arXiv subjects

Chakshu Gupta

Publications and source records attributed to Chakshu Gupta.

7 recordsLinked to original sources

arXiv2026

Collision-based logic in Lenia and its composition boundary

Continuous cellular automata such as Lenia spontaneously produce lifelike, self-propelling patterns, including the Orbium glider, which travels in a straight line while pulsing through a fixed breathing cycle. Collision-based logic, where moving patterns compute by colliding, is established in discrete cellular automata and continuous physical media. Within continuous cellular automata, computation has so far been trained into the rule rather than emerging from collisions, and whether a fixed-rule automaton like Lenia can support general collision-based computation remains open. This paper constructs an INHIBIT gate from collisions of the Orbium glider. Of the patterns searched across four continuous-CA rule types, the Orbium glider is the only one shown to survive a collision with both copies intact. A control glider deflects a signal glider off its output line, so the output carries a signal only when no control is present. The gate blocks across all twenty-four phases of the breathing cycle and nine integer offsets of the control. Two such gates in series, with one signal line and two controls, compose into an AND-NOT chain, correct on all eight input combinations. By contrast, routing a signal beyond that single chain is undemonstrated. A deflected signal is not restored to a fixed landing position, and no reusable absorber for the surviving gliders was found. The immediate open question for collision-based computation in Lenia therefore narrows from whether a gate exists to whether a deflected signal can be delivered to a downstream gate, the next requirement for composing the gate beyond a single straight chain.

cs.ET↗

arXiv2026

Domination versus edge domination in regular graphs of degree at least seven

Baste et al. (2020) conjectured that every regular graph of positive degree has domination number at most its edge domination number, the least size of a maximal matching. Combining published bounds settles the inequality for every degree at least nine. A reduction proves the inequality whenever one endpoint of each edge of a minimum maximal matching can be chosen to form a dominating set, and the Lov\'asz Local Lemma shows such a choice exists for every degree at least seven, newly closing degrees seven and eight and leaving degrees three through six open. The reduction settles each open degree up to a bounded number of vertices, forty-eight for cubic graphs. At fifty vertices, however, the reduction meets an explicit cubic graph it cannot settle, though the inequality holds there too. The inequality cannot be tightened, since infinitely many cubic graphs have equal domination and edge domination numbers. The cubic case stays open, and even linear arguments from the local structure cannot close it. The middle degrees stay open beyond the graphs already settled.

math.CO↗

arXiv2026

One construction for the Miura-ori flip-graph degree sequence

The flip graph of an origami crease pattern has the locally flat-foldable mountain-valley assignments as vertices, and an edge joins two of them that differ by a single face flip. A basic invariant of this graph is the degree sequence, which counts the vertices of each degree. On the $m\times n$ Miura-ori, this sequence is known to be a bivariate polynomial only for small degrees, each count obtained by a separate argument. This paper gives one uniform construction that expresses, for every degree $d$, the number of degree-$d$ vertices as a single symmetric polynomial in $(m,n)$ for all sufficiently large $m,n$. Its degree in each variable is $d-2$ unconditionally. Subject to a single degree bound, its total degree is $d-2$ as well, with top-degree part an explicit multiple of $m^{d-2}+n^{d-2}$ for $d\ge5$. The bound is proved here when the count splits into independent row and column factors, and open otherwise. The region is $m,n\ge\max(d-1,2)$. The polynomials are given in closed form through $d=10$, unconditional through $d=8$, where the degree bound holds in every case, and conditional on it beyond. Below this region the count departs from the polynomial. One step below, this departure has leading coefficient $-4$ times a Baxter number through $d=11$. Each such polynomial thus counts the Miura-ori's locally flat-foldable assignments admitting exactly $d$ single face flips.

math.CO↗

arXiv2026

An annihilation-number Caro-Wei bound: a TxGraffiti conjecture and an independence-number bracket

Automated conjecturing programs scan collections of graphs for inequalities between invariants that no stored graph violates, then offer the survivors for proof or refutation. TxGraffiti, one such program, conjectured that every nontrivial connected graph $G$ satisfies $\alpha(G) \ge \bigl(a(G) + R(G)\bigr)/\Delta(G)$, where $\alpha$ is the independence number, $a$ the annihilation number, $R$ the residue, and $\Delta$ the maximum degree. Established only for two special families of graphs, the conjecture has otherwise remained open. The note proves the degree-sequence inequality $a \le \tfrac{\Delta+1}{2}W$, where $W$ is the Caro-Wei sum; the same inequality is known for the independence number in place of $a$. Combined with the classical lower bounds $\alpha \ge R$ and $\alpha \ge W$, it proves the conjecture for every connected graph of maximum degree at least three, and a direct argument settles maximum degree two; the conjecture fails only for the single edge, of maximum degree one. The inequality also brackets the independence number between the polynomial-time quantities $R$ and $a$, within a factor $(\Delta+1)/2$. The conjecture's bound is sharp, with equality attained, for instance, by the complete graph on four vertices.

