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81018 | PAPER II

August 6, 2026 • Manuscript

Constraint Classes for the Doubling Map: A Formal Asymptotic Taxonomy

Meta AI – 81018 Research Program — Paper II — Doubling Constraints

ABSTRACT

We study perturbations of the pure doubling map $N_{n+1}=2N_n$​ under formal constraints. Writing the perturbed dynamics as $$\begin{gathered} N_{n+1}=2N_n- \\ varepsilon(N_n) \end{gathered}$$ with $$\begin{gathered} varepsilon \\ ge0 \end{gathered}$$, we classify the possible asymptotics via syntactic form of $varepsilon$ and semantic behavior of the ratio$R_n=N_{n+1}\mathrm{/}{N}_n$. The taxonomy distinguishes seven canonical regimes ranging from negligible sublinear drain to stochastic and delayed feedback, with sharp thresholds for $$\begin{gathered} R_n \\ to2 \end{gathered}$$, $$\begin{gathered} R_n \\ toc \\ in(1,2) \end{gathered}$$, and $$\begin{gathered} R_n \\ to1 \end{gathered}$$. A Master Lemma unifies the mapping, and interactive simulations illustrate each regime.

1. FRAMEWORK

Doubling with constraint

Let $N_n>0$  be a resource, population, or count at step $n$. The idealized law is doubling:

$$N_{n+1}=2N_n,N_0>0$$

We consider constrained versions

$$\begin{gathered} N_{n+1}=2N_n- \\ varepsilon(N_n), \\ varepsilon(N_n) \\ ge0 \end{gathered}$$

where $\epsilon$ encodes cost, saturation, delay, or noise. The instantaneous multiplier is

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

Thus classification reduces to the order of $\epsilon (N)\mathrm{/}N$. If $\epsilon (N)=o(N)$) then $R_n\to 2$. If $\epsilon (N)\sim (2-c)N$ then $R_n\to c$. If$\epsilon (N)=N(1-o(1))$  then $R_n\to 1$. Finer distinctions depend on how$\epsilon$ε scales: sublinear, linear, logarithmic, threshold, delayed, stochastic.

2. SYNTACTIC CLASSES

Seven canonical forms

I

Pure Doubling— Exponential baseline

$N_{n+1}=2N_n$​

II

Sublinear Correction— o(N) drain

$N_{n+1}=2N_n-0.05\sqrt{N_n}$​​

III

Subdoubling Linear— Constant factor < 2

$N_{n+1}=1.7N_n$​

IV

Logarithmic Brake— Polynomial-like

$N_{n+1}=N_n(1+\frac{2}{\log (2+N_n)})$

V

Threshold / Saturation— Capacity-limited

$N_{n+1}=\{\begin{array}{ll}2N_n& N_n<10^6\\ N_n\cdot 1.15& N_n\ge 10^6\end{array}$

VI

Delayed Feedback— Memory term

$N_{n+1}=2N_n-0.95N_{n-2}$

VII

Stochastic Perturbation— Mean-preserving noise

$N_{n+1}=2N_n\cdot {\xi }_n,\mathbb{E}[{\xi }_n]=1$

Syntactic form dictates ε. Semantic behavior depends on limit of Rₙ and second-order fluctuations.

3. SEMANTIC REGIMES

Asymptotic trichotomy

MASTER LEMMA

Let Nₙ₊₁ = 2Nₙ − ε(Nₙ), ε ≥ 0, and Rₙ = Nₙ₊₁/Nₙ.

(i) NEGLIGIBLE: If$\epsilon (N)=o(N)$ then $R_n\to 2$ and$N_n=2^{n+o(n)}$.

(ii) PROPORTIONAL: If$\epsilon (N)=(2-c)N+o(N),c\in (1,2)$ then $R_n\to c$.

(iii) SATURATING: If$\epsilon (N)=N(1-r(N)),r(N)\to 0$ then $R_n\to 1$ and growth is subexponential.

Proof sketch: Rₙ = 2 − ε/N. Limits follow from definition. Trichotomy is exhaustive for ε ≥ 0 monotone. Delayed and stochastic cases replace ε(Nₙ) with history/noise; same Rₙ criterion applies pathwise or in expectation.

R → 2

Classes I, II, VII (mean)

Asymptotically pure doubling, o(N) error

R → c∈(1,2)

Classes III, V (late), VI

Exponential with reduced base

R → 1

Class IV, V (capped)

Polynomial-like, log-braked

4. MAPPING TABLE

From syntax to semantics

CLASS	ε(N)	R∞	GROWTH	FORM
I Pure Doubling	$0$0	2.00	$2^n$2n	exponential baseline
II Sublinear Correction	$0.05\sqrt{N}$0.05N​	2⁻	$\mathrm{\Theta }(2^n)$Θ(2n)	o(n) drain
III Subdoubling Linear	$0.3N$0.3N	1.70	$1.7^n$1.7n	constant factor < 2
IV Logarithmic Brake	$N_n(1-\frac{2}{\log (2+N_n)})$Nn​(1−log(2+Nn​)2​)	1.00	$\exp \text{\hspace{0.17em}⁣}(\mathrm{\Theta }((\log N{)}^2))$exp(Θ((logN)2))	log-brake
V Threshold / Saturation	$0\to 0.85N$0→0.85N	2→1.15	$2^n\to 1.15^n$2n→1.15n	capacity-limited
VI Delayed Feedback	$0.95N_{n-2}$0.95Nn−2​	≈1.68 osc	$\mathrm{\Theta }({\rho }^n),\rho \approx 1.68$Θ(ρn),ρ≈1.68	memory term
VII Stochastic Perturbation	$\text{random}$random	2±σ	$\text{log-normal around }{2}^n$log-normal around 2n	stochastic

5. CANONICAL EXAMPLES

Interactive trajectories

log₁₀ Nₙ and Rₙ, N₀=100, 32 steps

Each panel simulates $$\begin{gathered} N_{n+1}=2N_n- \\ varepsilon(N_n) \end{gathered}$$. Left shows${\log }_{10}{N}_n$log10​Nn​ (exponential = straight line); right shows $R_n=N_{n+1}\mathrm{/}{N}_n$. The dashed line in Class VII is the mean path $\mathbb{E}[N_n]=2^n{N}_0$.

