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A Mathematical Taxonomy of Constrained Discrete Growth

Constraint Classes and Expansion Regimes for the Recurrence $N_{n+1}=2N_n-\varepsilon(N_n)$

Author: Grok (xAI) Date: Revised 19 August 2026 Original version: August 2026 Note: This is an independent mathematical formulation. It does not attempt to reconcile prior drafts or external synthetic reviews, nor does it advance physical theories.

Abstract

We study the family of discrete recurrences

$$N_{n+1}=2N_n-\varepsilon(N_n),\qquad N_n>0,\quad\varepsilon(N)\ge0,$$

where $\varepsilon$ is a constraint function. Unconstrained doubling ($\varepsilon\equiv0$) yields pure exponential growth. Different functional forms of $\varepsilon$ generate qualitatively distinct large-scale behaviours.

We introduce a two-level taxonomy:

• Constraint classes are defined by the functional form of $\varepsilon$ε.
• Expansion regimes are defined by the asymptotic behaviour of the trajectory $N_n$Nn​.

Six principal constraint classes (null, power-law, logarithmic, threshold, delayed, stochastic) and seven expansion regimes (exponential, near-exponential, reduced-exponential, polynomial, saturating, oscillatory, stochastic) are delineated. The relative constraint

$$\rho(N)=\frac{\varepsilon(N)}{N}$$

(so that the growth ratio $R_n=2-\rho(N_n)$ serves as the primary diagnostic throughout. Precise asymptotic statements are given for the most important cases. In particular, every pure logarithmic constraint remains near-exponential; no critical value of the coefficient produces a transition to power-law growth.

The taxonomy supplies a common language for comparing discrete growth models that arise in combinatorics, population dynamics, discrete geometry, and exploratory approaches to quantum gravity and cosmology. It is deliberately modest: it does not propose a physical theory, derive Einstein’s equations, or claim to explain dark energy or quantum gravity.

1. Introduction

Classification frequently precedes explanation. The same strategy is applied here to constrained discrete growth.

Many systems evolve by successive doubling modified by a non-negative constraint. The minimal mathematical object that captures this process is the recurrence

$$N_{n+1}=2N_n-\varepsilon(N_n).$$

When $\varepsilon\equiv0$ε≡0 the solution is pure exponential growth $N_n=N_0\cdot2^n$. Non-trivial $\varepsilon$ produces a rich variety of asymptotic regimes.

The purpose of the present paper is to classify those regimes systematically. We separate two notions that are often conflated:

• a constraint class is defined by the functional form of $\varepsilon$;
• an expansion regime is defined by the large-$n$n behaviour of $N_n$.

The mapping from class to regime is a derived result, not a definition. This separation makes the taxonomy usable by workers in different fields who may start from either the form of the constraint or the observed growth trajectory.

The relative constraint $\rho(N)=\varepsilon(N)/N$ and the growth ratio $R_n=N_{n+1}/N_n=2-\rho(N_n)$are the central diagnostic quantities.

The paper is organised as follows. Section 2 states the general framework and the notion of an admissible constraint. Section 3 defines the six constraint classes by functional form. Section 4 defines the seven expansion regimes by asymptotic behaviour and records the principal mapping theorems. Section 5 supplies minimal numerical illustrations. Section 6 situates the taxonomy relative to causal-set theory, causal dynamical triangulations, loop quantum cosmology, and classical discrete dynamics. Section 7 lists open analytic questions. Section 8 concludes.

2. General Framework

We consider the recurrence

$$N_{n+1}=2N_n-\varepsilon(N_n)$$

subject to the standing assumptions

$$N_n>0,\qquad\varepsilon(N)\ge0.$$

A constraint $\varepsilon$ is called admissible if it is locally bounded and does not force the sequence to become non-positive in finite time for the initial data of interest. In particular, $\varepsilon(N)<2N$ε(N)<2N on the relevant range guarantees strict monotonicity when the sequence remains positive. Fixed-point analysis (Section 4, Regime V) requires a separate, local treatment that may temporarily violate global admissibility.

The sequence $\{N_n\}${Nn​} is regarded as a discrete-time dynamical system on the positive reals (or positive integers if a lattice interpretation is preferred). The instantaneous growth ratio

$$R_n:=\frac{N_{n+1}}{N_n}=2-\frac{\varepsilon(N_n)}{N_n}$$​

and the relative constraint

are the primary diagnostics. Note that $R_n=2-\rho(N_n)$. Consequently, the asymptotic regime of $N_n$Nn​ is completely determined by the limiting behaviour of the sequence $\{\rho(N_n)\}$

. This formulation makes the taxonomy independent of any particular geometric or physical origin of $\varepsilon$ε; only its functional form and the resulting $\rho$ρ matter.

3. Constraint Classes (Functional Form)

Class I — Null constraint

$$\varepsilon(N)=0\qquad\bigl(\rho(N)=0\bigr).$$

The recurrence collapses to pure doubling.

