What are the Contexts of Complementarities?

3.1 The Minimal Tools

If scientific explanation requires observers embedded within physical theory, what could be a minimal condition for coherent explanatory contexts? To answer this, I begin with two foundational premises widely valued in both physics and philosophy of science.

Premise 1 (Spatiotemporal Embedding): Any physical observer occupies a specific region of spacetime. This region has finite extent: beyond certain boundaries, no information can be accessed or transmitted through classical channels.

Premise 2 (Theoretical Reconstruction): Observers reconstruct physical observables using the principles and mathematical tools provided by available theory. This reconstruction process is subject to fundamental limitations imposed by the theory’s structure.

Premise 1 reflects the fundamental locality constraints imposed by special and general relativity. No observer can receive information from spacelike-separated events or from regions beyond their causal horizon. This is both a classically practical limitation and a theoretical constraint built into our modern theories of cosmology. Any weaker assumption would fail to capture the locality constraints that generate complementarity phenomena. Premise 2 acknowledges that explanation requires active theoretical reconstruction of physical quantities. Without this, we have no bridge from context to explanatory content. The observer must use theoretical principles to extract meaningful information from available data, and this process is constrained by what the theory permits.

I conclude that these two premises capture a minimal set of assumptions in the sense that weaker versions of either may sacrifice explanatory coverage. For instance, a weaker version of Premise 1 that ignored causal horizons would fail to distinguish between observers inside and outside black hole event horizons, eliminating the very structure that generates complementarity. Similarly, a weaker version of Premise 2 that treated explanation as passive observation would provide no mechanism for connecting theoretical principles to observational constraints.

At first glance, it appears that we are on a track to capture complementarities. The premises encompass the possible geometry for general relativity, specifically local spacetime, as well as the nonlocality of quantum mechanical reconstruction. We can now derive some conditions for explanatory contexts through the following considerations:

• Step 1: Given spatiotemporal embedding, any observer’s explanatory capacity is bounded by their causal accessibility: the spacetime region from which they can receive or transmit information. This defines what information is in principle available for explanation.

• Step 2: Quantum entanglement enables correlations between spacelike-separated regions that cannot be explained by classical causal connections. An observer may therefore reconstruct information about distant quantum systems through these correlations, even when no classical signal could travel between the regions. This “spooky action at a distance” defines a second dimension of observational access beyond what spacetime geometry alone permits. An observer’s access to quantum correlations determines which degrees of freedom can be reconstructed nonlocally, independent of direct causal contact within a single region of spacetime.

• Step 3: Given that physical theories involve background structures and approximation schemes, the observer’s explanatory framework depends on boundary and frame conditions – the geometric, topological, gauge, and asymptotic assumptions that define which effective theory applies and which symmetries are available.

• Step 4: Given that reconstruction requires physical implementation, explanatory capacity is constrained by operational limitations. These may consist in the precision, energy scales, and measurement protocols actually available to the observer.

From our empirically motivated steps, it follows that any explanatory context in modern physics must include these four elements:

1. where the observer is located within spacetime;

2. what the observer can access nonlocally beyond spacetime bounds;

3. what kinds of structural or geometric bounds constrain the observer’s location;

4. what kinds of interventions and observational bounds constrain the precision, scope, and operational realizability of the reconstruction.

These four jointly define the explanatory arena available to any physical observer and constitute a minimal structure required to map theory to explanation in a physically coherent way.

3.2 The 4-Tuple Context

As mentioned earlier, I propose that the context for quantum mechanics and complementarities is a quadruple:

Where each element plays a distinct role in delimiting what can be explained. I stress that each element corresponds to each of the four elements numbered above. In Section 4, I present examples of these elements.

3.3 Motivations of Independence

I examine the prospects of each component contributing uniquely to explanatory structure. One can see that none can easily be derived from the others:

C ≠ B: Two observers may share the same causal patch but adopt different boundary conditions. For instance, two observers outside a black hole may both remain causally excluded from the interior (C _same ) but model the background as either (i) asymptotically flat or (ii) asymptotically AdS. These yield different sets of reconstructible observables due to differences in effective theory, boundary terms, and symmetry algebra. This demonstrates that boundary conditions cannot be derived from causal structure alone.

