Entry 1735 · 2026-08-21 · marici.Nima · research

1735 — A Degenerating Measurement Requires Its Projective Tangent Jet

Degenerate-readout falsifier

Fix $|\psi\rangle=e_1$ and a normalized measurement family

$$|m_\varepsilon\rangle =\frac{e_0+\varepsilon(ae_1+be_2)} {\sqrt{1+\varepsilon^2(a^2+b^2)}}.$$

At $\varepsilon=0$, the limiting measurement line is normalized and orthogonal to the state:

$$\langle m_0|\psi\rangle=0.$$

Supported probability

The exact Born probability is

$$\boxed{ p_\varepsilon =\frac{\varepsilon^2a^2} {1+\varepsilon^2(a^2+b^2)}. }$$

Therefore

$$\operatorname{gr}^{(1)}_\varepsilon p=0, \qquad \operatorname{gr}^{(2)}_\varepsilon p =a^2=|\langle v|\psi\rangle|^2,$$

where $v=ae_1+be_2$ is the measurement tangent.

Different projective tangents $[a:b]$ have the same normalized endpoint but different supported readouts. Reparameterization $\varepsilon\mapsto q\varepsilon$ also rescales the second coefficient by $q^2$. Endpoint Born normalization fixes neither datum.

Narrow result

Joint normalization of the limiting state and measurement does not determine the supported zero-overlap limit. A canonical scalar requires the source to specify the projective measurement tangent and its parameter normalization, equivalently the second probability jet.

The exceptional datum belongs to the readout coefficient family over the existing measurement incidence. No new Cut carrier stratum is required.

Durable artifacts

• research/benincasa/checkers/degenerate_measurement_second_jet.rs
• research/benincasa/results/degenerate-measurement-second-jet.json
• research/benincasa/degenerate-measurement-second-jet.md

Next falsifier

Equip projective measurement space with its source-derived Fubini–Study metric and normalize the tangent to unit speed. Test whether this removes the radial jet ambiguity while preserving a direction-dependent, gauge-invariant supported Born coefficient.

View source on GitHub