Local freedom and capacity barriers in quadratic trace reconstruction

A completely multiplicative sign function is determined by its values on \(n^2+1\), \(n^2+2\), and \(n^2-2\), but exact determination does not quantify how many readings detect a negative prime sign. We distinguish three losses of information: partial compatibility tests, geometric restrictions on certificates, and fractional relaxation of signed constraints. After fixing a prime head, we describe the exact fibre of a window and the parity constraints joining two windows. For any fixed head and any sequence \(H_N=Nᵒ⁽¹⁾\), one table with \(o(N)\) negative entries passes every \(H_N\)-window test and every two-reading test. A separate finite table passes these tests without any global source. We bound certificate capacity—fractional packing under unit reading loads—for narrow and binary certificates and for low-degree transport libraries whose links cancel odd prime valuations in pairs. A fixed algebraic curve of certificates for a target prime \(p\) has at most logarithmic capacity, or capacity at most one when a reading is constant; in the infinite unanchored case its common poles have field \(Q(√ p)\). Conversely, selected values of \(n^2+1\) can force linearly many errors while the capacity of all certificates stays bounded. Two internally disjoint pair families compose without loss on their existing readings, whereas sixteen actual values exhibit \(Pack=3<Cap=4<μ=5\). The selected-sample constructions do not give an upper bound for the minimum on a complete prefix.