Entropy and Routing Obstructions for the Boundary-Field Spin-1 Heisenberg Chain, with Exact AKLT Matrix Elements

We study the odd open spin-1 antiferromagnetic Heisenberg chain with positive endpoint fields through its reversible ground-state transform. The main results are two unconditional all-length obstructions to standard proof strategies. First, a high-magnetization projection recursion gives an explicit product bound on the fully polarized ground-state probability. For every positive field sequence \( h_N=o(N) \) this yields \( \alpha_N(h_N)\to 0 \), \( \qquad \limsup_{N\to\infty}\alpha_N(h_N)\log(N/h_N)\leq \frac{30}{7} \), for the ordinary logarithmic Sobolev constant of the ground-state process, with logarithmic tail coefficient at least (7/6). This does not imply collapse of either the Hamiltonian gap or the modified logarithmic Sobolev constant. Second, for the Diaconis–Stroock/Sinclair length-weighted congestion functional used in this paper, every routing with unit edge lengths has congestion at least linear in N; the same conclusion holds for uniformly comparable edge lengths, while routings forced through a polarized configuration obey a stronger rare-state obstruction. These are limitations of that comparison certificate, not upper bounds on the physical gap.As a complementary exact calculation, transfer matrices for the open AKLT ground space give both the singlet/triplet compression of \( W_m=\sum_{i=1}^{m-1}P_{1,i} \) and its coupling to the orthogonal complement. The compressed spectral diameter is \( O(m3^{-m}) \), whereas \( ||Q_mW_m\mathsf P_m||\sim\sqrt{2(m-1)}/3 \). This finite-block statement quantifies why the bare compressed splitting alone does not control the perturbation at the Heisenberg point; it is not a replacement for a locality-sensitive AKLT stability theorem. Exact rational inertia and interval arithmetic then provide supporting finite-volume certificates: full gaps and the complete AKLT–Heisenberg interpolation are certified for \( N=3,5 \), and at \( N=5,h=1 \) one obtains both a negative pointwise Bakry–Émery curvature witness and [ 27/25<\Delta_5(1)<567/500. \( 27/25<\Delta_5(1)<567/500 \)These finite-volume statements have no asserted local-to-global consequence. The fixed-field uniform boundary-gap problem remains open.