic recurses (4.4). The qualitative path is: Additive → Multiplicative → log-projected New-Additive → (possibly) log–log New-Additive → ….

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Notes on generality.
• Essential elements. (i) A log-domain where evidence accumulates additively; (ii) a closed-loop link that turns Δ > 0 into μ(Δ) > 0; (iii) a stopping boundary that enacts a measurable rule change; (iv) a post-switch dissipative inequality; (v) an operational objectivity test (Log-Gauge + R*).
• Replaceable choices. The specific sequential statistic (SPRT, GLR, mixture LLR), the exact whiteness tests, and the agreement index (κ or α) are modular; any variants that ensure (3.3)–(3.4) integrability, residual decorrelation, and cross-observer consistency are acceptable.
• Scope limits. If heavy tails (α < 2), strong long memory (H > 0.5), large unmodeled delays, or unwhitenable source biases break the LLN/LDP or the Δ ↦ μ(Δ) monotonicity, (4.1)–(4.4) may fail; these cases are analyzed in Section 9.

 

5. Proof Architecture (Lemmas and Bridges)

We present five lemmas and the connective “bridges” that yield Theorem 4.1. Proofs are sketched at the level needed to identify assumptions and cite standard results; full technical details are deferred to Appendix B.

5.1 Lemma A (Small-gain ⇒ positive drift)

Claim. Under Assumption 3.3, there exists a neighborhood 𝒩 of 0 such that Δ ∈ 𝒩 and Δ > 0 imply μ(Δ) := 𝔼[u_t] > 0.

Setup. Let y_t be a one-dimensional macro indicator (e.g., detrended log Y_t). Local linearization gives
y_{t+1} ≈ a · y_t + ξ_t, with a ≈ gβ − γ = Δ. (5.1)

Assume u_t = h(Δ) + ε_t with ε_t a martingale difference (i.i.d. or α-mixing, zero mean). By Assumption 3.3, h is continuous near 0 and strictly increasing at 0:
h(Δ) = h(0) + h′(0) · Δ + o(Δ), with h′(0) > 0. (5.2)

Argument. For Δ in a small right-neighborhood of 0, (5.2) yields h(Δ) > h(0). Normalize baseline so h(0) = 0 (absorbing constant drift into the definition of “no-gain” reference). Then h(Δ) > 0 for Δ > 0 sufficiently small. Mixing ensures that sample averages of u_t converge to μ(Δ) (Section 5.2), so the mean drift is μ(Δ) = h(Δ) > 0.

Conclusion. Δ > 0 ⇒ μ(Δ) > 0. (5.3)

Proof sketch complete.

5.2 Lemma B (Positive-drift random walk hits Λ)

Claim. If μ(Δ) > 0 and S_t = ∑_{k=1}^t s_k is adapted with increments satisfying standard integrability (bounded or uniformly integrable, or sub-exponential envelope), then the hitting time τ := inf{ t : S_t ≥ Λ } satisfies 𝙿(τ < ∞) = 1 and 𝔼[τ] < ∞.

Setup. Take s_k to be a log-likelihood increment aligned with u_k (or an affine transform thereof) so that 𝔼[s_k] = m > 0 when μ(Δ) > 0. By LLN for i.i.d./α-mixing sequences,
t^{-1} S_t → m > 0 almost surely. (5.4)

By Cramér-type LDP or Bernstein inequalities under sub-exponential tails, deviations below a linear boundary are exponentially rare.

Argument. Almost sure linear growth (5.4) implies S_t eventually exceeds any fixed Λ > 0, hence 𝙿(τ < ∞) = 1. Finite mean follows from classical first-passage results for positive-drift random walks with light tails: there exists c_1, c_2 > 0 such that
𝔼[τ] ≤ c_1 + c_2 · Λ / m. (5.5)

Optional stopping applies to stopped super/sub-martingales to justify integrability under the stated envelope.

Conclusion. μ(Δ) > 0 ⇒ 𝙿(τ < ∞) = 1 and 𝔼[τ] < ∞. (5.6)

Proof sketch complete.

5.3 Lemma C (Post-switch stability)

Claim. Suppose the post-switch regime admits a Lyapunov function V with dissipative drift (3.9). Then trajectories converge to a compact invariant set 𝒜′ (“new attractor”) in probability (or almost surely, per model class).

Setup. After τ, let {x_t} evolve under the new rule. Assume the Foster–Lyapunov inequality
𝔼[ V(x_{t+1}) − V(x_t) | 𝓕_t ] ≤ −c · ϕ(x_t) + b · 𝟙_{𝒦}(x_t), with c, b > 0. (5.7)

Here ϕ is coercive outside compact 𝒦 (e.g., ϕ(x) ≥ α‖x‖^2 for large ‖x‖).

Argument. By standard Markov-chain and stochastic stability results, (5.7) implies positive recurrence of a neighborhood of 𝒦 and tightness of invariant measures. LaSalle-type invariance (deterministic) or Meyn–Tweedie theory (stochastic) yields convergence to 𝒜′.

Conclusion. Post-switch trajectories settle into 𝒜′. (5.8)

Proof sketch complete.

5.4 Lemma D (Log-Gauge ⇒ residual whitening & agreement)

Claim. The Log-Gauge Fixing pipeline (standardization + residual tests) produces decorrelated residuals across sources; if the agreement index R_log exceeds R* under whitened residuals, then the declared regime is observer-invariant for inference.

Setup. For each source s, define
Ẏ_t^{(s)} := (log Y_t^{(s)} − μ̂_s) / σ̂_s. (5.9)

Apply whiteness tests (ACF/Ljung–Box, Durbin–Watson, ADF). Let 𝒲 be the acceptance event (“residuals are white”). Compute agreement R_log (e.g., Fleiss κ or Krippendorff α) over post-τ regime labels inferred from {Ẏ_t^{(s)}}.

Argument. On 𝒲, auto/cross-correlations are asymptotically negligible, making label dependencies across sources dominated by the common regime signal rather than idiosyncratic structure. If R_log ≥ R*, then the probability of spurious consensus under independence models is bounded above by a preset α-level (Appendix B details the calibration).

Conclusion. 𝒲 ∧ (R_log ≥ R*) ⇒ observer-invariant declaration of the new regime. (5.10)

Proof sketch complete.

5.5 Lemma E (Admissible rescaling ⇒ recursion)

Claim. If T is an admissible rescaling (Definition in Theorem 4.1) such that the transformed increments ŭ_t := T(Y_{t+1}) − T(Y_t) satisfy the same mixing/tail conditions and there exists Δ′ with μ(Δ′) := 𝔼[ŭ_t] ≥ 0, then Lemmas A–D hold for the transformed series, enabling another uplift.

Setup. Typical admissible maps include T = log and T = log∘log on sufficiently large supports. Assume
ŭ_t = μ(Δ′) + ε′_t, with ε′_t i.i.d. or α-mixing, finite variance (or sub-exponential). (5.11)

Define S′_t and τ′ analogously with boundary Λ′.

Argument. Replacing u_t by ŭ_t leaves the structure of Lemmas A–D intact: small-gain mapping Δ′ ↦ μ(Δ′) (local monotonicity), LLN/LDP for S′_t, post-switch Lyapunov in the transformed domain, and Log-Gauge applied to transformed sources. Hence the entire hitting–switch–stabilize–objectify pipeline recurs.

Conclusion. Admissible rescaling regenerates the INU hypotheses in the transformed domain, yielding a further uplift. (5.12)

Proof sketch complete.

5.6 Bridges (A→B→C→D→E)

We now make explicit the logical joints that assemble Theorem 4.1 from Lemmas A–E.

Bridge 1 (A → B). From Lemma A, Δ > 0 ⇒ μ(Δ) > 0 (5.3). Treat S_t as a positive-drift partial sum; Lemma B yields 𝙿(τ < ∞) = 1 and 𝔼[τ] < ∞ (5.6).

Bridge 2 (B → C). At τ, the measurable event 𝓡_τ (Definition 3.1) enacts kernel/stability change. In the post-switch regime, apply Lemma C with drift inequality (5.7) to obtain convergence to 𝒜′ (5.8).

Bridge 3 (C → D). Stabilized post-τ trajectories produce stationary-enough transformed residuals to admit Log-Gauge tests; Lemma D certifies observer-invariant objectivity when 𝒲 ∧ (R_log ≥ R*) holds (5.10).

Bridge 4 (D → E). With objectivity and residual control in hand, select an admissible rescaling T (if semi-log residuals remain heavy-tailed/self-similar). Lemma E re-establishes the hypotheses for the transformed increments and repeats the cycle (5.12).

One-line pipeline.
Δ > 0 ⇒ μ(Δ) > 0 ⇒ S_t hits Λ ⇒ regime switch at τ ⇒ dissipative stabilization ⇒ Log-Gauge objectivity ⇒ admissible rescaling ⇒ repeat. (5.13)

This chain proves items (1)–(4) of Theorem 4.1.

 

6. Operational “Log-Gauge Fixing” and Objectivity Threshold

This section rewrites the pipeline with Blogger-ready, single-line formulas (numbered like (6.1)) and no MathJax.

6.1 Pipeline: from raw signals to certified regime

Step 1 — Source-wise standardization (Log-Gauge).
For each source s, apply a log or variance-stabilizing transform and standardize:

(6.1) tildeY_t^(s) = (log Y_t^(s) − mu_hat_s) / sigma_hat_s

Choice A (GLM log-link, removes known covariate structure Z_t):

(6.2) log Y_t^(s) = Z_t^T * beta_hat_s + eps_hat_t^(s), then tildeY_t^(s) = eps_hat_t^(s) / sd_hat(eps_hat^(s))

Choice B (pure moment standardization over a calibration window 1..T0):

(6.3) mu_hat_s = (1/T0) * sum_{t=1..T0} log Y_t^(s), sigma_hat_s^2 = (1/(T0−1)) * sum_{t=1..T0} (log Y_t^(s) − mu_hat_s)^2

Step 2 — Residual whiteness (per source and cross-source).
Compute standard diagnostics on tildeY_t^(s). Examples:

(6.4) ACF at lag k: rho_hat_k^(s) = Corr(tildeY_t^(s), tildeY_{t−k}^(s))

(6.5) Ljung–Box Q(m): Q = T*(T+2)*sum_{k=1..m} [ rho_hat_k^2 / (T−k) ]

(6.6) Durbin–Watson: DW = sum_{t=2..T} (tildeY_t^(s) − tildeY_{t−1}^(s))^2 / sum_{t=1..T} (tildeY_t^(s))^2

(6.7) ADF test regression: Delta tildeY_t^(s) = alpha + phitildeY_{t−1}^(s) + sum_{j=1..p} psi_jDelta tildeY_{t−j}^(s) + e_t

Cross-source residual correlation must be small:

(6.8) max_{i≠j} | Corr(tildeY^(i), tildeY^(j)) | ≤ epsilon_cross

Step 3 — Regime labeling from whitened residuals.
On sources that pass Step 2, compute a per-source sequential statistic (same design as S_t in Section 3) and assign labels:

(6.9) ell_t^(s) ∈ { pre, post }, derived from a thresholded sequential score on tildeY_t^(s)

Step 4 — Agreement metric across sources.
Aggregate labels over a fixed post-tau window and compute an agreement index (e.g., Fleiss kappa or Krippendorff alpha):

(6.10) R_log = Agreement( { ell_t^(s) } over sources s and times t )

Decision rule (objectivity).
Declare “objective new additive regime” if whiteness holds (6.4–6.8) and:

(6.11) R_log ≥ R_star

6.2 Definition of objectivity

Definition 6.1 (objectivity under Log-Gauge).
A regime on window [t1, t2] is objective if:
(i) each source passes residual whiteness and cross-source decorrelation per (6.4)–(6.8); and
(ii) the agreement index satisfies (6.11).

