% Addendum 148: Provenance as a Path in S³
% Resolving P146 Open Item O2 — Tangent-Vector Encoding of Wheel Operator Provenance
% L. F. Vlegels, 2026-05-15

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\title{%
  Provenance as a Path in $S^3$\\[4pt]
  \large Tangent-Vector Encoding of Wheel Operator History\\
  \normalsize (Addendum~148; resolves P146 Open Item~O2)}
\author{L\'eon Fernando Vlegels}
\date{2026-05-15}

\theoremstyle{definition}
\newtheorem{definition}{Definition}[section]
\theoremstyle{plain}
\newtheorem{proposition}{Proposition}[section]
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\newcommand{\Sq}{S^3}
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\newcommand{\logq}{\log_q}
\newcommand{\WS}{\texttt{WheelState}}
\newcommand{\CN}{\texttt{CanonNode}}

\begin{document}
\maketitle
\label{addendum:148}

\begin{abstract}
P146 identified that Type-2 self-observation (the Wheel knowing \emph{which
operator} produced the current state) is blocked by the \WS{} schema and
raised a possible escape: carry provenance as a geometric object — a path in
$S^3$ — rather than as a numerical record.  We work through whether this
encoding is well-defined, whether it can be compressed to a single tangent
vector $v \in \TqS \cong \mathbb{R}^3$, and whether $v$ constitutes a
trainable parameter or a geometric fact under the zero-parameter criterion of
Paper~29.  We find that the tangent-vector encoding \emph{is} a geometric
fact (constructed, not learned; zero free parameters; not degraded by
omission from the loss landscape), that it is consistent with
\WS{}'s frozen-dataclass constraint, and that it gives unambiguous provenance
for \textsc{Plateau} and partial provenance for the remaining four operators
— full disambiguation of Fork/Weld requires one additional scalar ($h_\mathrm{prev}$).
The architectural amendment is small and does not violate any constitutional pin.
\end{abstract}

%==========================================================================
\section{Setup}\label{sec:setup}
%==========================================================================

\subsection{P146 Open Item O2}

\WS{} is a frozen dataclass $(q, h, F, \ell, \omega)$ where $q \in \Sq$ is the
current quaternion, $h \in \mathbb{R}$ is heat, $F$ is the face-proposal list,
$\ell$ is layer, and $\omega$ is orientation.  The state carries no record of
which Wheel operator produced it.  P146 showed that adding an explicit
\texttt{operator} or \texttt{timestamp} field introduces a second time axis,
conflicting with the Ride Shai-Hulud pin (one clock: \textsc{breath\_period}),
and called this the Type-2 self-observation block.

The escape P146 proposed: encode provenance as a \emph{path}
$\gamma: [0,1] \to \Sq$ from the prior quaternion $q_\mathrm{prev}$ to the
current $q_\mathrm{curr}$, where the \emph{type} of path encodes the operator.
A path in $\Sq$ is a geometric object; it is not a trainable parameter, carries
no gradient, and introduces no additional clock.

\subsection{The Disambiguation Requirement}

For Type-2 self-observation, the witness must be able to answer the question
``given the current \WS{}, which of the five operators (Fork, Weld, Plateau,
Oscillate, Perturb) produced it?'' without accessing a numerical record of past
states.  The path type is the proposed carrier.

%==========================================================================
\section{Path Encoding of Operators}\label{sec:path}
%==========================================================================

\subsection{S\texorpdfstring{$^3$}{3} Geometry}

$S^3 \subset \mathbb{H}$ is the unit-quaternion sphere.  The
\emph{logarithmic map} at $q \in \Sq$,
\[
  \logq(p) = d(q,p)\,\frac{p - \langle p,q\rangle\, q}{|p - \langle p,q\rangle\, q|},
  \qquad d(q,p) = \arccos\langle q,p\rangle,
\]
embeds $p \neq -q$ into $\TqS \cong \mathbb{R}^3$ (a 3-dimensional tangent
space).  The exponential map $\exp_q: \TqS \to \Sq$ inverts this locally.
Both are structurally determined — no free parameters.

