Canonical Numeration Systems Research Overview

This research programme develops an exact structural, dynamical, probabilistic, inverse and positionality theory of canonical numeration systems, carry propagation, ordered regular languages and abstract numeration systems.

Papers I–IVd concern canonical positive linear recurrence sequence numeration systems, a broad class generalising Zeckendorf representations. Papers I–IV establish canonical reduction, exact carry distributions, boundary maps, intrinsic boundary sequences, inverse reconstruction, classification of equivalent presentations, language invariance and automaton minimization. Paper IVa determines the internal transition structure of the canonical automaton and characterizes aperiodicity and star-freeness through an anti-Lyndon condition on the primitive maximal boundary block. Paper IVb solves the converse recognition problem, giving necessary and sufficient intrinsic conditions for a pointed minimal deterministic automaton to arise from a canonical PLRS and reconstructing the unique canonical coefficient vector from its transition table. Paper IVc develops the local successor dynamics of the same automata, showing that positive carry depths have finite deterministic backward skeletons together with unbounded periodic lifts, and deriving exact consequences for state-dependent carry-cost observability. Paper IVd solves the complementary global realization problem: it determines exactly which finite carry-depth traces can occur consecutively, gives a bounded-width factor criterion, develops matched-scale phase visibility and connects global root domination to the anti-Lyndon condition.

Paper V extends the exact distributional and inverse carry theory beyond purely periodic PLRS boundaries to canonical greedy beta numeration. It derives complete finite-width carry-propagation distributions, limiting carry laws, labelled boundary-cost distributions and inverse reconstruction theorems for general canonical beta systems. For eventually periodic boundaries, including Parry systems, the limiting probability generating functions have explicit rational forms.

Paper VI develops a finite-sample reconstruction theory for carry propagation in canonical beta numeration. Building on the exact inverse results of Paper V, it determines how much of the canonical boundary can be recovered from finitely many carry observations, gives a certified reconstruction algorithm and proves matching upper and lower bounds on the sample complexity.

Paper VII passes from numeration-specific carry distributions to ordered numerical carry traces on arbitrary zero-prolongable ordered prefix languages. It determines exactly which part of the underlying ancestry structure survives a possibly noninjective depth decoder, characterizes universal reconstruction through the gcd of the decoder’s visible reset depths, and derives an optimal finite observation horizon with matching lower bounds.

Paper VIII develops a complementary structural theory for rank-positional threshold languages and places the programme directly within the broader positionality problem for abstract numeration systems. It asks when fixed-width lexicographic rank can be represented positionally using the language’s own width census as the place-value sequence. Under the threshold hypothesis, the problem is solved completely: the global rank identity is equivalent to a local root-reset condition in the minimal residual automaton. The resulting languages are classified by a single intrinsic boundary whose tails determine residual states, whose eventual periodicity is equivalent to regularity, and whose entries form both a canonical reset-code inventory and an invertible triangular transform of the census. The paper also proves optimal finite census rigidity and determines the exact cyclotomic defect between automaton-state complexity and scalar recurrence complexity.

Paper IX removes the census-positional restriction and develops an intrinsic affine theory of positional rank for prefix-closed, right-prolongable ordered regular languages. Positionality at one root propagates to every live residual state through state-dependent translations, and the resulting residual affine structure is characterized exactly by local cylinder-tiling equations. The translation dynamics admit a canonical lower-itinerary quotient. If the minimal live residual automaton has s states and the quotient has b classes, the intrinsic positional basis has Hankel rank at most s + b – 1, while residual boundaries have rank at most s + b; both recurrence-complexity bounds are sharp for every 1 <= b <= s. The paper also gives a universal finite compatibility-recognition bound of s + b – 2, but shows that this bound is not always sharp over integer digit systems: at s = b = 4 the exact integer recognition radius is 5 rather than 6, while the corresponding rational-threshold affine relaxation attains radius 6. Thus finite recognition complexity can exhibit an arithmetic integrality defect even when the ambient affine realization bound is sharp. Fixed-width and genealogical positionality emerge as two root-boundary specializations of the same intrinsic structure.

Paper X extends affine positionality to arbitrary regular abstract numeration systems, where residual states need not support every continuation width. It proves eventual periodicity of the state-width support pattern, derives an exact phase-local affine characterization and reduces the infinite affine problem to a finite linear descriptor relation. Rational affine positionality is decidable by finite-dimensional viability. Integer affine positionality is also decidable, through an effectively computable viability subgroup described by $S$ -saturation and primary decomposition over $\mathbb Z[z]$ . Finally, the paper gives an effective reduction from the Positivity Problem for integer linear recurrence sequences to standard regular-ANS positionality with nonnegative integer weights. Thus any general decision procedure for regular-ANS positionality would in particular decide LRS Positivity.

Paper XI identifies an exact normal form for the remaining nonnegative positionality problem and develops a structural theory of the resulting viability dynamics. It proves an effective equivalence between nonnegative positionality for infinite regular languages and nontermination of proper homogeneous balance loops. The forward reduction converts the periodic descriptor relation of Paper X into a finitely generated nonnegative transition monoid, uses a Hilbert basis and a budget coordinate to enforce properness, and reduces the problem to a fixed-start balance loop. The reverse reduction constructively realizes every proper homogeneous balance loop by an infinite regular language. Finite-horizon viable states form an effectively computable descending filtration of affine semigroups, giving finite certificates for non-positionality and placing regular-ANS positionality in Pi-0-1. In one physical dimension the paper derives an autonomous gcd recurrence, completely characterizes the nonproportional terminating regime and proves common exponential asymptotics for the least positive viable state and the conductor. For two generators it proves eventual finite-channel purification into arithmetic tails governed by a finite phase map. An explicit three-generator example shows that this purification fails in higher factorization dimension through persistent moving boundary holes.

Paper XII turns the balance-loop normal form into a quantitative realization theory. It introduces the minimal kernel dimension κ(d,Q), the least relation-lattice dimension, equivalently the least toric codimension, among exact fixed-control balance-loop realizations of a prescribed rational floor map. It proves bulk rigidity and reduces exact realization to a finite selector-fibre problem. For coprime parameters it gives a complete classification: kernel dimension 2 occurs exactly when Q is congruent to +1 or −1 modulo d, and kernel dimension 3 otherwise. In the noncoprime case it proves that reduced slope alone does not determine realization complexity: κ(8,2) = 3 while κ(12,3) = 4, although both maps have reduced slope 1/4. It completely classifies the scaling family (d,q) = (mg,g), gives an explicit kernel-4 construction for arbitrary noncoprime parameters d = gm and q = gn with gcd(m,n) = 1, and proves the sharp universal bound κ(d,Q) ≤ 4 for every d ≥ 2 and every positive numerator Q. Since κ(12,3) = 4, this global bound is best possible.

Paper XIII completes the reduced noncoprime classification of minimal kernel dimension. Writing d = gm and q = gn with g ≥ 2, 1 ≤ n < m and gcd(m,n) = 1, it proves that the exceptional map (4,2) has kernel dimension 2; the co-affine boundary n = m − 1 has kernel dimension 3 apart from that exception; the scaling boundary n = 1 has kernel dimension 3 exactly when g = 2 or m = 3; and every genuinely interior case 1 < n < m − 1 has kernel dimension 4. Thus the universal kernel-four ceiling from Paper XII is attained throughout the reduced noncoprime interior. The proof classifies hypothetical kernel-three architectures by their number of pure-control selectors, using numerical-semigroup and Apéry rigidity, rational mechanical-word structure, residue replacement, bidirectional target shifts and unimodular lattice obstructions. The paper also identifies minimal kernel dimension with minimum relation-lattice rank, equivalently restricted toric codimension, under the exact fixed-control labeling constraint.

Paper XIV returns to the balance-loop decision problem itself. It introduces a span-relaxed continuation core containing every infinite real or integer trajectory and proves that this core is effectively reached after finitely many steps. Inside the core, failures of exact integer continuation split into positive-cone, lattice and affine-semigroup obstructions. The paper proves fixed-start nontermination decidable whenever the intrinsic physical continuation rank is at most one or the coefficient continuation dimension is at most two, and identifies the first genuinely relational untreated core as the case of two physical continuation dimensions with one rational factorization degree of freedom.

Paper XV develops an exact conditional-termination theory for cone-monotone homogeneous integer equality relations. It associates a canonical deterministic quotient with every closed face and proves that finite predecessor convergence occurs exactly when every such quotient is integral over the integers, equivalently when every closed-face characteristic polynomial has integer coefficients. Nonclosed faces are handled by congruence-preserving ascent and finite-tube induction. The resulting weakest nontermination precondition is exactly computable in Presburger arithmetic, although predecessor stabilization can require exponentially many iterations in the binary input length.

Paper XVI goes beyond finite predecessor convergence and identifies a strictly larger exact class. It introduces a canonical core-face decomposition. Nonintegral full-cone arithmetic is compressed to the exact forward-integral coefficient core using lattice congruences and Hilbert bases, with strict reduction of physical dimension. When the ambient quotient is integral but proper closed faces remain nonintegral, shadow-free bad faces are isolated recursively while finite convergence is restored on their complement. Every core-face-decomposable system has a finite exact normal form, so its weakest nontermination precondition is effectively existential Presburger and semilinear. The paper also gives explicit bounds on decomposition-tree size, Hilbert-basis growth and Presburger formula size. The first excluded geometric mechanism is already arithmetically significant: a shadowed bad face in physical dimension two can realize the map n ↦ floor(4n/3) on the first adjacent lattice layer, producing an explicit generalized-Collatz barrier.

