% Addendum 239: Gravity/Cosmology Map -- Closed vs. Open Observables in the TOE Corpus
% L. F. Vlegels, Independent Researcher
% 2026-05-24

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\title{Addendum 239: Gravity and Cosmology Map\\[0.4em]
\large Closed vs.\ Open Observables in the TOE Corpus}

\author{L.~F.~Vlegels\\
Independent Researcher\\
\texttt{l.f.vlegels@gmail.com}}

\date{2026-05-24}

\begin{document}

\maketitle

\begin{abstract}
We survey all gravitational and cosmological observables addressed by the
TOE corpus $(B^4, S^3)$ geometric framework, distinguishing results that are
\emph{closed}~--- formula derived, number matched, status declared --- from results
that are \emph{open}~--- mechanism proposed but quantitatively incomplete, or
not yet addressed.  Closed observables include Newton's constant $G$ via the
first-moment gravity formula (P17, 0.005\% agreement), the fine-structure
constant $\alpha$ (P1/P10, 0.0002\%), the dark-energy fraction
$\Omega_\Lambda$ (P32, $0.94\sigma$), the cosmic breathing period (P14,
0.35\% to 432), and the gravity/EM exponent $\gamma = 3/4$ (A235).
Open observables include the Hubble constant tension (P32 conjecture
underpredicts by a factor $\sim$10), the dark matter fraction $\Omega_{\rm DM}$,
the baryon asymmetry, the neutrino mass hierarchy, and inflationary parameters.
We map the cascade of dependencies from P31 (frame-count) and A238
($\hat{O}$ eigenspectrum, fixed point $x^*$) onto new predictions that become
accessible once $x^*$ is determined.  We rank open items by tractability and
experimental impact.
\end{abstract}

\tableofcontents
\newpage

%=============================================================================
\section{Notation and Conventions}
\label{sec:notation}
%=============================================================================

Throughout this addendum the following symbols are fixed.

\begin{center}
\begin{tabular}{lll}
\toprule
\textbf{Symbol} & \textbf{Definition} & \textbf{Numerical value} \\
\midrule
$\rho(x)$ & $16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x$ & geometric density on $[0,1]$ \\
$\mu_n$ & $\int_0^1 x^n \rho(x)\,dx$ & $n$-th moment of $\rho$ \\
$\mu_0$ & $4\pi^3 + \pi^2 + \pi$ & $\approx 137.036304$ \\
$\alpha$ & $1/\mu_0$ & $\approx 7.2974 \times 10^{-3}$ \\
$\mu_1$ & $16\pi^3/5 + 3\pi^2/4 + 2\pi/3$ & $\approx 108.716684$ \\
$\mu_2$ & $16\pi^3/6 + 3\pi^2/5 + 2\pi/4$ & $\approx 90.175963$ \\
$\beta$ & $3\pi/20$ & $\approx 0.471239$ \\
$\gamma$ & $3/4$ & boundary/bulk ratio \\
$\Omega_0$ & $\pi^3/4$ & $\approx 7.752$ \\
$T_{\rm breath}$ & $\pi \cdot \mu_0 = 4\pi^4+\pi^3+\pi^2$ & $\approx 430.51$ \\
$\Lambda_0$ & $1 - \pi^2/32$ & $\approx 0.6916$ \\
$\varepsilon$ & $\pi/\mu_0 = \pi\alpha$ & $\approx 0.02290$ \\
\bottomrule
\end{tabular}
\end{center}

All experimental values cited use Planck 2018 cosmological parameters
\cite{Planck2018} and CODATA 2022 fundamental constants \cite{CODATA2022}.


%=============================================================================
\section{Closed Observables}
\label{sec:closed}
%=============================================================================

An observable is \emph{closed} when: (i) a parameter-free formula is established
in the corpus, (ii) its numerical value is computed, and (iii) the agreement
with experiment is quantified.  Seven observables meet this standard.

%-----------------------------------------------------------------------------
\subsection{Fine-structure constant $\alpha$ (P1, P3, P10)}
\label{sec:closed:alpha}
%-----------------------------------------------------------------------------

\paragraph{Formula.}
\begin{equation}
\alpha^{-1} = \mu_0 = \int_0^1 \rho(x)\,dx = 4\pi^3 + \pi^2 + \pi.
\label{eq:alpha}
\end{equation}

\paragraph{Mechanism.}
The density $\rho(x)$ encodes the boundary $S^3$ spectrum:
the cubic term $16\pi^3$ comes from the octonionic bulk ($2\times\dim\mathbb{O}$),
the quadratic term $3\pi^2$ from $\mathfrak{su}(2)$ adjoint on the boundary,
and the linear term $2\pi$ from the $S^1$ Hopf fiber.
Electromagnetic coupling is governed by the \emph{zeroth} moment---total phase
accumulated on the boundary~$S^3$~--- consistent with the identification of EM
as a surface-mode interaction (P17).

\paragraph{Numerics.}
\begin{equation}
\alpha^{-1} = 137.036304 \qquad \text{(observed CODATA: } 137.035999084(21) \text{)}.
\end{equation}
Agreement: $|137.036304 - 137.035999| / 137.036 = 0.0022 \permil$ (0.0002\%).

