Breaking the high-degree barrier in Lehmer's problem with a human-AI discovery loop

Summary of Results

• The workflow combines local numerical and symbolic prototyping with distributed cloud searches, exact verification, and interactive human analysis of returned results.The paper presents this as a response to the need for infrastructure that can search and verify at scales beyond a single workstation.Source: §1 · pp. 2, 8

Introduction

• Lehmer’s problem asks whether an integer polynomial’s Mahler measure can exceed 1 by an arbitrarily small amount, or whether a definite gap exists above 1.The measure summarizes how far a polynomial’s roots stray from the unit circle.Source: §1 · p. 3

Summary of Results

• Across degrees d ≥30, independent search methods found empirical minima rising toward α1 from below, and 95.5% of their optima reduce to specializations of Boyd’s polynomial.These structural patterns enabled the generalized analytical construction.Source: §1 · pp. 5, 9
• The Search Ensemble established upper bounds across 824 even degrees, matching 85 and improving six of the 91 published Literature Optima.The six improvements include a degree within the previously systematically searched d ≤180 regime.Source: §1 · pp. 8–9
• Across 995 even degrees from 12 to 2000, the Generalized Boyd family stays strictly below α1, holds 99.6% of best-known records, and improves all degrees d ≥166.Its maximum measure is 1.25535826 at d = 1932; the complete verified benchmark dataset covers 996 even degrees from 10 to 2000.Source: §1 · p. 9
• The five-parameter construction eliminates the degree-skipping of the one-parameter Boyd family by solving for exponents that reach any target even degree.The earlier family’s cyclotomic factors cause discontinuous degree jumps; its 504 outputs skip 375 even degrees.Source: §1 · p. 9

2. Human–AI Discovery Loop

• The broader pattern is to compose machine search with human mathematical reasoning so each can correct the other’s blind spots.The paper surveys guided intuition, construction, program evolution, and neurosymbolic or formal systems as prior approaches.Source: §2 · pp. 9–10
• The workflow integrates Gemini with Google Antigravity to connect mathematical reasoning, algorithm design, and software engineering in an interactive discovery loop.Antigravity links the reasoning model to a persistent workspace with local tools and cloud-cluster launchers.Source: §2 · p. 10
• Unlike isolated black-box code-evolution systems, the workflow combines human-guided mathematical reasoning, algorithm design, and high-performance software engineering.It separates rapid local prototyping from distributed cloud sweeps and decouples numerical candidate generation from asynchronous exact verification.Source: §2 · pp. 10–11
• The workflow addresses two bottlenecks: rapid evaluation of polynomial roots across millions of candidates and exact irreducibility verification over Q.Its three pillars are asynchronous verification, complementary search algorithms scaled across cloud compute, and analytical synthesis through cyclotomic factoring.Source: §2 · p. 10
• The paper frames its approach as a fifth mathematical-AI paradigm: distributed cloud experimentation, scalable algorithm engineering, and symbolic hypothesis verification.The comparison distinguishes it from four surveyed paradigms and from human-steered local constraint-solving sessions such as [95].Source: §2 · pp. 9–11

3. Numerical Search Algorithms for Small Mahler Measure

• Because candidate counts grow exponentially with degree, the search focuses on sparse polynomials; Smyth’s theorem restricts candidates below Mahler measure 1.324718 to reciprocal polynomials of even degree.The leaderboard records the smallest known measure for each degree.Source: §3 · p. 11, §3.1 · p. 11

3.1. Efficient search

• Search workers stream candidates while asynchronous verification deflates degree-redundant polynomials, checks irreducibility, computes Mahler measure, and updates the leaderboard.The pipeline also removes cyclotomic polynomials and duplicates without pausing candidate search.Source: §3.1 · p. 12

3.2. Search Methods

• The search methods include sparse direct search, priority-based coefficient perturbations, reinforcement learning, evolutionary operations, and local coefficient refinement.Priority search starts from a cyclotomic polynomial; reinforcement learning updates coefficient probabilities using high-scoring sampled polynomials.Source: §3.2 · pp. 12–13
• Dense reinforcement-learning discoveries motivated periodic tiling with defects, which searches over a short repeating coefficient block and a symmetric central defect rather than all d/2 coefficients.The method scales to d = 2000 and motivates the analytical construction in Section 4.Source: §3.2 · p. 13

3.3. Numerical Search Results

• 824 even-degree optima were established across d ≤2000, with 6 strict improvements over prior bounds and upper bounds across 733 additional degrees.The prior literature covered 91 even degrees through d ≤250; the searches matched 85 and improved six.Source: Figure 3 · p. 14, §3.3 · p. 14