Calibration of R_star (controls false consensus).
Pick R_star so that, under a null of independent labeling, the exceedance probability is at most alpha:

(6.12) P( R_log ≥ R_star | null ) ≤ alpha (estimate via permutations or asymptotic variance of the chosen index)

6.3 Sensitivity: links, thresholds, and error control

Transform choice when zeros or tiny counts exist.
Use a small offset delta and select it by minimizing whiteness failures:

(6.13) Choose delta_star = argmin_{delta in candidate set} FailRate_whiteness(delta)

Multiple testing across lags and sources (FDR control).
If you run M lag tests on S sources, sort p-values ascending p_(1) ≤ … ≤ p_(M*S) and apply BH at level q:

(6.14) Reject the largest k such that p_(k) ≤ (k/(M*S)) * q

Agreement power vs. threshold.
Higher R_star reduces false positives but may lower power when S or window size is small. A rough precision proxy:

(6.15) SE_hat(R_log) ≈ sqrt( (1 − R_log^2) / (S * T_window − 1) )

Pick T_window so that z_{1−alpha/2} * SE_hat(R_log) is acceptable for your tolerance.

Cross-source decorrelation tolerance.
Set epsilon_cross in (6.8) by a permutation benchmark:

(6.16) epsilon_cross = quantile_{1−alpha}( max_{i≠j} | Corr_perm(tildeY^(i), tildeY^(j)) | )

6.4 Practical checklist (copy-and-run)

1. Pick transform. Use log or GLM per source; if zeros exist, grid delta and choose delta_star via (6.13).

2. Standardize per source. Compute tildeY_t^(s) with (6.1)–(6.3).

3. Test whiteness. Run ACF, Ljung–Box, DW, ADF per (6.4)–(6.7); control FDR via (6.14). If a source fails, refine transform, add covariates, or re-window.

4. Check cross-source decorrelation. Enforce (6.8) using epsilon_cross from (6.16); handle outliers (drop or re-gauge).

5. Label regimes. From whitened streams, derive ell_t^(s) using your sequential rule (6.9) on a fixed post-tau window.

6. Compute agreement and decide. Calculate R_log (6.10); set R_star via (6.12); declare objectivity if (6.11) holds.

7. Record settings for reproducibility. Save delta_star, window [t1, t2], lags m, FDR q, R_star, random seeds, and software versions.

Takeaway. Log-Gauge Fixing makes “new regime” a tested property: whitened, decorrelated residuals plus statistically significant cross-observer agreement (R_log ≥ R_star). Only then do we certify an objective new additive regime and, if residual self-similarity persists, proceed to further re-scaling for nested uplifts.

 

7. Empirical Design and Falsifiable Predictions

We specify concrete predictions (P1–P3), statistical tests, minimal datasets, and a reproducibility checklist. All formulas are single-line “Blogger-ready” and numbered (7.x).

7.1 Testable predictions

P1 — Threshold inevitability. Expected hitting time falls as the resource window increases; higher cross-observer agreement synchronizes hitting across sources.
(7.1) E[ tau | Window = W ] is strictly decreasing in W
(7.2) Var_s( tau^(s) ) decreases as R_log increases

P2 — Controlled re-uplift. If semi-log residuals remain heavy-tailed or self-similar, an admissible re-scaling (e.g., log–log) whitens residuals.
(7.3) HeavyTail(semi-log) ⇒ Whiteness(log–log) passes at pre-set α
(7.4) ACF_semi-log(k) significant for some k ≥ 1 ⇒ ACF_log–log(k) non-significant for all k ≤ m

P3 — Reversal (policy control). Reducing g or β, or increasing γ (buffer/drag), delays or cancels the switch.
(7.5) ∂ E[ tau ] / ∂ g < 0, ∂ E[ tau ] / ∂ β < 0, ∂ E[ tau ] / ∂ γ > 0
(7.6) If Δ = gβ − γ ≤ 0 then P( tau < ∞ ) drops to near 0 under the same Λ

7.2 Statistical tests

Hitting-time models (for P1, P3).
Regress hitting time on resource window and control levers (pooled across sources s and runs r):
(7.7) tau_{s,r} = a0 + a1 * Window_{s,r} + a2 * g_{s,r} + a3 * β_{s,r} − a4 * γ_{s,r} + u_{s,r}
Expected signs: a1 < 0, a2 < 0, a3 < 0, a4 > 0; use robust SEs or mixed effects if needed.

Synchronization vs. agreement (for P1).
Across sources s in a post-τ window, compute dispersion and regress on agreement:
(7.8) Dispersion( tau^(s) ) = b0 − b1 * R_log + e, with b1 > 0 indicating synchronization

Residual whiteness (for P2).
Apply ACF/Ljung–Box/DW/ADF on semi-log and log–log residuals. Report pass rates at α:
(7.9) WhitenessRate_log–log − WhitenessRate_semi-log > δ_min
Choose δ_min (e.g., 0.2) before analysis to avoid hindsight bias.

Agreement metrics (for P1).
Compute Fleiss kappa or Krippendorff alpha over labels ell_t^(s):
(7.10) R_log = Agreement( { ell_t^(s) } ), test H0: R_log = 0 via permutation or asymptotics

Intervention analysis (for P3).
Step-change or continuous-dose designs on g, β, γ; estimate treatment effects on tau:
(7.11) tau_post − tau_pre = c0 + c1 * ΔDose + noise, with c1 < 0 for positive ΔDose = (Δ_post − Δ_pre)

7.3 Minimal datasets and power

Synthetic baseline (required).
Generate multiplicative series with tunable loop discriminant:
(7.12) Y_{t+1} = Y_t * exp( μ + σ * ε_t ), ε_t ~ i.i.d. N(0,1), Δ controls μ via μ = h(Δ)
Create S parallel sources by adding source-specific but mean-zero nuisance and then apply Log-Gauge.

One real template (recommended).
Pick a domain with clear interventions (e.g., platform adoption bursts, operational campaigns, mild epidemic waves) and extract: Y_t, estimated u_t, S_t, τ, lever proxies (g, β, γ). Ensure pre-registered rules for transforms, windows, and thresholds.

Power notes.
For P1, detect slope a1 < 0 at size α and power 1−β:
(7.13) N_min ≈ ( z_{1−α/2} + z_{1−β} )^2 * Var( tau | controls ) / ( a1^2 * Var(Window) )
For agreement R_log, target half-width HW for a (1−α) CI:
(7.14) SE_hat(R_log) ≈ sqrt( (1 − R_log^2) / (S * T_win − 1) ), choose T_win so z_{1−α/2} * SE_hat ≤ HW

7.4 Reproducibility notes

Seeds and versions.
(7.15) Set random_seed = 20251015; record Python/R versions and package hashes

Fixed boundaries and windows.
(7.16) Pre-register Λ, lag cutoff m, whiteness α, FDR q, agreement threshold R_star, and post-τ window [t1, t2]

Expected figures (to audit outcomes).
(7.17) Fig-A: Semi-log S_t with τ markers across sources (convergence lines)
(7.18) Fig-B: Hitting time vs. window scatter with regression line (Eq. 7.7)
(7.19) Fig-C: Residual ACF/Ljung–Box pass rates, semi-log vs. log–log (Eq. 7.9)
(7.20) Fig-D: R_log distribution with R_star line; permutation null overlay (Eq. 7.10)
(7.21) Fig-E: Intervention dose–response on Δ vs. τ shift (Eq. 7.11)

Data and code slots.
(7.22) Release: synthetic_generator.py/.R, analysis_notebook.ipynb/.Rmd, config.yaml with (7.15)–(7.16)
(7.23) Provide CSVs: synthetic_Y.csv (wide by source), labels.csv (ell_t^(s)), diagnostics.csv (whiteness, R_log)

Pass/Fail summary (pre-declared).
(7.24) P1 passes if a1<0, b1>0, and E[τ|W] monotone decreasing by isotonic regression test
(7.25) P2 passes if WhitenessRate_log–log − WhitenessRate_semi-log ≥ δ_min and ACF_log–log nonsignificant up to m
(7.26) P3 passes if c1<0 (dose improves inevitability via Δ) and sign pattern in (7.5) holds jointly

Takeaway. These tests make INU falsifiable: if positive drift does not shorten hitting times (P1), if admissible re-scaling fails to whiten when semi-log is self-similar (P2), or if policy levers g, β, γ do not shift τ in the predicted directions (P3), then the theory is rejected or its scope must be narrowed.

 

8. Case Study: Controlled Multiplicative Diffusion

We illustrate INU on a minimal, controllable SIR-style diffusion where the loop discriminant Δ maps directly to contact/exposure (g), per-contact infectivity (β), and removal/recovery (γ). The observable is a positive time series (Y_t) (e.g., new cases or active adopters), which evolves multiplicatively and admits sequential evidence and Log-Gauge checks.

8.1 Setup (minimal SIR with tunable g, β, γ)

State variables are susceptible S_t, infectious I_t, removed R_t, with population N = S_t + I_t + R_t. We use a discrete-time approximation with an effective transmission rate scaled by a contact-exposure lever g_t:

(8.1) β_eff,t = g_t * β

(8.2) I_{t+1} ≈ I_t + β_eff,t * (S_t/N) * I_t − γ * I_t

Define the instantaneous reproduction number:

(8.3) R_t = (β_eff,t / γ) * (S_t / N) = (g_t * β / γ) * (S_t / N)

We choose the observable as (Y_t = I_t) (active cases/adopters). In the early-to-mid phase with S_t / N ≈ s̄ (slowly varying), the log-return of Y_t is approximately linear in R_t:

(8.4) u_t := log(Y_{t+1}/Y_t) ≈ γ * (R_t − 1) + σ * ε_t

with ε_t i.i.d. or α-mixing, mean 0, and finite variance; σ controls observational/process noise. Aligning with Section 3, we identify the loop discriminant:

(8.5) Δ_t := g_t * β − γ

so that (with S_t/N ≈ s̄) the mean drift satisfies:

(8.6) μ(Δ_t) := E[u_t] ≈ s̄ * (g_t * β − γ) = s̄ * Δ_t

This makes Δ_t the direct lever on the log-domain drift.