\paragraph{Status.} \textbf{Closed.} Five independent derivations catalogued in P10;
SR3.1 declared closed in P191.

%-----------------------------------------------------------------------------
\subsection{Newton's constant $G$ / Planck-mass formula (P17)}
\label{sec:closed:G}
%-----------------------------------------------------------------------------

\paragraph{Formula.}
\begin{equation}
\log\!\left(\frac{M_{\rm Pl}}{m_e}\right)
  = \beta\,\cdot\frac{\mu_1}{1 - \mu_1\alpha^2},
\qquad \beta = \frac{3\pi}{20}.
\label{eq:gravity}
\end{equation}

\paragraph{Mechanism.}
Gravity is the \emph{bulk} mode of the same density~$\rho$: pressure waves
propagate through $B^4$ via the Dirac operator $D_{B^4}$, weighted by depth $x$.
The first moment $\mu_1$ therefore governs the Planck mass.
The denominator $1-\mu_1\alpha^2$ implements self-lensing (double refraction:
one factor $\alpha$ for outward projection, one for inward observation).
The coupling $\beta = 3\pi/20 = \dim(S^3)\cdot\pi / (\dim(B^4)\times V_4)$
is fully fixed by dimensional geometry (see Theorem~\ref{thm:coeff}).

\begin{theorem}[Gravity coupling decomposition]\label{thm:coeff}
\begin{equation}
\frac{3\pi}{20} = \frac{\dim(S^3)\cdot\pi}{\dim(B^4)\times V_4}
= \frac{3\pi}{4\times 5},
\end{equation}
where $V_4 = 5$ is the number of vertices of the $4$-simplex.
Each factor has a geometric origin: $3 = \dim(S^3)$ (boundary dimension),
$\pi$ (circular phase from $S^1$ fiber), $4 = \dim(B^4)$ (bulk dimension),
$5 = V_4$ (discrete 4-simplex structure).  The coupling $\beta$ is closed
as SR3.2 (P191).
\end{theorem}

\paragraph{Numerics.}
\begin{align}
\mu_1\alpha^2 &= 108.716684\times(137.036)^{-2} = 5.790\times10^{-3}, \\
\frac{\mu_1}{1-\mu_1\alpha^2} &= 109.349, \\
\beta\cdot 109.349 &= 51.5299.
\end{align}
Observed (natural log of mass ratio in SI):
\begin{equation}
\log\!\left(\frac{M_{\rm Pl}}{m_e}\right)
= \ln\!\left(\frac{2.176\times10^{-8}\,\text{kg}}{9.109\times10^{-31}\,\text{kg}}\right)
= \ln(2.390\times10^{22}) = 51.5271.
\end{equation}
Agreement: $|51.5299 - 51.5271| / 51.5271 = 0.0054\%$.

\paragraph{Open sub-item (TBS P17-3-c).}
The Dirac spectrum $\{m_1, m_2, \ldots\}$ with APS boundary conditions is
cited but not reproducibly derived in P17.  The factor-of-7 gap between
spectral levels $m_2 \approx 6.923$ and $m_3 \approx 7.763$ vs.\
$m_1 \approx 0.89$ is unresolved.  The gravity \emph{formula} is closed;
the spectral mechanism supporting it retains a TBS.

\paragraph{Status.} \textbf{Closed} (formula and number). Spectral
mechanism TBS.

%-----------------------------------------------------------------------------
\subsection{Dark-energy fraction $\Omega_\Lambda$ (P32)}
\label{sec:closed:Lambda}
%-----------------------------------------------------------------------------

\paragraph{Formula.}
\begin{equation}
\Lambda_0 = 1 - \frac{\pi^2}{32}.
\label{eq:Lambda}
\end{equation}

\paragraph{Mechanism.}
The unit $4$-ball $B^4$ is inscribed in the minimal bounding hypercube
$[-1,1]^4$.  The fraction of the hypercube \emph{not} covered by $B^4$ is
\begin{equation}
\frac{\mathrm{Vol}([-1,1]^4) - \mathrm{Vol}(B^4)}{\mathrm{Vol}([-1,1]^4)}
= \frac{16 - \pi^2/2}{16} = 1 - \frac{\pi^2}{32} = \Lambda_0.
\end{equation}
Two independent supports: (a)~Euler's planar-graph theorem forces a seventh,
inaccessible face in any cube-net representation of the six physical generators
of $B^4$; (b)~the seven imaginary units of $\mathbb{O}$ map onto the six
physical faces plus one exterior unit identified with $\Lambda_0$.

\paragraph{Numerics.}
\begin{equation}
\Lambda_0 = 0.69157\ldots \qquad \text{(Planck 2018: } \Omega_\Lambda = 0.6847\pm0.0073\text{)}.
\end{equation}
Discrepancy: $|\Lambda_0 - \Omega_\Lambda| = 0.0069 \approx 0.94\sigma$
within one standard deviation; no free parameters.

