Ruqing Chen

The degree-28 polynomial of a genus-4 linearised operator, the quadratic-form torsor, and the symplectic monodromy Sp_6(F_2)

Let q = 2^k and let 𝔮_λ(x) = T(x) + 2 Tr(λ9 x⁹ + λ5 x⁵ + λ3 x³ + λ1 x) be the Z/4-valued quadratic form attached to the supersingular genus-4 curves y² + y = λ9 x⁹ + λ5 x⁵ + λ3 x³ + λ1 x over F_q, where T is the Witt-vector trace of the Teichmüller representative. Its polar opera...

Prime polynomials in Legendre intervals over F_{2^k}[t], III: the hexad of a quartic linearised polynomial, the symplectic group Sp_4(F_2), and the Witt sums behind the constant X

Third part of a series on the variance of prime polynomials in Legendre intervals {f^2 + s : deg s ≤ d} over F_q[t], q = 2^k (Open Problem 8.2 of Part I, DOI 10.5281/zenodo.23225896; Part II, DOI 10.5281/zenodo.23252762). Part II reduced the degree-6 constant X to sums W'(λ_3, λ_...

Prime polynomials in Legendre intervals over F_{2^k}[t], IV: point counts on hexad modular curves, Brown invariants of Witt sheaves, and the constant X

Let q = 2^k with k odd. Part III (DOI 10.5281/zenodo.23290724) reduced the closed form X = q^6 + q^5 of the second constant of the degree-6 Legendre variance to two statements about the sextic family B_λ = λ5² u⁶ + λ3² u² + λ5 u + 1, whose roots form the hexad of ker L'_λ: the po...

Even integers as sums of two sums of two squares with few prime factors

Let N ≡ 2 mod 4 be a sufficiently large integer. We show that the number of n ≤ N such that n and N − n are both sums of two squares and Ω(n(N − n)) ≤ 11 is at least (3/2) S(N) N / (log N)^2, where S(N) is the binary Goldbach singular series. With other parameters the method show...

On primes p for which N − p is an almost prime sum of two squares, and other class number one forms

Let N be a sufficiently large even integer. We show that the number of primes p ≤ N such that N − p is a sum of two squares having at most nine prime factors, counted with multiplicity, is at least (1/10) S(N) N / (log N)^2, where S(N) is the binary Goldbach singular series. The...

On primes p for which N − p is an almost prime sum of two squares

Let N be a sufficiently large even integer. We show that the number of primes p ≤ N such that N − p is a sum of two squares having at most nine prime factors, counted with multiplicity, is at least (1/10) S(N) N / (log N)^2, where S(N) is the binary Goldbach singular series. The...

On primes p for which N − p is an almost prime sum of two squares, and other class number one forms

Let N be a sufficiently large even integer. We show that the number of primes p ≤ N such that N − p is a sum of two squares having at most nine prime factors, counted with multiplicity, is at least (1/10) S(N) N / (log N)^2, where S(N) is the binary Goldbach singular series. The...

Two-channel duality, fold memory, and arithmetic blindness in polymer contact networks

The contact network of a collapsed polymer is a finite graph, and graph invariants of the kind used in algorithmic class field theory can be read off it exactly: the Laplacian, its sandpile group, the discrete Hodge decomposition, the Albanese metric of a periodic cover, and the...

Algorithmic Non-commutative Class Field Theory, Volume III: Higher-Rank Buildings, Non-Equilibrium Topology and Higher-Order Games

This monograph is the third volume of "Algorithmic Non-commutative Class Field Theory". It studies spectral, dynamical and arithmetic structures on finite quotients of Ã₂ (and, where possible, Ã_d) buildings, in particular on the complexes X₃, Y₂₁, …, Y₁₆₈ of Volumes I–II and on...

