When two quantum-entangled agents coordinate their actions, a third-party observer — one with no access to the quantum system — can experience environmental effects indistinguishable from classical communication, even though no information was transmitted. The Limited Communication Theorem (Ericksen 2026) proved this formally. This paper asks the next question: does that phenomenon have deeper mathematical structure that standard real-valued measures miss? We propose a complex-valued extension of environmental divergence: Δ_C = Δ_TV × e^{iφ}, where the phase encodes the quantum fill fraction — how far quantum coordination has closed the gap between classical local behavior and full classical communication. Four computational tests show that this construction traces structured, physics-determined curves in the complex plane, organizes coordination scenarios by physically meaningful fill fraction, produces interpretable real-time trajectories under noise and resource switching, and reveals that three independent construction pathways converge on the same geometric object: the square-root embedding of the probability simplex into CP¹, the complex projective line, equipped with the Fisher-Rao metric. We prove that the real-valued divergence and its complex magnitude are formal LOSR monotones — they can only decrease under local degradation, never increase. The phase angle and imaginary component behave as diagnostic indicators that are non-increasing under local degradation, isolating the portion of the observer's experience that is structurally indistinguishable from communication. The central result: two quantum strategies can produce identical outcome statistics under every classical measurement while being fully orthogonal in the complex Fubini-Study metric. When the environment evolves, this hidden separation becomes a measurable 13.1% total variation divergence — a predictive failure that no real-valued model can anticipate or resolve. The complex-projective representation is not an arbitrary formalism — it is the geometry required for predictive consistency when quantum coordination enters multi-outcome environments. This paper is a companion to "The Limited Communication Theorem" (Ericksen 2026, Zenodo DOI: 10.5281/zenodo.18868863). The core results of that paper are independent of this extension. If the results here hold under further scrutiny, the limited communication phenomenon is not merely a statistical anomaly — it carries its own irreducible geometric structure in the macroscopic world.
When two quantum-entangled agents coordinate their actions, a third-party observer — one with no access to the quantum system — can experience environmental effects indistinguishable from classical communication, even though no information was transmitted. The Limited Communication Theorem (Ericksen 2026) proved this formally. This paper asks the next question: does that phenomenon have deeper mathematical structure that standard real-valued measures miss? We propose a complex-valued extension of environmental divergence: Δ_C = Δ_TV × e^{iφ}, where the phase encodes the quantum fill fraction — how far quantum coordination has closed the gap between classical local behavior and full classical communication. Four computational tests show that this construction traces structured, physics-determined curves in the complex plane, organizes coordination scenarios by physically meaningful fill fraction, produces interpretable real-time trajectories under noise and resource switching, and reveals that three independent construction pathways converge on the same geometric object: the square-root embedding of the probability simplex into CP¹, the complex projective line, equipped with the Fisher-Rao metric. We prove that the real-valued divergence and its complex magnitude are formal LOSR monotones — they can only decrease under local degradation, never increase. The phase angle and imaginary component behave as diagnostic indicators that are non-increasing under local degradation, isolating the portion of the observer's experience that is structurally indistinguishable from communication. The central result: two quantum strategies can produce identical outcome statistics under every classical measurement while being fully orthogonal in the complex Fubini-Study metric. When the environment evolves, this hidden separation becomes a measurable 13.1% total variation divergence — a predictive failure that no real-valued model can anticipate or resolve. The complex-projective representation is not an arbitrary formalism — it is the geometry required for predictive consistency when quantum coordination enters multi-outcome environments. This paper is a companion to "The Limited Communication Theorem" (Ericksen 2026, Zenodo DOI: 10.5281/zenodo.18868863). The core results of that paper are independent of this extension. If the results here hold under further scrutiny, the limited communication phenomenon is not merely a statistical anomaly — it carries its own irreducible geometric structure in the macroscopic world.
