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.
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 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
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)**.
When does operational distinguishability contain sufficient information to identify physical structure? We address this question through a resource-relative framework of Operational Informational Distinguishability (OID), designed to investigate the mathematical consistency of identifiability across quantum states, information geometry, and physical processes. An operational observer is characterized by (\mathcal{D}=(\mathcal{A},\mathcal{M},\mathcal{R},\delta)), whose labelled outcome distributions define an operational signature and induce equivalence classes of physically indistinguishable realizations. We establish an exact reconstruction criterion: a proposed geometric assignment factors uniquely through the operational signature if and only if it is constant on these equivalence classes. This provides a necessary-and-sufficient condition for operational reconstruction, while separating identifiability from the additional requirements of local non-degeneracy and stability. For faithful finite-dimensional quantum-state manifolds, we establish the measurement–information-geometric hierarchy[g^{(\mathcal{M})}\preceq g^{\mathrm{SLD}}\preceq g^{\mathrm{BKM}}relating measurement-restricted Fisher information, the symmetric logarithmic derivative metric, and the Bogoliubov–Kubo–Mori metric. We clarify the directional attainability of the SLD bound and its distinction from simultaneous multiparameter attainability. As an explicit algebraic result, we characterize the operator span generated by diagonal phase modulation of cyclic-shift eigenstate projectors in the one-excitation sector: it is precisely the constant-diagonal Hermitian subspace, of dimension (N^{2}-N+1), leaving (N-1) independent population directions unresolved. Adjoining site-basis projectors restores informational completeness. This idealized construction is connected to experimentally realized entangled measurements of (W) states. We further extend operational distinguishability to quantum processes through tester-restricted signatures and divergences, explicitly separating signalling from intervention-defined causal influence. Finally, we introduce accumulated information-geometric path length as a candidate operational clock parameter, without identifying it with physical time. By relating these constructions to holographic information-distance bounds, bulk entanglement, conditional mutual information and recovery, and recent approaches to emergent geometry and informational time, we formulate a central testable question: can a common resource-relative distinguishability structure remain identifiable, stable, and compositionally consistent across states, processes, geometry, temporal ordering, and causality? The framework makes both constructive reconstruction and mathematically explicit obstructions possible, providing a falsifiable methodological program rather than presupposing the emergence or unification of physical structure.
📌 Overview This work develops a constructive framework in which arithmetic constraint systems serve as a non-dynamical substrate for information transmission. The central idea is to encode discrete messages into parameterised families of Diophantine conditions whose globally consistent solution sets exhibit structured projection asymmetries. These asymmetries can be accessed locally through partial observations and used to recover the encoded information. 🔑 Key Contributions Arithmetic signalling modelA formal framework in which messages are encoded into constraint families and recovered via projection. Global consistency principle (Arithmetic Realisability)A structural assumption enabling extension from local observations to globally consistent solutions. Decoding via admissibility structureA deterministic decoding functional based on fibre admissibility. Amplification mechanismIterated constraint composition yields asymptotic saturation of signalling capacity. Spacetime interpretationA reformulation of signalling as constraint-based inference, independent of propagation and compatible with relativistic causality. 🧠 Conceptual Contribution The paper proposes a shift in perspective: Information transfer need not be realised through dynamical propagation, but may instead arise from inference over globally constrained structure. Under this view, the apparent prohibition of faster-than-light communication reflects a restriction on propagation-based mechanisms rather than a fundamental limitation on information dependence. 📊 Numerical Illustration The paper includes a numerical exploration demonstrating: separation of admissible projections, convergence of decoding accuracy, rapid amplification under iteration. These results provide intuition for the geometric structure underlying the theoretical framework. 📁 Contents ftl.tex — LaTeX source references.bib — bibliography ftl.pdf — compiled manuscript 🏷️ Keywords arithmetic constraints, Diophantine equations, constraint satisfaction, nonlocal dependence, signalling theory, causality, block universe, information theory 📜 License This work is released under the CC BY 4.0 license.