Continuum Coulomb energy: hardness beyond an orbital basis

Source and verification

This article explains family 275 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

🏷️ The continuum electronic energy problem

An electronic Hamiltonian describes moving electrons in the field of fixed nuclei. In atomic units, with nuclear positions $R_{\alpha }\in {\mathbb{R}}^3$ and positive charges $Z_{\alpha }$, it is

$$\begin{array}{rl}H=& -\frac{1}{2}\sum _{i=1}^N{\Delta }_{x_i}\\ & -\sum _{i,\alpha }\frac{Z_{\alpha }}{|x_i-R_{\alpha }|}\\ & +\sum _{i<j}\frac{1}{|x_i-x_j|}.\end{array}$$

The terms are kinetic energy, electron–nucleus attraction, and electron–electron repulsion. Each electron has a two-dimensional spin space. The state space is ${\bigwedge }^N{L}^2({\mathbb{R}}^3;{\mathbb{C}}^2)$: wavefunctions change sign when the position and spin coordinates of two electrons are exchanged together. All spin sectors are allowed.

The operator is defined by its closed quadratic form on antisymmetric $H^1$ wavefunctions. Here $H^1$ means that the function and its first weak spatial derivatives are square integrable. The three-dimensional Hardy inequality controls the Coulomb singularities. For example, with other particle coordinates fixed and any $\eta >0$,

$$\begin{array}{rl}\int \frac{|u{|}^2}{|x-R|}& \le 2\Vert u{\Vert }_2\Vert \nabla u{\Vert }_2\\ & \le \eta \Vert \nabla u{\Vert }_2^2+{\eta }^{-1}\Vert u{\Vert }_2^2.\end{array}$$

Summing suitable such estimates makes the form lower bounded. Its spectral infimum $E_0=\inf \mathrm{spec}H$ is the infimum of the energy over normalized form-domain states. It need not be an attained eigenvalue.

The nuclei are clamped, and the convention here omits their mutual repulsion. No orbital basis or electronic wavefunction is supplied as input. The problem is to distinguish $E_0\le a$ from $E_0\ge b$ for rational thresholds $a<b$, under a promise excluding the interval between them.

🏷️ Hardness and the encoding of physical data

QMA is the class of promise problems that have polynomial-time quantum verification using a polynomial-size quantum witness, with a fixed completeness–soundness gap. A problem is QMA-hard if every such problem reduces to it by a deterministic classical polynomial-time map preserving the two promised answers. Hardness does not by itself establish membership in QMA.

The two manuscripts use distinct rational nuclear positions, a unary electron count, and binary rational thresholds and coordinates. Unary encoding prevents the electron count from being exponentially larger than its written description. If $L$ is the complete input length, the promise requires $b-a\ge L^{-1}$.

Two continuum Coulomb hardness results, as stated in the manuscripts

The electronic spectral-infimum problem is QMA-hard with positive integer nuclear charges encoded in binary. A companion proves the same conclusion with every nuclear charge equal to one. Both reductions have polynomial output length and can produce threshold separation at least one.

The binary-charge construction permits exponentially large charge values with polynomially many bits. (OpenAI, 2026) The unit-charge construction instead produces polynomially many nuclei, each of charge one. (OpenAI, 2026b) Splitting each exponentially large charge in the first construction into unit charges would lose polynomial output size; the second proof needs a different physical realization.

Neither theorem assumes neutrality, strict electron binding, existence of a ground eigenvector, or a molecular excitation gap. The threshold separation is a decision precision, not a spectral gap. The final inputs contain no magnetic field or freely adjustable external potential beyond the specified nuclei.

🏷️ Why a finite orbital calculation is insufficient

Choose finitely many one-electron orbitals and let $\Pi$ project onto the many-electron space built from them. The variational principle gives $E_0\le E_{\Pi }$, where $E_{\Pi }$ is the minimum of the compressed Hamiltonian. This transfers a low-energy YES witness. It does not transfer a NO lower bound: an omitted continuum state might lie below every selected-orbital state.

The essential analytic task is therefore a lower bound of the form $E_0\ge E_{\Pi }-ϵ$ on every antisymmetric continuum state. Both manuscripts build a gap on the entire one-particle complement, then use a quadratic estimate to control many-electron coupling to it. The finitely many orbitals are tools inside the proof, not a restriction imposed on the physical problem.

