The Condensate Replaced by a Number: Bogoliubov's Substitution and Its Self-Priced Error Bar

In 1947 Nikolai Bogoliubov founded the microscopic theory of superfluidity on a single operation: the zero-momentum mode's operators a_0, a_0^\dagger are replaced by the c-number \sqrt{N_0}, particle-number conservation is broken by hand, the Hamiltonian becomes quadratic, and its diagonalization returns the phonon spectrum E(k) = \sqrt{\epsilon_k^2 + 2gn\epsilon_k} on which superfluidity's microscopic basis stands. The paper's own warrant for the replacement is a size argument — the commutator 1 is small compared with a_0, a_0^* themselves — and a size argument is a regime statement, not a derivation. This paper evaluates the substitution with the theory's own arithmetic on declared inputs: the same diagonalization returns a quantum depletion of 1.682088 percent of the gas outside the condensate at the mildest declared coupling, 4.757664 percent at the central inputs, and 13.456707 percent at the strongest — the substitution's self-evaluated error bar, printed by its own computation — while the c-number discards the condensate mode's number fluctuations (relative spread 0.003333 at the declared central size) and commutator (1/N_0 = 0.000111) by decree, and the discarded third- and fourth-order terms return at the relative scale \sqrt{\gamma} = 0.031623. The spectrum, the sound speed, and superfluidity's basis are the 1947 paper's derivations and stand entire. What is counted here is the derivation status of the step they stand on: 1 c-number substitution carrying the discarded terms it makes necessary, 0 derivations in 1947 of the substitution. Census headline, unchanged: NOT SOLVED / PARTIAL · producing-count 0.