Invariant Structural Value: Fundamental Constants and Formulas as Invariant Relations

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

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:

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,

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

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

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,

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

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

has the unique analytic solution $f(x) = e^{x}$, hence $e = f(1)$. The

construction is explicit: the Picard iteration

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).