Architectures of Knowing (I): Revisiting the No-Go Theorems of Quantum Reality

In college I was drawn to a particular kind of knowledge frontier: results that don't necessarily explain everything, but rule out combinations of ideas that cannot all be true. The underdetermination boundaries that every discipline eventually meets. In quantum mechanics, Bell, Kochen–Specker, and Conway–Kochen's “Free Will” theorems are exactly that; each is a no-go statement proved with precision and minimal assumptions. The same pattern in the EMH/CAPM joint-hypothesis tangle first pulled me toward quant finance. Later, building models of markets, agents, and world-dependency graphs, I returned to the quantum theorems as stress tests for thought: clean structural frameworks against which any world-model can be checked for internal consistency. The shared core is a question that recurs in many domains: how to describe an entangled, self-referential system whose parts observe and influence one another, a system that interacts with its own predictions. Whether a model aligns with or diverges from these quantum constraints, the comparison is always productive: it exposes where our descriptions of interdependence, context, or agency begin to break.

This first essay lays out the reasoning itself: how each no-go theorem thinks. The final section, "Beyond Physics," sketches how the logical structures illuminate macro-scale world-modeling. Later pieces will expand on this connection, exploring how these reasoning can inspire architectures for reflexive models in markets, adaptive agents, and world causal graphs.

• Realism.
  Realism is the idea that physical properties such as position or momentum exist with definite values whether or not we look, i.e., the result of a possible measurement exists independently of the observer. Einstein, Podolsky, and Rosen (EPR) gave a more precise criterion of reality (quoted below in §III)
  Quantum mechanics challenges the assumption of realism.
• State & measurement.
  A state is how a system was prepared. A measurement is an operation with a well-defined set of outcomes. In classical physics, measuring simply reveals the state; in quantum physics, measuring usually changes it.
• Distant.
  Two events are distant (or spacelike separated) if no signal, influence, or causal effect traveling at or below the speed of light could go from one to the other in time to have an effect. In short, they are both spatially separated (physically far enough apart) and causally separated (neither lies in the other’s light cone—the region of spacetime that can be reached by light or slower signals).
• Locality (local causality).
  Locality is the principle that physical influences cannot act instantaneously across space: an event, instantaneously, can affect only what lies within its immediate causal neighborhood. This principle depends on two deeper conditions: special relativity (any signal or causal influence can travel no faster than light) and causality (an event can be influenced only by other events within its backward light cone).
  Under locality, once you condition on all information in the shared past of two distant events (call it $\lambda$), then, what happens “here” should not depend on which measurement is chosen “there,” i.e., the results at each site depend only on local settings. Formally, ( P(A,B|a,b,$\lambda$)=P(A|a,$\lambda$)P(B|b,$\lambda$) ).
• Spin.
  Spin is a quantum version of angular momentum, an intrinsic “twist” each particle carries. Measuring spin along any direction gives only certain discrete results.
  For a spin-½ particle (like an electron) the outcomes are $\pm 1$.
  For spin-1, the squared spin along a direction yields $0$ or $1$. If you measure a spin-1 particle along any three mutually orthogonal directions, you'll always find “two 1s and one 0.”
• Entanglement.
  Two particles prepared together can have joint properties not reducible to separate states; their properties are linked so that describing one requires describing both. In such settings, we say the two particles are entangled.
  For example, in a "singlet" pair, if one particle’s spin is measured as “up” along any chosen direction, it immediately fixes that the other will yield “down” in the same direction, regardless of distance between them. No message is sent between them; the correlation comes from how the pair was prepared.
• Measurement independence (“free settings”).
  When testing locality, we also assume the experimenters are "free" to choose how to measure - their choices of measurement settings $a,b$ are not themselves functions of, or statistically tied to, the hidden past information $\lambda$.
  If that independence failed (i.e., if the experimenters’ supposedly random choices were already linked to the particles’ hidden states), then “superdeterminism” could “fake” quantum predictions.
• Context.
  In quantum mechanics, some measurements can be performed together without interfering—those measurements are called compatible or commuting. A context is the full set of such mutually compatible measurements. Each context defines a local patch of reality that can be described consistently.
  Different contexts may overlap but cannot always be merged into one global description.
• Non-contextuality.
  A theory is non-contextual if the outcome for any given quantity is fixed and independent of which other compatible quantities are measured alongside it.
  Classically we take this for granted: the temperature of water doesn’t depend on whether we also measure its volume. However, the Kochen–Specker theorem shows that quantum mechanics cannot satisfy this assumption for all observables simultaneously.