🏷️ Spin interactions and virtual double occupancy

Both constructions start from a cited QMA-hard problem for signed Heisenberg interactions on a finite square-lattice subgraph. Write $h_{ij}={\mathbf{\sigma }}_i\cdot {\mathbf{\sigma }}_j$, using Pauli matrices rather than spin matrices divided by two. Its eigenvalues are $-3$ on the singlet and $1$ on the triplets.

A strongly coupled auxiliary singlet pair converts signed interactions into positive exchanges. Coupling the endpoint spins to opposite members of the pair gives one sign; coupling them to the same member gives the other. The second-order calculation retains an explicit scalar shift. Fixed additional subdivision stages provide a planar layout whose contact lengths can be adjusted independently. All finite gadget strengths and inverse nonzero strengths remain polynomially bounded.

The perturbative tool is elementary but quantitative. If $A\ge gQ$, $P=I-Q$ is its nonzero kernel projection, and $\Vert B\Vert \le g/4$, set $H_{\mathrm{eff}}=(PBP-PB{A}^{-1}BP){|}_P$. Then

$$\begin{array}{c}E(A+B)=E(H_{\mathrm{eff}})\\ +O(\Vert B{\Vert }^3/g^2).\end{array}$$

The inverse is zero on the kernel. Solving the complementary block equation at the lowest energy gives an inverse differing from $A^{-1}$ by $O(\Vert B\Vert /g^2)$; substituting back gives the lower estimate. A trial vector $p-A^{-1}Bp$ gives the matching upper estimate, including its normalization error. The first-order block $PBP$ need not vanish.

To realize positive spin exchange with electrons, put two spin modes on each of $m$ sites and use $m$ electrons. Let $n_i\in \{0,1,2\}$ be the occupancy. Hopping creates an empty site and a doubly occupied site virtually; a second hop can return to the singly occupied space. The singlet couples to the two double-occupancy states while the triplets do not. This is the superexchange mechanism.

🏷️ Retaining direct Coulomb interactions

The unit-charge construction cannot discard the long-range direct Coulomb terms at its polynomial scales. For localized real spatial modes ${\psi }_i$, define

$$\begin{array}{c}{v}_{ij}=\iint \frac{|{\psi }_i(y){|}^2|{\psi }_j(y^{\prime }){|}^2}{|y-y^{\prime }|}dyd{y}^{\prime },\\ U=v_{ii}.\end{array}$$

This section uses finite-fermion energy units, equal to $\lambda$ times the dilated energy; the Coulomb coefficients therefore have no $1/\lambda$ prefactor. The common same-site coefficient is $U$, while the off-site coefficients depend on separation. The manuscript proves the matrix bound $(v_{ij})\ge cI$ for a fixed $c>0$. To see the mechanism, test the signed density $F={\sum }_i{a}_i{\psi }_i^2$ against disjoint bump functions near the sites. Poisson’s equation and Cauchy–Schwarz bound the squared density–test pairing by its Coulomb energy times the test’s Dirichlet energy. The near-diagonal overlap matrix makes the pairing comparable to ${\sum }_i{a}_i^2$. This gives coercivity even for signed occupancy defects.

Set $s_i={\sum }_{j\ne i}{v}_{ij}$ and $d_i=n_i-1$. A one-body counterterm $-{\sum }_i{s}_i{n}_i$, created by small well-depth changes, completes the direct interaction into

$$\begin{array}{rl}& \frac{U}{2}\sum _i{n}_i(n_i-1)-\sum _i{s}_i{n}_i\\ & +\sum _{i<j}{v}_{ij}{n}_i{n}_j\\ & =\frac{1}{2}\sum _{i,j}{v}_{ij}{d}_i{d}_j-\sum _{i<j}{v}_{ij}.\end{array}$$

The identity uses ${\sum }_i{n}_i=m$. A nonzero integer defect vector with sum zero has squared length at least two, so coercivity gives a fixed gap above single occupancy. One hop along $\{i,j\}$ costs exactly $U-v_{ij}$. Thus hopping amplitude $t_{ij}$ produces the second-order spin operator $t_{ij}^2(h_{ij}-I)/(U-v_{ij})$. Choosing $t_{ij}=\tau \sqrt{K_{ij}(U-v_{ij})}$ realizes exchange ${\tau }^2{K}_{ij}$, with a computable scalar shift and cubic perturbative error. Here $\tau$ is a sufficiently small inverse polynomial. (OpenAI, 2026b)

🏷️ Two physical realizations of the wells

The binary-charge manuscript uses hydrogenic wells at a scale $Z_{\ast }\asymp e^D$, with $D$ polynomial in the source size. Each site has a large primary nucleus and a smaller positive secondary nucleus. In coordinates $y=Z_{\ast }x$, a secondary displacement tunes the isolated-well energy close to $-1/2$. The continuum energy comparison is made in units $E_0/Z_{\ast }+m{Z}_{\ast }/2$, and the onsite repulsion approaches $5/8$.

