Subradical and Radical Dominance

• Subradical and radical dominance are formal analytical concepts that distinguish asymptotic function growth and competing reaction pathways in chemistry and number theory.
• They provide a framework to set precise sifting thresholds and control analytic errors in Selberg–Delange sums through defined growth orders.
• In astrochemistry, these orders quantify temperature-dependent kinetic competition between hydrogen abstraction and radical–radical coupling reactions.

Subradical and radical dominance are formalized analytical notions introduced for classifying the asymptotic growth of functions and, in parallel, for describing competing mechanistic pathways in chemistry and number theory. In arithmetic contexts, these dominance orders offer a precise framework for comparing sifting thresholds in Selberg–Delange-type sums and for analyzing the boundary of analytic error control. In chemical contexts, particularly astrochemical surface chemistry, they quantify the kinetic competition between hydrogen abstraction (“subradical”) and radical–radical coupling pathways as temperature varies. This article presents rigorous definitions, algebraic properties, mechanistic illustrations, and analytic significance as developed in recent works (Alamoudi, 15 Jan 2026, Enrique-Romero et al., 2022).

1. Definitions of Subradical and Radical Dominance

Let $f(x)$ and $g(x)$ be positive real-valued functions on $[X_0,\infty)$. Subradical dominance is defined by $f<_{\sqrt[\forall]{}}g$ if for every $\epsilon>0$, $f(x)=o(g(x)^\epsilon)$ as $x\to\infty$. Equivalently, $f$ is subradical if $f=o(x^\epsilon)$ for all $\epsilon>0$. There exists $h(u)=o(u)$ such that $f(x)\ll \exp(h(\log x))$.

Radical dominance is defined by $f<_{\sqrt[\exists]{}}g$ if there exists some fixed $0<\epsilon<1$ for which $f(x)=o(g(x)^{1-\epsilon})$ as $x\to\infty$; $f$ grows strictly more slowly than a positive power of $g$.

2. Illustrative Examples and Algebraic Properties

Several archetype functions demonstrate these orders:

• Typical subradical examples: $\exp(\sqrt{\log x})<_{\sqrt[\forall]{}} x$ since $\exp(\sqrt{\log x})=o(x^\epsilon)$ for any $\epsilon>0$. More sharply, $$\exp\left(\frac{\log x}{(\log\log x)^{1+\delta}}\right)<_{\sqrt[\forall]{}}x$$ for all $\delta>0$. Such hierarchies are strict and chainable.
• Radical but not subradical: Pure powers $x^\alpha$ ($\alpha>0$) are not subradical; $x^\alpha<_{\sqrt[\exists]{}}x$.
• Incomparable or identical lower sets: For $<_{\sqrt[\forall]{}}$, the lower set for $x$ coincides with that for $x/\log x$, despite these functions being non-comparable; for $<_{\sqrt[\exists]{}}$, pure powers are distinguished, but $x$ and $x/\log x$ share the same radical lower set.

The subradical order is transitive and closed under sums, products, and positive powers (Corollary 6.2). Radical dominance distinguishes pure powers but does not separate $x$ from $x/\log x$.

Example Function	Subradical Order	Radical Order
$\exp((\log x)^{1-\delta})$	$<_{\sqrt[\forall]}$ next	$<_{\sqrt[\exists]} x$
$x^\alpha$	No	$<_{\sqrt[\exists]} x$
$x$, $x/\log x$	Incomparable	Same lower set

3. Main Structural Theorems

The principal characterization is as follows:

• Theorem 6.4: $\{h: h<_{\sqrt[\exists]}f\} = \{h: h<_{\sqrt[\exists]}g\}$ iff $\frac{f}{g}<_{\sqrt[\forall]}f$ and $\frac{g}{f}<_{\sqrt[\forall]}g$. The proof leverages the non-inclusion of small powers in one set when the other fails subradicality.
• Theorem 6.5: For $$Y(x)=Y_0\exp\bigg(\mathscr{p}\frac{\log x}{(\log\log(x+1))^{1+\epsilon}}\bigg)$$, $\log Y(x)$ is maximal among all $\log s(x)$ for subradical $s$, i.e., for any subradical $s$, either $\log s<_{\sqrt[\exists]}\log Y$ or $\log s$ and $\log Y$ are incomparable. This establishes a boundary for subradical sifting ranges.