E ≠ C: Consider a GHZ state shared among three parties in spacelike-separated labs. Each party’s causal access, and location in spacetime is locally restricted, but entanglement correlations exist across global Hilbert sectors. Observers may have the same causal patch (C) but differ in their access to global entanglement structure (E), affecting reconstructability (e.g., two parties can reveal nonlocal correlations only when jointly considered). Indeed, sharing a quantum state like a GHZ state often provides different observations regardless of whether a location is shared or not.

B ≠ O: Fix boundary conditions in a system (e.g., impose Dirichlet boundary on a scalar field in AdS). An observer’s choice of measurement sequence can render certain predictions infeasible even if the boundary geometry permits them. This operational component (O) governs what can actually be detected and verified, independently of B.

O ≠ E: Entanglement reach refers to the observer’s access to quantum correlations; operational configuration refers to measurement precision and apparatus configuration. Even with full entanglement access, a poorly-selected sequence of measurements may prevent precise extraction of information from entangled subsystems, showing the need for O as a separate component.

Hence I assert that there are empirical motivations to consider each component as a unique and non-derivable constraint on explanation in the scenarios considered. Any reduction risks conflate fundamentally distinct physical limitations.

3.4 Application to f _c and Structural Mapping

As mentioned in the introduction, this 4-tuple schema has its motivations partly in f _c , a context-dependent map for explanation. It assigns each physically defined context to a similarity subspace (S _i ): a subset of physical laws that yield consistent predictions within a particular set of conditions.^[12] In the original construction, such laws are assigned based on previous iterations of similar observations (Ryoo 2025). If we know that a double slit creates wave-like patterns, then a triple slit with the same experimental configuration will also generate wave-like interference. More precisely, contexts are deemed similar when they share the same 4-tuple structure, ensuring that the same theoretical constraints and physical limitations apply. Specific prescriptions for assigning similarity subspaces are found in Ryoo’s (2025) development. This mapping assumes that not all laws can be applied globally. Instead, laws become explanatory only when bound to a specific physical context. The mapping itself takes the following form:

Section 4 will show this mapping in action through two relevant case studies. The mapping effectively inputs the context 4-tuple of a particular complementary feature and outputs the set of laws which apply.

Ultimately, this formal structure builds on the idea that physical theories are not globally valid but must be applied in parts. This in part aligns with the patchwork view of science (Dupré 1993; Cartwright 1999), though rather than asserting that different domains need different models, f _c provides a rule-based method for assigning coherent laws to specific contexts. It is also worth noting that the similarity subspaces that result from f _c are not arbitrary. They are constrained by the internal structure of the theory, including its symmetries, causal order, and entanglement features. This mapping, applied to the 4-tuple context, identifies the specific subspace of physical laws that remain valid within that context, thereby structuring explanation according to the theory’s own internal constraints.

3.5 On Interpretation-Ladenness

An argument could be made that quantum mechanics under particular interpretations brings about a sense of consistency amongst the outcomes, hidden variables, branches, etc., of the wavefunction. This may render the mapping to be interpretation-laden. However, our current interpretations of quantum mechanics do not account for complementarities that arise across causal patches – relative to different observers’ local spacetimes. I contend that the 4-tuple framework maintains neutrality by focusing on operationally accessible features that remain invariant across major interpretational divides.

Causal Accessibility (C) remains well-defined across interpretations because it depends only on spacetime geometry and the light cone structure, which are interpretation-independent features of relativistic physics. Whether one adopts Copenhagen, Many-Worlds, or hidden variable theories, the causal patch accessible to an observer is determined by the metric structure of spacetime. This is invariant under the stance that spacetime may be emergent.

Entanglement Reach (E) presents a more subtle case. Of course, interpretations differ on the ontological status of entanglement. Whether it represents genuine nonlocal correlations (Copenhagen), branching correlations across many-worlds, or statistical correlations over hidden variables (Bohmian mechanics), they agree on the operational consequences. By this I mean that all interpretations predict the same correlation patterns in EPR-type experiments and agree on which observables can reveal nonlocal effects. The 4-tuple captures this operational structure without committing to any particular ontological account of what entanglement “really is.”

