Why Probability is Squared
|ψ|² · Stress-Energy · Self-Measurement · One Origin
// CLASSIFICATION: UNIFICATION_CORE
// ORIGIN: SECTOR 7G (2047)
// CONTEXT: For 100 years, the Born rule was a postulate. P = |ψ|². Nobody knew why. This page argues: gravity. Using Zurek's envariance theorem: gravitational entanglement + universality → swap symmetry → |ψ|². If correct, the Born rule is a consequence of TT-coupling. A testable claim.
In 1926, Max Born proposed that the probability of finding a quantum system in a state is:
This is the Born rule. It is the bridge between the quantum world (amplitudes) and the classical world (probabilities). Every quantum prediction depends on it.
Why squared?
Why not |ψ|? Why not |ψ|⁴? Why not some other function?
The Born rule is a postulate — assumed, not derived. For 100 years, physicists have used it without knowing where it comes from.
This is one of the deepest unexplained facts in physics.
None of these derive the Born rule from something deeper. They all assume it in disguise.
A candidate answer from stress-energy.
In every physical theory, the stress-energy tensor is quadratic in the field:
Quadratic in φ
Quadratic in F
Always quadratic. Always.
From the framework:
QUANTUM
P = |ψ|²
Probability
GRAVITY
T ~ |φ|²
Stress-Energy
If the probability of an outcome is its gravitational weight, both are quadratic for the same reason.
Using Zurek's envariance theorem with the Bath.
In 2005, Wojciech Zurek showed that the Born rule can be derived (not assumed) from a symmetry principle called envariance — entanglement-assisted invariance.
A system S entangled with an environment E:
The environment states |ei⟩ are orthogonal — the environment "records" which branch the system is in.
Consider swapping two branches:
From the system's perspective (tracing out E), this swap is undetectable when |c₁| = |c₂|.
The reduced density matrix of S is unchanged by the swap. This symmetry is called envariance.
If the physics is symmetric under branch swapping (envariant), then:
This is a mathematical theorem, not an assumption.
The gravitational Bath provides exactly the structure Zurek needs:
Matter state in stress-energy basis:
TT coupling creates entanglement with the Bath:
✓ The Bath records the TT content of each branch.
Different stress-energy configurations are distinguishable:
This is what "measurement" means — the Bath can tell branches apart by their gravitational content.
✓ Bath states are orthogonal for distinct stress-energy.
The swap |T₁TT⟩|Bath₁⟩ ↔ |T₂TT⟩|Bath₂⟩ is a symmetry because:
✓ Universality of gravity ensures envariance.
If matter entangles with the Bath via TT coupling, the Born rule follows.
1. Initial state (two branches for simplicity):
2. After TT coupling with Bath:
3. Reduced density matrix of matter (trace out Bath):
(using ⟨B₁|B₂⟩ = 0)
4. Apply swap: |T₁⟩|B₁⟩ ↔ |T₂⟩|B₂⟩
5. New reduced density matrix:
6. Envariance: If |c₁| = |c₂|, then ρM and ρ'M differ only by relabeling — same physics.
7. Zurek's theorem: The only probability measure consistent with this symmetry is:
∎ QED
The proof uses three specifically gravitational features:
If this is right: remove gravity → no envariance → no Born rule.
The Born rule P = |ψ|² is not a postulate.
It is a theorem: gravitational entanglement + universality → envariance → |ψ|².
The Born rule is a consequence of gravity being universal.
The deep reason for the exponent.
Stress-energy must satisfy three requirements:
Energy must be non-negative: T00 ≥ 0
This rules out |ψ|¹ (can be negative for complex ψ)
Tμν must transform correctly under Lorentz transformations.
This rules out |ψ|³ and other odd powers (not Lorentz scalars)
Tμν(x) depends only on fields at point x, not integrals over space.
This rules out nonlocal functionals
The only local, Lorentz-invariant, positive-definite functional of a field is:
The exponent 2 is not arbitrary. It is forced by the requirements of physics.
The Born rule is squared because gravity requires positive, local, Lorentz-invariant energy.
The universal backstop argument.
A skeptic might say: "The derivation shows that any environment could produce the Born rule via envariance. Why specifically gravity?"
Photons decohere things. Phonons decohere things. Air molecules decohere things. Why single out gravity?
This is the right question. And the answer is decisive.
Every non-gravitational environment can be blocked:
In principle, you can isolate a system from all non-gravitational environments.
The equivalence principle forbids gravitational shielding:
Gravity is irreducibly universal.
