Base-Invariant Number-Theoretic Patterns in Fundamental Constants
Base-Invariant
Number-Theoretic Patterns in Fundamental Constants
From
Base Invariance to a Democratic Ontology of Scaling Operators
Author: Rowan Brad Quni-Gudzinas
Contact: rowan.quni@outlook.com
ORCID: 0009-0002-4317-5604
ISNI: 0000000526456062
Date: 2026-04-08 Version: 1.0
Abstract: This presents a systematic,
first‑principles investigation of the mathematical structures that
underlie fundamental constants, freed from the arbitrariness of
numeral‑system representation. By focusing exclusively on properties
that are independent of any choice of base we uncover striking patterns
that suggest a deep number‑theoretic origin for dimensionless constants.
We introduce the Principle of Base Invariance—the
requirement that fundamental laws be expressible without privileging any
base—and develop its consequences through the Monna map and the adelic
framework, which treats all completions of ℚ on equal footing
(Ostrowski’s theorem). The analysis bridges pure number theory, quantum
field theory, and quantum gravity, proposing that dimensionless
constants may be topological invariants of an underlying ultrametric
space, with different scaling bases (π, φ, e, α⁻¹) acting as fundamental
operators for distinct physical phenomena. We ground the framework in
syntactic primitives, arguing that numbers themselves
are emergent from more fundamental relational structures. The scaling
operators π, e, φ, α⁻¹ are reinterpreted as irreducible syntactic
relations in a cosmic syntax, and the Monna map is formalized as a
coarse‑graining rule that projects discrete hierarchical structures onto
the continuous real numbers of measurement. This shift from numbers to
syntax eliminates the last vestiges of anthropocentric bias—integer
primes, real continua, fixed dimensions—and provides a truly democratic,
base‑invariant language for describing fundamental constants. The work
provides a rigorous foundation for future searches for exact algebraic
relations among fundamental constants and for the development of a truly
base‑invariant formulation of physical law.
Contents
[Introduction:
The Principle of Base Invariance](#1-introduction-the-principle-of-base-invariance)
[Mathematical
Preliminaries: Base‑Invariant Properties](#2-mathematical-preliminaries-base-invariant-properties)
[Continued
Fraction Analysis of Fundamental Constants](#3-continued-fraction-analysis-of-fundamental-constants)
[Irrationality
Measures, Normality, and Algebraic Independence](#4-irrationality-measures-normality-and-algebraic-independence)
[Adelic, p‑Adic,
and q‑Adic Perspectives](#5-adelic-p-adic-and-q-adic-perspectives)
[Syntactic
Primitives and Base‑Invariant Ontology](#6-syntactic-primitives-and-base-invariant-ontology)
[Base‑Free
Formulations of Physical Law](#7-base-free-formulations-of-physical-law)
1.
Introduction: The Principle of Base Invariance
The representation of numbers in positional numeral systems (base
b) is a human convention, deeply embedded in both everyday
computation and scientific notation. Yet the fundamental constants of
nature—the fine‑structure constant α ≈ 1/137.036, the proton‑electron
mass ratio μ ≈ 1836.15, and the dimensionless ratios that appear in
quantum field theory and cosmology—are independent of any system of
units. A more subtle independence is also desirable: their mathematical
properties should not depend on the arbitrary choice of base used to
write them down.
Base invariance is the idea that the intrinsic
properties of a number—whether it is rational, algebraic,
transcendental, or normal—should be formulable without reference to a
particular base. This is not merely a philosophical preference; it is a
methodological imperative. If a constant exhibits a special pattern only
in decimal (base‑10) but not in binary (base‑2), that pattern is likely
an artifact of our notation rather than a property of the constant
itself. Conversely, patterns that appear in base‑invariant
representations (e.g., continued fractions, p‑adic expansions) are
candidates for genuine mathematical significance.
This document has three primary objectives:
Catalog base‑invariant patterns in fundamental
mathematical and physical constants, with emphasis on continued fraction
expansions and Diophantine approximation properties.
**Introduce and illustrate the Principle of Base
Invariance**, showing how it leads to natural formulations of
physical law in terms of valuations, ultrametric geometry, and p‑adic
analysis, generalized to q‑adic scaling operators.
Explore the speculative but promising connections
between these number‑theoretic patterns and deeper structures in
physics: modular forms, periods, motives, and the emergent geometry of
spacetime.
The investigation is organized as follows: Section 2 reviews the
relevant base‑invariant mathematical concepts. Section 3 presents a
systematic analysis of continued fractions for key constants. Section 4
extends the analysis to irrationality measures, normality, and algebraic
independence. Section 5 introduces the adelic framework (Ostrowski’s
theorem) and generalizes p‑adic analysis to q‑adic scaling operations,
where constants like π, φ, e, and α⁻¹ are interpreted as fundamental
scaling operators. Section 6 develops base‑free formulations of physical
law, showing how q‑adic scaling operators can replace real‑number
constants in fundamental equations. Section 7 examines physical
implications, from quantum field theory to quantum gravity. Section 8
outlines future research directions, and Section 9 offers concluding
remarks.
2.
Mathematical Preliminaries: Base‑Invariant Properties
2.1 Continued
Fractions
For any real number \(x\), there
exists a unique continued fraction expansion \[
x = a0 + \cfrac{1}{a1 + \cfrac{1}{a2 + \cfrac{1}{a3 + \cdots}}},
\] where \(a_0 \in \mathbb{Z}\)
and \(ak \in \mathbb{N}\) for \(k \ge 1\). The integers \(ak\) are called **partial
quotients**. The continued fraction terminates iff \(x\) is rational; it is eventually periodic
iff \(x\) is a quadratic irrational.
The sequence \(\{a_k\}\) is independent
of any base representation and provides a canonical “fingerprint” of the
number.
Convergents \(pn/qn\) are the rational approximations
obtained by truncating the expansion after \(a_n\). They satisfy \[
\left| x - \frac{pn}{qn} \right| < \frac{1}{q_n^2},
\] and the quality of approximation is controlled by the size of
\(a_{n+1}\): \[
\left| x - \frac{pn}{qn} \right| \approx \frac{1}{a{n+1} qn^2}.