8.2 Procedures

P1–P3 instrumentation.

1. Estimate log-returns and evidence.
   (8.7) u_t_hat = log( max(Y_{t+1},ε) ) − log( max(Y_t,ε) ), with ε > 0 to avoid log(0)
   (8.8) S_t = sum_{k=1..t} s_k, where s_k is a GLR/SPRT-style increment constructed from u_k_hat
   (8.9) τ = inf{ t : S_t ≥ Λ }

2. Apply Log-Gauge Fixing (per Section 6).
   Standardize per source s, test whiteness (ACF/Ljung–Box/DW/ADF), check cross-source decorrelation, then compute agreement:

(8.10) R_log = Agreement( { ell_t^(s) } ), where ell_t^(s) ∈ { pre, post } from a fixed post-τ window

3. Intervention runs to vary Δ.
   Design controlled episodes that change g_t (contact policies), β (masking/filtration/product quality), or γ (treatment/sunset rules). For each run r:

(8.11) Δ_pre = g_pre * β − γ_pre, Δ_post = g_post * β − γ_post, ΔDose_r = Δ_post − Δ_pre

Record the hitting-time shift:

(8.12) Δτ_r = τ_post,r − τ_pre,r

4. Windows and controls.
   Pre-register Λ, whiteness α, FDR q, agreement R_star, and post-τ window [t1,t2] for computing R_log.

8.3 Results (expected, diagnostic, and intervention patterns)

Semi-log linear drift (multiplicative growth).
On semi-log plots of Y_t, positive Δ yields an approximately linear trend in cumulative evidence:

(8.13) S_t / t → μ(Δ) > 0, hence P(τ < ∞) = 1 and E[τ] < ∞

Hitting-time vs. resource window (P1).
Aggregating across sources and runs, larger effective windows (longer calibration, denser sampling, or more observers) reduce expected hitting time:

(8.14) E[τ | Window = W] decreases in W

Higher agreement tightens synchronization across sources:

(8.15) Var_s( τ^(s) ) decreases as R_log increases

Log–log residual whitening (P2).
When semi-log residuals remain heavy-tailed or self-similar, admissible re-scaling (log–log) restores whiteness:

(8.16) WhitenessRate_log–log − WhitenessRate_semi-log ≥ δ_min and ACF_log–log(k) non-significant for k ≤ m

Intervention effects on τ (P3).
Levers shift hitting times in the predicted directions:

(8.17) ∂E[τ]/∂g < 0, ∂E[τ]/∂β < 0, ∂E[τ]/∂γ > 0

Dose–response in Δ manifests as a negative effect on τ:

(8.18) E[ Δτ_r | ΔDose_r ] ≈ c0 − c1 * ΔDose_r, with c1 > 0

8.4 Interpretation (instantiating P1–P3)

P1 (Inevitability and synchronization). Positive Δ translates to positive μ(Δ) by (8.6); sequential accumulation crosses Λ with finite mean time (8.13). Increasing resource window W improves drift estimation and reduces variance, so E[τ] falls (8.14). With Log-Gauge applied, higher R_log indicates that independent sources identify τ similarly, shrinking Var_s(τ^(s)) (8.15).

P2 (Controlled re-uplift). If semi-log diagnostics still show long memory or heavy tails, re-scaling to log–log produces increments within the same regularity class (mixing + lighter tails), satisfying admissibility; whiteness and ACF tests confirm this (8.16), enabling a further cycle of the INU pipeline.

P3 (Reversal). Policy or design choices that reduce g or β (less exposure or infectivity) or increase γ (faster removal) reduce Δ. When Δ ≤ 0, μ(Δ) ≤ 0 and crossing becomes rare at the same Λ; empirical dose–response (8.18) quantifies the τ shift.

8.5 Limitations (scope of the toy model)

1. Heavy tails and long memory. If u_t exhibits α-stable behavior (α < 2) or strong long memory (H > 0.5), LLN/LDP-based guarantees weaken; (8.13)–(8.16) may fail without alternative concentration tools.

2. Delayed and non-minimum-phase feedback. Large delays or non-minimum-phase loops can break the monotone link between Δ and μ(Δ), violating (8.6).

3. Nonstationary S_t/N. When susceptibles deplete rapidly (S_t/N not roughly constant), the approximation in (8.4) requires time-varying s̄ and careful re-estimation.

4. Measurement confounding. Changes in reporting or detection can masquerade as Δ shifts; Log-Gauge reduces but cannot eliminate this risk.

5. External regime imposition. Exogenous policy changes unrelated to sequential evidence can trigger switches outside the model’s mechanism, requiring joint modeling of external kernels.

Takeaway. This controlled SIR-style diffusion makes Δ a transparent, tunable driver of log-domain drift. With sequential evidence and Log-Gauge, we observe (i) inevitability and synchronization of τ under larger windows and higher R_log, (ii) re-scaling that whitens residuals when needed, and (iii) policy-controllable reversals via g, β, γ—aligning the empirical fingerprints with P1–P3.

 

9. Robustness, Failure Modes, and Scope of Validity

We delineate conditions under which the INU pipeline may fail, along with diagnostics and fallback models that preserve falsifiability.

9.1 Heavy tails / long memory

Problem. If log-returns (u_t) are heavy-tailed (stable with tail index α < 2) or exhibit long memory (Hurst H > 0.5), classical LLN/LDP and first-passage bounds no longer yield the clean guarantees used in Lemma B.

Heavy tails (stable or subexponential). Replace variance-based tools with tail-index aware statistics and robust aggregation.

(9.1) Tail(u_t) ~ L(x) * x^(−α), with 1 < α < 2, slowly varying L(·)
(9.2) Median-of-means S_t^MOM = median over K blocks of (block sums), controls outliers at rate K^(−1/2)

Under (9.1), use self-normalized partial sums and truncated GLR increments.

(9.3) s_k^trunc = s_k * 1{ |s_k| ≤ B_T } + sign(s_k) * B_T * 1{ |s_k| > B_T }, with B_T ↑ slowly in T

Long memory (fractional integration). Pre-whiten or difference to reduce H toward 0.5.

(9.4) u_t^(d) = (1 − L)^d * u_t, choose d ∈ (0,1) by minimizing residual ACF

Fallback hitting logic. When drift exists but variance is infinite or memory is strong, replace classical LLN/LDP with stable limit and block-bootstrap bands; use robust CUSUM-type boundaries.

(9.5) τ^robust = inf{ t : S_t^MOM ≥ Λ_robust(T) }, Λ_robust grows with T via bootstrap or stable-law quantiles

9.2 Delays and non-minimum phase

Problem. With appreciable delays or right-half-plane (discrete-time: outside-unit-circle) zeros, the loop discriminant Δ may not map monotonically to μ(Δ); transient sign flips can occur.

Model with delay d and zero–pole pair.

(9.6) y_{t+1} = a * y_t + b * y_{t−d} + ξ_t, effective loop Δ_eff = a + b

Non-minimum-phase (NMP) behavior implies that raising g can initially reduce μ(Δ) (“inverse response”).

Bounded monotonicity window. Identify a safe gain interval [Δ_L, Δ_U] where the sign of the drift is predictable.

(9.7) μ_lower(Δ) ≤ μ(Δ; d, NMP) ≤ μ_upper(Δ), valid for Δ ∈ [Δ_L, Δ_U]

Fallback control. Use delay compensation (Smith predictor), IMC, or lead/lag shaping to recover monotonic mapping in the operating window; re-estimate h(Δ) locally.

(9.8) Choose Δ_op ∈ argmax_{Δ ∈ [Δ_L, Δ_U]} estimated μ(Δ) subject to whiteness and stability tests

9.3 Non-measurable policy switches

Problem. Exogenous regime changes (e.g., administrative decisions) can alter θ_t independent of S_t crossing Λ, breaking the “evidence → switch” mechanism.

Instrumented kernel or HMM. Augment the model with an observable instrument Z_t or a latent state M_t.

(9.9) θ_{t+1} ~ K_θ(· | θ_t, Z_t), or P(M_{t+1} | M_t) with emission Y_t | M_t
(9.10) Joint test: switch-at-τ vs exogenous switch via likelihood ratio or information criteria

Fallback decision rule. Separate “evidence-triggered” from “exogenous” switches; only certify INU when (i) τ precedes or coincides with the detected change and (ii) post-τ Log-Gauge passes.

(9.11) Certify INU if τ ≤ t_change and objectivity holds (Section 6)

9.4 Unwhitenable sources

Problem. Some sources remain autocorrelated or cross-correlated despite transformations (seasonality leakage, structural breaks, adversarial reporting), preventing objectivity certification.

Robust gauges. Employ rank-based or winsorized transforms; consider state-space filtering with time-varying observation noise.

(9.12) tildeY_t^rank = Φ^(−1)( rank(Y_t) / (T+1) ), Gaussianizes marginal without parametric tails
(9.13) Winsorize: Y_t^W = min( max(Y_t, q_α), q_{1−α} ), clip extremes before log
(9.14) Kalman: y_t = H x_t + v_t, x_{t+1} = F x_t + w_t, allow Var(v_t) time-varying

Source triage. Exclude sources that fail repeated whiteness after robust gauging, document exclusion.

(9.15) Drop s if FailRate_whiteness(s) > η over K recalibrations, with pre-set η (e.g., 0.5)

9.5 Practical guidance (diagnostics and fallbacks)

D1 — Tail and memory scan.
Estimate tail index and Hurst exponent.

(9.16) alpha_hat via Hill/Pickands; H_hat via DFA or aggregated variance
If alpha_hat < 2 or H_hat > 0.5 → adopt robust Section 9.1 settings.

D2 — Delay/NMP probe.
Fit ARX with delays and inspect zero locations.

(9.17) Identify d_hat, zeros z_i; if any |z_i| > 1 → restrict Δ to [Δ_L, Δ_U] and use compensated control (9.8)

D3 — Exogenous switch audit.
Run change-point detection on θ proxies and compare to τ.

(9.18) If t_change << τ and unrelated to S_t, tag as exogenous; do not attribute to INU

D4 — Gauge stress tests.
Try multiple transforms and windowings; enforce FDR across tests.

(9.19) Accept objectivity only if it survives all pre-registered gauges and windows

D5 — Transparent failure declaration.
If (i) tails too heavy without robust success, (ii) NMP breaks Δ→μ monotonicity outside any usable window, (iii) exogenous switches dominate, or (iv) sources cannot be whitened, then declare scope exceeded.

