#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:
| 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,
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).