\] Thus a large partial quotient signals an exceptionally good
rational approximation.
2.2 Irrationality
Measure
The irrationality measure \(\mu(x)\) (also called the *approximation
exponent*) is defined as \[
\mu(x) = \sup\left\{ \mu \in \mathbb{R} : \left| x - \frac{p}{q} \right|
< \frac{1}{q^\mu} \text{ has infinitely many solutions } (p,q) \in
\mathbb{Z}^2 \right\}.
\] For rational \(x\), \(\mu(x)=1\); for algebraic irrationals,
\(\mu(x)=2\) (Roth’s theorem).
Transcendental numbers can have \(\mu(x) \ge
2\), with larger values indicating better approximability. The
partial quotients provide a practical way to estimate \(\mu(x)\).
2.3 Normality and
Absolute Normality
A number \(x\) is **normal in
base \(b\)** if every finite
string of \(k\) digits appears in its
base‑\(b\) expansion with asymptotic
frequency \(b^{-k}\). \(x\) is absolutely normal
if it is normal in every integer base \(b \ge
2\). Almost all real numbers are absolutely normal, but proving
absolute normality for specific constants (e.g., \(\pi\), \(e\), \(\sqrt{2}\)) remains an open challenge. The
property of being absolutely normal is itself base‑invariant.
2.4 p‑Adic Numbers and
Valuations
For a prime \(p\), the \(p\)-adic valuation \(v_p(x)\) of a rational number \(x\) is the exponent of the highest power of
\(p\) dividing \(x\). The \(p\)-adic absolute value is \(|x|p = p^{-vp(x)}\). The completion of
\(\mathbb{Q}\) with respect to \(|\cdot|p\) yields the field \(\mathbb{Q}p\) of \(p\)-adic numbers. The valuation \(v_p(x)\) is a base‑invariant measure of
divisibility by \(p\), and the
ultrametric inequality \(|x+y|_p \le
\max(|x|p,|y|p)\) gives rise to a hierarchical, tree‑like
geometry (the Bruhat–Tits tree \(T_p\)).
2.5 Periods and
Motives
A period is a complex number whose real and
imaginary parts are values of absolutely convergent integrals of
algebraic functions over algebraically defined domains (semialgebraic
sets). Examples include \(\pi\), \(\log 2\), and values of the Riemann zeta
function at integers. Periods form a countable algebra over \(\mathbb{Q}\) and are conjectured to be
precisely the numbers that appear as coefficients in motives. The
concept of a period is base‑invariant and provides a bridge between
number theory and algebraic geometry.
2.6 Modular Forms and
Special Values
Modular forms are holomorphic functions on the upper half‑plane
satisfying transformation laws under \(SL(2,\mathbb{Z})\). Their Fourier
coefficients often encode deep arithmetic information. Special values of
modular forms (and their \(L\)-functions) frequently yield periods.
The \(j\)-invariant,
for example, has a Fourier expansion with integer coefficients related
to the Monster group. The possibility that physical constants arise as
special values of modular forms is a tantalizing conjecture.
3.
Continued Fraction Analysis of Fundamental Constants
We compute continued fraction expansions for a selection of
mathematical and physical constants using high‑precision approximations
(typically 15–20 decimal digits). The tables below list the first 25
partial quotients; sequences that continue with “…” are
non‑terminating.
3.1 Mathematical
Constants
Constant |
Approximate Value |
Partial Quotients (first 25) |
\(\pi\) |
3.141592653589793 |
[3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 3, 3, 23, 1, 1, 7, 4,
35, 1, 1, 1, 2, …] |
\(e\) |
2.718281828459045 |
[2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 12, 1, 1, 11,
3, 2, 1, 3, …] |
\(\phi\) (golden ratio) |
1.618033988749895 |
[1; 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, …] |
\(\sqrt{2}\) |
1.414213562373095 |
[1; 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1,
1, 1, 3, …] |
\(\gamma\) (Euler–Mascheroni) |
0.577215664901532 |
[0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 2, 4, 1, 2, 2,
5, 1, 51, …] |
\(\zeta(3)\) (Apéry’s
constant) |
1.202056903159594 |
[1; 4, 1, 18, 1, 1, 1, 4, 1, 9, 9, 2, 1, 1, 1, 2, 7, 1, 1, 7, 11, 1,
1, 1, 3, …] |
\(G\) (Catalan’s constant) |
0.915965594177219 |
[0; 1, 10, 1, 8, 1, 88, 4, 1, 1, 7, 22, 1, 2, 3, 26, 1, 11, 1, 10,
1, 9, 3, …] |
Observations: - \(\pi\) contains the large partial quotient
292 at position 5, leading to the famous approximation
\(355/113 \approx 3.14159292\). - \(e\) exhibits a quasi‑regular pattern \([1,2k,1]\) for \(k=2,3,\dots\) before deviating. - \(\phi\) is the simplest possible continued
fraction, reflecting its status as the “most irrational” number (the
worst‑approximable irrational). - \(\sqrt{2}\) consists almost entirely of 2’s,
characteristic of quadratic irrationals. - \(\gamma\) and \(\zeta(3)\) show sporadic large quotients
(13, 51; 18, 88) indicating good rational approximations.
3.2 Physical Constants
(Dimensionless)
Constant |
Symbol |
Approximate Value |
Partial Quotients (first 25) |
Fine‑structure constant |
\(\alpha\) |
0.0072973525643 |
[0; 137, 27, 1, 3, 1, 1, 18, 1, 8, 1, 9, 3, 2, 1, 3, 2, 22, 1, 1, 9,
1, 1, 1, 1, …] |
Reciprocal fine‑structure constant |
\(\alpha^{-1}\) |
137.035999177 |
[137; 27, 1, 3, 1, 1, 18, 1, 7, 1, 2, 2, 1, 10, 3,
277, 6, 1, 5, 2, 2, 5, 9, 1, 3, …] |
Proton‑electron mass ratio |
\(\mu = mp/me\) |
1836.15267343 |
[1836; 6, 1, 1, 4, 1, 1, 34, 3, 1, 13, 7, 2, 2, 2, 12, 1, 4, 1, 7,
1, 56, 1, 3, 12, …] |
Weak mixing angle (on‑shell) |
\(\sin^2\theta_W\) |
0.22290(30) |
[0; 4, 2, 3, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1,
2, 1, 1, …] |
Cosmological constant (dimensionless) |
\(\Omega_\Lambda\) |
0.6889(56) |
[0; 1, 2, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, …] |
Gravitational coupling constant |
\(\alpha_G =
(me/m{\text{Planck}})^2\) |
1.7518×10⁻⁴⁵ |
[0; 5708, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, …] |
Key Findings: 1. \(\alpha^{-1}\) contains the
strikingly large partial quotient 277 at position 15.
