Michael M. Ross | 2026 Publications
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Primes in Square Intervals: The Remaining Analytic Obstacle
The spine of the Jn analytic program. A five-stage Bombieri–Friedlander–Iwaniec dispersion chain reduces prime existence in Jn to a single averaged-Kloosterman hypothesis in the Deshouillers–Iwaniec style; the perfect-square centering at (2n)2 places a square factor in the second Kloosterman argument, producing a positive-proportion Salié subset.
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A Minimal Distributional Hypothesis for Primes in the Interval [4n²−n, 4n²+n]
Master reduction. States the minimal averaged hypothesis on the negative-square discrepancy and derives prime existence in Jn from it via Buchstab decomposition.
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Eliminating the Parity Obstruction in a Quadratic Interval
Removes the ultra-rough semiprime obstruction in Jn via the factorization gap (2n+1)2 > 4n2+n, establishing that every composite in Jn has a prime factor ≤ 2n.
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A Note on the Parity Obstruction in a Quadratic Interval
Companion note to the parity-elimination paper above; earlier and more compact treatment of the factorization-gap argument.
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The Quadratic Character Contribution to the Averaged Negative Square Discrepancy
Parallel angle on the master hypothesis via character decomposition. Verifies the real-character contribution Δ2(m) using Jutila and Heath–Brown zero-density estimates — unconditional, ineffective due to potential Siegel zeros.
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Reciprocal Phases and the Exponential Sum Approach to Jn
Phase-factorization and exponential-sum analysis at the Jn scale; develops machinery used in the dispersion chain.
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A Structural Barrier to Linear Sieve Methods in Quadratic Intervals
Shows the parity-style obstruction to linear sieves at the Jn scale: the linear-sieve lower-bound function f(s) vanishes for s ≤ 2 at realistic levels of distribution. Motivates the dispersion-method route taken in the centerpiece paper.
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Primes in Structured √x-Intervals and Density-One Quadratic Persistence
Transfers the BFI dispersion framework from Jn to the family Jn(k) = (n2/k, (n+1)2/k); under a density-one weakening of the transferred hypothesis, the failure set has natural density zero for k ∈ {2, 3}. The Salié coherence available for the parent Jn is unavailable here.
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Semiprimes 2p and 3p in Quadratic Intervals: A Legendre-Class Persistence Problem
Computational and structural verification that every Jn(k) for k ∈ {2, 3} contains a semiprime of the form 2p or 3p; verified through n = 107 with zero failures, framed via a Cramér-gap parsimony argument.
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A Multiplicity-Corrected Legendre Sieve for Primes in Consecutive Square Intervals
Modified Legendre sieve replacing the local density 1/p with d/p2 for large primes. Computational verification confirms the asymptotic ratio π(In)/E(d) ∼ (eγ/2)(1 + 1/ln d).
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Factor Rays and the Self-Conjugate Parabola: Deterministic Coverage Geometry in Square Intervals
Geometric coverage analysis of factor rays and the parabola n=k2 in the square-interval setting.
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Inverse Pairs and the Loop Structure of Collatz-Type Functions
Loop geometry in 3x+n maps; conjugacy theorem for 3x+3k; explicit inverse-pair inputs.
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Spike Structures, Finite-State Exclusion, and Bounded-Exponent Collatz Cycles
Modular spike structures and finite-state exclusion of bounded-exponent Collatz cycles; full computational search through L ∈ [50, 200] with no R=2 cycles found across ~7.5 billion candidates.
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The Goldbach Problem for Primorials: Singular Series, Type II Sums, and LPF Structure
Integrated treatment combining the singular-series analysis of primorial Goldbach partitions with the bilinear minor-arcs hypothesis and LPF allocation; positions Hypothesis H(k) at the current research frontier, strictly weaker than Elliott–Halberstam.
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Sieve Structures of Primorial Goldbach Complements
Earlier structural treatment of Goldbach-complement sieves at primorial moduli; precursor to the integrated paper above.