The Universal Hopf Envelope of Framed Digital Images: Ordered-Partition Antipodes and Coordinate Quotients

Framed digital images admit horizontal juxtaposition and a coproduct that splits active pixels while retaining the allocated frame. The resulting bialgebra has no antipode because its blank column is group-like and noninvertible. Identifying images with column words specializes Takeuchi’s free-Hopf construction: adjoining the inverse of the blank column gives the Hopf envelope. Right-normalized nonblank columns form a connected graded Hopf factor. For this model we give a cancellation-free ordered-partition antipode and an exact Hilbert series, including the scalar and one-generator exceptions to exponential growth. We characterize the commuting coordinate quotient Hfr=Sym(V)#K[Z] by column factorization and translated-pixel commutativity, classify its group-like and fixed-frame skew-primitive elements, and establish the lifting criterion for translation-equivariant color maps. Global character weights must be constant along horizontal translation orbits. On finite patches, a more general local weighted Boolean zeta transform admits arbitrary weights, Kronecker factorization, exact conditioning formulas, and O(n2n) inversion. Two exact-arithmetic programs perform 10,150 regression checks. Stored tests on nine public images include 61,440 binary patches, 552,960 pixels, 360 stability trials, and RGB equivariance controls. They validate local inversion and the translation criterion, with zero reconstructed-pixel errors; they do not establish restoration performance.