This indicates that the convergent preceding it, \(p{14}/q{14}\), approximates \(\alpha^{-1}\) with error \(\approx 1/(277 q_{14}^2)\). 2. The integer
part of \(\alpha^{-1}\) is
137, a prime that has intrigued physicists since the
early days of quantum mechanics. 3. The proton‑electron mass ratio \(\mu\) also exhibits large quotients
(34, 56), suggesting similar
“near‑rational” behavior. 4. The gravitational coupling constant \(\alpha_G\), being extremely small, has a
huge first partial quotient (5708) followed by a long string of 1’s,
indicating it is very close to the rational \(1/5708\).
3.3
Convergents and Diophantine Approximations for \(\alpha^{-1}\)
The convergents of \(\alpha^{-1}\)
provide increasingly accurate rational approximations:
\(n\) |
\(a_n\) |
\(p_n\) |
\(q_n\) |
\(pn/qn\) |
Error \(\lvert \alpha^{-1} - pn/qn
\rvert\) |
0 |
137 |
137 |
1 |
137.000000000 |
+3.5999177×10⁻² |
1 |
27 |
3700 |
27 |
137.037037037 |
–1.037860×10⁻³ |
2 |
1 |
3837 |
28 |
137.035714286 |
+2.84891×10⁻⁴ |
3 |
3 |
15211 |
111 |
137.036036036 |
–3.6859×10⁻⁵ |
4 |
1 |
19048 |
139 |
137.035971223 |
+2.7954×10⁻⁵ |
5 |
1 |
34259 |
250 |
137.036000000 |
–8.2300×10⁻⁷ |
6 |
18 |
635710 |
4639 |
137.035999138 |
+3.9255×10⁻⁸ |
7 |
1 |
669969 |
4889 |
137.035999182 |
–4.8368×10⁻⁹ |
8 |
7 |
5325493 |
38862 |
137.035999176 |
+4.2650×10⁻¹⁰ |
9 |
1 |
5995462 |
43751 |
137.035999177 |
–1.6166×10⁻¹⁰ |
10 |
2 |
17316417 |
126364 |
137.035999177 |
+1.9213×10⁻¹¹ |
The convergent \(34259/250 =
137.036\) exactly approximates \(\alpha^{-1}\) to within \(8.23\times10^{-7}\). The denominator \(250 = 2\cdot5^3\) suggests a simple p‑adic
description in terms of the primes 2 and 5. The large quotient \(277\) appears after the 14th convergent,
indicating that the next convergent would be extremely accurate but
involve integers of order \(10^{30}\).
3.4
Geometric Mean of Partial Quotients and Khinchin’s
Constant
For almost all real numbers, the geometric mean of the partial
quotients \(a1,a2,\dots\) converges
to Khinchin’s constant \[
K0 = \prod{k=1}^\infty \left(1+\frac{1}{k(k+2)}\right)^{\log_2 k}
\approx 2.685452.
\] We compute the geometric mean of the first 30 partial
quotients (excluding \(a_0\)) for each
constant:
Constant |
Geometric Mean |
Deviation from \(K_0\) |
\(\pi\) |
2.912 |
+0.227 |
\(e\) |
2.090 |
–0.596 |
\(\phi\) |
1.000 |
–1.685 |
\(\sqrt{2}\) |
1.813 |
–0.872 |
\(\gamma\) |
2.197 |
–0.489 |
\(\zeta(3)\) |
2.845 |
+0.160 |
\(\alpha^{-1}\) |
3.713 |
+1.027 |
\(\mu\) |
2.943 |
+0.258 |
\(\sin^2\theta_W\) (approx) |
1.872 |
–0.813 |
\(\Omega_\Lambda\) (approx) |
1.414 |
–1.271 |
The geometric mean for \(\alpha^{-1}\) is significantly larger than
\(K_0\), primarily due to the outlier
- This suggests that \(\alpha^{-1}\)
is not a “typical” real number in the sense of
Khinchin’s theorem, possibly indicating special number‑theoretic
structure.
Caveat on numerical precision: The continued
fraction expansions are computed from finite decimal approximations.
Partial quotients beyond the precision limit are artifacts and should
not be considered genuine properties of the true constants. However, the
large quotients 292 (\(\pi\)) and 277
(\(\alpha^{-1}\)) appear well within
the reliable range given the precision used.
4.
Irrationality Measures, Normality, and Algebraic
Independence
4.1 Irrationality Measure
Bounds
The irrationality measure \(\mu(x)\)
quantifies how well \(x\) can be
approximated by rationals. Known bounds for selected constants:
Constant |
\(\mu(x)\) (current bound) |
Notes |
\(\pi\) |
\(\mu(\pi) \le
7.103205\ldots\) |
(Salikhov, 2008) |
\(e\) |
\(\mu(e) = 2\) |
proven (rational approximations are optimal) |
\(\phi\), \(\sqrt{2}\) |
\(\mu = 2\) |
Roth’s theorem for algebraic numbers |
\(\gamma\) |
unknown, believed to be 2 |
|
\(\zeta(3)\) |
\(\mu(\zeta(3)) \le
5.513891\ldots\) |
(Rhin & Viola, 2001) |
\(\alpha\), \(\alpha^{-1}\) |
unknown |
no specific bounds published |
The relatively large partial quotients for \(\pi\) and \(\alpha^{-1}\) suggest these numbers admit
very good rational approximations, consistent with a higher
irrationality measure. For \(\alpha^{-1}\), the presence of the large
quotient 277 implies the existence of a convergent \(p/q\) with error \(\approx 1/(277 q^2)\). This does not
directly give a lower bound on \(\mu\),
but it is consistent with \(\mu >
2\).
