The Home Pairing in the Two-Boundary Shape: Completeness Cancels the Chosen Part, Not the Counted Part

Closes debt D-041-080-01, the only door left by house paper #41 (Two Boundaries in Time), whose section 6 asked for the home pairing to be stated in the two-boundary formalism's operator shape -- an operator built from an overlap and a pair of projectors -- on the vacuum ledger's own variables, and for a check of whether completeness there forces the same total cancellation the closed ledger forced in the formalism. The home pairing is stated in full: the ledger's N entries span a finite space, the pre-selected state is the counting state (each entry counted once, phases free and irrelevant), the entry signs form a diagonal charge observable Q whose trace is the unpaired excess sigma*U, each opposite-sign pair and each unpaired entry is a projector block (the blocks are complete by partition, so ABL probabilities over them sum to exactly one), and the pairing is the involution the blocks define. Proved and machine-checked: (1) pairing-symmetrising the charge, (Q + P Q P)/2, returns exactly the unpaired part of Q -- the paired part cancels entry by entry; (2) the completeness identity at home: summed over any complete set of future boundaries weighted by their post-selection probabilities, weak values of Q return the forward value sigma*U/N exactly (max error 3.13e-16 over random unitary completions), while the anomalous parts -- individually as large as 19.5 against a forward value of 1/3 -- cancel to zero; house paper #41's own worked case (delta = 0.1: weak value 19, partner -1/19, weighted sum 0) is re-run as a control and reproduces. (3) The corollary answering the door: completeness at home forces total cancellation if and only if U = 0. The produced fact is a placement: the home remainder r = U/M is a function of the threshold counts alone, defined before any future boundary is chosen, so in the operator's shape it sits in the forward part of the ledger, which completeness preserves by identity -- not in the anomalous part, which is the only part completeness cancels. The difference between the two ledgers is that the formalism's remainders are chosen and the home remainder is counted. #41's RHYME verdict on the translation stands untouched and its toll remains unpaid; the counts are spent as declared and never re-derived, every theorem being proved for general (N, M, U); the vacuum catastrophe remains NOT SOLVED. Graded DERIVED-INDEPENDENT: the completeness identity is literature, but the home operator in this shape, the iff-U=0 corollary, and the counted-vs-chosen placement theorem answer an either/or question the owing paper left as its only door, in the branch that paper itself priced as the first produced fact about what the home pairing is.