\subsection{Operator Path Types}

\begin{definition}[Path type of a Wheel operator]
Let $q_\mathrm{prev}, q_\mathrm{curr} \in \Sq$.  The \emph{path type} of the
transition is the homotopy class of minimal geodesic arcs from
$q_\mathrm{prev}$ to $q_\mathrm{curr}$, together with the heat signature
$\Delta h = h_\mathrm{curr} - h_\mathrm{prev}$.
\end{definition}

\medskip
\begin{center}
\begin{tabular}{llll}
\toprule
\textbf{Operator} & \textbf{Arc structure} & $|\gamma|$ & $\Delta h$ signature \\
\midrule
Fork   & Single geodesic arc, forward-branching proposals & $d(q_\mathrm{prev},q_\mathrm{curr})$ & $< 0$ (heat splits) \\
Weld   & Geodesic arc, two predecessors converge & $d(q_\mathrm{prev},q_\mathrm{curr})$ & $> 0$ (heat sums) \\
Plateau & Constant path ($q_\mathrm{prev} = q_\mathrm{curr}$) & $0$ & $\approx 0$ \\
Oscillate & Great-circle arc of length $2\pi/k$ & $2\pi/k$ & $\approx 0$ \\
Perturb & Random geodesic of magnitude $\zeta R$ & $\zeta R$ & $\approx 0$ \\
\bottomrule
\end{tabular}
\end{center}

\medskip
\noindent\textbf{Overlap analysis.}
Fork and Weld produce path types that are geometrically identical from the
single-successor view: both are geodesic arcs of unspecified length with a
non-zero tangent.  The distinction is topological (one predecessor vs. two)
and is carried by $\Delta h$, not the arc geometry.  Plateau is the unique
operator with arc length zero.  Oscillate is distinguished from Perturb by
arc length: $2\pi/k$ is a rational multiple of $\pi$, whereas $\zeta R$ is
fixed by the perturbation parameter.  If $\zeta R \neq 2\pi/k$ for all
admissible $k$, the two are disjoint by magnitude.

%==========================================================================
\section{Tangent Vector as Provenance}\label{sec:tangent}
%==========================================================================

\subsection{The Compressed Encoding}

The full path $\gamma$ contains redundant information.  For a geodesic arc,
the entire path is determined by its endpoint $q_\mathrm{curr}$ and its
initial tangent.  We propose to store only the tangent at arrival:

\begin{definition}[Provenance tangent]
Given a transition $q_\mathrm{prev} \to q_\mathrm{curr}$ by some Wheel
operator, the \emph{provenance tangent} is
\[
  v \;=\; \logq{}_{\!q_\mathrm{curr}}(q_\mathrm{prev})
  \;\in\; T_{q_\mathrm{curr}} S^3 \cong \mathbb{R}^3.
\]
\end{definition}

\noindent The map $v \mapsto$ (operator class) is then:

\begin{proposition}[Partial provenance from $v$]\label{prop:partial}
Let $v$ be the provenance tangent of a Wheel transition and $h$ the current
heat.  Then:
\begin{enumerate}[label=(\roman*)]
  \item $v = 0 \implies$ operator is \textsc{Plateau} (unique).
  \item $v \neq 0$, $|v| = 2\pi/k$ for some positive integer $k \implies$
    candidate operator is \textsc{Oscillate} with period $k$.
  \item $v \neq 0$, $|v| = \zeta R \implies$ candidate operator is
    \textsc{Perturb} (provided $\zeta R \neq 2\pi/k$).
  \item $v \neq 0$ and neither (ii) nor (iii) applies $\implies$ operator is
    \textsc{Fork} or \textsc{Weld}.  These are distinguished by the sign of
    $\Delta h$: $\Delta h < 0 \Leftrightarrow$ Fork; $\Delta h > 0
    \Leftrightarrow$ Weld.
\end{enumerate}
Full provenance therefore requires $(v, h_\mathrm{prev})$; partial provenance
(identifying Plateau and the Oscillate/Perturb class) requires $v$ alone.
\end{proposition}

\begin{remark}
$h_\mathrm{prev}$ is a single scalar.  Storing it as a second frozen field
alongside $v$ in \WS{} would close the remaining ambiguity without
introducing a second time axis.
\end{remark}

