Compact Exact Representation on the Integer Lattice: Characters, Divisors, and Cut-Cycle Coordinates for Grid and Mesh Problems

A large exact problem can often be replaced by a small exact object fixed by a boundary: the state that every admitted continuation reads, together with a separate lift that recovers source-coordinate witnesses. This paper develops that compact exact representation on the integer lattice, where the decisions are made by characters, divisors, unimodular cut-cycle coordinates, and prime-coded keys rather than by floating tolerances. For rational symmetric operators the Schur boundary algebra gives continuation-complete states with exact inertia and resolvent lift; the Schur matrix and the min-plus marginal shell are minimal for their grammars, and chordal, signed-weight, representation-size, and interval-admission results fix where the construction stays exact. For direct- and alternating-current networks, cut-cycle coordinates from a unimodular tree sweep give exact energy, dual, and port identities; decimal impedance data are unramified outside {2, 5}; sets of binding limits form the divisor lattice of a squarefree key; the knot cell of the cycle Gram yields a face conductor that divides the largest Smith invariant factor of the coupled face system; atlas certificates cover demand regions and every single-line outage of declared mesh families; and a Gaussian-integer presentation carries the alternating-current branch relations. For periodic slabs, characters of the translation group block-diagonalize the operator, mirror orbits weight the blocks, ramified characters keep a little group inside the condensed block, and the divisor tower answers every width dividing the period from one set of blocks; the implicit midpoint rule is a Möbius warp of frequency, contact reduces exactly to a recursion on the ports, and coupled thermoelastic characters extend the reduction to a nonsymmetric reciprocal pencil. Further parts treat min-plus shells for one and two currents, polynomial Chinese-remainder sector algebras, the Redwood direct limit with its Fock fibre, and the conductor-foliated frustration boundary. Representative measurements on named public cases are recorded apart from the mathematics. On a 1,000,000-bus DC mesh, 115 MW was certified firm across the full demand box and every one of 1,998,000 single-line outages by a certificate of at most 794 bytes, answered from 25.27 MiB in 0.0202 ms per query against 2,818.25 MiB and 605.607 ms for a warm HiGHS solver. On a 1,000,000-bus AC mesh, 27.529101 MW was guaranteed for every single-line outage against a proved upper bound of 28.000000 MW, by a 439-byte certificate. A periodic lattice-material slab of 1,064,448 quadratic tetrahedra was solved in 84.9 s and 1.7 GiB against 2,872.1 s and 53.4 GiB for MUMPS with METIS, and the same slab at 1,089,994,752 elements ran in 1.8 GiB on one desktop computer. A nonlinear finite-strain slab with reciprocal heat and two impacts ran 7.78 to 9.29 times faster than an independent MOOSE application on six paired grids, with every endpoint passing, and the same case at 1,019,464 elements completed in 13.057 hours and 5.385 GiB on one processor; MOOSE at that size is projected, not measured, and no speed ratio is claimed there. Every numbered result carries a grade (derived, replayed on manufactured or named public cases, open, or falsified). Exact counterexamples, retained as kills, mark where each construction stops; losses are printed beside the wins; open problems are stated precisely. The verification script for the mathematics is available from the author on reasonable request rather than posted with this record. Every definition, identity and worked value it checks is printed in full in the paper, so each check can be reimplemented independently from those statements alone. Patent reservation. The mathematics is disclosed here under CC BY 4.0; patent and trademark rights are not licensed. The engineering, algorithms, and apparatus that practice it are reserved under U.S. Provisional Patent Application 64/190,840 (Holonomic Boundary Compiler, filed November 3, 2026) and are not disclosed in this paper.