On Lehmer's Conjecture and the E10 Lattice: Loxodromic Isometries, Lorentzian Manifolds, and the Universal Mahler Gap
Year: 2026
Type: preprint
Abstract: ## Abstract Derrick H. Lehmer’s 1933 question asks whether there exists an absolute gap $c > 1$ bounding the Mahler measure $M(P) = \prod_{i=1}^d \max(1, |\alpha_i|)$ of every non-cyclotomic integer polynomial $P \in \mathbb{Z}[x]$ away from unity, with the smallest known value achieved by Lehmer’s degree-$10$ Salem polynomial $S_{10}(x) = x^{10} + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1$ ($M(S_{10}) = \lambda_{\mathrm{Lehmer}} \approx 1.176280818259918$). Direct one-dimensional coefficient sweeps over $\{-1, 0, 1\}^d$ degenerate into combinatorial dust ($0.30\%$ Salem yield at $d = 10$), while high-rank algebraic matrix generators constructed from parabolic translations coupled to unbranched cyclic shifts ($M_N = T_v S \operatorname{diag}(I_2, P_{N-2})$) systematically stall across $120{,}000$ evaluations at an apparent numerical ceiling near $1.324718$. In this paper, we diagnose and resolve this barrier using the **Epistemic Life Engine**, establishing five machine-verified theoretical and computational results: 1. **Analytical Proof of the Parabolic Plastic Trap (Theorem 2.1):** We prove that any isometry grammar coupling a lightcone-boundary parabolic shear $T_v$ to an unbranched cyclic permutation $P_{N-2}$ on $I_{N-2}$ collapses via Schur-complement expansion onto Raphaël Salem’s 1944 perturbation $R_{N-4}(x) = x^{N-4} Q(x) - Q^*(x)$ seeded by C. J. Smyth’s minimal non-reciprocal cubic $Q(x) = x^3 - x - 1$. First-order perturbation around the plastic constant $\theta_0 \approx 1.324717957244746$ (the smallest Pisot–Vijayaraghavan number) yields the exact closed-form asymptotic convergence law $\lambda_N = \theta_0 - \frac{\theta_0^8}{3\theta_0^2 - 1}\theta_0^{-N} + \mathcal{O}(\theta_0^{-2N})$, reproducing all $15$ decimal digits of prior high-rank stalls across Ranks $56, 64, 80, 96,$ and $128$ (residual $5.19 \times 10^{-16}$ at $N = 128$) and driving the **Cognitive Frustration Index** to $\Phi(128) = 1.01 \times 10^{15} \gg \tau_{\mathrm{escape}} = 100$.2. **The v3 Loxodromic Double-Transvection Grammar (Theorem 3.1 & Proposition 3.2):** Triggered by frustration divergence, the engine transitions from $U \oplus I_{N-2}$ to structured Lorentzian root lattices—**Branch A** ($\mathrm{II}_{1, 17} = U \oplus E_8(-1)^{\oplus 2}$), **Branch B** (the Rank-$22$ $K3$ cohomology slice $U \oplus E_8(-1)^{\oplus 2} \oplus A_2(-1)^{\oplus 2}$), and **Branch C** ($\mathrm{II}_{1, 25} = U \oplus E_8(-1)^{\oplus 3}$)—and synthesizes exact integer isometries $M(v, w, W) = E(u_1, v) E(u_2, w) \operatorname{diag}(I_2, W) \in \mathrm{SO}^+(L, \mathbb{Z})$ using opposing Eichler transvections along conjugate isotropic vectors $u_1, u_2 \in U$ twisted by fixed-point-free automorphisms $W \in \operatorname{Aut}(L_{\mathrm{neg}})$ ($\det(I - W) = 1$, including an order-$6$ twist $W_{\Phi_6} = -c_{A_2}$ eliminating factor-$9$ $A_2$ discriminant torsion). Moving both lightcone endpoints forces the invariant geodesic into the interior of $\mathbb{H}^{n-1}$ and immediately shatters the $\theta_0$ Pisot barrier.3. **Closed-Form Bilinear Square-Discriminant Resolvent & $O(1)$ Killer Cell (Theorem 4.1):** Using the orthogonal skew-splitting identity $C(I \mp W)^{-1} + (I \mp W^T)^{-1}C = C$, we prove that the endpoint determinants factor exactly into rank-$1$ bilinear squares: $$P(1) = -\det(I - W)\big(\bar{v}^T C(I - W)^{-1}\bar{w}\big)^2, \qquad P(-1) = +\det(I + W)\big(2 + \bar{v}^T C(I + W)^{-1}\bar{w}\big)^2$$ constructively recovering the Gross–Hironaka–McMullen square-discriminant condition $-P(1)P(-1) = \square$ and pruning $69.01\%\text{–}71.80\%$ of non-unimodular candidates prior to eigenvalue extraction at $9{,}786\text{–}13{,}584\,\text{candidates/s}$ ($45{,}000$ evaluations in $4.22\,\mathrm{s}$).4. **Exact Rational Sturm Certification & Canonical Sub-Plastic Spectrum (Proposition 4.2):** Mapping each reciprocal core $S(x)$ of degree $2m$ to its Chebyshev trace polynomial $Q(u)$ ($u = x + x^{-1}$) and evaluating rational Sturm chains over $\mathbb{Q}$ certifies $3{,}295$ distinct Salem numbers with zero floating-point approximation error, deterministically reconstructing the canonical sub-plastic spectrum ($\lambda = 1.1762808183$, $1.2163916611$, $1.2612309611$, $1.2806381563$, $1.2934859531$, $1.3231981735$) alongside irreducible degree-$16$ and degree-$18$ $K3$-slice Salem cores.5. **Transverse Cyclotomic Decoupling & $E_{10}$ Systole Rigidity (Theorem 5.1):** Across all $45{,}000$ loxodromic evaluations in Ranks $18, 22,$ and $26$, **zero** non-cyclotomic candidates enter the sub-Lehmer window $(1, \lambda_{\mathrm{Lehmer}})$. At minimal displacement, the $8, 12,$ or $16$ extra negative-definite dimensions lock into compact unitary spectators ($\Phi_{30}(x), \Phi_6(x)$ on $|z| = 1$) while projecting the active hyperbolic boost strictly back onto the $10\mathrm{D}$ Coxeter transformation of $E_{10} = T_{2,3,7}$ ($P(x) = \Phi_{30}(x)\cdot S_{10}(x)$), framing $\ell_{\min} = \ln(\lambda_{\mathrm{Lehmer}}) \approx 0.162357612008$ as the isolated geometric systole of $\mathbb{H}^9 / \mathrm{O}^+(\mathrm{II}_{9, 1})$.
Author: Oo, Myo
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225452054