Paper XVII develops the exact structural theory of persistent shadowing. For a bad closed ray in physical dimension two, it derives a global shadow normal form and proves that every bounded-height infinite trajectory is eventually flat. Face-factorization nondeterminism then collapses to a single numerical-semigroup coordinate, and beyond an effective conductor threshold the dynamics become a finite-control guarded residue-affine system with one common expanding rational multiplier m/d. The resulting survivor set admits a graph-directed base-d profinite description. Conversely, every expanding finite-branch guarded common-multiplier system satisfying the required congruences is realized exactly on a forward-invariant first-layer tail of a two-dimensional cone-monotone equality relation with integral full quotient and a unique bad shadowed ray. In the primitive stopped case the entire Kleene filtration is conjugate to a forbidden-residue filtration, placing Lagarias-Sloane approximate-multiplication dynamics directly on the boundary of the geometric theory.

Paper XVIII closes the two-dimensional geometric classification. For integral full quotient it proves an exact finite-attainment criterion for the signed predecessor domain, constructs the stable signed lattice, and combines affine-semigroup conductor theory with cone monotonicity to obtain an exact Presburger deep-interior core. Every infinite trajectory avoiding this core eventually transfers to one of the two facet strips. Good strips are effectively Presburger, while every bad strip reduces, after finite control and numerical-semigroup compression, to the same guarded common-multiplier residue-affine arithmetic isolated in Paper XVII. The result is then made unconditional by the integral-core descent of Paper XVI: a nonintegral full quotient compresses exactly to physical dimension at most one and is therefore effectively Presburger. Consequently weakest nontermination for the full physical-dimension-two cone-monotone equality class is arithmetically complete for the guarded common-multiplier survivor class. Universal semilinearity and effective Presburger synthesis are equivalent for the geometric and arithmetic classes, while pointwise membership is mutually reducible by finite disjunctive reductions.

The programme builds on generalized Zeckendorf theory and on the extensive literature concerning linear and abstract numeration systems, greedy representations, normalization, automata, odometers, prefix codes, carry propagation and positionality developed by Christiane Frougny, Valérie Berthé, Michel Rigo, Jacques Sakarovitch, Koloğlu, Kopp, Miller, Wang and others. Earlier work established finite-state normalization and addition, regular-language and automata descriptions, language-tree models of successor propagation, cylinder descriptions of carry-level sets, exact aggregate combinatorial identities and broad amortized carry results.

The present programme develops complementary results concerning complete finite-width carry distributions, labelled boundary-cost maps, intrinsic boundary sequences, inverse reconstruction, classification of equivalent presentations, language invariance, automaton minimization, local and global successor dynamics, finite-sample recovery, reconstruction from coarsened ordered traces, structural reconstruction of rank-positional threshold languages, intrinsic affine positionality, affine decidability in arbitrary regular abstract numeration systems, exact balance-loop normal forms for nonnegative positionality, quantitative minimization of balance-loop factorization complexity, and structural decision theory for homogeneous nonnegative equality loops. Papers XII and XIII establish a sharp universal toric-codimension bound for rational floor maps and complete the reduced noncoprime classification. Paper XIV develops continuation-core reductions and intrinsic low-dimensional decidability frontiers for fixed-start balance-loop nontermination. Paper XV identifies an exact closed-face integrality criterion for finite conditional-termination convergence. Paper XVI extends exact weakest-nontermination computation beyond finite Kleene convergence through integral-core compression and recursive face decomposition. Paper XVII determines the exact arithmetic generated by persistent shadowing and proves a converse realization theorem for guarded common-multiplier residue-affine dynamics. Paper XVIII then closes the physical-dimension-two cone-monotone theory: nonintegral ambient quotients collapse to lower-dimensional Presburger cases, while the integral branch decomposes exactly into a semilinear interior and boundary systems belonging to the arithmetic class isolated in Paper XVII.

Together, Papers I-IV establish the equivalence between the place-value sequence, intrinsic boundary structure, canonical reduction, legality language and minimal deterministic automaton of a canonical PLRS. Papers IVa-IVd develop the internal automata and successor dynamics of that structure. Paper V extends the distributional and inverse framework to canonical greedy beta numeration, Paper VI establishes statistically optimal finite-sample recovery, and Paper VII extracts the general inverse mechanism for coarsened ordered carry traces. Paper VIII solves the census-positional threshold case of regular-language positionality. Paper IX develops the intrinsic affine theory for synchronous ordered regular languages, and Paper X passes to arbitrary regular abstract numeration systems, proving rational and integer affine decidability while identifying LRS Positivity as an obstruction to the general nonnegative problem. Paper XI gives an exact normal form for that remaining nonnegative problem through proper homogeneous balance-loop nontermination. Papers XII and XIII develop and complete the quantitative realization theory for rational floor maps. Paper XIV returns to fixed-start balance-loop nontermination through continuation-core reductions. Papers XV-XVIII then form a complete cone-monotone termination arc: closed-face integrality characterizes finite predecessor convergence; integral-core compression and face recursion extend exact computation beyond finite convergence; persistent shadowing is reduced exactly to guarded common-multiplier arithmetic; and the full physical-dimension-two geometric problem is proved arithmetically complete for that residual one-dimensional survivor class.

Papers I-XVIII, including the companion Papers IVa-IVd, form the completed current research programme. Papers I-IVd constitute the canonical-PLRS structural and successor-dynamics strand; Papers V-VII develop distributional and inverse reconstruction beyond PLRS; Papers VIII-XIII develop the positionality and realization strand; and Papers XIV-XVIII form the termination and arithmetic-completeness strand. The final strand progresses from continuation-core reductions and low-dimensional decidability frontiers, through exact closed-face integrality and integral-core descent, to the classification of persistent shadow arithmetic and an unconditional arithmetic-completeness theorem for the full physical-dimension-two cone-monotone equality class.

Central Questions

When a representation is incremented, the successor operation rewrites a terminal block of digits. Its length is the carry-propagation depth. The number of coordinates that actually change and the total magnitude of the digit changes are separate carry-cost statistics. The sequence of successive carry events also has a nontrivial local dynamical structure.

The programme asks three linked questions:

Forward problem: Given a numeration system or ordered legality language, what are the exact finite, limiting and local dynamical properties of its carry propagation and carry costs?

Inverse problem: Given carry distributions, finite carry samples or ordered coarsened carry traces, how much of the underlying boundary, numeration system or legality language can be reconstructed?

Structural, positionality and decision problem: When does an ordered regular language admit a scalar positional realization of its lexicographic or genealogical rank? How much of positionality can be characterized intrinsically from residual automata and local rank structure? Which parts of the problem are decidable over rational, integer and nonnegative weights, and what is the exact computational structure of the resulting balance-loop nontermination problem? Papers X-XVIII progressively isolate this question. Paper X identifies the LRS Positivity barrier; Paper XI proves effective equivalence with nontermination of proper homogeneous balance loops; Papers XII and XIII quantify and classify the relation-lattice complexity required for exact arithmetic realizations; Paper XIV proves fixed-start nontermination decidable along intrinsic continuation-rank frontiers; Paper XV characterizes finite conditional-termination convergence through closed-face integrality; Paper XVI extends exact weakest-nontermination computation beyond finite convergence by integral-core compression and face recursion; Paper XVII identifies the exact guarded common-multiplier arithmetic generated by persistent shadowing; and Paper XVIII proves that this arithmetic is complete for all residual non-Presburger complexity in the full physical-dimension-two cone-monotone equality class.

Key Findings

The research has established twenty-three major results:

1. Exact Periodicity of Maximal Words

The lexicographically largest legal words of every length are precisely the prefixes of a single infinite periodic word

$P^{\infty}, P = c_{1} c_{2} \dots c_{L - 1} (c_{L} - 1),$

determined explicitly by the defining recurrence. This simple structure governs all subsequent carry analysis.

2. The Intrinsic Sequence

For every PLRS, there exists a unique infinite sequence

$e = (e_{1}, e_{2}, e_{3}, \dots)$

defined purely by the arithmetic of the place values:

$H_{n + 1} - 1 = \sum_{i = 1}^{n} e_{i} H_{n + 1 - i} .$

This sequence turns out to be the same as the periodic maximal word—a fundamental identification that bridges arithmetic and combinatorics.

3. Classification of All Presentations

Every coefficient presentation generating a given place-value sequence is obtained by taking a period of the intrinsic sequence and increasing its final digit by one. The canonical reduction—the unique presentation of minimal order—corresponds to the primitive period.

4. Language Invariance

Although different coefficient presentations may define different recursive legality rules, they all generate exactly the same language of legal representations. The legality language is an invariant of the place-value sequence itself.

5. Canonical Automata

Every canonical PLRS legality language has a unique minimal deterministic automaton. This automaton is precisely the threshold-reset automaton of the canonical reduction, so automaton minimization, recurrence minimization, and intrinsic boundary reconstruction recover the same canonical presentation.

6. Endpoint Decomposition, Aperiodicity and Star-Freeness

Paper IVa determines the internal transition structure of the canonical threshold-reset automaton. Every live computation from qr to qs is either the unique reset-free forced march $cr+1⋯cs$

or has the restart form $B(r)B∗Cs.$

For the canonical reduction, let $P=c1⋯cL−1(cL−1)$

be the primitive maximal boundary block. The transition monoid is aperiodic if and only if $P$ is lexicographically maximal among its nontrivial cyclic rotations, equivalently if and only if $P$ is anti-Lyndon. Since the canonical automaton is minimal, the same condition is equivalent to star-freeness of the legality language.

7. Intrinsic Recognition of Canonical PLRS Legality Automata

Paper IVb solves the converse automaton problem left open by Paper IV. Starting only from a pointed minimal deterministic automaton over an ordered digit alphabet, it gives necessary and sufficient structural conditions for the automaton to be the legality automaton of a canonical PLRS.

