Thinking In Structure

Category: Math

Mathematics

  • Shadowed Faces and Residue-Affine Dynamics in Cone-Monotone Integer Relations

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    This paper develops the exact arithmetic structure generated by shadowed bad faces in cone-monotone homogeneous integer equality relations. It identifies the first persistent boundary mechanism lying beyond finite Kleene convergence and shows that, in physical dimension two, this mechanism is governed by one-dimensional guarded residue-affine dynamics with a common expanding rational multiplier.

    The relations have the form Bλ = Aλ′, with nonnegative integer coefficient states and cone-monotone generator displacements. The analysis focuses on a bad closed ray whose canonical deterministic quotient is nonintegral. A rigidity theorem shows that every nontrivial such ray carries a unique common multiplier

    m / d,   with   m > d ≥ 2   and   gcd(m,d) = 1.

    A transverse height associated with the shadowed face is nondecreasing along every trajectory. Consequently, every infinite trajectory either escapes to unbounded transverse height or is eventually confined to a fixed height. The bounded-height branch admits a complete exact structural classification.

    At every fixed shadow layer, all nondeterminism arising from different factorizations along the face collapses to a single numerical-semigroup coordinate. Beyond an effectively computable conductor threshold, the resulting dynamics become a finite-control guarded residue-affine system of the form

    d n′ = m n + c,

    with finitely many branch constants and finite source-and-target residue guards. Thus apparently higher-dimensional factorization dynamics reduce exactly to a one-dimensional arithmetic survivor problem.

    The paper also gives a graph-directed base-d profinite description of the infinite flat survivor set. Finite arithmetic itineraries correspond to explicit residue cylinders modulo increasing powers of d, providing an exact connection between the geometric predecessor filtration and arithmetic congruence dynamics.

    A converse realization theorem shows that this arithmetic description is sharp. Every expanding finite-branch guarded common-multiplier residue-affine system satisfying the necessary congruence conditions can be realized exactly on a forward-invariant first-layer tail of a physical-dimension-two cone-monotone homogeneous equality relation. The construction can simultaneously preserve an integral full canonical quotient and produce a unique nonzero proper bad shadowed face.

    In the primitive factorization-independent case, the arithmetic specializes to the common-multiplier setting associated with the Lagarias–Sloane approximate-multiplication problem. For the stopped primitive construction, the complete Kleene predecessor filtration is linearly conjugate to a forbidden-residue survivor filtration. At depth ℓ, the surviving states form explicit residue cylinders modulo dℓ.

    The result therefore isolates persistent shadowing as an exact arithmetic boundary rather than merely providing an isolated Collatz-like example. The paper does not claim a general decision procedure for the resulting survivor problem. Instead, it identifies precisely which arithmetic dynamics are forced by a bad shadowed face and proves that the same arithmetic class can be realized conversely inside the geometric model.

    This classification supplies the arithmetic component used in the subsequent two-dimensional completeness theorem, where guarded common-multiplier survivor dynamics are shown to capture all residual non-Presburger complexity in the full physical-dimension-two cone-monotone equality class.

    Paper XVII of the Canonical Numeration and Positionality Research Programme.

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    Related companion papers:

    • Closed-Face Integrality and Exact Conditional Termination for Cone-Monotone Integer Equality Relations
    • Beyond Finite Kleene Convergence: Integral Cores and a Collatz Barrier for Cone-Monotone Integer Relations
    • Arithmetic Completeness for Two-Dimensional Cone-Monotone Integer Relations

  • Arithmetic Completeness for Two-Dimensional Cone-Monotone Integer Relations

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    This paper develops an exact structural theory of weakest nontermination for the full class of two-dimensional cone-monotone homogeneous integer equality relations of the form Bλ = Aλ′, with nonnegative integer coefficient states, pointed source cone and nonzero source columns.

