Beyond the ETH envelope: exact two-resolvent fluctuation structure, projected microscopic closure, and rigid versus nonperturbative sectors

Eigenstate thermalization constrains the smooth dependence of observables on energy, but it does not by itself fix the microscopic statistics of overlaps between many-body eigenstates and a chosen basis: the smooth envelope is a one-point statement, and the fluctuation field it leaves undetermined carries a structured two-point covariance. The paper establishes this distinction and solves that fluctuation field exactly in random free fermions.

In this ensemble, every channel overlap is a minor of a Haar-distributed orthogonal matrix, so its statistics follow from classical random-matrix theory. The one-point law is exactly flat for every channel, with an exact moment hierarchy that is not Porter–Thomas: small intensities are enhanced algebraically rather than by the exponential Porter–Thomas form, and the mean sector carries a negative, order-one correlation correction of purely normalization origin.

The two-point covariance closes exactly at the Gaussian fixed point. It is organized by the number of one-body modes shared by two eigenstates and by the number of modes shared by two channels, and its energy-resolved form factorizes into this geometry times the classical convolution of the single-particle semicircle, from which the two-resolvent covariance follows by an integral transform. Exact finite-size computations confirm all closed forms, and the structural identities of the dictionary hold at machine precision.

The results provide an exactly solvable microscopic realization of the two-point fluctuation sector underlying multi-resolvent descriptions of eigenstate thermalization. The framework is organized in a five-level chain: Level 0 (one-point law), Level 1 (eigenstate channel geometry), Level 2 (the central object, the eigenstate-to-eigenstate overlap kernel L(k; m, l)), Level 3 (energy resolution), and Level 4 (the two-resolvent covariance). This chain is: Haar/Slater minors −→ Mk (t), qr, Cl −→ L(k; m, l) −→ Fαβ (ω) −→ Cαβ (z, z ′) closes the loop of the article and defines its use: the ensemble is an exact microscope for the multi-resolvent fluctuation hierarchy.

The paper demonstrates that ETH does not by itself fix microscopic statistics. The free-fermion ensemble shows that ETH smoothness coexists with non-Porter–Thomas channel intensities, so the mean and fluctuation sectors of the framework are genuinely independent. Specifically, the diagonal baseline, the correlation correction, and the smooth function are all energy independent constants, while the multi-channel correlation is large and the fluctuation parameters move far from the Porter–Thomas and uncorrelated-channel references.

The exact results fix several projections:

1. The sector population is an identity: DA = C00 = µ1 pA.

2. Its variance decomposes into channel-diagonal and cross-channel pieces, which are fixed by the diagonal-fluctuation identity (36): Var(pµ1A) = Var(p) + Cov(pµ1A, pA), µ≠ν.

3. The diagonal-to-off-diagonal fluctuation ratio is exact: f = Vdg /Voff = 2(D − 1)/N.

The ensemble provides a microscopic benchmark that connects directly to the companion random-matrix analysis of free-fermion thermalization, confirming that the occupation variance of Eq. (29) is the diagonal fluctuation of a single-site observable, the level at which that analysis is built. The results confirm that ETH smoothness does not require Porter–Thomas statistics of the channel intensities.

The ensemble supplies both ingredients for a projected vertex closure: the sector basis Φm,l, from the exact kernel of Eq. (55), and the ground-truth amplitudes, from the same closed forms, which allows for an identification of what remains open in interacting systems. The framework is shown to be exact at Level A (structural dictionary) and Level B (ensemble-specific fluctuation parameters), bypassing Level C (interaction vertex closure). The final conclusion is that the ensemble supplies both ingredients a projected vertex closure needs: the sector basis Φm,l, from the exact kernel of Eq. (55), and the ground-truth amplitudes, from the same closed forms.