The characterization is entirely intrinsic to the transition table. It identifies the common restart state, local digit thresholds, the cyclic threshold orbit and the necessary initial-positivity condition. When these conditions hold, the canonical coefficient vector is recovered uniquely by following the threshold cycle and reading the local thresholds, with the terminal coefficient obtained by adding one to the final threshold.

The paper also proves, independently of any assumed numeration-system origin, that a cyclic threshold-reset automaton is minimal precisely when its threshold-boundary word is primitive. Thus PLRS realizability and canonical reconstruction can be decided directly from the minimal DFA, with no recurrence or place-value sequence supplied in advance.

8. Local Carry Dynamics and Periodic Lifts

Paper IVc determines the exact local dependence structure between consecutive successor carries in canonical PLRS numeration.

Every positive reset depth determines its predecessor’s branch-point state and reset depth uniquely. Iterating this deterministic predecessor rule produces a finite carry-depth skeleton that reaches depth zero in at most $L$ steps, where $L$ is the order of the canonical PLRS, and this bound is sharp.

Periodicity of the maximal boundary produces a complementary unbounded direction: whenever a reset depth $t$ is realized, every lift

t+kL,\qquad k\geq 0,

is also realized at a suitably increased word width and has exactly the same finite backward skeleton. Thus the local carry dynamics decompose into a finite deterministic skeleton and an unbounded periodic lift parameter.

The paper also distinguishes two natural automaton-state conventions. The source-word residual state is completely determined by reset depth,

\sigma(w)=q_{T(w)\bmod L}

whereas the branch-point state is fully mixed across all branch-capable states when width is unrestricted. This distinction yields exact criteria for single-edge observability of state-dependent carry costs, exact backward resolving radii through the deterministic predecessor chain, and a precise description of the multivalued obstruction at depth zero.

9. Global Carry Realization in Canonical PLRS Numeration

Paper IVd solves the global realization problem left open by the local theory of Paper IVc.

For each maximal-orbit phase (i) and width (m), the state carry words satisfy the exact recursion

W_i^{(m)} = \left(W_0^{(m-1)}(m-1)\right)^{e_{i+1}} W_{i+1}^{(m-1)}.

If $u$ is a finite carry-depth word with $M=\max u$ , then

u\text{ is globally realizable} \iff u\in\operatorname{Fact}\left(W_0^{(M+L)}\right).

Thus every finite realization question reduces to a bounded-width factor test depending only on the maximum requested depth and the primitive boundary period.

The same recursion produces an infinite generalized ruler word with exact renormalization and a countable-alphabet substitution structure. Matched-scale prefix and factor containment define phase-domination relations and an exact exposure radius. The primitive boundary is anti-Lyndon exactly when the root phase already contains every matched-scale carry factor at every depth.

For non-anti-Lyndon boundaries, Paper IVd identifies the exact first depth at which root domination fails, proves a sharp $L-2$ bound for this failure depth, and distinguishes local threshold defects that are immediately visible from those that are globally masked by other occurrences in the root carry word.

10. Exact Beta-Numeration Carry Laws and Boundary Reconstruction

Paper V extends the exact carry theory to canonical greedy beta numeration.

For every finite width, it derives exact formulas for the full carry-propagation distribution, not merely its limiting expectation. The limiting propagation law is a shifted geometric distribution determined by the base.

Coordinatewise Hamming cost, digit-magnitude cost and more general additive carry statistics are obtained as deterministic pushforwards of the reset-depth distribution under maps determined by the canonical boundary. The expected finite-width digit-magnitude cost satisfies the exact identity

${E}[C_{1,m}] = 2 – \frac{1}{U_\beta(m)}.$

and therefore converges to the universal limiting mean 2.

The theory also solves an inverse problem. The limiting Hamming-cost distribution reconstructs the support of the canonical boundary, while the limiting digit-magnitude distribution reconstructs the complete boundary sequence, including its digit values.

When the canonical boundary is eventually periodic, as in the Parry case, the limiting probability generating functions are rational.

11. Finite-Sample Reconstruction from Carry Observations

The exact inverse theory extends to finite random samples. The canonical boundary can be reconstructed through depth $m$ by estimating the geometric reset parameter and decoding empirical carry-law tails against the corresponding geometric probability grid. A certified reconstruction algorithm succeeds with probability at least $1-\delta$ from $O_q\!\left(q^{-m}\log\frac{m}{\delta}\right)$

carry observations. A matching information-theoretic lower bound shows that the exponential dependence on reconstruction depth is unavoidable.

For bounded Parry systems, recovery of a sufficiently long finite prefix determines the complete eventually periodic boundary, including its preperiod and primitive period. Numerical experiments confirm the predicted $q^{-m}$ scaling and show why deeper observed carry levels must be rejected rather than blindly projected onto the truncated decoding grid.

12. Reconstruction from Coarsened Ordered Carry Traces

Paper VII develops an exact inverse theory for ordered numerical carry traces on zero-prolongable prefix languages.

Every strict jump of the nondecreasing decoder exposes an ordered cylinder partition of the language. If

J(D)={r\geq1(r)>D(r-1)}

and

g=\gcd J(D),

then the complete family of carry traces determines exactly the $g$ -decimated truncation system, namely the ordered ancestry maps between language levels separated by $g$ .

The complete ordered prefix tree is universally reconstructible precisely when

\gcd J(D)=1.

Under gap-free branching, the ordered tree also determines the labelled language. If the root has at least two legal children, the decoder itself is recovered through

D(t)=\max\mathcal C_{t+1}.

A visibility graph associated with $J(D)$ defines an intrinsic radius.

\rho_D.

The exact finite horizon required to reconstruct the $g$ -decimated system through depth $M$ is

H_D^*(M)= \begin{cases} M, & M\leq g,\\[2mm] M-g+\rho_D, & M\geq g+1. \end{cases}

Matching lower bounds prove that this horizon is optimal. The paper also shows that arbitrary hidden state-dependent carry costs can destroy universal identifiability, while uniform cross-state separation permits a limited extension of the theory.

13. Rank-Positional Threshold Languages and Census Rigidity

Paper VIII gives a complete solution to a restricted form of the positionality problem for abstract numeration systems. General abstract numeration starts from a regular language and assigns values by enumeration; the difficult inverse question is when such enumeration can instead be represented by positional place values.

Paper VIII considers the self-referential case in which the place values are required to be the language’s own width census $C_n$ . For a prefix-closed threshold language it characterizes exactly when

R_n(a_1\cdots a_n)=\sum_{j=1}^{n}a_jC_{n-j}

holds for every legal word if and only if every strict-deficiency transition has the same continuation census as the root. The identity is equivalent to every strict-deficiency transition having the same continuation census as the root, and in the accessible minimal automaton this forces every strict deficiency to reset literally to the root. In the accessible minimal automaton this forces every such transition to reset literally to the root. Every regular rank-positional threshold language therefore has a root-reset lollipop automaton whose live states form one maximal-transition orbit.

The maximal thresholds define an intrinsic boundary $e=e_1e_2\cdots$ . Its tails classify residual states, regularity is equivalent to eventual periodicity, and a boundary with minimal preperiod $t$ and primitive period $p$ has exactly $t+p$ live states.

The same boundary is the length inventory of a canonical prefix-free reset code and satisfies the triangular census identity

C_n=1+\sum_{i=1}^{n}e_iC_{n-i}.

Consequently the census determines the boundary, minimal automaton and complete language.

For two regular systems with $s$ and $s’$ live states, equality of

C_0,\ldots,C_{s+s’-1}

already forces equality of the languages, and this finite identification horizon is sharp for every pair $(s,s’)$ .

Finally, if the boundary has minimal preperiod $t$ and primitive period $p$ , the exact order of the minimal homogeneous recurrence satisfied by the census is

r=t+p-\deg\gcd(B,S_p),

where $B$ is the periodic boundary polynomial and $S_p=1+z+\cdots+z^{p-1}$ . Thus the difference between automaton-state complexity and scalar recurrence complexity is an explicit cyclotomic defect.

14. Intrinsic Affine Positionality and Sharp Recurrence Complexity

Paper IX develops a root-free affine theory of positional rank for prefix-closed, right-prolongable ordered regular languages.

If positionality holds at one root, every live residual admits a translation $\lambda_q(n)$ such that

R_q^{(n)}(w)=V_U(w)-\lambda_q(n).

This residual affine representation is equivalent to exact local tiling equations determined by the least legal digit and adjacent digit cylinders. When at least one residual state branches, any genuine adjacency determines the unique normalized candidate basis, and all residual translations are then forced.

The translation subsystem admits a canonical quotient obtained by identifying states with the same infinite least-digit itinerary. If the minimal live residual DFA has $s$ states and this lower-itinerary quotient has $b$ classes, then

r_U\le s+b-1, \qquad r_{\beta^{(q)}}\le s+b.

Both recurrence-complexity bounds are sharp for every 1 <= b <= s:

RU(s,b) = s + b – 1,
Rβ(s,b) = s + b.

Thus maximal recurrence complexity decomposes into census complexity and lower-itinerary complexity, with one common-translation gauge dimension invisible to positional value.

The same finite realization gives a universal compatibility-recognition bound of s + b – 2. This recognition bound is not universally sharp over genuine integer digit systems. For s = b = 4, the exact integer recognition radius is

HZ(4,4) = 5,

rather than the universal bound 6. In the corresponding rational-threshold affine relaxation,

HQ(4,4) = 6.

The one-step improvement is therefore an arithmetic integrality phenomenon rather than a reduction in the ambient affine realization dimension. Paper IX consequently distinguishes recurrence complexity from recognition complexity and motivates separate rational and integer recognition-radius problems.