4.1. Collapsing Empirical Results via Cyclotomic Factoring

• Multiplying P600 by Φ30 yields F(x, −x^303), while multiplying P1846 by Φ12 yields F(−x^11, −x^914).These examples show how cyclotomic factors cancel repeated coefficient blocks.Source: §4.1 · p. 16
• 95.5% of dense best polynomials for d ≥30 collapse to sparse polynomials with 6 or 7 terms, and 99.7% of those arise from Boyd’s two-variable polynomial under sign choices.Cyclotomic multipliers remove repeating interior blocks without changing Mahler measure.Source: §4.1 · pp. 15–16

4.2. Five-Parameter Analytical Construction

• For each target degree, the construction varies parameters, checks that the cyclotomic divisor has its targeted degree, and selects the irreducible quotient with minimal Mahler measure.The five-parameter family extends the one-parameter construction, which skips degrees because its cyclotomic factor degree jumps.Source: §4.2 · pp. 16–17
• Every even degree from 12 to 2000 has a verified irreducible polynomial below α1; the Generalized Boyd algorithm holds 99.6% of best-known records across the extended parameter sweep.The construction approaches α1 from below at rate O(d^-3/2).Source: §4.2 · p. 17

• Minimizing log Mahler measure confines exterior roots inward and, through reciprocal symmetry, pushes interior roots outward toward the unit circle.The exterior roots incur a log |z| penalty; reciprocal symmetry pairs each interior root with an exterior partner.Source: §5 · p. 18
• Irreducibility makes roots distinct, while integer coefficients impose pairwise repulsion that supports increasingly uniform angular spacing on the unit circle.At finite degree, this balance places over 80% of roots on the circle, with mean spacing approximately 2π/(0.825d).Source: §5 · p. 18
• Across all 824 benchmark polynomials, 77.0%–88.0% of roots lie exactly on the unit circle, with a mean of 82.51%.Exact rational root counting verifies the unit-circle roots; the remaining approximately 17.49% lie off the circle in reciprocal pairs at log-distance O(1/d).Source: §5 · p. 18, Figure 4 · p. 18
• Representative root plots show off-circle roots either clustered near −1 or split into evenly spaced clusters, depending on the polynomial.The radial distances from the unit circle are magnified for visibility.Source: §5 · p. 18, Figure 5 · p. 19

• Each Generalized Boyd entry includes a five-integer parameter tuple and cyclotomic divisor, alongside polynomial representations that support inspection and computer-algebra use.The sparse quotient expression is recorded for 991 degrees where the family attains or ties the best-known record.Source: §6 · pp. 19–20
• The Generalized Boyd family supplies 991 of the 996 dataset records, accounting for 99.6% of records from d = 12 to 2000 and all records for d ≥166.Search methods provide the remaining five optima at d ∈ {10, 12, 16, 32, 164}.Source: §6 · p. 20
• The dataset covers all 996 even degrees from 10 to 2000, matching 85 known optima and improving six published benchmarks in the range d = 30 to 250.The six improved degrees are 178, 188, 190, 202, 242, and 250.Source: §6 · pp. 19–20

7. Discussion

• Across even degrees 12–2000, Boyd specializations yield verified irreducible polynomials with Mahler measure strictly below α1 ≈1.255434, approaching it from below at rate A·d^-3/2.The paper leaves open whether α1 is the true limit infimum or substantially smaller values can recur at arbitrarily large degrees.Source: §7 · p. 20
• At d = 1000, the seven-term numerator has Mahler measure ≈1.25513074, and cyclotomic division yields a 537-term irreducible polynomial.The displayed compact quotient can be evaluated without expanding it.Source: §7 · p. 20
• Large-scale numerical search exposed repeating coefficient patterns that researchers compressed through cyclotomic factoring into an exact analytical construction.The paper presents broad computational search, automated verification, and analytical synthesis as a practical human–AI discovery-loop template.Source: §7 · p. 21

A. Asymptotic Analysis of the Generalized Boyd Family

• For small a and growing degree d, b/a ≈ d/(2a) grows without bound, while cyclotomic factors leave Mahler measure unchanged because M(Φ)=1.The specialized polynomial’s degree is d = 2a + 2b − deg Φ ≈ 2a + 2b.Source: §A.1 · p. 22
• As b/a →∞, the one-dimensional specialization’s Mahler measure converges to the two-variable Mahler measure α1 = 1.255433866266608745189… .Boyd and Lawton’s limit theorems establish convergence because the path equidistributes on the unit two-torus.Source: §A.1 · p. 22
• Boyd’s two-variable measure α1 is identified as the smallest known limit point of Mahler measures.Here α1 = M(F(X,Y)) for Boyd’s polynomial F(X,Y).Source: §A.1 · pp. 21–22

