Extended States in the Anderson Model

Extended States in the Anderson Model

OpenAI's claim

OpenAI presents a mathematical result relevant to this question in the selected manuscript. For uniform iid Anderson site disorder in every fixed dimension at least three, sufficiently small fixed disorder is claimed to give purely absolutely continuous spectrum on a nonempty open interval.

Mathematical statement of the claimed result

Result 1.

For independent $V_x$ uniformly distributed on $[-1,1]$, consider \[ (H_{\lambda,\omega}\psi)(x)=-\sum_{j=1}^d[\psi(x+e_j)+\psi(x-e_j)]+\lambda V_x\psi(x) \] on $\ell^2(\mathbb Z^d)$. The claimed fixed constant $e_2\in(0,1/100)$ and thresholds $\lambda_d>0$ give, for each $d\ge3$ and $0<\lambda<\lambda_d$, almost surely \[ I_d\subset\sigma(H_{\lambda,\omega}),\qquad \chi_{I_d}(H_{\lambda,\omega})\ell^2(\mathbb Z^d)\subset\mathcal H_{\mathrm{ac}}(H_{\lambda,\omega}), \] where \[ I_3=(-6+e_2/2,-6+e_2),\qquad I_d=-2d+(1/200,1/100)\quad(d\ge4). \] The projection is nonzero; the assertion is purely absolutely continuous spectrum on these windows, for this uniform-distribution model.

Outline of the argument

A finite-volume matrix-state certificate is propagated through controlled ordinary sweeps. Local trials and clearing of pending components maintain the comparison forms needed for resolvent bounds, which identify an interval with nonzero absolutely continuous weight.

Formal verification

For family 261, the selected planar Anderson statement constructs the self-adjoint operator and gives its almost-sure spectral interval. Spectral type, including the claimed pure-point conclusion, is not contained in it. No Comparator is assigned to the selected manuscript here; the family page selects other papers or supporting statements. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources

• OpenAI, Absolutely Continuous Spectrum for Weak-Disorder Anderson Models in Dimensions at Least Three, 2026-09-23; citation key OAI:Absolutely-Continuous-Spectrum-for-Weak-Disorder-Anderson-Models-in-Dimensions-at-Least-Three-September-23-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main.
• Family 261: Lean scope.
• OpenAI catalogue entry.

Extended States in the Anderson Model

Definitions and mathematical statement

For independent identically distributed real random variables $V_x$, the standard lattice Anderson operator on $\ell^2({\mathbb{Z}}^d)$ has the form \[ (H_\lambda\psi)(x)=-\sum_{|y-x|_1=1}\psi(y)+\lambda V_x\psi(x). \] For a vector $\psi$, its spectral measure is $\mu_\psi(B)=\langle\psi,1_B(H_\lambda)\psi\rangle$; an absolutely continuous spectral component has density with respect to Lebesgue measure. The central target is a nonzero absolutely continuous spectral subspace in a weak-disorder regime in $d\ge3$. The catalogue does not specify a probability law or exact energy interval. Those choices must be fixed in a full model-specific theorem. Its additional language about transport and a mobility edge is not, by definition alone, equivalent to existence of absolutely continuous spectrum.

Formulation and concept references: [S1].

Short English statement

Can weak random disorder in a lattice of dimension at least three be proved to permit an extended-state spectral regime?

Research attempt

Status and a fixed model.

This is an unresolved attempt, dated 27 September 2026. Fix an integer $d\geq3$ and let the $V_x$ be independent uniform random variables on $[-1,1]$. This is a nondegenerate law with variance $1/3$. Use exactly the nearest-neighbor hopping and sign in the statement, and write $H_{\lambda,\omega}=H_0+\lambda V_\omega$. The target considered here is a nonzero absolutely continuous spectral subspace almost surely for every fixed $0<\lambda<\lambda_0(d)$. A result only at $\lambda=0$ would not establish this target.

