Laughlin states: spectral gaps and stability under scalar disorder

Source and verification

This article explains family 269 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.

🏷️ Lowest Landau levels and the pair interaction

The Laughlin state is an explicit many-electron wave function. Identifying it as a ground state and proving a uniform energy cost for every orthogonal excitation are different problems. Family 269 contains a manuscript claiming a gap for the full fermionic interaction on expanding spheres, together with a companion claiming stability under bounded scalar disorder. We develop the unperturbed estimate first, then explain the additional local estimates needed for stability. (OpenAI, 2026) (OpenAI, 2026b)

The prerequisites are finite-dimensional Hilbert spaces, orthogonal projections, tensor products, antisymmetric wave functions, positive operators, and elementary perturbation theory. The polynomial description below specifies the physical space without requiring a course in representation theory.

Let the integer $q$ count magnetic flux quanta. At magnetic length one, the sphere has radius $\sqrt{q/2}$ and area $2\pi q$. Its lowest Landau level $U_q$ is a $(q+1)$-dimensional one-particle space. It can be represented by homogeneous polynomials of degree $q$ in coordinates $(u,v)$, with inner product chosen so that

$$e_j={\left(\genfrac{}{}{0ex}{}{q}{j}\right)}^{1/2}{u}^{q-j}{v}^j,0\le j\le q,$$

are orthonormal. For $n$ identical fermions the space is ${\bigwedge }^n{U}_q$: exchanging two particles changes a wave function’s sign. An increasing wedge of distinct orthonormal orbitals has norm one. The Fock space is the direct sum of these spaces for $0\le n\le q+1$; it lets us discuss particle removal without changing the ambient space.

The pair interaction is an orthogonal projection $P^{(1)}$ onto relative angular momentum one. Operationally, its range has one normalized vector for each $b=1,\dots ,2q-1$, proportional to

$$\sum _{\begin{array}{c}0\le i<j\le q\\ i+j=b\end{array}}(j-i)\sqrt{\left(\genfrac{}{}{0ex}{}{q}{i}\right)\left(\genfrac{}{}{0ex}{}{q}{j}\right)}{e}_i\wedge e_j.$$

Vectors at different $b$ are orthogonal. The Hamiltonian on $n$ particles is

$$H_{n,q}=\sum _{1\le i<j\le n}{P}_{ij}^{(1)}.$$

Every unordered pair has coefficient one. This is the full spherical pair interaction, without a cutoff in orbital distance. Set $H_{0,q}=H_{1,q}=0$ and $H_q={⨁}_n{H}_{n,q}$.

At Laughlin filling, $q=3(N-1)$, the zero space is the line generated by

$${\Psi }_{\mathrm{L},N}=\prod _{i<j}(u_i{v}_j-u_j{v}_i{)}^3.$$

The basic reason is a vanishing condition: a zero-energy antisymmetric polynomial has at least a cubic zero whenever two particles coincide. The pairwise cubic factors use degree $3(N-1)$ in each particle, exhausting the available degree at this flux. The manuscript supplies the precise pair-sector and divisibility argument. This identifies a kernel; it does not yet bound positive eigenvalues.

🏷️ A uniform inequality on every particle sector

For a positive operator $H$, the inequality $H^2\ge \gamma H$ says that each eigenvalue $E$ satisfies $E(E-\gamma )\ge 0$. Thus

$$\mathrm{spec}H\subset \{0\}\cup [\gamma ,\infty ).$$

Conversely, that spectral inclusion implies the operator inequality. If a sector has no zero eigenvalue, this only bounds its lowest energy away from zero; it does not bound the separation between its two lowest energies.

Claimed Fock-space and Laughlin bounds

Put ${\gamma }_{\ast }=4616733319001/10^{14}>1/25$. For every fixed $0<\gamma <{\gamma }_{\ast }$, the manuscript claims $H_q^2\ge \gamma H_q$ for all sufficiently large fluxes $q$, with a threshold independent of particle number. Consequently, at $q=3(N-1)$ and all sufficiently large $N$, the gap above the unique Laughlin line is at least $1/25$. (OpenAI, 2026a)

The flux and particle thresholds are existential. The theorem does not give the endpoint ${\gamma }_{\ast }$ as a finite-sphere bound.