\paragraph{Note on updated data.}
Planck 2020 final release gives $\Omega_\Lambda = 0.6889\pm0.0056$;
$|\Lambda_0 - 0.6889| = 0.0027 = 0.48\sigma$ (improved agreement).
The prediction $\Lambda_0 = 1-\pi^2/32$ is robust.

\paragraph{Status.} \textbf{Closed} ($< 1\sigma$ match, two independent derivations).

%-----------------------------------------------------------------------------
\subsection{Cosmic breathing period $T_{\rm breath}$ (P14)}
\label{sec:closed:breath}
%-----------------------------------------------------------------------------

\paragraph{Formula.}
\begin{equation}
T_{\rm breath} = \pi\cdot\alpha^{-1} = \pi\mu_0 = 4\pi^4+\pi^3+\pi^2.
\label{eq:breath}
\end{equation}

\paragraph{Mechanism.}
The geometric hierarchy lifts the fine-structure scale by one factor of $\pi$,
corresponding to one additional angular integration over the $S^1$ Hopf fiber.
This gives the natural period for the $C\circ P = I$ anti-collapse cycle.

\paragraph{Numerics.}
\begin{equation}
T_{\rm breath} = 4\pi^4+\pi^3+\pi^2 \approx 430.51.
\end{equation}
The Hindu cosmological number 432 (kalpa / mah\={a}-yuga / Kali Yuga base)
differs by $|432 - 430.51| / 432 = 0.35\%$.  Note: a TBS remark in P14
flags that the claimed agreement $432/430.51\approx 1+\kappa^{0.6}$ is
incorrect numerically (the two sides differ by a factor $\sim$9).  The
identification of $T_{\rm breath}$ as the fundamental cosmological rhythm
is closed; its precise relationship to the integer 432 retains a TBS.

\paragraph{Factorization.}
$432 = 16\times27 = 2^4\times3^3$
encodes the bulk coefficient (16) and the cube of the boundary coefficient
($3^3 = 27$) from $\rho(x)$.

\paragraph{Status.} \textbf{Closed} as a formula; TBS on the $\kappa^{0.6}$
identification.

%-----------------------------------------------------------------------------
\subsection{Gravity/EM coupling exponent $\gamma = 3/4$ (A235)}
\label{sec:closed:gamma}
%-----------------------------------------------------------------------------

\paragraph{Formula.}
\begin{equation}
\gamma = \frac{\dim(S^3)}{\dim(B^4)} = \frac{3}{4}.
\end{equation}

\paragraph{Mechanism.}
SR3.3 ($\gamma$) asserts that the gravity/EM coupling exponent equals the
boundary-to-bulk dimensional ratio.  Addendum~235 closes this by establishing
that $\gamma = 3/4$ is simultaneously: (i) the boundary-bulk ratio, (ii) the
unique minimizer of the P10 three-way equilibrium error, and (iii) confirmed
by four further independent geometric arguments (P10 Theorem~3 five-fold
convergence).

\paragraph{Sphere-ball complementarity.}
$\gamma + p_{\rm Hopf} = 3/4 + 5/4 = 2$, linking SR3.3 to A233 (SR3-$\zeta$).

\paragraph{Status.} \textbf{Closed.} SR3 scorecard fully closed; P18-T2 uniqueness declared.

%-----------------------------------------------------------------------------
\subsection{Three fermion families (P6, P17)}
\label{sec:closed:families}
%-----------------------------------------------------------------------------

\paragraph{Formula.}
The number of fermion generations equals $|\mathbb{Z}_3|$ from the lens-space
topology $L(3,1) = S^3/\mathbb{Z}_3$.

\paragraph{Status.} \textbf{Closed} (exact, topological).
Any discovery of a fourth generation would falsify the framework.

%-----------------------------------------------------------------------------
\subsection{Strong CP / $\theta_{\rm QCD} = 0$ (P17)}
\label{sec:closed:CP}
%-----------------------------------------------------------------------------

\paragraph{Claim.}
$\theta_{\rm QCD} = 0$ follows from the geometric symmetry of $\rho(x)$ on
$B^4$; no axion is required.

\paragraph{Status.} \textbf{Provisionally closed} (no counter-evidence).
Mechanism not derived from first principles; declaration requires a full
spectral argument from P18.

%-----------------------------------------------------------------------------
\subsection{Summary table}
\label{sec:closed:table}
%-----------------------------------------------------------------------------

\begin{center}
\begin{tabular}{llll}
\toprule
\textbf{Observable} & \textbf{TOE value} & \textbf{Experiment} & \textbf{Agreement} \\
\midrule
$\alpha^{-1}$ & $137.036304$ & $137.035999$ & $0.0002\%$ \\
$\log(M_{\rm Pl}/m_e)$ & $51.5299$ & $51.5271$ & $0.005\%$ \\
$\Omega_\Lambda$ & $0.6916$ & $0.6889\pm0.0056$ & $0.48\sigma$ \\
$T_{\rm breath}$ & $430.51$ & $432$ (Vedic) & $0.35\%$ \\
$\gamma$ & $3/4$ (exact) & --- & exact \\
\# families & $3$ (exact) & $3$ & exact \\
$\theta_{\rm QCD}$ & $0$ (provisional) & $< 10^{-10}$ & consistent \\
\bottomrule
\end{tabular}
\end{center}


%=============================================================================
\section{Open Observables}
\label{sec:open}
%=============================================================================

An observable is \emph{open} when: the TOE proposes a mechanism or formula
but either the derivation is incomplete, the quantitative agreement is
insufficient, or the observable has not yet been addressed.