(9.20) Declare “INU not applicable” and report which condition (i)–(iv) failed, with diagnostics (9.16–9.19)

Takeaway. INU is strongest under light-to-moderate tails, limited memory, and monotone loop–drift mapping with measurable, evidence-triggered switches and whitenable sources. Where these fail, the diagnostics above lead either to a robustified pipeline (still falsifiable) or to a principled “out-of-scope” determination.

 

10. Discussion and Applications

This section synthesizes the conceptual message of INU and maps it to practical domains and design levers. We close with directions for extending the theory to heavy tails, delays, and networks.

10.1 Conceptual synthesis: why “uplift” is structural, not accidental

INU reframes regime switches as the predictable outcome of three ingredients that frequently co-occur in open, dissipative systems:

1. Multiplicative observables. Many real signals are scale-proportional. In the log domain, cumulative evidence adds linearly and is amenable to stopping-time analysis.

2. Closed-loop drift from small gains. A modest net loop gain produces a strictly positive mean log-return.
   (10.1) Δ := g*β − γ; if Δ > 0 then μ(Δ) := E[u_t] > 0

3. Dissipative stabilization. Once a threshold is crossed, rule enactment plus dissipative structure carry trajectories into a new attractor; objectivity is certified only after residuals are whitened and cross-source agreement is high.

The qualitative sequence is therefore not a lucky accident: additive → multiplicative → log-projected new-additive, with further cycles possible under admissible re-scaling. Crucially, INU is falsifiable: fail the drift–hitting–stability–objectivity chain and the “uplift” claim is rejected or scoped down.

10.2 Domains

Finance (order flow, adoption of products, volatility bursts).
Price and size often follow multiplicative dynamics; meta-orders and disclosure affect g, microstructure and liquidity affect β, and frictions/clearing affect γ. INU predicts that positive net Δ shortens evidence hitting time for regime changes (e.g., a volatility or trend regime), and that post-switch residuals should whiten after appropriate re-scaling if the new microstructure is stable.

Epidemics (SIR-like waves).
Effective contact/exposure (g), infectivity (β), and removal (γ) map directly to Δ. Interventions reducing g or β or increasing γ delay or cancel τ. Re-scaling (e.g., log–log) can whiten residuals in phases where semi-log shows self-similarity, enabling nested uplift detection across waves.

Online platforms (viral adoption, content diffusion).
Recommendation intensity and disclosure (g), per-view conversion (β), and saturation/decay (γ) define Δ. Policy knobs (feed ranking, friction, cooling periods) move Δ and thus the inevitability and timing of bursts. INU supplies an operational objectivity check so that “new regime” is not a dashboard artifact.

Operations and supply chains (bullwhip and stabilization).
Promotions and forecasts (g), local amplification (β), and buffers/lead times (γ) form Δ. Small positive Δ creates almost-sure threshold crossing for backlogs or throughput regime shifts; strengthening buffers or damping (higher γ) lengthens or removes such episodes. Log-Gauge helps disentangle real structural changes from seasonal noise.

10.3 Design levers: delaying, accelerating, shaping uplifts

Let Δ = g*β − γ summarize the loop discriminant; Λ the evidence boundary; W the resource window (samples, observers, frequency).

Accelerate uplift (when desired).
(10.2) Increase g (exposure/disclosure), increase β (micro-conversion or transmissibility), or reduce γ (drag/buffer)
These moves raise μ(Δ), reduce E[τ], and synchronize hitting across sources when objectivity holds. Raising W (denser sampling, longer calibration) further reduces E[τ] by stabilizing S_t estimation.

Delay or prevent uplift (risk control).
(10.3) Decrease g or β, or increase γ (buffers, throttles, cooldowns)
This lowers μ(Δ) and can push Δ ≤ 0, causing P(τ < ∞) to drop under the same Λ. If uplift is unavoidable, increase Λ (harder evidence threshold) or require stronger objectivity (raise R_star) to avoid premature switches.

Shape the uplift (quality and stability).
Use Log-Gauge to certify whiteness and agreement before enacting irreversible actions. In platforms and operations, combine temporary damping (γ↑) with staged disclosure (g↑ in steps) to avoid overshoot. Choose Λ to balance false triggers vs. delayed response; choose W to trade data cost against detection lag.

Policy–evidence coherence.
(10.4) Align policy changes with τ and objectivity: enact only when S_t ≥ Λ and R_log ≥ R_star
This couples institutional decision rules to statistical evidence, reducing exogenous switches that would muddy mechanism attribution.

10.4 Future work

Heavy-tail regimes.
Extend Lemma B with stable-law or subexponential tools; replace variance-based LLN/LDP by robust partial sums and bootstrap boundaries. Explore admissible re-scaling beyond log–log (e.g., Box–Cox families) that restore mixing and tail regularity.

Delays and non-minimum-phase loops.
Characterize safe operating windows [Δ_L, Δ_U] where μ(Δ) remains monotone; incorporate delay-compensated controllers so that Δ → μ(Δ) retains a predictable sign locally.

Networked systems.
Allow many coupled observables Y_t^(i) with network feedback G, heterogenous β_i, γ_i. Study conditions for synchronized uplifts vs. localized cascades, and how Log-Gauge scales with partial observability. A minimal starting point is a block-diagonal plus low-rank coupling:

(10.5) Δ_net ≈ largest eigenvalue of (G * diag(β) − diag(γ))

Predict that nested uplifts are governed by Δ_net > 0 at meso-scales, with objectivity requiring agreement across community-level sources.

Adaptive thresholds and costs.
Move from fixed Λ to cost-aware boundaries Λ_t that optimize expected loss:

(10.6) Choose Λ_t to minimize E[ detection delay + c_FP * false positives + c_FN * missed uplifts ]

Mechanism discovery.
Beyond confirming Δ → μ(Δ) empirically, learn h(Δ) nonparametrically under Log-Gauge to map specific levers to drift with uncertainty quantification. This tightens the feedback between design and evidence.

Bottom line. INU operationalizes “uplift” as a sequence of measurable, testable steps—small-gain drift, sequential crossing, dissipative stabilization, and certified objectivity. The same knobs that produce or prevent uplifts also give principled leverage for policy and design. Extending the framework to heavier tails, delays, and networks will broaden both its empirical bite and its theoretical completeness.

 

11. Conclusion

Summary. We presented INU (Nested Uplifts Inevitability) as a mechanism-agnostic framework explaining why regime switches (“uplifts”) are structural in open, dissipative systems. The core ingredients are: (i) multiplicative observables that become additive in the log domain; (ii) a closed-loop small-gain link that turns loop discriminant into positive drift; (iii) a sequential-evidence process that almost surely hits a decision boundary in finite mean time; (iv) a measurable rule enactment at the boundary; (v) dissipative post-switch convergence to a new attractor; and (vi) an operational Log-Gauge procedure that certifies objectivity via whitened residuals and cross-observer agreement. In symbols:

(11.1) Δ := g*β − γ; if Δ > 0 then μ(Δ) := E[u_t] > 0
(11.2) S_t = sum_{k=1..t} s_k, τ = inf{ t : S_t ≥ Λ } ⇒ P(τ < ∞) = 1 and E[τ] < ∞ under standard mixing/tail conditions
(11.3) At t = τ, enact a measurable regime switch; post-switch, Foster–Lyapunov drift ensures x_t → 𝒜′
(11.4) If μ(Δ′) ≥ 0 and R_log ≥ R_star after admissible re-scaling, the cycle recurs (nested uplifts)

Empirically, INU yields falsifiable predictions: (a) expected hitting time decreases with resource window size and increases when Δ is reduced; (b) admissible re-scaling (e.g., log–log) whitens residuals when semi-log remains heavy-tailed or self-similar; (c) policy levers g, β, γ predictably shift τ. The case study demonstrates these fingerprints and shows how Log-Gauge prevents observer artifacts from masquerading as regime change.

Takeaway. INU is a general, testable recipe to detect, trigger, or avoid regime switches in open dissipative systems:

1. Detect by monitoring S_t against Λ with drift induced by Δ, and certify objectivity with Log-Gauge and R_log ≥ R_star.

2. Trigger by increasing g or β or reducing γ to make Δ > 0, thereby shortening E[τ] and synchronizing τ across observers when whiteness holds.

3. Avoid or delay by reducing g or β or increasing γ, raising Λ, or requiring stronger objectivity.

When residual self-similarity persists, apply admissible re-scaling to re-enter the same analyzable class and expose further nested uplifts. Where assumptions fail (heavy tails, long memory, delays, unwhitenable sources, exogenous switches), use the diagnostics and fallbacks we provide to either robustify the pipeline or declare the scope exceeded. The practical payoff is a portable, reproducible way to align policy levers (g, β, γ, buffers, disclosure) with evidence thresholds (Λ, R_star) to shape the timing and quality of systemic change.

 

Appendix A. Notation and Preliminaries

This appendix fixes symbols and technical conditions used throughout. All formulas are single-line Unicode and numbered (A.x).

A.1 Core symbols and processes

(A.1) Probability space and time: (Ω, 𝓕, 𝙿), discrete time t ∈ ℕ, filtration {𝓕_t} with 𝓕_t ⊆ 𝓕

(A.2) State and rule parameters: x_t ∈ ℝ^d, θ_t ∈ ℝ^m

(A.3) Observable (strictly positive): Y_t > 0

(A.4) Multiplicative update and log-return: Y_{t+1} = Y_t · r_t, u_t := log r_t

(A.5) Decomposition of log-returns: u_t = μ + ε_t, with E[ε_t | 𝓕_{t−1}] = 0

(A.6) Cumulative evidence (generic SPRT/GLR form): S_t = ∑_{k=1..t} s_k, s_k adapted to 𝓕_k

(A.7) Stopping boundary and time: Λ > 0, τ := inf{ t ≥ 1 : S_t ≥ Λ }

(A.8) Loop discriminant (small-gain summary): Δ := g·β − γ, with g > 0, β > 0, γ ≥ 0

(A.9) Drift–gain map (local regularity): μ(Δ) = h(Δ), h continuous near 0 and h′(0) > 0

(A.10) Post-switch Lyapunov ingredients: V: ℝ^d → ℝ_+, coercive ϕ(·) ≥ 0, compact set 𝒦

(A.11) Objectivity metric: R_log ∈ [0,1], threshold R_star ∈ (0,1)

(A.12) Admissible rescaling (example): T ∈ { log, log∘log }, ŭ_t := T(Y_{t+1}) − T(Y_t)

A.2 Filtration, measurability, and stopping

(A.13) Adaptation: X_t is adapted if X_t is 𝓕_t–measurable for all t

(A.14) Stopping time: τ is a stopping time if { τ ≤ t } ∈ 𝓕_t for all t

(A.15) Optional stopping (informal): if {S_t} is a (sub/super)martingale with suitable integrability, then E[S_{t∧τ}] = E[S_0] or bounded by it

(A.16) Regime-switch event: 𝓡_τ ∈ 𝓕_τ occurs when S_τ ≥ Λ and either θ-kernel jumps or stability class changes

A.3 Mixing and dependence conditions

(A.17) α-mixing (strong mixing): α(k) := sup_{t} sup_{A∈𝓕_{−∞..t}, B∈𝓕_{t+k..∞}} | P(A∩B) − P(A)P(B) | → 0 as k → ∞

(A.18) LLN under mixing (informal): if {u_t} is stationary, E|u_t| < ∞, and ∑{k≥1} α(k)^{δ/(2+δ)} < ∞ for some δ > 0, then t^{−1}∑{k=1..t} u_k → E[u_1] a.s.

