Fragment 2-RDM Sheaf

A single network learns fragment-local two-electron reduced density matrices as a cellular sheaf with physics-fixed restriction maps, so that the sheaf’s consistency residual becomes a per-region estimator of electronic nearsightedness.

A single network learns transferable, fragment-local two-electron reduced density matrices (2-RDMs), organised as a cellular sheaf with operator-valued stalks and physics-fixed compression restrictions. The construction yields a diagnostic. Under a state-dependent metric the sheaf’s consistency residual is a per-region estimator of nearsightedness-of-electronic-matter at the model’s context radius. The narrative is in the electronic-structure article; this note records the construction and its results.

Which coordinate is spent (the design space)

A sheaf-type architecture is a functor , and specifying one requires four independent choices: the index category together with its topology, the fibre that defines a stalk, the morphism class allowed for restrictions that defines what transport may be, and which of these is learned. Most disagreements between such architectures are disagreements about which coordinate to spend freedom on. This construction fixes the fibre and the restrictions by physics and learns only the section.

Invertibility is the pivot

One coordinate has outsized consequences.

Proposition. Invertibility ⇒ local system

If every restriction is invertible, is a local system: transport is an isomorphism, no information is destroyed, and the comparison map from a global object to its restrictions is injective. A separation failure is the existence of distinct globals with identical restrictions, so such a failure can occur only with non-invertible restrictions.

Corollary. A nonlinear fibre forces invertibility

If is a category of manifolds rather than linear spaces, a structure-preserving morphism cannot fail to be injective while staying a morphism; transport must be a diffeomorphism, hence a group action. Architectures on -bundles, learned maps, versor actions, or congruence by an invertible matrix are all in the invertible regime and cannot represent a situation in which restricting genuinely discards information, which is exactly the situation of a quantum marginal.

The construction

• Stalks. For a fragment with one-particle space in an atom-localised orthonormal basis, , with a positive semidefinite (PSD) 2-RDM as a stalk element.
• Restrictions. For , orbital-block compression , the 2-RDM of the partial trace over modes outside . These restrictions are fixed by physics rather than learned, and they compose on the nose ().
• Base and cover. Objects are fragments, morphisms chemistry-preserving induced-subgraph monomorphisms; a cover is a family of -hop atomic neighbourhoods , whose nonempty intersections form the nerve. With , and sections .
• Two radii. Stalk radius and message-passing depth give a derived context radius , so a prediction depends only on atoms within graph distance of .

Proposition. A stalk is a Euclidean Jordan algebra; the restriction is a Peirce projection

is a euclidean-jordan-algebra with and ; its symmetric cone is the PSD 2-RDMs. Compression is the quadratic representation of the projector, , which by the Peirce fact is the orthogonal projection onto with kernel . It is positive but not a Jordan homomorphism, and above all singular, so the invertibility argument above applies. The restrictions are allowed to lose information because the Jordan product needs no inverses.

Fixing the restriction is what makes the residual a measurement

Quantities sit in three tiers. A hard quantity is exact by construction with zero parameters, a soft quantity is penalised and its residual reported, and a free quantity is learned. Positivity is correctly hard, since a Cholesky form gives it exactly and “how PSD is it” carries no chemistry; consistency is correctly soft.

Why the restrictions must be fixed

If restrictions were learned, then for any per-fragment invertible the pair has the same overlap residual, a gauge redundancy, and makes every family consistent. Whatever number the loss then reports is a statement about the model’s own learned geometry, not about the object. With restrictions fixed and non-invertible there is no gauge, the section space is external, and the residual is a comparison against a ruler that was not fit to the data. This is the precondition for the nearsightedness claim.

Assembly is Möbius inversion, cancellation-free

The intersection poset of the cover is a meet-semilattice. An additive observable’s local values assemble by left Kan extension along ; on the finite poset this is a latching colimit, and the value/increment passage is mobius-inversion. So the alternating inclusion–exclusion formula is not one option among many. It is the decategorified canonical assembly, with its coefficients forced.

Proposition. Cancellation-free assembly

Every pair index visible in some cover fragment has a unique minimal fragment in containing it, so the indices partition. Hence

where is on the indices minimally assigned to . It agrees with the alternating sum by Möbius inversion but is a sum of disjoint non-negative-weight contributions, so per-fragment errors add rather than amplify. Indices whose atoms lie in no cover fragment are left unassigned, and that set is exactly the separation gap below.

What descent fails, and in what degree

The nerve poset glues automatically, so sheafhood there is vacuous; the substantive question is descent for the Grothendieck topology on , and it fails. Let be the product of fragment restrictions; it lands in , and descent would assert is an isomorphism.

