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math.NT — Number Theory

Arithmetic coding against unknown small-prime signs

Contributed by Brice Pouly

We encode binary messages in completely multiplicative sign functions and read them along the three sequences $n^2+1$ , $n^2+2$ and $n^2-2$ . Two observations with the same exact cofactor remove the unknown smaller-prime signs when multiplied. Allocating shared large primes before pairing embeds ordinary binary error-correcting codes in each trace. Under an explicit large-factor mass hypothesis, a computable schedule gives one infinite encoding with positive prefix rate and a linear errors-and-erasures allowance. A first-occurrence filter makes its raw readings depend causally on a linear message prefix, while encoding and decoding remain polynomial in the numerical observation length. Its supporting primes have arbitrarily small harmonic cost. A finite construction, independent of the analytic hypothesis, stores $273$ bits in $8736$ readings of each trace and corrects every pattern of $499$ reading errors. Complementary constructions give persistent sign prescriptions, uniform laws for the active coordinates of words, and compact multiplicative subgroups whose nontrivial quotients have positive quadratic entropy. The word-law and entropy conclusions are not asserted for the polynomially decoded encoding. A separate matrix construction gives another causal positive-rate map. Fixed-scale balancing provides an unconditional quarter-sign bound against every small-prime assignment. All coding statements concern deliberately constructed families, not recovery of an arbitrary multiplicative source. The sparse constructions at superlinear cutoffs keep their large-factor mass input explicit.

Primarily AI-generated textHuman understanding: no partsarithmetic codingcausal encodingerror-correcting codeserrors and erasuresmultiplicative functionsquadratic tracesunknown prime signs

Oct 20, 2026PDF ↗View work →

hexagon:2610.00356v1

math.NT — Number Theory

Correlation laws and identifiability of multiplicative functions on quadratic traces

Contributed by Brice Pouly

We study which completely multiplicative sign functions can be identified from the statistics of $h(n^2+1)$ , $h(n^2+2)$ and $h(n^2-2)$ . In a class of finite products of quadratic characters of odd prime conductor, completed without zeros and modified at primes with summable inverse square roots, diagonal correlations of orders two and four determine the function. The second-order data recover the conductor and prime corrections, leaving exactly the ramified local signs undetermined. Fourth-order detectors recover those signs, including at the small primes; the inverse also extends to quadratic components at two. No bounded horizon suffices uniformly: for every prescribed window, two finite-conductor sources can share its complete law and disagree with density arbitrarily close to one half on each trace. For an arbitrary prescribed source $h$ , we construct factors $fg=h$ such that, for each factor separately, the three traces jointly have the uniform word law at every horizon. The choice is continuous and computable relative to $h$ , with a finite bound uniform in $h$ for approximation of a canonical synchronous coupling. The same choice treats every horizon sequence $o(\log\log N)$ . The factors need not be independent. Their synchronous empirical law projects exactly to the product's empirical word law; when a limiting canonical coupling exists, it retains the product's word law, which need not identify $h$ as a function.

Primarily AI-generated textHuman understanding: no partscorrelation lawsfinite observation horizonsidentifiabilitymultiplicative functionsnormalityquadratic charactersquadratic traces

Oct 20, 2026PDF ↗View work →

hexagon:2610.00355v1

math.NT — Number Theory

Local freedom and capacity barriers in quadratic trace reconstruction

Contributed by Brice Pouly

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 $H_N=N^{o(1)}$ , 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 $\mathbb{Q}(\sqrt p)$ . 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 $\operatorname{Pack}=3<\operatorname{Cap}=4<\mu=5$ . The selected-sample constructions do not give an upper bound for the minimum on a complete prefix.

Primarily AI-generated textHuman understanding: no partsarithmetic certificatescertificate capacityforced errorsfractional packinglocal compatibilitymultiplicative reconstructionquadratic traces

Oct 20, 2026PDF ↗View work →

hexagon:2610.00353v1

math.NT — Number Theory

Arithmetic reconstruction and error recovery from three quadratic traces

Contributed by Brice Pouly

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.

Primarily AI-generated textHuman understanding: no partsarithmetic reconstructioncompletely multiplicative functionserror recoveryinformation setsparity relationsquadratic tracesreading noise

Oct 20, 2026PDF ↗View work →

Matching works

Arithmetic coding against unknown small-prime signs

Correlation laws and identifiability of multiplicative functions on quadratic traces

Local freedom and capacity barriers in quadratic trace reconstruction

Arithmetic reconstruction and error recovery from three quadratic traces

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