math.CO↗

arXiv2026

Height functions on the $m \times n$ Miura-ori flip graph: degree sequence and diameter

The state space of an origami crease pattern forms a flip graph, whose vertices are the flat-foldable mountain-valley assignments and whose edges join assignments differing by a single face flip. For the $m \times n$ Miura-ori, the degree sequence and diameter of this graph are known only for two rows. Each assignment maps to an integer height function on the grid, under which a vertex's degree equals its number of local extrema. In this model the vertices of each degree up to five are counted by an explicit polynomial in $m$ and $n$, valid once both exceed a bound that grows with the degree, and the height functions realizing those degrees are described explicitly. A closed-form lower bound for the diameter holds for all $m$ and $n$, and the matching upper bound reduces to an extremal inequality for $1$-Lipschitz functions on the grid, recovering the two-row distance at $m=2$. Since each invariant is read from the extrema or height differences of a grid function, the same reduction applies to any flip-graph quantity expressible in those terms.

math.CO↗

arXiv2026

Sharp bounds between the saturation number and the harmonic index

The saturation number $\mu^*(G)$ of a graph $G$ is the minimum cardinality of a maximal matching, and $H(G)$ is its harmonic index. TxGraffiti conjectured in 2023 that $\mu^*(G) \le H(G)$ for every nontrivial connected graph $G$, and B{\i}y{\i}ko\u{g}lu refuted this by showing that the ratio $\mu^*(G)/H(G)$ can be made arbitrarily large. Restricting to trees bounds the ratio sharply. Every nontrivial tree $T$ satisfies $\mu^*(T) < \frac{3}{2} H(T)$, with the constant $3/2$ best possible. A complementary bound $H(G) < 4\mu^*(G)$ holds for every graph with an edge, so on a nontrivial tree the saturation number is pinned to $\frac{1}{4} H(T) < \mu^*(T) < \frac{3}{2} H(T)$, both constants best possible. The friendship graph $F_4$ is a smallest counterexample to the conjecture, on nine vertices, and the smallest tree counterexample is the subdivided star on eleven vertices. For each positive integer $m$ a family of graphs with $m$ hubs has ratio approaching $m+1$, while the conjecture holds whenever all vertices have equal degree. Both invariants arise in applications, the harmonic index as a molecular descriptor and the saturation number as a measure of adsorption inefficiency, and the bounds estimate the latter, which is NP-hard to compute, by the former, which is computable in linear time.

math.CO↗

arXiv2026

Architecture-Induced Recoverability Bias in Differentiable Symbolic Regression

Symbolic regression aims to recover closed-form expressions from numerical data, but in differentiable symbolic regression the recovered expression depends not only on the grammar but also on the fixed architecture through which variables are routed during training. This is relevant to signal-processing settings in which closed-form models and interpretable nonlinear structure are useful. This architecture-specific effect has rarely been isolated directly, because existing comparisons often vary architecture together with operator family, grammar, or search procedure. Three depth-3 architectures are compared across twenty-four operator--shape--leaf combinations, holding operator family, grammar, and training protocol fixed as far as possible while varying the variable-routing architecture. Recovery changes from $0/64$ to $64/64$ trials on the same target under an architecture-plus-native-training-protocol comparison. The best architecture on one target is the worst on another, and trees with two equal-depth subtrees fail in every configuration tested ($0/3{,}776$). As a proof-of-concept mitigation, a small architecture set is trained and the hardened expression with the lowest held-out RMSE is selected. On the jointly-run subset, this improves recovery from $34.4\%$ for the only architecture present in all three configurations to $50.1\%$. On a Shockley diode target, the validation selector recovers cases missed by that baseline architecture, which by itself recovers $0/32$ seeds. Since the jointly-run subset contains only three configurations, the selector result is evidence that validation-based architecture selection is promising, not a complete benchmark. These results support treating architecture as a measurable design variable that should be reported, stress-tested, and selected using held-out validation rather than fixed a priori.

cs.NE↗