Exponentially small tunneling is calibrated relatively, not merely by absolute proximity of an orbital to a hydrogen ground state. A certified polynomial Ritz-space calculation evaluates the one-well energy to exponentially small error using polynomially many bits. The construction computes primary positions from explicit surrogate integrals, fixes integer charges, and then tunes secondary displacements. Uniform estimates make this order independent of unknown many-electron eigenstates. The detailed oracle and relative tunneling proofs are substantive parts of that manuscript. (OpenAI, 2026a)

The unit-charge manuscript instead uses a polynomial scale $\lambda$, with $y=\lambda x$. After dividing energy by ${\lambda }^2$, each unit nucleus and electron–electron repulsion has coupling $1/\lambda$. Smooth planar wells have localized modes, while a broad positive-density slab provides transverse confinement. With $D=\log \lambda$, contact separations of order $D$ give exponentially decaying mode overlaps that are inverse powers of $\lambda$.

The continuous nuclear density has the form $q=\rho +\Delta V_{\ell }/(4\pi )$ inside a large box, where $V_{\ell }$ is a compactly supported well correction. A fixed sufficiently large background $\rho$ makes $q\ge \rho /2$. Its attractive Coulomb potential is exactly the slab potential plus $V_{\ell }$, by the fundamental-solution identity. The well depths include the direct-interaction counterterm above.

The hopping equation on each contact is $\lambda T(d)=\tau \sqrt{K_e(U-v(d))}$, with positive exponentially decaying $T$. Endpoint estimates supply a sign-change bracket. Approximate bisection finds a small residual using a Lipschitz bound; monotonicity and a derivative lower bound are unnecessary. The contact layout realizes all selected lengths simultaneously.

🏷️ From positive density to unit point charges

The density must be replaced by actual equal point charges without destroying the energy comparison. Interpolate $q_t=\rho +t\Delta V_{\ell }/(4\pi )$ and use velocity ${\mathcal{V}}_t=-\nabla V_{\ell }/(4\pi q_t)$. It solves the continuity equation. Its time-one flow transports uniform density to $q$, preserving the mass of every transported quadrature node.

Use eight equal-weight tensor-product Gauss nodes in each cube of side $h$. The rule is exact through total degree three in the original cube coordinates. Choose $\rho h^3/8=1/\lambda$ and move every node through the flow. Each has mass $1/\lambda$ in dilated coordinates, hence charge one physically. Rationally approximate the transported positions within a controlled error; separation and a Lipschitz inverse keep them distinct.

Two different estimates are needed for the change $V$ from continuous to point-nucleus potential:

$$\begin{array}{rl}|⟨u,Vu⟩|& \le p(n,D){\lambda }^{-2/3}\Vert u{\Vert }_{H^1}^2,\\ \Vert PVP\Vert & \le p(n,D){\lambda }^{-4/3}.\end{array}$$

Here $P$ is the one-particle mode projection and $p$ denotes a fixed polynomial bound, not a probability. The first inequality holds on all $H^1$ states. Near quadrature nodes, Hardy controls the singular kernel; away from them, fourth derivatives give an $O(h^2)$ summed form error. After compression, the orbital densities have smooth Coulomb fields, and degree-three exactness gives the stronger $O(h^4)$ error. Rational rounding is included in both estimates. (OpenAI, 2026b)

🏷️ The continuum complement and square completion

Spatial localization first proves a fixed gap outside the selected one-particle modes, together with a small residual on those modes. The $O(D^{-2})$ localization cost only needs to be small compared with that fixed gap; it is not used as the final energy-accuracy bound.

