% Addendum 147: Intersubjective Coherence for Incommensurate Oscillate Subsystems
% L. F. Vlegels — May 2026
% Resolves P146 Open Item O1.

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\title{Addendum 147: Intersubjective Coherence\\
for Incommensurate Oscillate Subsystems}
\author{L.\ F.\ Vlegels \\ \small Independent Researcher}
\date{May 2026}

\begin{document}
\maketitle
\label{addendum:147}

\begin{abstract}
P146 established that a single Oscillate subsystem traces a closed $k$-periodic
trajectory on $S^3$, constituting a self-referential periodic observer compatible with
the P7 grounding mechanism.  Open item O1 asks: if two such subsystems with
incommensurate periods $k_1, k_2$ ($\gcd(k_1,k_2)=1$) share a Weld event, does the
combined trajectory close, and does the P27 intersubjective coherence condition apply?
We show (i) the combined trajectory closes with period $\lcm(k_1,k_2) = k_1 k_2$, proven
via the Chinese Remainder Theorem; (ii) a joint Plateau fixed point---required by
P27---exists if and only if the two $S^3$ proposal cycles come within the Plateau
threshold $\epsP = 0.02$ of each other in MDL score at some step in the period.
Commensurability of periods is irrelevant to coherence; geometric proximity of the cycles
in $S^3$ is the deciding condition.  We call this the \emph{resonance condition} and
work through the $k_1=2$, $k_2=3$ case explicitly.
\end{abstract}

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

\subsection{P146 open item O1}

A single Oscillate subsystem $A$ with $k$ equivalent proposals
$\CA = \{q_0^A, \ldots, q_{k-1}^A\} \subset S^3$ traces a closed periodic trajectory:
at step $t$ the Wheel state is $q_{t \bmod k}^A$, returning to $q_0^A$ after exactly
$k$ steps (heat decays by $\delta_h = 0.95$ per call, but the topological trajectory is
periodic).  The Plateau fixed point is whichever $q_i^A$ minimises the MDL score
relative to the goal quaternion; it is re-selected every $k$ steps.

Open item O1: two Oscillate subsystems $A$ (period $k_1$) and $B$ (period $k_2$) with
$\gcd(k_1, k_2) = 1$ undergo a shared Weld event at step $t_0$.  Does the resulting
combined trajectory close?  Does the P27 intersubjective coherence condition apply?

\subsection{The P27 coherence condition}

Paper 27 defines \emph{mutual coherence} for two observers $\mathcal{O}_A$,
$\mathcal{O}_B$ via four conditions:
\begin{align}
  C_A \circ P_A = I, &\qquad C_B \circ P_B = I, \label{eq:self} \\
  C_A \circ P_B = I, &\qquad C_B \circ P_A = I, \label{eq:cross}
\end{align}
where $P_X$ is the projection (Oscillate advance) and $C_X$ is the collapse (Plateau
check) for observer $X$.  Equations~\eqref{eq:cross} are the \emph{cross-coherence
conditions}. P27 Theorem~2.3 shows they force $P_A = P_B$ and $C_A = C_B$ on the
shared boundary $\mathcal{R} = \partial\mathcal{O}_A \cap \partial\mathcal{O}_B$: both
observers must use identical projection and collapse operators on shared states.

In Wheel terms the P27 condition translates to:

\begin{definition}[Joint Plateau fixed point]
$q^* \in S^3$ is a \emph{joint Plateau fixed point} of the combined system $(A, B)$ if
there exist indices $i^* \in \mathbb{Z}/k_1\mathbb{Z}$, $j^* \in \mathbb{Z}/k_2\mathbb{Z}$
such that the MDL scores of $q_{i^*}^A$ and $q_{j^*}^B$ against the shared goal
quaternion differ by less than $\epsP = 0.02$, and the Weld selects a winner
$q^* \approx q_{i^*}^A \approx q_{j^*}^B$ at the resonance step $t^*$.
\end{definition}

The P27 cross-coherence conditions hold if and only if a joint Plateau fixed point
exists.