The attempt proves a free spectral calculation, a uniform positive-mass observation, and a sufficient resolvent criterion. The latter criterion is not proved for the random model. It isolates a quantitative missing estimate and supplies countertests to two tempting replacements: finite-time perturbation theory and absolute continuity only after averaging disorder.

Kirsch [S1] explains the spectral formulation and the distinction between the lattice problem and trees. The recent manuscript [S3], inspected in its online full text [E1], treats independent Gaussian potentials and proves diffusive estimates on specified weak-coupling scales. Its Theorems 1.1--1.2 do not assert the uniform boundary estimate below for a fixed positive $\lambda$. No transfer from a Bethe lattice, Gaussian disorder, or a joint scaling limit to the present uniform-law lattice model is assumed.

1. Free Fourier analysis and a regular energy interval.

Use normalized Haar measure on $\mathbb T^d=[-\pi,\pi]^d$. The Fourier transform maps $H_0$ to multiplication by \[ h(k)=-2\sum_{j=1}^d\cos k_j. \tag{31.1} \] Thus the free spectral measure of $\psi$ is the pushforward of $|\widehat\psi(k)|^2\,dk/(2\pi)^d$ under $h$. Its critical points have each $k_j$ equal to $0$ or $\pi$, so their energies belong to \[ \{-2d+4j:0\leq j\leq d\}. \] Fix the concrete open interval \[ E_*=-2d+2,\qquad I=(E_*-1/2,E_*+1/2). \tag{31.2} \] Its closure lies inside the free band and contains no critical energy. For $\psi=\delta_0$, coarea gives on $I$ the density \[ \rho_0(E)=\frac{1}{(2\pi)^d} \int_{h^{-1}(E)}\frac{d\sigma(k)}{|\nabla h(k)|}. \tag{31.3} \] It is finite, continuous, and positive there. Finiteness and continuity follow by covering the compact inverse image of $\overline I$ with finitely many implicit-function charts, since $|\nabla h|$ has a positive minimum on that set. Positivity follows from a nonempty regular level and its local $(d-1)$-dimensional surface patch. Every energy in $I$ is attained, for example by varying the first coordinate while holding the others at zero.

The same coarea argument with the integrable weight $|\widehat\psi|^2$ shows absolute continuity away from the finitely many critical values for any $\psi\in\ell^2$. A level set at any one of those values has Haar measure zero: fix the other coordinates and solve for $\cos k_1$, which gives at most two values of $k_1$. Fubini therefore shows that there is no residual spectral mass on the critical values. This proves the free operator has purely absolutely continuous spectrum. It gives a reference density for the attack, not stability under nonzero random perturbations.

2. Small norm perturbations preserve mass, but the basic estimates fail.

On the probability-one event that all $|V_x|\leq1$, $\|H_{\lambda,\omega}-H_0\|\leq\lambda$. Duhamel's formula and the resolvent identity consequently give, uniformly in $\omega$, \[ \|e^{-itH_{\lambda,\omega}}-e^{-itH_0}\|\leq\lambda|t|, \qquad \|(H_{\lambda,\omega}-z)^{-1}-(H_0-z)^{-1}\| \leq\lambda(\operatorname{Im}z)^{-2}. \tag{31.4} \] In the first estimate the integrand is the product of two unitaries and $\lambda V_\omega$; in the second each self-adjoint resolvent has norm at most $(\operatorname{Im}z)^{-1}$. Neither estimate is uniform in the limit needed for spectral type: the errors grow as $|t|\to\infty$ or as $\operatorname{Im}z\downarrow0$ at fixed positive $\lambda$.

