% Paper 13: Quintic Structure — TOE canon. Léon Fernando Vlegels, CC BY 4.0.
% Status: standing diagnosis (A293): the quintic structure is a schema, not a construction; conditions, map, coefficients, degree bound, and rationality premise all unsupplied or unproven.
% Registry: P013_1, P013_2, P013_3_c (confirmed-load-bearing) — ledger: Paper 40 and addenda/verify/tbs_registry.json.
% Verifier: verify_P013.py (historical series; includes expected-fail checks that document recorded gaps).


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\title{\textbf{Self-Referential Boundary Inversion and the Quintic Structure\\
of the Fine-Structure Constant}}
\author{L\'{e}on Fernando Vlegels\\
Independent Researcher}
\date{2025-12}

\begin{document}
\maketitle

\begin{abstract}
We derive the irreducible quintic fixed-point equation governing the emergence of 
the fine-structure constant $\alpha$ from the inversion of the self-referential observation operator 
defined in the geometric $(B^{4}, S^{3}, S^{1})$ framework. 
Previous papers established the energy equilibrium $E_{\mathrm{self}} = 13.177$, 
the oscillation law $\kappa = \alpha^{5/4}$, and the spectral correspondence between 
geometric and quantum beta functions. 
However, these results treated $\alpha$ as an external normalization parameter. 
In this work, we reverse the direction: we treat $\alpha$ as \emph{unknown} 
and impose the three global consistency conditions implied by the framework.
We prove that these constraints collapse to a single fifth-degree algebraic equation
whose unique positive real root is the physical $\alpha$.
This paper completes the missing structural link between the spectral papers
and the fermion-sector mass papers.
The corpus's standing diagnosis (Addendum 293, Paper 40) is that what follows is a schema rather than a completed construction: the consistency conditions and the elimination map are not specified to the point where the quintic can be computed or its claims verified.
\end{abstract}

\tableofcontents
\newpage
\section{Introduction}

The geometric framework established across  
\emph{Mathematical Foundations of Geometric Fundamental Physics},  
\emph{The Perfect Stable Sphere},  
\emph{Geometric Spectral Theory},  
and  
\emph{Self-Referential Observation and the Fermion Sector}  
determines much of low-energy physics from three irreducible geometric layers:  
bulk $B^{4}$, boundary $S^{3}$, and observational $S^{1}$ fibers.

These works showed that the cubic density
\[
\rho(x) = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x
\]
satisfies
\[
\int_0^1 \rho(x)\,dx = 4\pi^3 + \pi^2 + \pi \approx 137.036,
\]
reproducing $\alpha^{-1}$ to 0.0002\% precision.

But one major structural element remained unproven:

\begin{quote}
\emph{What determines $\alpha$ itself?}
\end{quote}

The answer is sought in the inversion of the self-referential observation operator.
Here we set out how this inversion would lead to an irreducible \emph{quintic} in $\alpha$; as the abstract and the Status note record, what follows is a schema for that construction, not the construction itself.

\section{Background and Prior Results}

\subsection{The cubic density and its moments}

The cubic density generates moments
\[
\mu_n = \int_0^1 x^n\rho(x)\,dx =
\frac{16\pi^3}{n+4} + \frac{3\pi^2}{n+3} + \frac{2\pi}{n+2}.
\]

Key values:
\[
\mu_0 = \alpha^{-1},\qquad
\mu_1 \approx 108.71668,\qquad
\mu_2 \approx 90.17596.
\]

\subsection{Self-lensing energy and oscillation parameter}

The self-lensing energy functional
\[
E_{\mathrm{self}} = \frac{\int_0^1 (\rho'(x))^2 dx}{2\mu_0^2}
\]
evaluates to $13.177$, creating the oscillation arena described previously.

The oscillation parameter satisfies
\[
\kappa = \alpha^{5/4}.
\]

\subsection{The observation operator}

The operator is
\[
\hat O_\alpha = -\frac{d^2}{dx^2} + 
\frac{(\rho'(x))^2}{2\mu_0^2} + 
E_{\mathrm{self}}\,x^2(1-x)^2.
\]

Its first eigenvalue $\lambda_1(\alpha)$ generates the electron mass scale.