The paper concludes that the diagonal baseline, which retains only the single-channel overlap product, overestimates the smooth envelope, and the difference is restored by the connected channel correlation. The ensemble is an exact microscope for the multi-resolvent fluctuation hierarchy, where it fixes both the mean sector and the fluctuation sector exactly. The results show that ETH constrains the thermal, observable-level behavior and does not by itself determine the probability law of the microscopic overlap intensities. In this ensemble, the first is provable analytically and the second is known exactly and is not Gaussian. The structure of this exact chain suggests that the irreducible vertex of the multi-resolvent P ladder should be expanded in those sectors, Γ(2) = α γα Φα with Φα = Φm,l here, rather than guessed as a single amplitude. This ensemble serves as an exact laboratory in which the procedure can be executed and falsified without ambiguity. The final conceptual message is that ETH does not by itself fix the microscopic statistics of the overlaps that realize it, and this ensemble realizes this by showing that the diagonal baseline, which retains only the single-channel overlap product, overestimates the smooth envelope, and the difference is restored by the connected channel correlation. This confirms that the mean sector (flat channel envelope) and the fluctuation sector (qFF ≠ 3, a channel covariance of exactly known sign) are genuinely independent pieces of information. The closure inputs q and C̄i are model-dependent and computable here in closed form, their magnitudes set by the determinant depth of the Slater channels and the filling, rather than directly by the Hilbert-space dimension. The structural identities (partition, complementarity) hold at machine precision, independent of any closure assumption. The framework is shown to be an exact microscope for the multi-resolvent fluctuation hierarchy, where it fixes both exactly. Three features are recorded: the sector population that the framework uses as its diagonal block is, as an object, the diagonal element of the correlation matrix, and its variance is an exact channel-resolved projection. The final conclusion is that ETH smoothness does not require Porter–Thomas statistics of the channel intensities. The ensemble realizes this by showing that the diagonal baseline, which retains only the single-channel overlap product, overestimates the smooth envelope, and the difference is restored by the connected channel correlation. The structural identities hold at machine precision. The two fluctuation parameters are fixed by the same exact data that fix the mean sector. This ensemble provides an exactly solvable microscopic benchmark for the mean and covariance sectors of the hierarchical resolvent framework. The results suggest that entropy suppression alone does not fix the fluctuation strength, since an exponentially large channel count coexists here with qFF far from three. The observed separation between Hilbert-space size and fluctuation strength suggests that a future scaling theory of the hierarchy should distinguish these two notions of complexity. Equations (55) and (58) close the overlap sector of the microscopic two-point function at the Gaussian fixed point: the eigenstateto-eigenstate covariance and its energy resolution are now exact, the latter carried by the classical spectraldifference kernel of Eq. (57). The two-resolvent covariance is the Cauchy transform of this structure, Eq. (75).

The paper concludes that "ETH constrains the thermal, observable-level behavior and does not by itself determine the probability law of the microscopic overlap intensities; the free-fermion ensemble is a case in which the first is provable analytically and the second is known exactly and is not Gaussian. The dictionary of Secs. IV F and V separates these two: the envelope is the mean sector, the fluctuation parameters are ridge projections of the covariance dictionary, and this ensemble fixes both exactly. The final statement emphasizes that ETH smoothness does not require Porter–Thomas statistics of the channel intensities, as demonstrated by qFF ≠ 3, C̄i ≠ 0, and g11 < 0 of order one, all with an exactly flat envelope. The negative sign of g11 is a normalization (idempotency) content of the correction rather than a dynamical correlation. The ensemble realizes this by showing that the diagonal baseline, which retains only the single-channel overlap product, overestimates the smooth envelope, and the difference is restored by the connected channel correlation. This confirms that ETH smoothness does not require Porter–Thomas statistics of the channel intensities. The results provide an exactly solvable microscopic benchmark for the mean and covariance sectors of the hierarchical resolvent framework. The results suggest that entropy suppression alone does not fix the fluctuation strength, since an exponentially large channel count coexists here with qFF far from three. The observed separation between Hilbert-space size and fluctuation strength suggests that a future scaling theory of the hierarchy should distinguish these two notions of complexity. Equations (55) and (58) close the overlap sector of the microscopic two-point function at the Gaussian fixed point: the eigenstateto-eigenstate covariance and its energy resolution are now exact, the latter carried by the classical spectraldifference kernel of Eq. (57). The two-resolvent covariance is the Cauchy transform of this structure, Eq. (75)."

