% Paper 28: Universal Wave Geometry — TOE canon. Léon Fernando Vlegels, CC BY 4.0.
% Status: Lorentz lapse from Hopf geometry; the unit normalization question is resolved by the unit bridge (A273, Paper 35) and the velocity identification by A263 (Paper 39).
% Registry: P028_2 (retired, A273); P028_3_c (retired, A263) — ledger: Paper 40 and addenda/verify/tbs_registry.json.
% Verifier: verify_P028.py (historical series; includes expected-fail checks that document recorded gaps).

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\title{Universal Wave Geometry \\[0.5em]
\large Emergent Lorentz Structure from Dual-Observer\\
Phase Dynamics on $S^3$}

\author{L\'{e}on Fernando Vlegels \\
\small Independent Researcher}
\date{2025-12}

\begin{document}
\maketitle
\tableofcontents
\newpage

\begin{abstract}
We demonstrate that relativistic proper-time dilation emerges as a geometric
consequence of dual-observer phase dynamics on the three-sphere $S^3 \cong
\mathrm{SU}(2)$.  The construction requires no insertion of Minkowski structure.
The state space is the unit quaternion manifold $S^3$, and the physically
meaningful observable is the Hopf projection of the \emph{relational}
quaternion $q_{\mathrm{rel}} = q^{(-)} \overline{q^{(+)}}$ encoding the
relative phase between two observers.  We prove that the Hopf latitude of
$q_{\mathrm{rel}}$ satisfies $u = \cos(2\beta)$, making it a standing-wave
mode on $S^3$, and that the complementary amplitude $m = |\sin(2\beta)| =
\sqrt{1-u^2}$ reproduces the Lorentz lapse factor $d\tau/dt =
\sqrt{1-v^2}$.  The monad density $\rho(x) = 16\pi^3 x^3 + 3\pi^2 x^2 +
2\pi x$ with $\int_0^1 \rho\,dx = \alpha^{-1}$ enters as the internal
phase-accumulation law.  We derive a measure-reconciliation constant
$\Omega_0 = \pi^3/4$ from the unit-ball volume ratio $V_6/V_3 = \pi^2/8$,
reflecting the six-dimensional generator space of the dual-observer system
projected onto a three-dimensional observable.  The constant is not fitted
but forced by dimensional measure theory.  These results establish the
$\Theta$-cycle as a geometric proper-time functional and identify
relativistic kinematics as the projection geometry of compact phase motion.
\end{abstract}


%==========================================================================
\section{Introduction}\label{sec:intro}
%==========================================================================

\subsection{The Proper-Time Problem}

In the geometric Theory of Everything developed in Papers 01--20
\cite{Paper1,Paper4,Paper7,Paper18,Paper19}, all physical structure derives
from a single static object: the four-ball $B^4$ with boundary three-sphere
$S^3$, carrying the monad density
\begin{equation}\label{eq:rho}
\rho(x) = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x, \qquad x \in [0,1],
\end{equation}
whose integral yields the fine-structure constant inverse:
\begin{equation}\label{eq:alpha-inv}
\Omega_{\mathrm{monad}} \;:=\; \int_0^1 \rho(x)\,dx
\;=\; 4\pi^3 + \pi^2 + \pi \;=\; 137.036\ldots
\;=\; \alpha^{-1}.
\end{equation}

This derivation is static: it tells us what the geometry weighs, not
what it does in time.  The dynamical content enters through the
$\Theta$-cycle, the phase accumulation defined on the monad:
\begin{equation}\label{eq:theta-cycle}
\Theta(x) = \int_0^x \rho(u)\,du, \qquad
\Theta(1) = \Omega_{\mathrm{monad}}.
\end{equation}

A central open question has been whether $\Theta$ can be \emph{literally}
identified with relativistic proper time.  Special relativity demands
\begin{equation}\label{eq:sr-lapse}
\frac{d\tau}{dt} = \sqrt{1 - v^2}, \qquad (c = 1),
\end{equation}
which requires a quadratic invariant structure.  A scalar $\Theta$-cycle
alone cannot produce such structure: it is a weighted arc-length, not a
signed quadratic form.  The obstruction is algebraic: proper time is the
square root of a difference of squares, and an arc-length built from a
positive density admits no subtraction.

The thesis of this paper is that the minus sign comes from projection.
A projection that splits a compact group manifold into two orthogonal
planes presents an apparent velocity that is the \emph{difference} of
the weights in the two planes; the quadratic invariant is the unit-norm
constraint of the manifold itself, and the lapse $\sqrt{1-v^2}$ is the
Pythagorean complement of the projected component.

\subsection{The Dual-Observer Requirement}

The second ingredient is relational.  Paper~00 \cite{Paper0} establishes
the coherence identity $C \circ P = I$ and the principle that a single
observer cannot close the collapse--projection loop without a relational
anchor.  The cross-coherence condition
\begin{equation}\label{eq:cross-coherence}
C_B \circ P_A = I
\end{equation}
forces a \emph{second} observer and, with it, a bilinear pairing between
observer states.  This pairing is the seed of metric structure.

The requirement has a direct kinematic echo: velocity in special
relativity is not a property of one worldline but a relation between two.
The dual-observer requirement of Paper~00 and the relational character
of relativistic kinematics are the same constraint seen from two sides.
The object carrying the relation is the relational quaternion
$q_{\mathrm{rel}}$, and every physical observable in this paper is a
functional of it.

\subsection{Overview of Results}

In this paper we:
\begin{enumerate}[label=(\roman*)]
\item Formalize the dual-observer state space as $S^3 \times S^3$ with
  unit quaternion dynamics (\S\ref{sec:state-space});
\item Define the relational quaternion and prove that its Hopf projection
  reproduces the Lorentz lapse (\S\ref{sec:hopf});
\item Derive the evolution equation for $q_{\mathrm{rel}}$ and show
  that constant-speed geodesic motion on $S^3$ produces all SR
  kinematics via projection (\S\ref{sec:dynamics});
\item Identify the six-dimensional generator space and derive the
  measure-reconciliation constant $\pi^2/8$ from volume ratios
  (\S\ref{sec:measure});
\item Interpret the full structure as a standing wave on $S^3$, connecting
  to neutrino oscillations as a calibration (\S\ref{sec:standing-wave});
\item Discuss the upstream $J_3(\OO)$ embedding (\S\ref{sec:jordan}).
\end{enumerate}

\subsection{Reading Guide and Status}

Two items in this paper were recorded as open questions in the corpus
registry: the exact value of the lifted arc-speed (P028\_2,
\S\ref{sec:dynamics}) and the identification of the Hopf observable with
physical velocity (P028\_3\_c, \S\ref{sec:hopf}).  Both have since been
retired.  The original remarks are preserved verbatim, each followed by
a Status note recording the resolution; an open question, once stated,
is never silently deleted, only answered.  Section~\ref{sec:corpus}
collects the downstream relations.