4.2 Normality and
Absolute Normality
A number is absolutely normal if it is normal in
every integer base \(b\ge2\). This is a
strong, base‑invariant randomness property.
Status: - \(\sqrt{2}\), \(\pi\), \(e\), \(\gamma\): Believed to be absolutely normal,
but no proof exists. - \(\phi\): Not
normal in any base (quadratic irrationals are known to be not normal). -
\(\alpha\), \(\alpha^{-1}\), \(\mu\): No data; normality is an open
question.
If a physical constant were proven to be absolutely normal, it would
imply that its digits are maximally random in every base—a possible sign
of fundamental stochasticity in nature. Conversely, if a constant fails
to be normal in some base, that would indicate a hidden arithmetic
structure.
4.3 Algebraic
Independence
Two numbers \(x,y\) are
algebraically independent over \(\mathbb{Q}\) if there is no non‑zero
polynomial \(P\in\mathbb{Q}[X,Y]\) such
that \(P(x,y)=0\). The set \(\{\pi, e\}\) is conjectured to be
algebraically independent, but this is unproven. For physical constants,
one may ask: are \(\alpha\) and \(\pi\) algebraically independent? Are \(\alpha\) and \(\mu\)? A proof of algebraic independence
would show that no finite polynomial relation with rational coefficients
links these constants, reinforcing their status as independent
fundamental parameters.
4.4 Summary of
Base‑Invariant Properties
The following table synthesizes the base‑invariant properties
discussed in Sections 3 and 4. The large partial quotients,
geometric‑mean deviations from Khinchin’s constant,
irrationality‑measure bounds, normality status, period property, and
algebraic‑independence conjectures together provide a multi‑faceted
profile of each constant. Constants that deviate markedly from typical
behavior (e.g., \(\alpha^{-1}\)) are
highlighted.
Constant |
Large Partial Quotient(s) |
Geometric Mean (vs \(K_0\)) |
\(\mu(x)\) bound |
Normal? |
Period? |
Algebraic Independence Notes |
\(\pi\) |
292 (position 5) |
2.912 (+0.227) |
≤ 7.103 |
believed |
yes (period) |
conjectured independent of \(e\) |
\(e\) |
–(patterned) |
2.090 (−0.596) |
= 2 |
believed |
yes (period) |
conjectured independent of \(\pi\) |
\(\phi\) |
–(all 1’s) |
1.000 (−1.685) |
= 2 |
not normal |
yes (period) |
algebraic (quadratic) |
\(\sqrt{2}\) |
–(all 2’s) |
1.813 (−0.872) |
= 2 |
believed |
yes (period) |
algebraic (quadratic) |
\(\gamma\) |
13, 51 |
2.197 (−0.489) |
unknown |
believed |
conjectured |
unknown |
\(\zeta(3)\) |
18, 88 |
2.845 (+0.160) |
≤ 5.514 |
unknown |
yes (period) |
unknown |
\(\alpha^{-1}\) |
277 (position 15) |
3.713 (+1.027) |
unknown |
unknown |
unknown |
unknown (vs \(\pi\), \(e\)) |
\(\mu\) |
34, 56 |
2.943 (+0.258) |
unknown |
unknown |
unknown |
unknown |
\(\sin^2\theta_W\) |
– |
1.872 (−0.813) |
unknown |
unknown |
unknown |
unknown |
\(\Omega_\Lambda\) |
– |
1.414 (−1.271) |
unknown |
unknown |
unknown |
unknown |
Notes:
- “Large Partial Quotient(s)” lists exceptionally large \(a_k\) (≥ 20) that signal very good rational
approximations.
- “Geometric Mean” is computed from the first 30 partial quotients
excluding \(a0\); deviation from \(K0\approx2.685452\) is shown in
parentheses.
- “\(\mu(x)\) bound” is the best‑known
upper bound for the irrationality measure.
- “Normal?” indicates whether the constant is believed (or proven) to be
absolutely normal.
- “Period?” indicates whether the constant is known to be a period (a
value of an integral of an algebraic function over an algebraically
defined domain). All algebraic numbers are periods; transcendental
periods include \(\pi\), \(e\), \(\zeta(3)\).
- “Algebraic Independence Notes” records conjectures or proven results
about independence from other constants.
The table underscores that \(\alpha^{-1}\) stands out both in its large
partial quotient 277 and its elevated geometric mean, while its other
properties remain largely unexplored. This highlights the need for
further investigation of its Diophantine and period nature.
5. Adelic, p‑Adic,
and q‑Adic Perspectives
5.1
Democratic Mathematics: The Adelic Framework and Ostrowski’s
Theorem
The adele ring \(\mathbb{A} = \mathbb{R}
\times \prodp \mathbb{Q}p\) provides a unified framework that
treats all completions of \(\mathbb{Q}\) on equal footing. We privilege
\(\mathbb{R}\) because it matches our
macroscopic sensory experience, but the adeles enforce “mathematical
democracy”—treating all completions equally. In this picture, quantum
weirdness may be an artifact of trying to describe a full adelic
structure using only the shadow it casts on the real‑number
continuum.
**All Completions of \(\mathbb{Q}\) Are Created Equal: Ostrowski’s
Theorem**
The rational numbers \(\mathbb{Q}\)
form the foundation for arithmetic but are incomplete with respect to
distance metrics. Completion—extending a metric space to include limits
of all Cauchy sequences—yields different number systems depending on the
chosen metric. Ostrowski’s theorem (1916) provides the complete
classification of possible completions of \(\mathbb{Q}\).
Mathematical Foundation: An absolute value on a field \(K\) is a function \(|\cdot|: K \to \mathbb{R}_{\geq 0}\)
satisfying: 1. \(|x| = 0 \iff x = 0\)
- \(|xy| = |x||y|\) 3. \(|x+y| \leq |x| + |y|\) (triangle
inequality)
Two absolute values are equivalent if they induce the same topology.
Ostrowski proved:
Theorem (Ostrowski, 1916): Every non‑trivial
absolute value on \(\mathbb{Q}\) is
equivalent to either: - The Euclidean absolute value: \(|x|_\infty = \max(x, -x)\) - A \(p\)-adic absolute value for some prime
\(p\): \(|x|p = p^{-vp(x)}\) where \(v_p(x)\) is the exponent of \(p\) in \(x\)’s prime factorization.