What a prime-programmed polymer can read: sieve composition, congruence traps, and arithmetic blindness

A self-avoiding chain whose bending stiffness follows the gaps of a set of primes (the prime-driven topological polymer, PDTP)registers the ordering of the prime gaps, but earlier work showed that this imprint is carried by the smooth prime-number-theoremenvelope at the small-pri...

Spectral Kakutani–Feldman–Hájek Criticality of Correlated Conformational Modes: Mode Amplitude Perturbation, Iso-Spectral Givens Rotation, and Tail-Increment Finite-Size Elimination

This is the third paper of the series "Statistical Pharmacology via Kakutani Dichotomy". Paper II (DOI 10.5281/zenodo.23012217) decomposed the covariance part of the correlated Kakutani index of two Gaussian conformational ensembles into a commuting amplitude part and a non-commu...

Statistical Pharmacology via Kakutani Dichotomy IV: KMS Inverse-Temperature Criticality β_c = 1, Pocket-to-Distal Tail Allostery, and the Four-Quadrant Ligand Classification

This is the fourth paper of the series Statistical Pharmacology via Kakutani Dichotomy, which transports the tail-event phase transition of Localized Bost–Connes systems (R. Chen, DOI 10.5281/zenodo.22883179) to the statistical geometry of protein conformational ensembles. Papers...

When does sequence order matter? An information-theoretic taxonomy of biological representation tasks and the limits of local encoding

Biological sequence models range from amino acid composition and k-mer counters to convolutional networks and billion-parameter language models. When is a local representation sufficient, and when is global order indispensable? This closing paper of an eight-part series answers t...

When does sequence order matter? An information-theoretic taxonomy of biological representation tasks and the limits of local encoding

Biological sequence models range from amino acid composition and k-mer counters to convolutional networks and billion-parameter language models. When is a local representation sufficient, and when is global order indispensable? This closing paper of an eight-part series answers t...

Symmetry-constrained spectral fibres: reverse-complement group actions on double-stranded DNA and the fibres of circular molecular fingerprints

A representation that depends on an object only through its local content is constant on the object's spectral fibre, the set of objects with the same multiset of local pieces. This paper treats two settings in which a symmetry enters. Double-stranded DNA. Reverse complementation...

FibreBench: stress-testing protein sequence encoders against exact combinatorial spectral invariants

Protein representation models are usually evaluated on test sets split by sequence identity. Such splits cannot reveal what an architecture is unable to distinguish by construction. FibreBench is a benchmark built from an exact invariant instead. Each test pair consists of a huma...

Phenotypic insufficiency of local protein representations: fibre-preserving rearrangements of human C2H2 zinc-finger arrays reorder DNA-recognition residues under an invariant 25-spectrum

A protein representation built from local sequence statistics assigns the same vector to all sequences with the same multiset of k-mers, the spectral fibre. Whether this blind spot matters depends on the phenotype. For C2H2 zinc-finger proteins, whose fingers read DNA in order, i...

The Wasserstein separation of the broken phases: a finite-dimensional reduction and the quadratic chaos bound

For a localized Bost-Connes system in its symmetry-broken phase, the Connes-Rieffel distance W_1 between the two extreme KMS states was previously squeezed between w_0 Pi_infinity(beta) and w_0. This paper sharpens both sides. A finite-dimensional reduction sandwiches W_1 between...

The adaptive coupling upper bound for the Wasserstein separation: microscopic flips against the macroscopic symmetry cost

For the extreme broken-phase KMS states of a localized Bost-Connes system, separated by the order parameter chi(g) prod_j epsilon_j(x_j), the only known upper bound on the Connes-Rieffel distance W_1 was the trivial group-flip cost w_0. This paper constructs an x-adaptive couplin...

The ray-class repair for affine systems: the class group as the index of definable scalings

For a non-principal prime p of a number field the affine Bost-Connes-type monoid is not well defined over the ring of integers. This paper proves that the obstruction is itself the invariant: with h the order of [p] in the class group, the globally definable scalings form exactly...