A.2. Universal Bound on Cyclotomic Factors (δ ≤14)

• For the reciprocal family k = 0, every cyclotomic factor has degree δ ≤14, with possible degrees {0, 4, 6, 8, 10, 14}.The maximum follows from mutually exclusive factor families contributing at most degrees 6 and 8.Source: §A.2 · pp. 22, 24

A.3. Existence of Self-Consistent Solutions for Every Even Degree

• For every even degree d ≥12, a self-consistent parameter tuple exists with δ=0, giving an exact-degree polynomial; allowing δ>0 enlarges the candidate pool for smaller Mahler measures.The construction chooses a coprime pair (a,b) and signs that avoid cyclotomic factors.Source: §A.3 · pp. 24–25

A.4. Oscillatory d−3/2 Asymptotics

• The square-root behavior at t = ±t0 makes those neighborhoods dominate the error term, while smooth intervals decay faster than any power of 1/n.The square-root singularity stops repeated integration by parts from eliminating the dominant contribution.Source: §A.4 · p. 26
• M(Pd) = α1 + A·d^-3/2 cos(ωd + ϕ) + O(d^-5/2) gives the oscillatory asymptotic expansion.The remainder comes from the next |t∓t0|^3/2 term.Source: §A.4 · p. 26
• Minimizing over (a,b,k,s0,s1) at each even degree selects phases with cos(ωd + ϕ) ≈ −1, so the minimum approaches α1 from below at rate A·d^-3/2.A fixed one-parameter sequence oscillates both above and below α1.Source: §A.4 · p. 26

A.5. Parameter Cutoff Analysis

• The family uses parameters a, b, k, s0, and s1, with a and b positive integers, 0 ≤ k < a, and s0, s1 ∈ {−1, +1}.The numerator is the polynomial in (43), with the cyclotomic factor removed to obtain Pd.Source: §A.5 · p. 27
• When gcd(a, b) = 1, the Fourier average vanishes unless a divides m, reducing the contributing terms to m = ra.The parameterization and averaging over θ lead to this divisibility condition.Source: §A.5 · p. 27
• The leading term matches the smooth unit-circle integrand, whose full-circle integral is zero, while the one-sided branch cut contributes to the reduction.The resulting asymptotic expression is proportional to cos(ωd + ϕa), with the factor of a canceling the 1/a prefactor.Source: §A.5 · pp. 27–28
• For large a, the integral is suppressed by O((d/a^2)^(3/2)), producing strictly smaller reductions below α1.The analysis attributes the suppression to decay within a single oscillation period.Source: §A.5 · p. 28
• For d ≤ 2000, the numerical optimum occurs at moderate a, and expanding the cutoff beyond a ≤ 20 leaves the lower envelope unchanged.The inner shift k already gives dense phase sampling for a ≤ 20.Source: §A.5 · p. 28

B.1. Exact Unit-Circle Root Counting via Sturm’s Theorem

• Reciprocal symmetry and w = z + z^-1 map unit-circle roots to real roots of Q(w) in (−2, 2), each corresponding to a conjugate pair.Irreducibility ensures no endpoint roots at w = ±2 and no repeated roots.Source: §B.1 · p. 28
• Sturm’s theorem over Q counts the real roots of Q(w) in (−2, 2) exactly, avoiding floating-point error in the unit-circle root count.The signed remainder sequence evaluates the variation difference at −2 and 2.Source: §B.1 · p. 28
• The unit-circle root fraction lies within [77.0%, 88.0%] across degree regimes, with a global mean of 82.51%.The reported statistics cover all 824 benchmark polynomials.Source: §B.1 · p. 28, Table 7 · p. 29

B.2. Off-Circle Root Structure

• The approximately 17.5% of roots off the unit circle occur in reciprocal pairs, and their radial distance contracts as O(1/d).The exterior-root logarithms sum to less than log α1, while their number grows proportionally to d.Source: §B.2 · p. 29
• Off-circle roots arise only where the specialized arc has |t| > t0, approximately 156.1°.For |t| < t0, both Y-roots lie on the unit circle; on the complementary arc they split radially into reciprocal moduli.Source: §B.2 · p. 29
• When a = 1, off-circle roots gather in one arc; when a > 1, they form a evenly spaced clusters separated by 2π/a.The cluster locations depend on s0: near θ = π for s0 = +1 and near θ = 0 for s0 = −1 when a = 1.Source: §B.2 · pp. 29–30