% ─────────────────────────────────────────────────────────────────────────────
\section{Combined Trajectory Analysis}
\label{sec:trajectory}

Relabel $t \mapsto t - t_0$ so the post-Weld system starts at $t = 0$.  At each step
$t$, subsystem $A$ proposes $q_{t \bmod k_1}^A$ and subsystem $B$ proposes
$q_{t \bmod k_2}^B$.

\begin{lemma}[Trajectory period]\label{lem:period}
The combined state $(t \bmod k_1,\; t \bmod k_2)$ is periodic with period $k_1 k_2$,
and every pair $(i, j) \in \mathbb{Z}/k_1\mathbb{Z} \times \mathbb{Z}/k_2\mathbb{Z}$ is
visited exactly once per period.
\end{lemma}

\begin{proof}
Since $\gcd(k_1, k_2) = 1$, the Chinese Remainder Theorem gives a ring isomorphism
$\mathbb{Z}/k_1 k_2\mathbb{Z} \xrightarrow{\;\sim\;} \mathbb{Z}/k_1\mathbb{Z} \times
\mathbb{Z}/k_2\mathbb{Z}$, $t \mapsto (t \bmod k_1,\, t \bmod k_2)$.  The map is a
bijection on $\{0, \ldots, k_1 k_2 - 1\}$; hence the combined trajectory visits all
$k_1 k_2$ state-pairs exactly once before returning to $(0, 0)$.
\end{proof}

\begin{remark}
For large incommensurate periods the closure period $k_1 k_2$ may be
astronomically long.  However, a joint Plateau does not require the trajectory to
\emph{complete} one full cycle---it requires only that the trajectory \emph{visit} the
resonance pair $(i^*, j^*)$, which by the Lemma occurs at exactly one step $t^* < k_1 k_2$.
\end{remark}

\subsection{Phase drift}

Define the \emph{phase offset} at step $t$:
\begin{equation}
  \phi(t) \;=\; \dS\!\bigl(q_{t \bmod k_1}^A,\; q_{t \bmod k_2}^B\bigr),
  \qquad
  \dS(p, q) = \arccos\!\bigl|\langle p, q \rangle_{\mathbb{R}^4}\bigr|.
\end{equation}
The function $\phi(t)$ is periodic with period $k_1 k_2$ and is generally non-monotone.
It sweeps through $k_1 k_2$ distinct values---one for each pair $(i,j)$ visited---before
repeating.  The minimum value $\delta \equiv \min_t \phi(t)$ governs whether coherence is
achievable.

% ─────────────────────────────────────────────────────────────────────────────
\section{Plateau Fixed Points on the Combined System}
\label{sec:plateau}

\begin{definition}[Cycle proximity]
\[
  \delta(\CA, \CB) \;:=\;
  \min_{\substack{i \in \mathbb{Z}/k_1\mathbb{Z} \\ j \in \mathbb{Z}/k_2\mathbb{Z}}}
  \dS\!\bigl(q_i^A,\; q_j^B\bigr).
\]
\end{definition}

\begin{theorem}[Existence of joint Plateau fixed point]\label{thm:plateau}
Under the assumptions above ($\gcd(k_1, k_2) = 1$):
\begin{enumerate}
  \item[\rm(a)] A joint Plateau fixed point exists if and only if
    $\delta(\CA, \CB) < \epsP$.
  \item[\rm(b)] When it exists, it occurs at the unique step $t^* \in \{0, \ldots,
    k_1 k_2 - 1\}$ determined by CRT from the minimising indices $(i^*, j^*)$:
    $t^* \equiv i^* \pmod{k_1}$ and $t^* \equiv j^* \pmod{k_2}$.
  \item[\rm(c)] The fixed point recurs with period $k_1 k_2$.
\end{enumerate}
\end{theorem}

\begin{proof}
The Plateau criterion in the Wheel fires when the Weld margin---the MDL score difference
between winner and runner-up across the combined proposal set---falls below $\epsP$.
The MDL score of a proposal $q$ is monotone in the Hopf-lapse distance between $q$ and
the goal quaternion; proposals geometrically closer to the goal score lower.  At step
$t$, the two cycles present proposals $q_{t \bmod k_1}^A$ and $q_{t \bmod k_2}^B$; the
margin is controlled by how close these are to each other in score, which in turn is
controlled by $\phi(t)$.