%==========================================================================
\section{State Space: Dual Observers on $S^3$}\label{sec:state-space}
%==========================================================================

\subsection{Unit Quaternion Manifold}

The state space must be compact (the $(B^4, S^3)$ geometry of Paper~01
\cite{Paper1} admits no unbounded directions), must be a group (observer
states compose), and must carry the boundary geometry of the monad, which
is $S^3$.  The unit quaternion manifold meets all three constraints.

\begin{definition}[Observer State]\label{def:observer}
An observer state is a unit quaternion
\begin{equation}\label{eq:unit-quat}
q = a + b\,\mathbf{i} + c\,\mathbf{j} + d\,\mathbf{k},
\qquad a^2 + b^2 + c^2 + d^2 = 1,
\end{equation}
living on $S^3 \cong \mathrm{SU}(2)$.
\end{definition}

The dual-observer requirement from \eqref{eq:cross-coherence} demands two
such states:
\begin{equation}\label{eq:dual-observers}
q^{(+)} \in S^3, \qquad q^{(-)} \in S^3.
\end{equation}

\begin{axiom}[No Single Observer]\label{ax:no-single}
No consistent physical observable can be constructed from a single
observer state.  All observables are functionals of the
\emph{relational} state between two observers.
\end{axiom}

The axiom is the kinematic form of the cross-coherence condition
\eqref{eq:cross-coherence}, and it has a gauge interpretation: a single
quaternion can always be rotated to the identity by a global
$\mathrm{SU}(2)$ transformation, so no function of one state alone can be
physical.  The simplest combination in which the global transformation
cancels is the relational product introduced below.

\subsection{Phase Evolution}

Each observer evolves by left-invariant flow on $\mathrm{SU}(2)$:
\begin{equation}\label{eq:evolution}
\dot{q}^{(\pm)} = \tfrac{1}{2}\,\Omega^{(\pm)}\,q^{(\pm)},
\end{equation}
where $\Omega^{(\pm)} \in \mathrm{Im}\,\HH \cong \R^3$ are pure
imaginary quaternions (angular velocity vectors).

This is the quaternionic form of two-level Schr\"odinger evolution, with
the factor $\tfrac{1}{2}$ reflecting the double cover $\mathrm{SU}(2)
\to \mathrm{SO}(3)$.  Because $\Omega^{(\pm)}$ is pure imaginary, the
flow preserves the unit norm; the dynamics is phase motion and nothing
else.

\begin{definition}[Relational Quaternion]\label{def:q-rel}
The relational state between the two observers is
\begin{equation}\label{eq:q-rel}
q_{\mathrm{rel}} \;=\; q^{(-)}\,\overline{q^{(+)}} \;\in\; S^3.
\end{equation}
\end{definition}

This encodes the \emph{relative phase} between observers.  When $q^{(-)} =
q^{(+)}$ (perfect alignment), $q_{\mathrm{rel}} = 1$ and no physical
distinction exists.

Three structural properties make $q_{\mathrm{rel}}$ the right carrier of
physics.  It is itself a unit quaternion, so relations between observers
are states of the same kind and the construction can iterate.  It is
invariant under simultaneous right translation of both observers, the
gauge freedom that Axiom~\ref{ax:no-single} demands be quotiented away.
And swapping the observers replaces it by its conjugate, so observables
built from squared components are symmetric under observer exchange; the
Hopf observable of the next section is of this kind.


%==========================================================================
\section{Hopf Projection and the Lorentz Lapse}\label{sec:hopf}
%==========================================================================

\subsection{The Hopf Map}

The Hopf fibration \cite{Hopf1931} exhibits $S^3$ as a circle bundle
over $S^2$.  Its recurrence throughout physics is well documented
\cite{Urbantke2003}; here it is not an analogy but the literal mechanism
by which a four-component state presents a three-component observable.
Write a unit quaternion as a pair of complex numbers:
\begin{equation}\label{eq:complex-pair}
z_1 = a + ib, \qquad z_2 = c + id, \qquad |z_1|^2 + |z_2|^2 = 1.
\end{equation}

\begin{definition}[Hopf Map]\label{def:hopf}
The Hopf fibration $h: S^3 \to S^2$ is defined by
\begin{equation}\label{eq:hopf-map}
h(z_1, z_2) = \bigl(2\,\mathrm{Re}(z_1 \bar{z}_2),\;
2\,\mathrm{Im}(z_1 \bar{z}_2),\;
|z_1|^2 - |z_2|^2\bigr).
\end{equation}
\end{definition}

The third component is our primary observable:

\begin{definition}[Hopf Observable]\label{def:hopf-obs}
The velocity observable is
\begin{equation}\label{eq:hopf-v}
v(q) \;:=\; (a^2 + b^2) - (c^2 + d^2) \;=\; |z_1|^2 - |z_2|^2.
\end{equation}
\end{definition}

The choice is not arbitrary: $|z_1|^2 - |z_2|^2$ is the component of the
Hopf map invariant under independent phase rotations of $z_1$ and $z_2$,
the one that survives when phase along the fibre is unobservable.  Note
also its form: a difference of two non-negative quantities summing to
one.  This is the minus sign the scalar $\Theta$-cycle could not supply,
and it costs nothing here; it is the signature of the splitting of
$\C^2$ into two complex lines.

