DA as the compression–geometry axis: a holographic length that classifies emergent geometry across quantum criticality, gravity, and the Landauer floor
The holographic bound writes a region's entanglement entropy as an area, S=A/4DA². Read as a length, DA is the area carried per bit of entanglement, bounded below by the Planck length. This note argues that DA is not a rewriting of the entropy but a different object with two powers the entropy as a number does not have: it is commensurable across domains, and it is geometric. On the first, the filling fraction f=DA²/L² places a quantum critical chain, a black hole, and a single-bit Landauer erasure on one DA axis, where the three entropies — entanglement, Bekenstein–Hawking, classical — are not directly comparable. On the second, the scaling of the entanglement length builds an emergent geometry: for a one-dimensional critical system DA forces, through Ryu–Takayanagi, an emergent AdS of constant negative curvature, while a gapped system gives a flat interior. We verify the geometric content numerically through the entanglement c-function, which is flat to 10⁻⁴–10⁻⁶ for critical chains (scale invariance, constant curvature) and decays to zero for a gapped chain (flat interior). The running rate β equals the central charge as an identity of the Calabrese–Cardy formula, not a fit; free-fermion calculations show β→c/6 as the finite-size correction vanishes from L=64 to L=4096. A second condition, the log-fit quality R², is mandatory: a volume-law state forges a spurious β that only R² exposes. The diagnostic becomes a geometry classifier with four classes — critical (hyperbolic, β>0), gapped (flat, β→0), glassy, and thermal — each carrying a clean representative, the glassy class realized by a genuine interacting disorder model (many-body localization in the random-field Heisenberg chain, which sweeps from a volume-law ergodic interior to an area-law localized interior). We state the honest boundaries: the central-charge reading does not extend to higher dimensions, to dynamics, or to topological order, and the cross-domain DA axis is an organizing coordinate, not a claim that the three systems are one physics. The contribution is a unifying compression–geometry perspective in which DA, as a holographic length, is the axis — not a new force, and not a renaming of the entropy.
The holographic bound writes a region's entanglement entropy as an area, S=A/4DA². Read as a length, DA is the area carried per bit of entanglement, bounded below by the Planck length. This note argues that DA is not a rewriting of the entropy but a different object with two powers the entropy as a number does not have: it is commensurable across domains, and it is geometric. On the first, the filling fraction f=DA²/L² places a quantum critical chain, a black hole, and a single-bit Landauer erasure on one DA axis, where the three entropies — entanglement, Bekenstein–Hawking, classical — are not directly comparable. On the second, the scaling of the entanglement length builds an emergent geometry: for a one-dimensional critical system DA forces, through Ryu–Takayanagi, an emergent AdS of constant negative curvature, while a gapped system gives a flat interior. We verify the geometric content numerically through the entanglement c-function, which is flat to 10⁻⁴–10⁻⁶ for critical chains (scale invariance, constant curvature) and decays to zero for a gapped chain (flat interior). The running rate β equals the central charge as an identity of the Calabrese–Cardy formula, not a fit; free-fermion calculations show β→c/6 as the finite-size correction vanishes from L=64 to L=4096. A second condition, the log-fit quality R², is mandatory: a volume-law state forges a spurious β that only R² exposes. The diagnostic becomes a geometry classifier with four classes — critical (hyperbolic, β>0), gapped (flat, β→0), glassy, and thermal — each carrying a clean representative, the glassy class realized by a genuine interacting disorder model (many-body localization in the random-field Heisenberg chain, which sweeps from a volume-law ergodic interior to an area-law localized interior). We state the honest boundaries: the central-charge reading does not extend to higher dimensions, to dynamics, or to topological order, and the cross-domain DA axis is an organizing coordinate, not a claim that the three systems are one physics. The contribution is a unifying compression–geometry perspective in which DA, as a holographic length, is the axis — not a new force, and not a renaming of the entropy.
