Parameterized Scanning Transforms and Families of Graph Invariants A Full-Acquisition and Projection Theory for Zero–One Scanning Tables

This paper develops a theorem-safe full-acquisition layer for the zero–onescanning tables introduced in the Wang framework. The earlier row-majorflattening of a scanning table is replaced first by an arbitrary deterministic cellorder and then by a still more general finite mechanism: choose a covariantvertex-label table, acquire numerical data from all required vertex pairs withany fixed amount of redundancy, impose any deterministic schedule on theacquired coordinates, and finally project the resulting numerical table to ascalar, vector, matrix, nested table, or operator packet. We prove that everysuch fixed rule gives graph-isomorphism invariants, that full acquisition israw-information-equivalent to the minimal complete pair table regardlessof redundancy, and that line sweeps and analytic curve scans are specialparameterized schedules with finite critical sets under explicit hypotheses.A central point is a boundary theorem. If a scan traverses every coordinateof a fixed raw table exactly once and the whole resulting word is retained,then changing the scan order cannot add information: every scan word isobtained from the row-major word by a fixed permutation. Likewise, ifan acquisition rule records every prescribed pair at least once, then extraduplicated coordinates are recoverable from, and determine no more than, theminimal complete pair table. New separating power can therefore arise onlyafter a non-invertible projection or coordinate-forgetting step, such as windowstatistics, autocorrelations, threshold geometry, scalar summaries, operatorcompression, or quotient packets. The paper formulates this distinctionprecisely and connects it to the ordered-pattern, meta-matrix, quotient-computation, Babai-interface, and finite-dimensional operator-algebra papers of the series