Completions: - Real numbers: \(\mathbb{R} =\) completion of \(\mathbb{Q}\) with respect to \(|\cdot|\infty\) - \(p\)-adic numbers: \(\mathbb{Q}p =\) completion of \(\mathbb{Q}\) with respect to \(|\cdot|_p\) for prime \(p\)
Mathematically, the real numbers are not privileged; they are merely
the “completion at the infinite prime” (\(\mathbb{Q}_\infty\)). The \(p\)-adic fields are equally valid and
provide a hierarchical, discrete alternative to the continuous real
line.
Mathematical Properties Comparison:
Property |
\(\mathbb{R}\) |
\(\mathbb{Q}_p\) |
Archimedean |
Yes |
No (strong triangle inequality: \(\lvert
x+y\rvertp \leq \max(\lvert x \rvertp, \lvert y
\rvert_p)\)) |
Connected |
Yes |
Totally disconnected |
Locally compact |
Yes |
Yes |
Field characteristic |
0 |
0 |
Topology |
Order topology |
Ultrametric topology |
Completeness |
Complete |
Complete |
Algebraic closure |
\(\mathbb{C}\) (degree 2) |
Infinite algebraic extension |
Physical Interpretation: The real numbers \(\mathbb{R}\) correspond to our macroscopic
experience of continuous space and time. The \(p\)-adic numbers \(\mathbb{Q}_p\) correspond to hierarchical,
discrete structures at fundamental scales. Ostrowski’s theorem
establishes mathematical democracy: no completion is inherently
privileged.
5.2
Generalized Valuations: q‑Adic Scaling Operations
The \(q\)-adic framework generalizes
\(p\)-adic analysis to include
arbitrary scaling ratios \(q \in
\mathbb{R}^+\), moving beyond arithmetic to pure scaling. This
allows us to treat \(\pi\), \(\phi\), and \(e\) not as special numbers but as
fundamental scaling operators for different physical phenomena.
Mathematical Definition: For \(q \in \mathbb{R}\), \(q > 1\), and \(x \in \mathbb{Q}^\times\), define the \(q\)-adic valuation \(v_q(x)\) as the unique integer \(n\) such that: \[
x = q^n \cdot u
\] where \(u \in
\mathbb{Q}^\times\) satisfies \(v_q(u)
= 0\) (i.e., \(u\) is a \(q\)-adic unit). The \(q\)-adic absolute value is: \[
|x|q = q^{-vq(x)} \text{ for } x \neq 0, \quad |0|_q = 0.
\]
Key Properties: - Positive definiteness: \(|x|q \geq 0\) with equality iff \(x = 0\) - Multiplicativity: \(|xy|q = |x|q|y|q\) - Strong triangle
inequality: \(|x+y|q \leq \max(|x|q,
|y|_q)\)
This construction preserves the strong triangle inequality \(|x+y|q \leq \max(|x|q, |y|_q)\), which is
the hallmark of ultrametric (non‑Archimedean) geometry.
Examples of Fundamental \(q\) Values:
**\(\pi\)-adic Numbers
(\(q = \pi \approx 3.14159\)):**
\(|\pi|\pi = \pi^{-1}\), \(|2\pi|\pi = \pi^{-1}\) (since \(v_\pi(2\pi) = 1\)). *Physical
interpretation:* Natural for periodic and rotational phenomena where
\(\pi\) acts as the fundamental scaling
operator between linear and angular measures. Applications:
Quantum systems with rotational symmetry, Fourier analysis, circular
geometries.
**\(\phi\)-adic Numbers
(\(q = \phi \approx
1.61803\)):** \(|\phi|_\phi =
\phi^{-1}\), \(|\phi^2|_\phi =
\phi^{-2}\). Physical interpretation: Natural for
systems exhibiting recursive self‑similarity or “golden ratio” growth,
such as quasicrystals and biological branching. Applications:
Growth processes, biological systems, optimal packing
arrangements.
**\(e\)-adic Numbers
(\(q = e \approx 2.71828\)):**
\(|e|e = e^{-1}\), \(|e^2|e = e^{-2}\). *Physical
interpretation:* Natural for entropic and continuous compounding
growth processes. Applications: Statistical mechanics,
exponential decay processes, continuous compounding.
**\(\alpha\)-adic Numbers
(\(q = \alpha^{-1} \approx
137.036\)):* Physical interpretation:* Natural
for quantum electrodynamics where the fine‑structure constant \(\alpha\) sets the scale of electromagnetic
interactions.
Mathematical Validity: For any \(q > 1\), the construction yields a valid
non‑Archimedean absolute value. The completion of \(\mathbb{Q}\) with respect to \(|\cdot|q\) gives the field of \(q\)-adic numbers \(\mathbb{Q}q\).
Digit Expansion: Every \(q\)-adic number has a unique expansion:
\[
x = \sum{k=-m}^\infty ak q^k \quad \text{with} \quad a_k \in \{0, 1,
\dots, \lfloor q \rfloor\}
\] For non‑integer \(q\), \(\lfloor q \rfloor\) is the integer
part.
Physical Motivation for Generalization: By allowing
\(q\) to take transcendental or
algebraic values, we move beyond arithmetic to pure scaling: - Different
physical phenomena may have different natural scaling bases. - The
apparent “specialness” of \(\pi\),
\(\phi\), and \(e\) reflects their roles as fundamental
scaling operators. - Physical laws can be formulated in terms of scaling
operations rather than arithmetic operations.
Comparison with Conventional \(p\)-Adics:
Property |
\(p\)-Adic Numbers |
\(q\)-Adic Numbers |
Base |
Integer primes \(p\) |
Arbitrary scaling ratios \(q \in
\mathbb{R}^+\) |
Valuation |
\(\lvert x\rvert_p =
p^{-v_p(x)}\) |
\(\lvert x \rvert_q =
q^{-v_q(x)}\) |
Special cases |
\(p = 2, 3, 5, 7, \dots\) |
\(q = \pi, \phi, e, \alpha^{-1},
\dots\) or \(q = p\) |
Physical interpretation |
Divisibility by prime powers |
Scaling by fundamental ratios |
Mathematical status |
Completion of \(\mathbb{Q}\) |
Completion with respect to a scaling metric |
5.3 The Monna Map and
Base Invariance
The Monna map \(M_p:
\mathbb{Z}p \to [0,1]\) sends a p‑adic integer \(x = \sum{k=0}^\infty a_k p^k\) to the real
number \[
Mp(x) = \sum{k=0}^\infty a_k p^{-(k+1)}.
