Harmonic Structures in Posets and Digit Polygons: Exact Valuations, Fluctuations, and Reconstruction

A periodic digit expansion draws a polygonal walk whose overlap and empty regions retain arithmetic information. Harmonic Structures in Posets and Digit Polygons develops an analytical account of these connections, bringing order constraints and geometric multiplicity into a common harmonic framework while keeping their probability laws and metric weights explicit. For finite posets, an ideal-frontier representation yields harmonic identities, sharp width bounds, and exact record covariances. At fixed width the variance grows at most logarithmically, while a family of ordinal sums has independent increments, an explicit dilogarithm variance constant, and a normal limit. For finite measured covers, overlap depth becomes a positive moment problem on reciprocal integers. Interpolation reconstructs every layer, and linear-programming duality determines sharp minimax errors for recovery from bounded-noise measurements. Applied to digit polygons, the theory identifies winding multiplicity with interval-order width. An exact rational algorithm evaluates all depth tails directly from the original edges, with a quadratic operation count apart from sorting, while integer coordinates provide explicit margins for noisy depth certification. Closed-section incidence preserves the information needed to detect connectivity and holes. Protected barriers produce period-79 orbits with at least two holes in infinitely many bases of both quadratic families, and a safe periodic closing theorem gives conditions for further constructions. The treatment also retains the boundary harmonic laws and prefix geometry of the predecessor, with explanatory figures and reproducible checks accompanying the proofs. Order-polytope formulas and path likelihoods identify the change of measure leading to uniform linear extensions. Uniform frontier choice is exact precisely for ordinal sums of antichains, and comparison certificates transfer under sharp density conditions. Optimal fixed-width variance constants, unbounded arithmetic hole counts, linear hole bounds, sharper arithmetic recovery, and general uniform-extension balance applications remain open. The finite edge evaluation does not imply bounded closed harmonic formulas in the base.