Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.32078

Oracle-Tree Forcing and Wilfs Inequality for Thirteen Left Elements

Michel Gaspar
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study filter-Laver forcing on omegaxomega with successor sets containing final quadrants. The observation retaining the minimum coordinate and orientation is a complete projection. Its finite nontrivial hidden-label quotients are Cohen, whereas the full offset quotient has Boolean density equal to the ground-model dominating number dV . For the associated grid-meager ideal I_box, we prove cov(I_box) = add(M) for the usual meager ideal M, and obtain further cardinal bounds. We also analyze threshold games, oracle degrees and the distinction between positive and filter-large certificate sets: recurrent filter-large acceptance admits refinements whose every branch succeeds. As an arithmetic application, we prove Wilfs inequality c \leq 13e for numerical semigroups with thirteen elements below the conductor c, where eis the embedding dimension. A complete arithmetic proof, including a bounded exact computation, is given in the appendix. Its terminating certificate procedures supply a computable quadrant tree and uniform refinements for every forcing condition. Finite support kernels describe persistence of the registered proofs; arithmetic absoluteness explains precisely why forcing organizes these certificates without replacing their arithmetic justification.

Explore related subjects

math.LO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Michel Gaspar. 2026-09-25. Oracle-Tree Forcing and Wilfs Inequality for Thirteen Left Elements. https://arxiv.org/abs/2609.32078

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Small additive groups with $A+ξA=\mathbb R$

We prove that, for every irrational $ξ$, any $F_σ$ additive subgroup of $\R$ has an $F_σ$ $\Q$-vector space extension of the same Hausdorff dimension satisfying $A+ξA=\R$. Iteration gives simultaneous surjectivity for any prescribed countable family of irrational multipliers. Such a space exists in every prescribed dimension $d\in[0,1)$. We also construct in ZFC a zero-dimensional lightface $Δ^0_3$ $\Q$-vector subspace $A$ for which $A+ξA=\R$ holds for every irrational $ξ$, answering both parts of Question~2 of Ye, Yu, and Zhao. This universal group is a difference of two $F_σ$ sets, but is neither $F_σ$ nor $G_δ$. Both constructions use direct inductions on finite binary strings and admit effective versions. As applications, we obtain zero-dimensional $F_σ$ groups whose Cartesian squares have Hausdorff dimension one, and relate their dilate sums to Marstrand's projection theorem.

math.LO↗
arXiv2026

Proof Theory for Non-Contingency Logic

{Non-contingency logic $(\KWL)$ replaces the usual necessity operator of modal logic with an operator expressing that a proposition is necessarily true or necessarily false. Besides its intrinsic logical interest, it admits natural interpretations as knowing whether in epistemic logic and as decidability under the arithmetical interpretation of provability logic. Although the semantics of non-contingency logic have been extensively studied, its proof theory remains comparatively underdeveloped. In this paper, we develop a uniform proof-theoretic framework for $\KWL$ over a broad class of frame conditions. Our approach is based on generalized path conditions (GPCs), a grammar-theoretic formalism that uniformly captures many standard modal frame properties. For every finite set $\gpc$ of GPCs, we construct a corresponding labelled sequent calculus. All calculi share a common set of logical rules and differ only by a single structural rule generated from $\gpc$, which captures the underlying frame conditions. We prove that these calculi are sound and complete with respect to their corresponding frame classes, thereby providing a uniform proof theory for a large family of non-contingency logics.}

math.LO↗
arXiv2026

Shelah's Main Gap and the generalized Borel-reducibility

We answer one of the main questions in generalized descriptive set theory, the Friedman-Hyttinen-Kulikov conjecture on the Borel reducibility of the Main Gap. We show a correlation between Shelah's Main Gap and generalized Borel reducibility notions of complexity. For any $κ$ satisfying $κ=λ^+=2^λ$ and $2^{\mathfrak{c}}\leqλ=λ^{ω_1}$, we show that if $T$ is a classifiable theory and $T'$ is a non-classifiable theory, then the isomorphism of models of $T'$ is strictly above the isomorphism of models of $T$ with respect to Borel-reducibility. We also show that the following can be forced: for any countable first-order theory in a countable vocabulary, $T$, the isomorphism of models of $T$ is either analytic co-analytic, or analytically-complete.

math.LO↗