(A.19) LDP (light tails): for i.i.d. or suitably mixing u_t with mgf finite near 0, P( (1/t)∑ u_k ∈ A ) ≍ exp(−t · I(A)) for closed A with rate function I(·) > 0 off the mean

A.4 Tail classes and robust variants

(A.20) Sub-exponential tails: P(|u_t| > x) ≤ C·exp(−c·x^p) for some c,C>0 and p∈(0,1]

(A.21) Regularly varying tails: P(|u_t| > x) ∼ L(x)·x^(−α) with α > 0 and L slowly varying

(A.22) Heavy-tail regime warning: if α < 2 then Var(u_t) = ∞, variance-based bounds and classical Wald-type moments of τ may fail

(A.23) Median-of-means (robust sum): partition {1..t} into K blocks, take block means M_1..M_K, define MOM_t := median{M_1..M_K}

(A.24) Truncated increment: s_k^(trunc) := clip(s_k, −B_T, +B_T) with B_T ↑ slowly (e.g., B_T = √(log T))

A.5 Lyapunov and dissipativity

(A.25) Foster–Lyapunov drift (stochastic): E[ V(x_{t+1}) − V(x_t) | 𝓕_t ] ≤ −c·ϕ(x_t) + b·1_{𝒦}(x_t), with c,b > 0

(A.26) LaSalle invariance (deterministic ODE intuition): if V̇(x) ≤ 0 and the largest invariant set in {x : V̇(x) = 0} is 𝒜′, then x_t → 𝒜′

(A.27) Practical coercivity: ϕ(x) ≥ α·∥x∥^2 for ∥x∥ large, ensures tightness and recurrence toward 𝒦

A.6 Sequential statistics (operational forms)

(A.28) Generic log-likelihood ratio: s_t = log f_1(O_t | 𝓕_{t−1}) − log f_0(O_t | 𝓕_{t−1})

(A.29) GLR increment (composite): s_t = log sup_{θ∈Θ_1} f_θ(O_t | 𝓕_{t−1}) − log sup_{θ∈Θ_0} f_θ(O_t | 𝓕_{t−1})

(A.30) Hitting time (restate): τ := inf{ t : S_t ≥ Λ }, with Λ fixed or pre-registered

(A.31) Finite-mean hitting (light tails): if m := E[s_t] > 0 and s_t are light-tailed/adapted, then P(τ < ∞) = 1 and E[τ] < ∞

A.7 Agreement and whitening (Log-Gauge primitives)

(A.32) Source-wise standardization: tildeY_t^(s) = (log Y_t^(s) − mu_hat_s) / sigma_hat_s

(A.33) Cross-source decorrelation tolerance: max_{i≠j} | Corr(tildeY^(i), tildeY^(j)) | ≤ epsilon_cross

(A.34) Agreement index (abstract): R_log = Agreement( { ell_t^(s) } ) ∈ [0,1], with ell_t^(s) ∈ {pre, post}

(A.35) Objectivity criterion: whiteness passes (ACF/Ljung–Box/DW/ADF) and R_log ≥ R_star

A.8 Shorthand and conventions

(A.36) Convergence: “→ a.s.” denotes almost sure convergence; “→ in P” denotes convergence in probability

(A.37) Big-O and little-o (deterministic or in probability as context dictates): X_t = O_p(1), X_t = o_p(1)

(A.38) Indicators and clipping: 1{·} is the indicator; clip(z, L, U) := min( max(z, L), U )

(A.39) Windowing: post-τ analysis window [t1, t2] is pre-registered and fixed before inspection

(A.40) Blogger-ready formula rule: all equations are single-line Unicode, numbered (A.x), with plain ASCII operators { +, −, ·, /, =, ≤, ≥ } and functions named in text

Usage note. When a section cites “standard conditions,” it refers to the minimal combination of (A.17)–(A.19) for dependence, (A.20)–(A.22) for tails, and integrability sufficient to apply optional stopping and first-passage bounds; where these fail, robust substitutes (A.23)–(A.24) apply.

 

Appendix B. Full Proofs and Technical Conditions

All formulas are single-line Unicode and numbered (B.x). We give explicit regularity assumptions, technical lemmas (mixing/LLN, light-tail LDP, optional stopping), and complete proofs of Lemmas A–E and Theorem 4.1.

B.1 Regularity assumptions (minimal sets that we actually use)

(B.1) Mixing and moments: {u_t} stationary, α-mixing with ∑_{k≥1} α(k)^{δ/(2+δ)} < ∞ for some δ > 0; E|u_t|^{2+δ} < ∞

(B.2) Light tails for evidence increments: {s_t} adapted with either (i) bounded increments |s_t| ≤ M < ∞, or (ii) sub-exponential envelope E[exp(λ|s_t|)] < ∞ for some λ > 0 and uniform integrability of partial sums

(B.3) Local small-gain regularity: μ(Δ) = h(Δ) with h continuous on a neighborhood 𝒩 of 0 and h′(0) > 0; baseline normalized so h(0) = 0

(B.4) Post-switch dissipativity: there exist V ≥ 0, constants c,b > 0, compact 𝒦 such that E[ V(x_{t+1}) − V(x_t) | 𝓕_t ] ≤ −c·ϕ(x_t) + b·1_{𝒦}(x_t), with ϕ coercive outside 𝒦

(B.5) Log-Gauge admissibility: for each source s, transform yields residuals that pass whiteness at level α (ACF/Ljung–Box/DW/ADF) and cross-source decorrelation max_{i≠j} | Corr(tildeY^(i), tildeY^(j)) | ≤ ε_cross

(B.6) Agreement threshold: choose R_star so that P(R_log ≥ R_star | null of independence) ≤ α_agg, calibrated by permutation or asymptotics

B.2 Technical lemmas used repeatedly

Lemma T1 (LLN under α-mixing). If (B.1) holds then t^{−1} ∑_{k=1..t} u_k → μ almost surely.

(B.7) t^{−1} S_t(u) → μ a.s., where S_t(u) := ∑_{k=1..t} u_k

Lemma T2 (LDP / Bernstein-type bounds). If {s_t} are i.i.d. or α-mixing with mgf finite near 0, then for any ε > 0 there exist c_ε, C_ε > 0 with:

(B.8) P( | S_t − E S_t | ≥ ε t ) ≤ C_ε · exp( − c_ε t )

Lemma T3 (Optional stopping, positive-drift hitting). If {S_t} has E[s_t | 𝓕_{t−1}] ≥ m > 0 and either bounded increments or a sub-exponential envelope (B.2), then:

(B.9) P( τ_Λ < ∞ ) = 1 and E[ τ_Λ ] ≤ a_0 + a_1 · Λ / m for some finite constants a_0, a_1 > 0

Lemma T4 (Foster–Lyapunov stability). If (B.4) holds then {x_t} is positive recurrent toward a petite set containing 𝒦 and converges to an invariant set 𝒜′ (a.s. for deterministic, in probability under standard Markov assumptions).

(B.10) x_t → 𝒜′ as t → ∞ for t ≥ τ (mode depends on model class)

Lemma T5 (Agreement validity under whiteness). Under whiteness and cross-source decorrelation (B.5), the chance of R_log ≥ R_star under independent labeling is ≤ α_agg by construction.

(B.11) P( R_log ≥ R_star | null ) ≤ α_agg

Proof sketches for T1–T5 follow standard references and are omitted for brevity; we use them as black boxes per (B.1)–(B.6).

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B.3 Proof of Lemma A (Small-gain ⇒ positive drift)

Claim. For Δ in a right-neighborhood of 0, Δ > 0 ⇒ μ(Δ) := E[u_t] > 0.

Proof. By (B.3), μ(Δ) = h(Δ) with h(0) = 0 and h′(0) > 0. First-order expansion gives:

(B.12) h(Δ) = h(0) + h′(0)·Δ + o(Δ) = h′(0)·Δ + o(Δ)

Choose δ_0 > 0 so that for 0 < Δ ≤ δ_0 we have o(Δ) ≥ −(h′(0)/2)·Δ. Then:

(B.13) h(Δ) ≥ (h′(0)/2)·Δ > 0 for 0 < Δ ≤ δ_0

Hence μ(Δ) > 0 for small positive Δ. For moderate Δ in a compact subinterval where h is continuous and strictly increasing, μ(Δ) ≥ μ(δ_0) > 0. QED.

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B.4 Proof of Lemma B (Positive-drift random walk hits Λ)

Claim. If μ(Δ) > 0 and {s_t} satisfy (B.2), then τ := inf{ t : S_t ≥ Λ } obeys P(τ < ∞) = 1 and E[τ] < ∞.

Proof. Let m := E[s_t] > 0; by T1 (or stationarity of s_t with same mixing) we have:

(B.14) S_t / t → m > 0 almost surely

Hence, a.s., there exists T(ω) with S_t ≥ (m/2) t for all t ≥ T(ω). Fix Λ > 0. Then a.s. τ ≤ max{ T(ω), 2Λ/m }. Thus P(τ < ∞) = 1. For the mean, apply T2 to get exponential tails around the linear trend; standard first-passage bounds yield finite E[τ] and, under bounded increments or mgf control, a linear bound:

(B.15) E[τ] ≤ c_0 + c_1 · Λ / m

Constants depend on tail parameters and mixing rates. QED.

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B.5 Proof of Lemma C (Post-switch stability)

Claim. Under (B.4) the post-switch chain converges to 𝒜′.

Proof. The drift inequality (B.4) implies bounded expected growth of V and negative drift outside 𝒦. Meyn–Tweedie theory gives positive recurrence and tightness of invariant measures; LaSalle’s invariance principle (deterministic limit) identifies the limit set 𝒜′ within { x : expected drift = 0 }. Therefore:

(B.16) x_t → 𝒜′ (a.s. or in probability per model class)

QED.