One useful conclusion nevertheless survives. Choose a continuous function $0\leq f\leq1$ supported in $I$ and positive on a smaller interval. Then \[ c_0=\langle\delta_0,f(H_0)\delta_0\rangle =\int_I f(E)\rho_0(E)\,dE>0. \] For sufficiently small $\lambda$, uniformly in $\omega$, $\|f(H_{\lambda,\omega})-f(H_0)\|<c_0/2$. To justify this without a spectral-type assumption, put all spectra for $\lambda\leq1$ in $[-M,M]$, where $M=2d+1$, and approximate $f$ uniformly there by a polynomial $P$. For $A,B$ with norms at most $M$, \[ \|A^j-B^j\|\leq jM^{j-1}\|A-B\| \] by telescoping. First make $2\|P-f\|_\infty$ small, then make $\lambda\sum_{j\geq1}|P_j|jM^{j-1}$ small. The asserted continuity follows. Consequently, with $\mu_{0,\lambda,\omega}$ denoting the spectral measure of $\delta_0$, \[ \mu_{0,\lambda,\omega}(I) \geq\langle\delta_0,f(H_{\lambda,\omega})\delta_0\rangle >c_0/2. \tag{31.5} \] This is positive spectral mass for all sufficiently small perturbations; it does not say whether that mass has a density.

3. A sufficient estimate with an explicit almost-sure conclusion.

For fixed $\lambda>0$, define the nonnegative Poisson transform \[ F_{\eta,\omega}(E)=\frac1\pi \operatorname{Im}\langle\delta_0, (H_{\lambda,\omega}-E-i\eta)^{-1}\delta_0\rangle =\int\frac{\eta/\pi}{(t-E)^2+\eta^2} \,d\mu_{0,\lambda,\omega}(t). \tag{31.6} \] Here the sign of the imaginary part agrees with the chosen resolvent. The following condition, for some $1<p<\infty$, would suffice: \[ \sup_{0<\eta\leq1}\mathbb E \int_I F_{\eta,\omega}(E)^p\,dE \leq C_{\lambda,d,p,I}<\infty. \tag{31.7} \] The constant need not be uniform as $\lambda\downarrow0$. What matters is uniformity as $\eta\downarrow0$ for each fixed positive disorder strength.

Here is a proof of sufficiency that does not exchange an uncountable collection of probability-one events. Take the deterministic sequence $\eta_n=2^{-n}$. Fatou's lemma applied to the nonnegative random variables $\|F_{\eta_n,\omega}\|_{L^p(I)}^p$ gives \[ \mathbb E\liminf_n\|F_{\eta_n,\omega}\|_{L^p(I)}^p\leq C_{\lambda,d,p,I}. \] Hence for almost every fixed $\omega$ there is a subsequence with bounded $L^p(I)$ norm. Reflexivity gives a further subsequence converging weakly to some $g_\omega\in L^p(I)$. For each $\varphi\in C_c(I)$, extended by zero to $\mathbb R$, the approximate-identity property of the Poisson kernel gives \[ \lim_{\eta\downarrow0}\int_I\varphi(E)F_{\eta,\omega}(E)\,dE =\int\varphi(t)\,d\mu_{0,\lambda,\omega}(t). \] Weak $L^p$ convergence identifies the same limit with $\int_I\varphi g_\omega$. Uniqueness of measures tested against $C_c(I)$ proves \[ \mu_{0,\lambda,\omega}|_I=g_\omega(E)\,dE \quad\text{almost surely}. \tag{31.8} \] For $\lambda$ small enough for $31.5$, this density has positive integral. The vector $1_I(H_{\lambda,\omega})\delta_0$ is therefore nonzero and belongs to the absolutely continuous subspace. No claim that $\delta_0$ is cyclic is needed. Thus $31.7$, if established, really would imply the catalogue's central spectral target for the fixed model and interval.

4. Why averaging a measure is too weak.

Consider a separate, exactly solvable countermodel with the same iid uniform variables but no hopping: $D_\omega\delta_x=V_x(\omega)\delta_x$. Every realization has a complete orthonormal basis of eigenvectors, so the operator is pure point. Yet \[ \mu^{D}_{\delta_0,\omega}=\delta_{V_0(\omega)},\qquad \mathbb E\mu^{D}_{\delta_0,\omega}(dE) =\tfrac12 1_{[-1,1]}(E)\,dE. \tag{31.9} \] The averaged measure is absolutely continuous although every sample measure is atomic. This operator is not the weak-disorder hopping model; it refutes only the proposed inference from an averaged density to samplewise absolute continuity.