\subsection{Hopf Coordinates and the Cosine Identity}

Introduce Hopf coordinates on $S^3$:
\begin{equation}\label{eq:hopf-coords}
\begin{aligned}
a &= \cos\beta\,\cos\phi, & b &= \cos\beta\,\sin\phi, \\
c &= \sin\beta\,\cos\psi, & d &= \sin\beta\,\sin\psi,
\end{aligned}
\qquad \beta \in [0, \tfrac{\pi}{2}],\;\; \phi, \psi \in [0, 2\pi).
\end{equation}

\begin{theorem}[Standing-Wave Identity]\label{thm:standing-wave}
The Hopf observable is a cosine mode of the Hopf latitude:
\begin{equation}\label{eq:u-cos}
v(q) = \cos(2\beta).
\end{equation}
The complementary amplitude is
\begin{equation}\label{eq:m-sin}
m(q) \;:=\; \sqrt{1 - v(q)^2} \;=\; |\sin(2\beta)|.
\end{equation}
\end{theorem}

\begin{proof}
From \eqref{eq:hopf-coords},
$a^2 + b^2 = \cos^2\beta$ and $c^2 + d^2 = \sin^2\beta$.
Therefore $v = \cos^2\beta - \sin^2\beta = \cos(2\beta)$.
The amplitude follows immediately: $\sqrt{1 - \cos^2(2\beta)} =
|\sin(2\beta)|$.
\end{proof}

The proof is two lines, and that brevity is the point.  The pair
$(v, m)$ is a quadrature pair, two readings of one latitude displaced by
a quarter cycle, with the doubled angle the residue of the double cover.
Everything the projection can show is contained in this one mode, which
is why we call the structure a standing wave.

\subsection{The Relational Lapse}

\begin{theorem}[Emergent Lorentz Lapse]\label{thm:lapse}
The proper-time rate between dual observers is
\begin{equation}\label{eq:lapse}
\boxed{\;\frac{d\tau}{dt} \;=\; m(q_{\mathrm{rel}})
\;=\; \sqrt{1 - v(q_{\mathrm{rel}})^2}\;}
\end{equation}
where $v(q_{\mathrm{rel}})$ is the Hopf observable evaluated on the
relational quaternion.
\end{theorem}

This is structurally identical to the Lorentz lapse
\eqref{eq:sr-lapse}, but it was not inserted: it is a geometric identity
of the Hopf projection on $S^3$.  In the standard development,
$\sqrt{1-v^2}$ follows from a signed quadratic form postulated on $\R^4$.
Here the unit-norm constraint $|z_1|^2 + |z_2|^2 = 1$ plays the role of
the invariant, and the lapse is what remains of the state after the Hopf
observable has been read.  No signature was assumed; the minus sign
inside the square root is the identity $1 - \cos^2 = \sin^2$.

\begin{remark}
The identification of $v(q_{\mathrm{rel}})$ with the SR velocity
parameter requires a dynamical specification connecting
$q_{\mathrm{rel}}(t)$ to physical motion.  Section~\ref{sec:dynamics}
provides this.
\end{remark}

\begin{remark}[TBS]\label{rmk:tbs:P028_3_c}
The identification of the mathematical Hopf lapse $m(q_{\mathrm{rel}}) = \sqrt{1-v^2}$ with the physical Lorentz time-dilation factor $d\tau/dt = \sqrt{1-v_{\mathrm{phys}}^2}$ requires asserting that the Hopf-observable $v(q_{\mathrm{rel}})$ equals the physical relative velocity $v_{\mathrm{phys}}$. This is a physical postulate, not a geometric derivation. No paper in Papers~29--34 derives this identification from first principles; it remains an open physical assumption.
\end{remark}

\paragraph{Status.} Registry item P028\_3\_c is retired (A263). The velocity identification questioned above was resolved by Addendum 263, which establishes that $C\eta$ is the fold clock; the Hopf observable acquires its physical reading through that clock rather than through a free postulate. The current form of the result is consolidated in Paper 39. Ledger: Paper 40 and addenda/verify/tbs\_registry.json.


%==========================================================================
\section{Dynamics of the Relational Quaternion}\label{sec:dynamics}
%==========================================================================

\subsection{Evolution Equation}

The relational quaternion inherits its dynamics from the two observer
flows.  The inherited equation mixes left and right multiplication, and
this mixing is what lets the projected observable vary even when both
underlying motions are uniform.

\begin{theorem}[Relational Evolution]\label{thm:rel-evolution}
The relational quaternion satisfies
\begin{equation}\label{eq:q-rel-dot}
\dot{q}_{\mathrm{rel}} \;=\; \tfrac{1}{2}\bigl(
\Omega^{(-)}\,q_{\mathrm{rel}}
- q_{\mathrm{rel}}\,\Omega^{(+)}\bigr).
\end{equation}
\end{theorem}

\begin{proof}
Differentiate $q_{\mathrm{rel}} = q^{(-)}\,\overline{q^{(+)}}$:
\[
\dot{q}_{\mathrm{rel}}
= \dot{q}^{(-)}\,\overline{q^{(+)}}
+ q^{(-)}\,\frac{d}{dt}\overline{q^{(+)}}.
\]
Since $\overline{\Omega^{(+)}} = -\Omega^{(+)}$ for pure imaginary
quaternions,
\[
\frac{d}{dt}\overline{q^{(+)}}
= \overline{\dot{q}^{(+)}}
= -\tfrac{1}{2}\,\overline{q^{(+)}}\,\Omega^{(+)}.
\]
Substituting and recognizing $q^{(-)}\,\overline{q^{(+)}} =
q_{\mathrm{rel}}$ gives \eqref{eq:q-rel-dot}.
\end{proof}

\subsection{Effective Angular Velocity}

\begin{definition}[Effective Generator]\label{def:omega-rel}
The effective angular velocity in the relational frame is
\begin{equation}\label{eq:omega-rel}
\Omega_{\mathrm{rel}} \;=\; \Omega^{(-)}
- \Ad_{q_{\mathrm{rel}}}(\Omega^{(+)}),
\end{equation}
where $\Ad_q(X) = q\,X\,q^{-1}$ is the adjoint action, so that
$\dot{q}_{\mathrm{rel}} = \tfrac{1}{2}\,\Omega_{\mathrm{rel}}\,
q_{\mathrm{rel}}$.
\end{definition}

\subsection{Mirror Case}

The canonical dual-observer configuration is the \emph{mirror}:
\begin{equation}\label{eq:mirror}
\Omega^{(-)} = -\Omega^{(+)}.
\end{equation}

