Can an SDP search improve the lower bound on the Laughlin spectral gap?

A bound on the operator, not the observed gap

The target is the V1 pseudopotential Hamiltonian, the simplest parent Hamiltonian of Laughlin's state at filling 1/3: every pair of fermions pays energy one in relative angular momentum one and nothing otherwise. The conjecture is that its positive spectrum stays uniformly away from zero. On the sphere with Q flux quanta, the paper works with H_Q, the sum of these pair projectors over all particle numbers, acting on the lowest-Landau-level Fock space.

The result is an operator inequality, H_Q squared greater than or equal to γ times H_Q, for every γ below γ*. Because the spectrum of H_Q is then contained in {0} together with [γ, ∞), and the zero mode at the Laughlin flux is unique, this yields the usual ground-state gap. The threshold flux Q_γ is existential and independent of particle number; the paper states that its margins are small, so the flux at which the inequality first holds is presumably very large, and the statements concern the thermodynamic limit.

Two corollaries carry the constant to concrete settings. On spheres with Q = 3(N−1), the Hamiltonian is at least (1/5)(I − P_L,N) for all N ≥ N_0, where P_L,N projects onto the Laughlin state. For the untruncated planar V1 Hamiltonian, (H^pl_N)² ≥ γ* H^pl_N holds uniformly in particle number and total angular momentum. A further corollary bounds the charge gap Δ_N below by N/[5(N−1)] for N ≥ N_0.

Three costs that shrank the old constant

The earlier proof balanced the constant as 1 − η − δ − 1222ε, with η = 0.93527, δ = 61/4096 = 0.01489, and 1222ε = 0.003666. Each term has a distinct job. η is the two-body cost of the pair-annihilator parts of the seven auxiliary operators. δ bounds the three-body spin blocks that those operators never reach, estimated by Cauchy–Schwarz with a factor z+1 in block z. The 1222ε term pays for an allowance εI in the four-body certificate, a Bessel estimate repeated in each of the 23 retained spin blocks.

The paper attacks each term separately. For the three-body tail, comparing a spin block with the rotation average of a single product vector replaces the factor z+1 by 1/max_p Bin(z, 1/3)(p), which behaves like (2/3)√(πz); the tail δ drops from 0.0149 to 0.0041. For the four-body allowance, the rotation average of one vector at total deficit T pays the allowance of every spin block D ≤ T at once, so an identity allowance costs 144ε rather than 1222ε, and block-dependent allowances cost a sum the paper calls the staircase price.

These two estimates alone, applied to the original seven rows without changing the certificate, lift the constant above 0.0601. The table in the paper records the sequence: 0.0461673 with the original data, 0.0494013 with the staircase price, 0.0569644 with the product-twirl tail, and 0.0601984 with both. Every entry is a rigorous constant; the ones using new auxiliary operators are certified by exact rational arithmetic, while the others reuse the original certificate unchanged.

Auxiliary operators become SDP variables

The larger gain comes from the auxiliary operators themselves. Every quantity in the certificate depends on these operators only through their Gram matrices, and linearly. Optimizing over them is therefore a semidefinite program: the search variables are the Gram matrices, and the constraints are the three-body inequalities and the four-body positivity conditions of the admissibility definition.

Within the original index sets, this optimization already lowers η to 0.92144 with no allowance at all. Enlarging the index sets is the main lever: it lowers η to 0.78097 and simultaneously pushes the tail beyond block 35, where it is negligible. The index bounds are written (p̄, j̄, ı̄, t̄); the original choice is (7,8,8,7), and the final certificate uses (17,18,18,17) with 64 rows.

The progression in the table shows where the leverage sits. Ten rows on the original index sets give 0.0636643 without allowance and 0.0744614 with the product-twirl tail. Twenty rows at (9,10,13,9) give 0.1882970 with no allowance needed. Thirty-seven rows at (14,15,15,14) give 0.2138574 with one block allowance, 47 rows at (16,17,17,16) give 0.2178926, and 64 rows at (17,18,18,17) give 0.2190037. Pricing the final allowances with the staircase price, still with c_z = z+1, raises that last figure to 0.2190169.

Numerics find the certificate, rationals verify it

Only the first step is numerical. The SDP is solved with a dense primal–dual interior-point method using the HKM search direction and Mehrotra's predictor–corrector scheme, after removing common null spaces of the constraint matrices. At every optimum the priced violation of the three-body inequalities was below 2×10⁻⁸, and for (7,8,8,7) and (9,10,10,9) solving with hard constraints gives the same optima to all digits shown.

