A companion paper, The Gravitational Baseline Principle, proposes a minimal pre-geometric quantum substrate and identifies two precise technical conditions that established theorems require before continuum geometry can be said to emerge from it: linear, Lorentz-consistent dispersion, and a mutual-information matrix that encodes a genuine metric distance. This paper tests both directly by exact diagonalization on the critical spin-1/2 antiferromagnetic Heisenberg chain, with a gapped or dimerized chain as a control. The dispersion condition is confirmed to high precision, matching the exact des Cloizeaux-Pearson result and decisively excluding a quadratic alternative. The metric-encoding condition is falsified in its originally hoped-for Euclidean form but not featurelessly: the specific functional form points toward a hyperbolic alternative, confirmed directly via the standard mathematical definition of hyperbolicity, the Gromov four-point delta, rather than a curve fit. An attempt to extend this to a 2D substrate is reported honestly as inconclusive at reachable system sizes, along with a genuine methodological finding: geometrically frustrated lattice shapes can have degenerate ground states that make naive mutual-information calculations non-reproducible unless explicitly checked. A final test sweeps several small dimerization strengths, checking whether the crossover away from hyperbolicity tracks the correlation length's own predicted field-theory scaling. The correlation length itself confirms the predicted scaling cleanly, but whether the geometric diagnostic's own breakdown tracks the same scale is only weakly supported and reported as an open result, not a positive one.
A companion paper, The Gravitational Baseline Principle, proposes a minimal pre-geometric quantum substrate and identifies two precise technical conditions that established theorems require before continuum geometry can be said to emerge from it: linear, Lorentz-consistent dispersion, and a mutual-information matrix that encodes a genuine metric distance. This paper tests both directly by exact diagonalization on the critical spin-1/2 antiferromagnetic Heisenberg chain, with a gapped or dimerized chain as a control. The dispersion condition is confirmed to high precision, matching the exact des Cloizeaux-Pearson result and decisively excluding a quadratic alternative. The metric-encoding condition is falsified in its originally hoped-for Euclidean form but not featurelessly: the specific functional form points toward a hyperbolic alternative, confirmed directly via the standard mathematical definition of hyperbolicity, the Gromov four-point delta, rather than a curve fit. An attempt to extend this to a 2D substrate is reported honestly as inconclusive at reachable system sizes, along with a genuine methodological finding: geometrically frustrated lattice shapes can have degenerate ground states that make naive mutual-information calculations non-reproducible unless explicitly checked. A final test sweeps several small dimerization strengths, checking whether the crossover away from hyperbolicity tracks the correlation length's own predicted field-theory scaling. The correlation length itself confirms the predicted scaling cleanly, but whether the geometric diagnostic's own breakdown tracks the same scale is only weakly supported and reported as an open result, not a positive one.
A companion paper (Range Versus Entanglement) built a graph Laplacian from the ground state mutual information matrix of the antiferromagnetic Heisenberg (XXZ) spin chain and found that AICc model selection on the lowest eight nonzero eigenvalues consistently prefers a linear dispersion over the quadratic dispersion predicted by a continuum Laplace-Beltrami operator. This paper traces that tension and finds it is substantially, though not entirely, a methodological artifact. The exact open chain dispersion relation has a curvature inflection at k=N/2, proved analytically and confirmed numerically; a fixed eight-mode fitting window straddles or approaches this inflection whenever N is 16 or smaller, biasing AICc toward the linear model for any locally supported adjacency matrix on an open chain, whether or not it is derived from entanglement. This is demonstrated directly on a content-free, unweighted nearest-neighbor chain, which reproduces the same crossover with no physics at all, and on two-body concurrence, which shows the same crossover as mutual information. Once the fitting window is defined adaptively to stay below the proven inflection, AICc prefers the quadratic model at every informative system size. A genuine residual survives this correction: the fitted curvature of mutual information's low-k spectrum has the opposite sign from concurrence's or the ideal chain's, indicating a real, still-unexplained feature specific to spin-spin correlation derived adjacency matrices.
