Arbitrary-order suppression of source mismatch in photonic entanglement swapping

We establish the minimum number of same-source photons needed for arbitrary-order suppression of source mismatch in photonic entanglement swapping and construct a static passive circuit family that attains it. The result applies to unencoded photons, optics that act identically on all internal states, complete photon-number-resolving detection, and local-unitary recovery of the retained qubits. For two pure source wave packets with squared overlap $u$, suppressing the bare mismatch error $\epsilon=(1-u)/2$ to order $\epsilon^r$ at nonzero ideal success probability requires at least $r$ photons from each source. A cyclic interferometer attains the minimum total of $2r$ photons at every order, with ideal acceptance $2^{2-3r}$. An exact transfer expression extends the calculation to distinct wave packets and correlated classical source noise. Near a common wave packet, the leading error is proportional to the summed delayed Hong--Ou--Mandel losses along a closed tour through each source's emissions. For equally spaced emissions, losses that increase concavely with delay select chronological wiring, whereas convex growth selects a folded order; independent fluctuations add the same leading penalty to every wiring. In a Gaussian drift model, folding lowers the smallest reachable error in the compared catalogs by about $23\%$ without adding photons. Including detection, storage, and routing losses, the photon-cost advantage depends on the target and is largest in a narrow target window; at target error $4.35\times10^{-4}$, the pooled cost is lower by a factor of about $5.6$ with equal acceptance and randomization options in both catalogs. A common spectral filter is cheaper in this finite-noise example; a separate pure-source analysis gives a regime with a better cyclic cost exponent.