This paper asks a question the quantum information literature has not addressed: what happens to the rest of the world when quantum coordination activates? When two parties coordinate using shared entanglement, the standard analysis stops at their joint measurement outcomes. This paper goes further — it formalizes what a third-party observer, one completely excluded from the quantum state, experiences in the downstream environment. The answer: environmental consequences that no classical physics can explain, and in the strongest case, consequences that are identical to what the observer would experience if the parties had secretly communicated. The paper introduces the excluded observer interface model, defines environmental divergence as an operational measure, and proves a leakage-performance separation theorem: quantum coordination simultaneously exceeds the classical performance ceiling while leaking zero information about local conditions to the excluded observer — breaking the classical Pareto frontier between coordination performance and operational security. Demonstrated across the CHSH game (Δ_TV = 0.104, distinguishable from communication) and the Mermin-Peres Magic Square game (Δ_TV = 0.111, indistinguishable from communication, ε = 0). This is Paper 1 in a four-paper series: Paper 1 (this paper) — Existence. Proves the excluded observer's environment diverges from classical baselines. Paper 2 — Structure. Extends the framework into complex-valued information geometry, demonstrating that the excluded observer's information space acquires a CP¹ structure and that complex phase tracking becomes operationally necessary for predictive consistency in multi-outcome environments. Zenodo DOI 10.5281/zenodo.18944106. Paper 3 — Detection. Shifts the observer partition inward, deriving a sequential detection architecture for the coordinating party to detect marginal bias using only local statistics. Proves the protocol is information-theoretically optimal (saturates the Shannon capacity bound). Zenodo DOI 10.5281/zenodo.19056735. Paper 4 — Action. Derives the organizational decision-theoretic consequences: optimal hierarchy depth as a computable function of adversary coordination quality, decision window, and stakes magnitude. The result generalizes beyond quantum coordination to any scenario with external time pressure, communication friction, and unforeseen disruption. Zenodo DOI 10.5281/zenodo.19057065.
Paper 3 in the Limited Communication series. The first two papers asked what an outside observer sees when quantum coordination affects their environment. This paper asks the opposite question: what does the coordinating party themselves see? If a physical mechanism exists that shifts the measurement statistics of an entangled partner — even slightly — the partner can detect that shift over time using sequential statistical testing, without any classical communication channel. This paper develops the complete detection architecture: how to detect the shift, how long it takes, how confident you can be, and how to verify pre-agreed signals using a codebook established before separation. The paper does not claim this capability exists today. It derives what follows if it becomes possible, using the mathematical framework established in Papers 1 and 2. The theory is built before the technology, so the formal tools are ready if the physics arrives. Technically: derives the detection-theoretic consequences when a coordinating party observes their own marginal statistics under a hypothesized non-linear bias mechanism. Develops sequential hypothesis testing (SPRT) architecture for detecting orthogonal divergence — environmental divergence at the entangled partner's location — without any classical communication channel. Proves the detection protocol saturates the Shannon capacity bound. Positions the conditional within a trilemma relative to Gisin's (1990) theorem on non-linear signaling. Extends the LCT framework inward, from the excluded observer to the coordinating party.
We explore whether quantum entanglement might be detectable through information-theoretic coordination patterns in measurement outcomes alone, without requiringquantum states, density matrices, or Hilbert space structure. Using Local DivergenceIntensity (LDI), a universal measure of symbolic sequence decorrelation [6], we ana-lyze three types of simulated quantum measurements: (1) Bell states (simulated per-fect correlation), (2) product states (simulated independence), and (3) quantum walks(simulated structured dynamics).In simplified probabilistic simulations that approximate expected quantum mea-surement statistics, we observe 95% separation between correlated and independentbitstrings (∆Φ = 0.953) with 2% added noise. The framework operates on discrete bi-nary sequences and reveals different scale-dependence patterns for different correlationtypes.This conceptual study suggests LDI analysis of measurement sequences could po-tentially complement traditional quantum information measures, but experimental val-idation on real quantum hardware is required before any claims about detecting actualquantum entanglement can be made.
