This paper establishes global laws governing operational signalling in measurement-dependent models under adaptive multi-round interaction. Unlike earlier analyses based on fixed profiles or one-step behaviour, the present work studies the full dynamical evolution of the underlying probability distribution. Previous papers in the series showed that signalling is not identifiable from finite data, that repeated interaction can nevertheless generate operational distinguishability, and that one-step amplification is governed by a precise structural coefficient. The adaptive setting introduced a Bellman-type recursion and revealed a persistent multiscale regime associated with harmonic-comparable profiles. However, a global theory of the resulting dynamics was not previously available. The present paper provides that global theory. It establishes a universal exponential control law showing that multi-round signalling capacity is governed by the cumulative one-step gain. It proves a complete classification of adaptive trajectories into three regimes: head-heavy dynamics with microscopic peeling, tail-heavy dynamics with macroscopic peeling, and a critical harmonic regime with logarithmic growth. It further shows that strong universality fails at the level of profile geometry: there is no finite family of asymptotic profile classes capturing all non-collapsing trajectories. These results demonstrate a fundamental dichotomy. Adaptive signalling is universal at the level of control behaviour, but not at the level of profile dynamics. While all systems obey the same exponential law, the underlying distributions exhibit a continuum of persistent geometries. This completes the operational signalling programme developed across six works, providing a unified description of adaptive dynamics and clarifying the limits of universality in measurement-dependent systems.
This paper establishes global laws governing operational signalling in measurement-dependent models under adaptive multi-round interaction. Unlike earlier analyses based on fixed profiles or one-step behaviour, the present work studies the full dynamical evolution of the underlying probability distribution. Previous papers in the series showed that signalling is not identifiable from finite data, that repeated interaction can nevertheless generate operational distinguishability, and that one-step amplification is governed by a precise structural coefficient. The adaptive setting introduced a Bellman-type recursion and revealed a persistent multiscale regime associated with harmonic-comparable profiles. However, a global theory of the resulting dynamics was not previously available. The present paper provides that global theory. It establishes a universal exponential control law showing that multi-round signalling capacity is governed by the cumulative one-step gain. It proves a complete classification of adaptive trajectories into three regimes: head-heavy dynamics with microscopic peeling, tail-heavy dynamics with macroscopic peeling, and a critical harmonic regime with logarithmic growth. It further shows that strong universality fails at the level of profile geometry: there is no finite family of asymptotic profile classes capturing all non-collapsing trajectories. These results demonstrate a fundamental dichotomy. Adaptive signalling is universal at the level of control behaviour, but not at the level of profile dynamics. While all systems obey the same exponential law, the underlying distributions exhibit a continuum of persistent geometries. This completes the operational signalling programme developed across six works, providing a unified description of adaptive dynamics and clarifying the limits of universality in measurement-dependent systems.
This paper studies operational signalling in measurement-dependent models under adaptive multi-round interaction. Unlike fixed-profile analyses, the underlying probability distribution is allowed to evolve dynamically across rounds, leading to fundamentally different behaviour. Previous work established the limitations of single-shot inference, the emergence of signalling under repeated interaction, and the structural theory governing one-step amplification. In particular, a complete one-step theory for the k-ary setting identified optimal signalling coefficients and extremal profile structures. However, these results assume that the probability profile remains fixed across rounds. The present work addresses the adaptive setting in which the profile evolves in response to interaction. Key results Persistence of signallingSignalling remains possible under adaptive interaction even when one-step coefficients are small. Level cascade mechanismA structural process is identified in which probability mass concentrates across rounds, driving amplification. Multiscale dynamicsMulti-round signalling capacity depends on the evolution of the