    The full canonical deterministic quotient gives the first structural dichotomy. If it is nonintegral, an exact integral-core compression preserves weakest nontermination while strictly reducing physical dimension. In physical dimension two, this reduces the problem to dimension at most one, where weakest nontermination is effectively Presburger-definable. Nonintegral full quotients therefore create no genuinely new two-dimensional arithmetic obstruction.

    For the remaining integral-full-quotient branch, the paper removes the nonnegativity constraint and studies the resulting signed predecessor sequence as the forward-domain sequence of an additive linear relation on Zn. Finite attainment is characterized exactly: the signed sequence stabilizes after finitely many steps if and only if the canonical rational deterministic quotient is integral over Z. In the integral case, the stable signed lattice is effectively computable.

    Intersecting this stable lattice with the nonnegative orthant produces an affine semigroup. Normalization and conductor theory, combined with cone monotonicity, then give an exact deep-interior saturation theorem: beyond an effectively computable translated cone, weakest nontermination is precisely membership in the stable signed lattice. The entire deep interior is therefore effectively Presburger-definable.

    In physical dimension two, the complement of the conductor region decomposes into two bounded facet strips and a finite corner. Good strips have effectively Presburger weakest-nontermination sets. Every remaining bad strip reduces exactly, after a finite prefix, to a finite-branch guarded residue-affine system with one common expanding multiplier:

    d n′ = m n + c

    where m > d ≥ 2 and m and d are coprime.

    This yields an unconditional arithmetic reduction for the entire two-dimensional cone-monotone equality class. Every weakest-nontermination set is a finite union of effectively Presburger pieces and inverse images of survivor sets for guarded common-multiplier residue-affine systems. Conversely, every system in this residual arithmetic class is realized exactly on a forward-invariant first-layer tail of a two-dimensional cone-monotone equality relation with integral full canonical quotient.

    Consequently, universal semilinearity and effective Presburger synthesis for two-dimensional geometric weakest-nontermination sets are equivalent to the corresponding problems for the residual arithmetic survivor class. Pointwise membership is mutually reducible by finite disjunctive reductions.

    The result isolates the complete residual source of non-Presburger complexity in physical dimension two. Nonintegral ambient quotients collapse to lower-dimensional Presburger cases; the deep interior and all good boundary regions are tame; and every remaining arithmetic obstruction is forced onto a bad shadowed face and reduces to one-dimensional guarded common-multiplier dynamics. Within the stated cone-monotone equality class, no further two-dimensional geometric case remains to be classified.

    Paper XVIII of the Cone-Monotone Integer Relations research programme.

    Related companion papers:

    • Closed-Face Integrality and Exact Conditional Termination for Cone-Monotone Integer Equality Relations
    • Beyond Finite Kleene Convergence: Integral Cores and a Collatz Barrier for Cone-Monotone Integer Relations
    • Shadowed Faces and Residue-Affine Dynamics in Cone-Monotone Integer Relations: Global Shadow Normal Forms, Kleene Conjugacy, and a Lagarias-Sloane Barrier

  • Beyond Finite Kleene Convergence: Integral Cores and a Collatz Barrier for Cone-Monotone Integer Relations

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    Beyond Finite Kleene Convergence: Integral Cores and a Collatz Barrier for Cone-Monotone Integer Relations

    This paper studies exact weakest nontermination for homogeneous nondeterministic integer relations of the form Bλ = Aλ′ under a cone-monotonicity condition.

    A companion paper characterized exactly when the descending predecessor sequence stabilizes after finitely many steps. Paper XI goes beyond that boundary. It identifies a strictly larger, effectively recognizable class for which the exact weakest nontermination set remains computable even when finite Kleene convergence fails.

    The main construction is a canonical core-face decomposition. When the full-cone deterministic quotient is nonintegral, the relation is compressed to its exact forward-integral coefficient core, retaining the lattice congruences required by infinite computations while strictly reducing physical dimension. When the ambient quotient is integral but nonintegral proper closed faces remain, shadow-free bad faces are isolated recursively and finite convergence is restored on their complement.