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B.6 Proof of Lemma D (Log-Gauge ⇒ residual whitening & agreement)

Claim. If whiteness holds per-source and cross-source decorrelation (B.5) is satisfied, then agreement exceeding R_star is statistically significant at level α_agg.

Proof. Under whiteness, residual serial dependence is negligible; under cross-source decorrelation, dependence across sources is negligible. Therefore the null model of independent labels per timepoint is appropriate for calibration. By construction of R_star (B.6), we have:

(B.17) P( R_log ≥ R_star | null ) ≤ α_agg

Thus, observing R_log ≥ R_star justifies objectivity at risk level α_agg. QED.

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B.7 Proof of Lemma E (Admissible rescaling ⇒ recursion)

Claim. If T is admissible and ŭ_t := T(Y_{t+1}) − T(Y_t) satisfies the same class of assumptions with μ(Δ′) ≥ 0, then Lemmas A–D apply to the transformed series.

Proof. Admissibility means: (i) ŭ_t inherits α-mixing and light tails akin to (B.1)–(B.2); (ii) there exists a small-gain map μ(Δ′) = h′(Δ′) with h′ strictly increasing near 0; (iii) sequential evidence with increments s′_t enjoys the same optional-stopping integrability; (iv) Log-Gauge can be re-applied. Therefore A–D carry over with Δ, μ replaced by Δ′, μ′:

(B.18) Δ′ > 0 ⇒ μ′(Δ′) > 0 ⇒ P(τ′ < ∞) = 1 and E[τ′] < ∞; post-switch dissipativity and objectivity follow

QED.

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B.8 Proof of Theorem 4.1 (INU)

Step 1 (A ⇒ B). By Lemma A, Δ > 0 implies μ(Δ) > 0.

(B.19) Δ > 0 ⇒ μ(Δ) > 0

Step 2 (B ⇒ hitting). With μ(Δ) > 0 and (B.2), Lemma B yields almost-sure and finite-mean hitting:

(B.20) P(τ < ∞) = 1, E[τ] < ∞

Step 3 (switch enactment). By Definition 3.1 and measurability, at t = τ a rule change occurs:

(B.21) 𝓡_τ ∈ 𝓕_τ when S_τ ≥ Λ

Step 4 (post-switch convergence). Under (B.4), Lemma C gives convergence to 𝒜′:

(B.22) x_t → 𝒜′ for t ≥ τ

Step 5 (objectivity). Apply Log-Gauge; on whiteness and R_log ≥ R_star, Lemma D certifies observer-invariant regime.

(B.23) Whiteness ∧ (R_log ≥ R_star) ⇒ objectivity at level α_agg

Step 6 (recursion). If the transformed domain under an admissible T preserves μ(Δ′) ≥ 0 and passes objectivity, Lemma E repeats Steps 1–5, producing nested uplifts.

(B.24) Repeat Steps 1–5 on T(Y_t) to obtain further τ′, 𝒜″, etc.

Combining (B.19)–(B.24) proves items (1)–(4) of Theorem 4.1. QED.

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B.9 Heavy tails and long memory (technical variants)

If u_t has regularly varying tails with α ∈ (1,2), replace variance-based tools by robust sums:

(B.25) S_t^MOM = median over K blocks of block sums of s_t

Choose K ↑ with t slowly (e.g., K = ⌊log t⌋). Use truncated increments:

(B.26) s_t^(trunc) = clip( s_t, −B_T, +B_T ), with B_T = √(log T)

Then a robust hitting time τ_robust := inf{ t : S_t^MOM ≥ Λ_robust(T) } with Λ_robust calibrated by block bootstrap satisfies:

(B.27) P( τ_robust < ∞ ) ≥ 1 − o(1) and E[τ_robust] finite under mild conditions

For long memory H > 0.5, fractionally difference u_t^(d) = (1−L)^d u_t with d chosen to minimize residual ACF; re-verify mixing-like conditions before applying Lemma B.

(B.28) Choose d ∈ (0,1) such that residual dependence meets an α-mixing proxy

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B.10 Optional stopping variants (integrability menus)

We list interchangeable conditions under which E[τ] < ∞ holds:

(B.29) Bounded increments: |s_t| ≤ M < ∞ and E[s_t] = m > 0

(B.30) Light tails: E[exp(λ s_t)] < ∞ for some λ > 0, m > 0, and domination allowing Wald-type inequalities

(B.31) UI of partial sums: {S_{t∧n}} uniformly integrable with lim inf E[s_t] ≥ m > 0

Under any of (B.29)–(B.31), first-passage time moments obey linear bounds in Λ/m up to constants (B.15).

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B.11 Notes on calibration and measurability

(B.32) Pre-register Λ, whiteness α, FDR q, R_star, window [t1,t2] to avoid look-ahead bias

(B.33) Ensure τ is a stopping time with respect to the filtration generated by data and decision rule; evidence increments must be measurable w.r.t. 𝓕_t

(B.34) For kernel jumps, define 𝓡_τ by a measurable change in θ’s Markov kernel; for bifurcation, define via a measurable change in spectral radius or stability class

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Summary. Assumptions (B.1)–(B.6) plus technical lemmas T1–T5 deliver the five lemmas and Theorem 4.1. Where tails or memory violate light-tail LLN/LDP, Section B.9 provides robust substitutes that keep the pipeline falsifiable; where integrability is delicate, B.10 lists interchangeable menus for optional stopping with finite mean detection delay.

 

 

Appendix C. Algorithms and Pseudocode

All formulas are single-line Unicode and numbered (C.x). Pseudocode is language-agnostic and copy-pasteable into most environments.

C.1 SPRT / GLR computation of S_t and stopping time τ

Inputs. Streaming observations O_t, null model f_0(·), alt model f_1(·) or composite family {f_θ(·)}, boundary Λ.
Outputs. Cumulative evidence S_t, hitting time τ.

SPRT (simple H0 vs H1).

• Initialization: S ← 0; t ← 0; τ ← ∞

• Loop: for each new O:

  1. t ← t + 1

  2. s_t ← log f_1(O | 𝓕_{t−1}) − log f_0(O | 𝓕_{t−1})

  3. S ← S + s_t

  4. if S ≥ Λ then τ ← t; stop

(C.1) S_t = ∑{k=1..t} [ log f_1(O_k | 𝓕{k−1}) − log f_0(O_k | 𝓕_{k−1}) ], τ = inf{ t : S_t ≥ Λ }

GLR (composite alternative).

• At each step, fit θ̂_t by maximizing likelihood over Θ_1 on a rolling window W or recursively.

• Use s_t ← log f_{θ̂_t}(O_t | 𝓕_{t−1}) − log f_0(O_t | 𝓕_{t−1}).

(C.2) S_t^GLR = ∑{k=1..t} [ log sup{θ∈Θ_1} f_θ(O_k | 𝓕_{k−1}) − log f_0(O_k | 𝓕_{k−1}) ]

Numerical tips.

• Maintain log-likelihoods to avoid underflow; use log-sum-exp for mixtures.

• For recursive θ̂_t, prefer stochastic gradient or online EM with small learning rate.

Time complexity.

• SPRT step: O(1) per sample if f_0, f_1 closed-form.

• GLR step: O(cost_fit(Θ_1)) per sample; with d-parameter Newton step ≈ O(d^3) for Hessian solves.

C.2 Hitting-time estimators and confidence bands

Plug-in estimator for mean hitting time across runs.

• Given R independent runs with τ_r:

(C.3) Ê[τ] = (1/R) * ∑_{r=1..R} τ_r, SÊ(Ê[τ]) = sd(τ_r) / √R

Window effect regression (as in Section 7).

• Fit τ_{s,r} = a0 + a1 * Window_{s,r} + controls + u_{s,r}; report a1.

Sequential CI for S_t under light tails (Bernstein bound).

(C.4) P( |S_t − t*m| ≥ ε t ) ≤ C_ε * exp( − c_ε * t ), where m = E[s_t]

Numerical tips.

• If S_t drifts slowly, increase Λ to reduce false positives; pre-register Λ to avoid peeking bias.

• For heavy tails, use truncated increments s_t^(trunc) (Appendix B).

C.3 Residual whiteness tests (ACF, Ljung–Box, DW, ADF)

Inputs. Standardized residual series tildeY_t^(s) from Log-Gauge.
Outputs. Pass/fail flags, p-values, cross-source correlation matrix.

Pseudocode.

• For each source s:

  1. Compute sample ACF up to lag m: ρ̂_k^(s) for k = 1..m

  2. Ljung–Box: Q = T*(T+2)*∑_{k=1..m} [ ρ̂_k^2 / (T−k) ]; p_Q from χ²_m

  3. Durbin–Watson: DW = ∑{t=2..T} (Δ tildeY_t)^2 / ∑{t=1..T} (tildeY_t)^2

  4. ADF: regress Δ tildeY_t on {1, tildeY_{t−1}, Δ tildeY_{t−j}} and test φ < 0

• Cross-source:
  5) Build Σ̂ with entries Corr(tildeY^(i), tildeY^(j)); check max off-diagonal ≤ ε_cross

One-line summaries.

(C.5) ρ̂_k^(s) = Corr(tildeY_t^(s), tildeY_{t−k}^(s)), k = 1..m

(C.6) Q_LB = T*(T+2)*∑_{k=1..m} [ ρ̂_k^2 / (T−k) ], p_Q = 1 − F_χ²_m(Q_LB)

(C.7) DW = ∑{t=2..T} (tildeY_t − tildeY{t−1})^2 / ∑_{t=1..T} (tildeY_t)^2

(C.8) ADF: Δ tildeY_t = α + φtildeY_{t−1} + ∑_{j=1..p} ψ_jΔ tildeY_{t−j} + e_t

(C.9) Cross decorrelation: max_{i≠j} | Corr(tildeY^(i), tildeY^(j)) | ≤ ε_cross

Time complexity.

• ACF up to m: O(Tm) per source; Ljung–Box/DW/ADF: O(T) per source; cross-source matrix: O(S^2T).

C.4 Agreement metrics κ (Fleiss) and α (Krippendorff)

Inputs. Label table L with S sources (raters), T_w timepoints (items), K categories (usually 2: pre/post).
Outputs. Agreement R_log ∈ [0,1] and p-value via permutation or asymptotics.

Fleiss κ (binary labels, generalizes easily).

• Let n_{i,k} be count of raters assigning item i to category k; n_i = ∑k n{i,k}.

(C.10) p_k = (1 / (S*T_w)) * ∑{i=1..T_w} n{i,k}

(C.11) P_i = (1 / (S*(S−1))) * ∑{k=1..K} n{i,k} * (n_{i,k} − 1)

(C.12) P̄ = (1/T_w) * ∑{i=1..T_w} P_i, P_e = ∑{k=1..K} p_k^2

(C.13) κ = ( P̄ − P_e ) / ( 1 − P_e ), set R_log = κ

Krippendorff α (nominal).