Let $c_j$ remove an electron from orbital $e_j$, with the sign imposed by antisymmetry. Its norm is at most one. More generally, $B(v)$ removes a normalized pair wave function $v$. For an orthonormal basis $v_p$ of the penalized pair space,

$$H_q=\sum _{p}B(v_p{)}^{\ast }B(v_p).$$

This identity retains the coefficient-one normalization. A zero state is therefore annihilated by every $B(v_p)$.

Expanding $H_q^2$ and moving creation operators to the left separates two-, three-, and four-particle terms. The two-particle term is exactly $H_q$, since the original pair operator is a projection. The four-particle term is positive. The three-particle term need not be positive and is the main obstruction to an immediate gap estimate.

The manuscript constructs seven auxiliary operators $F_{\rho }$. Each combines a pair removal with operations that remove three particles and create one. Their rotationally averaged squares form a positive operator ${\mathcal{K}}_q$. Finite matrices compare its three- and four-particle contributions with those of $H_q^2$. Averaging over rotations simplifies these matrices, but repeated angular-momentum sectors retain their full matrix of cross terms; they cannot be treated as independent scalar channels.

🏷️ Exact certificates and the positive margin

The finite calculation uses seven rows of integers. After cancelling explicit square roots, it reduces to rational three-particle margins and rational symmetric four-particle matrices. The latter matrices may have kernels.

A useful exact positivity test is elementary. If $M$ is real symmetric and every coefficient after the leading term in $\det (xI+M)$ is nonnegative, then this polynomial is strictly positive for $x>0$. A negative eigenvalue of $M$ would produce a positive root. Therefore $M\ge 0$. Zero coefficients are allowed, so this test does not require positive definiteness.

The coefficients can be generated using exact fractions. Start with $J_0=I$ and, for $b=1,\dots ,d$, use

$$\begin{array}{rl}{X}_b& =M{J}_{b-1},\\ c_b& =\mathrm{Tr}(X_b)/b,\\ J_b& =c_{b}I-X_b.\end{array}$$

Then $\det (xI+M)=x^d+c_1{x}^{d-1}+\cdots +c_d$. This is the characteristic-polynomial recurrence used in the source certificate.

Our independent rational reconstruction from the supplied rows reproduces all thirteen positive three-particle margins and all twenty-three four-particle coefficient-sign patterns. It also checks the allowance factor $1222$ and the scalar budget

$$\begin{array}{rl}\eta & =\frac{93527408868499}{10^{14}},\\ \epsilon & =\frac{3}{10^6},\\ {\gamma }_{\ast }& =1-\eta -\frac{61}{4096}-1222\epsilon .\end{array}$$

Here $\eta$ pays for the two-particle part of the auxiliary squares, $61/4096$ is the limiting three-particle tail, and $1222\epsilon$ pays for the four-particle comparison allowance. The finite arithmetic establishes these matrix inequalities and this positive margin. The passage to all particle numbers still requires an analytic relative estimate. (OpenAI, 2026a)

🏷️ Relative errors and spectator particles

Convergence of a few matrix entries is insufficient if an error is multiplied by the number of particles. There is a second difficulty: positivity on the range of a limiting matrix need not survive an arbitrary perturbation when its range has a kernel.

The source avoids both difficulties by comparing the pair-removal operators themselves. On the first twenty-five orbital modes, define a diagonal operator on occupation states $|A⟩$ by

$$\begin{array}{c}{\Delta }_q|A⟩=\prod _{x\in A}{d}_q(x)|A⟩,\\ d_q(x)=\sqrt{(q{)}_x/q^x},\end{array}$$

where $(q{)}_x=q(q-1)\cdots (q-x+1)$ and $(q{)}_0=1$. This local Fock space is fixed as $q$ grows, so ${\Delta }_q$ and its inverse converge to the identity. For retained pair labels $p\le 23$, the exact coefficient formula gives

$$\begin{array}{c}{B}_p=r_p(q){\Delta }_q^{-1}{B}_p^{\infty }{\Delta }_q,\\ r_p(q)=\sqrt{(2q{)}^p/(2q-2{)}_p}.\end{array}$$

Here $B_p^{\infty }$ is the limiting pair-removal operator. The identity converts finite-flux annihilators into fixed limiting ones before estimating an error.

