Essays on structure, culture, and systems
Paper XII. Minimal Kernel Dimension for Fixed-Control Balance-Loop Realizations of Rational Floor Maps. Preprint, August 2026. This paper introduces the minimal kernel dimension of an exact fixed-control balance-loop realization of a rational floor map. For the partial map x ↦ floor(Qx/d), defined on integers not divisible by d, the invariant κ(d,Q) measures the least relation-lattice [ ] The post…
Let L ⊆ {0, 1, , c}* be an infinite regular language, ordered genealogically, and let GL(w) denote the genealogical rank of w ∈ L. We ask whether there exists a sequence of nonnegative integer weights U = (Un)n≥0 such that GL(a1⋯am) = ∑i=1m aiUm−i (a1⋯am ∈ L). The companion paper [5, Theorem 6.1] proves [ ] The post Nonnegative Positionality in Regular Abstract NumerationSystems: Balance-Loop…
We study positionality for abstract numeration systems built from arbitrary infinite regular languages over finite ordered digit alphabets. Unlike the synchronous prefix-closed setting, residual languages need not support every continuation width, so fixed-width rank data live on partial state-width supports. We show that these supports are eventually periodic and derive an exact phase-local…
Let L be an infinite prefix-closed, right-prolongable regular language over a finite ordered alphabet of nonnegative integer digits. We ask when the lexicographic ranks of its fixed-width cross-sections, or more generally unit-spaced graded ranks, admit a normalized scalar positional representation VU(a1 ··· an) = ∑j=1n ajUn-j, U0 = 1. Positionality at one root propagates [ ] The post Intrinsic…
Let P = e1 ⋯ eL be the primitive maximal boundary block of a canonical positive linear recurrence sequence (PLRS) numeration system. Earlier local carry theory determines individual successor depths but not which such depths can occur consecutively. We solve this global realization problem. For each maximal-orbit phase i and width m, the state carry [ ] The post Global Carry Realization in…
For a canonical positive linear recurrence numeration system with threshold-reset legality automaton and maximal boundary block P = p1 pL, we give an exact local grammar for the reset-depth sequence produced by lexicographic succession. Every positive reset depth determines its predecessor s branch-point state and depth exactly (Theorem A); every carry-depth excursion collapses to depth 0 in [ ]…
Let L be a prefix-closed language over a finite ordered digit alphabet, and assume that at every reachable residual the legal next digits form an initial interval {0, 1, …, d(q)}. Write Cn = L ∩ Σn , C0 = 1, and let Rn(w) be the zero-based lexicographic rank of w ∈ L ∩ Σn. [ ] The post Rank-Positional Threshold Languages: ResetStructure, Census Rigidity, and Finite Reconstruction appeared first on…
Paper IV showed that every presentation of a canonical positive linear recurrence sequence (PLRS) numeration system determines a threshold-reset automaton, and that this automaton minimizesto the threshold-reset automaton of the canonical reduction. Left open there was the converse recognition problem: given an arbitrary minimal deterministic automaton over an ordered digitalphabet, with no…
Canonical positive linear recurrence sequence numeration systems admit a natural deterministic threshold-reset automaton for their legality languages. Paper IV showed that, after minimization, every presentation of a fixed system yields the automaton of the canonical reduction. The present paper studies the internal transition structure of that canonical automaton. We introduce anchored…
This paper develops an exact inverse theory for ordered carry traces on zero-prolongable prefix languages. It asks how much of an underlying legality language can be reconstructed when the reset depth between consecutive legal words is observed only through a possibly noninjective numerical decoder. The paper proves that every strict jump of the decoder reveals [ ] The post Reconstruction of…