full profile, not just scalar coefficients. Control formulationThe problem is formulated as a controlled process on the probability simplex, revealing a Bellman-type recursive structure. Significance These results show that adaptive evolution fundamentally alters the nature of operational signalling. Static optimality does not determine dynamic capacity; the ability to reshape the underlying distribution across rounds introduces new mechanisms that are invisible in one-step analyses. This paper extends a broader research programme on operational signalling under measurement dependence, providing the first systematic treatment of adaptive profile dynamics. Keywords: measurement dependence, operational signalling, adaptive interaction, Bellman recursion, probability simplex, multiscale dynamics
Description This paper develops a non-degenerate k-ary extension of a binary Bellman-type model for operational signalling under measurement dependence. In earlier work, operational signalling was shown to emerge under repeated interaction even when no single round exhibits identifiable control. However, extending this framework beyond the binary setting introduces structural degeneracies: naive k-ary generalisations collapse or overestimate the signalling coefficient. This paper resolves these issues by introducing a block-overlap principle, which leads to a corrected one-step signalling coefficient and a complete structural characterisation of the model. The coefficient admits a max-linear formulation over block scales, revealing an underlying piecewise-linear geometry and enabling exact analysis. Main results Exact one-step coefficientA corrected k-ary signalling coefficient Γₖ(p) defined via block overlap, admitting a max-linear representation. Sharp bounds 1Hk−1≤Γk(p)≤1,\frac{1}{H_{k-1}} \le \Gamma_k(p) \le 1,Hk−11≤Γk(p)≤1, where Hk−1H_{k-1}Hk−1 is the harmonic number. Harmonic optimalityThe harmonic profile pr∝1/rp_r \propto 1/rpr∝1/r uniquely minimises the signalling coefficient. Geometric structureΓₖ(p) is piecewise-linear and determined by block-scale extrema. Multi-round amplification (conditional)Under structural assumptions, signalling amplifies according to an(p)=1−(1−Γk(p))n.a_n(p) = 1 - (1 - \Gamma_k(p))^n.an(p)=1−(1−Γk(p))n. Significance This work establishes the correct k-ary formulation of the operational signalling model and identifies the harmonic profile as the critical extremal structure. It provides the structural foundation for subsequent developments in the series, including adaptive dynamics, persistence of multiscale structure, and global control laws. Keywords operational signalling; measurement dependence; Bellman recursion; k-ary models; harmonic distribution; max-linear structure; information amplification; multiscale systems Notes This paper forms part of a series on operational signalling under measurement dependence. See accompanying repositories for related works in the series.
This work studies repeated measurement-dependent models under the constraint that all one-round marginals coincide across messages. In the single-shot setting, this condition precludes signalling detectable at the level of marginals. We show that this limitation does not persist under repeated interaction: adaptive strategies enable operational signalling that is encoded entirely in correlations. We construct explicit two-round protocols in asymmetric binary shared-latent models that produce distinguishable joint distributions while preserving identical marginals at every round. This establishes the existence of purely correlational signalling and demonstrates a strict separation between adaptive and non-adaptive strategies. We introduce the one-probe-and-lock protocol, which separates a branching step from deterministic amplification across subsequent rounds. In the canonical symmetric model, this yields the scalingC⁽ⁿ⁾ = 1 − 2¹⁻ⁿ,showing that distinguishability increases monotonically with the number of rounds and approaches unity. We solve the three-round problem exactly in the canonical model, proving that the optimal value is 3/4. For general asymmetric models, we develop a complete structural reduction of the three-round problem, reducing the space of adaptive strategies to a finite family of branch continuation laws and admissible orientation patterns. In the monotone regime, we show that all nontrivial extremisers lie in a single opposite-crossing configuration and derive a closed-form expression for the branch-lock protocol. We then establish the sharp universal boundsup C⁽³⁾ = 3/4for all binary shared-latent models. These results demonstrate that repeated measurement-dependent interactions support a structured form of operational signalling that is invisible at the marginal level. The analysis reduces the multi-round problem to a minimal analytic core and provides a complete characterisation of the three-round setting.