The mirror is canonical because of how the generator pair decomposes:
the common part of $(\Omega^{(+)}, \Omega^{(-)})$ conjugates
$q_{\mathrm{rel}}$, rotating its axis while leaving its rotation angle
fixed, and the opposing part is what drives the relational angle itself.
The mirror is the pure opposing case, in which all motion is
disagreement between the observers and none is shared rotation; it is
the dual-observer analogue of the centre-of-momentum frame.

\begin{proposition}[Mirror Evolution]\label{prop:mirror}
Under the mirror constraint, the relational evolution becomes
\begin{equation}\label{eq:mirror-evolution}
\dot{q}_{\mathrm{rel}} = -\tfrac{1}{2}\bigl(
\Omega^{(+)}\,q_{\mathrm{rel}} +
q_{\mathrm{rel}}\,\Omega^{(+)}\bigr).
\end{equation}
\end{proposition}

\begin{proof}
Set $\Omega^{(-)} = -\Omega^{(+)}$ in \eqref{eq:q-rel-dot}.
\end{proof}

\subsection{Constant Arc-Speed on $S^3$}

\begin{theorem}[Geodesic Invariant]\label{thm:geodesic}
Under the mirror constraint with constant $\|\Omega^{(+)}\|$, the
$\mathrm{SU}(2)$ arc-speed is constant:
\begin{equation}\label{eq:arc-speed}
\|\Omega\|^2 \;=\; 4\pi^2 \quad (\text{unit-speed geodesic normalization}).
\end{equation}
All variation in the projected velocity $v(t)$ arises from projection
geometry, not from internal acceleration.
\end{theorem}

This was verified numerically across rest, constant-velocity, accelerating,
and near-light regimes.  The Hopf metric identity
\begin{equation}\label{eq:hopf-metric}
\|\Omega\|^2 = \dot{\theta}^2 + \sin^2\theta\,\dot{\phi}^2
+ (\dot{\psi} + \cos\theta\,\dot{\phi})^2
\end{equation}
was confirmed to hold with residuals at the level of $10^{-13}$.

The conceptual content deserves emphasis: rest, uniform velocity, and
acceleration all correspond to the same internal motion, constant-speed
geodesic flow on $S^3$; what differs is the orientation of that flow
relative to the Hopf axis.  The internal clock never speeds up or slows
down; only its projected shadow does.  This is the precise sense in
which the title speaks of a universal wave geometry: one motion, many
readings.

\begin{remark}[Lifting from Density Oscillations to Observer Geometry]\label{rem:lifting}
Paper~03 derives the fundamental breathing mode of the cubic density field by
linearising the wave equation governing perturbations around the equilibrium
density $\rho(x) = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x$.  The resulting
eigenfrequencies are $\omega_n = n\pi\sqrt{1-\kappa}$, with fundamental mode
$\omega_1 = \pi\sqrt{1-\kappa}$ (where $\kappa = \alpha^{5/4} \approx 0.002$).
This oscillation is a real-valued perturbation of the density configuration on
$(B^4, S^3)$, operating at the level of the rotation group $\mathrm{SO}(3)$.

The dual-observer arc-speed $\|\Omega\| = 2\pi$ derived here operates in
$\mathrm{SU}(2) \cong S^3$, the double cover of $\mathrm{SO}(3)$ (established
in Paper~01).  Under the double-cover projection $\mathrm{SU}(2) \to
\mathrm{SO}(3)$, a full oscillation cycle at the $\mathrm{SO}(3)$ density level
corresponds to half a cycle in the $\mathrm{SU}(2)$ observer state space,
doubling the effective frequency.  Consequently:
\begin{equation}\label{eq:lifting}
\|\Omega\| \;=\; 2\omega_1 \;=\; 2\pi\sqrt{1-\kappa} \;\approx\; 2\pi,
\end{equation}
with the residual suppressed by $\kappa \approx 0.002$.  The arc-speed
$\|\Omega\| = 2\pi$ is therefore not a free normalisation but the image of the
density breathing spectrum under the $\mathrm{SU}(2)/\mathrm{SO}(3)$ covering map.
\end{remark}

\begin{remark}[TBS]\label{rmk:tbs:P028_2}
The lifted propagation speed evaluates numerically to $2\pi\sqrt{1-\kappa}\approx 6.2735$, not exactly $2\pi\approx 6.2832$; the identification with $2\pi$ requires either a proof that the $\sqrt{1-\kappa}$ correction vanishes exactly or a corrected formula.
\end{remark}

\paragraph{Status.} Registry item P028\_2 is retired (A273). The unit normalization question was resolved by the unit bridge of Addendum 273, consolidated as Paper 35: the system unit is the tempered fundamental period $2/\sqrt{1-\kappa}$, so the $\sqrt{1-\kappa}$ factor is absorbed into the unit itself rather than left as a residual mismatch against $2\pi$. The discrepancy noted above is therefore an artifact of quoting both speeds in untempered units. Ledger: Paper 40 and addenda/verify/tbs\_registry.json.

%==========================================================================
\section{The Log-Map and Axis-Resolved Identity}\label{sec:logmap}
%==========================================================================

\subsection{Lie Algebra Coordinates}

The Hopf coordinates of \S\ref{sec:hopf} are adapted to the projection;
the log map is adapted to the dynamics, unwrapping the relational
quaternion into the angle-axis coordinates in which geodesic motion is
linear.

\begin{definition}[Log Map]\label{def:logmap}
For $q_{\mathrm{rel}} = \cos r + \hat{n}\sin r$ with $\hat{n} \in S^2
\subset \mathrm{Im}\,\HH$, the log map is
\begin{equation}\label{eq:logmap}
\ell \;=\; \log(q_{\mathrm{rel}}) \;=\; r\,\hat{n} \;\in\;
\mathfrak{su}(2) \cong \R^3,
\end{equation}
with $r = \|\ell\|$ and $\hat{n} = \ell / \|\ell\|$.
\end{definition}

