Strongly Admissible Triplets

• Strongly admissible triplets are defined by rigorous conditions that guarantee every orbit in Collatz-type maps becomes periodic and that quantum states over CAR algebras satisfy saturation of entropy inequalities.
• Explicit families provide concrete cycle structures and algorithmic criteria, including continued-fraction based bounds to verify cycle length and convergence behaviors.
• Applications span discrete dynamical systems and quantum information theory, generalizing classical problems like the 3n+1 conjecture and modeling quantum Markov processes.

A strongly admissible triplet is a central concept in two distinct mathematical contexts: dynamical systems generalizing the Collatz problem, and quantum information theory on states over the CAR (canonical anticommutation relation) algebra. Despite differences in domain, the unifying theme is the characterization of triplets (either of integers or subalgebras) with robust admissibility and maximal regularity properties—in the first case, global convergence to periodicity of orbits in a non-linear map, and in the second case, saturation of entropy inequalities, implying a strong conditional independence structure.

1. Strongly Admissible Triplets in Collatz-Type Dynamical Systems

Given fixed integer parameters $d\ge 2$, $\kappa_0 = \pm 1$, and coprime integers $\alpha > d$, $\beta$, a Collatz-type map $T:\mathbb{N}\to\mathbb{N}$ is defined by

$$T(n) = \begin{cases} n/d & \text{if } n\equiv 0\pmod d, \ \bigl(\alpha n + \beta[\kappa_0 n]_d\bigr)/d & \text{otherwise,} \end{cases}$$

where $[x]_d$ denotes the remainder of $x$ modulo $d$. The well-definedness and map invariance require the admissibility conditions

$$\alpha + \kappa_0\beta > \frac{\kappa_0 - 1}{2}\beta d, \qquad \alpha + \kappa_0\beta \equiv 0\pmod d.$$

A triplet $(d,\alpha,\beta)_{\pm}$ satisfying these is called admissible. Further, it is:

• Weakly admissible if $T$ admits finitely many distinct cycles,
• Strongly admissible if every orbit under $T$ is eventually periodic with no divergence, that is, every $n\in \mathbb{N}$ is eventually absorbed in a finite cycle (Bouhamidi, 24 Jan 2026).

This definition encapsulates, and generalizes, the classical Collatz $(2,3,1)_+$ problem, now recast as the conjecture that $(2,3,1)_+$ is strongly admissible.

2. Illustrative Families, Structure, and Concrete Examples

Infinite families of strongly admissible triplets have been constructed with explicit cycle structure. For instance, given integers $d\ge 2$, $\nu_0,\nu_1\ge 1$, $1\le\delta\le d-1$, $\kappa_0,\kappa_1=\pm 1$, define

$$\alpha = d^{\nu_1} - \kappa_1\delta,\qquad \beta = \kappa_0(d^{\nu_0} - \alpha).$$

Then $(d, \alpha, \beta)_\pm$ is admissible and supports explicit "trivial" cycles of lengths $\nu_0$ or $\nu_1$, with strong admissibility under positivity conditions.

Another family generalizes Jones–Lagarias's $(3x+k)$ cycles, using

$$\alpha = d^{\nu_1} + 1,\qquad \beta = d^{2\mu_0+\nu_1} - \alpha^2,\quad 2\mu_0 > \nu_1 \ge 1,$$

yielding $\mu_0(d-1)^2$ distinct cycles each of prescribed length, again with strong admissibility under mild hypotheses.

Representative examples include:

Triplet $(d,\alpha,\beta)_\pm$	Cycle Structure	Strong Admissibility Property
$(2,3,1)_+$	$\Omega(1)=(1\to2\to1)$	Conjectured (the $3n+1$ conjecture)
$(10,12,8)_+$	$\Omega(4)=(4\to8\to16\to24\to32\to40\to4)$	Verified, unique cycle, no divergence
$(5,6,4)_+$	$\Omega(4)=(4\to8\to12\to16\to20\to4)$	Verified, unique cycle, no divergence
$(3,4,1)_-$	$\Omega(1)$ and $\Omega(7)$, both explicit finite cycles, no divergence	Verified, two cycles

In all strongly admissible cases, $\mathbb{N}$ is partitioned into disjoint basins of attraction for each cycle, absorbing all orbits (Bouhamidi, 24 Jan 2026).

3. Partition, Cycle Length Bounds, and Algorithmic Criteria

For a strongly admissible triplet, the positive integers decompose as a disjoint union over the cycle basins $G_{\omega}$: $$\mathbb{N} = \bigsqcup_{\omega\in \mathrm{Cycl}} G_{\omega}.$$ Lower bounds on the length $L$ of cycles are given in terms of parameters $(d,\alpha,\beta)$, minimal orbit value $M_0$, and properties of continued fractions. Specifically, for $gcd(d,\alpha) = 1$, $\beta > 0$, and cycle $\Omega$ with length $L$ and $\overline K$ nontrivial iterates: $$0 < \overline K \log_d\bigl(1+\frac{\beta}{\alpha\ \max\Omega}\bigr) < L - \overline K\xi < \sum_{n\in\overline\Omega}\log_d\bigl(1+\frac{\beta(d-1)}{\alpha n}\bigr),\quad \xi = \log_d(\alpha).$$ Hurwitz-type bounds yield $L \ge \mu_0\sqrt{M_0}$, with $\mu_0 = \sqrt{\alpha\log d/[\beta(d-1)\sqrt{5}]}$, and sharper cycle lower bounds are computable via continued-fraction expansions of $\xi$ (Bouhamidi, 24 Jan 2026).