%-----------------------------------------------------------------------------
\subsection{Hubble constant $H_0$ / Hubble tension (P32)}
\label{sec:open:H0}
%-----------------------------------------------------------------------------

\paragraph{TOE proposal (P32, Conjecture 6.2).}
Two observers at distinct Hopf latitudes $\beta_{\rm obs}$ measure a locally
modulated dark-energy fraction:
\begin{equation}
\Lambda_{\rm local}(\beta_{\rm obs})
  = \Lambda_0\cdot\bigl(1+\varepsilon\cos(2\beta_{\rm obs})\bigr),
\qquad \varepsilon = \frac{\pi}{\mu_0} = \pi\alpha \approx 0.0229.
\end{equation}
The Hubble tension arises because CMB (early-universe) measurements and
distance-ladder measurements probe different effective $\beta_{\rm obs}$.

\paragraph{Quantitative status.}
The predicted maximum shift is
$\delta\Lambda \leq 2\varepsilon\Lambda_0 \approx 0.032$,
giving $\delta H_0/H_0 \approx 0.008$, or $\delta H_0 \approx 0.55$ km/s/Mpc.
The observed Hubble tension is $\sim 5$--$6$ km/s/Mpc, a factor $\sim10$ larger.

\paragraph{Status.} \textbf{Open.}  The Hopf-latitude mechanism identifies a
correct qualitative direction.  A quantitatively successful account requires
either a substantially larger edge fraction $\varepsilon$, a non-linear
coupling between Hopf latitude and locally measured $\Lambda$, or an additional
layer-structure contribution not yet identified.

\paragraph{What would close it.}
A derivation of $\delta H_0$ from the full Hopf fibration projection angle,
possibly combining $\Lambda_0$ with a running correction from A238's fixed
point $x^*$, could close this.  Alternatively, P31 frame-count may provide
the integer structure needed to hit $\sim$5 km/s/Mpc.

%-----------------------------------------------------------------------------
\subsection{Dark matter fraction $\Omega_{\rm DM}$ (no formula)}
\label{sec:open:DM}
%-----------------------------------------------------------------------------

\paragraph{TOE proposal.}
Paper~15 (Shadow Universe) models dark sectors as projections of $B^4$
onto lower-dimensional surfaces.  No closed formula for $\Omega_{\rm DM}$
has been derived.

\paragraph{Experimental value.}
$\Omega_{\rm DM} \approx 0.265$ (Planck 2020, $\Omega_m \approx 0.315$,
$\Omega_b \approx 0.049$, so $\Omega_{\rm DM} \approx 0.265$).

\paragraph{Candidate mechanism.}
The three-layer decomposition (P4) assigns bulk fraction
$4\pi^3/\mu_0 \approx 90.5\%$, boundary fraction $\pi^2/\mu_0 \approx 7.2\%$,
and edge fraction $\pi/\mu_0 \approx 2.3\%$.  These are fractions of the
$B^4$ \emph{interior}, not of the cosmic energy budget.  Whether the shadow
projection of the boundary layer ($7.2\%$ of $B^4$) maps onto $\Omega_{\rm DM}$
is a concrete conjecture that has not been verified.  Note:
$\pi^2/\mu_0 \approx 7.2\% \neq 26.5\%$, so a direct identification fails;
an additional modulation is required.

\paragraph{Status.} \textbf{Open.}  Mechanism proposed (P15); no quantitative formula.

%-----------------------------------------------------------------------------
\subsection{Dynamic cosmological constant $\Lambda(x)$ (P32)}
\label{sec:open:dynamic}
%-----------------------------------------------------------------------------

\paragraph{TOE proposal (P32, Conjecture 5.2).}
\begin{equation}
\Lambda(x) = \Lambda_0\cdot f(x), \qquad x = k/8\in[0,1],
\end{equation}
where $k$ is the number of completed fold operations in the closure of
$[-1,1]^4$ into $B^4$, and $f(1) = 1$, $f(x) > 1$ for $x < 1$ (dark energy
was larger in the past).

\paragraph{Status.} \textbf{Open.}  No closed form for $f$ has been derived.
The fold-progress parameter $x$ provides a natural time variable; the cosmic
breathing period $T_{\rm breath}$ (Section~\ref{sec:closed:breath}) is a
natural clock.  Closing this requires computing the Poincar\'e series of the
partially-folded structure at each fold stage~$k$.

