% On the Causality of Natural Numbers % Aurey Hyppa, An M. Rodriguez* % September 3, 2025 ## Abstract We define a canonical era function on the natural numbers by withholding each integer until it can be generated from already admitted integer labels. Powers are taken as primitive generation events, and multiplicative recombination is allowed only across coprime blocks. This yields a recursive map $\tau:\mathbb N\to\mathbb Z_{\ge 1}$, where $\tau(n)$ is the least era in which $n$ can appear. The resulting structure is a well-founded dependency order, distinct from the usual order by size. Primes are forced to appear at their own eras, while composites may appear earlier than their numerical value when their generators are already present. We give the formal definition, derive basic properties, and compute the first era sets. ## One-Sentence Summary Natural numbers can be ordered by the earliest era in which they can be generated from admitted integer labels using powers and coprime recombination. ## Keywords natural numbers, causality, constructive order, prime powers, generations, dependency order ## Introduction The natural numbers $\mathbb N=\{1,2,3,\dots\}$ are usually ordered by size. That order is static: every factorization is valid at once. Here we instead introduce a generational order. Integers are withheld until the integer labels needed to generate them have appeared. The intended examples are: - era $1$: $1^1$ - era $2$: $2, 2^2$ - era $3$: $3, 3^2, 2^3, 3^3$ - era $4$: $2^4, 3^4, 4^2, 4^3, 4^4$ The key point is that powers are primitive generation events. An integer such as $64$ need not wait for the representation $2^6$; it can already appear in era $4$ as $4^3$. More generally, once a number has appeared, it may later act as a base when its own label-era arrives. To pass from examples to mathematics, we define a canonical era function $\tau(n)$. ## Theory: Era Function ### Power cost For $n\ge 1$, define the power cost $$ \kappa(n) = \min\{\max(a,b): n=a^b,\ a,b\in\mathbb N,\ b\ge 1\}. $$ The trivial representation $n=n^1$ is always allowed, so $\kappa(n)\le n$. Examples: $$ \kappa(4)=2, \qquad \kappa(8)=3, \qquad \kappa(16)=4, \qquad \kappa(64)=4, $$ since $$ 4=2^2,\quad 8=2^3,\quad 16=2^4=4^2,\quad 64=2^6=4^3=8^2. $$ ### Era function Define $\tau:\mathbb N\to\mathbb Z_{\ge 1}$ recursively by $$ \tau(1)=1, $$ and for $n>1$, $$ \tau(n) = \min\left( \kappa(n), \min_{\substack{ab=n\\1 --- - [Preferred Frame Pre-Prints's on GitHub.com](https://github.com/siran/research) - Back to main journal: [Preferred Frame](https://preferredframe.com) (built: 2026-03-25 16:26 EDT UTC-4)