The frozen universe: the maths

The Page–Wootters picture: a 12-reading clock entangled with a spin-½, in one stationary state of the whole. Exact, computed live.

Equations

|Ψ⟩ = (1/√N) Σ_k |k⟩_clock ⊗ U^k|↑⟩_spin: The whole universe's state: clock reading k paired with the spin turned k steps. It doesn't change.
θ_k = 2π f k / N: The spin's direction when the clock reads k (f turns per clock cycle).
P(up) = cos²(θ_k/2): The chance the spin is measured 'up the page', given that reading.
|⟨Ψ|W|Ψ⟩|² = cos²(πf/N): Without entanglement, applying one tick to the whole changes it by this overlap (1 would mean unchanged).

N	12	clock readings
f	1–3	spin turns per clock cycle

Clock reads 6, f = 1: spin direction (cos θ)	-1	θ = π: pointing the other way
Not entangled, f = 1: overlap after one tick	0.933	cos²(π/12): the whole has changed

The app's test suite puts the lab in each setup, reads the lab's own result, and fails if it strays from these values.

A toy universe: one clock, one spin. The real proposal concerns the whole universe.
The clock is ideal and discrete.

As a model of our universe, what counts as a 'clock' and how it's chosen are open questions — the reason it's tagged contested.

Sources: D. N. Page & W. K. Wootters, Phys. Rev. D 27, 2885 (1983); E. Moreva et al., Phys. Rev. A 89, 052122 (2014).