Let $p$ be a fixed prime number (2, 3, 5, …). For any nonzero rational number $x \in \mathbb{Q}$, write it uniquely as
$$ x = p^{n} \frac{a}{b}, $$
where $a, b \in \mathbb{Z}$ are integers not divisible by $p$, and $n \in \mathbb{Z}$. The p‑adic absolute value (or p‑adic norm) is defined by
$$ |x|{p} = p^{-n}, \qquad |0|{p} = 0. $$
Properties:
$$ |x+y|{p} \le \max(|x|{p}, |y|_{p}). $$
The ordinary Archimedean absolute value $|x|$ satisfies $|x+y| \le |x|+|y|$; the strong triangle inequality is strictly stronger. It implies that all triangles are isosceles: for any three points $a,b,c$, the two largest distances among $|a-b|{p}, |b-c|{p}, |a-c|_{p}$ are equal.
The field of p‑adic numbers $\mathbb{Q}{p}$ is the completion of the rational numbers $\mathbb{Q}$ with respect to the metric $d{p}(x,y) = |x-y|_{p}$. Every p‑adic number can be written uniquely as a Laurent series in $p$:
$$ x = \sum{k=n}^{\infty} a{k} p^{k}, \qquad a{k} \in \{0,1,\dots , p-1\},\; a{n} \neq 0. $$
The integer $n$ is the p‑adic valuation $v{p}(x) = n$; then $|x|{p}=p^{-n}$. The p‑adic integers $\mathbb{Z}_{p}$ are those with $n \ge 0$ (no negative powers of $p$).
Example ($p=2$): $$ \frac{1}{3} = 1 + 2 + 2^{2} + 2^{4} + 2^{5} + \cdots \quad\text{(repeating pattern)}, $$ so $\left|\frac{1}{3}\right|_{2}=1$.
Arithmetic in $\mathbb{Q}_{p}$ proceeds digit‑by‑digit with carries, exactly as in base‑$p$ arithmetic but with infinite expansions allowed to the left (for integers) and to the right (for fractions).
For a prime $p$, the Bruhat‑Tits tree $T_{p}$ is an infinite, connected, cycle‑free graph (a tree) where every vertex has degree $p+1$. It can be constructed in several equivalent ways:
Properties of $T_{p}$:
Visualization: For $p=2$, the tree is a binary tree (each vertex has three neighbours). For $p=3$, it is a ternary tree (four neighbours), etc.
The Monna map (also called the Minkowski question‑mark function for $p=2$) is a continuous, measure‑preserving bijection $M{p}:\mathbb{Q}{p} \to \mathbb{R}$. It is defined by “flipping” the p‑adic expansion:
If $$ x = \sum{k=n}^{\infty} a{k} p^{k}, \qquad a{k} \in \{0,1,\dots ,p-1\}, $$ then $$ M{p}(x) = \sum{k=n}^{\infty} a{k} p^{-k}. $$
Notice the exponent changes sign: $p^{k}$ becomes $p^{-k}$. This turns the p‑adic metric (where higher powers of $p$ are smaller) into the real metric (where higher powers of $p$ are larger).
Key properties:
Example ($p=2$): The dyadic rational $x = 0.1011{2} = 2^{-1}+2^{-3}+2^{-4}$ in real notation corresponds to the 2‑adic number $x = \dots 000.1011{2}$ (since in 2‑adics, higher powers are smaller). Under $M{2}$, we simply read the digits as a real binary expansion: $M{2}(x) = 0.1011_{2} = \frac{11}{16}$.
Physical role: In the STC, the Monna map is the coarse‑graining that projects the discrete, hierarchical Bruhat‑Tits tree onto the continuous spacetime we perceive. It explains why continuous physics works so well at macroscopic scales, while preserving the discrete, syntactic foundation.
A system is discrete scale invariant if it is invariant under rescaling by a fixed factor $q>1$. That is, if $f(x)$ is an observable, then
$$ f(qx) = \lambda f(x) $$
for some constant $\lambda$. The general solution of this functional equation is
$$ f(x) = x^{\alpha} P\!\left(\frac{\ln x}{\ln q}\right), $$
where $\alpha = \ln\lambda / \ln q$ and $P$ is a periodic function with period 1. Writing the periodic function as a Fourier series gives log‑periodic oscillations:
$$ f(x) = x^{\alpha} \left[ A{0} + \sum{n=1}^{\infty} A{n} \cos\!\left( \frac{2\pi n}{\ln q} \ln x + \phi{n} \right) \right]. $$
The Bruhat‑Tits tree is self‑similar under scaling by $p$; hence any observable derived from it (e.g., correlation functions, mass ratios, CMB power spectrum) will exhibit log‑periodicity with scale factor $q = p$.
| Symbol | Meaning | Typical value |
|---|---|---|
| $p$ | Prime underlying the tree | |
| $q$ | Discrete scale factor | |
| $\ln q$ | Log‑periodic period | |
| $v_{p}(x)$ | p‑adic valuation | |
| $\ | x\ | |
| $M_{p}(x)$ | Monna map |
Relation to Planck scale: The fundamental length scale in the STC is the Planck length $\ell_{P} = \sqrt{\hbar G/c^{3}} \approx 1.616 \times 10^{-35}\ \text{m}$. The depth of the tree at a given physical scale $L$ is roughly
$$ \text{depth} \sim \frac{\ln(L/\ell_{P})}{\ln p}. $$
Thus, the observable universe (size $\sim 10^{27}\ \text{m}$) corresponds to a depth of about
$$ \frac{\ln(10^{27} / 10^{-35})}{\ln 2} \approx \frac{\ln(10^{62})}{\ln 2} \approx \frac{142.7}{0.693} \approx 206 $$
levels. This large depth ensures that the discrete structure is effectively continuous at cosmological scales, but its log‑periodic imprint remains detectable.