Let $\Pi$ now denote the many-particle projection requiring every electron to occupy the mode space. The sum of one-particle complementary projections dominates $I-\Pi$. Positivity of electron repulsion, a kinetic lower bound, and the general form-replacement estimate give a coercive bound on the complementary many-particle space. The mixed block obeys

$$\begin{array}{c}|⟨q,\mathcal{H}w⟩|\le \beta \Vert w\Vert \Vert q{\Vert }_{H^1},\\ \beta =p(n,D){\lambda }^{-2/3},\end{array}$$

for $w\in \mathrm{Ran}\Pi$, $q\in \mathrm{Ran}(I-\Pi )$. The shifted, dilated Hamiltonian is $\mathcal{H}$. After subtracting its compressed minimum $E_{\Pi }$, the complementary form retains a positive coefficient $\alpha$, whose reciprocal is polynomially bounded. Decomposing a normalized state as $u=w+q$ gives

$$\begin{array}{rl}& ⟨u,\mathcal{H}u⟩-E_{\Pi }\\ & \ge \alpha \Vert q{\Vert }_{H^1}^2-2\beta \Vert w\Vert \Vert q{\Vert }_{H^1}\\ & \ge -\frac{{\beta }^2}{\alpha }\Vert w{\Vert }^2.\end{array}$$

This is why the larger general form error is admissible: coupling contributes quadratically, at scale ${\lambda }^{-4/3}$. Multiplying by $\lambda$ to reach the finite-fermion units costs only ${\lambda }^{-1/3}$. Using the general form error linearly would instead produce a growing error ${\lambda }^{1/3}$.

The compressed one- and two-body coefficients are compared with the finite fermion Hamiltonian $H_d$. Although its many-body dimension is exponential, there are only $O(m^2)$ one-body and $O(m^4)$ two-body creation–annihilation monomials, each of norm at most one. Thus coefficient errors accumulate only polynomially. Combining the compression and complement estimates gives

$$\begin{array}{c}|\lambda \inf \mathrm{spec}\mathcal{H}-E(H_d)|\\ \le p(n,D){\lambda }^{-0.1}.\end{array}$$

The first manuscript has the analogous quadratic exclusion mechanism at exponential scales. Its complement leakage is at most $e^{-1.2D}$ in dilated units, still negligible after multiplication by $Z_{\ast }\asymp e^D$. Neither argument assumes that the full continuum infimum is attained or that the minimizing state lies in a chosen spin sector.

🏷️ Thresholds and polynomial output length

For the unit-charge construction, undoing the dilation and all scalar shifts gives an affine prediction $F(E_{\mathrm{in}})$ for the physical energy, with positive slope $s=\lambda {\tau }^2$. If the source thresholds are $a_s,b_s$, set $\gamma =\min (1,b_s-a_s)$. The construction makes the total energy error at most $s\gamma /8$.

Approximate $F(a_s+\gamma /4)$ and $F(b_s-\gamma /4)$ by rational output thresholds, each within $s\gamma /32$. These margins preserve both YES and NO. Their separation is at least $7s\gamma /16$, which exceeds one after requiring $s\gamma \ge 8$.

All amplification factors in the unit-charge construction are polynomial. Fixing a sufficiently large constant exponent in $\lambda$ therefore meets the finite error budgets with polynomially many nuclei, rational bits, quadrature evaluations and flow steps. The large slab offset is computed to an appropriately smaller inverse-polynomial accuracy. The binary-charge construction instead stores its exponentially large offset and charges with polynomial bit length and requires certified exponential accuracy in the one-well calculations. Neither reduction queries a many-electron ground energy as a subroutine.

🏷️ Key steps and proof outlines

An absolute orbital error cannot calibrate exponential tunneling

In the binary-charge construction, an overlap of size $e^{-D}$ must be known with small relative error. An $O(D^{-1})$ absolute approximation of a normalized orbital says little about its tail at distance $D$. The source repairs this by building a pointwise comparison function adapted to the slowly varying field of the other wells.

For one center $X_i$, let $r_i=|y-X_i|$, and let $W_i$ denote its capped background field. Define the ray integral

$$F_i(y)={\int }_0^{r_i}{W}_i(X_i+t(y-X_i)/r_i)dt.$$

It satisfies ${\partial }_{r_i}{F}_i=W_i(y)$ exactly. Consequently, substituting

$$f_i(y)={\pi }^{-1/2}{e}^{-r_i+F_i(y)}$$

into the one-well equation cancels the leading background contribution. With the isolated energy tuned to within $e^{-1.01D}$ of $-1/2$, the residual divided by $f_i$, on a suitable annulus, is bounded by

$$C\left(\frac{m}{D{r}_i}+\frac{m^2}{D^2}\right),$$

where $m$ is the number of wells.