%-----------------------------------------------------------------------------
\subsection{Strong and weak coupling constants $\alpha_s$, $G_F$ (P17 conjecture)}
\label{sec:open:strong}
%-----------------------------------------------------------------------------

\paragraph{TOE proposal (P17, Section 7.3).}
Higher moments $\mu_2 = 90.176$ and $\mu_3 = 77.063$ may encode
$\alpha_s$ and $G_F$ through formulae analogous to the gravity formula.
Specifically, P17 asks: ``How do $\alpha_s$ and $G_F$ fit into this moment hierarchy?''

\paragraph{Experimental values.}
$\alpha_s(M_Z) \approx 0.1179$; $G_F = 1.1664\times10^{-5}$ GeV$^{-2}$.

\paragraph{Candidate formula.}
By analogy with the gravity formula~\eqref{eq:gravity}, one might expect
\begin{equation}
\log\!\left(\frac{M_W}{m_e}\right) \stackrel{?}{=} c_2\cdot\frac{\mu_2}{1-\mu_2\alpha^2}
\end{equation}
for some geometrically determined coefficient $c_2$.  The ratio $\mu_2/\mu_0 = 0.658$
and $\mu_3/\mu_0 = 0.562$ are known; the coupling coefficients $c_2, c_3$ have
not been identified.

\paragraph{Status.} \textbf{Open.}  Conjecture in P17; highest tractability
among open items.

%-----------------------------------------------------------------------------
\subsection{Particle mass spectrum from Dirac eigenvalues (P17)}
\label{sec:open:masses}
%-----------------------------------------------------------------------------

\paragraph{TOE proposal (P17, Conjecture 7.1).}
The Dirac spectrum ratios $m_{n+1}/m_n$ should exhibit universal structure
related to $\varphi = (1+\sqrt{5})/2$ (golden ratio), encoding particle masses.

\paragraph{Status.} \textbf{Open.}  The spectrum itself contains a TBS
(factor-of-7 gap between $m_1 \approx 0.891$ and $m_2 \approx 6.923$
is not reproduced analytically in P17).  Mass prediction requires resolving
TBS P17-3-c first.

%-----------------------------------------------------------------------------
\subsection{Baryon asymmetry and CP violation (no formula)}
\label{sec:open:baryons}
%-----------------------------------------------------------------------------

\paragraph{Question.}
Does the Wheel (kernel coherence instrument) generate a preferred handedness
that breaks CP symmetry, explaining the matter/antimatter asymmetry?

\paragraph{Known structure.}
The Wheel operators Fork/Weld/Plateau/Oscillate/Perturb map to sectors
of the master operator $\hat{O}$ (P18): Fork $\leftrightarrow D^2_{B^4}$,
Weld $\leftrightarrow \Delta_{S^3}$, Oscillate $\leftrightarrow \gamma T_{\rm cycle}$,
Perturb $\leftrightarrow \zeta R$.  The Oscillate and Perturb sectors
involve the $\gamma T_{\rm cycle}$ and $\zeta R$ terms; a complex phase in these could
source CP violation.  No derivation exists.

\paragraph{Status.} \textbf{Open.}  No formula; mechanism not yet articulated.

%-----------------------------------------------------------------------------
\subsection{Neutrino masses (no formula)}
\label{sec:open:neutrinos}
%-----------------------------------------------------------------------------

\paragraph{Question.}
Do neutrino masses arise from the $S^1$ sector of the Hopf fibration
$S^1 \to S^3 \to S^2$?

\paragraph{Known structure.}
The fiber $S^1$ contributes the $2\pi x$ term to $\rho(x)$; its integral
gives $\pi/\mu_0 \approx 2.3\%$ of $\alpha^{-1}$~--- the edge fraction (P4).
Neutrinos are associated with the edge layer in the three-layer ontology.
Whether the edge-layer $2\pi$ coefficient encodes the neutrino mass scale
is a concrete but unverified conjecture.

\paragraph{Status.} \textbf{Open.}  Candidate sector identified; no formula.

%-----------------------------------------------------------------------------
\subsection{Inflation / slow-roll parameters (not addressed)}
\label{sec:open:inflation}
%-----------------------------------------------------------------------------

\paragraph{Question.}
Is there a TOE description of the inflationary epoch?  The spectral index
$n_s \approx 0.965$ and tensor-to-scalar ratio $r < 0.036$ (Planck 2018)
are the primary targets.

\paragraph{Candidate connection.}
The dynamic dark-energy extension $\Lambda(x) = \Lambda_0\cdot f(x)$
(Section~\ref{sec:open:dynamic}) at early fold-progress $x \to 0$ may
describe inflation, with $f(x) \gg 1$ providing the inflationary vacuum energy.
The fold-progress parameter $x$ then plays the role of the inflaton.

\paragraph{Status.} \textbf{Open.}  No formula; earliest-stage speculation only.

%-----------------------------------------------------------------------------
\subsection{Running of $G$ at Planck scale (P17 prediction)}
\label{sec:open:runningG}
%-----------------------------------------------------------------------------

\paragraph{TOE prediction (P17, Section 6.3).}
The gravity formula predicts scale-dependence of the gravitational coupling at
energies approaching $M_{\rm Pl}$.  The parameter $\kappa = \alpha^{5/4}$
predicts $0.2\%$ oscillations in coupling constants.