Both failures are one comparison map's defect, in degree zero

Separation (injectivity). By the Peirce fact , the pair entries whose modes sit on atoms more than apart. This kernel is structural and unconditional but soft, since the discarded weight is small for gapped systems and shrinks with without closing. It exists because the restrictions are non-invertible.
Gluing (surjectivity). A consistent family need not come from any -electron state; deciding this is the overlapping quantum marginal problem, QMA-complete. No small-parameter rescue. Both live at ; nothing here is .

Remark. N-representability is orthogonal to the sheaf

The sheaf apparatus (, , projections) is a functor of the Hilbert layer alone. N-representability structures the cone layer, which that apparatus does not see. A family can be perfectly consistent and grossly non-representable, or representable per fragment and inconsistent. Identifying N-representability with a cohomological obstruction merges independent axes and wrongly predicts that improving consistency improves representability.

The state-dependent metric

State-dependence cannot enter through the cochain spaces; it enters through the inner product, which fixes the adjoint and hence and every projection. With the covariance of the model’s prediction error on the overlap ,

The unweighted loss mis-calibrates (it treats a well-determined core orbital and a diffuse one alike, and up-weights large fragments); the -orthogonal projection onto is the minimum-variance consistent repair. The space is state-blind; the metric is not.

The residual estimates nearsightedness

Observation. A section always exists

The true family satisfies exactly. Residual is therefore not caused by the separation gap.

The network computes from the -neighbourhood only; its Bayes-optimal value is . On an overlap the two neighbours predict the same true object from different conditioning sets.

Proposition. Residual = nearsightedness

At the Bayes-optimal readout,

which vanishes for all edges iff is a.s. determined by , that is, iff nearsightedness-of-electronic-matter holds at scale on that overlap. (Compression is linear and commutes with conditional expectation, so each side equals its conditional; subtract.)

The decay test. Track the calibrated residual across context radii and sample sizes. Statistical and optimisation error decays with , whereas structural error is a floor independent of , decaying only in , and not even in where locality genuinely fails. A residual plateauing in both is a localised certificate of misspecification, computable from the model’s own outputs with no reference calculation, precisely because the ruler was fixed by physics.

A second orthogonal residual from cover change

The residual has a blind spot. Since is a section under any cover, cannot see the separation gap. The categorical structure supplies the missing measurement. With the intersection posets of two covers () and , the cancellation-free partition makes the difference exactly the indices assigned in but unassigned in , which is the increment of the separation gap between the radii.

diagnostic	measures	blind to
(within cover)	whether conditioning at radius suffices, the nearsightedness of the prediction	the separation gap
(between covers)	the long-range content the smaller cover structurally discards	misspecification both covers share

Hypothesis. decays exponentially in at the density-matrix decay length on nearsighted systems and plateaus where locality fails. It is a second, independent test of the same physics, sensitive to exactly what cannot see.

Realising it

The readout must be variable-size, PSD, -antisymmetric, and -equivariant. Writing into a latent rank space carrying a unitary -representation and setting makes PSD automatic (and deliberately rank-deficient, so the stalks live on where only the Jordan route reaches), antisymmetry from the codomain, and equivariance automatic because the unitary action cancels in the product; the residual gauge is the compact commutant of . Training minimises , with spin enforced globally by inclusion–exclusion and used only as an objective, never as a propagation operator (linear diffusion preserves neither the PSD nor the N-representable cone). There is no wavefunction and no variational SDP bound, so the assembled energy brackets nothing.

What would falsify it

The two claims are independent. The within-cover claim dies if plateaus alike on nearsighted and non-nearsighted systems, if its floor merely tracks existing locality probes without being cheaper, or if the -plateau disappears under larger models at fixed (capacity confound). The cover-change hypothesis dies if shows no rate difference between the two regimes. Losing all of them leaves the design-space organisation intact.

Fibers in the recent sheaf-network literature

The stalk here is a matrix-valued, positive semidefinite fiber whose dimension is set by the fragment basis, which is exactly the fiber-design axis the 2026 sheaf-network literature has opened. Sheaf networks native to the SPD manifold carry a covariance matrix on each fiber and reach state of the art on molecular benchmarks (Peng et al.), and Hilbert-bundle formulations take the fiber to be a function space whose sheaf Laplacian converges to the connection Laplacian under refinement, which is the basis-set-independent limit a fragment density matrix wants. Both read the choice of stalk, not just the base, as the substantive design decision.

At the mean-field rung the connection is tighter still. Harish shows the positive-semidefinite-shifted single-particle molecular Hamiltonian, in a localised atomic-orbital basis, is itself the Laplacian of an -equivariant cellular sheaf whose -steerable restrictions recover Slater–Koster theory. The object this construction learns as a correlated two-electron marginal therefore appears one rung down as the very operator being diagonalised, and its sheaf counts non-bonding orbitals.