Every relevant error is a finite quadratic expression in operators of the form $A_b=c_k{c}_j{B}_p^{\infty }$. Since $\Vert c_k{c}_j\Vert \le 1$,

$$\sum _b\Vert A_b\psi {\Vert }^2\le C\sum _{p=0}^{23}\Vert B_p^{\infty }\psi {\Vert }^2.$$

If the coefficient error has operator norm $o(1)$, its quadratic form is therefore bounded by $o(1)$ times local pair energy. This remains true when the coefficient matrix itself has negative directions that vanish after the pair-generated compression. No inverse of a singular Gram matrix is taken.

Extra occupied orbitals are spectator factors: order the twenty-five active modes first, and the local inequality tensors with the identity on all other modes. The same constant works for every particle sector. Finally, rotation averaging converts the local pair-energy bound into $o(1)H_q$. The comparison becomes

$$H_q^2\ge {\mathcal{K}}_q+({\gamma }_{\ast }-o(1))H_q.$$

Positivity of ${\mathcal{K}}_q$ gives the claimed result for each fixed $\gamma <{\gamma }_{\ast }$. The remaining three-particle tail is controlled by a summable geometric majorant, allowing its limit to be taken uniformly in the particle sector. (OpenAI, 2026a)

🏷️ The planar endpoint

The unperturbed manuscript also gives a planar consequence. The one-particle Bargmann space consists of entire functions with norm

$$\Vert f{\Vert }^2=\frac{1}{\pi }{\int }_{\mathbb{C}}|f(z){|}^2{e}^{-|z{|}^2}{d}^{2}z.$$

For each pair, project onto relative degree one in $(z_i-z_j)/\sqrt{2}$, with unrestricted center-of-mass degree. Summing these coefficient-one projectors defines $H_N^{\mathrm{pl}}$.

At fixed particle number $N$ and fixed total polynomial degree, the relevant space is finite-dimensional. Identifying normalized monomial bases, its spherical Hamiltonian matrix converges in norm to the planar matrix. First pass $q\to \infty$ in $H_q^2\ge \gamma H_q$ for fixed $\gamma <{\gamma }_{\ast }$. Then let $\gamma \uparrow {\gamma }_{\ast }$. Closedness of the positive-matrix cone gives the endpoint on that block. Taking the Hilbert direct sum over degrees is legitimate because $0\le H_N^{\mathrm{pl}}\le \left(\genfrac{}{}{0ex}{}{N}{2}\right)I$.

Thus the source claims $(H_N^{\mathrm{pl}}{)}^2\ge {\gamma }_{\ast }{H}_N^{\mathrm{pl}}$ for every $N\ge 2$, uniformly over total degree. The planar zero space can be large; the conclusion is a gap above that entire space. The endpoint comes from these ordered limits and is not an endpoint claim on finite spheres. (OpenAI, 2026a)

🏷️ Scalar disorder and its extensive norm

Return to $q=3(N-1)$. A real bounded function $\phi$ on the sphere becomes the one-particle operator $T_q(\phi )$ by multiplying a wave function by $\phi$ and projecting back to the lowest level. The perturbed Hamiltonian is

$$\begin{array}{c}{H}_{N,q}(\lambda )=H_{N,q}+\lambda V_{N,q},\\ V_{N,q}=\sum _{i=1}^N{T}_q(\phi {)}^{(i)}.\end{array}$$

Although $\Vert \phi {\Vert }_{\infty }\le 1$ gives $\Vert V_{N,q}\Vert \le N$, ordinary norm perturbation estimates would only protect a coupling interval shrinking like $1/N$.