Classical mechanics assumes realism: a system’s properties exist whether or not we look; measurement reveals rather than creates. Given a classical state, the future is determined by the equations of motion. If two distant systems are correlated, a shared cause in their past suffices to explain it; influences do not outrun signals.

In quantum mechanics, measurement is not passive. A measurement generally changes the state. Operationally, the post-measurement state depends on both the prior state and the measurement operator we choose. As a result, the most we can predict are probabilities for outcomes, not certainties.

In the two-slit experiment, particles are sent toward a barrier with two narrow openings. When both slits are open and no one checks which slit each particle goes through, the particles land on the screen in a pattern of bright and dark bands. That’s an interference pattern—the kind made by waves overlapping; each particle behaves as if it travels through both slits at once, interfering with itself. But if a detector is placed to find out “which slit did it actually go through?”, the pattern changes completely. The interference vanishes, and we see two simple clusters, as if each particle took only one path. The mere act of measuring which path destroys the wave-like behavior. The lesson is simple: in quantum theory, what you can say about a system depends on how you look—and the act of looking changes what is there to be said.

The departure from classical certainty led Einstein, Podolsky, and Rosen (EPR, 1935) to argue that if both locality and realism hold, then quantum mechanics must be an incomplete description of physical reality. They proposed a criterion of reality:

If, without disturbing a system, one can predict with certainty (probability 1) the value of a physical quantity, then there is an element of reality corresponding to that quantity.

(In their setup, “without disturbing” is enforced by locality and spacelike separation: the two measurements are far enough apart in spacetime that no influence moving at or below light speed could connect them in time.)

EPR’s conclusion follows from the thought experiment below.

Prepare two particles in an entangled state and separate them (by, say, 20 light-years) so their measurements are spacelike separated. Because of the entanglement preparation, measuring a certain quantity on particle A lets us predict with certainty (i.e., determine) the corresponding quantity on particle B.

• 1.Locality.
  Assuming locality holds, then the act of measuring particle A cannot have a simultaneous effect on particle B (no influence outside B’s light cone). Learning A’s result does not disturb B.

• 2.Perfect prediction ⇒ element of reality (EPR’s criterion).
  Assuming EPR’s realism holds, if we are able to determine the value of a quantity (say, the position or momentum of B), it would imply that the quantity already exists prior to its measurement, independently of the observer.
  Since measuring A lets us determine B’s result without disturbing B, EPR says B’s quantity exists—it had a definite value whether or not we measured it.

• 3.Choice of what to measure at A.
  We could choose to measure one of two incompatible quantities on A (e.g., position or momentum; or, in the spin version, measurements along different axes). Either choice would let us determine the matching quantity at B, because of entanglement.

• 4.The Paradox and Conclusion.
  Because we could have determined either quantity of B (e.g., position or momentum) without disturbing B, B must already have definite values for both of these quantities simultaneously, under EPR's definition of reality. This would contradict the Heisenberg Uncertainty Principle which states that the position and momentum of a particle cannot be simultaneously known. Hence the paradox: it appears that quantum theory fails to describe some physical quantities that already exist and could, in principle, be determined.

Holding locality and EPR’s reality criterion fixed, EPR concluded that quantum mechanics is incomplete: the quantum state does not include all the elements of reality. They proposed the possibility of hidden variables: additional, currently unseen variables that would “fill in” those definite values and restore determinism.

A hidden-variable theory posits some extra information $\lambda$ such that the outcome of any measurement (e.g., each result at A and B) is a (deterministic) function of $\lambda$ and the local setting. In a two-particle experiment, $\lambda$ would pre-determine the pair of outcomes for any pair of measurement choices. If such a theory respected locality and free settings (choices $a,b$ are independent of $\lambda$), it would provide the kind of “complete” description EPR hoped for.

EPR’s argument sets the stage and shows the tension among assumptions: locality, a precise criterion of reality, and the hidden variable hypothesis. The next theorems test this program:

• Bell asks: can local hidden variables with free settings reproduce the observed pattern of correlations? (Answer: no.)
• Kochen–Specker asks: can non-contextual hidden variables pre-assign values independent of measurement context? (Answer: no, in dimension ≥ 3.)
• Conway–Kochen asks: with the above structure and relativistic independence, can only the experimenters’ choices be “free” while particles are fully fixed by the past? (Answer: no, "free will" must be symmetric.)