Abstract Whether the spatial geometry implied by a quantum state can be recovered from its entanglement structure is a central question in the "space from Hilbert space" program. We study the sharpest tractable version of this question: can equal-time mutual information (MI) detect metric inhomogeneity, as opposed to topology and dimensionality alone? Working with inhomogeneous critical free fermions, whose emergent curved-CFT metric is known in closed form, we derive an exact expression for the Weyl-dressed conformal separation between two points, D/r = e^([ϕ(a)+ϕ(b)]/2) ⟨e^(−ϕ)⟩ with ϕ = ln v, and expand it in cumulants of ϕ. The leading term is δ1 = 1/2 [ϕ(a) + ϕ(b)] − ⟨ϕ⟩, which vanishes identically for linear ϕ and is small for many gently varying monotonic profiles; when it vanishes, the leading response degrades to 1/2 Var(ϕ). This explains a practical obstruction: the standard linear-v and linear-ln v ramp experiments are second-order-only probes of the emergent metric, and we verify numerically that it improves predictive variance over plain lattice distance by only ∆R^2 ≈ 4 × 10^(−3). The criterion is constructive. Profiles that place the inhomogeneity in the interval interior while matching endpoint velocities restore first-order sensitivity. Exact free-fermion computations on chains of N = 240 sites confirm the resulting predictions: MI is suppressed across a velocity valley and enhanced across a velocity bump at identical lattice separation, a sign reversal no lattice-distance model can produce; and 42 profiles drawn from four functional families collapse onto a single line ln[I_profile/I_flat] = −X_resp δ1 with X_resp = 1.64 and R^2 = 0.9969, improving to R^2 = 0.9991 when the second-order cumulant is included. We conclude that equal-time MI does encode the emergent conformal geometry, but that identifiability—not signal strength—is the binding constraint, and we give a design rule for experiments that respect it. Repository Contents & Replication Scripts This repository provides the complete, standalone codebase used to execute the exact free-fermion correlation-matrix computations and regenerate all numerical data tables, out-of-sample family cross-validations, and data collapse figures presented in the text. manuscript.pdf: Complete unblinded research article text. free_fermion_sim.py: Simulates the inhomogeneous hopping (XX) chains across N=240 site geometries and computes the exact fermionic mutual information matrices via Peschel's correlation matrix method. cumulant_analysis.py: Extracts the response coordinate coordinates, maps the profile variants (Gaussian, sech², Lorentzian, raised cosine), and handles the multi-family data collapse regression profiles. Environment Requirements: Python 3.8+ with standard scientific computing installations (numpy, scipy, matplotlib). Double-Blind Review & Anonymity Compliance Author names, institutional affiliations, contact emails, and regional funding details are permanently withheld from these files to guarantee total anonymity. Content processing and textual structuring were partially optimized using conversational generative AI models, with all underlying mathematical derivations, code architecture, and numerical assertions independently validated by the author.
Mutual information across the commensurate--incommensurate crossover and near the Kosterlitz--Thouless point of the spin-21J1--J2 Heisenberg chain
What quantum mutual information adds to conventional correlation functions is often unclear. We study the spin-1/2 J1-J2 Heisenberg chain, using DMRG on infinite chains and periodic rings and an SU(2) decomposition of the mutual information into correlation channels. For singlet ground states the two-site mutual information is an exact function of the spin correlator, so any additional content must come from blocks. In the dimerized phase, blocks resolve bond-energy correlations in the SU(2)-scalar channel without an independent asymptotic length: singlet and triplet correlation lengths agree to about 1%. The scalar channel dominates block mutual information at large distances in the commensurate regime up to J2/J1=0.40, but falls to a few percent beyond the disorder point, which coincides numerically with the Majumdar-Ghosh point J2/J1=1/2; the structure-factor peak splits only at the Lifshitz point J2/J1=0.520622(2). On the gapless side we compare block mutual information on rings extrapolated to infinite size with the two-interval prediction of SU(2)1 conformal field theory across the Kosterlitz-Thouless point J2/J1=0.241167. There, for block lengths l≤6, the correction decays as l−p with p=1.96-2.07, and any logarithmic term is at least 60 times smaller than in the Heisenberg chain. Expanding in the marginal coupling (from level spectroscopy) the first-order coefficient is consistent with l−2 scaling and the second-order coefficient is positive for x≤1/4, suggesting that the mutual information relative to the field-theory value becomes stationary at the KT point for large blocks. The scalar-channel share approaches 1/4 at small cross ratio x, as expected from SO(4) symmetry; under two stated assumptions its first-order response to the coupling is (3/8)lnx−5/8 at small x, matched by the l=4 data within 1-3% at x≤1/9.
This paper builds a graph Laplacian from the ground state mutual information matrix of the isotropic antiferromagnetic Heisenberg (XXZ) spin chain and uses it as a case study in what this class of construction does and does not establish. A first-pass analysis reproduces several textbook-looking signatures, including a statistically significant nonzero finite-size intercept in the lowest nonzero eigenvalue that persists and grows under periodic boundaries, naively suggestive of an entanglement-induced mass scale. This reading does not survive scrutiny: the intercept's magnitude varies by an order of magnitude under different adjacency normalizations and can change sign under an AICc-preferred finite-size model with logarithmic corrections. Repeating the construction with the classical, non-entanglement, connected spin correlator reproduces the same intercept phenomenon more cleanly; toggling only quantum coherence on and off via discord leaves it almost unchanged; and restricting to an unambiguous entanglement-only measure, two-body concurrence, collapses the intercept to a value statistically indistinguishable from zero. Across every construction tested, the intercept tracks whether the underlying matrix has long-range support, not whether it is quantum in origin.