    For every core-face-decomposable system, the recursion terminates after at most the physical dimension and flattens to a finite exact normal form. The weakest nontermination precondition is therefore effectively definable in existential Presburger arithmetic and is effectively semilinear. The earlier finite-Kleene class appears exactly as the depth-zero case.

    The paper also develops explicit quantitative bounds for the size of the decomposition tree, the number and size of Hilbert-basis generators introduced during core reduction, and the syntactic size of the resulting Presburger formulas. An exhaustive exact census of 4,608 small two-dimensional systems shows that the new core-reduction and shadow-free mechanisms solve 646 of the 1,320 systems rejected by finite Kleene convergence, reducing the unresolved fraction by 48.94%.

    The boundary of the method is arithmetically significant. In physical dimension two, a shadowed bad face can already realize the integer map n ↦ floor(4n/3) on the first adjacent lattice layer. Exact nontermination for a specific explicit relation is thereby equivalent to an unresolved generalized-Collatz reachability problem. Physical dimension two is minimal for this obstruction.

    Main results:

    • canonical finite core-face decomposition;
    • exact dimension-reducing integral-core compression preserving lattice congruences;
    • relative finite convergence outside shadow-free bad faces;
    • finite exact normal form for the weakest nontermination set;
    • effective existential-Presburger and semilinear representation;
    • explicit decomposition-tree, Hilbert-basis and formula-size bounds;
    • complete classification in physical dimension one;
    • exhaustive 4,608-system computational census;
    • a dimensionally minimal generalized-Collatz barrier in physical dimension two.

    Verification material: The accompanying repository contains the exact-arithmetic verifier used to reproduce the 4,608-system census reported in Section 15. The computational results are validation data only; no theorem depends on the census.

  • Fixed-Start Nontermination in Homogeneous Nonnegative Equality Loops

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    Fixed-Start Nontermination in Homogeneous Nonnegative Equality Loops: Continuation Cores, Arithmetic Obstructions, and Decidable Frontiers develops a structural and arithmetic theory for deciding whether a prescribed initial state in a homogeneous nonnegative equality loop can generate an infinite integer trajectory. The problem arises naturally from balance-loop representations in numeration systems, but also belongs to the broader theory of integer linear-constraint loops and termination.

    The paper introduces a span-relaxed continuation core, an effectively computable rational polyhedral cone that contains every infinite real or integer trajectory and stabilizes after at most rank(A) + 1 iterations. Within this core, failure of an exact integer successor is classified into three distinct mechanisms: a positive-cone obstruction, a lattice obstruction, or an affine-semigroup hole. A Hilbert-basis formulation also gives finite certificates for total viability of the continuation core.

    The principal decidability result proves fixed-start nontermination decidable whenever the intrinsic physical continuation rank is at most one, regardless of the ambient matrix dimensions. In the difficult expanding case, a maximal-slope Apéry construction gives uniformly efficient predecessors; once the viability gcd stabilizes, numerical-semigroup conductors become uniformly bounded, forcing exact finite stabilization of the integer viability filtration. The paper also proves decidability whenever the coefficient continuation dimension is at most two.

    Deterministic continuation cores are further connected to the Positivity Problem for integer linear recurrence sequences through an invariant orbit lattice. The resulting framework separates linear consistency, positive geometry, lattice congruences and semigroup holes, and identifies the first genuinely relational untreated case as a two-dimensional physical continuation core with one rational factorization degree of freedom.

  • Closed-Face Integrality and Exact Conditional Termination for Cone-Monotone Integer Equality Relations

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    This paper gives an exact characterization of conditional termination for a class of nondeterministic homogeneous integer equality relations over nonnegative integer coefficient vectors.

    The central result identifies precisely when the descending predecessor sequence reaches the weakest nontermination precondition after finitely many steps. Under pointedness of the source cone, nonzero source generators and cone-monotone displacement, finite convergence occurs if and only if the canonical deterministic quotient associated with every closed face is integral over the integers. Equivalently, every closed-face quotient must have an integer characteristic polynomial.