Boundary Conditions (B) and Operational Configuration (O) are similarly interpretation-neutral. Boundary conditions concern the geometric and gauge-theoretic assumptions required to define a well-posed physical problem: features that remain constant across interpretational frameworks. Operational configuration captures measurement protocols and their fundamental limitations, which manifest identically regardless of whether one thinks measurements cause collapse, reveal pre-existing values, or select branches.

I maintain that complementarities manifest at the level of operational predictions rather than the metaphysical commitments of interpretations. In black hole complementarity, both Copenhagen and Many-Worlds interpretations agree that external and infalling observers cannot operationally access each other’s descriptions simultaneously (see Section 4.1 for details). They differ on whether this reflects fundamental incompatibility (Copenhagen) or correlations across different branches, but they make identical predictions about what each observer can measure and verify.

This operational focus allows the framework to remain neutral while still capturing the structural features that generate explanatory fragmentation. The 4-tuple identifies the physical constraints that determine explanatory boundaries without taking a stance on the underlying metaphysics of quantum mechanics.

4.1 Black Hole Complementarity

Black hole complementarity was proposed to resolve the black hole information paradox, which arises when unitarity in quantum mechanics clashes with the classical nature of Hawking radiation. From the perspective of an external observer, information appears to be encoded in the event horizon. Meanwhile, an infalling observer experiences a smooth passage through the horizon, consistent with the equivalence principle from general relativity. Both views appear valid, but they cannot be globally reconciled with a single global description – hence forming a complementarity (Susskind et al. 1993; Almheiri et al. 2013).

The f _c framework captures this conflict as a case of incompatible contexts. The external observer’s context includes late-time access to Hawking radiation, semiclassical spacetime near the horizon, and long-duration measurement of correlations. We can examine the elements of the 4-tuple as so:

Causal Accessibility (C):

For the external observer, causal accessibility is limited to the region outside the event horizon. Signals from the interior cannot reach this observer, so explanations must be constructed from data available at or beyond the horizon. This restricts the observer to semiclassical analysis of outgoing Hawking radiation. In contrast, the infalling observer maintains causal access to regions inside the horizon up to the singularity. This access defines a different causal patch and yields different explanatory constraints.

Entanglement Reach (E):

The external observer analyzes long-range entanglement across the radiation field. This includes correlations between early and late Hawking quanta, which support the idea of unitary information recovery. The infalling observer, however, accesses near-horizon entanglements between local modes. These do not capture the global pattern of information flow and are limited by the short proper time before reaching the singularity.

Boundary Conditions (B):

Each observer relies on a different background assumption. The external observer models the spacetime as asymptotically flat and treats the horizon as a semiclassical surface. The infalling observer assumes a smooth local geometry governed by general relativity. These boundary conditions yield different effective field theories and shape what kinds of regularity or anomalies are expected.

Operational Configuration (O):

Operational configuration captures the structural constraints on what kinds of measurements can be defined, implemented, and sustained within a given physical regime. For black hole complementarity, this includes the finite proper time available to infalling observers before reaching the singularity, the impossibility of sustaining long-duration measurements behind the horizon, and the incompatibility between local quantum field interactions and nonlocal entanglement extraction protocols. For the external observer, meaningful measurements require semiclassical detectors operating very far away, which are incapable of probing Planck-scale curvature or interior geometry (Dey et al. 2019).

Putting this formally, the external context is:

It maps to a similarity subspace.

defined by quantum field theory on curved spacetime. This subspace explains information recovery through long-range entanglement. On the other hand, the infalling observer’s context is.