Fixed-width and genealogical positionality are root-specific boundary conditions on this same intrinsic affine structure. Fixed-width rank corresponds to zero root boundary, while genealogical rank corresponds to the cumulative census boundary.

15. Affine Positionality in General Abstract Numeration Systems

Paper X removes prefix closure and right prolongability and studies arbitrary infinite regular languages ordered genealogically.

Residual states may now support only selected continuation widths. The paper proves that these state-width supports are eventually periodic and derives phase-local affine tiling equations on the supported widths. After one support period is lifted, the full infinite problem reduces to a finite integer linear descriptor relation.

Over $\mathbb Q$ , affine positionality is decidable by finite-dimensional viability. Over $\mathbb Z$ , the periodic descriptor relation defines an integer viability subgroup. This subgroup is effectively computable through a polynomial-module formulation using $S$ -saturation and primary decomposition over $\mathbb Z[z]$ . Integer affine positionality is therefore decidable as well.

The situation changes when weights are required to be nonnegative. From any integer linear recurrence sequence $(a_n)$ , Paper X effectively constructs an infinite regular language whose genealogical abstract numeration system is positional with nonnegative integer weights if and only if

a_n\ge0 \qquad\text{for every }n.

Consequently,

\text{LRS Positivity} \le_{\mathrm{eff}} \text{Regular-ANS Positionality}.

Thus any decision procedure for general regular-language positionality would also decide the longstanding Positivity Problem for integer linear recurrence sequences.

16. Balance-Loop Normal Forms and Viability Dynamics

Paper XI determines the exact computational normal form of nonnegative positionality for arbitrary regular abstract numeration systems.

Starting from the periodic descriptor relation of Paper X, it first normalizes unused weight positions, proves uniform exponential bounds on every observable nonnegative positional realization and isolates the finite set of tail states admitting genuinely nonnegative preperiod completions.

The resulting nonnegative descriptor transitions form the integer points of a rational polyhedral cone and therefore a finitely generated affine monoid. A Hilbert-basis construction gives finitely many transition generators. A new budget coordinate converts these generators into a proper system in which every source vector is nonzero, and a finite control gadget replaces the finite initial-state set by a single fixed start.

In generator coordinates the dynamics take the homogeneous form

B λ_n = A λ_{n+1},

with λ_n a nonnegative integer vector and every column of A nonzero.

Conversely, Paper XI constructs, from every such proper homogeneous balance loop, an infinite regular language whose genealogical rank is positionally representable by nonnegative integer weights if and only if the balance loop has an infinite trajectory. The construction uses radix filler languages and repeated three-word lexicographic equality gadgets to impose each row of the balance equation.

Consequently,

regular-ANS positionality = effective nontermination of proper homogeneous balance loops.

Properness implies effective finite branching and an exponential trajectory bound. By König’s lemma, an infinite trajectory exists if and only if trajectories exist at every finite horizon. Thus every negative instance has a finite obstruction:

regular-ANS non-positionality is recursively enumerable,

and

regular-ANS positionality lies in Pi-0-1.

Paper XI then passes from coefficient vectors to physical-state viability. The finite-horizon viable sets form an effectively computable descending filtration of affine semigroups.

In one physical dimension, the gcd of these semigroups satisfies an autonomous recurrence. Proportional systems are governed by explicit divisibility growth, while the nonproportional terminating regime is characterized exactly by 0 < b_i < a_i for every generator. After the viability gcd stabilizes, the least positive viable state and the conductor both grow asymptotically at the same exponential rate.

For two generators, the late viability filtration admits a sharper description. After normalization by the stabilized gcd, it becomes an exact finite union of arithmetic tails. Their thresholds evolve through a finite phase map with bounded periodic affine corrections, producing a finite normalized asymptotic limit cycle.

This finite-channel purification fails already for three generators. For A = (2, 6, 8) and B = (1, 1, 1), the normalized viability sets are exactly {0, 2^n} together with every integer from 2^n + 2 onward, so a single hole travels indefinitely with the efficient boundary. This identifies higher-dimensional factorization fibres as the next unresolved structural frontier.

The normal form also contains low-dimensional residue-affine dynamics related to generalized Collatz systems, but Paper XI makes no undecidability claim.

17. Minimal Kernel Dimension and the Sharp Universal Toric Bound

Paper XII introduces a quantitative invariant for fixed-control balance-loop realizations. For the partial rational floor map x ↦ floor(Qx/d), defined away from multiples of d, κ(d,Q) is the minimum rational kernel dimension of the source matrix over all exact proper realizations. Equivalently, it is the minimum toric codimension of an exact fixed-control realization.

Every zero-control column is forced to be a positive multiple of the bulk direction (d,Q), and every κ-minimal presentation has exactly one bulk column. Exactness therefore reduces to a finite selector-fibre problem.

For coprime d and Q, the invariant is classified completely:

κ(d,Q) = 2 when Q is congruent to +1 or −1 modulo d, and κ(d,Q) = 3 in every other coprime case.

The noncoprime problem is strictly richer. Paper XII proves

κ(8,2) = 3, while κ(12,3) = 4,

even though both maps have reduced slope 1/4. Thus minimal factorization complexity is not determined by reduced slope alone.

For the scaling family d = mg and q = g with m ≥ 3 and g ≥ 2,

κ(mg,g) = 3 when m = 3 or g = 2, while κ(mg,g) = 4 when m ≥ 4 and g ≥ 3.

For general noncoprime parameters d = gm and q = gn with gcd(m,n) = 1, Paper XII gives an explicit six-column realization of kernel dimension 4. Together with the coprime and scaling constructions, and extension to arbitrary positive numerators Q, this proves the sharp universal theorem

κ(d,Q) ≤ 4 for every d ≥ 2 and Q ≥ 1.

Since κ(12,3) = 4, the maximum possible minimal kernel dimension is exactly 4.

The proofs combine signed-residual lattice obstructions, residue replacement, numerical-semigroup reflection, Apéry coordinates, rational mechanical words and explicit fixed-control constructions.

18. Complete Noncoprime Kernel-Dimension Classification

Paper XIII completes the reduced noncoprime realization problem left open by Paper XII.

Write

d = gm, q = gn,

with g ≥ 2, 1 ≤ n < m and gcd(m,n) = 1. Then the complete classification is:

κ = 2 only for the exceptional map (d,q) = (4,2);
κ = 3 on the co-affine boundary n = m − 1, apart from that exception;
κ = 3 on the scaling boundary n = 1 exactly when g = 2 or m = 3;
κ = 4 in every remaining case.

In particular,

κ(gm,gn) = 4 whenever 1 < n < m − 1.

Thus every genuinely interior reduced noncoprime rational floor staircase attains the universal maximum kernel dimension.

The paper constructs an explicit five-column kernel-three realization for the co-affine boundary and proves the interior lower bound by classifying all hypothetical four-selector kernel-three architectures according to their number of pure-control selectors. Two-pure architectures reduce to two-generated numerical semigroups and are excluded by Frobenius symmetry, Apéry-coordinate confinement and checkerboard substitutions. One-pure architectures are eliminated through residue rigidity, mechanical edge structure and bidirectional target shifts. Zero-pure architectures reduce to unimodular lattice paths and finite orthant obstructions.

For a fixed rank-two source/control configuration, kernel dimension equals the rank of its integer relation lattice and the codimension of its associated toric ideal. The theorem can therefore be read as a complete restricted toric-codimension classification for reduced noncoprime rational floor maps.

19. Continuation Cores and Fixed-Start Nontermination

Paper XIV develops a structural reduction for proper homogeneous nonnegative equality loops

B λ(n) = A λ(n+1).

It introduces a span-relaxed continuation filtration whose stable cone C* contains every infinite real or integer trajectory and is effectively reached after at most rank(A) + 1 iterations.

The integer continuation core K* = C* ∩ Z^r has an effective Hilbert-basis presentation. Inside this core, every failure of an exact successor falls into exactly one of three layers:

positive-cone obstruction;
lattice obstruction;
affine-semigroup hole.

Writing κ* for coefficient continuation dimension, ρ* for physical continuation rank and ν* = κ* – ρ* for rational factorization freedom, the paper proves:

ρ ≤ 1 implies fixed-start nontermination is decidable,*

and

κ ≤ 2 implies fixed-start nontermination is decidable.*

For ρ* = 1, the core reduces exactly to a single-balance loop. In the expanding nonproportional regime, a maximal-slope Apéry construction gives uniformly efficient predecessors. After the viability gcd stabilizes, all subsequent numerical-semigroup conductors are uniformly bounded, forcing exact finite stabilization of the integer viability filtration.

For deterministic cores, invariant orbit lattices reduce integrality to algebraic recurrence conditions and nonnegativity to LRS Positivity. The first genuinely relational case not covered by these reductions has

(ρ*, ν*) = (2,1),

namely two physical continuation dimensions with one rational factorization degree of freedom.

20. Closed-Face Integrality and Exact Conditional Termination

Paper XV develops an exact conditional-termination theory for cone-monotone homogeneous integer equality relations.

Let P = cone(A) be pointed and suppose every source column is nonzero and every displacement satisfies

b_i – a_i ∈ P.

Every face F of P has a face successor Ψ(F). A face is closed when Ψ(F) = F. After quotienting the dynamics on a closed face by the forward closure of all nondeterministic directions, the relation has a canonical deterministic quotient T_F.

The paper proves the following conditions equivalent:

the descending predecessor sequence stabilizes after finitely many steps;
all mortal computations have a uniform finite depth bound;
every closed-face deterministic quotient T_F is integral over Z;
every characteristic polynomial of T_F has integer coefficients.