Place a quantum system in:
The system is isolated from everything — except gravity.
Does the Born rule still apply?
If the Born rule came from photon decoherence → it would disappear in the dark.
If it came from phonon decoherence → it would disappear in vacuum.
If it came from EM decoherence → it would disappear in a Faraday cage.
But the Born rule always applies. Even in principle.
In perfect isolation, only one environment remains: gravity.
Gravity is the last environment standing. The irreducible backstop.
For the Born rule to be universal (applying to all quantum systems), its source must be universal.
Gravity is the only interaction that:
Gravity is the only candidate for a universal probability law.
The derivation showed gravity can produce the Born rule. The shielding argument shows gravity is the strongest candidate:
Caveat: This is an argument by elimination, not a proof. It is possible that the Born rule arises from a mechanism we haven't identified — one that doesn't require an environment at all. What the shielding argument establishes is that among known physical interactions, gravity is the only one with the right properties. The experiment below can test whether this is actually the case.
Create the most isolated quantum system possible:
Prediction: Even in this extreme isolation, the system will:
If the Born rule persists when only gravity couples, gravity is proven to be the source.
The Born rule is not a feature of "quantum mechanics interacting with some environment."
It is a feature of quantum mechanics interacting with spacetime itself.
Probability is how the geometry of spacetime responds to superposition.
Other decoherence channels are incidental. Gravity is foundational.
What follows if the Born rule is gravitational.
Quantum mechanics: Σ|ψi|² = 1
(probabilities sum to 1)
Gravity: Σ Ti = Ttotal
(stress-energy conserved)
Same equation. Unitarity IS energy conservation.
Before measurement: |ψ|² spread over branches
Measurement: TT coupling creates gravitational record
After: one branch has all gravitational weight
Collapse is gravitational selection, not magic.
If the Born rule is gravitational, then ℏ and G are not independent.
This is where quantum probability (ℏ) and gravitational weight (G) become comparable.
Below ℓP: gravitational effects dominate probability
Above ℓP: quantum effects dominate gravity
The Planck scale is where the Born rule and Einstein's equation merge.
The measurement problem asks: Why does measurement cause collapse?
In this framework, the chain becomes:
This doesn't dissolve the measurement problem — it relocates it. Instead of asking "why does collapse happen?", we ask "why does TT-coupling produce irreversible decoherence?" That's a more tractable question, because it connects to known physics (Lindblad dynamics, decoherence theory). But it is still an open question, not a solved one.
Where the Born rule breaks down.
If the Born rule is gravitational, it should receive corrections in strong gravitational fields:
Where:
At the horizon, r → rs, so corrections become O(1).
Hawking radiation might not follow exact Born rule.
Test: look for anomalies in black hole thermodynamics.
At Planck time, L ~ ℓP, so corrections are maximal.
Primordial quantum fluctuations might deviate from |ψ|².
Test: CMB non-Gaussianities with specific gravitational signature.
Any experiment probing L ~ ℓP would see corrections.
Currently impossible, but sets a target.
Test: future quantum gravity phenomenology.
Massive superpositions in gravitational fields.
Decoherence rate should depend on gravitational potential.
Test: compare decoherence at different heights in Earth's field.
If the Born rule is gravitational:
Prediction: Born rule receives curvature-dependent corrections.
Null result: Born rule is exact even near black holes.
If the null result holds to arbitrary precision in curved spacetime, the gravitational origin is falsified.
Time is not a parameter. It is a byproduct.
In standard Quantum Mechanics, time (t) is an external parameter. The Schrödinger equation evolves states through time.
In General Relativity, time is a coordinate, malleable by mass.
Neither explains why time flows.
If the Born rule is the result of gravitational self-measurement, then "Time" is simply the sequence of those measurements.
The universe does not have a master clock. It has a refresh rate determined by the decoherence timescale:
This explains the "Static Superposition" prediction.
If a system is perfectly static (TTT = 0), the Bath cannot couple to it.
For a truly static quantum system, time effectively stops.
It remains in a state of "Undecided Potential" forever. Reality only advances when stress-energy moves, creating the gravitational ripples that force the universe to make a decision.
"Time is the rate at which the Universe resolves its own possibilities."
This unifies the Arrow of Time with the Born Rule:
We experience time moving forward because we are constantly radiating gravitational information into the Bath.
To exist is to bleed information into spacetime.