# Operational Signalling is Not Identifiable from Finite Measurement-Dependent Data **Author:** Bob Jefferson **ORCID:** https://orcid.org/0009-0003-1460-055X --- ## 📌 Overview This work develops a finite, explicit framework for analysing **operational signalling** in measurement-dependent latent-variable models. It establishes a sharp distinction between: - **Formal signalling** — dependence of observed distributions on remote settings - **Operational signalling** — the existence of a usable communication channel The central result is that these notions do not coincide: apparent signalling in observational data does not determine whether communication is possible. --- ## ✨ Main Results ### 1. Non-identifiability Operational signalling **cannot be inferred from finite observational data alone**. The same family of conditional distributions {μxy} can be realized both:- with zero signalling capacity, and - with nonzero signalling capacity --- ### 2. Exact capacity law The maximal operational signalling strength satisfies: Cymax(Γ)=ΓDy where:- Dy = observable distinguishability (total-variation diameter) - Γ = control parameter (message-to-setting influence) This factorises signalling into:> **observable distinguishability × latent control** --- ### 3. Binary structural constraint In a restricted symmetric binary subclass: Γ+Δ≤1 where:- Δ = latent measurement dependence This gives the operational bound: Cy≤(1−Δ)Dy --- ### 4. Higher-dimensional limitation An explicit counterexample shows: - the binary tradeoff is **not universal** - for larger setting spaces, control and measurement dependence can both be maximal --- ## 🧠 Significance These results show that: - Observational data alone cannot determine whether a communication channel exists - Operational signalling depends on **latent causal structure**, not just observed correlations - Even so, structural constraints can emerge in restricted settings This has implications for: - Bell-type scenarios - measurement dependence - quantum communication and cryptography - the operational interpretation of signalling --- ## 📂 Files - **OperationalSignalling.pdf** — final paper - **OperationalSignalling.tex** — LaTeX source - **references.bib** — bibliography --- ## 📜 License This work is licensed under **Creative Commons Attribution 4.0 (CC BY 4.0)**.
The leading asymptotic growth of Lanczos coefficients is controlled by spectral tails and furnishes a coarse classification of Krylov dynamics. We show that the \textit{global topology} of the spectral measure, specifically the number of connected components, the gap structure, and the behaviour at gap-closing transitions, encodes a finer hierarchy of dynamical invariants invisible to tail-based arguments. Using the Riemann-Hilbert formulation of orthogonal polynomials and Deift-Zhou steepest descent, we recover the Freud growth laws bn∼n1/β for single-cut measures and determine their sub-leading corrections from endpoint data. Gapped spectra produce quasiperiodic Lanczos oscillations at a frequency fixed by the filling fraction of the spectral bands alone, and hence predictable from the band edges. We verify this in the SSH chain and its next-nearest-neighbour deformation. At a gap-closing transition the oscillation amplitude is governed by the Hastings-McLeod solution of Painlevé II, decaying as n−1/3 at criticality and interpolating between the gapped and merged phases, so that the topology change of the spectral curve is realised as a Krylov phase transition. We also demonstrate that, while in the conformal limit of SYK the operator scaling dimension is invisible in the leading rate α=πT, it can be extracted from the subleading offset b0=πT(Δ−21). These results establish a refined notion of universality in operator growth, classified by spectral topology rather than spectral tails alone.
We develop a mathematically rigorous and experimentally testable framework for the classification of physical regimes based on the modular structure of quantum states. The central object of the theory is the modular generator K = −log ρ, which provides a unified language for spectral, geometric, and information-theoretic diagnostics. We introduce a universal instrument panel (MSRO: Modular–Spectral RG Observables) that integrates spectral quantiles, commutator-based response functions, partition-based locality diagnostics, and information-geometric structures. A key requirement of the framework is strong portability: the same diagnostic protocol applies across multiple domains—including open quantum systems, information geometry, entanglement-based models, and RG-like flows—without domain-specific retuning. The main result of the work is the derivation of a universal and experimentally testable scaling law for the modular response signal: ν(λ) ∼ 1 log λ, which characterizes critical regimes in a wide class of quantum systems. We prove that this scaling emerges from spectral asymptotics of the modular operator and establish a strict equivalence between spectral, geometric, and information-theoretic descriptions of criticality: ν ↔ k(q) ↔ I(λ) ↔ K ↔ R ↔ Δ. The framework is supported by functional analytic results (including Fr´echet differentiability of log ρ), operator inequalities (Golden–Thompson, monotonicity of relative entropy), and stability estimates. We further provide a concrete experimental protocol for quantum simulators, including statistical error bounds and Fisher information constraints, making the prediction directly testable. Beyond regime classification, the results suggest a new perspective on criticality as a geometric phenomenon associated with vanishing modular curvature. The approach opens pathways toward large-N limits, quantum field extensions, and the discovery of new classes of non-standard critical behavior.