Prime-Pair Incidence Matrices: Almost-All Higher Prime Correlations and Pascal Transforms

Abstract

This paper studies the binary prime-pair incidence matrix defined by (Aₙ)ₘ,ᵣ = 1ℙ(m − r)1ℙ(m + r), whose row index records the midpoint of a prime pair and whose column index records its half-gap. Rows therefore encode unequal Goldbach representations, while columns count prime pairs having prescribed even gap 2r. This matrix formulation brings prime-pair correlation theory, Goldbach structure, structured matrix analysis, Pascal transforms, p-adic dynamics, and singular-series arithmetic into a common finite-dimensional framework. The principal arithmetic results are unconditional almost-all column theorems obtained by transferring modern prime-correlation results directly to the incidence matrix. For every ε > 0 and shift range satisfying X⁸⁄³³⁺ε ≤ H ≤ X¹⁻ε, the expected Hardy–Littlewood asymptotic holds for the dyadic increment of all but o(H) half-gap columns. Thus almost every individual column in this range exhibits the predicted prime-pair density at a substantially shorter scale than is available from classical fixed-gap asymptotics. The column formalism also captures higher prime correlations. For every fixed ℓ ≥ 3, a vertical product of entries within a single column records a length-ℓ arithmetic progression of primes with common difference 2r. Applying the 2026 higher-correlation theorem of Matomäki, Radziwiłł, Shao, Tao, and Teräväinen shows that, when H ≥ X¹⁄³⁺ε, all but o(H) half-gaps have the predicted Hardy–Littlewood ℓ-point dyadic asymptotic. The same incidence column therefore simultaneously supports the pair-correlation problem and arbitrary fixed-length prime arithmetic progressions. Several additional unconditional matrix-level distribution laws are established. The prime number theorem gives the global Frobenius asymptotic ‖Aₙ‖²_F ∼ n²/(log n)² and asymptotic equality of Frobenius mass between the two mod-4 parity blocks. Merikoski's averaged Hardy–Littlewood theorem yields, for every 7/12 < θ < 1, ∑ᵣ≤⌊nᶿ/2⌋ Cₙ(r) ∼ n¹⁺ᶿ/(log n)², together with corresponding mean-column and growing-band Frobenius laws. A short-average lower bound further forces a positive linear-in-n/log n number of incidences into the first C log n columns whenever C > 1/2. The finite matrix itself has strong exact structure. The paper derives Goldbach row identities, fixed-gap column identities, support constraints, a mod-4 parity decomposition with spectral decoupling, a Toeplitz–Hankel Hadamard representation, prime-chain formulas for powers, and a sharpened periodic nilpotency envelope. These identities expose deterministic matrix structure beneath the arithmetic distribution of the nonzero entries. Weighted generalized Pascal matrices act on the incidence matrices by exact two-sided transforms. Their semigroup, exponential, differential, entrywise inversion, and finite-section properties are developed explicitly. The action extends compatibly to a projective p-adic Pascal flow, with analytic deformation formulas and a Lucas-theoretic digitwise kernel modulo p. A banded triangular-core approximation model is also studied. Existence of constrained minimizers, explicit gradient identities, exact normal equations, convexity properties, and Karush–Kuhn–Tucker conditions are derived for the resulting structured optimization problem. For the Hardy–Littlewood prime-pair singular series, the paper proves a squarefree divisor convolution and an exact Dirichlet-series factorization, together with meromorphic continuation to Re(s) > −1/2 and an exact finite-cutoff deficit identity. Under the fixed-gap Hardy–Littlewood conjecture, normalized finite column Dirichlet polynomials converge, together with all derivatives on fixed windows, to the corresponding singular-series Dirichlet function. The resulting theory gives a unified matrix representation of Goldbach counts, fixed-gap prime pairs, almost-all higher prime correlations, singular-series arithmetic, generalized Pascal dynamics, and structured finite-dimensional transforms.