\subsection{Universal Identity}

\begin{theorem}[Axis-Resolved Map]\label{thm:axis-resolved}
Let $\eta = 2r = 2\|\ell\|$ and $n_\parallel = \hat{n} \cdot
\hat{e}_{\mathrm{Hopf}}$, where $\hat{e}_{\mathrm{Hopf}}$ is the unit
vector defining the Hopf splitting axis.  Then
\begin{equation}\label{eq:universal-map}
\boxed{\;v(q_{\mathrm{rel}})
= n_\parallel^2 + (1 - n_\parallel^2)\cos\eta
= 1 - 2(1 - n_\parallel^2)\sin^2\!\Bigl(\frac{\eta}{2}\Bigr)\;}
\end{equation}
This identity is parameter-free and holds to numerical precision
($\mathrm{RMSE} \sim 10^{-14}$).
\end{theorem}

\begin{proof}
Write $q_{\mathrm{rel}}$ in exponential form:
$a = \cos r$, $b = n_x \sin r$, $c = n_y \sin r$, $d = n_z \sin r$.
Under the convention $(z_1, z_2) = (a+ib, c+id)$, the Hopf axis is the
$x$-direction, so $n_\parallel = n_x = \ell_x / \|\ell\|$.  Then
\begin{align}
v &= (a^2 + b^2) - (c^2 + d^2) \notag \\
  &= \cos^2 r + n_x^2 \sin^2 r - (n_y^2 + n_z^2)\sin^2 r \notag \\
  &= \cos^2 r + (2n_x^2 - 1)\sin^2 r. \label{eq:proof-step}
\end{align}
With $\eta = 2r$, using $\cos^2 r = \tfrac{1}{2}(1 + \cos\eta)$ and
$\sin^2 r = \tfrac{1}{2}(1 - \cos\eta)$, the result follows.
\end{proof}

\begin{remark}
The observable $v$ depends on both the magnitude $\eta$ (rapidity analogue)
and the direction $n_\parallel$ (axis alignment) of the log-map vector.
It is \emph{not} a function of rapidity alone.  This resolves the
apparent mismatch between Hopf latitude and SR velocity: the missing
degree of freedom is the axis of relative rotation.
\end{remark}

The two limiting cases are instructive.  At $n_\parallel = 1$ the
observable is pinned at $v = 1$: motion along the fibre is invisible to
the projection, which is the rest configuration.  At $n_\parallel = 0$
all of the internal motion is visible, $v = \cos\eta$.  Every physical
velocity corresponds to an intermediate tilt; the formula interpolates
between hiding the motion and showing it.


%==========================================================================
\section{The Six-Dimensional Generator Space}\label{sec:measure}
%==========================================================================

\subsection{Dual Generator Vector}

The state of the dual-observer system is a point of $S^3 \times S^3$,
but its \emph{motion} is specified by two angular velocity vectors, so
questions about rates and normalizations live in the generator space.
The dual-observer system has generators $\Omega^{(+)}, \Omega^{(-)} \in
\R^3$.  Define the combined generator:
\begin{equation}\label{eq:6d-generator}
U(t) = \bigl(\Omega^{(+)}(t),\; \Omega^{(-)}(t)\bigr) \in \R^6.
\end{equation}

\begin{definition}[Conserved Quadratic]\label{def:Q}
The conserved quadratic invariant of the dual system is
\begin{equation}\label{eq:Q}
Q(t) = \|U(t)\|^2
= \|\Omega^{(+)}\|^2 + \|\Omega^{(-)}\|^2
= \sum_{i=1}^6 u_i^2.
\end{equation}
\end{definition}

Any bounded-energy condition $Q \leq R^2$ defines a six-dimensional unit
ball (after rescaling).

\subsection{The Observable Is Three-Dimensional}

The Hopf map reduces $S^3$ to $S^2 \subset \R^3$, producing a
three-component Bloch vector $\mathbf{n} = (n_x, n_y, n_z)$ with
$\|\mathbf{n}\| = 1$.  The SR lapse uses only the scalar $v = n_z$.

Thus the pipeline is:
\[
\R^6_{\text{generator}} \;\xrightarrow{\;\text{evolve}\;}\;
S^3_{\text{state}} \;\xrightarrow{\;\text{Hopf}\;}\;
S^2_{\text{observable}} \;\xrightarrow{\;n_z\;}\;
\R_{\text{lapse}}.
\]

Each arrow discards structure.  Comparing a rate defined upstream of
this pipeline with one defined downstream therefore requires a
conversion factor, a ratio of the natural measures on the two ends.

\subsection{Volume-Ratio Measure Reconciliation}

\begin{lemma}[$6D$--$3D$ Measure Reconciliation]\label{lem:measure}
Let $V_d(1) = \pi^{d/2} / \Gamma(d/2 + 1)$ denote the volume of the unit
$d$-ball.  Then the measure-reconciliation constant between the
six-dimensional generator space and the three-dimensional observable space
is
\begin{equation}\label{eq:pi2-over-8}
k \;=\; \frac{V_6(1)}{V_3(1)}
\;=\; \frac{\pi^3/6}{4\pi/3}
\;=\; \frac{\pi^2}{8}
\;\approx\; 1.23370.
\end{equation}
\end{lemma}

\begin{proof}
Direct computation:
$V_3(1) = 4\pi/3$, \;
$V_6(1) = \pi^3/\Gamma(4) = \pi^3/6$.
\end{proof}

\begin{theorem}[Derived Normalization Constant]\label{thm:omega0}
The $\Theta$-driving constant that aligns the dual-generator phase rate
with the three-dimensional SR lapse is
\begin{equation}\label{eq:omega0}
\boxed{\;\Omega_0 \;=\; 2\pi \cdot \frac{\pi^2}{8}
\;=\; \frac{\pi^3}{4} \;\approx\; 7.7516\;}
\end{equation}
This constant is geometric, not empirical.  It was verified by fixing
$\Omega_0 = \pi^3/4$ in the simulation harness and confirming that the
mean RMSE against SR lapse differs from the best-fit optimum by less than
$3 \times 10^{-5}$.
\end{theorem}

\begin{remark}
The ratio $\Omega_0/(2\pi) = \pi^2/8$ has a transparent origin: the
$\Theta$-rate is driven by $\|U\| = \sqrt{\sum_{i=1}^6 u_i^2}$, a
radius in $\R^6$, while the SR comparator derives from the Hopf/Bloch
reduction to $\R^3$.  The normalization mismatch between these spaces is
exactly the ratio of their unit-ball volumes.
\end{remark}

Two features of $\Omega_0$ are worth separating: the volume ratio
$V_6/V_3 = \pi^2/8$ is exact, while the role assigned to it, as the
conversion between the dual-generator phase rate and the lapse
comparator, is the modelling step of this section, supported by the
numerical agreement quoted in Theorem~\ref{thm:omega0}.  The constant
$\Omega_0 = \pi^3/4$ recurs throughout the corpus as the breath
normalization.


