Few new reals

We introduce a new method for building models of CH, together with $\Pi_2$ statements over $H(\omega_2)$, by forcing. Unlike other constructions in the literature, our construction adds new reals, although only $\aleph_1$-many of them. Using this approach, we prove that a very strong form of the negation of Club Guessing at $\omega_1$ known as Measuring is consistent together with CH, thereby answering a well-known question of Moore. The construction works over any model of ZFC + CH with a regular cardinal $\kappa>\omega_2$ such that $2^{{<}\kappa}=\kappa$ and such that there is a partial square sequence on $\kappa\cap \text{cf}(\omega_1)$. It can be described as a finite-support forcing construction of length $\kappa$ with side conditions consisting of suitable symmetric systems of models with markers. The CH-preservation is accomplished through the imposition of copying constraints on the information carried by the condition, as dictated by selected pairs of models with markers coming from the side condition. A minor variation of the main construction produces a forcing notion giving rise to a model of Measuring together with $2^{\aleph_0}$ being arbitrarily large. This answers another question of Moore.