Arithmetic reconstruction and error recovery from three quadratic traces

We study completely multiplicative sign functions \(h\) through their values at positive integers of the forms \(n^2+1\), \(n^2+2\) and \(n^2-2\), with observed indices \(1\le n\le N\). First occurrences of prime factors give a canonical information set and a complete, prefix-compatible family of parity relations, each involving at most \(\lfloor\log_2N\rfloor+3\) readings. These relations yield exact bounded-distance repair, including corrupted information coordinates, in time \(N^{O(1)}(4+\lfloor\log_2N\rfloor)^t\) for \(t\) errors; erasures enter through an explicit rank parameter. We then construct arithmetic redundancy without estimating a quadratic regulator. If \(h(p)=-1\) and \(K=\log N/\log\max(3,p)\), the three traces contain at least \(\exp(\exp(c\log K/\log\log K))\) negative readings for every fixed \(0<c<\log2\) and sufficiently large \(K\), uniformly in the prime \(p\) and all other prime signs. The same number of each sign occurs on one trace between \(N^\alpha\) and \(N\), for each fixed \(\alpha<1\). The proof combines short reconstruction certificates, Lehmer divisibility and balanced divisors. Coprime blocks give compact access to these families without factoring large recurrence terms. Under independent reading noise with fixed erasure probability \(0\le a<1\) and conditional flip probability \(0\le b<1/2\), every source prefix \(P=\lfloor\exp((\log N)^\delta)\rfloor\), for fixed \(0<\delta<1\), is recoverable from \(O(P)\) readings with polynomially small failure probability in \(P\) and preparation polynomial in the numerical \(P\) and \(\log N\). A fixed positive erasure rate nevertheless leaves exponentially many assignments of observed large-prime signs. No positive-density inverse for unrestricted sources is asserted.