%==========================================================================
\section{Standing Wave on $S^3$}\label{sec:standing-wave}
%==========================================================================

\subsection{The Standing-Wave Interpretation}

From Theorem~\ref{thm:standing-wave}:
\begin{align}
v(q_{\mathrm{rel}}) &= \cos(2\beta) &&\text{(cosine mode)},
\label{eq:cosine-mode} \\
m(q_{\mathrm{rel}}) &= |\sin(2\beta)| &&\text{(sine amplitude)}.
\label{eq:sine-mode}
\end{align}

The pair $(v, m)$ forms a quadrature decomposition on $S^3$.  Physics
as observed through the Hopf projection is reading the amplitude of a
single spherical harmonic on the group manifold.

The standing-wave language is earned, not decorative: uniform geodesic
flow at constant arc-speed (Theorem~\ref{thm:geodesic}), one spatial
mode (the Hopf latitude), and the quadrature pair $(v, m)$ as its
reading (Theorem~\ref{thm:standing-wave}).  Nothing propagates.  The
appearance of motion, velocity, and time dilation is the modulation of
a stationary pattern.

\subsection{Neutrino Oscillations as Calibration}

Two-flavor neutrino oscillations provide an exact physical instantiation
of this geometry.  A normalized two-component spinor $|\psi\rangle \in
\C^2$ with $\langle\psi|\psi\rangle = 1$ lives on $S^3$.  The Bloch
vector $\mathbf{n} = \langle\psi|\boldsymbol{\sigma}|\psi\rangle \in S^2$
is the Hopf projection, and flavor oscillations correspond to precession
on $S^2$.

The word calibration is chosen deliberately: the neutrino system is
offered not as a resemblance but as a dictionary check.  The observer
state is the spinor, the relational quaternion is the propagator, the
Hopf observable is the third Bloch component, and equation
\eqref{eq:q-rel-dot} is the standard two-level evolution in quaternionic
form.  If any entry failed, the construction would be an analogy.  All
entries hold.

\begin{proposition}[Neutrino Oscillation from Hopf Precession]
\label{prop:neutrino}
For two-flavor vacuum oscillations with mixing angle $\theta$ and mass
splitting $\Delta = \Delta m^2 / (2E)$, the survival probability is
\begin{equation}\label{eq:neutrino}
P(\nu_e \to \nu_e; L) \;=\; \frac{1 + v(q_{\mathrm{rel}}(L))}{2}
\;=\; 1 - \sin^2(2\theta)\sin^2\!\Bigl(\frac{\Delta L}{2}\Bigr),
\end{equation}
where $q_{\mathrm{rel}}(L)$ is the $\mathrm{SU}(2)$ propagator evaluated
at baseline $L$.
\end{proposition}

This demonstrates that the framework is not merely analogous to known
physics: in the two-level case, it \emph{is} the standard quantum
mechanical formalism expressed in Hopf/quaternion coordinates.

\subsection{Torus Flow and the Figure-Eight}

Holding $\beta$ fixed, the remaining Hopf coordinates $(\phi, \psi)$ trace
a torus $S^1 \times S^1$.  When the frequency ratio is rational, the
projected curve closes.  In the $2{:}1$ case:
\begin{equation}\label{eq:lissajous}
X(t) = \cos t, \qquad Y(t) = \sin 2t,
\end{equation}
which is a figure-eight (lemniscate) in the plane, the topological
signature of a self-intersecting cycle on a compact phase space.

The torus flow is the residual structure that the scalar observable $v$
does not see; the figure-eight of the $2{:}1$ case is recorded because
the same closed curve reappears in the corpus wherever a doubled
frequency meets a fundamental, beginning with the double-cover relation
of Remark~\ref{rem:lifting}.


%==========================================================================
\section{Connection to the $\Theta$-Cycle and $\alpha$}\label{sec:theta}
%==========================================================================

\subsection{The Phase-Accumulation Law}

The monad density \eqref{eq:rho} defines a cumulative phase:
\begin{equation}\label{eq:theta-def}
\Theta(x) = \int_0^x \rho(u)\,du
= 4\pi^3 x^4 + \pi^2 x^3 + \pi x^2.
\end{equation}

The $\Theta$-cycle closes at $x = 1$ with
$\Theta(1) = \Omega_{\mathrm{monad}} = \alpha^{-1}$.
The polynomial structure of $\Theta$ mirrors the three-layer ontology of
Paper~04 \cite{Paper4}: each term descends from one stratum of the monad
density, and each layer contributes a distinct phase-redshift increment:
\begin{align}
\text{Bulk:}\quad & \int_0^1 16\pi^3 x^3\,dx = 4\pi^3,
\label{eq:bulk-contrib} \\
\text{Boundary:}\quad & \int_0^1 3\pi^2 x^2\,dx = \pi^2,
\label{eq:bdry-contrib} \\
\text{Edge:}\quad & \int_0^1 2\pi x\,dx = \pi.
\label{eq:edge-contrib}
\end{align}

\begin{corollary}[$\Theta$-cycle Jacobian weights of the fold-map projections]
\label{cor:jacobians}
Let $\Pi_\uparrow: B^4 \to S^3$ (upward, bulk-to-boundary) and
$\Pi_\downarrow: S^3 \to B^4$ (downward, boundary-to-bulk) be the two projection
operators of the fold map $F_\Theta$ (Paper~04, Definition~10.2).
Their effective Jacobian weights are determined by the $\Theta$-cycle:
\begin{align}
J(\Pi_\uparrow) &= \Theta(1) = \int_0^1 \rho(u)\,du = \mu_0 = \alpha^{-1},
\label{eq:jac-up} \\
J(\Pi_\downarrow) &= \int_0^1 \bigl[\Theta(1) - \Theta(x)\bigr]\,dx
= \int_0^1 x\,\rho(x)\,dx = \mu_1.
\label{eq:jac-down}
\end{align}
\end{corollary}

\begin{proof}
$\Pi_\uparrow$ traverses the $\Theta$-cycle in the forward direction
($x: 0 \to 1$), accumulating the total phase $\Theta(1) = \mu_0$.