Choose a radial correction

$$g(r)=A_0\left[\frac{m}{D}(1+\log (r/10))+\frac{m^{2}r}{D^2}\right].$$

Its derivative has precisely the size needed to dominate that residual. For fixed sufficiently large $A_0$, $f_i{e}^g$ and $f_i{e}^{-g}$ become upper and lower barriers. A tiny additive constant handles the outer boundary; the positive zeroth-order coefficient permits the maximum principle on the annulus. At radii at most $2D$, the relative effect of that constant is exponentially small and $g=O((m\log D+m^2)/D)$. Hence

$${\psi }_i(y)=f_i(y)\left(1+O\left(\frac{m\log D+m^2}{D}\right)\right).$$

This controls the positive tunneling integrals multiplicatively and makes their deterministic calibration meaningful. It is a separate analytic obligation from approximating the one-well eigenvalue (OpenAI, 2026a).

The unit-charge wells are designed from a positive resolvent

The unit-charge companion begins with a planar bump $f(r)=(1-|r{|}^2{)}_{+}^{16}$ and sets

$$\varphi =(-{\Delta }_r+1{)}^{-1}f,W=-\frac{f}{2\varphi }.$$

The heat-kernel representation of the resolvent shows $\varphi >0$, with exponential tails. Since $(-\Delta +1)\varphi =f$,

$$\left(-\frac{1}{2}\Delta +W\right)\varphi =-\frac{1}{2}\varphi .$$

Thus the desired orbital is built into the potential rather than obtained from an uncontrolled numerical fit. For a test function $u$, the shifted quadratic form is

$$⟨u,\left(-\frac{1}{2}\Delta +W+\frac{1}{2}\right)u⟩=\frac{1}{2}\int {\varphi }^2|\nabla (u/\varphi ){|}^2.$$

It is nonnegative and has only the ground-state direction in its kernel. Compact support of $W$, local compactness, and the positive exterior shift yield a fixed gap above that direction.

The same modes must also impose an occupancy penalty for signed charge defects. Let $F={\sum }_i{a}_i{\psi }_i^2$, with arbitrary real $a_i$, and choose disjoint translated bumps ${\chi }_i$. For $w={\sum }_i{a}_i{\chi }_i$, the density–test overlap matrix is a fixed positive diagonal matrix plus exponentially small off-diagonal terms. Therefore

$$\int Fw\ge c\sum _i{a}_i^2,\int |\nabla w{|}^2\le C\sum _i{a}_i^2.$$

Poisson’s equation gives

$${\left|\int Fw\right|}^2\le \frac{1}{4\pi }\left(\iint \frac{F(y)F(y^{\prime })}{|y-y^{\prime }|}dyd{y}^{\prime }\right)\int |\nabla w{|}^2.$$

Combining these proves the uniform Coulomb coercivity $(v_{ij})\ge c^{\prime }I$. This supplies the defect gap and the positive denominator $U-v_{ij}$ in superexchange. Positivity of individual Coulomb coefficients alone would not establish either conclusion for signed defects (OpenAI, 2026b).

🏷️ Connections and proof status

Coulomb atoms and ionization discusses how many electrons can remain bound and how atomic energy behaves. The present complexity problem allows arbitrary promised nuclear inputs and requires no binding premise; an ionization estimate does not supply the all-state continuum comparison used here.

Quantum parity lower bounds concerns the power of a restricted circuit model. QMA-hardness instead compares decision problems through efficient reductions; it is not a direct lower bound against every algorithm for molecular energies.

This exposition follows the stated continuum theorems and selected finite-model, quadrature, complement-elimination and threshold mechanisms. The quantitative source hardness theorem, detailed well/oracle constructions, and several geometric estimates remain attributed inputs. No finite numerical example establishes the uniform complexity reductions, and this selected-source review is not a verification of every analytic estimate.

🏷️ Manuscripts and verification scope

• Continuum Coulomb hardness with binary nuclear charges — pinned manuscript and companion source.
• QMA-hardness of continuum Coulomb energy with unit nuclear charges — pinned manuscript and companion source.

The family scope page describes selected formalization coverage. The page describes both binary-charge and unit-charge hardness in the full spinful fermionic continuum, with rational positions, unary electron count, deterministic polynomial-time reductions and separated thresholds. This stated coverage does not turn the present selected-source reading into an independent audit of either complete reduction. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026a. Continuum Coulomb hardness with binary nuclear charges.

🐻  OpenAI 2026b. QMA-hardness of continuum Coulomb energy with unit nuclear charges.