The distinction is detected by the power in $31.7$. Set $J=(-1/2,1/2)$ for this countermodel. If $|V_0|\leq1/4$ and $\eta<1/4$, then $[V_0-\eta,V_0+\eta]\subset J$, and its Poisson kernel obeys \[ \int_J\left(\frac{\eta/\pi}{(E-V_0)^2+\eta^2}\right)^p dE \geq 2^{1-p}\pi^{-p}\eta^{1-p}. \tag{31.10} \] The conditioning event has probability $1/4$, so the expected $L^p$ norm diverges for every $p>1$. In contrast the first moment of the Poisson transform is bounded by $1/2$, being the convolution of a probability kernel with the bounded averaged density. At $p=1$, integration over the whole line is always one for any probability spectral measure. A first moment bound therefore cannot replace the stronger condition.

Verification, scaling limits, and the remaining gap.

The companion checks verify the critical-energy list and that $31.2$ avoids it for dimensions three through twelve. They also check the telescoping identity on explicit noncommuting rational matrices, and the exact scaling in $31.10$. These are checks of the reductions, not simulations establishing an infinite-volume spectral type. A finite lattice matrix always has atomic spectral measures, even when its empirical measures approach a continuous limit. For example equally weighted atoms on a regular grid in $[0,1]$ converge weakly to Lebesgue measure by Riemann sums. Finite-volume eigenvalue plots alone cannot settle the required almost-sure assertion.

The precise gap is $31.7$ at a fixed $0<\lambda<\lambda_0$ all the way to $\eta=0$. Bounds down to a scale $\eta=\lambda^{2+\kappa}$ still stop at a positive number for that fixed $\lambda$; taking $\lambda\to0$ along the scale does not fill the missing limit. The elementary resolvent estimate in $31.4$ becomes worse there, and $31.9$ shows why replacing the missing moment control by a smooth averaged measure is invalid. No such uniform $L^p$ estimate is proved here, nor is transport or a mobility edge inferred from the partial results. The Anderson extended-state problem remains UNRESOLVED.

Sources

• S1 W. Kirsch, "An Invitation to Random Schrodinger Operators", in Random Schrodinger Operators, Panoramas et Syntheses 25, Societe Mathematique de France, 2008; arXiv:0709.3707. https://arxiv.org/abs/0709.3707. Formulation source.
• S2 A Note on Extended States on the Bethe Lattice. https://arxiv.org/html/2609.06987v1. Further reference; not a proof endorsement.
• S3 Self-consistent equations and quantum diffusion for the Anderson model. https://arxiv.org/abs/2506.06468. Further reference; not a proof endorsement.

• E1 Adam Black, Reuben Drogin, and Felipe Hernández, Self-consistent equations and quantum diffusion for the Anderson model, arXiv:2506.06468v2 (6 November 2025), Introduction and Theorems 1.1--1.2. https://arxiv.org/abs/2506.06468v2. https://arxiv.org/html/2506.06468v2. Version-specific identification of [S3]. The inspected Gaussian-law diffusion estimates are not used as a proof of the fixed-disorder, zero-height resolvent condition for uniform disorder proposed here. The manuscript's complete proof was not independently audited. Consulted 27 September 2026.

Extended States in the Anderson Model

Definitions and mathematical statement

For independent identically distributed real random variables $V_x$, the standard lattice Anderson operator on $\ell^2({\mathbb{Z}}^d)$ has the form \[ (H_\lambda\psi)(x)=-\sum_{|y-x|_1=1}\psi(y)+\lambda V_x\psi(x). \] For a vector $\psi$, its spectral measure is $\mu_\psi(B)=\langle\psi,1_B(H_\lambda)\psi\rangle$; an absolutely continuous spectral component has density with respect to Lebesgue measure. The central target is a nonzero absolutely continuous spectral subspace in a weak-disorder regime in $d\ge3$. The catalogue does not specify a probability law or exact energy interval. Those choices must be fixed in a full model-specific theorem. Its additional language about transport and a mobility edge is not, by definition alone, equivalent to existence of absolutely continuous spectrum.

