Nonuniqueness of Remote Outcomes

• Nonuniqueness of remote outcomes are phenomena where asymptotic states in quantum mechanics, dissipative PDEs, and topological spaces are not uniquely determined by initial or local data.
• In quantum nonlocality, dropping outcome uniqueness reveals loopholes in Bell inequalities derivations, motivating interpretations like Many-Worlds that preserve local causality under no-signalling.
• In PDE and topological frameworks, techniques like convex integration and end-cohomology demonstrate that multiple admissible solutions or manifold ends can emerge from identical local conditions.

Nonuniqueness of remote outcomes encompasses a spectrum of phenomena in mathematical physics and topology wherein the asymptotic, distant, or causally separated "results" of a process are not determined uniquely by initial or local data. This concept emerges with striking clarity in three major contexts: the derivation of Bell inequalities in quantum theory (and the role of remote measurements), the theory of admissible weak solutions in dissipative PDE systems (where remote states are spacetime endpoints of solutions), and the topology of noncompact manifolds (where "ends" encode the remote structure of a manifold). Across these domains, nonuniqueness signals profound limitations on determinism, selection principles, or the reconstructibility of global behavior from local information.

1. Remote Outcomes in Quantum Nonlocality

Bell's local causality (LOC) principle posits that, given a specification of all relevant beables $\lambda$ from the shared past of two spacelike regions, the probability of an outcome in one region is independent of settings or measurements in the other. Mathematically, this is expressed as

$$P(a,b\mid x,y,\lambda) = P(a\mid x,\lambda)\,P(b\mid y,\lambda)$$

where $a$ and $b$ are outcomes, $x$ and $y$ are measurement settings for Alice and Bob. Further decomposed, this encapsulates parameter independence ($P(a\mid x,y,\lambda)=P(a\mid x,\lambda)$) and outcome independence ($P(a\mid x,y,b,\lambda)=P(a\mid x,y,\lambda)$).

The standard derivation of the CHSH inequality uses the assumption that remote outcomes—e.g., Bob's $b$ when viewed from Alice's spacetime region—are unique for each run: $$\sum_{b\in\{\pm1\}} P_{\rm occ}(b\mid y,\lambda) = 1,\qquad P_{\rm occ}(b_1\,\text{and}\,b_2\mid y,\lambda) = 0,\ (b_1\ne b_2)$$ This uniqueness is essential for probability decompositions underpinning the derivation.

2. Loophole in Bell's Derivation: Nonuniqueness of Remote Outcomes

If the assumption of uniqueness is dropped—specifically if, in a given run, multiple $b$ can occur concurrently—the conditional probabilities $P(a\mid x,y,b,\lambda)$ are no longer well-defined. In the Many-Worlds (Everett) interpretation, measurements do not select single outcomes but result in superpositions (branches), and both $b=+1$ and $b=-1$ can coexist in a single universal state.

In such settings, the standard decomposition,

$$P(a\mid x,y,\lambda) = \sum_{b} P(a\mid x,y,b,\lambda) P(b\mid y,\lambda)$$

fails: there is no single $b$ to condition upon. As a result, the logic leading from outcome independence to full factorization—and hence the CHSH inequality—is blocked. Nevertheless, LOC in its unconditional form is preserved; the probability for an outcome in one spacetime region is not altered by operations in the other, as enforced by quantum no-signalling.

This loophole is specific to formulations of local causality based on probabilities for single outcomes. The original Bell principle is circumvented only if uniqueness is abandoned, as in Everettian quantum theory (Saunders, 29 Nov 2025).

3. PDE Models: Infinitely Many Remote States as Solutions

In the theory of partial differential equations, particularly for multi-dimensional compressible systems with dissipation (e.g., Aw–Rascle models), nonuniqueness manifests as "remote outcome" multiplicity: for given compatible initial and final data, there exist infinitely many admissible weak solutions $(\rho,u)$

$(\rho(0),u(0)) \to (\rho(T),u(T)),$

all satisfying the energy inequality. These solutions connect spatially remote endpoints, even under dissipative mechanisms such as density-gradient regularizations (Chaudhuri et al., 2022).

