Michael M. Ross | 2026 Publications
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Audited Censuses of Two Generalized Fermat Families: Coverage Ledgers and Large Datasets
Exhaustive, machine-auditable censuses of two generalized Fermat families — exactly the seven known coprime solutions to 1016, and none among 193,776 solutions to 1030 — with descent-based closure audits and two disclosed defects.
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Coprimality Density and the Proper-Solution Deficit in the Generalized Fermat Family {2, 3, m}
A closed-form, per-anchor coprimality density and an unconditional 2-descent census of 670 Mordell classes, proving part of the proper-solution deficit dead and locating the remainder on positive-rank curves.
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The Hall Near-Miss Census to 1011: Exact Families, a Depleted Residual Population, and a Preregistered Scaling Test
Ninety billion cubes and their nearest squares resolved into exact parametrized families and a residual population depleted for the second consecutive preregistered decade — real depletion, arriving shallower than the registered law.
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One-Step Fermat Factorization and Goldbach Representations with Almost-Equal Summands in Square-Centered Intervals
Inside the square-centered interval, instant Fermat factorization and near-equal Goldbach representation are the same event — provable on average, open pointwise at window exponent ½, and last absent through 10⁸ at n = 1,884,296.
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Exceptional Sets for Semiprimes in Quadratic Intervals
This note proves three exceptional-set theorems for semiprimes in the intervals (n², (n+1)²) by applying the computational investigation of the companion papers to the analytic findings of Matomäki–Teräväinen and Gafni–Tao.
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Exact Semiprimes in Short Intervals: Prime Supply, Factor-Pair Multiplicity, and the Parity Barrier
A 10¹¹-term computation showing semiprimes are far denser and more evenly spaced than the √x scale would require, reframing "a semiprime between every pair of squares" as a parity problem rather than a Legendre one.
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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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An Unconditional Golden-Ratio Lower Bound for Coprime Adjacent Divisors of Squarefree Integers
An unconditional proof that the maximum number of coprime adjacent divisor pairs of a squarefree integer with k prime factors grows at least like φ^k/√k, improving the Erdős–Simonovits lower bound (√2 + o(1))^k.
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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%.
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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: Obstructions to the Square-Centered Dispersion Route
This note withdraws the earlier square-centered dispersion program by proving that its V-type hypotheses carry no conditional content and that its trilinear Kloosterman hypothesis has no useful instance. It replaces this with a precise obstruction framework: the exact Poisson object, its unavoidable slow phase, and the non-sparse Ramanujan coefficient structure that a future approach must overcome.
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A One-Hypothesis Reduction for Primes in [4n²−n, 4n²+n]
An unconditional bound is proved for the semiprime tail by pushing the linear sieve to the Fouvry–Iwaniec level x^{5/8} against prime moduli exceeding x^{1/2}. Reduces the existence of primes in [4n²−n, 4n²+n] to a single medium-range Buchstab lower bound at any cutoff beyond (2n)^{0.9393}.
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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 Modified Legendre Product for Primes in Consecutive Square Intervals
A modified Legendre product for primes in consecutive square intervals: a two-term Mertens-type expansion predicts the mean count ratio out-of-sample to 1.8×10⁻⁵ at n = 10⁹, and an exhaustive census locates the last pointwise violation of the mean-level upper bound at n = 77,433.
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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.
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The Distribution of Two-Sided Fortunate Numbers and the Goldbach Amplification Constant for Primorials
Pending publication.
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Sifted Symmetry and Sieve Structures of Primorial Goldbach Complements
Proves that the Goldbach problem for primorials p_k# with small summand is identically a prime-gap problem at the Cramér scale (log N)² — and that below this threshold the problem is provably invisible to every sieve functional and mean-square method, even under GRH.
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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 and 2-adic Transition Laws in the Accelerated Collatz Map
Modular spike structures and 2-adic transition laws for the accelerated Collatz map; a cycle-equation search (≈ 7.5×10⁹ candidates, L ∈ [50, 200]) found no R = 2 cycle but under incomplete enumeration, so the exclusion remains conjectural.
The Jacobian Conjecture Counterexample: Cubic Corollary and Uniqueness of j=1
Two notes on the Jacobian conjecture counterexample of Levent Alpöge (with Claude Fable 5), analyzed through the construction described in Terence Tao's digestion of the counterexample.
Empirical Structure of the Gilbreath Decay Constants
Exact values of c₄–c₆ and a binary digit-sum decay law for the Gilbreath constants of Chase, Hunter & Tao (arXiv:2607.08712), with Monte Carlo evidence that the growth threshold is exponential rather than linear.
BTW Is Gilbreath’s conjecture garden-variety numerology?
Supersedes...