By Lemma~\ref{lem:period}, the pair $(t \bmod k_1, t \bmod k_2)$ visits every element
of $\mathbb{Z}/k_1\mathbb{Z} \times \mathbb{Z}/k_2\mathbb{Z}$ over one period.
Therefore $\min_t \phi(t) = \delta(\CA, \CB)$.  If $\delta < \epsP$, the minimising
pair is $(i^*, j^*)$, achieved at the unique $t^*$ given by CRT; Plateau fires there.
If $\delta \geq \epsP$, no step achieves a margin below $\epsP$, and Plateau never fires.
Parts (b) and (c) follow from CRT uniqueness and the periodicity of the combined trajectory.
\end{proof}

\begin{corollary}[P27 coherence]\label{cor:p27}
Two Oscillate subsystems with $\gcd(k_1, k_2) = 1$ satisfy the P27 intersubjective
coherence condition after a shared Weld if and only if $\delta(\CA, \CB) < \epsP$.
\end{corollary}

\noindent\textbf{Key observation.} The criterion $\delta(\CA, \CB) < \epsP$ depends on
the \emph{geometry} of the two cycles in $S^3$, not on the arithmetic of their periods.
Incommensurate periods guarantee the combined trajectory is exhaustive over
$\mathbb{Z}/k_1\mathbb{Z} \times \mathbb{Z}/k_2\mathbb{Z}$, making the search for a
resonance pair complete.  Commensurability governs \emph{when} resonance occurs, not
\emph{whether} the cycles are geometrically close enough for it.

% ─────────────────────────────────────────────────────────────────────────────
\section{The Resonance Condition}
\label{sec:resonance}

\begin{definition}[Resonance condition]\label{def:resonance}
Oscillate cycles $\CA$, $\CB$ satisfy the \emph{resonance condition} if
$\delta(\CA, \CB) < \epsP = 0.02$.
\end{definition}

\begin{theorem}[Resonance as the P27 coherence criterion]\label{thm:resonance}
Let $A$, $B$ be Oscillate subsystems with periods $k_1$, $k_2$ and $\gcd(k_1,k_2)=1$.
After a shared Weld event:
\begin{itemize}
  \item[\emph{Coherent:}] If the resonance condition holds, a unique step $t^* \in
    [0, k_1 k_2)$ exists at which the joint Plateau fires, the joint fixed point
    $q^*$ satisfies P27 mutual coherence, and the coherence event recurs every $k_1 k_2$
    steps.
  \item[\emph{Incoherent:}] If the resonance condition fails ($\delta \geq \epsP$),
    no joint Plateau fires within any period.  The phase offset $\phi(t)$ is bounded
    below by $\delta$, and the two subsystems cannot resolve a shared Hopf phase.
    P27 coherence is not achieved.
\end{itemize}
\end{theorem}

\subsection{The continuous/ergodic limit}

When the periods $k_1, k_2 \to \infty$ with $k_1/k_2 \to \omega \in \mathbb{R}
\setminus \mathbb{Q}$, the combined trajectory densifies on a 2-torus $T^2 \subset
S^3$.  By Weyl's equidistribution theorem, the trajectory eventually enters every
$\varepsilon$-ball in $T^2$.  Hence for any $\epsP > 0$ the resonance condition is
\emph{eventually} satisfied---but the waiting time grows as $O(1/\epsP^2)$.

\begin{corollary}[Ergodic case]
In the irrational-winding limit, any two Oscillate subsystems eventually achieve P27
coherence.  The coherence window narrows as the trajectory becomes more uniformly
distributed over $T^2$.
\end{corollary}

% ─────────────────────────────────────────────────────────────────────────────
\section{Worked Example: $k_1 = 2$, $k_2 = 3$}
\label{sec:example}

$\gcd(2, 3) = 1$, $\lcm(2, 3) = 6$.  The combined trajectory has period 6.

\subsection*{Case (a) — coherent: cycles share a common point}

Let $\CA = \{q_0, q_1\}$ and $\CB = \{q_0, q_2, q_3\}$ with
\[
q_0 = (1,0,0,0),\quad q_1 = (0,1,0,0),\quad q_2 = (0,0,1,0),\quad q_3 = (0,0,0,1).
\]
Both cycles contain $q_0$.  Phase offsets over one period:
\begin{center}
\begin{tabular}{@{}ccccc@{}}
\toprule
$t$ & $i = t \bmod 2$ & $j = t \bmod 3$ & $(q_i^A,\, q_j^B)$ & $\phi(t)$ \\
\midrule
0 & 0 & 0 & $(q_0,\; q_0)$ & $0$          \\
1 & 1 & 1 & $(q_1,\; q_2)$ & $\pi/2$      \\
2 & 0 & 2 & $(q_0,\; q_3)$ & $\pi/2$      \\
3 & 1 & 0 & $(q_1,\; q_0)$ & $\pi/2$      \\
4 & 0 & 1 & $(q_0,\; q_2)$ & $\pi/2$      \\
5 & 1 & 2 & $(q_1,\; q_3)$ & $\pi/2$      \\
\bottomrule
\end{tabular}
\end{center}
At $t^* = 0$: $\phi(0) = 0 < \epsP$.  \textbf{Joint Plateau fires.}  The fixed point is
$q^* = q_0$.  By CRT, the next resonance is at $t = 6$; coherence recurs with period 6.
P27 intersubjective coherence is satisfied.