\section{Forward vs. Inverse Problem}

The previous papers solve the \emph{forward} problem:
\[
\alpha \;\to\; \rho(x) \;\to\; \mu_n \;\to\; V(x;\alpha)
\;\to\; \hat O_\alpha \;\to\; \lambda_1 \;\to\; m_e.
\]

Here we solve the \emph{inverse} problem:
\[
m_e \;\to\; \lambda_1(\alpha) \;\to\; \hat O_\alpha 
\;\to\; V(x;\alpha) \;\to\; \mu_0(\alpha)
\;\to\; \alpha.
\]

This inversion creates the quintic structure.

\section{Deriving the Quintic Consistency Equation}

\subsection{Condition 1: $\kappa$-law}

From $\kappa = \alpha^{5/4}$ and
\[
\kappa_{\mathrm{geom}}(\alpha)
= C_1 \frac{\mu_2}{\mu_0^2}
+ C_2 \frac{\mu_1}{\mu_0}
+ C_3,
\]
we equate:
\[
\kappa_{\mathrm{geom}}(\alpha) = \alpha^{5/4}.
\]

Raising to the fourth power eliminates fractional exponents:
\[
\kappa_{\mathrm{geom}}(\alpha)^4 = \alpha^5.
\]

\subsection{Condition 2: eigenvalue constraint}

Let $F(\lambda_1(\alpha),\beta(\alpha)) = 0$ encode the electron mass scale.

Both $\lambda_1$ and $\beta=\mu_1/\mu_0$ depend on $\alpha$ rationally.

\begin{remark}[TBS]\label{rmk:tbs:P013_3_c}
The claim that $\lambda_1(\alpha)$ depends rationally (or even algebraically) on $\alpha$ is asserted without proof. The first eigenvalue of the Sturm--Liouville operator $\hat{O}_\alpha = -d^2/dx^2 + V(x;\alpha)$ with $\alpha$-dependent potential is a transcendental function of $\alpha$ in general; establishing any algebraic dependence requires a separate theorem that is not provided here or elsewhere in the corpus.
\end{remark}

\subsection{Condition 3: normalization}

$\mu_0 = \alpha^{-1}$.

\subsection{Consolidation}

After algebraic elimination of intermediate quantities:
\[
N(\alpha)^4 - \alpha^5 D(\alpha)^4 = 0,
\]
where $N$ and $D$ are polynomials in $\alpha$ from the moment structure.

Clearing denominators yields:
\[
P(\alpha) =
a_5\alpha^5 + a_4\alpha^4 + a_3\alpha^3 +
a_2\alpha^2 + a_1\alpha + a_0 = 0.
\]

\begin{remark}[TBS]\label{rmk:tbs:P013_2}
The claim that denominator-clearing of the three consistency conditions yields exactly degree 5 rather than a higher degree is asserted but not proved; a complete Gröbner-basis or resultant computation demonstrating the degree bound is required.
\end{remark}

This is the quintic.

\begin{remark}[TBS]\label{rmk:tbs:P013_1}
The coefficients $a_5,a_4,a_3,a_2,a_1,a_0$ of the claimed irreducible quintic are never explicitly computed from the three consistency conditions; only their existence is asserted. The explicit coefficient expressions must be derived and verified.
\end{remark}

\paragraph{Status.} Registry items P013\_1, P013\_2, and P013\_3\_c are all confirmed load-bearing (A293), and together they constitute the corpus's standing diagnosis of this paper, also stated in Paper 40: the quintic structure is a schema, not a construction. Conditions 1 through 3 and the elimination map $F$ are unspecified, so the coefficients $a_5,\ldots,a_0$ are uncomputable as stated and the degree-5 bound is unverifiable until the construction is supplied. The rationality premise behind Condition 2 is unproven, and is generically false for Sturm--Liouville eigenvalues. The schema stands as the record of the intended structure; none of its quantitative claims is established. Ledger: Paper 40 and addenda/verify/tbs\_registry.json.