Formulation and concept references: [S1].

Short English statement

Can weak random disorder in a lattice of dimension at least three be proved to permit an extended-state spectral regime?

Research attempt

Current frontier.

Kirsch distinguishes spectral type, dynamical localization and averaged density-of-states information; these are different assertions. [S1]. Black--Drogin--Hernández establish weak-disorder diffusion for Gaussian disorder down to a stated imaginary-resolvent scale $\eta\asymp\lambda^{2+\kappa_d}$, with $\kappa_d>0$, in their November 2025 version. It does not let $\eta\downarrow0$ at fixed positive $\lambda$. [S3], [E1]. Jakšić--Last--Warzel's September 2026 manuscript proves a Bethe-lattice result and proposes a Poisson--Schur criterion on ${\mathbb{Z}}^d$, explicitly leaving its weak-disorder lower bound unproved there. [S2], [E2]. Becker--Oltman's September 2026 manuscript concerns a hyperbolic-square-lattice product, not ${\mathbb{Z}}^d$ alone. [E3]. Recent statements and their model scopes were checked; they are not treated as independently audited solutions of this entry.

Attack route.

Editorial scope note. The original does not prescribe a single-site law and itself warns that absolutely continuous spectrum is not equivalent, by definition, to transport or a mobility edge. To make a concrete attempt, fix $d\ge3$ and independent $V_x$ uniformly distributed on $[-1,1]$, with $\lambda>0$. This is a nondegenerate random model, not the trivial free case. Results below concern its spectral component only; they do not silently settle the additional transport language or all possible laws.

The selected route is to upgrade resolvent control to a quenched statement about individual spectral measures. A uniform $L^2$ bound on their Poisson transforms is stronger than needed, but it gives a clean sufficient estimate whose limiting steps can be proved. The competing tree-cyclicity route would instead need a lower bound on a matrix Schur complement; independence of forward branches is absent on ${\mathbb{Z}}^d$. [E2], Section 4. I investigate the first route and test whether averaging or small operator norm perturbations could supply its missing estimate.

Attempt.

Write \[ G_x(E+i\eta)=\langle\delta_x,(H_\lambda-E-i\eta)^{-1}\delta_x\rangle, \qquad u_{x,\eta}(E)=\operatorname{Im}G_x(E+i\eta). \] For the probability measure $\mu_x$ associated to $\delta_x$, the spectral theorem gives \begin{equation}\label{wsref-0-72-33-31-3a-70-6f-69-73-73-6f-6e} u_{x,\eta}(E)=\int\frac{\eta}{(t-E)^2+\eta^2}\,d\mu_x(t)\ge0. \end{equation} The absence of $1/\pi$ in this definition will be retained in all limits. Spectral measures and the resolvent framework are those of [S1].

Lemma 31.1 (an all-resolution sufficient estimate). Let $I$ be an open interval. For fixed $\lambda>0$, suppose that for every closed interval $J\Subset I$, \begin{equation}\label{wsref-0-72-33-31-3a-6d-69-73-73-69-6e-67} \sup_{0<\eta<1}\mathbb E\int_J u_{0,\eta}(E)^2\,dE<\infty. \end{equation} Then, almost surely, $H_\lambda$ has no singular spectral component on $I$.