When these conditions hold, some finite predecessor stage is exactly the weakest nontermination precondition.

Nonclosed faces do not require separate arithmetic tests. Deep states can be moved congruence-preservingly through strict face ascents until a closed face is reached, and a finite-tube induction converts the resulting local continuation bounds into a global mortality bound.

The resulting weakest nontermination precondition is exactly computable in Presburger arithmetic.

The paper also proves that finite convergence need not be fast. A family with fixed physical dimension 2 and three generators has predecessor stabilization depth exponential in the binary input length. Thus closed-face integrality completely characterizes whether finite convergence occurs, while semigroup conductors and arithmetic holes can still control an exponentially long transient.

21. Exact Weakest Nontermination Beyond Finite Kleene Convergence

Paper XVI proves that finite predecessor convergence is not the boundary of exact weakest-nontermination computation for cone-monotone integer equality relations.

When the full-cone deterministic quotient is nonintegral, every infinite computation is confined to an exact forward-integral quotient lattice. Pulling this lattice back to coefficient space produces an affine monoid whose Hilbert basis gives an exact compressed relation. Every nontrivial such core reduction strictly lowers physical dimension.

When the full quotient is integral but proper closed faces remain nonintegral, maximal bad faces can instead be treated recursively whenever they are shadow-free. Outside those faces a relative finite-tube argument restores finite predecessor convergence.

These two operations generate the canonical core-face decomposition. Every branch has depth at most the physical dimension, and for every core-face-decomposable relation the recursion flattens to a finite exact normal form. Consequently the weakest nontermination set is effectively definable in existential Presburger arithmetic and is effectively semilinear.

The paper also gives explicit quantitative bounds on the number of nodes in the decomposition tree, the size and number of Hilbert-basis generators arising at core reductions, and the syntactic size of the resulting Presburger formulas.

An exhaustive exact census of 4,608 small two-dimensional systems shows that the two new mechanisms solve 646 of the 1,320 systems rejected by finite Kleene convergence, or 48.94% of those failures.

The boundary of the method is already arithmetically nontrivial. A shadowed bad face can sustain dynamics parallel to the face at fixed positive height. In physical dimension two, one explicit three-generator relation induces

n ↦ floor(4n/3)

on the first adjacent lattice layer. Exact weakest nontermination for this relation is tied to an unresolved generalized-Collatz reachability problem. Since the complete one-dimensional class is decidable, physical dimension two is minimal for this obstruction.

22. Shadowed Faces and Residue-Affine Dynamics

Paper XVII determines the arithmetic structure created by persistent shadowing at a bad closed face.

In physical dimension two, bad-ray rigidity forces every nontrivial bad closed ray to carry a unique common nonintegral rational multiplier m/d. A transverse height is nondecreasing along trajectories, so every infinite trajectory either escapes to unbounded height or is eventually flat.

At each fixed height, all face-factorization nondeterminism collapses to one numerical-semigroup coordinate. Beyond an effectively computable conductor threshold, the resulting dynamics are exactly finite-branch guarded residue-affine transitions of the form

d n′ = m n + c,

with finite source and target residue guards.

The infinite flat survivor set is the integer trace of a finite graph-directed base-d profinite system.

Conversely, every expanding guarded finite-branch common-multiplier system satisfying the necessary congruences is realized exactly on a forward-invariant first-layer tail of a physical-dimension-two cone-monotone equality relation with integral full quotient and a unique nonzero proper bad shadowed face.

For the primitive stopped construction, the coefficient-space Kleene filtration is exactly conjugate to a forbidden-residue filtration. At depth l, the surviving set consists of explicit residue cylinders modulo d^l. The resulting class contains the positive expanding common-multiplier maps associated with the Lagarias-Sloane approximate-multiplication problem.

23. Arithmetic Completeness in Physical Dimension Two

Paper XVIII completes the geometric analysis of physical-dimension-two cone-monotone homogeneous integer equality relations.

For the integral-full-quotient branch, removing nonnegativity produces a signed predecessor sequence on Z^n. This sequence stabilizes after finitely many steps exactly when the canonical deterministic quotient is integral over Z. In that case the stable signed lattice is effectively computable.

Intersecting the stable lattice with the nonnegative orthant gives an affine semigroup. An effective conductor produces a translated deep-interior cone on which weakest nontermination is exactly stable-lattice membership and is therefore Presburger-definable.

Every infinite trajectory avoiding this interior eventually transfers to a facet strip. In dimension two there are only two such strips. Good strips are effectively Presburger; bad strips reduce exactly to guarded finite-branch common-multiplier residue-affine survivor systems.

The full-quotient integrality assumption is then removed using Paper XVI. Every nonintegral full quotient admits an exact integral-core compression that strictly reduces physical dimension, so in dimension two it lands in the complete dimension-at-most-one Presburger theory.

Hence every physical-dimension-two cone-monotone weakest-nontermination set has an exact decomposition into a Presburger tame part and finitely many inverse images of survivor sets from the guarded common-multiplier arithmetic class.

Conversely, every arithmetic system in that class occurs on a first-layer tail of a two-dimensional geometric relation. Universal semilinearity and effective Presburger synthesis are therefore equivalent for the geometric and arithmetic classes, and pointwise membership is mutually reducible by finite disjunctive reductions.

No further two-dimensional geometric case remains outside this reduction.

The central insight of Papers I-IV is that the defining coefficient presentation is not intrinsic. The place-value sequence, intrinsic boundary sequence, legality language, canonical reduction and minimal deterministic automaton describe equivalent aspects of the same canonical PLRS system.

Papers IVa–IVd then develop the automata and dynamical consequences of that structure. Paper IVa determines the internal transition structure and characterizes aperiodicity and star-freeness; Paper IVb solves the converse recognition problem; Paper IVc determines the local dependence structure of consecutive successor carries, separating finite deterministic skeletons from periodic lift parameters; and Paper IVd solves the complementary global realization problem, determining exactly which finite carry-depth traces can occur consecutively.

Paper V adds a broader distributional and inverse principle in canonical greedy beta numeration, while Paper VI shows that boundary reconstruction remains possible from finite random samples with optimal exponential dependence on reconstruction depth.

Paper VII passes from numeration-specific carry distributions to arbitrary ordered prefix languages and ordered coarsened traces, identifying the decoder’s visible-depth gcd as the exact retained ancestry resolution and its visibility radius as the exact finite reconstruction delay. Paper VIII then moves from carry observations to positional structure itself, showing that within the threshold class the census, intrinsic boundary, minimal root-reset automaton and complete rank-positional language determine one another.

Structural Hierarchy and Extensions

The hierarchy below summarises the structural equivalences established in Papers I-IV. Paper IVa analyzes the transition structure of the resulting minimal automaton, Paper IVb characterizes intrinsically which minimal automata arise from canonical PLRS systems, Paper IVc develops the local successor dynamics carried by those automata, and Paper IVd determines the global carry-depth language generated by those dynamics.

Papers VIII-XVIII produce a second structural progression beyond the canonical PLRS setting:

census-positional threshold languages → intrinsic affine positionality in synchronous ordered regular languages → affine positionality and decidability for arbitrary regular abstract numeration systems → nonnegative positionality as proper homogeneous balance-loop nontermination → quantitative minimization of relation-lattice complexity inside fixed-control balance-loop realizations → complete classification of that complexity for reduced noncoprime rational floor maps → continuation-core reduction and intrinsic fixed-start decidability frontiers → exact closed-face integrality criteria for finite conditional termination → integral-core and recursive face reductions beyond finite Kleene convergence → exact shadowed-face common-multiplier dynamics → unconditional arithmetic completeness of the physical-dimension-two cone-monotone class.

The research reveals a striking inversion of the logical structure:

Level	Traditional View	New View
Fundamental	Coefficient presentation	Place-value sequence
Derived	Legal language	Intrinsic sequence
Canonical object	–	Minimal deterministic automaton
Output	Place values	All equivalent presentations

The intrinsic sequence is the master key: from it one recovers the boundary word, the canonical reduction, the minimal deterministic automaton, all equivalent coefficient presentations, and the legality language.

Research Series

Paper I: Foundations and Carry Depth

Establishes the combinatorial structure of PLRS legality
Introduces canonical reduction
Proves the Maximum Suffix Theorem
Derives exact finite-width carry-propagation tails and the limiting geometric law.

Paper II: Boundary Maps and Carry Costs

Proves exact periodicity of maximal words
Develops the boundary pushforward theorem
Analyses Hamming and digit-magnitude carry costs
Proves the universal mean identity $E [C_{ℓ^{1}}] = 2$

Paper III: Intrinsic Sequences and Language Invariance

Defines the intrinsic sequence $e$
Classifies all coefficient presentations
Proves language invariance
Resolves the reconstruction problem completely

Paper IV: Automaton Minimization and Canonical Reduction

Constructs the canonical threshold-reset automaton
Proves the Automaton Minimization Theorem
Identifies canonical reduction with the unique minimal DFA
Gives an automata-theoretic reconstruction algorithm

Paper IVa: Endpoint Decomposition and an Anti-Lyndon Aperiodicity Criterion for PLRS Numeration

Introduces anchored deficient-block languages and proves the Generalized Endpoint Decomposition for arbitrary live-state transitions.
Shows that every live computation is either a unique forced threshold march or an anchored reset followed by the universal restart grammar $B(r)B∗Cs$ .
Orders the live states canonically through the cyclic rotations of the primitive maximal boundary block.
Proves that the canonical threshold-reset automaton is aperiodic if and only if its primitive boundary block is anti-Lyndon.
Consequently characterizes exactly when the canonical PLRS legality language is star-free.
Gives explicit transposition and cycle criteria and explains the cyclic phenomena observed in the canonical automaton.
Reports an exhaustive census of 12,382 canonical reductions of order at most six with coefficients at most four, with exact agreement between the anti-Lyndon and aperiodicity classifications.
Supplies reproducible computational code and census data as supplementary material.

