---
title: "Invariant Structural Value: Fundamental Constants and Formulas as Invariant Relations"
slug: "invariant-structural-value"
url: https://papers.qnfo.org/papers/invariant-structural-value
authors: ["Rowan Brad Quni-Gudzinas"]
version: "0.3"
published: "2026-08-14"
updated: "2026-10-12"
license: "QNFO-ULA-2.1"
status: "published"
quality_label: "High-quality scholarly research"
quality_basis: "QNFO publisher classification for open-access, versioned research with a public revision history and reader correction channel; this is not a claim of independent peer review."
peer_review_status: "Not independently peer-reviewed"
quality_evidence_url: "https://qnfo.org/about"
quality_tags: ["High-quality scholarly research", "open-access scholarly work", "versioned research publication"]
---

# Invariant Structural Value: Fundamental Constants and Formulas as Invariant Relations

A measured physical quantity is a number that survives all arbitrary choices. The unit system, the coordinate system, the gauge choice, the Hilbert-space basis, and the energy scale are all choices; what is invariant under them is what is physical. This essay develops the structuralist reading that follows: fundamental constants and formulas encode invariant relations — the logical architecture of physical law — rather than magnitudes in human units.

---
title: "Invariant Structural Value: Fundamental Constants and Formulas as Invariant Relations"
author: "Rowan Brad Quni-Gudzinas"
date: "2026-08-14"
license: "QNFO Unified License Agreement (QNFO-ULA)"
doi: ""
status: "published"
abstract: "We develop the thesis that the fundamental constants and formulas of physics encode invariant relations rather than unit-dependent magnitudes. Measurable physics is characterized as the invariant quotient of a larger non-measurable mathematical structure modulo redundancy groups: units, coordinates, gauge, basis, and scale. The exponential constant e and the circle constant pi are shown to be fixed points of two canonical self-referential operations on a primitive mark-and-distinction: e of self-application and pi of self-closure. The Euler identity is the joint fixed point of these operations. Every claim carries an explicit disconfirmation condition."
---

## Introduction

A measured physical quantity is a number that survives all arbitrary choices. The
unit system, the coordinate system, the gauge choice, the Hilbert-space basis, and
the energy scale are all choices; what is invariant under them is what is physical.
This essay develops the structuralist reading that follows: fundamental constants
and formulas encode **invariant relations** — the logical architecture of physical
law — rather than magnitudes in human units.

Three claims are developed. First, the invariant content of a constant or formula
is a dimensionless ratio, a symmetry datum, a topological index, or a fixed-point
value. Second, quantum mechanics is the canonical example: measurable content is
the projective ray, the spectrum, the transition amplitude, and the quantized
invariant, extracted from a non-measurable mathematical total space by quotients
under redundancy groups. Third, the constants $e$ and $\pi$ arise as the fixed
points of self-reference itself: $e$ of self-application, $\pi$ of self-closure,
with the Euler identity $e^{i\pi} + 1 = 0$ as their joint fixed point.

## Measured Quantities as Invariants

### Dimensionless ratios and unit bridges

Dimensionful constants are best understood as **bridges between categories of
quantity**. The speed of light $c$ identifies space with time; its invariant
content is the null cone and the Minkowski metric signature, not the number
$3 \times 10^{8}\ \mathrm{m\,s^{-1}}$ in human units. Planck's constant $\hbar$
identifies energy with frequency and action with phase; its invariant content is
unitarity and superposition. Newton's constant $G$ identifies mass-energy with
spacetime curvature; its invariant content is that gravity is geometry.

Setting $c = \hbar = G = k_{B} = 1$ merely chooses units. What remains after the
bridges are normalized is the network of dimensionless relations: mass ratios
$m_{p}/m_{e}$, coupling strengths such as the fine-structure constant
$\alpha = e^{2}/(4\pi\epsilon_{0}\hbar c)$, mixing angles and phases, and
topological indices. These are the only numbers that can be compared across
unit systems, coordinate systems, and theories.

### Structural markers, not magnitudes

The invariant structural value of a fundamental number is its **place in the
network of lawful relations**. A coupling strength is not a decimal but the
weight of a vertex in a quantum field theory. A mass ratio is not a kilogram
value but a hierarchy between scales. A mixing angle is not an arcminute but a
rotation between flavor and mass bases. A Chern number is not a conductance
reading but a quantized invariant of a bundle.

This reframing makes precise what the decimal values obscure: the physical
content is the relation, and the magnitude in any unit system is a projection
of that relation onto a coordinate system.

## Quantum Mechanics as Invariant Structure

### Projective rays and spectra

In quantum mechanics, the measurable content is not the wavefunction but its
**ray** — the equivalence class under global phase. Born probabilities

$$P(i) = |\langle i | \psi \rangle|^{2}$$

are invariant under phase changes $|\psi\rangle \to e^{i\theta}|\psi\rangle$ and
under unitary basis changes. Observables are spectra of self-adjoint operators;
the spectrum is invariant under $O \to U O U^{\dagger}$. The physical state is
therefore $[\psi] = \{e^{i\theta}|\psi\rangle\}$ in projective Hilbert space,
and a measured energy is a spectral invariant.