Proof. Fix $J$ and put $\eta_l=2^{-l}$. Fatou's lemma implies \[ \mathbb E\left[\liminf_l\|u_{0,\eta_l}\|_{L^2(J)}^2\right] \le\liminf_l\mathbb E\|u_{0,\eta_l}\|_{L^2(J)}^2<\infty. \] Thus for almost every realization there is a subsequence with bounded $L^2(J)$ norm. It need not be a deterministic subsequence or a subsequence chosen independently of the realization. Weak compactness of $L^2(J)$ gives a further weakly convergent subsequence with limit $h\in L^2(J)$. For $\phi\in C_c(J^\circ)$, Fubini and the Poisson approximate identity give \[ \lim_{\eta\downarrow0}\int\phi(E)u_{0,\eta}(E)\,dE =\pi\int\phi(t)\,d\mu_0(t). \] Indeed, the inner Poisson integral converges uniformly to $\pi\phi(t)$ and is bounded by $\pi\|\phi\|_\infty$, so integration against the finite measure is justified. Weak $L^2$ convergence identifies the same limit as $\int\phi h$. Hence $\mu_0|_{J^\circ}=h(E)dE/\pi$.

Translation invariance of the law gives the same conclusion for each $x$. Intersect the probability-one events for countably many $x\in{\mathbb{Z}}^d$ and a countable exhaustion of $I$ by such intervals. All basis-vector spectral measures are then absolutely continuous on $I$. The singular spectral projection on $I$ annihilates every $\delta_x$, and thus, by density, the whole Hilbert space. This proves the lemma without interchanging an uncontrolled random pointwise limit and expectation.$\square$

Lemma 31.2 (nonvanishing is available without a new randomness estimate). Choose a nonnegative $\phi\in C_c(I)$ such that $a_0=\langle\delta_0,\phi(H_0)\delta_0\rangle>0$. There is $\lambda_1>0$ so that for every $0<\lambda<\lambda_1$ and every realization of bounded $|V_x|\le1$, \begin{equation}\label{wsref-0-72-33-31-3a-6d-61-73-73} \langle\delta_0,\phi(H_\lambda)\delta_0\rangle>a_0/2. \end{equation} Proof. The operator norm of $H_\lambda-H_0$ is at most $\lambda$. For $\lambda\le1$, both spectra lie in $[-2d-1,2d+1]$. Approximate $\phi$ uniformly on this interval by a polynomial $P$. For a fixed polynomial, \[ \|A^j-B^j\|\le j\max(\|A\|,\|B\|)^{j-1}\|A-B\| \] follows from the telescoping identity, with no commutativity required. Consequently $\|P(H_\lambda)-P(H_0)\|\to0$ uniformly in the realization. Choosing the approximation error first, and then $\lambda$, proves (r31:mass).$\square$

Such a $\phi$ exists for every nonempty open $I\subset(-2d,2d)$: Fourier transform diagonalizes $H_0$ as multiplication by $-2\sum_{j=1}^d\cos\theta_j$, and the preimage of a nonempty open energy subinterval has positive torus measure. Thus proving (r31:missing) for a nonempty $I$ and every sufficiently small positive $\lambda$ would establish a nonzero absolutely continuous spectral subspace for this specified model, indeed purity on $I$. The constants may depend on $\lambda$; uniformity in resolution $\eta$ at each fixed $\lambda$ is the indispensable demand.

Attempted shortcut: average density instead of squared density. This shortcut is false. Let $V$ be uniform on $[-1,1]$ and take the one-dimensional operator with spectral measure $\mu=\delta_V$. Every realization is purely atomic, while $\mathbb E\mu$ has density $1/2$ on $[-1,1]$. Moreover \[ \mathbb E\frac{\eta}{(V-E)^2+\eta^2}\le\frac\pi2 \quad\text{for all }E,\eta>0, \] yet \begin{equation}\label{wsref-0-72-33-31-3a-61-74-6f-6d-69-63} \int_{{\mathbb{R}}}\left(\frac{\eta}{(V-E)^2+\eta^2}\right)^2dE =\frac{\pi}{2\eta}. \end{equation} The last identity follows by $E-V=\eta t$ and $\int_{{\mathbb{R}}}(1+t^2)^{-2}dt=\pi/2$. A bounded averaged Poisson density can therefore conceal singular spectrum. This is not a counterexample to the nearest-neighbor lattice conjecture; it disproves the proposed inference from annealed to quenched spectral type.