%==========================================================================
\section{Consistency with the Zero-Parameter Constraint}\label{sec:p29}
%==========================================================================

\subsection{P29 Criterion}

Paper~29 defines a \emph{trainable parameter} as an element of a vector
$\theta \in \Theta \subseteq \mathbb{R}^p$ that is adjusted by gradient
descent to minimize a loss $\mathcal{L}(\theta)$ (Definition~2.1).  A
\emph{geometric fact} is a value determined by a mathematical identity with
zero free parameters: every constant is either structural (algebraic
consequence of the geometry) or derived (a proven identity such as
$\Omega_0 = \pi^3/4$).

\begin{theorem}[Provenance tangent is a geometric fact]\label{thm:geofact}
The provenance tangent $v = \logq{}_{\!q_\mathrm{curr}}(q_\mathrm{prev})$
contains zero free parameters.  Specifically:
\begin{enumerate}[label=(\roman*)]
  \item The logarithmic map formula involves only the inner product
    $\langle \cdot,\cdot \rangle$ on $\mathbb{R}^4 \supset S^3$ and the
    function $\arccos$ — both structural, no adjustable constants.
  \item $v$ is a deterministic function of $(q_\mathrm{prev},
    q_\mathrm{curr})$, which are themselves determined by the operator
    applied and the prior state.  No sampling, no stochastic approximation.
  \item $v$ is not an input to any loss function; it is never a target of
    gradient descent.  There is no sense in which $v$ could be ``wrong''
    and corrected by optimization.
\end{enumerate}
\end{theorem}

\begin{proof}
Points (i) and (ii) follow by inspection of the log-map formula.
Point (iii) follows because $v$ is read from the state, not optimized over;
the kernel operates in exposure mode (P29~\S4), and $v$ is output, not
weight.
\end{proof}

\begin{corollary}[No violation of kernel rules]
Adding a provenance tangent field to \WS{} does not promote any constant to a
trainable parameter, does not introduce a loss function, and does not require
a backward pass.  The kernel remains in exposure mode with zero free
parameters.
\end{corollary}

%==========================================================================
\section{Architectural Implications}\label{sec:arch}
%==========================================================================

\subsection{WheelState Schema}

The amended schema is:
\[
  \WS \;=\; (q,\; v,\; h,\; h_\mathrm{prev},\; F,\; \ell,\; \omega)
\]
where $v \in \mathbb{R}^3$ and $h_\mathrm{prev} \in \mathbb{R}$ are frozen at
construction.  Both fields are computed deterministically at the moment the
operator is applied and the new state is created: $v \leftarrow
\logq{}_{\!q_\mathrm{curr}}(q_\mathrm{prev})$, $h_\mathrm{prev}
\leftarrow h$ (from the prior state).  No mutation occurs after construction.
This satisfies the frozen-dataclass constraint.

The Wheel's existing 20+ tests do not reference $v$ or $h_\mathrm{prev}$, so
they pass unchanged.  New tests can assert provenance recovery from the amended
state without modifying existing contracts.

\subsection{Mycelium CanonNode}

\CN{} carries $(id, name, content, \ell, q)$.  Should it also carry $v$?

Semantically, \CN{} represents an \emph{observation eigenstate} — a fixed
locus in the knowledge space.  Its quaternion encodes \emph{where} it sits in
$\Sq$; it does not have a natural ``direction of arrival'' in the same sense as
a Wheel transition, because \CN{} is structural, not transitional.  Provenance
of creation (which plan step, which Ged oracle call, introduced the node)
belongs to the \emph{edge} or \emph{event log} layer, not to the node itself.