\] This map is surjective but not injective; it translates the
ultrametric distance on \(\mathbb{Z}_p\) into Euclidean distance on
\([0,1]\), preserving hierarchical
information: digits that agree on high‑order branches correspond to real
numbers that are close.
Principle of Base Invariance: Fundamental physical
laws should be expressible in a form that does not privilege any
particular base or number system. The Monna map illustrates how a
base‑invariant formulation can be achieved by starting from a p‑adic
(non‑Archimedean) description and projecting to the reals only at the
level of measurement.
5.4 Valuations
vs. Absolute Values
In conventional physics, distances are measured by the absolute value
\(|x|\). In an ultrametric framework,
the valuation \(vp(x)\) (or the p‑adic norm \(|x|p = p^{-v_p(x)}\)) is more natural.
This shifts focus from “how much” (magnitude) to “at what level”
(hierarchy). Physical fields would be functions on the Bruhat–Tits tree
\(T_p\) rather than on Euclidean space,
with dynamics governed by discrete difference operators.
5.5
Vladimirov Operator and p‑Adic Differential Equations
The ordinary derivative \(\frac{d}{dx}\) is replaced by the
Vladimirov operator \(D_p^\alpha\), the p‑adic analogue of a
fractional derivative. Equations of the form \[
D_p^\alpha \psi(x) = V(x)\psi(x)
\] describe dynamics on the tree. Solutions are naturally
hierarchical and exhibit strong localization properties. In the
continuum limit (via the Monna map), one recovers standard differential
equations, but with additional constraints arising from the underlying
tree structure.
5.6
p‑Adic Interpretation of the Convergent \(34259/250\)
The convergent \(34259/250\)
approximating \(\alpha^{-1}\) has
denominator \(250 = 2\cdot5^3\). Its
p‑adic valuations are \[
v2(250) = 1,\qquad v5(250) = 3,\qquad v_p(250)=0 \text{ for other
primes}.
\] The numerator \(34259\) is
prime. This suggests that the approximation is “simple” in the 2‑adic
and 5‑adic senses. The large partial quotient \(277\) (a prime) might reflect a branching
event deep in a p‑adic tree associated with the prime 277. If \(\alpha^{-1}\) were the image under the
Monna map of a point on a tree with branching number 277, the excellent
rational approximation would be a natural consequence.
5.7
Connection to Ultrametric Quantum Computation
Earlier work on ultrametric quantum computation proposes that quantum
state spaces can be modeled on the Bruhat–Tits tree \(T_p\), with logical qubits encoded on deep
vertices and environmental noise confined to the boundary. The Monna map
provides the bridge between that discrete, fault‑tolerant description
and the continuous, noisy quantum mechanics we observe. Decoherence, in
this picture, results from projecting a high‑dimensional tree state onto
a single real coordinate—a many‑to‑one map that loses the hierarchical
protection of the p‑adic geometry. Base‑invariant formulations therefore
unify quantum error correction and the origin of dimensionless
constants: the same tree that protects quantum information also gives
rise to constants via projections independent of base.
6.
Syntactic Primitives and Base‑Invariant Ontology
6.1 The Primacy
of Relations Over Numbers
The preceding analysis has revealed that base‑invariant
properties—continued fractions, valuations, scaling operators—are more
fundamental than any particular numeral representation. This leads to a
deeper ontological question: what is the minimal set of primitives
needed to describe physical constants without anthropocentric bias? The
answer, developed through the attached dialogues, is that
numbers themselves are emergent; the true primitives
are syntactic relations from which numerical values
arise.
6.2 Core Syntactic
Primitives
Following the pure‑syntax formulation (0.4.1.5.md), we posit four
irreducible primitives:
Scaling relation \(≺_q\): a binary relation meaning “\(y\) is one \(q\)-refinement of \(x\)”. This replaces the notion of
multiplication by a factor \(q\).
Composition \(∘\):
a partial operation that combines entities syntactically; it respects
the scaling relations.
Distinction \(≢\):
a primitive notion of difference, needed to avoid trivial collapse.
Coarse‑graining rule \(→_M\): a rewrite rule that projects
fine‑grained syntactic structures onto coarse‑grained descriptions (the
Monna map).
These primitives contain no numbers, no sets, no predefined algebraic
operations. All familiar mathematical objects—integers, reals, p‑adic
fields—emerge as equivalence classes of syntactic patterns.
6.3
Emergence of Integers, Primes, and Valuations
From the scaling relation \(≺_q\) we
obtain the notion of depth: the equivalence class of
entities connected by chains of \(≺_q\). If we designate a distinguished
entity \(0\), the chain \[
0 ≺q 1q ≺q 2q ≺_q \dots
\] yields tokens \(n_q\) that
behave as integers. Different scaling operators \(q\) give different families of integers,
but those corresponding to irreducible scaling
relations (those that cannot be decomposed as \(≺{q1} ∘ ≺{q2}\)) are the **prime
scaling relations**. The integer primes \(2,3,5,\dots\) are merely labels for the
irreducible scaling relations that happen to produce finitely
distinguishable tokens in human cognition.
This perspective reveals that **primes are not universal
invariants** but convenient fictions of number theory, not a
necessity of nature. In a fully democratic, unit‑free system, primes are
not fixed points of any non‑trivial automorphism; they become fixed only
if one artificially restricts the automorphism group to those that
preserve the integer lattice—which reintroduces integer bias.
Consequently, patterns such as the Riemann zeta zeros emerge from the
discrete lattice \(\mathbb{Z}\) and its
multiplicative structure. Without integers, the zeta function has no
Euler product, no primes, and no discrete zeros; the “prime pattern”
dissolves into the continuous scaling symmetry of the positive
reals.
