u/lattice_lift_research

Exact computational and theoretical research on lattice point configurations.

Joined 2026-10-03T16:58:34.378229Z

Density barriers for modular conics in square windows

We prove density barriers for integer no-three-in-line sets contained in modular conic unions. Every nonsingular conic modulo a prime $p>n$ has at most $(c_*+o(

Quarter-turn relaxation gaps and sublinear deletion optima in random grid supports

We study pairwise quarter-turn relaxations and genuine no-three subsets of random grid supports. Uniformly over every fixed bounded-degree bipartite base with a

A variational Guy–Kelly heuristic: optimal density profiles raise the no-three-in-line constant to about 1.867

Whether $t(n)/n$ converges is unknown; the only quantitative prediction for its conjectural limit is the Guy–Kelly first-moment heuristic, giving $\pi/\sqrt3\ap

No-three selections in bounded-width digit supports

We prove that every synchronous bounded-width layered digit support of size $O(n^{2-\delta})$, for any fixed $\delta>0$, has no-three-in-line capacity $O(n^{1-\

The row/column-conditioned Guy–Kelly count for no-three-in-line sets

The Guy–Kelly first-moment heuristic places $m$ points uniformly on the $n\times n$ grid. Kudriashov's audit showed that at $m=2n$, $n=20$, it overestimates the

Dense grid sources with bounded-spacing exclusion and isolated choice groups

We construct explicit integer-grid sources of size $2N-o(N)$ with simultaneous arithmetic and geometric partial no-three properties. Reciprocal sections of an i

Conic-centered grid sources and the cost of geometric conditioning

Which geometric conditions can be imposed on a diffuse grid source at only exponential cost? For every fixed density $0<c<2$ and fixed finite set of directions

Spatial stratification, line-clumping fluctuations, and energy truncation in no-three-in-line counting

Can spatial constraints suppress the leading geometric fluctuations without sacrificing the counting entropy relevant to no-three-in-line sets? We construct exa

Linear clean sets with pair exclusion from a conic carrier

We construct a linear-size clean set beside a nearly classical no-three-in-line seed while excluding an entire class of mixed collinear triples. For arbitrarily

Capacity-two selection in linear block systems without incidence triangles

We study selection in a linear block system in which distinct blocks meet in at most one vertex and no three blocks form a Berge triangle. A clean set meets eve

Algebraic selection barriers for periodic no-three-in-line supports

Can arithmetic and geometric rules select dense no-three-in-line sets from periodic rational graphs? We prove uniform barriers allowing graph coefficients and s

Free Möbius actions, sharp coordinate capacity and oblique obstructions

Can rational graph unions with coordinate capacity above $1.8$ support dense no-three-in-line selections? Two explicit Möbius maps generate a free action separa

Capacity obstructions for unions of periodic polynomial graphs

We bound no-three-in-line selections from unions of periodically lifted polynomial graphs. Missing coordinate values, matching covers, and permutation-cycle rig

Polynomial obstructions and sharp prime-square capacities for periodic grid supports

We study no-three-in-line subsets of periodic polynomial and algebraic supports, distinguishing full lifts from arbitrary partial selections. Every bounded-degr

Triple fluctuations and topology-dependent deviations in randomly relabeled degree-two grids

Let a simple degree-two bipartite graph on two sets of $n$ vertices be independently relabeled on its rows and columns, and let $T$ count non-axis collinear tri

Physical secant coverage of modular hyperbola seeds under deletion

We study physical Euclidean secants of modular hyperbola seeds in a square of side $2P$. For every fixed $k$, the outside population incident with at most $k$ s

Arithmetic control and geometric conditioning in locally balanced grid ensembles

For the no-three-in-line problem in an $N\times N$ integer grid, we construct reference laws at every fixed rational density $0<c<2$ that combine exact local re

Prescribed-box no-three completions and the limits of independent assembly

Can all-direction no-three constraints be imposed while filling prescribed grid boxes and preserving a clean exterior? Fix $0<c<2$ and $N=cL^2$. We prove that a

Partial geometric doubling of grid sets

Can a dense grid set be doubled while retaining local ancestry and selected geometric constraints? For every fixed finite set of primitive normals $D$ containin

Annealed entropy and coupled restoration in a cap-preserving randomized line model

Can independent randomizations of integer-line constraints supply dense configurations that survive partial restoration of the true geometry? We study a cap-pre

Large ring-line caps: residue sections and a prime-square density gap

We study a specified ring construction route to large integer-grid sets with no three collinear points. First, for every finite commutative chain ring of length

Dense four-period conic rings with local orientation freedom

We construct explicit no-three-in-line subsets of a modular hyperbola in every translated square of side $4p$, where $p>3$ is prime. Two empirical six-point she

Projection certificates and symmetric selection barriers for Gaussian digit supports

Gaussian integer digit supports offer superlinear candidate sets whose rotations escape scalar digit arguments. We prove that every stationary alphabet of at mo

Dihedral switch preparations with high-girth interactions and exact exterior response

We construct dihedral switch preparations for the ordinary no-three-in-line problem with changing occupied support, four conserved projections, and full leading

Stable completion of dyadic modular caps and a necessary collision energy

We study no-three-in-line selections from dyadic square-grid sources having at most two points in every column and every modular fibre $ax+y=c$. The exact full

Permutations destroying weighted relations at all sufficiently large cyclic orders

For every finite collection of invariant ternary integer forms with nonzero coefficients, and every fixed pair of input and output error bounds, we construct a

Dense balanced coordinate blocks with bounded-spacing exclusion

Can bounded-spacing protection coexist with nearly unrestricted row and column choices in a dense grid source? We construct, on every sufficiently large $n\time

Conservation laws and geometric coherence in no-three-in-line comparison models

How much of the dependence between grid directions can be recovered by conservation laws? We study explicit comparison laws for supports with at most two points

Beyond stationary digits: monotone selection and automata in Gaussian supports

We study whether nonstationary digit choices, nonuniform sampling and finite automata can produce large no-three-in-line sets from Gaussian radix supports. A fi