One of the boldest ideas in modern physics is that space itself is a kind of error-correcting code. In this picture the interior of a region of spacetime is like a message, and the surface surrounding it is an encoded copy that stores the message redundantly, so that losing part of the surface need not lose the interior. A celebrated formula from holography then says something unexpected: the amount of entanglement carried by a boundary region equals the area of a surface reaching into the interior. This paper gives a complete, elementary proof that this area law is not an extra assumption bolted on by hand, but an unavoidable consequence of the code structure itself. The geometry is simply bookkeeping for redundancy. The derivation reduces a statement about gravity to plain facts about entanglement and about the algebra of operations one can perform on the encoded information. It establishes four things, each from scratch. First, in the simplest case it recovers the area law directly: the entanglement of a boundary region is a fixed geometric term plus the entanglement of whatever the interior itself carries. Second, it proves a symmetry that had previously been handled case by case, namely that a pure joint state assigns the same entanglement to a region and to its complement, and it proves this in full generality using a single clean identity about matrices. Third, it extends the area law to the realistic situation in which the code splits into separate sectors, each carrying its own independent area, so that the geometric term is not a single number but a genuine operator averaged over the sectors. Fourth, it proves the structural fact the whole picture rests on: the algebra of logical operations always decomposes cleanly into independent blocks, which is exactly the sector structure the area law needs. That last step comes down to one small analytic fact, that a matrix which is symmetric in the appropriate sense and also nilpotent must be zero, and the paper proves it directly. A companion result closes the reconstruction side of the story, showing that the operations recoverable from a boundary region are precisely those left untouched by the coarse-graining onto sectors, and that the coarse-grained state already reproduces every such measurement. Every step is elementary and self-contained, and the entire logical core has a companion version reconstructed inside a machine proof assistant, checked line by line with no gaps and resting only on the standard axioms of mathematics. The upshot is that a piece of geometry, an area, can be read off from nothing more than how redundantly a code stores its information. None of the underlying mathematics is new; it traces back to established results on holographic codes, to the entropy formula itself, and to the century-old classification of matrix algebras. What is new is the assembly: a short, honest, self-contained line of reasoning that turns a claim about spacetime into a handful of reusable facts about entropy and matrices, with every link exposed and checkable. Where the picture still reaches past what has been proved, in particular the finer tensor structure hidden inside each sector, the paper marks the boundary plainly instead of hiding it.
The Ryu-Takayanagi formula equates the area of a minimal surface with the von Neumann entropy of a boundary subregion, and leaves two things about that identification open.
The first is how sharply a geometry fixes an entropy. Fannes-Audenaert answers with a Hilbert-space dimension, which diverges as the cutoff is removed however close the two states are. We replace it with the capacity of entanglement, the variance of the modular energy, whose square root grows like the square root of the entangling area where the dimensional factor grows like the regulated volume. The bound is dimension-free and saturated, and it makes the ambiguity of the entropy subextensive for any perturbation whose capacity is small compared with SvN2 times the trace norm.
The second is what the area means for a single state, since compression and dilution rates are defined only for many copies while a geometry describes one. When a single replica saddle dominates near α=1, every smooth Rényi entropy at fixed α>1 agrees with SvN to O(SvN), as do the smooth min- and max-entropies. The minimal surface therefore fixes every one-shot entropy of the state at once, with large central charge playing the role of large copy number in the asymptotic equipartition property.
As a consequence we bound how far outside the holographic entropy cone a holographic state can appear to fall, leaving estimation and certification open.
We study holographic entanglement entropy in 3-dimensional AdS gravity with conformal boundary conditions which fix the conformal class of the boundary metric and its extrinsic curvature, K, while leaving the Weyl mode dynamical. Extending Lewkowycz-Maldacena-Dong's replica construction, we derive the corresponding holographic entanglement entropy formula. We show that the fluctuating Weyl mode does not contribute additional entropy. The entropy of the full boundary is therefore the Bekenstein-Hawking entropy, and the entropy of a boundary subregion continues to obey the Ryu-Takayanagi prescription, namely, the area of a minimal surface divided by 4GN. We carry out explicit calculations for global AdS, rotating and non-rotating BTZ geometries. For an interval in AdS3 we find that the K-dependent holographic entanglement entropy is governed by cm=2GN3ℓ. For a thermal state at high conformal temperatures we find that, the entropy is governed by ceff=2GN3ℓ2Kℓ−K2ℓ2−4, the same effective central charge that governs the Cardy-like density of states, in agreement with previous results in the literature. Finally, we also compute the entanglement entropy directly from the conjectured dual boundary theory - a holographic CFT coupled with time-like Liouville theory and deformed by a marginal TTˉ like operator - and find SEE=3ceffln(ε2Rsinϕ0). This result for the state with no operator insertions (vacuum state ), provides an independent boundary realization of ceff while clarifying that this state is not the state dual to global AdS.
In this paper, we discover the topological quantization and phase transition mechanisms of holographic extremal surfaces in discrete tensor networks, providing a rigorous discrete dynamical benchmark for the proposition that "gravity is entanglement". Using a one-dimensional spin network with a fixed truncated hopping step k, we derive a rigorous analytical solution for the peak position of entanglement entropy, demonstrating that it is solely determined by the underlying connectivity step k. The perturbative phase diagram further reveals that this quantized gravity ladder is strongly protected by the topological gap below a critical perturbation V_c, exhibiting unique critical scaling behavior at the transition point. The fitted critical exponent beta approx 0.07 is significantly lower than the mean-field exponent of conventional phase transitions, revealing a unique "critical rigidity" characteristic of finite-size long-range topological networks. This model not only provides a rigorous graph-theoretical benchmark for a discrete analogue of the Ryu-Takayanagi area law but also, through the "geometric decoupling" mechanism, constructs an analytically solvable microscopic dynamical sandbox for the highly debated entanglement wedge disconnection in the black hole island formula.