4. Analytic Significance in Sifted Sums and Selberg–Delange Theory

In the analytic study of sums such as

$$M_{k,\omega}(x,y) = \sum_{\substack{n\leq x \ p_1(n)>y}} \mu(n) \binom{\omega(n)-1}{k-1},$$

the size of $y=y(x)$ dictates the tractability of generating function estimates under Selberg–Delange machinery. Classical results permit $y$ up to $\exp(c\sqrt{\log x})$, a subradical range, but Alamoudi establishes that for any $\epsilon>0$, $$y\leq Y_0\exp(\mathscr{p}\log x/(\log\log x)^{1+\epsilon})$$ is asymptotically at the boundary of subradical ranges accessible by analytic methods using $<_{\sqrt[\forall]{}}$ order (Alamoudi, 15 Jan 2026).

This framework enables comparison and optimization of sifting thresholds, directly impacting the precision and reach of error estimates. Alladi’s higher-order dualities further anchor the centrality of such sums—since larger subradical $y$-ranges strengthen duality-based results for $k$-th prime factors.

5. Mechanistic Pathway Dominance in Radical Chemistry

In astrochemical surface chemistry, the dominance of radical–radical coupling (Rc) versus hydrogen abstraction (dHa) is determined by kinetic competition governed by activation barriers and temperature (Enrique-Romero et al., 2022). Rc occurs when two radicals form a covalent bond; dHa involves direct transfer of a hydrogen atom. The effective rate constants are derived via the Eyring expression,

$$k(T)=\frac{k_B T}{h}\exp\bigg(-\frac{ΔG^{‡}}{k_B T}\bigg),$$

where $ΔG^{‡}\simeq ΔH^{‡}$ at low temperatures.

Dominance varies:

• At $T\leq 20$ K, barrierless or $\Delta H^{‡}<2$ kJ Rc reactions dominate (large $k$), with dHa suppressed if its barrier exceeds 3 kJ.
• At $20-60$ K, Rc with barrier $<5$ kJ and dHa with barrier near 2 kJ are both competitive.
• At $T\geq 100$ K, nearly all channels with $\Delta H^{‡}<10$ kJ become fast, but high-barrier pathways remain slow.

Minor or subradical channels, such as secondary abstraction from CH$_3$O (13.2 kJ) or CH$_2$OH (39 kJ), are kinetically irrelevant at interstellar temperatures. Coupling with large barriers (e.g., CH$_3$O+CH$_3$O, 20.1 kJ) is essentially shut off.

6. Product Distribution and Pathway Branching

Distinct categories of product distribution arise from mechanistic dominance:

• Category 1 (Rc only): Ethane, methylamine, ethylene glycol are favored; dHa forbidden or too slow.
• Category 2 (Rc vs dHa competition): Glyoxal, methyl formate, glycolaldehyde, formamide; rates of Rc and dHa closely compete.
• Category 3 (Rc plausible, dHa only at low T via tunneling): Dimethyl ether, ethanol; at $T<30$ K, dHa may become relevant.
• Category 4 (Rc suppressed, only dHa via tunneling): Acetaldehyde, dimethyl peroxide; both pathways are slow except at high temperature.

System	Rc Barrier (kJ)	dHa Barrier (kJ)	Dominant Channel (10–100 K)
CH$_3$ + NH$_2$	≈0	n/a	Rc
HCO + CH$_2$OH	1.7	≈0 (barrierless)	dHa
CH$_3$O + CH$_3$O	20.1	11.7	Subradical (dHa)

This supports a temperature-dependent paradigm, with subradical abstraction and radical coupling pathways dynamically shifting in dominance as environmental conditions change.

7. Connection to Analytic Machinery and Partial Order Frameworks

The subradical and radical dominance orders intersect analytic error analysis in Selberg–Delange-type sums and mechanistic astrochemistry. In number theory, these orders demarcate the maximal range achievable for sifting thresholds under the present analytic formalism. The Hankel–gamma–derivative approach, generalized in (Alamoudi, 15 Jan 2026), extends the classical gamma-pole machinery, enabling refined extraction of higher-order terms and error bounds via combinatorial formulae in terms of Bell and Stirling numbers.

A plausible implication is that the introduction of strict partial order language ($<_{\sqrt[\forall]{}}$, $<_{\sqrt[\exists]{}}$) for growth comparison provides an invariant, transferably useful paradigm for both analytic number theory and physical chemistry, ensuring rigorous demarcation of feasible mechanisms—whether sieving in $M_{k,\omega}(x,y)$ or channel selection in astrochemical networks.

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Subradical and radical dominance formalize both a hierarchy of growth rates and a mechanism selection principle. They sharpen analytic comparisons, set precise boundaries in sifting and error control, and clarify pathway prevalence in kinetic chemistry, thus constituting foundational orders in both mathematics and physical sciences (Alamoudi, 15 Jan 2026, Enrique-Romero et al., 2022).