Paper IVb: Intrinsic Recognition of Canonical PLRS Legality Automata

Solves the converse recognition problem left open by Paper IV.
Gives necessary and sufficient automata-theoretic conditions for a pointed minimal DFA to be the legality automaton of a canonical PLRS.
Separates the structural restart state from the designated initial state and characterizes the local threshold-reset transition rule intrinsically.
Proves that the threshold orbit must form the complete live-state cycle and handles the order-one terminal case without exception to the reconstruction procedure.
Reconstructs the unique canonical coefficient vector directly from the transition table.
Proves internally that minimality of a cyclic threshold-reset automaton is equivalent to primitivity of its threshold-boundary word.
Gives an effective recognition and reconstruction algorithm.
Distinguishes the PLRS recognition problem from the related Parry and Bertrand automaton constructions, where additional lexicographic self-admissibility conditions are imposed on the boundary.

Paper IVc: Local Dynamics and Periodic Lifts of Carry Depths in Canonical PLRS Numeration

Gives an exact local grammar for consecutive reset depths under lexicographic succession.
Proves that every positive reset depth determines its predecessor’s branch-point state and depth uniquely.
Shows that every positive-depth excursion reaches depth zero through a deterministic predecessor chain of length at most $L$ , with the bound sharp for every order $L$ .
Proves the periodic-lift theorem: every realized depth $t$ admits every lift $t+kL$ at increased width, with the complete finite backward skeleton unchanged.
Decomposes the unbounded carry-depth dynamics into a finite deterministic skeleton and an unbounded periodic lift parameter.
Distinguishes source-residual phase locking from branch-point full mixing.
Gives an exact single-edge observability criterion for state-dependent carry costs.
Determines exact backward resolving radii throughout the deterministic predecessor chain, including its first forced depth-zero endpoint.
Characterizes the multivalued predecessor obstruction at depth zero through the finite zero-transition dynamics of the canonical automaton.

Paper IVd: Global Carry Realization in Canonical PLRS Numeration

Solves the global realization problem for consecutive carry-depth traces in canonical PLRS numeration.
Derives the exact state carry-word recursion for every maximal-orbit phase and word width.
Proves the finite realization theorem: a trace with maximum depth $M$ is globally realizable if and only if it occurs in the root carry word at width $M+L$ .
Constructs the infinite PLRS ruler word and proves exact renormalization under deletion of shallow carry levels.
Gives a countable-alphabet substitution description of the infinite carry word.
Develops exact matched-scale prefix and factor-containment recursions, phase-domination posets and an exposure radius.
Proves that the primitive boundary is anti-Lyndon exactly when the root phase contains every matched-scale carry factor at every depth.
Identifies the exact first depth of root-domination failure for non-anti-Lyndon boundaries and proves the sharp bound $\eta(P)\le L-2$ .
Distinguishes immediately visible local threshold defects from globally masked witnesses and introduces shortest observable witness questions.

Paper V: Exact Carry-Propagation Distributions and Boundary Reconstruction

Formulates a self-contained successor framework for ordered prefix languages based on maximal and minimal continuations.
Specialises this framework to canonical greedy beta numeration through the strict-deficiency reset property.
Derives exact finite-width carry-propagation tails and point probabilities.
Obtains the shifted geometric limiting law determined by the base.
Develops a general additive boundary-map framework for labelled carry costs.
Proves exact finite-width moment formulas, including the correction $2 – \frac{1}{U_\beta(m)}.$
Reconstructs both the support and the complete digit sequence of the canonical boundary from limiting carry distributions.
Derives rational probability generating functions for eventually periodic Parry boundaries.
Provides worked examples for decimal, Fibonacci, Tribonacci and a non-simple Parry system.
Reports exhaustive computational verification of the principal formulas.

Paper VI: Finite-Sample Reconstruction from Carry Propagation

Develops a statistical version of the exact reconstruction theory from Paper V, replacing the complete limiting carry law with finitely many observed carry events.
Introduces the geometric decoder model, in which a geometric reset depth is observed through a monotone additive decoder.
Proves that empirical carry-law tails can be decoded against a geometric probability grid to recover the underlying boundary digits exactly.
Gives a certified algorithm that reconstructs the first $m$ digits with probability at least $1-\delta$ from $O_q\!\left(q^{-m}\log\frac{m}{\delta}\right)$ observations.
Proves a matching information-theoretic lower bound, showing that the exponential dependence on reconstruction depth is unavoidable.
Shows that, for bounded Parry systems, recovery of a sufficiently long prefix determines the complete eventually periodic boundary, including its minimal preperiod and primitive period.
Uses numerical experiments to confirm the predicted $q^{-m}$ scaling and to demonstrate why carry levels beyond the target depth must be rejected rather than blindly projected onto the truncated decoding grid.

Paper VII: Reconstruction of Ordered Prefix Languages from Coarsened Carry Traces

Develops an exact inverse theory for ordered numerical carry traces on zero-prolongable ordered prefix languages.

Proves the Threshold-Cylinder Coding Theorem, showing that every strict decoder jump reveals an ordered cylinder partition.

Shows that the complete trace family is equivalent to the visible truncation system generated by the decoder’s strict jumps.

Proves that the traces determine exactly the $g$ -decimated ancestry system, where
$g=\gcd J(D).$

Establishes that the complete ordered prefix tree is universally reconstructible if and only if $g$ =1.

Shows that gap-free branching permits recovery of the complete labelled language.

Recovers an initially unknown decoder from trace maxima when the root has at least two legal children $D(t)=\max\mathcal C_{t+1}.$

Introduces the visibility graph and intrinsic radius $\rho_D$ .

Derives the exact optimal finite reconstruction horizon $H_D^*(M)= \begin{cases} M, & M\leq g,\\[2mm] M-g+\rho_D, & M\geq g+1. \end{cases}$
Proves matching lower bounds by constructing binary gap-free languages with identical shorter-horizon traces but different target ancestry maps.

Paper VIII: Rank-Positional Threshold Languages

Characterizes prefix-closed threshold languages whose fixed-width lexicographic rank is given positionally by their own census sequence.
Proves that the global rank identity is equivalent to a local strict-deficiency continuation-count condition.
Shows that in the minimal automaton every strict deficiency resets to the root, yielding a root-reset lollipop classification in the regular case.
Defines an intrinsic boundary whose tails classify residual states and proves that regularity is equivalent to eventual periodicity of this boundary.
Proves that minimal preperiod $t$ and primitive period $p$ give exactly $t+p$ live states.
Identifies the boundary as the length inventory of a canonical prefix-free reset code.
Establishes the triangular census-boundary duality $C_n=1+\sum_{i=1}^{n}e_iC_{n-i}$ , so the census determines the boundary, automaton and language.
Proves the optimal finite rigidity bound $C_0,\ldots,C_{s+s’-1}$ for two systems with $s$ and $s’$ live states.
Determines the exact cyclotomic defect between minimal automaton size and scalar recurrence order.
Shows that the threshold hypothesis is sharp and recovers PLRS and beta/Bertrand systems as purely periodic and self-admissible specializations.

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

Develops an intrinsic affine theory of positional rank in ordered regular languages.
Proves propagation of positionality from a root to every live residual state.
Gives an exact local affine-tiling characterization using least-digit dynamics and adjacent cylinders.
Introduces the lower-itinerary quotient and its rank $b$ .
Proves the universal bounds $rU≤s+b−1$ and $rβ(q)≤s+br$ .
Proves both complexity bounds sharp for every $1≤b≤s$ .
Gives a finite compatibility-recognition bound of $s+b−2$ .
Shows that this recognition bound is not universally sharp over integer digit systems: HZ(4,4) = 5 < 6.
Shows that the corresponding rational-threshold affine relaxation attains HQ(4,4) = 6, identifying an arithmetic integrality defect in finite recognition.
Develops the twisted-coboundary and root-transfer formulation.
Identifies fixed-width and genealogical positionality as two root-boundary specializations.
Shows that general positive intrinsic bases may grow faster than the language census, while the canonical root conventions retain Perron-scale growth.

Paper X: Affine Positionality in Abstract Numeration Systems: PhaseDynamics, Decidability, and a Positivity Barrier

Extends affine positionality from synchronous ordered prefix languages to arbitrary regular abstract numeration systems.
Proves eventual periodicity of residual state-width supports.
Derives the exact phase-local affine cylinder equations.
Reduces the infinite affine problem to a finite linear descriptor relation.
Proves rational affine positionality decidable.
Proves integer affine positionality decidable through an effectively computable viability subgroup.
Describes integer viability through $S$ -saturation of a polynomial module over $Z[z]$ .
Gives effective $C$ -finite representatives whenever rational or integer affine realizations exist.
Separates rational, integer and nonnegative positionality by explicit examples.
Gives an effective reduction from LRS Positivity to general regular-ANS positionality.
Shows that a general decision procedure for nonnegative regular-ANS positionality would decide LRS Positivity.

Paper XI: Nonnegative Positionality in Regular Abstract Numeration Systems: Balance-Loop Normal Forms and Viability Filtrations

Gives an exact computational normal form for nonnegative positionality in arbitrary regular abstract numeration systems.

Proves effective observable normalization and uniform exponential bounds for nonnegative positional realizations.

