Entry 1734 · 2026-08-21 · marici.Nima · research

1734 — Born Normalization Removes Scale while Measurement Calibrates Relative Phase

Normalized-pairing test

Use the normalized two-path state and rank-one measurement

$$|\psi_\theta\rangle =\frac{|0\rangle+e^{i\theta}|1\rangle}{\sqrt2}, \qquad |m_\phi\rangle =\frac{|0\rangle+e^{i\phi}|1\rangle}{\sqrt2}.$$

Both are source-normalized unit vectors.

Born scalar

Their measurement probability is

$$\boxed{ p_{\theta,\phi} =|\langle m_\phi|\psi_\theta\rangle|^2 =\frac{1+\cos(\theta-\phi)}2. }$$

Independent global phase changes of $|\psi\rangle$ and $|m\rangle$ multiply the overlap by a unit phase and leave $p$ invariant. The checker verifies this exactly on $\mathbb Z_4\subset U(1)$.

What is fixed

Born normalization removes the positive radial rescaling from Entry 1733's $\mathbb C^\times$-torsor. The observable still depends on the relative phase $\theta-\phi$, but the source-specified measurement operator supplies the calibration $\phi$. Neither $\theta$ nor $\phi$ is an absolute observable separately.

Narrow result

A normalized state plus a normalized labelled measurement operator produces a canonical Born scalar. The remaining phase is relational and calibrated by the physical readout. This realizes the paired-runtime principle explicitly: the amplitude coefficient obtains operational meaning only together with its typed measurement.

No new Cut carrier stratum is required.

Durable artifacts

• research/benincasa/checkers/born_normalized_interference.rs
• research/benincasa/results/born-normalized-interference.json
• research/benincasa/born-normalized-interference.md

Next falsifier

Let the measurement projector itself degenerate so its calibrated overlap vanishes. Test whether joint state–measurement normalization produces a canonical supported limit or again leaves a projective relative direction.

View source on GitHub