# Exposure

- 2026-08-30 [Proved] The exposed faces of a design's level power obey `V(level + 1) = occ V(level) - 2 sum over the axes of P S^level`, with `occ` the tile's filled cells, `P` its adjacent filled pairs along the axis and `S` the cross positions whose two end cells are both filled, because two adjacent blocks bury one face per spanning position and the spanning positions multiply by `S` a level; so `V(level)` is a sum of the powers `occ^level` and `S^level` in every dimension, the carpet perimeter closes as `(4*8^level + 16*3^level)/5` (A381517) and the sponge surface as `2*20^level + 4*8^level` (A332705), and all four counts fold from the residue corners without rendering the tile. Witness: mrlymath::formulas::surface `prediction_matches_census_on_every_cube_code` and `the_corners_fold_what_the_tile_shows`, [sequences](../sequences.md).