    The proof separates the problem into three structural layers. On closed faces, continuation lattices stabilize exactly when an integral serial lattice exists, which is equivalent to integrality of the deterministic quotient. Nonclosed faces are handled by automatic ascent through the face map until a closed face is reached. Deep-rescue arguments and a finite-tube induction then convert these local arithmetic conditions into a global uniform bound on mortality depth.

    The theorem is effective. The closed faces can be enumerated, their deterministic quotients computed by exact rational linear algebra, and the integrality condition checked directly. When the criterion holds, repeated Presburger predecessor computation eventually produces the exact weakest nontermination precondition.

    The paper also develops a quantitative theory of convergence. For a closed face with integral serial lattice K, the continuation-lattice filtration stabilizes after at most the base-2 logarithm of the lattice index. Propagating conductor, residue and tube bounds through the proof gives a computable global upper bound on predecessor convergence. Conversely, an explicit fixed-dimensional family has exact mortal depth m + 1 and shows that predecessor convergence can grow exponentially in the binary input length even when the deterministic quotient is trivial.

    The resulting class is distinct from previously known exact conditional-termination classes such as octagonal relations and finite-monoid affine systems. It permits genuinely nondeterministic relations, arbitrary physical dimension and infinite ambient matrix monoids. The paper also places the result within the wider literature on linear-constraint loops, affine counter systems and deterministic linear-loop termination.

    Two geometric figures illustrate the closed-face mechanism and the finite-tube induction. A companion exact-arithmetic verification package reproduces the principal examples, the long-transient family and the exhaustive small-system computational checks reported in the paper.

    Topics: conditional termination, weakest nontermination preconditions, integer equality relations, affine semigroups, cone monotonicity, closed faces, lattice stabilization, Presburger arithmetic, nondeterministic systems and termination analysis.

  • Complete Noncoprime Classification of Minimal Kernel Dimension for Rational Floor Maps

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    Paper XIII completes the noncoprime classification of minimal kernel dimension for rational floor maps in the fixed-control balance-loop model.

    Let d = gm and q = gn, where g ≥ 2, 1 ≤ n < m, and m and n are coprime. The paper determines exactly when the partial floor map Fd,q(x) = floor(qx/d), with multiples of d deleted, admits an exact proper realization of kernel dimension 2, 3 or 4.

    The complete classification is remarkably rigid. The exceptional map (d,q) = (4,2) has kernel dimension 2. Kernel dimension 3 occurs on the co-affine boundary n = m − 1, apart from that exception, and on the scaling boundary n = 1 when either g = 2 or m = 3. Every genuinely interior reduced noncoprime case, 1 < n < m − 1, has kernel dimension 4. Thus the universal kernel-four upper bound is attained throughout the interior.

    The principal new result is this interior maximality theorem. After bulk compression, any hypothetical kernel-three realization has four positive-control selectors. The proof classifies all such architectures according to the number of pure-control selectors. Two-pure architectures are reduced to two-generated numerical semigroups and excluded using Frobenius symmetry, Apéry coordinates, mechanical-word return structure and checkerboard substitutions. One-pure architectures are eliminated using residue rigidity, flat and rising edge structure, and bidirectional target shifts. Zero-pure architectures reduce to unimodular lattice paths and finite orthant obstructions.

    The paper also gives an explicit five-column kernel-three realization for the previously unresolved co-affine boundary n = m − 1, together with computational verification of representative constructions and the finite lattice classifications used in the proof.

    Conceptually, the invariant has a natural algebraic interpretation. For a fixed rank-two source/control configuration, kernel dimension equals the rank of its integer relation lattice, equivalently the codimension of the associated toric ideal. The problem can therefore be viewed as determining the minimum relation-lattice freedom, or restricted toric codimension, compatible with exact labeling of an entire fixed-control fibre by a rational floor map.

