Computational Horizons: Response to Bell's Theorem

Addendum to Section 4 - Quantum Mechanics as Routing

---

4.15 Bell's Theorem and the Measurement-Dependent Correlation

A sophisticated objection arises from Bell's theorem: if entanglement is merely "two pointers to the same node," how can measurement-choice-dependent correlations emerge?

In a Bell test, Alice and Bob each choose measurement settings (angles θ_A and θ_B for polarization measurement). The correlations between their outcomes depend on which angles they chose—not just on the shared state. The correlation function is cos²(θ_A - θ_B), where the angles are chosen after the particles separate.

Bell proved that no assignment of pre-existing values to a shared source can reproduce this correlation. This rules out hidden variable theories. The math is airtight.

4.15.1 Why This Framework Is Not Hidden Variables

The objection assumes the shared node contains a pre-existing value—a definite state waiting to be read. This is precisely what the routing framework denies.

The node is unresolved. It is not "spin up" or "spin down" sitting there waiting to be read. It is a superposition—an unresolved pointer to multiple possible outcomes.

Hidden Variable Theory	Routing Framework
Value λ exists before measurement	No value exists before measurement
Measurement reveals pre-existing λ	Measurement forces resolution
Correlation from shared λ	Correlation from shared unresolved state
Angle selects which aspect of λ to read	Angle is parameter in the query
Bell inequality must hold	Bell inequality does not apply

Bell's theorem proves: no pre-existing value can produce the observed correlations.

The framework replies: there IS no pre-existing value. The value emerges from the interaction between the query and an unresolved superposition.

4.15.2 The Cat, Mr Mittens, and the Damned Thing

Consider an intuitive formulation:

The node doesn't contain "spin up" or "spin down." The node contains cat—an unresolved entity that becomes specific only when queried.

• Bob's query: "Is this Mr Mittens?" (measurement at angle θ_B)
• Alice's query: "Is this one of those damned things?" (measurement at angle θ_A)

They are asking different questions about the same unresolved entity.

The cat is not "Mr Mittens" until Bob asks "Is this Mr Mittens?" The cat is not "a damned thing" until Alice asks "Is this a damned thing?"

The questions create the answers. The correlation between Alice's and Bob's results depends on the geometric relationship between their questions, not on any pre-existing property of the cat.

4.15.3 The Correlation as Question Overlap

The cos²(θ_A - θ_B) correlation is the overlap between their queries:

• If "Mr Mittens" and "damned thing" refer to the same category (θ_A ≈ θ_B): high correlation. When Bob sees Mr Mittens, Alice sees a damned thing.
• If their questions are orthogonal (θ_A ⊥ θ_B): no correlation. Bob's answer tells you nothing about Alice's.
• If their questions are opposite (θ_A = θ_B + 180°): anti-correlation. Bob sees Mr Mittens; Alice sees "not a damned thing."

The measurement angle IS the question. The superposition IS the cat. The correlation IS how much their questions overlap.

4.15.4 The Round-Trip Structure

The framework's round-trip structure makes this precise:

Alice's measurement:

OUTBOUND: Query(θ_A) travels to shared node
RESOLVE:  Node forced to produce outcome relative to θ_A
RETURN:   Result travels back to Alice

Bob's measurement:

OUTBOUND: Query(θ_B) travels to same shared node
RESOLVE:  Node produces outcome relative to θ_B (consistently with Alice's)
RETURN:   Result travels back to Bob

The amplitude for Alice's outcome is:

α_A = ⟨θ_A | ψ⟩

Her probability is |α_A|² = α_A × α_A* (the round-trip weight).

Bob queries the same superposition |ψ⟩ at angle θ_B:

α_B = ⟨θ_B | ψ⟩

The correlation between their outcomes emerges from how their query angles relate to each other AND to the shared superposition state. For the singlet state, this produces cos²(θ_A - θ_B).

4.15.5 Graph Locality vs Spacetime Locality

A remaining puzzle: if Alice and Bob are spacetime-separated, how can the node "know" both their angles?

The resolution lies in the distinction between graph topology and spacetime geometry:

• The shared node is not located IN spacetime
• Alice and Bob are both graph-adjacent to the node
• They are spacetime-separated but graph-local

     Alice ←────→ [Shared Node] ←────→ Bob
           (graph-adjacent)    (graph-adjacent)
         
     |<-------- spacetime-separated -------->|

Nonlocality in spacetime = locality in the graph.

The graph topology is not embedded in spacetime. Rather, spacetime emerges from graph structure. Two entities can be graph-neighbors while being light-years apart in the emergent metric. This is consistent with the holographic interpretation developed in Section 5: addresses ARE distances, routing topology IS geometry.

