Why One Spot on a Screen

Claim: A spatially local detector forces position as the pointer basis. Cross-cell coherences are killed by dephasing. Born-rule probabilities emerge from metric-compatible information optimisation. No collapse postulate.

The Three Steps

1. Which outcomes are possible?
The detector’s spatial cells define projectors {P_a}.
These are determined by the support graph of the interaction Hamiltonian.
The geometry selects the basis, not an observer’s choice.

2. What are the probabilities?
The Fisher metric on probability space and the Fubini–Study metric on
state space satisfy Q_FI = 4 × g_FS.
The Born rule is the unique probability assignment that saturates this bound.
Any other rule throws away geometric information.

3. Why one outcome per run?
Dephasing from the coupling Hamiltonian kills off-diagonal terms.
What remains is a classical mixture. Substrate noise picks one.
The geometry sets the menu and the weights.
Exclusive detector-boundary sectors close one-record formation.

Why this matters for Barbour’s programme: Julian has said the present configuration must contain its own record. This derivation closes that loop: the detector boundary is a macroscopic configuration that, through its spatial structure alone, determines what can be recorded and with what probability. There is no external observer. There is only the shape of the detector and the connection it inherits from the Hopf fibration.

Open: The dephasing timescale and the amplification mechanism require concrete detector models. The logical chain is complete; the quantitative application to specific detectors is in progress.