It includes local access near the horizon, smooth general relativistic geometry, and finite-duration detectors. This maps to.

a subspace defined by the equivalence principle and the local vacuum structure. The two subspaces cannot be jointly applied, due to the aforementioned inconsistency of the two theories and the laws within.^[13]

4.2 Entanglement Wedge Reconstruction

In AdS/CFT, the boundary conformal field theory encodes information about a higher-dimensional spacetime, referred to as a bulk spacetime.^[14] But the encoding is not uniform: different regions of the boundary reconstruct different regions of the bulk. The key object is the entanglement wedge: Dong et al (2016) showed that an observer with access to a boundary region A can reconstruct operators in the corresponding bulk region E(A), but not beyond it. This reconstruction depends on causal access and entanglement structure. We can examine the components of the tuple as so:

Causal Accessibility (C):

An observer with access to a boundary subregion A can manipulate and measure only operators localized within that region. The causal structure of the boundary restricts their ability to influence or reconstruct operators outside A. This causal limitation defines which bulk operators can be accessed through entanglement wedge reconstruction.

Entanglement Reach (E):

The observer’s entanglement reach is determined by the subregion they control. According to the Ryu–Takayanagi prescription and its extensions, access to region A enables reconstruction of the bulk entanglement wedge E(A). As one would expect, operators outside this wedge cannot be reconstructed without access to a larger boundary region. It is easy to see that explanatory power depends on how much of the boundary is entangled with the relevant bulk degrees of freedom.

Boundary Conditions (B):

The background geometry is fixed by the AdS spacetime and the conformal symmetry of the boundary theory. These conditions ensure that the mapping between bulk and boundary observables is well-defined. Changes in boundary geometry or symmetry assumptions would change the reconstruction map and the set of valid explanations.

Operational Configuration (O):

Measurements depend on the control of the boundary theory. An observer in region A can only access a subset of the full operator algebra. Their operational setup constrains what they can reconstruct and verify. These limitations follow from the structure of the theory and the distributed nature of the encoding.

Formally, the observer’s context is.

defined by the boundary region A, the accessible operator algebra A _A , the boundary conditions B _A (e.g. AdS spacetime), and the operational configuration O _A . The f _c mapping sends this context to a similarity subspace.

which contains only those bulk observables reconstructible from A. If two observers access disjoint regions A and B, their contexts.

will map to different subspaces.

Neither observer can reconstruct the full bulk. Only the union A ∪ B may give access to the interior region E(A ∪ B).

As Harlow (2017) affirms, bulk locality is preserved only within the entanglement wedge associated with the accessible boundary region. Explanation, in this case, is tied to reconstruction capacity, and the f _c framework models this directly. Each observer’s context maps to a subset of explanatory content allowed by the structure of the theory. Global descriptions are unavailable because the encoding is redundant and distributed (Pastawski et al. 2015).

This matches the context-dependence seen in black hole complementarity. In both cases, explanation is limited by causal and entanglement structure. f _c makes this limitation formal. It treats explanation as valid only within structurally defined boundaries that follow from the physical theory, beyond modeling choice or observer ignorance.

4.3 A Discussion of Minimality and Extensibility

Within our 4-tuple schema, these four components appear capable of capturing the contexts relevant to complementarity in contemporary spacetime physics.^[15] The case studies in black hole physics and holographic duality suggest that contexts differing in any of these dimensions yield different explanatory subspaces, while contexts agreeing on all four dimensions yield consistent explanations.

Thus far, I have argued in support of the following aspects of the 4-tuple through these considerations:

1. Each component follows necessarily from spatiotemporal embedding + theoretical reconstruction.

2. They are independent of each other, or at least not easily derivable from each other.

3. Case studies show comprehensive coverage of complementarity phenomena.

The next natural concern is: Could additional components be necessary? Potentially, but the burden of proof lies on critics to identify specific cases where the 4-tuple structure fails to capture relevant contextual differences. The framework is designed to be extensible. Indeed, additional components could be added if empirical cases demand them. Future developments may require additional components depending on the theoretical structure. For instance, spontaneous symmetry breaking (central in cosmological inflation and electroweak theory) introduces context-sensitive vacuum selection, which might demand a fifth axis representing dynamically emergent background structures. Likewise, in quantum information theory, explanatory contexts may depend on computational resource bounds: what can be reconstructed or simulated depends on access to entanglement distillation, error correction, or algorithmic feasibility. These cases suggest that the tuple may be nested within or extended to a higher-dimensional space of explanatory parameters, should empirical or theoretical pressure warrant it. Until these cases are formalized, the current 4-tuple stands as a minimally and empirically grounded structure for explanation. I leave such explorations for future work.

