Corpus record: PFT:ALLEN_HEX_GEODESIC_LAW_ON_THE_ALLEN_ORBITAL_LATTICE_AOL
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Allen Hex–Geodesic Law on the Allen Orbital Lattice (AOL)
November 2025
We state and justify a one–page rule connecting shortest paths on the Allen Orbital Lattice (AOL) with the lattice’s concentric hex-rings. Empirically, the total lattice energy/power accumulated on ring \(r\) follows a unimodal profile that peaks at a small \(r=r_\star\) and decays monotonically thereafter. The Allen Hex–Geodesic Law ties this spectral fact to geometry: on a hex lattice, the geodesic between two sites is realized by the axis-aligned straight path that crosses the fewest ring boundaries and maximizes overlap with rings at or below \(r_\star\).
Basel constant \(\pi^{2}/6\) on
the central hexagon (AOL).
PatternFieldTheory.com
Law 1 (Allen Hex–Geodesic Law (A, B, C)). Let the AOL be indexed by concentric hex-rings \(r=0,1,2,\dots\) around a source. Suppose the ring power \(P(r)\coloneqq\sum_{i\in \text{ring }r} C_i\) (with node cost/weight \(C_i\ge 0\)) satisfies a single peak at \(r=r_\star\) and is nonincreasing for \(r\ge r_\star\).
(Ring monotonicity \(\Rightarrow\) straightness) Among all lattice paths between two sites with fixed axial displacement, the axis-aligned straight path that crosses each ring at most once minimizes total cost and crossings; any detour increases cumulative \(r\) and hence accumulates larger \(\sum P(r)\).
(Hex Manhattan distance) The geodesic length equals the hex distance \(d_{\hexagon}\), i.e. the minimal number of ring crossings; thus the discrete geodesic on the AOL coincides with the standard axial straight path.
(Spectral–geometric equivalence) If \(P(r)\) is unimodal with a unique maximum, the energy-minimizing path and the shortest lattice path coincide. Therefore, on the AOL, the shortest distance between two points is the axis-aligned straight path.
Sketch of justification. On a hex lattice in axial coordinates, any path can be decomposed into signed steps along the three lattice axes. A detour that revisits a ring necessarily adds a nonnegative increment to the multiset of traversed ring indices. With \(P(r)\) nonincreasing for \(r\ge r_\star\), majorization implies the straight axial path—which visits the smallest possible set of ring indices in nondecreasing order—minimizes the path integral \(\sum_{r\ \text{crossed}} P(r)\). When node weights are uniform, this reduces to minimizing ring crossings, i.e. hex distance.
Remark 1 (Practical use). For planning, routing, or energy funnels on the AOL: (i) choose coordinates in the axial basis; (ii) move along the two axes that connect source to target with signs matching the displacement; (iii) avoid ring revisits. If a nonmonotone \(P(r)\) is observed (e.g. near a forced resonance), pre-warp by reweighting \(C_i\) so that the effective \(P(r)\) regains unimodality.
Note.
If you have the figure file, replace the placeholder box with
\includegraphics[width=0.86\linewidth]{AoL_ring_power_plot.png}.
The logo file path can be updated in \maketitle
(pi_hex_logo.png).