A numerical optimum sits on many tight constraints and cannot be rounded directly. The paper re-solves with constraints tightened by margins of order 10⁻⁹ to 10⁻⁷, adds a small penalty on the rank of each matrix, and extracts low-rank rows from the spectral decomposition, keeping eigenvectors whose eigenvalue exceeds 10⁻³ times the largest. The rows are then rounded to integers over a denominator P between 10¹⁰ and 10¹⁵, and the pair coefficients are re-optimized to minimize η subject to three-body margins of order 10⁻¹⁰ to 10⁻⁸. The smallest admissible allowances on the grid 1/P are found by exact bisection.

Admissibility is then checked in exact rational arithmetic: every three-body margin is computed as a rational number and checked positive, and every four-body condition is checked through a congruent matrix, both by exact symmetric LDLᵀ elimination and by the signs of characteristic coefficients via the Leverrier–Faddeev recurrence. Each certificate was verified by at least two independent exact implementations, and all of them reproduce the earlier constant 4616733319001/10¹⁴ exactly from its seven rows. In negative tests, perturbing all integers by a relative 10⁻⁶, or dropping any row, makes the check fail.

What the final certificate actually contains

The main certificate has (p̄, j̄, ı̄, t̄) = (17,18,18,17) and 64 rows over P = 10¹³, with 14,417 nonzero integers a_pj. It is too long to print, so it is distributed as an ancillary file with a SHA-256 hash, together with a self-contained verification script that uses only the Python standard library, reads only the certificate file, recomputes every condition in exact integer and rational arithmetic, and runs in under a minute.

Its data are specific: rows per shift t = 0,…,17 are 3,3,4,4,2,2,2,3,4,3,3,3,3,1,5,2,2,15; allowances ε_D are zero except at D = 13, 14, 15, 17, 20, 37, and 39; T_3 = 35 and T_4 = 53, with local modes 0 through 54 in the transfer lemma; all 33 three-body margins are positive, the smallest 1.99999×10⁻⁸ at z = 5; and no characteristic coefficient is negative in any block D ≤ 53.

The transfer from this finite data to the Fock-space inequality rests on Theorem 2.4, which restates the earlier comparison argument for arbitrary finite index sets and arbitrary block-dependent allowances. Its proof follows the earlier argument line by line, recording only where index ranges enter, and it reduces the operator inequality to finitely many limiting matrix inequalities. As an independent check of the limiting formulas on the enlarged ranges, coupled bases at finite flux were built from SU(2) ladder operators; for Q between 10³ and 10⁵ and a sample of blocks with z ≤ 35 and D ≤ 53, the finite-flux traces and matrices converge to the limiting data at rate O(1/Q).

How far this family of operators can go

The paper is explicit that 0.219 is not the true gap. A heuristic extrapolation of the SDP optima suggests that auxiliary operators of the present form can give at most about 0.222, while exact diagonalization of finite systems places the true gap near 0.4. The certified constant is therefore close to the ceiling of this operator family, not close to the physical gap.

A separate test probes whether the pair-to-particle operators of the kind used in the three-body analysis could help. At the optimum for (14,15,15,14), the largest value of the relevant test quantity over the covered blocks is 0.62, below the threshold of 1, so such operators cannot improve the certificate beyond removing the already negligible tail.

The stated route forward is a larger family of auxiliary operators, which could push the bound further toward the true gap, and the SDP formulation makes such families easy to test. The paper also records that the proof of the improvement and the SDP search were completed by the AI model Claude Opus 5.5 (Anthropic) under the author's direction, and that the human input was recognizing that the earlier certificate appeared hand-picked rather than optimized.

Read the certified constant, not the gap estimate

The most decision-relevant comparison in this paper is between three numbers that answer different questions. γ* = 0.2190037845063196 is a rigorously certified lower bound on the rescaled operator inequality, valid for all sufficiently large flux and uniform in particle number. The roughly 0.222 figure is a heuristic extrapolation of where this family of auxiliary operators saturates. The roughly 0.4 figure comes from exact diagonalization of finite systems and describes the true gap. Treating any one of these as a substitute for another would misstate what the paper establishes.

Within the certified column, the ordering matters too. The two new estimates alone move 0.0461673 to 0.0601984 without touching the certificate, while the SDP search over enlarged index sets moves it to 0.2190037. The staircase price applied to the final allowances adds only 0.2190169, so the dominant lever is the size of the auxiliary-operator index sets, not the pricing refinements.

The practical reading is that the bound is now above 1/5 on large spheres and positive and uniform on the plane, while the remaining distance to the true gap is attributed to the restricted form of the auxiliary operators rather than to the comparison architecture. Anyone extending this work should treat the SDP formulation as the testbed: the paper's own evidence says a larger operator family is what would move the certified number, and that the present family is already near its apparent ceiling.

Sources used in this answer