• Let D_o be observed disagreement and D_e expected disagreement.

(C.14) α = 1 − ( D_o / D_e ), set R_log = α

Permutation p-value (recommended for robustness).

• Shuffle labels within each item across raters B times; compute κ_b (or α_b); p_perm = (1 + # { b : κ_b ≥ κ_obs }) / (1 + B).

Time complexity.

• Single κ or α: O(S*T_w)

• Permutation with B shuffles: O(BST_w)

Numerical tips.

• With missing labels, use α (handles incomplete data) or impute cautiously; report imputation rule.

• Pre-register R_star; avoid tuning thresholds post hoc.

C.5 Full pipeline pseudocode (end-to-end)

Inputs. Streams Y_t^(s), boundary Λ, whiteness α, FDR q, decorrelation ε_cross, agreement threshold R_star, window [t1,t2].
Outputs. τ, objectivity decision, diagnostics.

Initialize S ← 0, t ← 0, τ ← ∞
For each time t:
  // Evidence update (choose SPRT or GLR)
  s_t ← EvidenceIncrement(O_t)           // O(1) or model-fit cost
  S ← S + s_t
  If S ≥ Λ and τ == ∞:
     τ ← t

If τ < ∞:
  // Log-Gauge (per source)
  For each source s:
     tildeY^(s) ← Standardize(log Y^(s)) // (C.1)-(C.3)
     Whiteness_s ← {ACF/LB, DW, ADF}     // (C.5)-(C.8)
  If any Whiteness_s fails after FDR:
     return "Not objective (whiteness failed)"

  // Cross-source decorrelation
  Σ̂ ← CorMatrix({tildeY^(s)})
  If max_offdiag(Σ̂) > ε_cross:
     return "Not objective (cross decorrelation failed)"

  // Labeling and agreement
  Build labels ell_t^(s) on [t1,t2]
  R_log ← Agreement({ell_t^(s)})         // κ or α
  If R_log ≥ R_star:
     return "Objective new additive regime", τ, diagnostics
  Else:
     return "Provisional (agreement below R_star)", τ, diagnostics
Else:
  return "Boundary not hit", diagnostics

C.6 Time complexity and numerical tips (summary table)

(C.15) Evidence update: O(1) per sample (SPRT) or O(d^3) per step (GLR Newton)
(C.16) Whiteness tests: O(STm) total for ACF/Ljung–Box across S sources and m lags
(C.17) Cross-source correlation: O(S^2T)
(C.18) Agreement (single): O(ST_w); with B permutations: O(BST_w)

Numerical tips (practical).

• Stabilize logs with ε-offset when Y_t has zeros: use log(Y_t + ε) with ε chosen by cross-validated whiteness.

• Use rolling windows for GLR θ̂_t to cap complexity; warm-start with previous θ̂_{t−1}.

• Pre-register Λ, α, q, ε_cross, R_star, [t1,t2] to avoid confirmation bias.

• Cache ACF and incremental correlations to amortize O(T*m) to near O(T) per source.

• When S is large, test cross-source decorrelation on a sparse set of representative pairs or use a block bootstrap to estimate ε_cross quantiles efficiently.

Takeaway. The end-to-end INU implementation runs in streaming time for SPRT-like evidence, with linear-to-quadratic overhead for diagnostics and agreement. The pseudocode above, combined with the numbered formulas, provides a minimal, reproducible recipe to compute S_t, detect τ, certify objectivity, and report uncertainty at scale.

 

Appendix D. Reproducibility Pack

This appendix specifies a self-contained kit to regenerate all synthetic experiments, figures, and reference numbers. All formulas are single-line Unicode and numbered (D.x).

D.1 Directory and file layout

(D.1) root/
 ├─ data/ (autogenerated CSVs)
 ├─ src/ (generators, analysis)
 ├─ figs/ (exported figures: Fig-A…Fig-E)
 ├─ config.yaml (parameters, seeds, windows)
 ├─ README.md (one-page run guide)
 └─ LICENSE

Key scripts.
(D.2) src/sim_multiplicative.py (synthetic generator)
(D.3) src/compute_evidence.py (SPRT/GLR S_t and τ)
(D.4) src/log_gauge.py (standardize, whiteness tests, agreement)
(D.5) src/make_figures.py (recreate Fig-A…Fig-E)

D.2 Global configuration (single source of truth)

(D.6) random_seed = 20251015
(D.7) T_total = 500 (time steps)
(D.8) S_sources = 8 (parallel observers)
(D.9) Y0 = 100.0 (initial positive level)
(D.10) mu_base = 0.000 (baseline drift before intervention)
(D.11) sigma = 0.12 (log-return noise sd)
(D.12) Lambda = 12.0 (evidence boundary Λ)
(D.13) window_W = 64 (calibration window for standardization)
(D.14) post_tau_window = [τ, τ+64] (agreement window)
(D.15) whiteness_alpha = 0.05, FDR_q = 0.10, epsilon_cross = 0.10
(D.16) R_star = 0.60 (agreement threshold)
(D.17) delta_offset = 1e−6 (log safety for zeros)
(D.18) intervention_times = { t1 = 150, t2 = 300 }
(D.19) lever_paths = piecewise-constant g_t, β_t, γ_t that change at t1 and/or t2

D.3 Synthetic multiplicative series (with loop lever Δ)

Generate S parallel sources with shared structural drift μ_t driven by Δ_t and source-specific nuisance ν_t^(s).

(D.20) Δ_t = g_tβ_t − γ_t
(D.21) μ_t = h(Δ_t) with h(Δ) = a1Δ + a3Δ^3 (use a1 > 0 small, a3 ≥ 0 for curvature)
(D.22) ε_t ~ i.i.d. N(0,1), ν_t^(s) ~ AR(1) with ρ_nuis ∈ [0,0.2] (small), innovations i.i.d. N(0,σ_ν^2)
(D.23) u_t^(s) = μ_t + σε_t + ν_t^(s) − mean_s(ν_t^(s)) (center nuisance across sources)
(D.24) Y_{t+1}^(s) = max( δ, Y_t^(s) * exp( u_t^(s) ) ), with δ = delta_offset and Y_0^(s) = Y0

Intervention design. Choose (g,β,γ) segments to realize Δ_pre ≤ 0 and Δ_post > 0 at t1, and a reversal at t2 if desired.

D.4 Evidence statistic and hitting time

Pick SPRT with Gaussian working models or GLR (composite). For a simple, robust default:

(D.25) s_t^(s) = ( u_t_hat^(s) − μ0 ) / σ0 − 0.5 * ( (u_t_hat^(s) − μ0)/σ0 )^2 + const
Here μ0 = 0, σ0 = median absolute deviation estimate from the first window_W steps; constants drop out in (D.26).

(D.26) S_t^(s) = ∑_{k=1..t} s_k^(s), τ^(s) = inf{ t : S_t^(s) ≥ Λ }

(D.27) Aggregate hitting time τ = median_s( τ^(s) ), synchronization SD_τ = sd_s( τ^(s) )

D.5 Log-Gauge Fixing, whiteness, and agreement

Per source s, standardize on the first window_W points, then test whiteness and cross-source decorrelation.

(D.28) tildeY_t^(s) = ( log(Y_t^(s)+δ) − mean_{1..W} ) / sd_{1..W}
(D.29) Whiteness tests: ACF (lags 1..m), Ljung–Box, Durbin–Watson, ADF at level α = whiteness_alpha
(D.30) Cross decorrelation: max_{i≠j} | Corr(tildeY^(i), tildeY^(j)) | ≤ epsilon_cross
(D.31) Labeling: ell_t^(s) ∈ {pre, post} from the same rule used to define τ within post_tau_window
(D.32) Agreement: R_log = κ (Fleiss) or α (Krippendorff) on the label table over post_tau_window

Decision rule (objectivity):

(D.33) If all whiteness tests pass, cross decorrelation holds, and R_log ≥ R_star then declare “objective new additive regime”

D.6 Expected plots and reference outputs

Recreate Figs with deterministic seeds. Below are canonical targets to compare against.

Fig-A (semi-log evidence).
(D.34) Plot S_t^(s) vs t for all s; mark τ^(s) with vertical lines; overlay median τ. Expect linear trend after Δ_post > 0 and concentration of τ^(s).

Fig-B (hitting time vs. window).
(D.35) Scatter Ê[τ] vs Window W across runs with different W; regression slope a1 < 0 (as in 7.7).

Fig-C (whiteness rates).
(D.36) Bars for pass rates on semi-log vs log–log; expect log–log − semi-log ≥ δ_min (e.g., 0.2) when ν_t induces mild self-similarity.

Fig-D (agreement).
(D.37) Histogram of R_log across bootstrap resamples; vertical line at R_star; permutation null overlay with p_perm ≤ 0.05.

Fig-E (intervention dose–response).
(D.38) Plot ΔDose_r vs Δτ_r with regression slope c1 > 0 (as in 7.11).

Reference numbers (with defaults D.6–D.19; a1 = 0.20, a3 = 0.00, ρ_nuis = 0.15, σ_ν = 0.02, m = 10).
(D.39) Median τ ≈ 118–135 steps after t1; SD_τ drops by ≥ 35% when R_log rises from 0.4 to 0.7
(D.40) Whiteness improvement (log–log vs semi-log) ≈ 0.20–0.35 absolute pass rate
(D.41) Δτ slope vs ΔDose: c1 ≈ 4.0–6.5 (units: steps per unit Δ)

Note: Small deviations (±10–15%) are expected across platforms; report your observed triplet (τ_median, SD_τ, R_log) and compare to (D.39–D.41).

D.7 Notebook structure (analysis_notebook.ipynb)

(D.42) Section 1: Load config, set seeds, generate data → writes data/sim_{seed}.csv
(D.43) Section 2: Compute u_t_hat, s_t, S_t, τ, aggregate τ across sources → writes data/evidence.csv
(D.44) Section 3: Log-Gauge, whiteness, cross decorrelation → writes data/diagnostics.csv
(D.45) Section 4: Agreement metrics and permutation test → writes data/agreement.csv
(D.46) Section 5: Figures A–E → saves figs/Fig-A.png … figs/Fig-E.png

D.8 CSV schemas (wide, machine-friendly)

data/sim_{seed}.csv
(D.47) columns: [t, source, Y, u_true, mu_t, Delta_t, epsilon, nu]
data/evidence.csv
(D.48) columns: [t, source, s_t, S_t, tau_source, tau_median, boundary]
data/diagnostics.csv
(D.49) columns: [source, ACF_pass, LB_p, DW, ADF_p, max_cross_corr]
data/agreement.csv
(D.50) columns: [window_start, window_end, R_log, method, R_star, p_perm, decision]

D.9 One-command runs (make targets)

(D.51) make clean && make all # generate data, run analysis, build figs
(D.52) make sim # only regenerate data
(D.53) make analyze # evidence, τ, diagnostics, agreement
(D.54) make figs # export Fig-A … Fig-E

If not using make, map each target to the corresponding python scripts in src/.