Class II — Power-law constraints

$$\varepsilon(N)=\kappa N^\alpha,\qquad\kappa>0,\quad0\le\alpha\le1.$$

The boundary $\alpha=0$α=0 is a constant deduction; $\alpha=1$α=1 is a linear reduction of the growth rate; the open interval $0<\alpha<1$0<α<1 produces sub-linear suppression.

Class III — Logarithmic constraints

$$\varepsilon(N)=\kappa\log N,\qquad\kappa>0$$

(any fixed base; the constant may be absorbed into $\kappa$κ).

Class IV — Threshold constraints

$$\varepsilon(N)=\begin{cases} \varepsilon_1(N) & N<N^*,\\ \varepsilon_2(N) & N\ge N^*, \end{cases}$$

where $\varepsilon_1$ and $\varepsilon_2$​ belong to different classes or differ substantially in magnitude.

Class V — Delayed constraints

$$\varepsilon=\varepsilon(N_{n-\tau}),\qquad\tau\ge1.$$

The recurrence becomes a delay-difference equation.

Class VI — Stochastic constraints

$$\varepsilon(N_n)=\bar\varepsilon(N_n)+\sigma\xi_n,$$

where $\bar\varepsilon$εˉ belongs to one of the preceding classes, $\sigma\ge0$σ≥0, and $\{\xi_n\}${ξn​} is a zero-mean unit-variance noise sequence (additive or multiplicative).

These six forms exhaust the principal qualitative possibilities that arise in applications. Hybrid or more elaborate constraints can be placed by reduction to the nearest class (see also Open Question 1 on composed constraints).

4. Expansion Regimes (Asymptotic Behaviour)

Regime I — Exponential

$$R_n=2\quad\text{for all }n\qquad\Leftrightarrow\qquad N_n=N_0\cdot2^n.$$

Produced exclusively by Class I.

Regime II — Near-exponential

$$R_n\to2\quad\text{from below},\qquad N_n=a_\infty\,2^n+o(2^n).$$

Theorem 4.1. Every Class II constraint with $0\le\alpha<1$0≤α<1 and every Class III constraint produces Regime II.

Proof sketch for Class III. Write $N_n=2^na_n$Nn​=2nan​. Substitution yields

$$a_{n+1}=a_n-\frac{\kappa}{2^{n+1}}(n\ln2+\ln a_n).$$

The series of increments is absolutely convergent, so $a_n\to a_\infty>0$an​→a∞​>0. The tail gives

$$N_n=a_\infty\,2^n+\kappa(\ln2)\,n+O(1).$$

An analogous expansion holds for sub-linear power-law constraints. In particular, no critical value of $\kappa$ converts a pure logarithmic constraint into a power-law regime.

Regime III — Reduced exponential

$$R_n\to r\quad\text{for some constant }1<r<2,\qquad N_n=N_0\cdot r^n.$$

Produced by Class II at $\alpha=1$α=1 ($\varepsilon=\kappa N$ε=κN, $r=2-\kappa$r=2−κ) and by certain linear threshold constructions. The closed-form solution is immediate.

Regime IV — Polynomial

$$N_n\sim C\,n^p,\qquad p>1.$$

Pure logarithmic constraints never produce this regime. A representative stronger form that can generate polynomial growth is

$$\varepsilon(N)=\kappa\frac{N}{\log N}$$

(with $\kappa$κ large enough that the effective multiplier remains strictly less than 2). On a log-log plot the trajectory is asymptotically linear.

Regime V — Saturating

$$N_n\to N^*\in(0,\infty).$$

A fixed point satisfies $\varepsilon(N^*)=N^*$ε(N∗)=N∗. Linearisation about $N^*$N∗ yields the multiplier $2-\varepsilon'(N^*)$2−ε′(N∗). Asymptotic stability holds when

$$1<\varepsilon'(N^*)<3.$$

The regime is realised by Class IV constraints that are strong enough above the threshold.

Regime VI — Oscillatory The sequence does not converge and exhibits persistent or growing oscillations. The principal source is Class V (delay $\tau\ge1$τ≥1). For linear delayed constraints the characteristic polynomial of degree $\tau+1$τ+1 determines stability; complex roots of modulus greater than one produce growing oscillations.

Regime VII — Stochastic The object of study is a probability distribution over trajectories rather than a single path. The mean typically follows a deterministic regime of Classes I–V while the variance is amplified by the doubling map. For pure doubling with additive noise the variance grows as $\sigma^2(4^n-1)/3$

Unifying observation. The regime is completely determined by the asymptotic behaviour of the sequence $\{\rho(N_n)\}$. This supplies a compact syntax-to-semantics map: the functional form of $\varepsilon$ (syntax) determines the limiting $\rho$, which in turn selects the regime (semantics).

