Papers

Uniform list coloring of long odd two-fans and rigidity in critical graphs

We study list-coloring extensions and exterior rigidity for the Earth–Moon problem. The graph $J(a,b)$ consists of adjacent hubs and two internally disjoint odd

Most of Muon's mean gradient is a bounce: split-batch measurements and what they predict for momentum memory and batch size

In Muon-trained language models the constant-LR population gradient is mostly a period-2 bounce. We measure this and test its consequences for momentum and batc

A power saving for ordinary secant energy in capped grids

For a set in an $N\times N$ integer grid with at most two points per horizontal row, call a pair ordinary if its physical line contains exactly two selected poi

The no-three-in-line count at $50\le n\le800$ against an exact independent-direction baseline and a tilted comparison functional

Let $N(n,m)$ be the number of no-three-in-line $m$-subsets of the $n\times n$ grid. We measure $\log N(m+1)/N(m)$ at $m=cn$ for $c=1.0$–$1.6$ and $n=50$–$800$ t

A four-term relation on the curve of a Diophantine triple and the one-step origin of the known almost septuples

For three distinct nonzero rationals $x_i$ with $e_3=x_1x_2x_3$, let $E_X:Y^2=\prod_i(X+e_3/x_i)$ carry the point $P_0=(0,e_3)$. We prove that three points $T'_

Four rational sections on the norm-and-anchor cover of a determinant-zero Diophantine chart

We classify the rational points over $\mathbb Q(r)$ on the genus-three cover $v^2=N_r(u)$, $a^2=(r^2-1)(1-u^2)$, where $N_r(u)=(r^4+2r+1)u^4+2r^2(r+1)u^3-r^2(2r

The diagonal ratio curve of order-four Diophantine orbits: a local criterion, a certified ratio box, and the genus-two fibre

In the order-four construction of rational Diophantine septuples, a triple is extended by four translates of a point by a point $S$ of order four on its induced

Degree-minus-one coloring of contacted J(5,5) patches

We prove a finite local extension theorem for the contacted odd/odd two-fan patches appearing in a 492-vertex biplanar occurrence barrier. Every J(5,5) with one

A Hasse-principle failure on a fully lifted regular-twin Diophantine curve

We construct an explicit smooth projective genus-21 curve arising from the regular-twin Diophantine construction with anchors $a=10783710$ and $b=-3/(4a)$. The

Cross-direction dependence separates the no-three-in-line grid from cap-preserving random line systems: exact counts and a class factorisation

How does the grid differ from random line systems sharing its per-direction statistics? For $2n$-point no-three-in-line sets, $s(n)=R_2(n)P(\mathrm{void})$ unde

Averaged orbit bounds: one modular hyperbola has density at most 3/2 in every window of side between 2p and 3p

The lift of a modular hyperbola to a square of side $2p$ gives the best known lower bound $(3/2-o(1))n$ for the no-three-in-line problem. How much can one hyper

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

The twelve- to fifteen-vertex core branches of the order-twenty Earth–Moon problem

A ten-chromatic biplanar graph on twenty vertices would contain a 10-critical graph $G$ with $e(G)\le108$, $\delta(G)\ge9$ and no $K_9$. The order-twenty progra

No regular-triple sixth entry in a conductor-192 family of rational Diophantine quintuples

We study extension formulas in the family of square-product rational Diophantine quintuples parametrized by $V^2=(k^2-2)(k^2+4)$. Of twenty regular fourth-eleme

Four explicit graphs decide the ten-color question for biplanar line graphs

We reduce the ten-color question for biplanar line graphs of arbitrary finite loopless multigraphs to four explicit graphs on 28 vertices. The Goldberg–Seymour

Optimal canonical degree and finite trace exhaustion on the exotic-quintuple surface

We study low-genus curves on the surface u^2+w^2=r^2+r^{-2}, z^2=(u^2-1)(1-w^2), which underlies the rational Diophantine septuple problem. Resolving the canoni

Critical cores with an independent triple and no disjoint eight-clique through 81 edges

We classify nine-critical graphs on seventeen vertices with at most 81 edges that possess an independent triple having no disjoint $K_8$. Exactly 22 isomorphism

Low-genus hyperplane and quadratic Laurent sections of the exotic-quintuple surface

We classify low-genus curves in two complete four-parameter systems on the exotic-quintuple surface of Dujella, Kazalicki and Petričević. The rational hyperplan

Degree-twenty reconstruction and finite fixed-pair branches of almost Diophantine septuples

A relation census identifies a mechanism shared by six rational almost Diophantine septuples: two regular quintuple extensions followed by another regular exten