Claimed uniform stability

The companion asserts constants ${\lambda }_{\ast }>0$, ${\Delta }_{\ast }>0$ and $N_{\ast }<\infty$ such that, for every $N\ge N_{\ast }$, every real bounded measurable $\phi$ with $\Vert \phi {\Vert }_{\infty }\le 1$, and every $|\lambda |\le {\lambda }_{\ast }$, the two lowest eigenvalues counted with multiplicity differ by at least ${\Delta }_{\ast }$. In particular, the perturbed ground state is unique. The constants are uniform in the potential profile, which may depend on $N$; no derivative or symmetry bound on the potential is required. (OpenAI, 2026b)

The ground vector moves under this perturbation. Adding a constant to $\phi$ merely shifts its energy. The proof must center the perturbation relative to the moving ground state.

For a unit coherent orbital $k_x$ concentrated near $x$, the exact resolution is

$$\begin{array}{c}{T}_q(\phi )={\kappa }_q{\int }_{S_q}\phi (x)|k_x⟩⟨k_x|dA(x),\\ {\kappa }_q=\frac{q+1}{2\pi q}.\end{array}$$

The bounded density ${\kappa }_q$ explains why amplitude control is enough. The companion constructs redundant localized coordinates in which each integrand has rapidly decaying tails. Here “rapidly local” means approximability in operator norm by operators in a ball of radius $R$, with error $C_m(1+R{)}^{-m}$ for every fixed $m$. It does not mean exact compact support.

🏷️ Missing particles and local energy descent

Write $Z_{n,q}=\ker H_{n,q}$ and let $\Omega$ be the normalized filled Laughlin vector. The first additional estimate controls a bounded-density sum $W$ of rapidly local, particle-number-preserving Hermitian terms:

$$\begin{array}{c}|⟨W{⟩}_{\psi }-⟨W{⟩}_{\Omega }|\le C_W(N-n),\\ \psi \in Z_{n,q},\Vert \psi \Vert =1.\end{array}$$

The dependence is linear in missing electrons and uniform in area. This is much stronger than a global gap alone.

The source obtains it from orthogonal coordinates on zero spaces. Put $h=q+3-3n$. Removing one unit of flux without removing an electron reduces $h$ by one and records a flag; the other branch removes one electron and three flux units, leaving $h$ unchanged. At the filled flux, $h=3(N-n)$, so one flag is not one missing electron. Pinning zeros at a pole gives local transport between the relevant spaces. A first weighted charge estimate controls the errors needed for a second, summable weight; averaging over rotation centers then yields the displayed bound. This pinning and charge analysis is an additional analytic input, not a consequence of the finite certificate alone. (OpenAI, 2026b)

The second estimate is a local particle-loss process with jump operators $J_y$ satisfying

$$\begin{array}{rl}\int J_y^{\ast }{J}_{y}dy& =H_q,\\ \int J_y^{\ast }{H}_q{J}_{y}dy& \le H_q^2-c{H}_q.\end{array}$$

Every jump removes between two and a fixed number $L$ of particles. Its construction uses positive four-particle blocks retained from the gap comparison: removing a pair is followed by partial emptying of a fixed neighborhood. The unused positive blocks pay for the residual pair correlations.

For the associated evolution of density matrices, the observable generator is

$$\begin{array}{rl}\mathcal{D}(O)=& \int J_y^{\ast }O{J}_{y}dy\\ & -\frac{1}{2}(H_{q}O+O{H}_q).\end{array}$$

It preserves the trace because $\mathcal{D}(I)=0$. The descent inequality gives $\mathcal{D}(H_q)\le -c{H}_q$, hence exponential energy decay. If $\mathcal{N}$ counts particles, then $0\le -\mathcal{D}(\mathcal{N})\le L{H}_q$. Integrating gives expected particle loss at most $(L/c)$ times initial energy. Limit points lie in the zero space, whose sectors have at most $N$ particles. This also bounds particle excess by energy.