Quantum mechanics is the study of uncertainty. But what does uncertainty mean within the structure of physical reality? From Newton onward, mathematical physics has largely approached motion by specifying a state, identifying the laws governing its evolution, and determining the subsequent behavior of the system. Fourier extended this program to heat, Maxwell to electromagnetic fields, Navier and Stokes to fluids, Hamilton and Lagrange to variational and canonical mechanics, Einstein to spacetime, and quantum mechanics to the evolution and measurement of the quantum state. Across these theories, increasingly powerful mathematics has described what a physical system may do and how its state may evolve. This work asks a logically prior question: Does everything that is mathematically accessible participate equally in physical realization? We propose that it does not. The central distinction developed in this work is between accessibility, admissibility, participation, and realization. A mathematical representation may contain a large space of possible states, trajectories, modes, or outcomes. Physical realization, however, need not occupy that space uniformly. We propose that relational structure establishes an admissible geometry within the accessible space; that geometry determines which possibilities participate strongly or weakly; and probability subsequently quantifies the residual uncertainty among physically supported realizations. The resulting ordering is: Potentiality → Relations → Structure → Geometry → Admissibility → Participation → Realization → Evolution. Under this interpretation, probability is not abolished. Superposition is not abolished. Quantum mechanics is not rejected. Rather, their physical ordering is reconsidered. Superposition describes what the mathematical representation can contain; relational geometry determines how strongly those possibilities can participate; probability describes the uncertainty remaining within that structured support; and observation records the realization that occurs. The NIST loophole-free two-photon Bell experiment provides the principal empirical test developed in this work. Beginning from the experimentally characterized entangled state—not from subsequently observed outcome counts—we construct a relational geometry and the corresponding outcome-dependent participation and information-cost landscapes. The NIST measurement configurations then become experimental sampling coordinates on a geometry determined independently from the later event records. The resulting analysis identifies several mutually reinforcing features: strong inequality among the participation of mathematically accessible Bell outcomes; a corresponding hierarchy of geometric information cost; concentration of experimental variation into reduced effective dimensions; agreement between predicted geometric support and held-out experimental records; preservation of Bell-inequality violation on the reduced realization geometry; compatibility with no-signalling; and robustness tests against multinomial, temporal, detector, and other alternative explanations. The interpretation is not a restoration of Bell-local hidden variables and does not require superluminal signalling. Bell's theorem continues to exclude the relevant locally factorized descriptions. Instead, the observed correlations are interpreted as local measurements of a common non-separable relational structure. Spatial separation does not imply relational or informational separability. This distinction also reframes the quantum measurement problem. Wave-function collapse need not be regarded as the event that creates the physical structure responsible for an outcome. The framework instead proposes that admissibility constrains participation prior to observation, with measurement revealing a realization from an already structured physical domain. A definitive replacement for the standard measurement postulate will require discriminating predictions beyond those already shared with conventional quantum mechanics. The broader implication concerns the theory of motion itself. Newtonian mechanics, Hamiltonian dynamics, Maxwell's equations, Fourier transport, Navier–Stokes flow, relativity, and quantum mechanics employ very different evolution equations, yet all operate upon structured physical domains. The admissibility framework proposes that their deeper commonality lies not in a single dynamical equation but in an ontological ordering: the laws of motion govern how reality evolves; admissibility governs the structure within which reality can become realized. This has consequences beyond quantum foundations. In engineering and computational physics, enormous mathematical state spaces are routinely explored through CFD, finite-element simulation, optimization, Monte Carlo methods, and digital twins. If physically significant behavior occupies a substantially smaller admissible geometry, computation can be redirected from exhaustive exploration toward the structures in which realizable behavior is concentrated. This provides a theoretical basis for reduced-order physical prediction, targeted simulation, transport localization, and deterministic Physics Intelligence. In AI infrastructure and high-performance computing, the same principle suggests that workloads, thermal transport, energy utilization, routing, scheduling, and hardware states need not be treated as equally relevant points within enormous combinatorial spaces. Identifying admissible low-dimensional structure before exhaustive computation offers a route toward reduced search complexity, improved