Bell’s theorem (1964) shows that even if two distant experiments share hidden common causes (hidden variables), there is a strict mathematical ceiling on how strongly their results can correlate if we insist on two classical intuitions: that nothing acts faster than light (locality) and that each experimenter’s setting choice is statistically independent of the system’s hidden state (free settings). Quantum experiments break that ceiling. Hence any complete explanation of the data must abandon at least one of those assumptions.

Two distant experimenters, Alice and Bob, each receive one particle from an entangled pair prepared in the singlet state. They independently choose which direction to measure their particle’s spin, noted $a$ for Alice and $b$ for Bob. Each measurement yields one of two results, $A,B\in -1,+1$. Suppose the pair share some hidden information $\lambda$ determined at the source.

Bell assumes three conditions:

• 1.Locality (local causality): Given $\lambda$, the outcome at one site does not depend on the setting at the other: ( P(A,B|a,b,$\lambda$)=P(A|a,$\lambda$)P(B|b,$\lambda$) ).

• 2.Measurement independence (free settings): The measurement choices $a,b$ are statistically independent of hidden variable $\lambda$.

• 3.Hidden variables exist: Each outcome is a definite function of its local setting and $\lambda$: ( A=A(a,$\lambda$), B=B(b,$\lambda$) ).

Under these assumptions, the joint probability of outcomes factorizes. If the universe obeys locality, then separated measurements act like independent coin tosses conditioned on the shared cause $\lambda$.

Bell considered correlations across several pairs of settings. For any local-hidden-variable model, basic probability algebra implies a constraint on how strongly results can be correlated across different measurement choices. In the CHSH form,

where ( E(a,b) ) is the expected correlation. This bound is a mathematical consequence of local factorization; it does not depend on the details of quantum theory.

Quantum mechanics predicts that for entangled particles the correlation between directions $a,b$ is ( E(a,b)=-$\cos \theta$ ), where $\theta$ is the angle between measurement axes. For certain choices of angles, the quantum value of the left-hand side reaches $2\sqrt{2}$, violating the inequality. Experiments have confirmed this.

If hidden variables exist, you cannot keep both locality and free settings and match the observed correlations. At least one must go: abandon locality (allow nonlocal dependence), or abandon free settings (accept superdeterminism), or abandon hidden variable theories (deny predetermined outcomes).

Kochen–Specker (1966) proves that for quantum systems rich enough (three or more dimensions), it is mathematically impossible to assign pre-existing, measurement-independent values to all observables while preserving the internal consistency of the theory. In other words, there is no single global “ledger” of truth: what can be true for one set of jointly measurable quantities may be inconsistent with what must be true for another. The theorem rules out non-contextual hidden-variable models—those in which each property has a definite value regardless of the measurement frame. The point is structural: any adequate description must carry its context—the set of compatible questions—within its definition of state. Attempting to erase that dependence leads to a contradiction.

Can every measurable property be assigned a pre-existing value that is independent of context—that is, independent of which other compatible measurements we choose to perform alongside it—while remaining consistent with quantum constraints?

Begin with a simple property of a spin-1 particle: if you measure the squared spin along three perpendicular directions, you always obtain two 1s and a 0. These three measurements can be performed in any order because the corresponding observables commute; measuring one does not disturb the others. This “two-1s-and-a-0” rule, often called the 101 property, is an experimental fact of quantum mechanics.

Now imagine that there exists a hidden-variable description in which every possible measurement already has a definite result. Then a single hidden variable $h$ would have to assign to each direction in space a value ( v(l)$\in 0,1$ ) such that every orthogonal triple of directions ( (l_1,l_2,l_3) ) obeys the 101 rule: the set of values ( {v(l_1),v(l_2),v(l_3)} ) is always two 1s and one 0. In other words, one should be able to build a consistent map

so that any direction appearing in different triples always carries the same assigned value.

Kochen and Specker proved that no such global map can exist. Using a finite, carefully chosen set of directions, any attempt to assign 0s and 1s consistent with all the required triples inevitably leads to a contradiction—some direction must be assigned 0 in one triple and 1 in another. Therefore, the assumption that a particle’s measurement results are predetermined and independent of context cannot hold.