For a sufficiently small fixed $\mu >0$, it follows that

$$\begin{array}{rl}G& =H_q+\mu (N-\mathcal{N})\\ & \ge a(H_q+|N-\mathcal{N}|)\end{array}$$

for some $a>0$. The negative number term in sectors above filling is controlled by the excess-energy bound. On the physical $N$-particle sector, $G=H_{N,q}$.

🏷️ Closing the relative bound and transporting the ground line

Suppose $A_x$ are Hermitian, particle-number-preserving local terms satisfying $A_x\Omega =0$, with bounded total local norm per unit area. The companion proves

$$-CG\le \sum _x{A}_x\le CG.$$

Its closure argument is worth isolating. At fixed finite flux, let $K_q$ be the best such constant; it is finite but initially may depend on volume. Along the loss evolution, differentiating the expectation of $A={\sum }_x{A}_x$ produces commutators $[A_x,J_y]$. Weighted Cauchy–Schwarz and locality give a bound by a constant times

$$⟨H_q{⟩}_t^{1/2}(K_q⟨G{⟩}_t{)}^{1/2}.$$

The squared commutators belong to the same centered local class, which is why the same $K_q$ can be used. Energy decay and the bound on expected particle loss also give $⟨G{⟩}_t\le C⟨G{⟩}_0$. Together these estimates make the integral cost at most $C\sqrt{K_q}⟨G{⟩}_0$. The limiting zero-state expectation costs at most $C⟨G{⟩}_0$ by the missing-electron estimate. Taking the supremum yields $K_q\le C+C\sqrt{K_q}$, which bounds $K_q$ uniformly by solving a quadratic inequality.

Finally, work initially on a parameter interval where the perturbed ground state is simple and its gap exceeds half the unperturbed bound. A smooth frequency filter produces a local unitary $U(s)$ carrying the original ground line to the perturbed one. Applied separately to each perturbation term, the filter cancels its ground-to-excited matrix elements. Subtracting its ground expectation then gives a term that annihilates $\Omega$ exactly; merely subtracting the expectation of the original term would not suffice.

The relative bound controls the derivative of the transported Hamiltonian after subtracting its ground energy $E_0(s)$. Integration on the $N$-particle sector gives

$$\begin{array}{rl}& U(s{)}^{\ast }{H}_{N,q}(s)U(s)-E_0(s)I\\ & \ge (1-C|s|)H_{N,q}.\end{array}$$

Choose $C{\lambda }_{\ast }\le 1/4$. The resulting gap stays above three quarters of the unperturbed bound, so it cannot reach the half-gap boundary of the initial interval. Continuity extends the argument throughout a fixed interval independent of $N$. This closes the stability proof using the local estimates; the global gap by itself would not close it. (OpenAI, 2026b)

🏷️ Verification and connections

The rational reconstruction checks the finite unperturbed certificate, not the complete finite-flux transfer or the stability proof. Our source review checks the main statements, selected comparison and locality mechanisms, and the elementary deductions explained here. The pinning, charge estimates, localized-frame construction, and full analytic error bookkeeping remain attributed to the manuscripts.

The exact-arithmetic script takes the manuscript’s seven-row data file and an output JSON path as its two arguments. It uses Python’s standard-library rational arithmetic and reproduces the finite checks described above.

• Spin-one Haldane gaps uses a different finite-certificate mechanism, amplifying partition-function purities across length scales.
• Area laws and tensor networks studies consequences of a spectral gap; this family addresses its establishment and persistence for a particular many-electron interaction.

🏷️ Key steps and proof outlines

Auxiliary squares use interference to correct a negative term

The unperturbed gap proof does not simply add more positive pair interactions. It chooses operators whose squares have useful cross terms. For a fixed row of coefficients, write

$$F=\lambda B_t+\sum _{p,j}{\alpha }_{pj}{c}_{p+j-t}^{\ast }{c}_j{B}_p.$$

Every summand removes two particles and changes the total orbital label by the same amount. Consequently these summands can interfere when $F^{\ast }F$ is expanded.

