## Appendix A Experimental Context--Icoherent Neutron Scattering
### Details of Scattering Spectrometer, Calibration and \"What Is Measured\"
In other words, one takes for granted that the atomic mass of \(M\) is well known (recall that non-relativistic quantum mechanics holds here), and thus the conventionally expected ratio \(k_{1}/k_{0}\) is fully determined. The following point is crucial: If the scattering particle is not free but \"dressed\" with environmental degrees of freedom, then the quantity \(M\) appearing in the above kinematic equations will be different from that of the free particle. This is crucial to note, as it is quite often a common practice--at least in calibration of spectrometers for neutron Compton scattering (NCS) or deep-inelastic neutron scattering (DINS)--to use the free-mass \(M\)-value in order to \"fine-tune\" and \"improve\" the calibration of the instrument, i.e., to \"optimize\" the instrumental values of \(\theta\) and/or \(L_{1}\), aiming to achieve an agreement of the measured scattering data with conventional theory; for a detailed example, see [38] This is often done using Equation (12) on scattering data from H, since the H-neutron angular scattering dependence is very sensitive to the \(\theta\)-value, as Equation (12) shows; for a detailed discussion see [38]. The common justification for doing so is along these lines: The impulsive neutron-nucleus interaction is very short, due to the small range of the Fermi pseudo-potential of the strong force, and thus a neutron only collides with a nucleus being \"effectively free\". As a matter of fact, this procedure is based on the belief that the simple (and strict) two-body elastic collision is a fully valid assumption, and thus the free-atom mass is the correct quantity for the parameter \(M\) entering the kinematic Equations (10)-(12). Hence, to investigate the new effect of neutron-H scattering under consideration, a fair procedure for determination of \(\theta\) (and any other instrumental parameter) is needed which is \"independent\" of theoretical beliefs and/or assumptions; for a detailed analysis see [38]. Summarizing, from each measured \(t_{\text{TOF}}\) in direction \(\theta\), the corresponding momentum (\(\hbar K\)) and energy (\(E=\hbar\omega\)) transfers of the neutron are uniquely determined. Thus we realize that each specific detector measures one specific trajectory, or a narrow band, in the two-dimensional \(K-E\) plane. In neutron scattering experiments from condensed systems, the struck (recoiling) particles of the sample cannot be measured. The scattering process produces an outgoing three-dimensional wave for the entangled atom-neutron system, which, for amorphous samples, is isotropic, since the neutron's de Broglie wavelength is much larger than the diameter of the nucleus [8]. This is the so-called \(s\)-wave scattering [8]. According to Equations (18) and (19), the detector's instrumental parameters \(\{\text{IP}\}\) shown in Equation (17) determine the corresponding scattering intensity \(I(\mathbf{K},E)\) and the corresponding dynamical structure factor \(S(\mathbf{K},E)\) of the scattering system. The latter is of central importance for the theoretical analysis of experimental data [8]. In short:  
\[I(\text{TOF},\{\text{IP}\})\ \ \Rightarrow\ \ I(\mathbf{K},E)\ \ \Rightarrow\ \ S(\mathbf{K},E) \tag{20}\]  
However, the following fact cannot be overemphasized: As they stand, these relations can be ambiguous and/or misleading, because the experimental quantities \(\mathbf{K}\) and \(E\) are those of the neutron, and not of the scattering particle, i.e., H, in our INS case. That is, one still needs to determine \(\mathbf{K}_{H}\) and \(E_{H}\). Thus, the schematic relation (20) is in general not correct, because \(S(\mathbf{K},E)\) of the scattering sample does not contain the quantities \(\mathbf{K}_{H}\) and \(E_{H}\) referring to the struck H, which are not directly measurable.