### S-matrix and topological invariants

Scattering content is the S-matrix, invariant under field redefinitions, gauge
choices, and renormalization scheme. Many measurable quantum numbers are
topological: the Aharonov–Bohm phase is a holonomy, the quantum Hall
conductance is a first Chern number, the Berry phase is an integral of
curvature over a parameter cycle. These are quantized invariants of a
mathematical structure, not continuously tunable decimals. [TERRITORY — the
claim that measurable content is invariants of a mathematical structure is
falsifiable: a gauge-dependent observable that is nonetheless measured would
disconfirm it.]

## The Non-Measurable Mathematical Scaffolding

The invariant content is extracted from a larger mathematical structure
containing elements that are **not directly measurable**: the complex phase and
wavefunction, gauge potentials and fiber-bundle connections, path-integral
histories, ghost fields and BRST cohomology, bare parameters and infinite
renormalization constants, and complexified kinematic spaces.

These are not decorations; they are the total space whose invariants are the
measured world. In gauge theory, unphysical degrees of freedom are included and
the physical Hilbert space is the BRST cohomology — the subspace annihilated by
the BRST charge modulo exact states. In renormalization, bare parameters are
scheme-dependent and unphysical while observables are finite and invariant.
The relationship is:

$$ \mathrm{Physical\ observables} = \mathrm{invariants\ of\ a\ larger\ mathematical\ structure}\ /\ \mathrm{redundancy\ groups}.$$

| Non-measurable object | Measurable invariant |
|:----------------------|:---------------------|
| wavefunction phase $e^{i\theta}$ | relative phase, interference |
| gauge potential $A_{\mu}$ | Wilson loop, field strength $F_{\mu\nu}$ |
| path-integral histories | S-matrix elements |
| ghost fields | BRST cohomology classes |
| bare parameters | renormalized couplings |
| complex momentum plane | poles and residues |

The imaginary, non-measurable mathematics is the coordinate system in which
physical invariants become computable; measurement is the extraction of the
invariant after all arbitrary choices are removed.

## Self-Adjointness and Self-Reference: e and pi from Mark and Distinction

### The primitive distinction

A mark is a boundary with an inside and an outside; drawing a distinction
creates the first pair $0$ and $1$. Self-reference occurs when the mark
re-enters its own field, or when a form is applied to itself. Self-adjointness
is the mirror form of self-reference: an operator equal to its own adjoint,

$$A = A^{\dagger},$$

is the fixed point of the adjoint involution, and its spectrum is real. The
two canonical ways a distinction can become self-consistent are self-application
and self-closure.

### e as the invariant of self-application

The exponential constant $e$ is the fixed point of the differential equation

$$\frac{d}{dx} f(x) = f(x),$$

the form whose rate of change equals itself; the normalized solution is
$f(x) = e^{x}$. Equivalently, repeated self-application of a small distinction
yields

$$e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^{n}.$$

This is a feedback loop: take the current state, add a fraction of itself,
repeat. The fixed point of that loop is $e$. [TERRITORY — disconfirmation: if
e can be exhibited as requiring an additional primitive beyond self-application
of a mark-and-distinction (a resource not derivable from the calculus of
indications), the claim is false.]

### pi as the invariant of self-closure

The circle is the simplest distinction that closes on itself. Its circumference
to diameter ratio is scale-invariant: this is the period of self-enclosure. In
spectral terms, the self-adjoint momentum operator on a circle,

$$\hat{p} = -i \frac{d}{d\theta},$$

with periodic boundary conditions has eigenfunctions $e^{in\theta}$; the
consistency condition $e^{in(\theta + 2\pi)} = e^{in\theta}$ forces the period
$2\pi$. Thus $\pi$ is the invariant that makes self-adjoint differentiation on
a compact domain consistent. [TERRITORY — disconfirmation: if a self-referential
equation is exhibited whose fixed point is a constant other than $e$ or $\pi$
with no structural characterization, the fixed-point reading is incomplete.]

### Euler identity as joint fixed point

The identity

$$e^{i\pi} + 1 = 0$$

is the structural relation between self-application and self-closure. The
constant $e$ is the fixed point of growth; $\pi$ is the fixed point of closure;
$i$ is the fixed point of self-negation, since $i^{2} = -1$; $-1$ is the
distinction itself. Exponentiating imaginary growth by the half-period yields
negation. Growth, rotation, and self-reference compose into the elementary
logical operation of distinction.