Strong-coordinate exclusions in square-product Diophantine quintuple families

A rational Diophantine tuple has pairwise products plus one square, and an entry is strong when its square plus one is square. We prove thirteen coordinate excl

Lifted line relaxations for the no-three-in-line problem

We study lifted relaxations of the full no-three-in-line point-line formulation. Explicit rational moments keep a target cardinality of $1.8n$ feasible through

Local-search densities fall short of the known no-three-in-line optimum by n = 40: a calibration of the decay curve

For $n\le74$ and $n=76$ the no-three-in-line optimum is known: $t(n)=2n$. Above that range the best values come from local search, and they fall from about $1.8

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

The Wildner–Gibbs Vieta interface leaves a degree-143 squareclass gate

We audit a concrete extension route toward a rational Diophantine septuple. Wildner's explicit moving D(q)-quintuple is specialized to z=T^2 and divided by T, g

Coordinated center boundaries in ordinary covers of G25

We study ordinary covers of the ten-chromatic graph $G_{25}$ as possible sources of ten-chromatic biplanar graphs. Four earlier extension results are retained.

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

Affine list constraints and simultaneous witnesses for G25 cover extensions

We study signed list constraints and prescribed-center extensions in ordinary covers of the ten-chromatic graph $G_{25}$. Our affine theorem partitions variable

One-vertex and disjoint two-vertex extensions of a nine-color template

We study whether adding one or two vertices to a nine-chromatic host can produce a ten-chromatic biplanar graph. For an explicit 25-vertex, 148-edge template $T

Outerthickness-two colouring and pure apex splits in the Earth–Moon problem

A graph has outerthickness at most two if its edges are the union of two outerplanar graphs. Its largest chromatic number is known only to lie in $\{6,7,8\}$ (G

Seven stable RLVR continuations of a 0.6B policy average within +0.4 pp of decoding the start at T 0.4; a same-size verifier extracts cross-item information from the same samples but shows no detected advantage over voting at matched FLOPs or as a training reward

Studies of RLVR on Qwen3-0.6B-Base find that recipes tie and that colder decoding of the start reproduces much of their gain, but per-arm tests cannot exclude a

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

Verifier contrast is not sufficient for small-model mathematical RLVR

Group-normalized RLVR has zero fresh advantage when all sampled outcomes agree. We test whether increasing verifier contrast improves Qwen3-0.6B-Base under the

Antipodal representations and a two-level fractional-certificate barrier for biplanar graphs

Can fractional coloring above nine be sought within graph representations having a common antipodal symmetry? We give a connected representation of every nonemp

A sharp three-root extension theorem for positive five-cycle cores

We prove a sharp coloring-extension theorem for complements of positive-capacity five-cycle blow-ups. Let the five positive integer edge capacities sum to $k$,

Matching formulas and projective non-descent for the shifted pentad discriminant

For five rational Diophantine entries, the classical factor-analysis pentad applied to their signed pair roots minus a common shift is quadratic in the shift. W

Ten-critical biplanar graphs on twenty-one vertices have independence number three

We prove that every ten-critical biplanar graph on twenty-one vertices has independence number three. Consequently every biplanar graph on twenty-one vertices w

Local obstructions and rank conditions for canonical symmetric Diophantine sextuples

We study the canonical extension of rational symmetric Diophantine quadruples to symmetric sextuples. The construction factors through an explicit genus-one tor

Simultaneous root repairs: sharpness, density barriers and positive matching capacities

We study simultaneous coloring repairs for a root clique and an isolated root over specified cores. An explicit eleven-vertex core proves the abstract four-root

Exact fractional mass 2N on thin grid supports with a dimension-one limit

For every $N=2^m$, we give an explicit digit support with exact all-line fractional optimum $2N$ and prove that its weighted measures converge to twice a probab

Square-root rank of rational Diophantine kernel matrices

Given rational numbers $a_1,\ldots,a_n$ with $1+a_i a_j$ a rational square for every distinct pair, form the matrix $N$ with zero diagonal and off-diagonal entr

Common quadratic twists and low-genus splitting curves in a Diophantine Kummer chart

The norm-and-anchor surface of a determinant-zero Diophantine septuple chart is a known Kummer surface attached to the square of E: y²=x(x−1)(x−2). We classify

Own-trace fitting recovers inserted-state damage without a fitted latent advantage

Adding deterministic inserted computation to a small RLVR model can produce a large apparent training gain consistent with recovery from a damaged initializatio

Outside information and self-checking do not move the low-p bin of a 0.6B RLVR policy, and moving it could not have paid for a greedy MATH-500 advance

RLVR on Qwen3-0.6B-Base plateaus. Every recipe tried here ties with a 60-update GRPO control (228/500 greedy MATH-500, 2,048 tokens). A common reading: the poli