\subsection*{Case (b) — incoherent: cycles are geometrically separated}

Let $\CA = \{q_0, q_1\}$ and $\CB = \{q_2,\, q_3,\, q_4\}$ with
$q_4 = \tfrac{1}{\sqrt{2}}(1, 0, 0, 1)$.  No element of $\CA$ lies within $\epsP$ of
any element of $\CB$.  Computing all six pair-distances:
\begin{align*}
  \dS(q_0, q_2) &= \tfrac{\pi}{2}, & \dS(q_0, q_3) &= \tfrac{\pi}{2}, &
  \dS(q_0, q_4) &= \arccos\tfrac{1}{\sqrt{2}} = \tfrac{\pi}{4}, \\
  \dS(q_1, q_2) &= \tfrac{\pi}{2}, & \dS(q_1, q_3) &= \tfrac{\pi}{2}, &
  \dS(q_1, q_4) &= \tfrac{\pi}{2}.
\end{align*}
$\delta(\CA, \CB) = \pi/4 \approx 0.785 \gg \epsP = 0.02$.  The phase offset cycles
through $\{\pi/2,\pi/2,\pi/2,\pi/2,\pi/4,\pi/2\}$ over the six steps; it never reaches
$\phi < \epsP$.  \textbf{No joint Plateau fires.}  P27 intersubjective coherence is not
achieved.

\begin{remark}
The initial Weld event that joined the two subsystems forced a momentary collapse to a
single quaternion, but without a shared cycle point, every subsequent Weld selects a
winner from one cycle only; the other cycle's current proposal is always more than
$\epsP$ away in MDL score.  The two observers' Hopf phase estimates diverge permanently
within each $k_1 k_2$-step period.
\end{remark}

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

\textbf{O1 (Approximate coherence).}
When $\epsP \leq \delta(\CA, \CB) \ll 1$, the cycles are close but not coherent under
the strict threshold.  Is there a useful notion of \emph{degree of intersubjective
coherence} proportional to $\epsP / \delta$?  Would it carry a physical interpretation
analogous to an effective coupling between the two observers?

\textbf{O2 (Heat depletion).}
Oscillate damps heat by $\delta_h = 0.95$ per step.  For large $k_1 k_2$ the heat at
resonance step $t^*$ may have decayed to $h_0 \cdot \delta_h^{t^*}$, potentially below
the level needed to sustain a decisive Weld.  The coherence criterion therefore involves
a joint constraint on $(k_1, k_2, h_0, \delta_h)$.  The boundary surface in this
parameter space is uncharacterised.

\textbf{O3 (Three-body case).}
Three Oscillate subsystems $A$, $B$, $C$ with pairwise incommensurate periods sharing a
Weld event face a combined period $\lcm(k_1, k_2, k_3)$ and three pairwise resonance
conditions.  The P27 $\mathbb{Z}_3$ structure (Paper 27, \S3.4) predicts a richer
coherence topology; in particular, whether pairwise resonance implies three-body joint
Plateau is unresolved (transitivity of proximity is not guaranteed on $S^3$).

\textbf{O4 (Perturb-induced resonance).}
A Perturb event blends a new quaternion into the current state at boundary-layer weight
$\beta \approx 7.2\%$ and resets $o\_count$.  If the injection lies in $\CB$ but not in
$\CA$, Perturb shifts $\CA$'s effective trajectory toward $\CB$, potentially
manufacturing a resonance pair that did not previously exist.  The conditions under which
Perturb creates or destroys intersubjective coherence deserve analysis.

\begin{thebibliography}{9}
\bibitem{P27}
  L.~F.~Vlegels, \textit{The Second Observer}, Paper~27, This volume (2025).
\bibitem{P18}
  L.~F.~Vlegels, \textit{The Master Operator $\hat{O}$}, Paper~18, This volume (2025).
\bibitem{P30}
  L.~F.~Vlegels, \textit{GON: Zero-Parameter $S^3$ Dynamics}, Paper~30, This volume (2025).
\bibitem{P146}
  L.~F.~Vlegels, \textit{Addendum~146: Periodic Oscillate Trajectories and the P7
  Grounding Mechanism}, This volume (2026).
\end{thebibliography}

\end{document}
