The same occurrence at every use
Abstract
Physical binary subset arithmetic
The increasing labels decoder is permutation equivalent to the accepted chainMask decoder at every bounded mask. It therefore reuses chainMask_sum for exact integer weighted masses instead of re-proving the binary expansion. Its pointwise positivity follows from the original ladder values. Nested masks split by literal bitwise xor into disjoint physical occurrences. These helpers concern physical labels even when distinct labels carry the same numeric source value. They are internal ingredients of the chamber constructor and supply no positive fixed-instance endpoint.