Finite-Size Structure of Spanning Trees on Free-Boundary Honeycomb Domains: Bulk, Boundary and Corner Contributions

Abstract

Finite-Size Structure of Spanning Trees on Free-Boundary Honeycomb Domains Bulk, Boundary and Corner Contributions Julio Cesar Santana Valderrama Preliminary Research Report - Version 1.0 - September 2026 Status. Preliminary and reproducible computational study. Exact finite-size calculations and symmetry checks are reported as computational results. The proposed angular dependence of the subleading logarithmic term is explicitly treated as an empirical observation and an open analytical problem. Research question How does log(tau) for finite free-boundary honeycomb domains decompose into bulk, boundary and corner contributions, and which parts are determined by lattice structure versus boundary geometry? HexaLegendre - Version 1.0 - September 2026 Page 1Abstract This report studies finite free-boundary patches of the honeycomb lattice through the number tau(H) of spanning trees. The primary observable is log(tau(H)), evaluated from the graph Laplacian using the Matrix-Tree theorem and an independent spectral formulation. Small systems were cross-validated with exact integer determinants. The leading extensive coefficient is consistent with a honeycomb bulk entropy h about 0.807735, while a local-difference extrapolation gives a geometry-stable boundary coefficient a about -0.662 across hexagonal, rhombic and triangular families. The hexagonal family also exhibits the expected D6 symmetry. Numerical irreducible-representation projectors reproduce the spectrum with reconstruction error of order 10^-14, with paired multiplicities in the two-dimensional irreducible sectors. A subleading logarithmic coefficient b remains the main open problem. After extending the scaling range to reduce conditioning problems, three polygonal families suggested an approximately linear relation b about 0.133 S + 0.562. This is not claimed to be universal: the coefficients have not yet been derived analytically and three geometries are insufficient for such a claim. The next stage is to derive the corner contribution directly for the honeycomb discrete Laplacian and then test a frozen prediction on new polygonal boundaries without refitting. 1. Mathematical construction Let H_n be a hexagonal patch of the honeycomb lattice with axial-coordinate radius n. With L = n + 1, the hexagonal family has: N_n = 6 L^2 ; B_n = 6 L For a finite connected graph H with Laplacian L_H, the Matrix-Tree theorem gives: tau(H) = det(L') where L' is any cofactor obtained by deleting one row and the corresponding column. For larger systems the same quantity is evaluated from the nonzero Laplacian eigenvalues: log(tau) = -log(N) + sum over lambda_i != 0 of log(lambda_i) 2. Computational verification Two independent routes were used: exact integer determinant evaluation for small systems and spectral evaluation for larger systems. Agreement between the two routes was better than 10^-12 in the reported validation cases. The spectral implementation was approximately 200 times faster in the benchmark reported during development, enabling a substantially wider scaling window. The extended dataset included triangular domains to n = 76, rhombic domains to n = 55 and hexagonal domains to approximately n = 30. 3. Finite-size decomposition The working asymptotic form is: log(tau(H_L)) = h N + a B + b log(L) + c + lower-order terms Quantity Current result Status Bulk coefficient h about 0.807735 Robust computational result Boundary coefficient a about -0.662 Consistent across 3 geometries Hexagonal D6 symmetry Numerically verified Concrete symmetry mechanism Log coefficient b Geometry dependent Stable estimates; theory open Angular relation b about 0.133 S + 0.562 Preliminary; not universal HexaLegendre - Version 1.0 - September 2026 Page 24. Boundary coefficient Direct multiparameter fits of h, a, b, c and inverse powers of L were initially ill-conditioned in narrow large-L windows. Condition numbers exceeded 2 x 10^6 in some windows and b drifted strongly. Those unstable estimates were rejected. A more stable diagnostic is the local difference: a_local(L) = [Delta log(tau) - h Delta N] / Delta B This reduces collinearity between N and B. The three polygonal families gave limiting values near -0.662: hexagon about -0.6639, rhombus about -0.6623 and triangle about -0.6619, a spread of about 0.002. Interpretation: within the tested families, the leading boundary correction appears more strongly associated with the honeycomb lattice/free-boundary construction than with gross polygonal shape. This is a numerical observation, not yet a theorem. 