\section{Irreducibility}

If the construction were completed as claimed, with explicit coefficients $a_5,\ldots,a_0$ computed from Conditions 1--3 and the elimination map (see the Status note above), the intended argument would run:

\begin{itemize}
\item no rational root would exist, by the Rational Root Test applied to the computed coefficients;
\item the Galois group would be shown to be $S_5$, implying insolubility by radicals;
\item the coefficients would reflect the five geometric contributions:
\begin{align*}
&\text{(1) bulk term } x^3,\\
&\text{(2) boundary term } x^2,\\
&\text{(3) edge term } x,\\
&\text{(4) self-lensing correction},\\
&\text{(5) oscillatory exponent } 5/4.
\end{align*}
\end{itemize}

None of these steps can currently be carried out: the coefficients are uncomputable as stated (Remark~\ref{rmk:tbs:P013_1}), so neither the Rational Root Test nor any Galois-group computation can be performed. The claim that the quintic is geometrically irreducible is conditional on completing the construction.

\section{Physical Interpretation}

The quintic structure reflects:

\begin{enumerate}
\item Five-layer constraint network.
\item Five degrees of curvature freedom.
\item Five distinct contributions to boundary feedback.
\item Non-reducibility of the three-layer ontology extended by two dynamical laws.
\end{enumerate}

The number $5$ appears because the feedback loop has five independent terms.

\section{Numerical Verification Program}

To confirm $\alpha$, once the construction is supplied:

\begin{enumerate}
\item Compute $P(\alpha)$ numerically near $\alpha^{-1}=137.036$.
\item Use Newton iteration on $P(\alpha)$.
\item Verify uniqueness of positive root.
\item Cross-check with full eigenvalue computation.
\end{enumerate}

None of these steps can be executed as the paper stands: until Conditions 1--3 are made explicit and the constraint $F$ is supplied, there is no computable $P(\alpha)$ to evaluate, no polynomial for Newton iteration, no coefficients against which to isolate the root, and no eigenvalue problem to cross-check. The program is the verification protocol for the completed construction, not verification that has been performed.

\section{Conclusion}

We have proposed a schema for the structural equation of the geometric framework: if the three consistency conditions and the elimination map were supplied, the inversion of the self-referential observation operator would determine $\alpha$ as the positive root of a quintic.

Per the standing diagnosis (Addendum 293, Paper 40), the schema is not yet a construction. The coefficients $a_5,\ldots,a_0$ are uncomputable as stated, the degree-5 bound is unverified, and the rationality premise behind Condition 2 is unproven. Completing the construction, by specifying Conditions 1--3 and the constraint $F$, computing the coefficients, and establishing the degree and irreducibility claims, is the open task this paper leaves to the corpus; only then would the theoretical circle begun in the previous papers close.

\begin{thebibliography}{99}

\bibitem{paper01}
L.~F.~Vlegels, ``The Perfect Stable Sphere: A Geometric Foundation for Fundamental Physics,'' This volume, Paper 1 (2025).

\bibitem{paper02}
L.~F.~Vlegels, ``Geometric Spectral Theory of the Fine-Structure Constant,'' This volume, Paper 2 (2025).

\bibitem{paper03}
L.~F.~Vlegels, ``Mathematical Foundations of Geometric Fundamental Physics,'' This volume, Paper 3 (2025).

\bibitem{paper10}
L.~F.~Vlegels, ``The Three-Fourths Correction: Equilibrium Formula for the Fine-Structure Constant,'' This volume, Paper 10 (2025).

\bibitem{paper12}
L.~F.~Vlegels, ``Five-Fold Convergence of a Geometric Coefficient,'' This volume, Paper 12 (2025).

\bibitem{abel1826}
N.~H.~Abel, ``Beweis der Unm\"oglichkeit, algebraische Gleichungen von h\"oheren Graden als dem vierten allgemein aufzul\"osen,'' \textit{J. Reine Angew. Math.} \textbf{1}, 65--84 (1826).

\end{thebibliography}

\end{document}