The theorem needs no reference to distance or communication between particles; it is indifferent to locality or non-locality. The conflict arises within a single system: quantum mechanics allows definite outcomes only within a given measurement context, and there is no way to extend those outcomes into a single, context-free ledger of truths.

Non-contextual hidden-variable theories are ruled out. Global, context-independent value assignment is impossible given the quantum structure; consistency is confined to the chosen commuting context.

The Conway–Kochen theorem (2006) extends the Kochen–Specker logic: Grant informational freedom to the experimenters, and you must grant an equal under-determination to the particles. In other words, if the experimenters are not a deterministic function of their past, then neither are the particles.

• SPIN: For spin-1, squared-spin along any orthogonal triple yields two 1s and one 0 (these observables commute).
• TWIN: For appropriately entangled pairs, the two particles’ squared-spin outcomes match when measured along the same direction (perfect entangled agreement).
• Relativistic independence (MIN/FIN): Choice of settings at one lab cannot influence the other in time to affect its outcome (spacelike separation of choices).
• Measurement independence (“free settings”): Each experimenter’s choice of directions is not a deterministic function of the physical information in her past light cone.

Begin by combining two earlier ingredients. The Kochen–Specker theorem forbids assigning fixed, context-independent values to all possible spin directions: a spin-1 particle measured along any three orthogonal directions must always yield two 1s and one 0, and no single hidden variable can pre-assign such triplets consistently.

Conway and Kochen then use an entangled pair in a singlet state so that the squared-spin of particle 1 along any direction is identical to that of particle 2 in the same direction (TWIN). Each experimenter—Alice and Bob—chooses measurement directions far enough apart in spacetime that neither choice can influence the other (relativistic independence). Alice measures along three orthogonal directions of her choosing; Bob measures along one direction of his choosing.

Now assume the particles have no “free will”—that their responses are determined by their past. Then each particle’s squared-spin along any direction must be a function of that past information, denoted $h$. Because the particles are entangled, Bob’s results determine Alice’s, so her entire triple of outcomes must also be functions of the same $h$. This means a single hidden variable would have to assign values to all possible orthogonal triples of directions while respecting the “two-1s-and-one-0” rule.

But the Kochen–Specker theorem already proves that no such assignment $h$ can exist. The assumption that the particles’ responses are fully determined by their past therefore leads to contradiction. The only escape is symmetry: if the experimenters’ choices of directions are not fixed by their past, then the particles’ responses cannot be either. Under these premises, one-sided determinism collapses—the independence of the experimenters’ settings and the indeterminacy of the particles’ outcomes must stand or fall together.

In the authors’ phrasing: if the experimenters have this kind of freedom, then so do the particles.

These theorems replace a single, universal ledger with a geometry of overlapping, internally consistent frames, and precise rules for where and why they cannot be fused.

We cannot directly extend quantum mechanics into the macro world, but these logical geometries carry inescapable parallelism and analogies. The structural relationships among independence, correlation, and consistency are not uniquely quantum. They are statements about systems with interacting observers and observables, systems that model themselves. Every world-model built on separability, exogenous choice, and global consistency, composed of mutually observing parts, will, beyond a point, encounter its own Bell-type contradiction. Markets, adaptive agents, level-k game theory, and causal graphs all live on that frontier, where context cannot be erased and observation loops back into the system. More practically, what can be inherited from the quantum debates is the discipline of constraint: a way of world-modeling that begins from what cannot all be true at once.

With this perspective in mind, let's take another look: What are the possible projections of the quantum debate onto the macro world?

• Bell: Locality + independence means bounded correlation; observed correlations exceed the bound.
  ⇒ Abstraction: Any system where observed coordination exceeds what separable local models predict.
  ⇒ Analogue: Markets or networks that show emergent coherence despite assumed independence.

• Kochen–Specker: Context-free value assignment impossible.
  ⇒ Abstraction: Any system where measurement, evaluation, or reward depends on the frame of reference or context.
  ⇒ Analogue: Model calibration between global vs local regimes.

• Conway–Kochen: Symmetry of free will on both sides.
  ⇒ Abstraction: Any system with reflexive feedback between observers and environment.
  ⇒ Analogue: Adaptive markets or reflexive agent systems where the actions, or even measurement, shape the data they learn from; environment changes in response to the aggregate behavior of agents (in the field of game theory and RL).

In practice: don’t chase a grand unified model. Instead, build an atlas with explicit gluing rules, and treat every “impossible” correlation as a signal where the map must be redrawn.