EP4-C documents the results of EP4A — a Lindblad Morse Gradient Test on the 3-qubit XXX Heisenberg spin chain using published ibmq_jakarta hardware parameters (T1=119 μs, T2=113 μs) — and identifies a structural consequence extending to the BCS superconductivity experiment (EP5-BCS). Core findings: (1) The Morse correlation dS = λ · D²_B is confirmed at 14.7× signal-to-noise on the Lindblad analytic trajectory. All thresholds were pre-declared before simulation. Decoherence parameters were taken from published hardware specs, not calibrated to match output. The Heisenberg revival F(|110⟩, U(π)|110⟩) = 1.000000 is verified. (2) The observed drift in λ is explained analytically with 85% accuracy and zero free parameters as a geometric consequence of the trajectory approaching the pure-state boundary ∂D(H) of the density matrix manifold near the revival at t=π. The drift is carried entirely by D_B shrinking as rotating-frame steps become geometrically short near the turning point; dS remains flat (CV=0.175). This is not a post-hoc explanation — the prediction follows from local eigenvalue structure of ρ at each step with no fitting. (3) Two independent aspects of the computation both indicate that the effective gravitational degree of freedom in EFT — f = |∇S|_B — is locally determined on D(H). First: λ(t) is predicted from local quantities only, with no non-local terms entering the drift explanation. Second: the Bures metric singularity near pure states is controlled entirely by the local eigenvalue spectrum of ρ at that point, directly analogous to how spacetime curvature in GR is determined by the local stress-energy tensor. (4) The BCS experiment (EP5-BCS, pending CAN Superconductors disc samples) has the identical geometric structure: the BCS ground state at T=0 is a pure state on ∂D(H). The clean Morse window — where the Pearson correlation between dS and D²_B is predicted to be high (> 0.85) and λ to track the zero-parameter geometric prediction — is 0.2 < T/T_c < 0.8, derived from EFT geometry before any BCS measurement. Boundary drift zones near T=0 and T=T_c are predicted as positive confirmations of the same structure. The document includes: full pre-declared threshold tables, per-step trajectory data, the zero-parameter λ prediction, all figures, reproducible Python code (NumPy/SciPy/Matplotlib only, no Qiskit required, runs in under 30 seconds), and the complete reasoning chain from initial results through geometric diagnosis to the locality finding. Verification chain:- EP4A (this document): Lindblad analytic trajectory — COMPLETE, 14.7× signal- EP4B: IBM hardware tomography at steps 5, 10, 20, 30, 35 — PENDING (script ready)- EP5-BCS: Superconducting disc, CAN Superconductors — PENDING, clean window 0.2 < T/T_c < 0.8 Related work:- EFT standalone: DOI 10.5281/zenodo.18963986- EQ(vT): DOI 10.5281/zenodo.18917317- EP4A (Morse Gradient Test, Lindblad): DOI 10.5281/zenodo.18969071- BCS pre-registration: DOI 10.5281/zenodo.18964362
Can spatial distance and proper time be defined operationally — by how a system leaks information to its environment — rather than postulated as primitive geometric quantities? This paper answers affirmatively for both, within Markovian open quantum dynamics, building on the survival-expansion framework of the companion paper "Local Curvature of Quantum Information" (doi:10.5281/zenodo.18792424). Spatial distance is defined as the excess irreversibility generated by superposing two states: when a system is superposed over two locations, the environment learns which at a rate growing with their separation. This distance is proven to be a genuine metric — it satisfies the triangle inequality via an isometric embedding into environmental "shadow space", recovers the Euclidean metric of R³ for independent position channels, and becomes undefined before decoherence even for orthogonal states, so that geometry and classicality emerge together. Proper time is defined as elapsed background time minus the minimal dissipation needed to traverse a worldline. The reversed triangle inequality of relativity (the twin paradox) then follows automatically from the subadditivity of optimal costs, and the exact Minkowski interval corresponds to a dissipation rate that saturates at high transport velocity: the light cone as a bound on irreversibility production. Both Lorentzian signatures are traced to a single convex-analytic root: spatial distance is directly a subadditive object (a norm), while proper time is an additive duration minus a subadditive optimal cost, hence superadditive. In an exactly solvable driven damped-oscillator model, the information deficit coincides identically with Clausius entropy production, the minimal transport deficit obeys Λ∗=2kBTbγmTΔx2, and the emergent light speed is c2=γmκkBTb=κD, with D the Einstein diffusion coefficient. The universality of the emergent light cone is shown to be equivalent to a Compton-clock scaling of the dissipation–time conversion rate (κ=ℏ2mc2), with the quantum–classical crossover condition kBTb=2ℏγ arising as an internal consistency requirement rather than a postulate. A junction-cost theorem establishes that velocity discontinuities incur strictly positive excess dissipation, so the turning twin ages strictly less even at equal speeds. All results are verified numerically on lattice decoherence models, exact linear–quadratic optimal control, and Fock-space computations; the verification scripts accompany the manuscript. Locality of the environmental couplings is assumed, not derived, and the construction has analog-gravity status: a Lorentzian interval in a preferred frame defined by the bath.