Michael M. Ross | 2026 Publications
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A Reduction of the Squarefree Coprime Adjacent Divisor Problem and a Conditional Golden-Ratio Lower Bound
A balanced-split identity and Fibonacci layer reduction for Erdős #1100, giving a conditional golden-ratio lower bound g_sf(k) ≥ (φ + o(1))^k that improves the Erdős–Simonovits floor of √2.
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Congruence Entropy and Prime-Rich Proximate-Prime Quadratics
A dataset of 2,221 "proximate-prime polynomials" (1,973 distinct) — quadratics through consecutive prime quadruples whose gaps fall in arithmetic progression — shows that their prime-richness is almost entirely explained by local congruence structure — the single nonresidue count S(D) accounts for ~60% of the variance, the nine Legendre symbols used individually ~76%, and a full local-obstruction model ~96%.
Stacked Proximate-Prime Quadratics and the Covering of Square Intervals
We stack prime-rich monic PPPs and ask whether the union of their prime values covers every square interval [k², (k+1)²). We find empirically that always-odd curves n²+n+c cover as independent Cramér draws — giving an explicit covering law K(M) ≈ 2(log M)² / C-bar with no correlation correction.
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Rough and Smooth Numbers in Square-Centered Intervals: First-Order Structure and Second-Order Genericity
Places Jn at the high-threshold endpoint of the rough-numbers-in-short-interval spectrum, where roughness and primality coincide — yet is invisible to the second-order statistics of the prime count.
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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; the precise analytic consequence of that structure is sharpened in the continuation paper.
Primes in Quadratic Intervals and a Square-Centered Kloosterman-Fraction Problem
Takes the second-level Kloosterman sums of the dispersion chain and, via a recombination identity, shows the completed-Kloosterman presentation to be an invertible detour; restates the obstacle as Hypothesis R, a precise square-centered trilinear Kloosterman-fraction estimate, and locates it relative to the Bettin–Chandee and Drappeau theory. The square centering yields a quadratic-character stratification.
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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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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
An independent exponential-sum route to Jn. Treats the Archimedean reciprocal phase e(A/d) by the van der Corput second-derivative method and a Type I / Type II decomposition, reaching a balanced bilinear endpoint parallel to — and linked by reciprocity to — the Kloosterman-fraction obstacle of the dispersion route.
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Two Coefficient Regimes: Where Well-Factorable Machinery Attaches in the Legendre and Goldbach Programs
Technical note delimiting the reach of the recent well-factorable level-of-distribution results of Bombieri–Friedlander–Iwaniec, Maynard, Lichtman, and Pascadi.
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The Coefficient-Concentration Obstruction: Why the Square-Centered Sieve Weight Resists the Large Sieve
Why the remaining obstacle resists: the sieve coefficient is frequency-spread precisely where its mass lives, placing the required cancellation beyond current large-sieve methods.
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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 quadratic-character structure available for the parent Jn, from its exact square centering, is weaker for these shifted centers.
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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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Empirical Diagnostics for a Shadow-Sieve Approach to Primes in Short Intervals
A computational study of a shadow-sieve framework for primes in the quadratic intervals Jn = [4n2 - n, 4n2 + n], in which π(Jn) is expressed exactly as the difference between B-rough survivors and composites caught by a bilinear "shadow" map, evaluated across square-centered, half-shifted, and non-square subintervals to confirm shape-invariance of the empirical ratio.
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The Goldbach Problem for Primorials: Singular Series, Type II Sums, and LPF Structure
Integrated treatment of primorial Goldbach via the singular series and a bilinear minor-arcs hypothesis H(k); proves H(k) ⟹ G(p_k#) ≥ 1, situates H(k) below Elliott–Halberstam in the WBLMH framework, and shows the natural LPF-allocation alternative yields no lower bound.
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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.
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Loop Structure of Collatz-Type Functions 3x+n: A Conjugacy Theorem and Powers of Three
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.