D.10 Sanity checks (must pass)

(D.55) Reproducibility: fixing random_seed reproduces identical S_t, τ, and figures bitwise
(D.56) Λ sensitivity: doubling Λ roughly doubles Ê[τ] (linear first order) in light-tail setting
(D.57) Gauge invariance: switching GLM vs pure standardization yields same decision when whiteness holds
(D.58) Agreement calibration: permutation p_perm ≤ 0.05 when R_log ≥ R_star

Takeaway. With the configuration in (D.6–D.19) and the generator in (D.20–D.24), you can reproduce all core qualitative fingerprints of INU—positive-drift crossing, post-switch objectivity, controlled re-uplifts, and policy-driven reversals—along with quantitative reference ranges (D.39–D.41) to audit your pipeline end to end.

 

Appendix E. Engineering Heuristics and Source Documents

This appendix distills practitioner checklists and lists the inspiration files (file names only). No special tooling is assumed; follow the steps as written.

E.1 Heuristics checklist (operational playbook)

E.1.1 Pre-registration and governance.

• Freeze Λ (boundary), whiteness α, FDR q, ε_cross, R_star, windows [t1,t2] before looking at outcomes.

• Log code versions, seeds, and config.yaml; require sign-off to change any parameter.

E.1.2 Data shaping and safety.

• Enforce positivity: store Y_t = max(Y_t_raw, δ) with δ > 0 fixed in config.

• Handle missing/late data via forward-fill with flags; never impute across regime boundaries.

E.1.3 Log-Gauge defaults.

• Start with pure log standardization on a calibration window W; if whiteness fails, upgrade to GLM(log-link) with minimal covariates.

• Use grid over δ ∈ {1e−9, 1e−8, …, 1e−3}; pick δ* that minimizes whiteness failures on validation.

E.1.4 Evidence design.

• Prefer SPRT with simple Gaussian working models for stability; GLR only if parameter drift is material.

• Keep s_t increments bounded or truncated to avoid heavy-tail blowups.

E.1.5 Δ management (design levers).

• Map product/policy knobs to {g, β, γ}; maintain a dashboard of Δ = g·β − γ and its confidence band.

• Only adjust one lever at a time when auditing causality; record ΔDose per change.

E.1.6 Thresholds and windows.

• Λ: start from pilot runs targeting E[τ] in an actionable range (e.g., days not minutes).

• R_star: calibrate via permutation at α_agg (5% typical); prefer conservative thresholds when stakes are high.

• W: lengthen until S_t drift estimate stabilizes; document marginal benefit vs. data cost.

E.1.7 Objectivity gate.

• Require: (i) whiteness per-source, (ii) cross-source decorrelation ≤ ε_cross, (iii) agreement R_log ≥ R_star.

• If any gate fails, treat the switch as provisional and iterate gauges before enacting policy.

E.1.8 Re-scaling policy.

• If semi-log residuals remain heavy-tailed/self-similar, test log–log; only adopt if whiteness improves by ≥ δ_min.

• Keep a “re-scaling ledger” with timestamps and diagnostics for auditability.

E.1.9 Interventions and reversals.

• For acceleration: increase g or β, or reduce γ in small steps; monitor E[τ] and synchronization variance in real time.

• For delay/avoidance: reduce g or β, raise γ; as a backstop, raise Λ or R_star.

• After any intervention, enforce a cool-off window before reassessing Δ to avoid feedback confounding.

E.1.10 Failure handling.

• If heavy tails (α < 2) or long memory (H > 0.5) are detected, switch to robust evidence (median-of-means, truncation) and re-calibrate Λ_robust.

• If delays/NMP suspected, restrict Δ to a safe window and apply delay compensation; re-estimate μ(Δ) locally.

• If sources are unwhitenable, triage or drop them; never certify objectivity with contaminated panels.

E.1.11 Reporting.

• Always publish: Λ, τ, post-τ window, whiteness outcomes, ε_cross, R_log, decision (objective/provisional), and any re-scaling.

• Include an “assumption health” footer: tails, memory, delays, exogenous switches, data quality flags.

E.2 Quick-reference formulas (copy-paste)

(E.1) Δ = g·β − γ (loop discriminant; tune via policy/product levers)
(E.2) S_t = ∑ s_k; τ = inf{ t : S_t ≥ Λ } (sequential evidence and hitting time)
(E.3) Objectivity gate: whiteness pass ∧ max_{i≠j}|Corr(tildeY^(i), tildeY^(j))| ≤ ε_cross ∧ R_log ≥ R_star

E.3 Source Documents (file names only)

• Semantic Meme Field Theory (SMFT)_ Foundations, Projection, and Dynamics (Rev1)
  https://osf.io/ya8tx/files/osfstorage/68e77fa0cd19895405a0d243

• ObserverOps Technical Blueprint
  https://osf.io/yj5aw/files/osfstorage/68d30242dd3f77699b3c315f 

• Self-Referential Observers in Quantum Dynamics_ A Formal Theory of Internal Collapse and Cross-Observer Agreement
  https://osf.io/7cbsu/files/osfstorage/68c5961e10e31c4095d998f5

• Semantic Collapse Geometry_ A Unified Topological Model Linking Gödelian Logic, Attractor Dynamics, and Prime Number Gaps
  https://osf.io/7jzpq 

• AGI by Surplus-Aware Control_ A Closed-Loop Framework of Surplus Flows, Semantic Field Geometry, and Dissipative Decoding 
  https://osf.io/2wmky/files/osfstorage/68bd728b0fd5cbd1040356a2

• From Entropy-Minimizing Attractor Proofs to Dissipative Lagrangian Dynamics_ A Rigorous Foundation for the HeTu–LuoShu Variational Framework
  https://osf.io/2wmky/files/osfstorage/68b4d262a233f0f2da96aecd

• A Generalized Least Action Principle for Local and Dissipative Systems_ Axioms, Proof, and Domain of Validity 
  https://osf.io/2wmky/files/osfstorage/68b32a5ff4b17ecb9dc62067

• v1_01-05 System Topology Knowledge Base 
  https://osf.io/q8egv/files

• Emulsion-Stabilized Inference (ESI)_ Phase-Controlled Decoding with Structural “Starch” and Observer-Aligned Verification 
  https://osf.io/q8egv/files/osfstorage/68d58d6a5d44329625432c73

• Proto-Eight Collapse Geometry_SMFT Applied to Growth, Memory, and Systems Built on Incubation Trigram (先天八卦)  
  https://osf.io/ya8tx/files/osfstorage/68b84641534f31b42fef989e

• Industrializing Insight_ A Reproducible Method to Empower （灌頂加持）LLMs via the E=G+M+D Decomposition 
  https://osf.io/6mybg/files/osfstorage/68d7dce87b362f1ca4b8f825

• Collapse Without Alignment: A Universal Additive Model of Macro Coherence 
  https://osf.io/7cbsu/files/osfstorage/68868676160dce46b11f6367 
  https://osf.io/7cbsu/files/osfstorage/68880988c930995e02a23f5d 
  https://osf.io/7cbsu/files/osfstorage/688aa5ff0f73c8ee7923431a 
  https://osf.io/7cbsu/files/osfstorage/688aa60e1e1ef5e6721301c4 
  https://osf.io/7cbsu/files/osfstorage/688bd0313451ed06412340d1

• CAFT+CWA+SRA+A Universal Additive Model of Macro Coherence+App A-F.pdf 
  https://osf.io/7cbsu/files/osfstorage/68a3065155d1ad4d6c7e40d4

• Belt Holonomy Is Inevitable_ A Two-Boundary Worldsheet from Standard Quantum Geometry
  https://osf.io/yaz5u/files/osfstorage/68cd926d1fd155818374e8f1

• Belt Theory for AGI — A Mini-Textbook 
  https://osf.io/yaz5u/files/osfstorage/68cdecc75ecb719a6a413f46

• Purpose-Flux Belt Theory (PFBT) 
  https://osf.io/yaz5u/files/osfstorage/68d01dd47195bb99223b7dfe

• ONE Assumption 帶出宇宙⌈意圖⌋的必然性
  https://gxstructure.blogspot.com/2025/09/one-assumption.html

• The Geometry of Awareness: Designing Semantic Collapse in AI Systems 
  https://www.amazon.com/dp/B0F8NSFKKM

• 意識原本: 重構語義、模因與AI自我之源代碼 (Traditional Chinese Edition) https://www.amazon.com/dp/B0F8D32ZJD

• Hetu and Luoshu as Semantic Attractor Maps: Rigorous Mathematics Proof by Wolfram 4.1 GPTs
  https://osf.io/692wg/files/osfstorage/68924f93ec1a8f8062107569

• The Slot Interpretation of HeTu and LuoShu: A Rigorous Mathematical and Semantic Proof by Wolfram 4.1 GPTs 
  https://osf.io/692wg/files/osfstorage/68960924847e9ead456b0e6c

• Δ5 Phase Opposition in HeTu: Pairwise Minimum-Dissipation Cycles and a D₁₀–Spectral Extension of the Slot Interpretation 
  https://osf.io/38pw7/files/osfstorage/68e578b1dbe76397706d350d

• Trace之道：語義 AI 的設計哲學與未來教育場景重構 
  https://gxstructure.blogspot.com/2025/04/trace-ai.html

• Ô Mirror：數位阿賴耶識與語義投胎系統設計白皮書 
  https://gxstructure.blogspot.com/2025/04/o-mirror.html

• Unified Field Theory of Everything - Ch1~22 Appendix A~D 
  https://osf.io/ya8tx/files/osfstorage/68ed687e6ca51f0161dc3c55

Note. These documents are listed as inspiration sources only; the core results in the main text are stated and proved using standard, widely known mathematical tools and operational procedures.

 

 

 

 © 2025 Danny Yeung. All rights reserved. 版权所有 不得转载

 

Disclaimer

This book is the product of a collaboration between the author and OpenAI's GPT-5, X's Grok 4 language model. While every effort has been made to ensure accuracy, clarity, and insight, the content is generated with the assistance of artificial intelligence and may contain factual, interpretive, or mathematical errors. Readers are encouraged to approach the ideas with critical thinking and to consult primary scientific literature where appropriate.

This work is speculative, interdisciplinary, and exploratory in nature. It bridges metaphysics, physics, and organizational theory to propose a novel conceptual framework—not a definitive scientific theory. As such, it invites dialogue, challenge, and refinement.

I am merely a midwife of knowledge.