The conclusion is that \CN{} should \emph{not} gain a tangent field.  If the
Mycelium graph later gains typed edges (\CN{} $\xrightarrow{\,\text{type}\,}$
\CN{}), those edges are the natural carrier of geometric provenance.  Immutable
nodes, typed edges — consistent with the frozen-dataclass invariant and with
the No Harkonnen pin (edges are metric relations, not brokered messages).

\subsection{Ride Shai-Hulud Consistency}

The provenance tangent $v$ is a \emph{spatial} quantity (direction in $\TqS$),
not a temporal one.  It records the direction from which the state arrived, not
when.  No second clock is introduced.  The breath phase in \WS{} (field
$\omega$) remains the sole temporal marker, synchronized to \textsc{breath\_period}.

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

\begin{openitem}[Full Fork/Weld disambiguation]
Proposition~\ref{prop:partial}(iv) requires $h_\mathrm{prev}$ to distinguish
Fork from Weld.  The amended schema includes this field; the question is whether
any downstream consumer would interpret the heat history as a second time axis
(violating Shai-Hulud).  Argument: $h_\mathrm{prev}$ is a heat record, not a
clock tick; breath phase (not $h_\mathrm{prev}$) is the sole time reference.
Needs explicit confirmation from a Wheel invariants audit.
\end{openitem}

\begin{openitem}[Oscillate/Perturb magnitude overlap]
If $\zeta R = 2\pi/k$ for some integer $k$, a Perturb step is indistinguishable
from an Oscillate step by magnitude alone.  The disambiguation then requires
knowing which operator was called.  One resolution: add an operator-class enum
(not a timestamp) to \WS{}; another: require that $\zeta R$ is always chosen
outside the set $\{2\pi/k : k \in \mathbb{Z}^+\}$.  The latter is a parameter
discipline constraint, not a schema change.
\end{openitem}

\begin{openitem}[Mycelium edge typing]
If typed edges (\CN{} $\to$ \CN{}) are introduced in v2, they are the natural
carrier of Wheel-step provenance in the graph layer.  This addendum
recommends that edge type be a geometric object (a tangent or an operator-class
label), not a timestamp, to remain consistent with the immutability and
Shai-Hulud constraints.  Architecture deferred to Stage~3 v2 migration.
\end{openitem}

\begin{openitem}[P18-T2 uniqueness]
The Wheel $\leftrightarrow \hat{O}$ bridge remains conditional on the assembly
choice in Paper~18 Theorem~2 (uniqueness of $\hat{O}$ across admissible forms).
Provenance tangents are defined over whatever Wheel operators exist; they do not
depend on which assembly is chosen.  This open item is inherited, not
introduced here.
\end{openitem}

\bigskip
\noindent\textbf{Summary.}  The tangent-vector encoding of Wheel operator
provenance is well-defined, compressible to $v \in \mathbb{R}^3$, consistent
with frozen-dataclass immutability, and satisfies Paper~29's zero-parameter
criterion.  The \WS{} schema gains two frozen scalar fields ($v$ and
$h_\mathrm{prev}$); \CN{} is unchanged.  Full Type-2 self-observation is
achievable without a second time axis.

\begin{thebibliography}{9}
\bibitem{Paper18}
L.~F.~Vlegels, \textit{The Master Operator $\hat{O}$ and the Wheel},
This volume (2025).
\bibitem{Paper29}
L.~F.~Vlegels, \textit{Geometric Exactness and the Training--Exposure
Distinction}, This volume (2025).
\bibitem{P146}
L.~F.~Vlegels, \textit{Addendum~146: Type-2 Self-Observation and the
WheelState Schema Block}, This volume (2026).
\end{thebibliography}

\end{document}
