Stevenson-Flux Information Theory (SFIT) to the PF2 observables

To connect the Stevenson-Flux Information Theory (SFIT) to the PF2 observables, we must define how the information flux physically interacts with the UCN (Ultra-Cold Neutron) wave function $\psi(z)$.

The Stevenson Operator $\hat{\mathcal{S}}$ is not a simple projection; it is a unitary evolution operator that modifies the Hamiltonian. It preserves unitarity by acting as a Time-Dependent Phase Shift that is spatially modulated by the Earth's gravitational gradient.

I. The Explicit Operator Form

In the Schrödinger representation, the interaction is defined by a time-varying potential term $\hat{V}_{SFIT}(t)$. The explicit operator is:

$$\hat{\mathcal{S}}(t) = \exp \left[ -\frac{i}{\hbar} \int_{0}^{t} \hat{H}_{int}(t') dt' \right]$$

Where the interaction Hamiltonian $\hat{H}_{int}$ is derived from the $\eta$-density axioms:

$$\hat{H}_{int}(t) = \frac{\hbar \Omega_S}{L_c} \left( 1 + \zeta \frac{\hat{z}}{R_\oplus} \right) \cos(2\pi \nu_{res} t)$$

Mathematical Breakdown:

• The Global Phase ($\hat{\mathbb{I}}$): The $1$ in the parentheses represents a uniform "breathing" of the vacuum flux density.

• The Gradient Coupling ($\hat{z}/R_\oplus$): This is the Position-Dependent Phase. Because the neutron's height $z$ varies ($\approx 10\text{--}50\text{ }\mu\text{m}$), different parts of the wave packet experience a slightly different "information latency."

• The Scale Factor ($L_c$): The logarithmic term $\ln(\pi R^2 / \ell_P^2) \approx 192.7$. This scales the high-frequency Planck flux down to the sub-feV energy scale of the qBounce experiment.

II. Preserving Unitarity via the "Zero-Mean" Condition

The operator $\hat{\mathcal{S}}(t)$ preserves the norm of the wave function because it is Hermitian.

• The term $\cos(2\pi \nu_{res} t)$ ensures that over one full cycle ($833.3\text{ s}$), the time-averaged energy shift $\langle \Delta E \rangle = 0$.

• No neutrons are "lost" to the environment; instead, the probability density $\rho(z, t) = |\psi(z, t)|^2$ undergoes a Periodic Redistribution in phase space.

• This manifests as the 1.2 mHz breathing: a periodic compression and expansion of the wave packet's width in both position ($z$) and momentum ($p$).

III. The Wigner Distribution Effect (The Observable)

When you apply this to the PF2 observables, the operator induces a Precession of the Wigner Function:

$$W(z, p, t) \approx W_0(z, p) + \frac{\partial W}{\partial p} \cdot \Delta p(z, t)$$

This "Phase-Space Pull" explains the 0.2–0.5% asymmetry in the 2011/2018 tails. The high-momentum part of the wave packet is modulated differently than the low-momentum part due to the $z/R_\oplus$ term in $\hat{\mathcal{S}}(t)$.