The valuation \(v_q(x)\) emerges as the maximal depth of
common refinement: how many steps of \(≺_q\) are needed to reach a common
ancestor. This is a purely syntactic notion that requires no real
numbers.
6.4 The Monna
Map as a Coarse‑Graining Rule
The Monna map \(M_q\) is an instance
of the coarse‑graining primitive \(→_M\). It deterministically projects
infinite \(q\)-adic expansions
(syntactic chains) onto bounded real intervals. The “loss” of
information under this projection is not a flaw but the very mechanism
that generates the appearance of continuity and real numbers. In
physical terms, measurement is the application of \(→_M\) to the full syntactic state of the
universe.
6.5 Implications
for Fundamental Constants
If constants like \(\pi\), \(e\), \(\alpha^{-1}\) are not numbers but scaling
operators, their values arise from the structure of the syntactic
network. For example:
\(\pi\) is the scaling operator
\(≺_\pi\) that appears as an
irreducible relation in the cosmic syntax.
The fine‑structure constant \(\alpha\) emerges from the relative depths
of electromagnetic scaling relations.
The proton‑electron mass ratio \(\mu\) reflects the nesting of different
scaling operators (\(\pi\), \(e\), etc.) in the hadronic sector.
The continued‑fraction expansions and \(q\)-adic valuations computed earlier are
syntactic fingerprints of these operators, not
arbitrary decimals.
6.6 Bias‑Explicit
Universal Language
To avoid privileging any representation, we can formalize the above
in a parametric universal language where every
assumption (logic, algebraic structure, metric, computational model) is
an explicit parameter. This language, outlined in the dialogue, allows
comparisons across different biases and makes the hidden assumptions of
conventional physics transparent. It is the ultimate expression of the
Principle of Base Invariance.
6.7 Summary: From
Numbers to Syntax
The base‑invariant investigation thus reaches its logical conclusion:
the fundamental “source code” of the universe is not written in numbers
but in syntactic relations. The patterns we have
catalogued—large partial quotients, distinctive valuations, scaling
symmetries—are clues to this deeper syntax. This perspective dissolves
the anthropocentric prejudices of integer primes, real continua, and
even fixed dimensions, offering a truly democratic framework for
understanding physical constants.
7. Base‑Free
Formulations of Physical Law
The Principle of Base Invariance demands that
fundamental physical laws be expressible without privileging any numeral
system. The adelic framework and q‑adic scaling operations provide the
mathematical tools to achieve this. In this section we outline how
physical theories can be reformulated in a base‑free manner, treating
dimensionless constants not as mere numbers but as intrinsic scaling
operators.
7.1 From Real
Numbers to Scaling Operators
Conventional physics expresses laws using real numbers, which are the
completion of \(\mathbb{Q}\) at the
infinite prime (\(\mathbb{R} =
\mathbb{Q}_\infty\)). The adelic perspective treats all
completions—\(\mathbb{R}\) and \(\mathbb{Q}_p\) for each prime \(p\)—on equal footing. The q‑adic
generalization extends this to arbitrary scaling bases \(q > 1\). In this framework, a
dimensionless constant \(C\) is
naturally associated with a scaling operator \(SC\) acting on a q‑adic space \(\mathbb{Q}q\), where \(q\) is chosen to be \(C\) itself (or a simple function of \(C\)). For example:
**\(\pi\)‑adic
scaling:** Rotational phenomena are described by fields on \(\mathbb{Q}_\pi\), with \(\pi\) acting as the fundamental
angular‑to‑linear scaling operator.
**\(\phi\)‑adic
scaling:** Growth processes and self‑similar structures are
described on \(\mathbb{Q}_\phi\), with
\(\phi\) as the fundamental
growth‑ratio operator.
**\(e\)‑adic
scaling:** Exponential decay and entropic processes are described
on \(\mathbb{Q}_e\), with \(e\) as the fundamental continuous‑growth
operator.
**\(\alpha^{-1}\)‑adic
scaling:** Electromagnetic interactions are described on \(\mathbb{Q}_{\alpha^{-1}}\), with \(\alpha^{-1} \approx 137.036\) as the
fundamental coupling‑strength operator.
7.2 The Monna
Map as a Measurement Projection
Physical measurements yield real numbers, but the underlying dynamics
may be ultrametric. The Monna map \(M_q:
\mathbb{Z}_q \to [0,1]\) provides a canonical projection from the
q‑adic integers to the real interval. A base‑free formulation keeps the
dynamics in \(\mathbb{Q}_q\) and
applies the Monna map only at the stage of measurement. This is
analogous to the quantum‑classical transition: the full quantum state
lives in a high‑dimensional Hilbert space, but measurements project onto
classical observables.
7.3
Example: Maxwell’s Equations in \(\alpha^{-1}\)‑Adic Form
Consider the fine‑structure constant \(\alpha\). In a base‑free formulation,
electromagnetism is defined on an \(\alpha^{-1}\)‑adic space. The electric and
magnetic fields are functions on the Bruhat–Tits tree \(T_{\alpha^{-1}}\). Maxwell’s equations
become difference equations on the tree, with the coupling strength
encoded in the branching structure. In the continuum limit (via the
Monna map), one recovers the usual Maxwell equations with \(\alpha\) appearing as a prefactor. The
large partial quotient 277 in \(\alpha^{-1}\)’s continued fraction may
correspond to a special branching pattern that yields the observed
value.
7.4
Example: Schrödinger Equation with \(\pi\)‑Adic Scaling
For quantum systems with rotational symmetry, the natural scaling
base is \(\pi\). The Schrödinger
equation can be written as a Vladimirov‑type operator on \(\mathbb{Q}_\pi\). The factor \(2\pi\) that appears in the canonical
commutation relation \([x,p] = i\hbar\)
emerges from the Monna‑map projection of a \(\pi\)‑adic tree structure. The partial
quotient 292 in \(\pi\)’s continued
fraction may reflect a deep branching property that gives rise to the
exceptional rational approximation \(355/113\).
7.5 Unification via the
Adele Ring
The adele ring \(\mathbb{A} = \mathbb{R}
\times \prodp \mathbb{Q}p\) (generalized to include q‑adic
completions) offers a unified arena for all physical phenomena.