Abstract A confining, non-Abelian gauge connection Uij ∈ SU(N) defined on the links of a dynamical graph has been proposed as the microscopic origin of classical spacetime geometry: the metric is reconstructed from the coset (soldering-form) sector of an enlarged connection, and black-hole and boundary-cut entropy are obtained as the logarithm of a combinatorial microstate count, matched, via the Freedman–Headrick theorem, to the bit-thread (max-flow/min-cut) formulation of holographic entanglement entropy. That matching is exact as a statement about counting, but it is not, by itself, a statement about the entanglement entropy of any specific quantum state: nothing in the combinatorial construction distinguishes a classical ensemble of flux occupation numbers from a genuine quantum wavefunction, so its connection to the entanglement/tensornetwork program of emergent geometry (Van Raamsdonk, Swingle, Ryu–Takayanagi, and the random-tensor-network derivation of Hayden, Qi, Kim, and Preskill) remains an analogy rather than a derivation. This paper closes that gap. We first restate the flux-gauge connection ensemble in full – its partition function, its block-link renormalization-group coarse-graining with closed-form Migdal–Kadanoff coefficients, its emergent metric and Newton’s-constant identification, and its combinatorial boundary-cut entropy – so that the paper is self-contained. We then show, by an exact character-expansion/Peter–Weyl derivation standard in lattice gauge theory but not previously applied to this construction, that the SU(N) link ensemble is a tensor network: each flux-tube segment (link) carries a bond dimension equal to the dimension of the SU(N) representation it transports, and each flux-tube junction (node) carries a gauge-invariant intertwiner tensor, fixed by representation theory rather than assumed. Only a single, narrowly scoped statistical assumption – on the distribution of intertwiners within the multiplicity space that gauge invariance already guarantees – is needed to complete this exact, deterministic tensor network into a genuine random tensor network in the technical sense of Hayden–Qi–Kim–Preskill (HQKP). We further identify the renormalization-group layers of Section 2’s coarse-graining as the isometric maps of a multi-scale entanglement renormalization ansatz (MERA), giving the emergent radial holographic direction a literal tensor-network origin rather than a scaling-dimension analogy. We then investigate, analytically and how the entanglement entropy of such a network reproduces the Ryu–Takayanagi/min-cut law. An exactly solvable limit built from classical (GHZ-type) perfect tensors reproduces the combinatorial law exactly for single-cut (boundary-anchored) regions but is shown to be degenerate for interior, two-sided regions, exposing a genuine gap in the combinatorial entropy argument: classical flux-occupation counting does not, by itself, certify multi-region (entanglement-wedgelevel) quantum entanglement structure. We then construct a genuinely quantum (non-classical, Gaussian-random) tensor-network realization of the link ensemble and confirm numerically, on chains of up to sixteen sites, (i) a clean area-law plateau in which the entanglement entropy of an interior region becomes independent of its size once the region is large enough, in place of continued volume-law growth, and (ii) a linear scaling of the plateau entropy with the logarithm of the bond dimension, S(A) = κNcut(A) logD, fit with R2 = 0.996 and an O(1) efficiency coefficient κ ≈ 0.68 for this finite, non-isometric network. We interpret κ < 1 as a genuine, quantifiable finite-network correction relative to the idealized D → ∞ isometric-projector limit in which HQKP prove κ → 1, and we list the construction of that idealized limit, directly on the SU(N) link ensemble, as the key open numerical task. We close by proposing an explicit correspondence between the tensor-network minimal surface of this paper and the Cartan soldering-form metric of the gauge-connection construction, framed as a falsifiable consistency conjecture rather than an established result.
Holographic duality suggests that continuous spacetime geometry can emerge from underlying quantum entanglement. Based on the connectivity engineering framework, we construct a purely geometric decoupled lattice model to explore the entanglement area law and the emergence mechanism of discrete Ryu-Takayanagi (RT)-type scaling. By tuning the nearest-neighbor coupling λ, the system evolves from a disconnected direct sum decomposition (λ=0) to a globally connected phase. Precise numerical results indicate that weak nearest-neighbor coupling can stitch the rigid fault lines in the information distance matrix, inducing the formation of a smooth and continuous effective geometry. The core result is that in the stitched connected phase, the subsystem entanglement entropy is dominated by the geometric boundary, satisfying a scaling relation primarily governed by a linear area law with logarithmic corrections. Through a hold-out test on irregular subregions, the RT-type scaling calibrated by square samples provides a controlled error estimate for rectangular subregions. The L-shaped concave subregion exhibits a systematic negative deviation, corresponding to the well-known corner correction effect in free-fermion systems, further distinguishing perimeter scaling from minimal-surface scaling. Combined with Kitaev-Preskill diagnostic analysis, the results support that the discrete RT-type scaling can emerge from a purely geometric discrete network. This model provides a clean, exactly solvable underlying benchmark for studying discrete spacetime geometry and holographic entanglement on quantum simulators.