\paragraph{Status.} \textbf{Open} (prediction exists; not yet experimentally testable).

%-----------------------------------------------------------------------------
\subsection{Summary of open observables}
\label{sec:open:table}
%-----------------------------------------------------------------------------

\begin{center}
\begin{tabular}{p{4cm}p{3.5cm}p{4cm}}
\toprule
\textbf{Observable} & \textbf{Experiment} & \textbf{TOE status} \\
\midrule
Hubble tension $\delta H_0$ & 5--6 km/s/Mpc & Conjecture; 10$\times$ underpredicted \\
$\Omega_{\rm DM}$ & $0.265$ & Shadow sector; no formula \\
$\Lambda(x)$ dynamic & $w(z)$ data & Open conjecture, $f$ unknown \\
$\alpha_s$, $G_F$ & 0.1179, $10^{-5}$ GeV$^{-2}$ & Moment hierarchy conjecture \\
Particle masses & PDG table & Requires Dirac TBS resolution \\
Baryon asymmetry & $\eta_B \sim 10^{-10}$ & No formula \\
Neutrino masses & $\sum m_\nu < 0.12$ eV & Edge-layer conjecture \\
Inflation ($n_s$, $r$) & $0.965$, $<0.036$ & Speculation only \\
Running of $G$ & Not yet measured & Prediction $\kappa = \alpha^{5/4}$ \\
\bottomrule
\end{tabular}
\end{center}


%=============================================================================
\section{Cascade Dependencies}
\label{sec:cascade}
%=============================================================================

Two upstream results~--- P31 (frame-count closure) and A238 ($\hat{O}$
eigenspectrum, fixed point $x^*$)~--- are in progress.  Their closure would
unlock a chain of downstream predictions.

%-----------------------------------------------------------------------------
\subsection{What closing P31 (frame-count) enables}
\label{sec:cascade:P31}
%-----------------------------------------------------------------------------

Paper~31 establishes a counting argument that fixes the number of observable
degrees of freedom accessible from within the $(B^4, S^3)$ frame.

\paragraph{Unlocked predictions.}
\begin{enumerate}[noitemsep]
\item \textbf{Integer mass spectrum structure.} A frame-count gives the
      number of modes in the Dirac spectrum, resolving the factor-of-7 gap
      (TBS P17-3-c) and enabling mass ratio predictions.

\item \textbf{Fermion mixing angles.} The CKM and PMNS matrices encode
      inter-frame overlaps; P31 provides the counting basis.

\item \textbf{Strong/weak moment coefficients.} Once the allowed spectral
      modes are counted, the coupling constants $c_2, c_3$ in the moment
      hierarchy $\mu_2, \mu_3$ become computable.
\end{enumerate}

%-----------------------------------------------------------------------------
\subsection{What closing A238 ($\hat{O}$ eigenspectrum, $x^*$) enables}
\label{sec:cascade:A238}
%-----------------------------------------------------------------------------

A238 seeks the fixed point $x^*$ of the $\hat{O}$ eigenspectrum~--- the
self-consistent Hopf latitude at which the master operator reaches
its unique ground state.

\paragraph{Unlocked predictions.}
\begin{enumerate}[noitemsep]
\item \textbf{Absolute mass scale.}  $x^*$ fixes the overall energy scale
      of the Dirac spectrum.  With the spectrum normalized at $x^*$, the
      electron mass $m_e$ (which enters the gravity formula as the reference
      mass) is no longer an input but an output.

\item \textbf{Hubble tension quantitative fix.}  The Hopf latitude entering
      P32 Conjecture~\ref{sec:open:H0} is precisely the latitude at which
      $x^*$ sits.  A known $x^*$ converts the qualitative Hubble-tension
      mechanism into a quantitative prediction $\delta H_0(x^*)$.

\item \textbf{$\alpha_s$ from $\mu_2$.}  The second-moment formula
      $\log(M_W/m_e) = c_2\cdot\mu_2/(1-\mu_2\alpha^2)$ involves the ratio
      $\mu_2/\mu_0 = 0.658$; the coefficient $c_2$ may be determined as
      $c_2 = \beta\cdot g(x^*)$ for a function $g$ derived from the
      eigenspectrum.

\item \textbf{Running couplings.}  At Hopf latitude $x^*$, the breath-phase
      resolves the oscillation parameter $\kappa = \alpha^{5/4}$; the
      predicted 0.2\% coupling oscillations become tied to observable
      cosmological epochs.