Efficient algorithms for computing such lower bounds are established, exploiting the continued-fraction structure—Algorithm 1 maximizing over the index of convergents, Algorithm 2 using sign-alternation of sequence $D_{n}(M)$—with both running in time logarithmic in $M$.

4. Strongly Admissible Triplets in Quantum Information: CAR Algebra and Strong Subadditivity

Consider the CAR algebra $\mathfrak{A}$ generated by $\{a_i, i\in Z\}$ with canonical anticommutation relations, and triplets of subalgebras $\mathfrak{A}_1,\mathfrak{A}_2,\mathfrak{A}_3$ associated to disjoint finite sets $A,B,C\subset Z$. The von Neumann entropy of a faithful state $\varphi$ on $\mathfrak{A}_{123}$ with density $\rho_{123}$ is

$$S(\varphi) = -\operatorname{Tr}\,\rho_{123}\log\rho_{123}.$$

The strong subadditivity (SSA) inequality is

$$S(\rho_{123}) + S(\rho_{B}) \le S(\rho_{12}) + S(\rho_{23}),$$

or, equivalently, $$\Delta\mathrm{SSA}(\varphi) := S(\rho_{12}) + S(\rho_{23}) - S(\rho_{123}) - S(\rho_B) \ge 0$$.

A state $\varphi$ is strongly additive (sometimes termed strongly admissible) on the triplet $(A,B,C)$ if equality holds: $\Delta\mathrm{SSA}(\varphi) = 0$ (Jencova, 2010).

5. Characterization Theorems and Quantum Markov Triplets

The paper establishes that, for a faithful state $\varphi$ on $\mathfrak{A}_{123}$, the following conditions are equivalent:

• $\Delta\mathrm{SSA}(\varphi) = 0$ (strong additivity/strong admissibility),
• The inclusion $\mathfrak{A}_{12}\subset\mathfrak{A}_{123}$ is sufficient for $\{\varphi, \varphi\circ E_{23}\}$, with $E_{23}$ the trace-preserving conditional expectation onto $\mathfrak{A}_{23}$,
• The density matrix factorizes as $\rho_{123} = x y$, with $x\in\mathfrak{A}_{12}^+$, $y\in\mathfrak{A}_{23}^+$.

Furthermore, a state $\varphi$ on $(A,B,C)$ is a Markov triplet if and only if there exists a unital completely positive (CP) map $E:\mathfrak{A}_{123}\to\mathfrak{A}_{12}$ satisfying module, invariance, and image conditions. For even states (with respect to parity automorphism), Markov property and strong additivity are equivalent (Jencova, 2010).

6. Structure, Parity, and Explicit Decompositions

For even strongly admissible (Markov) states, the density $\rho_{123}$ admits a canonical block-diagonal form along minimal central projections in $\mathfrak{A}_{12}$, with each block a tensor of positive elements $x_j\otimes y_j$, both even. This mirrors the classical Petz–Hayden–Jozsa–Winter decomposition,

$$\rho_{123} = (D_{AB}\otimes I_C)\cdot (I_A\otimes D_{BC}),\quad D_{AB}\in \mathfrak{A}_{12}^+, D_{BC}\in \mathfrak{A}_{23}^+.$$

In physical terms, this algebraic equality embodies strong conditional independence: the fermionic analogue of a quantum Markov chain, with $A$ and $C$ conditionally independent given $B$ (Jencova, 2010).

7. General Conjectures and Experimental Evidence

The framework of strongly admissible triplets enables generalizations of the Collatz conjecture. The conjecture that $(2,3,1)_+$ is strongly admissible recasts the $3n+1$ problem; analogous conjectures are posed for further families, e.g., $(10,12,8)_+$, or $(2^p+2^q,2^p+2^{q+1},2^p)_+$. Empirical computations, including large-scale enumerations and cycle-length tabulations, confirm strong admissibility for many cases up to high values of parameters and initial data. Cycle-length lower bounds derived from the established algorithms have, for instance, forced any putative nontrivial $3n+1$ cycle to be extraordinarily large if it exists (Bouhamidi, 24 Jan 2026).

A plausible implication is that the concept of strong admissibility provides a robust paradigm for both understanding dynamical phenomena in nonlinear integer maps and for characterizing exact conditional independence structures in quantum states, thus serving as a key notion in both discrete dynamics and quantum statistical inference.