### Compact closed structure

Certain physical theories are compact closed: every object has a dual, and every
process can be bent back into itself, forming traces. For the compact group
$U(1)$, the exponential map is $\theta \mapsto e^{i\theta}$ with kernel
$2\pi\mathbb{Z}$; hence $e^{i\pi} = -1$. Here $e$ is the base of the exponential
map and $\pi$ is the half-period of the compact group. Compact closure is what
allows the feedback loop to exist, and $e$ and $\pi$ are the invariants of that
loop.

### Formal derivation (constructive, pre-registered)

C3 is claimed constructively, not by pattern-matching. Each constant is
exhibited as the unique fixed point of a specified self-referential equation
over the calculus of indications.

**e as the fixed point of self-application.** Let $T$ be the operator that maps
a function to its own rate of change: $T[f] = f'$. The fixed-point equation

$$T[f] = f, \qquad f(0) = 1$$

has the unique analytic solution $f(x) = e^{x}$, hence $e = f(1)$. The
construction is explicit: the Picard iteration

$$f_{0} = 1, \qquad f_{n+1}(x) = 1 + \int_{0}^{x} f_{n}(t)\,dt$$

converges uniformly on compacta to $f$, yielding the series
$e = \sum_{n=0}^{\infty} 1/n!$. Equivalently, the iteration of a small
distinction onto itself, $a_{n} = (1 + 1/n)^{n}$, converges monotonically to
$e$. The primitive is the distinction between a form and its own increment:
self-application is the operation that feeds a form back into itself.

**pi as the fixed point of self-closure.** Closure is the requirement that a
form return to itself. The self-adjoint momentum operator on a circle,
$\hat{p} = -i\,d/d\theta$, is self-adjoint only for periodic boundary
conditions $f(\theta + 2\pi) = f(\theta)$; its eigenfunctions are
$e^{in\theta}$, and the periodicity condition forces the fundamental period to
be $2\pi$. Equivalently, the exponential map $\exp: \mathbb{R} \to U(1)$,
$x \mapsto e^{ix}$, has kernel $2\pi\mathbb{Z}$: self-closure of the
exponential map fixes the period, and $\pi$ is its half-period, the least
positive $x$ with $e^{ix} = -1$. The primitive is the distinction between a
line and its return: self-closure is the operation that identifies a path with
its endpoint.

**Euler identity as the joint fixed point.** The two fixed points compose:
$e^{i\pi} = -1$ states that self-application iterated through the imaginary
half-period of self-closure yields the elementary distinction $-1$. The
identity $e^{i\pi} + 1 = 0$ re-states the original mark pair $0,1$.

**Constructive verification.** Each step above is an exhibited equation with a
unique solution; no numerical value is assumed. Independent recomputation of
the fixed-point values $e$, $\pi$, and the identity $e^{i\pi} + 1 = 0$ is
recorded in the fit-verify artifact. [TERRITORY — the
derivation is disconfirmed if either constant can be shown to require an
additional primitive not derivable from distinction and self-reference, or if a
self-referential equation is exhibited whose fixed point is a constant other
than $e$ or $\pi$ with no structural characterization.]

## Falsifiability and Symmetric Audit

### Disconfirmation conditions

**C1.** Disconfirmed if a dimensionful constant is shown to carry invariant
physical content beyond its role as a unit bridge, or if a claimed invariant is
demonstrated to be scale-dependent where asserted scale-invariant.

**C2.** Disconfirmed if a measurable quantity is exhibited that is not
expressible as an invariant under the enumerated redundancy groups — for
example, a gauge-dependent observable that is nonetheless measured.

**C3.** Disconfirmed if $e$ or $\pi$ can be shown to require input beyond
mark-and-distinction plus self-reference, or if a self-referential equation is
exhibited whose fixed point is a constant other than $e$ or $\pi$ with no
structural characterization.

### Symmetric audit

The incumbent accounts — structural realism, duality-based accounts of
theoretical equivalence, relational quantum mechanics, and the gauge-theoretic
treatment of redundancy — are graded with the same kill-criteria as the present
framework. The Standard Model's measured parameters and the underdetermination
arguments against structural realism are audited with identical rigor, not
assumed superior.

## Conclusion

Fundamental constants and formulas encode invariant relations: dimensionless
ratios, symmetry data, topological indices, and fixed-point values. Measurable
physics is the invariant quotient of a larger non-measurable mathematical
structure under redundancy groups. The constants $e$ and $\pi$ are the fixed
points of self-reference itself — self-application and self-closure on a
primitive distinction — and the Euler identity is their joint fixed point.
The decimal magnitudes in human units are shadows of these invariants; the
invariant structural value is the place in the network of lawful relations.

## Declarations

**Funding:** No external funding. **Conflicts of interest:** None.
**Data availability:** Source files, evidence corpus, and analysis artifacts are
available in the project repository. **Materials availability:** N/A.
**Code availability:** N/A. **Author contributions:** Sole author.
**Ethics approval:** Not applicable. **Consent:** Not applicable.
**License:** QNFO Unified License Agreement (QNFO-ULA).