4.4 Relation to Philosophical Theories of Explanation

As developed, the f _c mapping is intended to extend primarily the Kairetic and Deductive Nomological accounts of explanation when consistency breaks down. As Ryoo (2025) describes, the subspaces of laws obtained in the scenarios described above can be utilized within such accounts as premises of a deductive model.

On the other hand, in causal regimes, the subspaces may support localized causal modeling (e.g., à la Woodward 2003), though the framework does not assume the global applicability of interventionist structures. Instead, it clarifies when causal relations are definable within a given context. One might suggest that f _c identifies the contextual boundaries within which Woodwardian causal relations are well-defined, while Woodward’s framework provides the causal structure internal to each context. Woodward assumes that causal relations are objective features of the world, independent of observational standpoint, whereas f _c implies that causal structure is context-dependent. Naturally, this raises the question of whether causal claims are discoveries about the world or constructions constrained by the observer’s location within it. In cases like black hole complementarity or AdS/CFT duality, causal relations may be well-defined within a given context but fail to extend across contexts. The obvious example is that causal claims relating infalling matter to outgoing Hawking radiation involve variables accessible to different observers, undermining the global coherence of interventionist causation.

Moreover, Woodward’s framework requires that interventions be possible in principle, yet many relevant quantities in quantum gravity are not physically manipulable (e.g. spacetime geometry and entanglement structure). It may even be that the basic features like transitivity become problematic: causal chains spanning multiple contexts do not necessarily yield valid global claims. While one could respond by relativizing causation to context or restricting causal talk to locally manipulable variables, either strategy may require significant revision of Woodward’s commitments. Any successful integration of f _c and interventionism may require more careful integrations or modifications.

Similarly, while unificationist accounts treat explanation as the compression of phenomena into general patterns, most prominently in Kitcher’s model of minimizing the number of argument schemata required to derive our beliefs (Kitcher 1981). f _c localizes explanatory patterns to physically defined contexts. One might initially interpret f _c as extending unificationism by systematizing explanation within bounded observational regimes. However, Kitcher’s framework prizes breadth of scope, minimal pattern diversity, and high stringency; yet f _c implies that different contexts often require fundamentally distinct explanatory schemata that cannot be unified without contradiction. Going back to black hole complementarity, these patterns are internally coherent but mutually incompatible and cannot be integrated into a single unifying argument without violating the structural assumptions of at least one explanatory regime. This raises several philosophical tensions. Does fragmentation across contexts signify the failure of unificationist ideals, or does it reveal the inapplicability of global unification in domains where the theory itself prohibits consistent overlap? Moreover, if explanation is modular, should each context-bound argument pattern be counted separately, undermining Kitcher’s pattern minimization criterion?

Kitcher affirms the requirement that explanatory derivations be deductively strong and epistemically constrained, calling it “stringency.” There is a need to examine whether this condition can be preserved when explanatory structure is localized. One may adopt a form of hierarchical or meta-level unification. While object-level patterns may fragment, the mapping itself could serve as a higher-order schema for systematizing which laws apply where. This would entail a structured framework for explanatory organization without enforcing global coherence. Alternatively, one might reconceive stringency and pattern scope as context-relative rather than absolute, preserving unificationist motivations in a pluralist framework. These possibilities suggest that f _c can be thought of as both an extension of the unificationist tradition, as well as a modification of it, that aims to preserve its core virtues of systematicity and explanatory economy. Simultaneously, it is important to note that it might relax the assumption of global applicability. A credible integration may therefore require reconceptualizing unification itself: from global compression to context-sensitive systematization, and from universal argument patterns to a structured mapping of local explanatory regimes. This reconfiguration retains the philosophical ambitions of unificationism while adapting them to the physical and structural constraints revealed by contemporary physics.

I conclude that the f _c framework as a context-sensitive epistemic tool may intuitively be compatible with and extending familiar causal, unificatory, and structural (contextuality) ideals into settings where explanation must be localized. It may be worthwhile to aim for a fuller development of its philosophical implications in the future.