• K. Harish, equivariant cellular sheaves for molecular electronic structure (arXiv:2608.23571).
• Peng et al., sheaf neural networks on the SPD manifold (arXiv:2604.20308).
• Hilbert bundles and cellular sheaves (arXiv:2605.06395); F. Barbero et al., Sheaf Neural Networks with Connection Laplacians (arXiv:2206.08702).

Transport across geometry

The construction above spends its structure on consistency at fixed geometry, but a second, orthogonal direction lets the fragmentation stay fixed while the geometry moves and asks the learned 2-RDM to transport smoothly as the molecule deforms or the active space slides. Consistency there is not a hard physical law but a transfer prior, so its natural operator is a connection-laplacian rather than the sheaf Laplacian . The same stalk assignment carries both readings: indexed over the fragment lattice of a fixed conformer it wants agreement on intersections (, the consistent families, a constraint because it holds for the true 2-RDM), and indexed over a manifold of geometries it wants parallelism (, the covariantly constant families, a bias that holds only as far as the electronic structure is transferable).

Take the base to be the conformation manifold of nuclear coordinates modulo . The one-particle space moves with , so the stalks form a bundle over ; tracking an orthonormalisation of the moving basis gives an isometric transport, a connection with unitary parallel maps. The 2-RDM is then a marginal transported along a path in nuclear configuration space, penalised by

This targets exactly the failures a fixed-geometry model cannot see, namely extrapolation off the training manifold and the force discontinuities that spoil ab-initio molecular dynamics. When only the active space moves, a -plane inside a fixed , the base is instead the grassmannian with the canonical connection on its tautological bundle, where restriction and parallel transport look like the same operation. They coincide only where the moved subspace stays orthonormal to the discarded complement, because Grassmann transport is an isometry while the compression is deliberately singular; where they disagree the transport picture must fall back to . This is open.

In the transport direction the connection is computed from geometry rather than learned, and only the fibrewise update is learned, so a message-passing layer reads with fixed by the moving frame and the weights. As with , the connection Laplacian can also serve merely as a smoothness regulariser on the predictor’s output rather than as the propagation operator.

The two directions compose. Indexing fragments inside each conformer and conformers over the deformation manifold gives an iterated grothendieck-construction , a total object fibred over conformers with sheaf-fibres.

Property. The two consistencies separate by a spectral sequence

Within-conformer (sheaf) consistency and across-conformer (transport) consistency are the two filtration degrees of the double complex on the iterated fibration, so a spectral sequence separates them and they can be diagnosed independently. Both remain orthogonal to N-representability, which lives on the cone layer neither Laplacian sees. In practice build the transport direction first as a smooth predictor over geometry, then add the fragment sheaf as the inner refinement.

Equation card

A compressed reference.

Setup

Cellular sheaf of fragment 2-RDMs: stalk (a positive semidefinite (PSD) 2-RDM), restriction = compression , coboundary , sections , Laplacian . Context radius .

Restriction = Peirce projection. On the euclidean-jordan-algebra , , the orthogonal projection onto , which is positive, singular, and not a Jordan homomorphism.

Invertibility pivot. Invertible restrictions ⟹ local system ⟹ no separation gap; a quantum marginal needs non-invertible restrictions, so the compressions are fixed and singular by design.

Assembly (cancellation-free). → mobius-inversion, kan-extension, replacing the many-body expansion

State-dependent metric. with the predictive error covariance; the space is state-blind, the metric is not.

Residual = nearsightedness. At the Bayes-optimal readout : → nearsightedness-of-electronic-matter

vanishing for all edges iff nearsightedness holds at scale . Decay test: a residual flat in both and is a localised certificate of misspecification.

Thesis

Fix the restrictions by physics and the consistency residual stops being a property of the model and becomes a measurement of the physics, namely a per-region estimator of nearsightedness at the context radius, with an independent cover-change companion for the long-range content it cannot see.

References

The ingredients are classical; the contribution is the compatibility condition with untouchable restrictions and the residual’s identification with nearsightedness.

• W. Kohn, Density functional and density matrix method scaling linearly with the number of atoms, Phys. Rev. Lett. 76 (1996) 3168; E. Prodan, W. Kohn, Nearsightedness of electronic matter, PNAS 102 (2005) 11635.
• J. Hansen, T. Gebhart, Sheaf neural networks (2020); C. Bodnar et al., Neural sheaf diffusion, NeurIPS, 2022.
• D. A. Mazziotti (ed.), Reduced-Density-Matrix Mechanics, Adv. Chem. Phys. 134, 2007 (N-representability, PQGT).
• J. Faraut, A. Korányi, Analysis on Symmetric Cones, Oxford, 1994 (Euclidean Jordan algebras, Koecher–Vinberg).
• S. Mac Lane, Categories for the Working Mathematician, 2nd ed., Springer, 1998 (Kan extensions); G.-C. Rota, On the foundations of combinatorial theory I, Z. Wahrsch. 2 (1964) 340 (Möbius inversion).