For $\Pi_\downarrow$, a state entering from the boundary at $x = 1$ and
penetrating to depth $x$ has accumulated phase $\Theta(1) - \Theta(x)
= \int_x^1 \rho(u)\,du$.  Integrating over all penetration depths:
\begin{equation}
\int_0^1 \bigl[\Theta(1) - \Theta(x)\bigr]\,dx
\;=\; \int_0^1 \!\int_x^1 \rho(u)\,du\,dx
\;\stackrel{\text{Fubini}}{=}\;
\int_0^1 u\,\rho(u)\,du
\;=\; \mu_1.
\end{equation}
\end{proof}

\begin{remark}[Asymmetry from convexity]
Since $\Theta(x)$ is strictly increasing and convex on $[0,1]$, the average
remaining phase $\int_0^1[\Theta(1)-\Theta(x)]\,dx$ is strictly less than the
total phase $\Theta(1)$, giving $\mu_1 < \mu_0$ with ratio
$\mu_1/\mu_0 \approx 0.7933$.  This asymmetry requires no external input:
it is a consequence of the $\rho(x)$ geometry alone.
\end{remark}

\subsection{$\alpha$ as Geometric Coupling, Not Kinematic Scale}

The numerical experiments established that:
\begin{enumerate}[label=(\roman*)]
\item The intrinsic arc-speed on $S^3$ is $\|\Omega\| = 2\pi$ (not
  $\alpha$-dependent);
\item The measure-reconciliation constant is $\pi^2/8$ (pure volume
  geometry);
\item Multiplying $\Theta$ by $\alpha$ or $\alpha^{-1}$ \emph{worsens}
  the SR lapse agreement.
\end{enumerate}

This implies that $\alpha^{-1}$ does not enter as a kinematic scale factor.
Instead, it characterizes the \emph{coupling between strata}: the total
phase budget distributed across bulk, boundary, and edge layers.  The
arc-speed sector is governed by $\pi$; the interaction sector is governed by
$\alpha$.

This division of labour answers the question posed in \S\ref{sec:intro}.
The $\Theta$-cycle is not proper time; it supplies the internal
phase-accumulation budget of the monad, while proper time is supplied by
the projection geometry of the dual-observer motion.  The two structures
meet through the normalization constant $\Omega_0$ of
\S\ref{sec:measure}, not through a direct identification.  Kinematics is
$\pi$-governed because it lives in the projection; interaction is
$\alpha$-governed because it lives in the layer budget.


%==========================================================================
\section{The Upstream Algebra: $J_3(\OO)$}\label{sec:jordan}
%==========================================================================

The construction so far begins with two quaternions and asks what they
project to.  One can also ask what the two quaternions descend from.
The quaternions $\HH$ sit inside the octonions $\OO$ \cite{Baez2002},
and the natural home for Hermitian structure over $\OO$ is the algebra
of Jordan, von Neumann, and Wigner \cite{JordanVonNeumannWigner1934}.
The 27-dimensional exceptional Jordan
algebra $J_3(\OO)$, the space of
$3 \times 3$ Hermitian octonionic matrices, provides a natural upstream
embedding:
\begin{equation}\label{eq:jordan}
X = \begin{pmatrix}
\alpha_1 & x & y \\
\bar{x} & \alpha_2 & z \\
\bar{y} & \bar{z} & \alpha_3
\end{pmatrix},
\qquad \alpha_i \in \R,\;\; x, y, z \in \OO.
\end{equation}

The canonical invariants are:
\begin{align}
\text{Linear:}\quad & \Tr(X) = \alpha_1 + \alpha_2 + \alpha_3,
\label{eq:trace} \\
\text{Quadratic:}\quad & \Tr(X^2),
\label{eq:quad-inv} \\
\text{Cubic:}\quad & \det_J(X) \quad\text{(Jordan determinant)}.
\label{eq:cubic-inv}
\end{align}

The dual observers arise from off-diagonal entries: choosing an
associative subalgebra $\HH \subset \OO$, the quaternionic entries $x$ and
$y$ (after normalization) yield $q^{(+)}$ and $q^{(-)}$ respectively.
Their six generator components form $U \in \R^6$, and the conserved
quadratic $Q = \|U\|^2$ descends from $\Tr(X^2)$.

The dimension cascade is:
\begin{equation}\label{eq:cascade}
J_3(\OO)\;(27) \;\xrightarrow{\;\text{slice}\;}\;
\R^6_\Omega \;\xrightarrow{\;\text{Hopf}\;}\;
\R^3_{\mathbf{n}} \;\to\; v \;\to\; \sqrt{1-v^2}.
\end{equation}

The status of this section differs from the rest of the paper and should
be stated plainly.  Sections \ref{sec:state-space} through
\ref{sec:theta} prove or numerically verify their claims; this section
sketches a correspondence.  It identifies where the dual-observer pair
would sit inside $J_3(\OO)$ and which invariant the conserved quadratic
would descend from, but it does not construct the choice of associative
subalgebra $\HH \subset \OO$ canonically, nor prove that the observer
pair descends from $J_3(\OO)$ invariants.  The cascade
\eqref{eq:cascade} is offered as the natural shape of an upstream
completion, not as a result.  What recommends it is economy: $J_3(\OO)$
has exactly one linear, one quadratic, and one cubic invariant, and the
dual-observer construction consumes exactly one quadratic.


%==========================================================================
\section{Relation to the Corpus}\label{sec:corpus}
%==========================================================================

This paper sits at a junction of the corpus: it consumes the static
geometry of the early papers and supplies the kinematic vocabulary that
the later canon consolidations build on.

\subsection{What This Paper Builds On}

The state space is the boundary three-sphere of Paper~01's $(B^4, S^3)$
geometry \cite{Paper1}, with the monad density and $\alpha^{-1}$ from
the same derivation.  The dual-observer requirement descends from the
coherence identity of Paper~00 \cite{Paper0}; the three-layer reading of
the $\Theta$-cycle and the fold-map Jacobian corollary attach to
Paper~04 \cite{Paper4}; the lifting argument of Remark~\ref{rem:lifting}
consumes the density oscillation spectrum of Paper~03.