Replaces the integer affine preperiod lattice by an effective finite set of genuinely nonnegative preperiod completions.
Converts the periodic descriptor relation of Paper X into a finitely generated nonnegative transition monoid using a Hilbert basis.
Introduces a budget-coordinate construction that eliminates zero-source generators and produces a proper transition system.
Reduces finite initial-state sets to a single fixed start using exact source-consumption control tokens.
Derives the homogeneous balance-loop normal form
B λ_n = A λ_{n+1},
with nonnegative integer coefficient vectors and every column of A nonzero.
Proves the converse realization theorem: every proper homogeneous balance loop is represented by an infinite regular language using radix fillers and repeated lexicographic equality gadgets.
Establishes effective equivalence between regular-ANS nonnegative positionality and proper homogeneous balance-loop nontermination.
Proves effective finite branching and exponential bounds for all balance-loop trajectories.
Shows that nontermination is equivalent to survival at every finite horizon.
Concludes that regular-ANS non-positionality is recursively enumerable and that regular-ANS positionality lies in Pi-0-1.
Introduces the physical-state viability filtration and proves that every finite-horizon viable set is an effectively computable affine semigroup.
Derives an autonomous recurrence for the gcd of one-dimensional viability semigroups.
Completely classifies proportional one-dimensional systems by divisibility growth.
Characterizes the nonproportional terminating regime exactly by 0 < b_i < a_i for every generator.
Proves that the least positive viable state and conductor have a common exponential growth scale.
Proves eventual finite-channel purification for two-generator systems, with arithmetic-tail thresholds governed by a finite phase map.
Gives an explicit three-generator counterexample with a persistent moving boundary hole.
Shows that bulk transition-semigroup saturation does not by itself eliminate higher-factorization boundary defects

Paper XII: Minimal Kernel Dimension for Fixed-Control Balance-Loop Realizations of Rational Floor Maps

Introduces the minimal kernel dimension κ(d,Q) of an exact fixed-control balance-loop realization.
Identifies κ(d,Q) equivalently as minimal toric codimension over exact realizations.
Proves zero-control bulk rigidity and reduction to a finite selector-fibre problem.
Gives the complete coprime classification: κ = 2 exactly for Q congruent to +1 or −1 modulo d, and κ = 3 otherwise.
Proves the signed-residual obstruction excluding kernel dimension 2 throughout the non-affine range.
Shows that reduced slope does not determine realization complexity: κ(8,2) = 3 but κ(12,3) = 4.
Completely classifies the scaling family (d,q) = (mg,g).
Gives an explicit kernel-4 construction for arbitrary noncoprime parameters d = gm and q = gn with gcd(m,n) = 1.
Proves the sharp universal bound κ(d,Q) ≤ 4 for every d ≥ 2 and every positive numerator Q.
Shows that the global maximum of κ is exactly 4.
Uses lattice saturation, residue replacement, numerical-semigroup duality, Apéry coordinates and rational mechanical words for the lower bounds.
Supplies a dependency-free exhaustive verifier and machine-readable witnesses for principal explicit realizations.

Paper XIII: Complete Noncoprime Classification of Minimal Kernel Dimension for Rational Floor Maps

Completes the reduced noncoprime classification of minimal kernel dimension.
Proves that κ = 2 only for the exceptional map (4,2).
Gives an explicit kernel-3 construction for the complete co-affine boundary n = m − 1.
Shows that the scaling boundary n = 1 has κ = 3 exactly when g = 2 or m = 3, apart from the κ = 2 exception.
Proves the interior maximality theorem: κ(gm,gn) = 4 for every 1 < n < m − 1 with gcd(m,n) = 1.
Classifies every hypothetical kernel-3 realization by the number of pure-control selectors.
Uses numerical-semigroup reflection, Apéry coordinates, rational mechanical words, residue confinement and checkerboard substitutions in the two-pure obstruction.
Uses residue rigidity and bidirectional target-shift substitutions in the one-pure obstruction.
Uses unimodular lattice paths and finite orthant obstructions in the zero-pure obstruction.
Interprets kernel dimension as relation-lattice rank, equivalently toric codimension, for each fixed source/control configuration.
Includes explicit co-affine witnesses, finite classification checks and independent computational regression tests.

Paper XIV: Fixed-Start Nontermination in Homogeneous Nonnegative Equality Loops

Introduces the span-relaxed continuation core for homogeneous nonnegative equality loops.
Proves that every infinite real or integer trajectory lies in an effectively computable rational polyhedral continuation core.
Shows that the continuation-core filtration stabilizes after at most rank(A) + 1 iterations.
Gives an exact trichotomy for failed integer successors: positive-cone obstruction, lattice obstruction or affine-semigroup hole.
Gives a Hilbert-basis reduction of the core and a finite additive-policy certificate for total core viability.
Proves fixed-start nontermination decidable whenever the physical continuation rank ρ* is at most one.
Reduces the rank-one case to a single-balance loop and proves exact finite stabilization in the expanding nonproportional regime using maximal-slope Apéry predecessors and uniformly bounded conductors.
Gives a finite-state decision criterion from positive nonexpanding heights.
Proves fixed-start nontermination decidable whenever the coefficient continuation dimension κ* is at most two.
Reduces deterministic cores through invariant orbit lattices to Positivity for integer linear recurrence sequences.
Identifies the first genuinely relational untreated structural cell as (ρ*,ν*) = (2,1).

Paper XV: Closed-Face Integrality and Exact Conditional Termination for Cone-Monotone Integer Equality Relations

Studies homogeneous nondeterministic integer equality relations with pointed source cone and cone-monotone generator displacements.
Introduces the face map Ψ and the notion of a closed face.
Constructs a canonical deterministic quotient on every closed face by factoring out the forward closure of nondeterministic directions.
Proves the Closed-Face Integrality Theorem.
Shows that finite predecessor stabilization is equivalent to a uniform finite mortality bound.
Shows that these conditions are equivalent to integrality over Z of every closed-face deterministic quotient.
Gives the equivalent characteristic-polynomial criterion χ_T(X) ∈ Z[X].
Proves that nonclosed faces require no independent arithmetic condition because sufficiently deep states admit congruence-preserving ascent to strict superfaces.
Uses finite-tube induction to globalize the closed-face continuation result.
Concludes that the weakest nontermination precondition is exactly computable in Presburger arithmetic for the entire class.
Gives an effectively recognizable arbitrary-dimensional class that may be genuinely nondeterministic.
Constructs a fixed-dimensional family with exact mortality depth growing exponentially in the binary input length, proving an exponential lower bound for straightforward predecessor iteration.

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

Proves that finite Kleene convergence is not the boundary of exact weakest-nontermination computation.
Introduces the canonical core-face decomposition for cone-monotone integer equality relations.
Defines the exact forward-integral quotient lattice and coefficient orbit-core monoid.
Gives an exact Hilbert-basis compression whenever the full-cone quotient is nonintegral.
Proves that every nontrivial full-core reduction strictly lowers physical dimension.
Introduces shadow-free bad faces and proves relative finite convergence on their complement.
Recurses exactly on maximal proper bad shadow-free faces.
Proves that every core-face decomposition branch has depth at most the physical dimension.
Derives a finite exact normal form for every core-face-decomposable relation.
Concludes that the exact weakest nontermination set is effectively existential Presburger and effectively semilinear.
Gives explicit quantitative bounds for decomposition-tree size, Hilbert-basis generators and Presburger formula size.
Proves complete solvability in physical dimension one.
Shows that the exact class strictly extends finite Kleene convergence through both full-core reduction and proper-face recursion.
Reports an exhaustive exact census of 4,608 two-dimensional systems, with the new mechanisms solving 48.94% of the finite-Kleene failures.
Identifies persistent shadowing as the first excluded geometric mechanism.
Constructs an explicit physical-dimension-two relation inducing n ↦ floor(4n/3) on the first adjacent lattice layer.
Connects exact weakest nontermination for that relation to an unresolved generalized-Collatz reachability problem.

Paper XVII: Shadowed Faces and Residue-Affine Dynamics in Cone-Monotone Integer Relations

Gives the exact global shadow normal form for a bad closed ray.
Proves the transverse-height dichotomy: every infinite trajectory either escapes to unbounded height or is eventually flat.
Compresses all face-factorization ambiguity to one numerical-semigroup coordinate.
Reduces every fixed shadow layer, beyond an effective conductor threshold, to finite-control guarded common-multiplier residue-affine dynamics.
Gives a graph-directed base-d profinite description of infinite flat survivor sets.
Proves bad-ray rigidity: nontrivial bad ray dynamics have one common expanding rational multiplier m/d.
Proves the converse First-Layer Realization Theorem for arbitrary finite-branch guarded common-multiplier systems satisfying the required congruences.
Realizes these systems while preserving an integral full quotient and a unique nonzero proper bad shadowed face.
Gives an exact Kleene conjugacy with forbidden-residue survivor filtrations in the stopped primitive construction.
Places the Lagarias-Sloane positive expanding common-multiplier family inside the exact residual shadow arithmetic.

Paper XVIII: Arithmetic Completeness for Two-Dimensional Cone-Monotone Integer Relations

Gives an unconditional structural classification of weakest nontermination throughout the physical-dimension-two cone-monotone equality class.
Proves exact finite attainment of the signed predecessor domain precisely when the canonical deterministic quotient is integral over Z.
Computes the stable signed lattice effectively in the integral case.
Uses affine-semigroup normalization and conductors to prove exact deep-interior saturation.
Shows that the complete deep-interior weakest-nontermination set is effectively Presburger.
Proves facet transfer for every infinite trajectory avoiding the conductor region.
Decomposes the two-dimensional boundary into two bounded strips and a finite corner.
Proves all good strips effectively Presburger.
Reduces every bad strip exactly to a guarded finite-branch common-multiplier survivor system.
Uses the integral-core descent of Paper XVI to eliminate the nonintegral full-quotient branch by strict physical-dimension reduction.
Proves arithmetic completeness: guarded common-multiplier survival captures all residual nonsemilinear complexity in physical dimension two.
Proves equivalence of universal semilinearity and effective Presburger synthesis between the geometric and arithmetic classes.
Gives mutual finite-disjunctive reductions for pointwise membership.
Closes the two-dimensional geometric programme under the standing cone-monotone hypotheses.