This effect is proved using convex integration methods. The technique demonstrates that wild solutions occupy a large, functionally diverse family. This breakdown of classical uniqueness demonstrates that strong selection criteria (e.g., energy admissibility) do not suffice to determine a unique remote outcome for the system's time evolution. The phenomenon is robust: even when both endpoints are trivial (constant density and velocity), nontrivial, dissipative flows realizing those endpoints are nonunique.

4. Topological Nonuniqueness: Ends of 4-Manifolds

Remote outcomes also arise in the topological setting of noncompact manifolds, where the notion of an "end" formalizes the structure at infinity. The operation of end-sum (connected sum at infinity), introduced for one-ended, open 4-manifolds $M$ and $N$ along rays $r\subset M$, $s\subset N$, produces new manifolds

$S_r := (M,r)\,\natural\,(N,s)$

whose proper homotopy type (and thus topology at infinity) is sensitive to the choice of rays. Calcut, Guilbault, and Haggerty proved that, by varying $r$ in an uncountable family, the resulting manifolds $S_r$ realize uncountably many distinct proper homotopy types—extreme nonuniqueness of remote outcomes (Calcut et al., 2020).

This nonuniqueness is detected via the end-cohomology algebra

$$H^*_e(X;R) = \varinjlim_{K\subset X,\,\text{compact}} H^*(X\setminus K;R)$$

and refined by ray-fundamental classes. The King theorem shows that the end-cohomology of an end-sum is uniquely determined by those of the summand manifolds together with the ray-fundamental classes. Distinct choices of ray typically yield nonisomorphic end-cohomology algebras, hence, inequivalent remote outcomes for the topological type of $S_r$.

5. Physical and Mathematical Significance Across Domains

The existence and utilization of nonuniqueness of remote outcomes is not a mere mathematical pathology, but an actively exploited mechanism in modern theory:

• Quantum foundations: Everettian probability (2-MANY microstate-counting) both accepts and leverages nonuniqueness. Physical probability is assigned by microstate counting over branches in Hilbert space, exactly reproducing quantum (Born rule) probabilities without invoking collapse. The approach maintains strict locality and measurement independence, while directly enabling observed violations of Bell inequalities and recasting them as empirical support for branching multiverses (Saunders, 29 Nov 2025).
• Continuum mechanics: Nonuniqueness in dissipative fluid-like PDEs demonstrates that even strong dissipative mechanisms and admissibility conditions do not necessarily restore determinism at the level of remote system behavior. Further structural or physical constraints (e.g., higher-order viscosity, regularity thresholds) may be necessary to select unique flows (Chaudhuri et al., 2022).
• Topology: The nonuniqueness of the ends of manifolds under end-sum operations underscores the subtlety and richness of topological classification in dimensions where exotic smooth structure abounds (notably $d=4$). Different choices of "connection at infinity" yield distinct infinite-topology manifolds undetectable by finite (local) probes (Calcut et al., 2020).

6. Open Problems and Perspectives

The recognition and detailed analysis of nonuniqueness of remote outcomes prompts several open directions:

• In quantum theory: whether alternative formulations of local causality can preclude this loophole, and the further implications for interpretations beyond Everett.
• In PDE and continuum mechanics: whether sharpened admissibility criteria or regularity conditions suffice to select unique, physically meaningful solutions.
• In manifold topology: the extent to which other high-dimensional cobordism and infinite constructions share similar remote nonuniqueness, and the classification of possible ends via invariants such as end-cohomology.

The paper of nonuniqueness of remote outcomes thus situates itself at the intersection of analysis, geometry, and quantum foundation, serving as a technical and conceptual fulcrum in the understanding of determinism and selection principles at a distance.