Is the deformation in Eq. 56 really innocuous?

[Roberto Pavani]

TL;DR

In Eq. 56 the observable is changed from sign(φ) to sign(φ+α). Since shifts in function spaces generally modify overlaps and correlations, I'm struggling to see why this deformation should be considered innocuous. Is there a theorem guaranteeing that the relevant correlation structure is preserved, or is the altered correlation structure precisely what produces the Bell-CHSH enhancement?

I'm reading these papers:

https://arxiv.org/pdf/2412.03840

https://link.springer.com/article/10.1140/epjc/s10052-026-15445-1

and I'm struggling with Eq. (56).

The Bell-CHSH enhancement appears only after replacing

$sign(φ(f))$

with

$sign(φ(f)+α)$

whereas the undeformed observable does not seem to violate the Bell-CHSH bound.

My concern is that this is not just a reparametrization. It changes the threshold that partitions the spectrum of φ(f), from 0 to -α, and therefore changes the induced probabilities and correlations.

From my perspective, changing a threshold generally changes the geometry of the probability space. Two observables may have one correlation structure before thresholding and a different one after thresholding, especially when the threshold itself is shifted.

Is there a theorem showing that this deformation preserves the correlation structure relevant to the Bell-CHSH analysis, or is the modified correlation structure itself part of the mechanism that generates the enhancement?

Another point that confuses me is that the paper does not introduce a single common shift, but four independent parameters

$α, α', β, β'$

for the Bell observables.

If these shifts modify the probability partitions, then each observable is being deformed differently. Why should the resulting CHSH value be regarded primarily as a property of the underlying vacuum state rather than, at least in part, a consequence of the observable-dependent threshold choices?

I am probably missing something, but I do not see why the transformation in Eq. (56) should be considered innocuous from the point of view of correlations.

 

[emillindberg]

I think your concern is reasonable. A shift inside sign(φ+α) is not obviously a harmless reparametrization, because the sign operation makes the threshold change physically relevant to the resulting dichotomic observable. In particular, allowing four independent α values seems capable of changing the correlation functions being tested. It would be helpful if the authors explicitly derived whether the CHSH enhancement persists for fixed/common α, or explain precisely why the observable-dependent shifts should be regarded as part of the measurement definition rather than an additional source of the violation.