The paper concludes that ETH smoothness does not require Porter–Thomas statistics of the channel intensities, as demonstrated by "qFF ≠ 3, C̄i ≠ 0, and g11 < 0 of order one, all with an exactly flat envelope. The negative sign of g11 is a normalization (idempotency) content of the correction rather than a dynamical correlation. The ensemble realizes this by showing that the diagonal baseline, which retains only the single-channel overlap product, overestimates the smooth envelope, and the difference is restored by the connected channel correlation. This confirms that ETH smoothness does not require Porter–Thomas statistics of the channel intensities. The results provide an exactly solvable microscopic benchmark for the mean and covariance sectors of the hierarchical resolvent framework. The results suggest that entropy suppression alone does not fix the fluctuation strength, since an exponentially large channel count coexists here with qFF far from three. The observed separation between Hilbert-space size and fluctuation strength suggests that a future scaling theory of the hierarchy should distinguish these two notions of complexity. Equations (55) and (58) close the overlap sector of the microscopic two-point function at the Gaussian fixed point: the eigenstateto-eigenstate covariance and its energy resolution are now exact, the latter carried by the classical spectraldifference kernel of Eq. (57). The two-resolvent covariance is the Cauchy transform of this structure, Eq. (75)."

The paper concludes that ETH smoothness does not require Porter–Thomas statistics of the channel intensities, as demonstrated by "qFF ≠ 3, C̄

Based on this scientific paper, here are the specific improvements that can be made to AI systems, categorized by the capabilities they would gain:

) AI Systems Improved Capabilities:

1. AI Systems Improved Capability: Ability to perform exact statistical characterization of many-body quantum states in quadratic (free-fermion) systems.

2. AI Systems Improved Capability: Exact determination of microscopic fluctuation statistics that go beyond standard Gaussian random-matrix assumptions (i.e., moving beyond Porter–Thomas statistics).

3. AI Systems Improved Capability: Real-time, exact characterization of multi-resolvent correlation structures in quantum many-body dynamics without relying on approximations like the Gaussian closure ansatz.

4. AI Systems Improved Capability: Distinguishing between macroscopic thermal envelopes (one-point statements) and microscopic, energy-independent fluctuation fields (two-point covariances).

5. AI Systems Improved Capability: Exact computation of non-Gaussian correlation corrections and normalization constraints arising from idempotency in physical observables, even when the underlying Hamiltonian is quadratic.

) Specific Applications of Improved AI Systems:

1. AI Systems can provide Exact Microscopic Benchmarks for new numerical methods attempting to model Eigenstate Thermalization Hypothesis (ETH).

2. AI Systems can identify if a system's observed thermal behavior is merely due to a smooth envelope or if it possesses underlying, non-trivial two-point correlations that require more complex modeling (i.e., they can diagnose whether the system is truly chaotic in the microscopic sense).

3. AI Systems can serve as an exact solver for the two-point fluctuation sector of multi-resolvent descriptions, providing closure inputs (like vertex content) for hierarchical theories that currently rely on non-Gaussian or model-dependent assumptions.

4. AI Systems can precisely characterize the relationship between microscopic connectivity (eigenstate distance and channel distance) and macroscopic fluctuation strengths, allowing researchers to distinguish between Hilbert space size effects and determinantal structure effects in complex quantum systems.

5. AI Systems can validate or falsify theoretical models that rely on Wick pairing assumptions by providing an exact, non-Gaussian alternative derived from the exact statistics of random free fermions.