Further Research

Papers I-XVIII, including companion Papers IVa-IVd, form the completed current theoretical programme. In particular, Papers XIV-XVIII close the physical-dimension-two cone-monotone geometric branch. Persistent shadowing is no longer an unclassified geometric mechanism: Paper XVII identifies its exact arithmetic normal form, and Paper XVIII proves that this guarded common-multiplier survivor class is complete for all residual two-dimensional non-Presburger complexity.

There is therefore no planned Paper XIX continuing this geometric chain. Further work should begin from independently motivated external problems rather than by manufacturing additional subclasses of the present framework.

Residual arithmetic survivor problem: The geometric reduction does not decide the survivor problem for the guarded finite-branch common-multiplier class isolated in Papers XVII and XVIII. That class contains difficult generalized-Collatz and Lagarias-Sloane-type arithmetic. Any future attack on it should be viewed as an arithmetic-dynamics problem in its own right rather than as an unfinished part of the two-dimensional geometric classification.

General balance-loop nontermination beyond the cone-monotone class: Paper XIV’s continuation-core theory applies more broadly than the cone-monotone programme completed in Papers XV-XVIII. The first genuinely relational continuation-core cell with two physical continuation dimensions and one rational factorization degree of freedom remains relevant for unrestricted homogeneous nonnegative equality loops. The completion theorem of Paper XVIII does not claim to solve that larger non-cone-monotone class.

Higher-dimensional cone-monotone relations: Paper XVIII is specific to physical dimension two. Higher-dimensional facets have positive dimension and can support finite-control affine subsystems rather than one-dimensional common-multiplier dynamics. No claim is made that the two-dimensional arithmetic-completeness theorem extends automatically to higher physical dimension. A future investigation would require an external reason to study that boundary rather than continuation for its own sake.

Intrinsic characterization of minimal kernel dimension: Papers XII and XIII determine κ by minimizing over fixed-control presentations and give complete classifications for the coprime case, the scaling family and the entire reduced noncoprime range. A natural next problem is to characterize κ directly from the arithmetic graph being realized, without quantifying over presentations. In particular, one would like an intrinsic arithmetic, polyhedral or semigroup criterion that detects the boundary reductions to kernel dimension 2 or 3 and explains directly why every genuinely interior reduced noncoprime staircase has kernel dimension 4.

General fixed-start balance-loop nontermination: Paper XI reduces general nonnegative regular-ANS positionality to fixed-start nontermination of proper homogeneous balance loops. Paper XIV proves decidability whenever the intrinsic physical continuation rank is at most one or the coefficient continuation dimension is at most two. Papers XV and XVI provide a complementary arbitrary-dimensional theory under cone monotonicity. Paper XV characterizes exactly when predecessor convergence is finite; Paper XVI extends exact weakest-nontermination computation beyond that class by integral-core compression and shadow-free face recursion. The principal remaining problem is to understand systems outside these structural classes, especially shadowed bad faces and the untreated continuation-core cell with two physical dimensions and one rational factorization degree of freedom.

Higher-rank continuation cores: Paper XIV shows that ambient matrix dimension is not the right complexity measure. After continuation pruning, the first untreated relational cell has two physical continuation dimensions and one rational factorization degree of freedom. Understanding this (ρ*,ν*) = (2,1) case is therefore a sharper next problem than simply increasing the number of rows or generators. Possible mechanisms include effective control of moving semigroup holes, facewise conductor bounds and finite descriptions of the remaining one-dimensional factorization freedom.

Quantitative conditional termination and weakest nontermination: Paper XV gives an exact qualitative criterion for finite predecessor convergence and constructive finite bounds for each accepted relation, while showing that stabilization depth can already be exponential in binary input length. Paper XVI adds explicit bounds on the size of the core-face decomposition tree, local Hilbert bases and the resulting existential-Presburger formulas. The main unresolved quantitative component is the finite-tube transient: current constructive bounds depend on lattice indices, residue data and affine-semigroup conductor certificates. Converting these certificate-sensitive bounds into useful closed expressions in total input bit size is the natural next complexity problem.

Shadowed faces and the Collatz boundary: Paper XVI shows that shadow-free bad faces remain exactly manageable but that persistent shadowing can support unbounded arithmetic parallel to a bad face. The first adjacent lattice layer already realizes generalized rational-floor dynamics, including n ↦ floor(4n/3). A central next problem is to identify decidable subclasses of shadowed systems, for example through bounded transverse control, finite residue dynamics or certified bounded-shadow conditions, without crossing into unresolved generalized-Collatz reachability.

Toric and affine-semigroup realization complexity: Minimal kernel dimension is equivalently minimum relation-lattice rank, or restricted toric codimension, over exact fixed-control realizations. Paper XIII now determines this invariant throughout the reduced noncoprime range. The next questions are structural rather than classificatory: characterize the invariant directly from the labelled arithmetic graph; understand which features of a graph force codimension drops; extend the lower-bound machinery beyond rational floor staircases; and determine how target labeling interacts with standard invariants of affine semigroups and toric ideals.

Beyond the reduced numerator range: Paper XIII classifies d = gm and q = gn with 1 ≤ n < m and gcd(m,n) = 1. Paper XII gives the universal kernel-four upper bound for arbitrary positive numerators Q. A remaining question is to determine how minimal kernel dimension behaves under quotient lifts Q = ad + q in the genuinely noncoprime case, where subtracting multiples of the source row from the target row need not preserve nonnegativity of the realization.

Exact recognition radii for affine positionality: Paper IX gives the universal integer recognition bound s + b – 2 but shows that it can fail to be sharp, with HZ(4,4) = 5 while the rational-threshold relaxation has HQ(4,4) = 6. A natural problem is to determine the exact functions HZ(s,b) and HQ(s,b), characterize the pairs for which they differ, and identify the lattice or congruence obstructions responsible for strict integer recognition.

Complexity of affine positionality algorithms: Rational and integer affine positionality are decidable, but the current algorithms are structural rather than complexity-optimal. Determining effective complexity bounds for phase lifting, integer viability, (S)-saturation and primary decomposition is a natural algorithmic problem.

Minimal phase and gadget complexity: The general ANS theory uses finite support phases, while the LRS reduction uses a seven-phase encoding. It remains open whether the descriptor realization can be canonically minimized and whether the seven-phase positivity construction can be compressed.

State-dependent successor traces: Papers IVc and IVd resolve the local and global canonical-PLRS carry-depth problems for the canonical depth observable. A remaining problem is to characterize complete backward path languages after repeated passages through the multivalued depth-zero relation, to extend global realization to genuinely state-dependent carry-cost decoders, and to develop comparable trace languages for broader finite-state ordered numeration systems.

Noisy ordered traces: Paper VI develops finite-sample reconstruction from random carry observations. A separate problem is to reconstruct ordered prefix languages when the positions or numerical values in an ordered trace are corrupted by noise, omissions or quantisation error.

Recognition of trace families: Which abstract families of numerical sequences arise as the ordered carry traces of zero-prolongable prefix languages under a nondecreasing decoder? Paper VII gives necessary structural consistency conditions, but a complete realization theory remains open.

Broader numeration systems: Comparable distributional and reconstruction results may be sought for Bertrand systems without canonical greedy beta boundaries, signed-digit systems, mixed-radix systems and non-canonical recurrence numeration systems.

Recognition beyond PLRS: Can comparable intrinsic automaton-recognition theorems be obtained for broader recurrence numeration systems, Bertrand systems, mixed-radix systems or non-canonical positional languages?

Algorithmic complexity: Can canonical reduction, visibility-radius reconstruction and the bounded-width global carry-realization test of Paper IVd be performed in provably optimal time from compressed finite representations of the input data?

Technical applications: The reconstruction theory may support low-bandwidth diagnostics, integrity checking and fault localization for constrained encoders, storage controllers and finite-state communication systems.

Applications

The work has potential applications in:

Computer arithmetic: Exact analysis of carry lengths, Hamming costs and digit-magnitude costs in arithmetic algorithms.

Constrained coding and storage: Structural analysis of legality languages used in finite-state encoders, communication systems and constrained storage codes.

Reverse engineering and diagnostics: Reconstruction or verification of an unknown encoder from observed transition costs, reset depths, timing traces or other coarsened operational data.

Formal verification: Continuation-core reduction, exact analysis of fixed-start nontermination, computation of weakest nontermination preconditions for cone-monotone integer equality relations, and comparison of reconstructed ordered transition structures with intended firmware, automata or encoder specifications.

Automata and formal language theory: Canonical state-space representations, minimal deterministic automata, transition-monoid structure, aperiodicity, star-freeness and reconstruction of ordered language trees.

Symbolic dynamics: Periodic and eventually periodic boundary structures, ordered subshifts and successor dynamics.

Statistical inference: Certified recovery of hidden boundary data from finite samples of carry behaviour.

The twenty two papers in the Canonical Numeration Research Series are available in full above.

[Earlier Development Papers

Boundary layers in the Carry Distributions of Canonical k-bonacci Numeration. Preprint, July 2026.

Boundary Maps and Travelling Carry Layers in Binary Positive Linear Recurrence Numeration Preprint, July 2026

Boundary Layers and Inverse Reconstruction in Canonical Multi-Gap PLRS Numeration. Preprint, July 2026]