    The result sits at the intersection of integer factorization, affine and numerical semigroups, toric geometry, Christoffel and mechanical words, and mixed-integer formulation complexity. General floor-map representability is easy in broader Presburger and mixed-integer models; the difficulty here comes from the much stronger requirement that every nonnegative factorization on a fixed-control slice must produce the correct target.

  • Minimal Kernel Dimension for Fixed-Control Balance-Loop Realizations of Rational Floor Maps

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    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 dimension, equivalently the least toric codimension, required by any proper realization.

    The paper first proves a rigidity theory for minimal presentations. Every zero-control column is forced to lie in the bulk direction (d,Q), and every kernel-minimal realization contains exactly one bulk column. This reduces the infinite arithmetic realization problem to a finite selector-fibre problem.

    For coprime parameters the invariant is classified completely. Kernel dimension 2 occurs exactly when Q is congruent to +1 or −1 modulo d; every other coprime case has kernel dimension 3.

    The noncoprime case exhibits genuinely new behaviour. In particular, κ(8,2) = 3 while κ(12,3) = 4, even though both maps have the same reduced slope 1/4. The scaling family (d,q) = (mg,g), with m ≥ 3 and g ≥ 2, is classified completely: the kernel dimension is 3 when m = 3 or g = 2, and 4 when m ≥ 4 and g ≥ 3.

    The paper then proves a general kernel-4 realization theorem for arbitrary noncoprime parameters d = gm and q = gn with gcd(m,n) = 1. Combined with the coprime classification and the scaling constructions, this yields the sharp universal bound

    κ(d,Q) ≤ 4

    for every d ≥ 2 and every positive numerator Q. Since κ(12,3) = 4, the universal bound is best possible.

    The lower bounds combine lattice saturation, signed-residual arguments, residue replacement, numerical semigroups, Apéry coordinates and rational mechanical words. The upper bounds are supplied by explicit fixed-control constructions. The results show that minimal factorization complexity is universally bounded, but is not determined by reduced rational slope alone; it also depends on the arithmetic structure of the excluded congruence class.

    The paper also identifies a toric interpretation of the invariant: κ(d,Q) is the minimum height of the toric ideal associated with the source matrix, taken over all exact fixed-control realizations of the prescribed floor map.

  • Nonnegative Positionality in Regular Abstract NumerationSystems: Balance-Loop Normal Forms and Viability Filtrations

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    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 that, after an effectively computable support preperiod τ and period p, the periodic tail can be stacked into a macrostate Xk ∈ ℤD satisfying

    𝒜Xk + ℬXk+1 = 0.

    It also shows that the nonnegative problem already contains the Positivity Problem for linear recurrence sequences. Here we identify an exact normal form for that remaining nonnegative problem.

    We define a proper homogeneous balance loop to be a system

    Bλn = Aλn+1,    λn ∈ ℕr,

    where A and B are nonnegative integer matrices and every column ai of A is nonzero, together with a fixed initial vector λ0. Properness gives ‖ai‖1 ≥ 1, which in turn makes the successor relation effectively finitely branching.

    Our main theorem proves an effective equivalence between nonnegative positionality for infinite regular languages and nontermination of proper homogeneous balance loops. The forward reduction passes from the periodic descriptor relation to its nonnegative integer transition monoid, uses a Hilbert basis, and adds a budget coordinate to remove zero-source generators. The reverse reduction is constructive: a regular language built from radix fillers and finitely many repeated lexicographic equality gadgets realizes any balance loop.

    The normal form has immediate consequences. Proper balance loops are effectively finitely branching, every trajectory has an effective exponential envelope, and nontermination is equivalent to survival at every finite horizon. Consequently, non-positionality of a regular abstract numeration system is recursively enumerable.

    The result isolates the positivity barrier as an infinite nonnegative integer viability problem within a restricted homogeneous class of linear-constraint loops.