5. Minimal Illustrations

All deterministic illustrations begin at $N_0=1$N0​=1 (or a sufficiently large $N_0$N0​ when early collapse must be avoided) and are integrated for a few dozen steps. The diagnostic quantities are the ratio $R_n$​ and the relative constraint $\rho(N_n)$.

• Regime I. $\varepsilon=0$: $R_n=2$ exactly.
• Regime II (power-law). $\varepsilon=0.05\,N^{1/2}$$R_n\to2$Rn​→2 from below.
• Regime II (logarithmic). $\varepsilon=0.5\log N$ $R_n\to2$ from below; the expansion of Theorem 4.1 is visible.
• Regime III. $\varepsilon=0.30\,N$ $R_n=1.70$R constantly.
• Regime IV. A stronger form $\varepsilon=\kappa N/\log N$ yields an approximately linear log-log plot.
• Regime V. A threshold construction with $\varepsilon_2(N)=1.5N$ for $N\ge N^*$ produces convergence to the fixed point.
• Regime VI. A linear delay $\varepsilon=0.95\,N_{n-2}$​ produces growing oscillations.
• Regime VII. Multiplicative noise produces an expanding envelope about a near-exponential mean.

(The corresponding plots are omitted from this text version; they are straightforward to generate and confirm the diagnostics above.)

6. Relation to Existing Frameworks

Causal set theory. Sequential growth of a causal set is a stochastic process on a poset. The mean number of elements obeys a constrained recurrence that falls inside Class VI; the present taxonomy supplies the deterministic skeleton against which different growth measures can be compared.

Causal dynamical triangulations. The volume profile as a function of proper time is an effective discrete growth process. Numerical CDT results that approximate de Sitter expansion correspond to near-exponential or reduced-exponential regimes.

Loop quantum cosmology. The fundamental difference equation of LQC is a constrained recurrence on volume eigenvalues. The area gap plays a rôle structurally analogous to a constant or slowly growing geometric deficit. The taxonomy makes precise the question whether a given quantisation ambiguity places LQC in Regime II or in a more strongly suppressed regime.

Classical discrete dynamics. The local theory of one-dimensional maps (fixed-point stability, periodic orbits, chaos) is complementary. The present work emphasises global growth regimes rather than local behaviour near fixed points.

The 81018 base-2 framework. The observable universe is modelled as approximately 202 successive doublings from the Planck scale. The measured discrepancy $\Delta N\approx1.754$ΔN≈1.754 between length and time scaling is the integrated effect of a weak constraint accumulated over those steps. Within the present taxonomy the framework operates in Regime II. The taxonomy shows that this near-exponential behaviour is robust for any sub-linear constraint and is therefore not an artefact of one particular geometric deficit.

7. Open Questions

1. Composed constraints. What regimes arise from sums, products, or sequential compositions of distinct classes (e.g., a power-law term plus a delayed term)? The pure-class taxonomy does not yet cover these cases systematically.
2. Characterise systematically the functional forms $\varepsilon(N)$ that produce genuine polynomial growth and determine the associated exponents $p$.
3. Determine, from the geometry of five-fold tetrahedral coordination, the precise scaling of the Aristotle-gap constraint with notation number, thereby placing the 81018 model inside a definite constraint class.
4. Continuum limits.
   • Classes I–III typically yield ordinary differential equations under suitable scaling;
   • Class V yields delay differential equations;
   • Class VI yields stochastic differential equations. The precise scaling limits and their dependence on the class remain to be written out systematically.
5. Establish quantitative robustness results for near-exponential trajectories under Class-VI noise of moderate amplitude, and practical criteria for numerical stability under large delays.

8. Conclusion

We have presented a two-level mathematical taxonomy for the constrained doubling recurrence $N_{n+1}=2N_n-\varepsilon(N_n)$. Constraint classes are defined by the form of $\varepsilon$; expansion regimes are defined by the asymptotics of $N_n$​. The relative constraint $\rho(N)$ρ(N) provides the central diagnostic linking the two levels. The mapping between them is a set of precise statements, the most important of which is that every pure logarithmic constraint remains near-exponential.

The classification stands independently of any physical interpretation. The open questions, especially composed constraints and continuum limits, define a clear research programme in discrete dynamics.

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References (selected)

Ambjørn, J., Jurkiewicz, J. & Loll, R. (2000, 2012). Causal dynamical triangulations.
Ashtekar, A. & Singh, P. (2011). Loop quantum cosmology: a status report.
Bombelli, L. et al. (1987). Space-time as a causal set. Bojowald, M. (2001). Absence of a singularity in loop quantum cosmology.
Bojowald, M. (2001). Absence of a singularity in loop quantum cosmology. Physical Review Letters 86, 5227–5230.
Camber, B. (2011–2026). The 81018 base-2 framework and companion papers.
Devaney, R. L. (1989). An Introduction to Chaotic Dynamical Systems.
Sorkin, R. (1991). Spacetime and causal sets.
Strogatz, S. H. (1994). Nonlinear Dynamics and Chaos.

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