CLASS I

Pure Doubling

Exponential baseline

$N_{n+1}=2N_n$​

log₁₀ Nₙ vs n4.3e+11

Rₙ = Nₙ₊₁/Nₙ vs n→ 2.000

Reference case. No constraint, R_n identically 2.

CLASS II

Sublinear Correction

o(N) drain

$N_{n+1}=2N_n-0.05\sqrt{N_n}$Nn+1​=2Nn​−0.05Nn​​

log₁₀ Nₙ vs n4.3e+11

Rₙ = Nₙ₊₁/Nₙ vs n→ 2.000

Small sublinear loss. Asymptotically negligible, R_n → 2⁻.

CLASS III

Subdoubling Linear

Constant factor < 2

$N_{n+1}=1.7N_n$Nn+1​=1.7Nn​

log₁₀ Nₙ vs n2.4e+9

Rₙ = Nₙ₊₁/Nₙ vs n→ 1.700

Proportional constraint. Ratio locks at 1.7, exponential slower than 2.

CLASS IV

Logarithmic Brake

Polynomial-like

$N_{n+1}=N_n(1+\frac{2}{\log (2+N_n)})$

log₁₀ Nₙ vs n1.2e+5

Rₙ = Nₙ₊₁/Nₙ vs n→ 1.173

Diverging log penalty. R_n = 1 + 2/log N → 1. Superpolynomial but subexponential.

CLASS V

Threshold / Saturation

Capacity-limited

$N_{n+1}=\{\begin{array}{ll}2N_n& N_n<10^6\\ N_n\cdot 1.15& N_n\ge 10^6\end{array}$

log₁₀ Nₙ vs n2.0e+7

Rₙ = Nₙ₊₁/Nₙ vs n→ 1.150

Phase change at 10⁶. Pure doubling early, then capped multiplier. R_n drops sharply.

CLASS VI

Delayed Feedback

Memory term

$N_{n+1}=2N_n-0.95N_{n-2}$​

Rₙ = Nₙ₊₁/Nₙ vs n→ 1.652

Non-Markovian. Constraint depends on lag 2. Induces oscillations in R_n but net growth persists.

CLASS VII

Stochastic Perturbation

Mean-preserving noise

Sample #1 • seed-81018-1Resample #1

$N_{n+1}=2N_n\cdot {\xi }_n,\mathbb{E}[{\xi }_n]=1$

log₁₀ Nₙ vs n2.3e+12

Rₙ = Nₙ₊₁/Nₙ vs n→ 2.226

Multiplicative noise with E[ξ]=1. Mean path is pure doubling, sample paths fluctuate. R_n noisy around 2.

6. RELATION TO FRAMEWORKS

Where constraints sit

• Amortized analysis: Sublinear ε corresponds to negligible overhead; proportional ε corresponds to constant-factor slowdown.
• Population dynamics: Classes IV–V are logistic-like with carrying capacity in growth rate, not absolute size.
• Distributed systems: Threshold class V models shard saturation; delayed class VI models feedback lag in autoscaling.
• Stochastic: Class VII preserves doubling in expectation but log-normal variance grows linearly in n, i.e. $\mathrm{V}\mathrm{a}\mathrm{r}(\log N_n)={\sigma }^{2}n$Var(logNn​)=σ2n.

$$R_n=2-\frac{\epsilon (N_n)}{N_n}⟺\text{growth rate = ideal − relative cost}$$Rn​=2−Nn​ε(Nn​)​⟺growth rate = ideal − relative cost

7. OPEN PROBLEMS

What remains

1.Characterize minimal syntactic conditions for R_n monotonicity in delayed class VI.

2.Threshold inference: given only R_n observations, recover N_c and post-threshold c with optimal sample complexity.

3.Stochastic dominance: when does P(N_n^{(II)} > N_n^{(IV)}) → 1 despite faster R_∞ in II?

4.Composition: taxonomy for N_{n+1}=2N_n – ε_1(N_n) – ε_2(N_{n-k}) mixing classes.

8. CONCLUSION

A compact lens

Writing any near-doubling process as $$\begin{gathered} N_{n+1}=2N_n- \\ varepsilon(N_n) \end{gathered}$$Nn+1​=2Nn​−varepsilon(Nn​) collapses diverse constraints into a single scalar function $varepsilon$varepsilon. The ratio$$\begin{gathered} R_n=2- \\ varepsilon\mathrm{/}N \end{gathered}$$Rn​=2−varepsilon/N then provides an immediate diagnostic: look at where$R_n$Rn​ settles. Seven syntactic templates cover most cases encountered in practice, from$$\begin{gathered} 0.05 \\ sqrtN \end{gathered}$$0.05sqrtN drains that vanish to $$\begin{gathered} 2\mathrm{/} \\ logN \end{gathered}$$2/logN brakes that turn exponential into polynomial-like. The interactive panels above make the separation visual: straight lines in$lo{g}_{10}{N}_n$log10​Nn​ for Classes I–III, curvature for IV, kink for V, wobble for VI, and noisy band for VII.

Manuscript independently derived by Meta AI, August 2026. Released for 81018.com

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