Quantum Fisher Information as a Spectral Entanglement Witness: the Projective Stability Gap and the Origin of Scale-Free Growth

Abstract

A recent inelastic-neutron-scattering experiment on the heavy-fermion strange metal $\mathrm{Ce}_3\mathrm{Pd}_{20}\mathrm{Si}_6$ reports a scale-free increase of the quantum Fisher information (QFI) density as the strange-metal state forms, witnessing high multipartite entanglement. We ask whether this signature is reproduced by the non-injective projection framework of Cosmochrony, and we isolate the structural condition under which it is. We first reduce the QFI density to the projective spectral data: using linear response, the QFI density is a weighted integral of the imaginary part of the dynamical susceptibility, whose infrared behaviour is controlled by the projective stability gap ${\Delta_\Pi}$ and the edge spectral weight of the projective stability operator ${L_\Pi}$. This yields a clean dichotomy. A finite gap forces the QFI density to saturate; a gap that closes with an anomalous edge density of states of exponent $a$ forces ${f_Q}(T)\sim T^{1-a}$, so that scale-free growth requires $a>1$. We then evaluate the two ingredients by exact diagonalisation. The single winding sector $w=2$ reduces to the critical almost-Mathieu (Harper) operator at flux $2/q$; its gap closes as ${\Delta_\Pi}\sim q^{-1}$ but its edge spectral dimension is $d_s\to 2$, giving $a\to 1$, the marginal boundary, hence only logarithmic growth. The full Cayley substrate of ${\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})}$, whose homogeneous growth dimension is $D=4$, instead has an edge spectral dimension $d_s\approx 4$, giving $a\approx 2$ and ${f_Q}(T)\sim T^{-1}$, a genuine power law. The conclusion is structural: in this framework the scale-free QFI growth is a property of the full nilpotent substrate and not of any isolated composite, consistent with the collective, multipartite character of the measured entanglement. The required thermal closure ${\Delta_\Pi}(T)\sim T$ is obtained at the level of the effective heat-kernel response: the framework identifies the diffusion scale with inverse temperature, and the induced linear-response flow gives ${\Delta_\Pi}(T)=\tfrac{d_s}{2}T$, so that the chain from the nilpotent substrate to the linear prediction ${f_Q}(T)\sim T^{-1}$ closes within the effective response regime. The predicted exponent is of the right order but steeper than observed. Numerically, the Born–Infeld bounded-response correction can soften the effective exponent from the linear value $\alpha_{\mathrm{th}}=1$ towards the observed $\alpha_{\mathrm{exp}}\approx 0.7$, with a size-stable crossing at a partial substrate saturation. The asymptotic exponent and the saturation level realised by the material remain open.