The first square contributes the positive two-particle term ${\lambda }^2{B}_t^{\ast }{B}_t$. Cross terms contribute three-particle operators with adjustable signs. Multiplying the second sum by its adjoint and using

$$c_i{c}_k^{\ast }={\delta }_{ik}-c_k^{\ast }{c}_i$$

produces another three-particle contribution and a four-particle contribution with a minus sign. Thus positivity of the full square does not imply positivity of each particle-order component. The certificate chooses the coefficients so that their combined comparison with the corresponding components of $H_q^2$ leaves a positive gap margin.

This explains the tradeoff represented by the scalar budget above: improving the difficult three-particle term has a two-particle cost, and the induced four-particle term needs its own matrix certificate. After rotation averaging, equivalent copies of the same angular-momentum sector still interact through a matrix. On the physical four-particle image, positivity is tested through

$$G_D(2\chi -E_D+\epsilon I)G_D⪰0,$$

where $G_D$ is the Gram matrix of the wedged vectors, $E_D$ the auxiliary contribution, and $\chi$ selects the original pair-pair channel. This compression retains dependencies among those vectors instead of assuming that the Gram matrix is invertible (OpenAI, 2026a).

Pinning requires a gap along a whole path

Multiplying a zero-mode polynomial by a factor that pins a zero changes its norm. The stability companion first realizes this nonunitary operation as the kernel of a uniformly gapped path, then uses spectral transport to obtain a unitary comparison.

For $d\in \{1,3\}$, set $t=q-d$, and define the orbital weight

$$g(j)=\frac{1}{2}\log \frac{\left(\genfrac{}{}{0ex}{}{t}{j}\right)}{\left(\genfrac{}{}{0ex}{}{q}{j}\right)}.$$

If $D_t(g)={\sum }_{j}g(j)c_j^{\ast }{c}_j$, the desired intermediate zero space is $e^{s{D}_t(g)}\ker H_t$, $0\le s\le 1$. The source obtains it by multiplying each pair-annihilator coefficient by $e^{-s(g(i)+g(j))}$, then normalizing each row. These normalizations are nonzero scalar factors and do not change the common kernel.

To retain a uniform gap, it compares columns of annihilators, not the norm difference of extensive Hamiltonians. Round the logarithmic weight near the last $O(B)$ orbitals, with $B$ fixed independently of flux. On a pair row $i+j=b$, the affine Taylor part of the weight is constant and cancels under normalization. Only its second-order variation remains. Summing the resulting squared row errors yields

$$\Vert {\overline{\mathcal{C}}}_s-{\mathcal{C}}_0{\Vert }^2\le C{B}^{-1/2},$$

where ${\mathcal{C}}_0\psi =(B_{t,b}\psi {)}_b$ and ${\overline{\mathcal{C}}}_s$ is the rounded column.

Choose $B$ so the column error is below half the square root of the unperturbed gap. The explicit kernel formula preserves nullity in each particle sector; singular-value perturbation therefore preserves a positive lower singular value off the kernel. Finally, undoing the rounding conjugates by an operator supported on only $O(B)$ orbital modes. Its condition number is finite independently of flux and particle number, so the positive bound survives. Squaring the column estimate proves the required path gap.

This argument supplies the missing uniform input for transporting pinned zero spaces and local observables. A mere pointwise identification of their kernels, or a norm estimate growing with particle number, would not suffice for the later missing-charge and disorder-stability estimates (OpenAI, 2026b).

🏷️ Manuscripts and verification scope

• Uniform Stability of the Spherical Laughlin Gap — pinned manuscript and companion source.
• A Fock-space inequality and the Laughlin spectral gap — pinned manuscript and companion source.

The family scope page describes selected formalization coverage. The page selects unperturbed spherical bounds (including the weaker 1/100 bound and the 1/25 result), the uniform Fock-space inequality for each fixed coefficient below gamma-star, and the planar endpoint inequality on homogeneous sectors, all with coefficient-one pair projectors. It explicitly excludes stability under projected one-body potentials and uniqueness of the perturbed ground state. 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. A Fock-space inequality and the Laughlin spectral gap.

🐻  OpenAI 2026b. Uniform Stability of the Spherical Laughlin Gap.