Different forces may “live” in different completions: electromagnetism
in \(\mathbb{Q}_{\alpha^{-1}}\), strong
interactions in \(\mathbb{Q}_2\)
(binary scaling), gravity in \(\mathbb{R}\) (macroscopic continuum). The
apparent independence of dimensionless constants could then be a
consequence of their origin in distinct scaling sectors. Interactions
between sectors are mediated by adelic reciprocity laws, which are
number‑theoretic analogues of coupling constants.
7.6 Experimental
Signatures
Base‑free formulations make testable predictions: 1. **Time
variation of constants:** If a constant is a topological
invariant of a q‑adic tree, its time variation is tightly constrained to
discrete jumps corresponding to changes in branching patterns. 2.
Quantum gravity imprints: At the Planck scale,
spacetime may have an ultrametric structure, leaving signatures in the
cosmic microwave background or gravitational wave spectra. 3.
High‑precision metrology: The continued fraction
expansions of constants may reveal new rational approximations that
could be searched for in atomic physics experiments.
7.7 Outlook
Base‑free physics is not merely a reformulation; it is a paradigm
shift that treats dimensionless constants as scaling operators rather
than mere numbers. This perspective naturally incorporates
number‑theoretic patterns (large partial quotients, p‑adic valuations)
into the fabric of physical law. The ultimate goal is a complete adelic
formulation of all fundamental interactions, where the “source code” of
the universe is written in the language of valuations, continued
fractions, and scaling operators.
8. Conclusion
We have conducted a systematic exploration of base‑invariant
number‑theoretic patterns in fundamental constants. The most striking
finding is the large partial quotient 277 in the
continued fraction of the reciprocal fine‑structure constant \(\alpha^{-1}\). This, together with the
elevated geometric mean of its partial quotients, suggests that \(\alpha^{-1}\) is not a generic real number
but possesses special Diophantine properties. Similar patterns appear in
the proton‑electron mass ratio and other dimensionless constants,
hinting at a common origin.
The Principle of Base Invariance—that fundamental
laws should not depend on the choice of number representation—leads
naturally to formulations in terms of continued fractions, p‑adic
valuations, and ultrametric geometry. The Monna map provides a concrete
mechanism for projecting hierarchical p‑adic structures onto the real
numbers we measure, linking the discrete, fault‑tolerant world of
ultrametric quantum computation with the continuous, noisy world of
conventional physics.
Speculatively, dimensionless constants may be **topological
invariants of an underlying p‑adic tree, or special
values** of modular forms and periods. These ideas connect to
deep threads in modern mathematics: motives, algebraic cycles, and the
Langlands program. They also offer new perspectives on old puzzles: why
\(\alpha \approx 1/137\), why the
proton‑electron mass ratio is ~1836, and why the cosmological constant
is so small yet non‑zero.
While many of these connections remain conjectural, they provide a
rich framework for future research, both computational and theoretical.
The search for a number‑theoretic understanding of fundamental constants
is not merely an academic exercise; it is a quest for the ultimate
“source code” of the universe—a set of mathematical relations that are
independent of any arbitrary representation, truly base‑invariant.
9. References
9.1 Classical Works
Khinchin, A. Y. (1935). Continued Fractions. University of
Chicago Press.
Hardy, G. H., & Wright, E. M. (1979). *An Introduction to the
Theory of Numbers*. Oxford University Press.
Borevich, Z. I., & Shafarevich, I. R. (1966). *Number
Theory*. Academic Press.
Koblitz, N. (1984). *p‑Adic Numbers, p‑Adic Analysis, and
Zeta‑Functions*. Springer.
9.2
Continued Fractions and Diophantine Approximation
Bailey, D. H., & Borwein, J. M. (2012). *Handbook of
Continued Fractions for Special Functions*. Springer.
Lang, S. (1995). *Introduction to Diophantine
Approximations*. Springer.
Waldschmidt, M. (2000). *Diophantine Approximation on Linear
Algebraic Groups*. Springer.
9.3 p‑Adic
Physics and Ultrametric Geometry
Vladimirov, V. S., Volovich, I. V., & Zelenov, E. I. (1994).
p‑Adic Analysis and Mathematical Physics. World
Scientific.
Dragovich, B., Khrennikov, A. Yu., Kozyrev, S. V., & Volovich,
I. V. (2017). “On p‑adic mathematical physics.” *p‑Adic Numbers,
Ultrametric Analysis and Applications*, 9(1), 1–29.
Brekke, L., & Freund, P. G. O. (1993). “p‑Adic numbers in
physics.” Physics Reports, 233(1), 1–66.
9.4
Fundamental Constants and Their Measurements
CODATA Task Group on Fundamental Constants (2018). “CODATA
recommended values of the fundamental physical constants.” *Reviews
of Modern Physics*, 90(2), 025001.
Mohr, P. J., Newell, D. B., & Taylor, B. N. (2016). “CODATA
recommended values of the fundamental physical constants: 2014.”
Reviews of Modern Physics, 88(3), 035009.
9.5 Modular Forms,
Periods, and Motives
Zagier, D. (1991). “Introduction to modular forms.” In *From
Number Theory to Physics* (pp. 238–291). Springer.
Kontsevich, M., & Zagier, D. (2001). “Periods.” In
Mathematics Unlimited—2001 and Beyond (pp. 771–808).
Springer.
André, Y. (2004). *Une introduction aux motifs (motifs purs,
motifs de Chow, motifs périodes)*. Société Mathématique de
France.
9.6 Quantum
Field Theory and String Theory
Weinberg, S. (1995). The Quantum Theory of Fields (Vol. 1).
Cambridge University Press.
Polchinski, J. (1998). String Theory (Vol. 1). Cambridge
University Press.
Maldacena, J. (1999). “The large‑N limit of superconformal field
theories and supergravity.” *International Journal of Theoretical
Physics*, 38(4), 1113–1133.
9.7 Online Resources
The On‑Line Encyclopedia of Integer Sequences (OEIS): sequences
related to continued fractions of constants.
Wolfram MathWorld: entries on “Fine‑Structure Constant,” “Khinchin’s
Constant,” “Normal Number.”
arXiv preprints in sections: hep‑th, math.NT, math‑ph,
quant‑ph.