arXiv subjects

Tigran Nersissian

Publications and source records attributed to Tigran Nersissian.

2 recordsLinked to original sources

arXiv2026

Diagonal Bases and Diagonal Periods of Elementary Cellular Automata

Which cellular-automaton diagonal families form bases in every finite window? For canonical polynomial lifts of elementary rules, two truth-table bits determine triangularity, and units on the matrix diagonal determine invertibility. Exactly 24 rules give universal binary bases; all remain universal over every modulus. Among triangular binary coordinate maps, the Pascal transform is uniquely characterized by converting OR convolution into pointwise multiplication, while increment becomes strict prefix summation. Explicit inverses and coordinate comparisons distinguish sparsity from evaluation cost. A Rule 30 polynomial construction gives Fibonacci bounds on interpolation order and prime-modulus periods. Exact additive periods anchor a finite census modulo two and three. These results separate all-window basis classification from optimization of a representation and from period patterns observed in finite windows.

nlin.CG↗

arXiv2026

Diagonal Periods and Newton Supports of Rules 30, 86 and 135

The single-seed Rule 30 cone has unbounded least diagonal periods in both directions. On the finite-support side, first contact fixes the least Newton index and an integer polynomial lift gives a Fibonacci ceiling. For reflected Rule 86, a backward map on periodic tail profiles retains the zero boundary and bounds periods independently of transients. At depth $m$, the Rule 30 period is at least $\lfloor m/2\rfloor+1$; for natural Rule 86 cuts through depth $m$, the largest period is at least $\tfrac12\log_2(m+2)$. This proves infinitely many reset/integration period doublings. The physical Rule 135 orbit is a translated complement of Rule 30 after its first step, yielding finite or cofinite supports. A separately initialized auxiliary recurrence has exact index-by-index stabilization. Residue tables and block formulas describe the remaining structure and permit evaluation from supplied representations. Construction and minimization remain separate costs.

nlin.CG↗