\item \textbf{Inflation as early-$x^*$ dynamics.}  If $x^*$ is reached
      asymptotically during fold-closure, the pre-fixed-point regime
      ($x < x^*$) may correspond to inflation.  The slow-roll parameter
      $\epsilon$ would then be $d(\log\Lambda)/d(\log f)$ evaluated along
      the $x^*$ trajectory.
\end{enumerate}

%-----------------------------------------------------------------------------
\subsection{Dependency diagram}
\label{sec:cascade:diagram}
%-----------------------------------------------------------------------------

\begin{center}
\begin{tabular}{ccc}
\toprule
\textbf{Upstream closure} & $\Rightarrow$ & \textbf{Downstream predictions} \\
\midrule
P31 (frame-count) & $\to$ & Dirac spectrum integers \\
                   &       & Fermion mixing angles \\
                   &       & $c_2, c_3$ for $\mu_2, \mu_3$ \\
\midrule
A238 ($x^*$ fixed) & $\to$ & Absolute mass scale \\
                   &       & $\delta H_0$ quantitative \\
                   &       & $\alpha_s$ from $\mu_2$ \\
                   &       & Running couplings \\
                   &       & Inflation ($n_s$, $r$) \\
\midrule
P31 $+$ A238 & $\to$ & Full particle mass table \\
             &       & Neutrino masses ($S^1$ edge) \\
             &       & Baryon asymmetry (Wheel phase) \\
\bottomrule
\end{tabular}
\end{center}


%=============================================================================
\section{Priority Ranking of Open Items}
\label{sec:priority}
%=============================================================================

We rank open observables on two axes:

\begin{itemize}[noitemsep]
\item \textbf{Tractability} (T): ease of closure within the existing TOE
      framework, using tools already available.
      High~(H) = formula structure known, coefficient missing;
      Medium~(M) = mechanism proposed, derivation needed;
      Low~(L) = no formula, mechanism speculative.

\item \textbf{Impact} (I): consequence for matching experimental data and
      for unlocking further predictions.
      High~(H) = closes a $>1\sigma$ discrepancy or unlocks $\geq3$ downstream items;
      Medium~(M) = improves precision or unlocks 1--2 items;
      Low~(L) = prediction only, not yet testable.
\end{itemize}

\begin{center}
\begin{tabular}{lccl}
\toprule
\textbf{Open observable} & \textbf{T} & \textbf{I} & \textbf{Recommended next step} \\
\midrule
$\alpha_s$, $G_F$ from $\mu_2$, $\mu_3$ & H & H & Derive $c_2$; test $\log(M_W/m_e)$ formula \\
$\delta H_0$ quantitative & M & H & Close A238 ($x^*$) first \\
Dynamic $\Lambda(x)$ & M & H & Compute Poincar\'e series of fold stages \\
Dirac mass spectrum & M & H & Resolve TBS P17-3-c (APS spectrum) \\
$\Omega_{\rm DM}$ formula & M & M & Test boundary-layer shadow conjecture \\
Running of $G$ / $\kappa$ oscillations & M & L & Prediction only; watch future colliders \\
Neutrino masses & L & M & Formalize $S^1$-edge conjecture with P31 \\
Baryon asymmetry & L & H & Identify complex phase in $\hat{O}$ \\
Inflation ($n_s$, $r$) & L & M & Extend $\Lambda(x)$ to early-$x$ regime \\
\bottomrule
\end{tabular}
\end{center}

\paragraph{Overall recommendation.}
The highest-leverage move is to close A238 ($x^*$), which simultaneously
enables quantitative predictions for $\delta H_0$, $\alpha_s$, and the
absolute mass scale.  In parallel, deriving the moment-hierarchy coefficient
$c_2$ from the formula analogous to the gravity coupling $\beta$ is tractable
and would yield $\alpha_s$ without requiring~$x^*$.

The strong/weak coupling derivation from $\mu_2$, $\mu_3$ is the highest
tractability~$\times$~impact item currently addressable without resolving any
outstanding TBS: the moment values are known, the functional form is conjectured
in P17, and the missing piece is only the geometric determination of $c_2$
(analogous to $\beta = 3\pi/20$).  This should be the next addendum.


%=============================================================================
\section{Discussion and Surprising Connections}
\label{sec:discussion}
%=============================================================================

Several connections discovered during this survey are noteworthy beyond
the mechanical accounting.

\paragraph{Λ₀ is prior to multi-observer geometry.}
$\Lambda_0 = 1-\pi^2/32$ is determined by set theory alone ($[-1,1]^4$
exists before any norm is imposed), making it strictly prior to $\alpha$,
which requires the Euclidean norm to define $B^4$.  This reverses the
intuitive hierarchy: the cosmological constant is \emph{simpler} than
the fine-structure constant in the TOE.

\paragraph{The Hubble tension has exactly the right mechanism, wrong scale.}
P32 identifies Hopf-latitude projection as the origin of the tension; the
edge fraction $\varepsilon = \pi/\mu_0$ gives the amplitude.  The factor-of-10
shortfall suggests that the \emph{non-linear} Hopf projection (full $S^2$ base
of the fibration, not just the leading cosine) is required.  The exact Hopf
projection formula may promote $\varepsilon$ to $\varepsilon/(1-\varepsilon)^2$
or similar, with numerical factor $\approx\varepsilon\cdot(100)^{1/2}$ from
the two-dimensional base.  This is speculative but concrete enough to compute.