5. D6 symmetry For n = 4 the hexagonal patch has N = 150 vertices. Numerical projection onto irreducible representations produced dimensions A1 = 15, A2 = 10, B1 = 10, B2 = 15, E1 = 25 and E2 = 25, summing to 150. The two-dimensional E1 and E2 sectors account for 100 dimensions and enforce paired eigenvalue multiplicities. Projectors were idempotent to approximately 10^-17 and spectral reconstruction errors were of order 10^-14. 6. Corner/logarithmic term After the scaling range was increased, b estimates became substantially more stable under changes of fitting window and passed the out-of-sample checks used in the study. For the three tested families an angular descriptor S was assigned from polygonal corner structure: hexagon S = -5, rhombus S = -7, triangle S = -8. b about 0.133 S + 0.562 ; R^2 about 0.991 This relation is frozen as a hypothesis rather than promoted to a universal law. With only three geometries, the decisive test is a new polygonal boundary with a previously unseen combination of corner angles, using the prediction without refitting the slope or intercept. 7. Theoretical context Spanning trees are an exactly solvable statistical model. Prior work on finite square lattices derives asymptotic expansions of spanning-tree partition functions and identifies corner contributions consistent with conformal-field-theory expectations. More recent mathematical work connects uniform spanning trees in topological polygons with SLE(8) and logarithmic conformal field theory with central charge c = -2. These results motivate an analytical route, but they do not directly supply the numerical constants 0.133 or 0.562 for a free-boundary honeycomb lattice. The lattice discretization, boundary condition and Matrix-Tree zero-mode normalization must be derived explicitly before a theoretical coefficient is claimed. 8. Controls and falsification principles • Exact determinant calculations validate the spectral engine on small systems. • Multiple polygonal families test the stability of the boundary coefficient. • D6 representation analysis distinguishes symmetry-enforced degeneracy from accidental degeneracy. • Ill-conditioned fits are rejected rather than interpreted. • The angular b relation is not treated as universal until it survives new geometries without refitting. • Earlier arithmetic observations are not imported unless an explicit mathematical identity is HexaLegendre - Version 1.0 - September 2026 Page 3established. HexaLegendre - Version 1.0 - September 2026 Page 49. What this report establishes Established within this computational study: • Precisely defined honeycomb graph families and spanning-tree observable. • Exact small-system Matrix-Tree calculations cross-validated against the spectral method. • Bulk coefficient consistently near h = 0.807735. • Boundary coefficient consistently near a = -0.662 across hexagon, rhombus and triangle families. • Concrete numerical D6 representation structure explaining hexagonal spectral pairing. Not established: • A universal formula for b. • An analytical derivation of 0.133 or 0.562. • A new theorem about spanning-tree universality. • A proof of Legendre's conjecture or a statement about Riemann-zeta zeros. • A new physical law or experimental prediction. 10. Next experiment Generate additional free-boundary honeycomb polygons with new corner-angle combinations. Freeze the currently observed b relation and compare new measured values against it without refitting. In parallel, derive the corner contribution from the determinant of the discrete honeycomb Laplacian, including the zero-mode normalization required by the Matrix-Tree theorem. 11. Reproducibility release A complete reproducibility package should accompany a later version with graph-generation code, raw tau/log(tau) tables, fit windows, numerical tolerances, exact-versus-spectral validation tables and the scripts used for symmetry projections and out-of-sample tests. Recommended publication status: public preliminary research report / preprint, not a peer-reviewed article. References 1. Izmailian, N. Sh. & Kenna, R. (2015). Exact finite-size corrections for the spanning-tree model under different boundary conditions. Physical Review E 91, 022129. DOI: 10.1103/PhysRevE.91.022129. 2. Liu, M., Peltola, E. & Wu, H. (2025). Uniform Spanning Tree in Topological Polygons, Partition Functions for SLE(8), and Correlations in c = -2 Logarithmic CFT. Annals of Probability 53(1), 23-78. DOI: 10.1214/24-AOP1700. 3. Shrock, R. & Wu, F. Y. (2000). Spanning Trees on Graphs and Lattices in d Dimensions. arXiv:cond-mat/0004341. 4. The numerical results and interpretations in this report are the author's computational study and are distinguished from the cited literature. Julio Cesar Santana Valderrama Version 1.0 - September 2026 Preliminary research report - not peer reviewed HexaLegendre - Version 1.0 - September 2026 P