orchestration, higher computational and energy efficiency, and physics-informed predictive control. In aerospace, energy, advanced manufacturing, climate systems, and data-center infrastructure, the framework similarly motivates identifying the geometry of physically participating transport before expending computational resources resolving the entire mathematically accessible domain. The work therefore proposes a common principle extending from fundamental physics to computational engineering: Mathematical possibility is larger than physical realization. Uncertainty may remain genuine at the level of an individual event without implying an absence of antecedent structure. The results presented here provide empirical evidence consistent with the proposition that physical realization is organized by relational geometry prior to probability, while identifying the further tests required to establish whether this ordering constitutes a fundamental law of quantum and classical motion. The next decisive tests include prospective Bell predictions, multipartite and contextuality experiments, Lorentz-covariant formulation, recovery of the classical limit, and extension to quantum field theory and curved spacetime. Nature need not explore everything mathematics permits. Motion is the realization of what physical structure admits. That description captures the actual architecture of the finished 130-page work: it preserves Bell's theorem rather than attempting to evade it, explicitly separates mathematical accessibility from physical participation, and treats the NIST comparison as evidence for the proposed framework while acknowledging that a distinctive prediction beyond standard quantum mechanics remains necessary for a physical replacement of collapse. Principal Implications What This Means for Bell’s Inequality Bell's theorem remains intact. The present framework does not attempt to restore Bell-local hidden variables, weaken Bell's result, or explain the observed correlations through faster-than-light signalling. Bell establishes that the correlations observed between Alice and Bob cannot be reproduced by the relevant locally factorized description. The question developed here begins after that result: if Bell-local separability fails, what physical structure carries the correlations? The proposed answer is relational geometry. Alice and Bob remain spatially separated and perform local measurements, but their outcomes arise from different interrogations of a common non-separable quantum structure. Spatial separation therefore does not imply informational or relational separability. The correlation need not be interpreted as something transmitted from Alice to Bob at the moment of measurement; it can instead be understood as a property of the joint structure to which both measurements belong. The NIST analysis gives this interpretation a quantitative form. The characterized entangled state determines the relational geometry before the subsequent detector records are used for evaluation. Alice's and Bob's measurement settings sample different regions of that geometry. The resulting Bell outcomes exhibit strongly unequal participation, while Bell violation and no-signalling are preserved. The proposed implication is therefore: Bell violation demonstrates the failure of Bell-local factorization; admissibility proposes a geometry for the non-separable physical structure that remains. This reframes the familiar intuition of "action at a distance." The deeper question may not be how information crosses the distance between Alice and Bob, but why spatial distance should require the relational information defining their joint state to become separable in the first place. What This Means for Wave-Function Collapse The framework does not discard wave-function collapse as an operational rule for updating the quantum state after an observation. It changes the proposed ontological role assigned to collapse. In the conventional collapse-centered picture, the quantum state contains multiple possible outcomes and measurement marks the transition to a definite result. This leaves a foundational question: what physically distinguishes the outcome that becomes real from those that do not? Admissibility inserts a structural stage before observation. The accessible quantum
This paper introduces the induced divergence, a new quantum divergence measure that replaces the hypothesis testing divergence in position-based decoding, simplifying the analysis of quantum communication and state redistribution while yielding tighter achievability bounds. Derived from a parent quantum relative entropy, it retains key properties such as data processing inequality and Löwner monotonicity. Like the hypothesis testing divergence, it depends on a smoothing parameter and interpolates between the parent relative entropy (as the smoothing parameter approaches one) and the min-relative entropy (as it approaches zero), the latter holding when applied to the sandwiched Rényi relative entropy of order α∈[0,2]. This framework refines the position-based decoding lemma, extending its applicability to a broader class of states and improving decoding success probabilities. Two key applications are considered: classical communication over quantum channels, where the induced divergence improves lower bounds on the distillable communication rate, and quantum state redistribution, where it leads to sharper bounds on communication costs. These results provide new insights into fundamental single-shot quantum information protocols and enhance existing analytical techniques.