4.15.6 What Bell Actually Proved

Bell proved: No pre-existing value λ can produce correlations that depend on measurement choices made after particle separation.

Bell did not prove: No unresolved superposition can produce such correlations.

The routing framework posits unresolved superpositions (cats), not hidden variables (pre-existing Mr Mittens assignments). The measurement angle is part of the query that forces resolution, not a selector for which pre-existing value to read.

Bell proved you cannot have a pre-existing Mr Mittens that explains the correlations.

He did not prove you cannot have a cat.

4.15.7 Formal Requirements

To make this response rigorous, the framework must demonstrate:

• Graph topology specification: How entangled states correspond to single nodes with multiple pointers
• Query parameterization: How measurement angles enter the fetch operation
• Correlation derivation: How round-trip weights produce cos²(θ_A - θ_B) specifically
• Metric emergence: How graph-local correlations manifest as spacetime-nonlocal correlations in the emergent metric

The third requirement is addressed by the round-trip structure: the correlation IS the geometric overlap of two queries to the same unresolved state, weighted by the Born rule (which itself emerges from round-trip path geometry, per Section 4.5).

The fourth requirement connects to the holographic principle (Section 5): if the routing graph is fundamental and spacetime is emergent, then "nonlocal" correlations in spacetime are simply local correlations in the underlying graph that happen to manifest at spacetime-separated locations in the emergent metric.

4.15.8 Bell's Theorem is Yoneda for Physics

The deepest connection has been hiding in plain sight since the 1950s.

The Yoneda Lemma (category theory, 1954): An object has no intrinsic identity. Identity IS the collection of arrows (morphisms) pointing at it. What a thing "is" is exactly how other things relate to it. There is nothing underneath.

Bell's Theorem (physics, 1964): A quantum state has no hidden variables. There is no λ underneath that measurements reveal. The correlations come from the measurement relationships, not from pre-existing properties.

These are the same theorem in different notation.

Yoneda	Bell
Objects have no intrinsic identity	States have no hidden variables
Identity = collection of morphisms into it	Properties = collection of measurements of it
What X "is" = how other things see X	What ψ "is" = how measurements query ψ
No object-in-itself	No state-in-itself

Yoneda says: Mr Mittens doesn't exist until Bob asks "Mr Mittens?" The cat's identity is constituted by the queries into it.

Bell says: Spin-up doesn't exist until you measure spin. The state's properties are constituted by the measurements of it.

The cat-in-itself does not exist. There is only cat-as-queried.

4.15.9 Category Theory Already Unified This

The framework presented in this paper is not analogizing physics to computation. It is recognizing that category theory already unified them—physics just hasn't noticed yet.

The Yoneda perspective dissolves the Bell "paradox":

• There is no paradox in correlations depending on measurement choice
• There WOULD be a paradox if hidden variables existed (Yoneda: they can't)
• The correlation IS the morphism structure
• Measurement angles ARE the arrows
• The cos²(θ_A - θ_B) IS the composition of morphisms

Bell proved that quantum mechanics is Yoneda-compliant: no hidden essence, only relational structure.

Einstein's objection ("God does not play dice") was, in categorical terms, an insistence that objects have intrinsic identity independent of morphisms. Yoneda says they don't. Bell proved they don't. The universe is categorical.

4.15.10 Implications for the Framework

If Bell = Yoneda, then:

• The routing graph IS the category: Nodes are objects, edges are morphisms
• Fetch IS hom-functor application: Querying a node = applying Hom(-, X)
• The Born rule IS naturality: The consistency of round-trip weights across different query paths
• Entanglement IS shared object: Two morphisms targeting the same object
• Measurement IS morphism: The angle θ is the specific arrow you're shooting

The framework doesn't need to "justify" why measurement creates outcomes rather than revealing them. Yoneda already proved this is the only coherent ontology. Objects don't have identity except through their morphisms. States don't have properties except through their measurements.

Physics spent 60 years confused about Bell because it was thinking in terms of sets (elements with intrinsic properties) rather than categories (objects with relational identity).

Category theory got there in the 1950s. We're just translating.

---

Provenance

Document

• Status: 🔴 Unverified
• Added: Response to Bell's theorem objection

North

slots:
- slug: computational-horizons-section-4
  context:
  - Parent section on quantum mechanics as routing

West

slots:
- context:
  - Follows main quantum mechanics section
  slug: computational-horizons-section-4
- context:
  - Linking Bell response as addendum to section 4
  slug: computational-horizons-section-4