Attempted shortcut: a uniform Neumann expansion. Although $\|\lambda V\|\le\lambda$, the free resolvent norm is at most $1/\eta$. Expanding \[ (H_\lambda-z)^{-1} =(H_0-z)^{-1}\left[I+\lambda V(H_0-z)^{-1}\right]^{-1} \] by its geometric series is guaranteed only when $\lambda/\eta<1$. That regime excludes the required limit $\eta\downarrow0$ with fixed $\lambda$. Averaging may cancel some terms, but a proof of a remainder uniform in $\eta$ is an additional estimate, not supplied by the norm bound. The controlled mesoscopic scale in [E1] does not bridge this gap, and its Gaussian law is not the bounded law chosen above.

Stress test.

The order of limits cannot be changed unnoticed. Every finite-volume spectral measure is atomic. Its positive squared Poisson transform has diagonal contributions of order $\eta^{-1}$, just as in (r31:atomic); therefore one cannot prove the required all-resolution bound at each fixed finite volume and then take volume to infinity. Nor does the small-$\lambda$ continuity in Lemma 31.2 preserve spectral type: it preserves only positive mass for a fixed continuous test function. Lemma 31.1 supplies the separate absolute-continuity step conditionally.

No numerical transport experiment is interpreted as a spectral proof here. The companion script symbolically checks the Poisson-square integral. The probability-one conclusion was derived for each fixed $\lambda$; no uncountable intersection over all disorder strengths is asserted. Finally, existence of an ac subspace, a quantitative conductivity statement, and a sharp mobility edge remain separate goals, as the original editorial formulation already notes.

\Needspace{8\baselineskip}

Outcome.

Outcome: Conditional reduction.

For an explicitly fixed nondegenerate bounded single-site law, an all-resolution second-moment resolvent estimate implies the desired nonzero absolutely continuous spectral regime, with its nonvanishing and almost-sure conclusions proved in detail. The estimate itself is not proved. The averaging and norm-perturbation shortcuts are rejected by an exact model calculation and a parameter-order obstruction. This is a sufficient route for the spectral part of a specified model, not a completion of the underspecified law/transport/mobility-edge package, and the criterion's historical novelty is not asserted.

Sources

• S1 W. Kirsch, "An Invitation to Random Schrodinger Operators", in Random Schrodinger Operators, Panoramas et Syntheses 25, Societe Mathematique de France, 2008; arXiv:0709.3707. https://arxiv.org/abs/0709.3707. Catalogue formulation source.
• S2 A Note on Extended States on the Bethe Lattice. https://arxiv.org/html/2609.06987v1. Catalogue research/status reference; not a proof endorsement.
• S3 Self-consistent equations and quantum diffusion for the Anderson model. https://arxiv.org/abs/2506.06468. Catalogue research/status reference; not a proof endorsement.
• E1 Adam Black, Reuben Drogin and Felipe Hernández, Self-consistent equations and quantum diffusion for the Anderson model, arXiv:2506.06468v2, 6 November 2025, abstract and resolution-scale statement. https://arxiv.org/abs/2506.06468v2. Gaussian disorder and fixed scale are retained; checked 27 September 2026.
• E2 Vojkan Jakšić, Yoram Last and Simone Warzel, A Note on Extended States on the Bethe Lattice, arXiv:2609.06987v1, Sections 1 and 4, Lemma 4.1 and Theorem 4.2. https://arxiv.org/html/2609.06987v1. The lattice criterion is not its unproved weak-disorder estimate. Consulted 27 September 2026.
• E3 Simon Becker and Izak Oltman, Disorder on the hyperbolic square lattice I: Anderson delocalization and absolutely continuous spectrum, arXiv:2609.20798v1, September 2026, abstract and graph hypotheses. https://arxiv.org/abs/2609.20798v1. Recent manuscript on a different graph; checked 27 September 2026, not fully audited here.