Competing Informational and Dynamical Criteria for Subsystem Structure: Relational Emergence Model — Revised Version 5

Abstract

REM_lambda Version 4 Version 4 presents a corrected numerical implementation of the variational selection framework used in the Relational Emergence Model (REM). The model evaluates candidate Hilbert-space factorisations through competition between informational differentiation and dynamical boundary cost. $$\Phi(F)=\Phi_S(F)-\lambda C_H(F)$$ The informational term is $$\Phi_S(F)=I(A:B)$$ and the sign-definite quadratic dynamical cost is $$C_H(F)=\mathrm{Tr}(\rho H_{\partial F}^{2})=\langle H_{\partial F}^{2}\rangle$$ Numerical correction in Version 4 Version 4 corrects two implementation issues in the numerical analysis accompanying REM_lambda Version 3. The discrete-cut implementation previously calculated $$ \langle H_{\partial F}\rangle^{2} $$ instead of the intended quadratic boundary-fluctuation cost $$ \langle H_{\partial F}^{2}\rangle $$ The previous continuous-factorisation routine transformed the state and the full Hamiltonian together, making the dynamical cost invariant under the candidate unitary transformation. Consequently, the Version 3 optimiser effectively maximised mutual information rather than the complete REM functional. Version 4 corrects both issues and recomputes all affected numerical results. Three-qubit XY-chain benchmark The numerical benchmark uses the asymmetric three-qubit XY-chain Hamiltonian $$H=J_{12}(X_1X_2+Y_1Y_2)+J_{23}(X_2X_3+Y_2Y_3)+h\sum_i Z_i$$ The parameters are $$J_{12}=1.5,\qquad J_{23}=0.6,\qquad h=0.2$$ The ground state is obtained through exact diagonalisation. For the two site-contiguous candidate partitions, the corrected dynamical cost is evaluated as $$C_H=\langle H_{\partial F}^{2}\rangle$$ At $$\lambda=0.2$$ the best site-contiguous partition achieves $$\Phi=0.560$$ The crossover between the information-dominated and dynamics-dominated contiguous partitions remains $$\lambda^*\approx0.165$$ Continuous factorisation optimisation For each candidate unitary $U$, the rotated Hamiltonian is $$H_U=UHU^\dagger$$ It is decomposed into local and interaction components: $$H_U=H_A(U)+H_B(U)+H_{\partial}(U)$$ The local components are obtained through partial-trace projection with an identity-component correction. The continuous optimisation evaluates the complete REM functional: $$\Phi(U)=I(A:B)_{U\psi}-\lambda\langle\psi_U|H_{\partial}(U)^2|\psi_U\rangle$$ A search over a four-parameter generator submanifold recovers $$\Phi=1.05$$ This exceeds the best site-contiguous result: $$\Phi=0.560$$ The corrected numerical improvement is therefore $$0.560\rightarrow1.05$$ The previously reported Version 3 result $$0.68\rightarrow1.32$$ is withdrawn because it was obtained using an incorrect objective function. Factorisation-space dimension The full factorisation space is represented by $$U(8)/(U(4)\otimes U(2))$$ The tensor-product representation of $U(4)\times U(2)$ has a one-dimensional kernel. The effective local-unitary subgroup therefore has dimension $$16+4-1=19$$ Since $$\dim U(8)=64$$ the quotient manifold has real dimension $$64-19=45$$ The present numerical implementation explores a four-parameter generator submanifold of this 45-dimensional quotient manifold. The recovered factorisation is a higher-scoring solution within the explored submanifold. It is not claimed to be the global optimum over the full factorisation manifold. Finite-size scaling Finite-size calculations were performed for $$N=3,4,6,8$$ under weak-center, linear, and step coupling profiles. All finite-size values were recomputed using the corrected quadratic dynamical cost $$C_H=\langle H_{\partial F}^{2}\rangle$$ Validation and reproducibility The accompanying reproducibility package includes: exact diagonalisation for asymmetric XY chains; evaluation of site-contiguous candidate cuts; factorisation-dependent reconstruction of the interaction Hamiltonian; continuous optimisation on a four-parameter generator submanifold; finite-size scaling calculations; deterministic benchmark values; 16 automated tests; GitHub Actions continuous integration for Python 3.12 and 3.13. The regression tests verify: exact agreement with $\langle H_{\partial F}^{2}\rangle$; the distinction between $\langle H_{\partial F}^{2}\rangle$ and $\langle H_{\partial F}\rangle^{2}$; factorisation dependence of the reconstructed boundary cost; consistency at the identity transformation; reproduction of corrected benchmark values; successful finite-size and output-generation workflows. Scope Version 4 supersedes the numerical results reported in Version 3. The theoretical REM functional and the use of the sign-definite quadratic boundary-fluctuation cost remain unchanged. Source and reproducibility GitHub repository: https://github.com/malko73/rem Version 4 DOI: https://doi.org/10.5281/zenodo.21427776