ABSTRACT Mathematical expressibility is a necessary but not sufficient condition for physical admissibility. A model may be written in well-defined symbols, embedded in formal geometry, or rendered algebraically consistent within a selected framework, and yet fail as physics if it entails unobserved violations of conservation, causal ordering, entropy, memory, measurement stability, or ordinary spacetime structure. This paper proposes the Structural Reality Compatibility Test (SRCT) as a general methodological constraint on proposed physical theories. The test requires that any posited hidden structure — whether additional temporal dimensions, branching ontologies, nonlocal pilot structures, collapse mechanisms, hidden variables, higher-dimensional manifolds, or purely formal histories — must account for the phenomenon it is introduced to explain without producing consequences inconsistent with the established behavior of the observable world. The paper first states the SRCT in plain terms and then develops it formally through scalar-field conservation, Lorentz invariance, Noether symmetry, and the Einstein field equations. Noether's theorem is treated as central: continuous symmetries of the action correspond to conserved quantities, with temporal-translation symmetry grounding energy conservation. Extra-temporal models, many-world branching, and related exotic structures are evaluated under the test. The difficulty is not that such models are mathematically unwritable; it is that they must give a structurally coherent account of all consequences of their additional structures. A second temporal dimension, for example, must explain why conservation, causal order, entropy, memory, and observable time evolution remain stable despite the enlarged temporal symmetry structure. A branching-world ontology must give a physically meaningful account of branching without invoking unobserved thermodynamic or causal consequences. The Information Boundary Principle (IBP) and Minimum Informational Boundary (MIB) are then considered as an example of a theoretical framework attempting to satisfy the SRCT. In that model, nonlocal quantum correlation is treated not as superluminal signaling, extra-temporal communication, or world multiplication, but as a projection-limited manifestation of higher-dimensional geometric convergence constrained by informational compatibility conditions. The point is not that mathematical novelty is forbidden. The point is that new structure must be reality-compatible. The burden of a proposed theory is not merely to explain the targeted anomaly. It must explain the anomaly without breaking the world. Modern theoretical physics has repeatedly advanced by allowing mathematics to reveal structures not obvious to ordinary perception. Relativity required a new geometry of spacetime. Quantum mechanics required a formalism in which probability amplitudes, operators, Hilbert space, and measurement outcomes replaced older mechanical pictures. Gauge theories, curved manifolds, path integrals, and field equations have all reinforced the same lesson: intuition alone cannot govern fundamental physics. The reverse error is, however, equally dangerous. A mathematical structure does not become physically real merely because it can be written down. Formal systems are permissive; physical reality is constrained. A proposed physical theory must do more than resolve an isolated equation or render a single anomaly formally tractable. It must remain compatible with the observed structure of the world. It must preserve conservation behavior, causal ordering, entropy, memory, measurement stability, and non-signaling unless it provides a precise and empirically adequate account of why those structures are modified. A theory that explains one phenomenon by introducing consequences that are nowhere observed has not solved the problem. It has relocated the problem into formalism. This distinction is particularly important in quantum foundations. Quantum entanglement presents a genuine explanatory challenge. The observed correlations between entangled systems are not reproducible through ordinary local separability. Bell-type results establish that the classical combination of locality and certain realist assumptions cannot recover quantum predictions. The natural theoretical response is to introduce hidden structure: branching worlds, nonlocal pilot waves, retrocausal influence, extra temporal dimensions, observer-relative facts, or higher-dimensional geometry. Some of these proposals are mathematically sophisticated. Some may prove partially correct. But all must satisfy the same basic requirement: they must account for the quantum anomaly without producing unobserved consequences across the rest of observable physics. This paper calls that requirement the Structural Reality Compatibility Test (SRCT). The test may be stated in its simplest form as follows: "A proposed physical theory must not merely be mathematically expressible. It must preserve the observed stability of physical reality unless it provides a precise, testable, and structurally coherent account of why that stability continues to appear preserved."