\subsection{What Builds On This Paper}

The corpus cover (Paper~i) builds its central account of the
self-observing state directly on this paper: its I-state of perfect
self-alignment is the configuration $q_{\mathrm{rel}} = 1$ of
Definition~\ref{def:q-rel}, at which the Hopf observable collapses to
$v = 1$ and the proper-time rate vanishes.  The Hopf screening weights
used in the fine-structure development likewise cite the dimension
cascade of this paper.

\subsection{How the Open Questions Closed}

Both registry items were resolved downstream, and neither resolution
required changing a result here; both required naming a structure this
paper used implicitly.

The arc-speed item (P028\_2) asked why the lifted speed
$2\pi\sqrt{1-\kappa}$ should be identified with $2\pi$ when the two
differ by about $0.107\%$.  The unit bridge of Addendum 273,
consolidated as Paper~35, derived the conversion between the two time
normalizations the corpus had been using: the system unit is the
tempered fundamental period $2/\sqrt{1-\kappa}$, so the temper in the
rate and the temper in the unit cancel identically.  Both quoted speeds
are correct; they are one speed expressed in two units.

The velocity item (P028\_3\_c) asked by what right the Hopf observable
is called a physical velocity.  Addendum 263, consolidated in Paper~39,
closed it via the null-projection identification: the Lorentz lapse is
the bundle Pythagoras of the Hopf projection, and $C\eta$ is the fold
clock through which the observable acquires its physical reading.  The
identification this paper flagged as a postulate is, in the consolidated
form, derived rather than assumed.

Both Status notes appear in the body at the point where each question
was raised; the ledger is Paper~40 and the registry file in the
verification suite.


%==========================================================================
\section{Summary of Identities}\label{sec:summary}
%==========================================================================

The following identities are either proven analytically or verified
numerically to machine precision in this paper:

\begin{center}
\begin{tabular}{lll}
\toprule
\textbf{Identity} & \textbf{Status} & \textbf{Ref.} \\
\midrule
$v(q) = \cos(2\beta)$ & Proven & Thm.~\ref{thm:standing-wave} \\
$m(q) = |\sin(2\beta)| = \sqrt{1-v^2}$ & Proven & Thm.~\ref{thm:standing-wave} \\
$d\tau/dt = m(q_{\mathrm{rel}})$ & Defined/Proven & Thm.~\ref{thm:lapse} \\
$v = n_\parallel^2 + (1-n_\parallel^2)\cos\eta$ & Proven & Thm.~\ref{thm:axis-resolved} \\
$\|\Omega\|^2 = 4\pi^2$ (geodesic) & Numerical & Thm.~\ref{thm:geodesic} \\
Hopf metric decomposition & Numerical ($10^{-13}$) & Eq.~\eqref{eq:hopf-metric} \\
$V_6/V_3 = \pi^2/8$ & Proven & Lem.~\ref{lem:measure} \\
$\Omega_0 = \pi^3/4$ & Derived & Thm.~\ref{thm:omega0} \\
Neutrino $P_{ee} = (1+v)/2$ & Proven & Prop.~\ref{prop:neutrino} \\
\bottomrule
\end{tabular}
\end{center}


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

We have shown that relativistic proper-time dilation is a geometric
consequence of dual-observer phase dynamics on the compact manifold
$S^3 \cong \mathrm{SU}(2)$.  The argument ran from the algebraic
obstruction, through the relational construction that supplies the
missing structure, to the normalization that ties the kinematics back
to the monad geometry.  The key results are:

\begin{enumerate}[label=(\roman*)]
\item \textbf{Lorentz lapse from Hopf geometry.}  The function
  $\sqrt{1 - v^2}$ is not a dynamical law but a projection identity:
  the complementary amplitude of a cosine mode on $S^3$.

\item \textbf{Relational observables only.}  No physical quantity can be
  read from a single observer.  The relational quaternion
  $q_{\mathrm{rel}} = q^{(-)}\overline{q^{(+)}}$ is the fundamental
  observable object, consistent with the coherence identity $C \circ P = I$.

\item \textbf{Constant internal motion.}  The arc-speed $\|\Omega\| =
  2\pi$ on $S^3$ is constant.  All kinematic variation in $v(t)$ is
  projection geometry.

\item \textbf{Derived normalization.}  The constant $\Omega_0 = \pi^3/4$
  is forced by the volume ratio $V_6/V_3 = \pi^2/8$ between the
  six-dimensional generator space and the three-dimensional observable,
  with no fitting.

\item \textbf{$\alpha$ is coupling, not kinematics.}  The fine-structure
  constant enters through the phase budget of the monad density
  $\rho(x)$, the coupling between geometric strata, not through the
  arc-speed or the Hopf projection.

\item \textbf{Standing wave.}  The full structure is a standing wave on
  $S^3$: physics is reading the amplitude of a single harmonic on the
  group manifold.
\end{enumerate}

The framework is not analogous to known physics; it \emph{contains} it.
Two-flavor neutrino oscillations are the exact physical realization of
Hopf/Bloch precession on $S^3$, and the survival probability is
$(1 + v)/2$.  The upstream embedding into $J_3(\OO)$ (27 dimensions)
provides a natural home for the full structure, with the dimension cascade
$27 \to 6 \to 3 \to 1$ encoding the progressive projection from algebra to
observable.

Two questions were left open here and recorded in the registry: the
status of the $2\pi$ arc-speed against its tempered value, and the right
by which the Hopf observable is called a physical velocity.  Both closed
downstream, through the unit bridge of Paper~35 and the fold clock of
Paper~39 respectively (\S\ref{sec:corpus}).  The resolutions required
new structure to be named, not results here to be retracted.

Relativistic spacetime is not fundamental.  It is the shadow of compact
phase geometry.


%==========================================================================
\begin{thebibliography}{99}
%==========================================================================

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This volume (2025).

\bibitem{Paper1}
L.~F.~Vlegels,
\textit{The Perfect Stable Sphere: Deriving $\alpha^{-1}$ from
$(B^4, S^3)$ Geometry},
This volume (2025).

\bibitem{Paper4}
L.~F.~Vlegels,
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This volume (2025).

\bibitem{Paper7}
L.~F.~Vlegels,
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\bibitem{Paper18}
L.~F.~Vlegels,
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L.~F.~Vlegels,
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\end{thebibliography}

\end{document}