  • Affine Positionality in Abstract Numeration Systems: Phase Dynamics, Decidability, and a Positivity Barrier

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    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 affine characterization of positionality in terms of active lexicographic cylinders. After lifting one support period, the full affine problem becomes a finite linear descriptor relation.

    Over the rationals, affine positionality is decidable by finite-dimensional viability and every realization admits a C-finite representative. Over the integers, the periodic tail defines a viability subgroup for an integer linear relation. We identify this subgroup with the constant part of an explicit S-saturated polynomial module, where S consists of polynomials with unit constant term. Primary decomposition over ℤ[z] therefore gives an effective decision procedure and an effective C-finite replacement.

    The nonnegative problem encounters a substantially stronger barrier. We construct, from any integer linear recurrence sequence (an), an infinite regular language whose genealogical abstract numeration system is positional with nonnegative integer weights if and only if an ≥ 0 for every n. Thus LRS Positivity reduces effectively to general regular-ANS positionality. The results separate the finite affine and integral layers from the remaining infinite sign constraint and show that any decision procedure for general regular-ANS positionality would in particular decide LRS Positivity.

  • Intrinsic Affine Positionality in Ordered Regular Languages: Fixed-Width and Genealogical Rank

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    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 to every live residual state. There are residual translations λq(n) such that

    Rq(n)(w) = VU(w) – λq(n),

    and this residual affine form is equivalent to local tiling equations. If aq is the least legal digit, p(q) = δ(q,aq), and adjacent digits have targets r,s, then

    λq(n+1) = aqUn + λp(q)(n),    Un = Nr(n) + λr(n) – λs(n).

    With at least one branching residual state, any genuine adjacency determines the only possible normalized basis; the basis and all residual translations are then unique and integral whenever compatibility holds.

    The translation subsystem has a canonical finite quotient. Two states are equivalent when their infinite least-digit threshold itineraries agree. If the minimal live residual DFA has s states and this lower-itinerary quotient has b classes, census dynamics require s coordinates while translations require only b. The resulting (s + b)-dimensional realization has one common-translation gauge mode invisible to positional value. Hence

    rU ≤ s+b-1,    rβ(q) ≤ s+b,

    where r denotes Hankel rank and β(q) = λq is the residual grading boundary. Both recurrence-complexity bounds are sharp for every 1 ≤ b ≤ s.

    RU(s,b) = s+b-1,    Rβ(s,b) = s+b.

    The same finite realization gives a compatibility-recognition bound of s + b − 2: if compatibility fails, some genuine adjacency detects the failure by that continuation depth.

    This recognition bound is not universally sharp for genuine integer digit systems. At s = b = 4, the exact integer recognition radius is 5 rather than the universal bound 6. In the corresponding rational-threshold affine relaxation the radius is 6. Thus the one-step improvement is caused by an arithmetic integrality obstruction rather than by a smaller ambient realization dimension.

    Equivalently, finite recognition complexity has two distinct sources: linear realization complexity and the integer lattice imposed by the digit thresholds. This motivates separate integer and rational recognition-radius problems beyond the universal s + b − 2 bound.

    Thus maximal positional recurrence complexity splits into census complexity plus lower-itinerary complexity, with one dimension removed by the gauge, while finite recognition can be strictly shorter because of additional arithmetic constraints.

    Fixed-width and genealogical positionality are root-specific specializations of the same intrinsic structure. Fixed-width rank occurs exactly when the chosen root boundary is zero. Genealogical rank occurs exactly when it equals the cumulative census

    Bn = ∑m<n |L ∩ Σm|,

    in which case the least root digit is 1 and

    U(z) = C(z) / [(1-z)g*(z)].

    The paper also develops a twisted-coboundary formulation of residual transfer, giving a root-change law and a path-independent description of affine boundary transport across the automaton.

    For positive bases, both canonical root conventions have Perron-scale growth, while general intrinsic boundaries may support strictly faster positive growth.