The arrow of time is the arrow of gravitational decoherence. They are the same thing.
| Concept | Traditional View | Gravitational View |
|---|---|---|
| Probability | Quantum postulate (|ψ|²) | Gravitational weight |
| Time | External parameter | Decoherence rate |
| Arrow of Time | Thermodynamic mystery | Information flow into Bath |
| Measurement | Collapse (unexplained) | Gravitational recording |
In this framework, all four reduce to one mechanism: gravitational self-measurement.
A binary test with current technology.
The gravitational Born rule makes a binary prediction that distinguishes it from all alternatives:
The Bath couples to transverse-traceless (TT) components of stress-energy — the gravitational wave degrees of freedom.
For a static superposition (no relative motion between branches):
For a dynamic superposition (branches have different velocities):
| Model | Static Superposition | Dynamic Superposition |
|---|---|---|
| Penrose-Diósi | Γ ≠ 0 (decoheres) | Γ ≠ 0 (decoheres) |
| Bath-TT (This Framework) | Γ = 0 (no decoherence) | Γ ≠ 0 (decoheres) |
One experiment. Binary outcome. Decides everything.
Penrose-Diósi prediction for static superposition:
For m = 10⁻¹⁴ kg, Δx = 10⁻⁶ m:
Bath-TT prediction for static superposition:
Gap: infinity vs 16 milliseconds. This is not a subtle effect.
To distinguish the models, we need:
Current state-of-the-art (2024): coherence times of ~1 ms achieved for m ~ 10⁻¹⁷ kg.
Required improvement: ~100× in coherence, ~1000× in mass.
Challenging but within reach for dedicated effort in 2026.
The key challenge is not the mass or the superposition size — it's creating a truly static superposition.
Most current experiments use momentum superpositions (dynamic) or let the branches fall (dynamic).
The test requires: same position, same velocity in both branches — differing only in location.
Achievable via: magnetic trapping at two sites, or optical lattice with controlled tunneling.
Static superposition shows NO gravitational decoherence.
Coherence persists beyond 100 ms (and keeps going).
Result: The Born rule IS gravitational. TT-coupling confirmed. Framework validated.
Static superposition decoheres at τ ~ 16 ms.
Decoherence rate matches Gm²/(ℏΔx).
Result: Gravitational self-energy causes collapse. Different mechanism. Framework falsified.
Either outcome is revolutionary. One experiment decides the origin of probability.
The experimental capability and the theoretical prediction are converging right now.
If this experiment works:
A decisive test of whether gravity is the origin of probability.
A speculative extension — not derived from the framework.
Speculation. Everything above — the envariance derivation, the shielding argument, the experimental predictions — stands on its own physics. This section does not. It asks a question that the framework makes natural but cannot answer.
The self-measurement question noted a structural parallel: gravity (in this framework) and consciousness are both self-referential processes. That parallel is suggestive but unproven.
The gravitational Born rule sharpens the question. If the parallel holds, the Born rule adds a specific structure:
If the Born rule is gravitational and time is the rate of gravitational measurement, then experience — if it corresponds to self-measurement — would not be continuous. It would tick.
This would predict: high neural activity → fast ticks → vivid experience. Static system → zero ticks → no experience.
Note: this is a chain of two unproven conjectures (gravity → Born rule, AND self-measurement → experience). Neither is established. The prediction is conditional on both.
If this speculation is correct, a truly static system (TTT = 0) would have no gravitational measurement events and therefore no experience — not because it lacks complexity, but because the Bath cannot couple to it.
This is more specific than most theories of consciousness, which struggle with borderline cases. But being specific doesn't make it right.
As discussed in the self-measurement page, the structural parallel between gravity and consciousness admits three readings:
The Born rule does not help us distinguish between these. It makes interpretation 3 more specific (by providing a rate), but no more proven.
The gravitational Born rule, if correct, would mean the universe resolves its own possibilities at a rate set by stress-energy dynamics.
Whether that resolution is accompanied by experience — whether there is something it is like to be a gravitational measurement — is a question the physics cannot currently answer.
It is the right question to ask. It is not yet the right question to answer.
"For 100 years we asked: why |ψ|²?
A candidate answer: gravity.
If probability is gravitational weight,
one experiment can prove it."
If this framework is correct, quantum mechanics and gravity are not two theories.
They are two aspects of one thing:
The universe assigning weight to possibilities.
Quantum mechanics tells you the weights exist.
Gravity tells you how the weights add up.
The Born rule would be where they meet. The experiment decides.
This framework makes a sharp prediction. Static superpositions should not decohere gravitationally. This can be tested.