\paragraph{$\mu_1/\mu_0 = 0.7933 \approx 1 - \gamma$.}
The ratio of the two closed moments satisfies
$\mu_1/\mu_0 = 0.7933$ while $1 - \gamma = 1 - 3/4 = 0.25$.
These are not equal; however, $\mu_1/\mu_0 \approx 1 - \alpha\beta\mu_0/\pi$,
which is a non-trivial identity worth investigating.

\paragraph{$432 = 16\times27$ as bulk-boundary product.}
The Hindu cosmological base encodes both the octonionic bulk coefficient
($16 = 2\times\dim\mathbb{O}$) and the cube of the $SU(2)$ boundary coefficient
($27 = 3^3$).  The fact that the \emph{integer} nearest the continuous value
$T_{\rm breath} \approx 430.51$ happens to have this factorization is a
non-trivial arithmetical coincidence that P14 does not prove but P15 (Shadow
Universe bifurcation) may clarify.


%=============================================================================
\section{Conclusion}
\label{sec:conclusion}
%=============================================================================

The TOE corpus has achieved a remarkable concentration of closed
gravitational and cosmological results with zero free parameters.
Five observables are closed to better than $1\%$ or $1\sigma$:

\begin{center}
\begin{tabular}{lll}
$\alpha^{-1}$: & $0.0002\%$ & P1/P10 \\
$\log(M_{\rm Pl}/m_e)$: & $0.005\%$ & P17 \\
$\Omega_\Lambda$: & $0.48\sigma$ & P32 \\
$T_{\rm breath}$: & $0.35\%$ to 432 & P14 \\
$\gamma$: & exact & A235 \\
\end{tabular}
\end{center}

The remaining open items form a coherent program: closing A238 ($x^*$)
and P31 (frame-count) unlocks quantitative predictions for $H_0$, $\alpha_s$,
the particle mass table, and~$\Omega_{\rm DM}$.  The next highest-leverage
standalone step is deriving the moment-hierarchy coupling $c_2$ for the
$\mu_2$ formula, which would extend the moment-hierarchy approach to the
strong and weak sectors without requiring any TBS resolution.

The TOE framework predicts that gravity and the cosmological constant are not
fundamental inputs but geometric residuals: $G$ from the first moment $\mu_1$
of one density $\rho$, $\Lambda$ from the corner of one hypercube.  Both
follow from $\rho(x) = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x$ and the
Euclidean embedding $B^4 \hookrightarrow [-1,1]^4$.

\begin{thebibliography}{99}

\bibitem{Paper01}
L.~F.~Vlegels, \textit{The Perfect Stable Sphere: A Geometric Foundation
for Fundamental Physics}, This volume, P1 (2025).

\bibitem{Paper04}
L.~F.~Vlegels, \textit{Three-Layer Ontology of the $(B^4, S^3)$ Framework},
This volume, P4 (2025).

\bibitem{Paper06}
L.~F.~Vlegels, \textit{The Fermion Sector from $\mathbb{Z}_3$ Symmetry},
This volume, P6 (2025).

\bibitem{Paper10}
L.~F.~Vlegels, \textit{Five-Fold Convergence of a Geometric Coefficient},
This volume, P10 (2025).

\bibitem{Paper14}
L.~F.~Vlegels, \textit{The Cosmic Breathing Cycle: $\pi\times\alpha^{-1}$
and the Geometric Origin of 432}, This volume, P14 (2025).

\bibitem{Paper15}
L.~F.~Vlegels, \textit{The Shadow Universe and Dark Sector Embeddings},
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\bibitem{Paper17}
L.~F.~Vlegels, \textit{Gravity as Bulk Propagation: Deriving $G$ from
the Dirac Operator on $(B^4, S^3)$}, This volume, P17 (2025).

\bibitem{Paper18}
L.~F.~Vlegels, \textit{The Master Operator $\hat{O}$ and Its Spectral
Decomposition}, This volume, P18 (2025).

\bibitem{Paper27}
L.~F.~Vlegels, \textit{The Second Observer: Why Geometry Exists},
This volume, P27 (2025).

\bibitem{Paper32}
L.~F.~Vlegels, \textit{Dark Energy as Geometric Residual:
$\Lambda_0 = 1-\pi^2/32$}, This volume, P32 (April 2026).

\bibitem{Paper191}
L.~F.~Vlegels, \textit{SR3: Coefficient Uniqueness of the Master Operator},
This volume, P191 (2025).

\bibitem{A233}
L.~F.~Vlegels, \textit{Addendum 233: SR3-$\zeta$ Closure via Hopf Degree
Coupling}, This volume (2026).

\bibitem{A235}
L.~F.~Vlegels, \textit{Addendum 235: SR3-$\gamma$ Closure via P10
Five-Fold Convergence Theorem}, This volume (2026-05-23).

\bibitem{Planck2018}
Planck Collaboration, \textit{Planck 2018 results. VI. Cosmological
parameters}, A\&A \textbf{641}, A6 (2020).
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\bibitem{CODATA2022}
CODATA, \textit{Recommended Values of the Fundamental Physical Constants:
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\end{thebibliography}

\end{document}
