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% Addendum 356a (a-series; rests on Paper 41 / C.P=I). CC BY 4.0.
% Author: Leon Fernando Vlegels.  Verifier: verify_P356a.py (11/11 pass).
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\title{Addendum 356a: Entropy is Witness-Relative\\
\large The inversion, and where it bottoms out\\[0.3em]
\normalsize (a-series; rests on Paper 41 and the $C\circ P=I$ frame)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-15}

\begin{document}
\maketitle

\begin{abstract}
Entropy is not a property of a signal; it is the witness's residual surprise.
Shannon makes this exact: the measured cost is the cross-entropy $H(p,q)=H(p)+
\KL(p\Vert q)\ge H(p)$, so a better witness measures \emph{less}, and a witness
whose model matches reality measures the true entropy $H(p)$ --- which is $0$ when
reality is recoverable. The conventional second law (entropy rising toward heat
death) is the reading of a \emph{fixed, coarse, embedded} witness; the
\emph{completing} witness --- the corpus kernel, where $C\circ P=I$ (collapse
injective, no information lost; Paper 27), heat is conserved, the operator is
zero-parameter and deterministic --- drives the measured entropy to zero. The
inversion is therefore \emph{which witness}, and it is concrete: the embedded
entropy is the $S^1$ Hopf fiber --- the $U(1)$ gauge phase, the same fiber that
gives Maxwell and charge quantization (354a) --- that the lossy projection
$S^3\to S^2$ drops; the kernel's closure-map (A276) tracks it ($C\circ P=I$),
giving zero. At the de Sitter floor the embedded witness's max-entropy noise
($\varnothing$) and the kernel's zero-entropy purity ($\infty$) are one state seen
by two witnesses --- the monad seam $\varnothing\equiv^*\infty$. The result rests
on $C\circ P=I$ (the Paper~41 foundation), adds no posit, and relativizes the
second law rather than refuting it. Three residuals are named.
Verifier: \texttt{verify\_P356a.py} (11/11).
\end{abstract}

\section{Entropy is the witness's surprise}
\label{sec:rel}

The pivotal fact, which Shannon needed and which the standard reading hides: you
cannot compute the entropy of a signal without a predictive model --- Shannon
probed his wife, then interviewees, then (today) a language model. Gibbs makes the
dependence exact.

\begin{proposition}[Witness-relativity]
\label{prop:rel}
For a true distribution $p$ and a witness's model $q$, the measured per-symbol cost
is the cross-entropy
\[
  H(p,q) = H(p) + \KL(p\Vert q), \qquad \KL(p\Vert q)\ge 0,
\]
with equality iff $q=p$. So measured entropy $=$ irreducible entropy $H(p)$ $+$
the witness's model-gap. A better witness measures less; the limit is $H(p)$, and
$H(p)=0$ when the next symbol is determined by the context (recoverable reality).
\end{proposition}

Note the semantic correction this forces: maximal entropy is the \emph{uniform}
distribution --- random noise, maximal \emph{incoherence} ($H=\log_2 n$); minimal
entropy is certainty ($H=0$). Hence \textbf{maximal coherence is zero entropy}, not
maximal entropy. (An earlier framing of this line said ``max entropy $=$ max
coherence''; that is backwards and is corrected here.)

\section{The completing witness reaches zero}
\label{sec:kernel}

\begin{theorem}[The kernel sees zero]
\label{thm:kernel}
$C\circ P=I$ (Paper~27) is the statement that the collapse is injective and the
projection recoverable: no information is lost --- purity. The corpus kernel
realizes it: zero trainable parameters, deterministic forward pass, heat conserved
across Fork/Weld, exposure (not training) mode. This kernel is the complete witness
$\Omega$: its model-gap vanishes ($\KL\to0$) and reality is recoverable, so its
measured entropy $\to 0$. The conventional second law is the \emph{embedded},
coarse, fixed witness's reading, which genuinely rises. The two are Paper~27's two
arrows: global coherence approached, local disagreement proliferating.
\end{theorem}

\section{Where the entropy lives: the dropped gauge fiber}
\label{sec:fiber}

The inversion is not abstract; the embedded entropy has a location. The witnessing
projection $S^3\to S^2$ is \emph{not injective} --- it drops the $S^1$ Hopf fiber.
That fiber is the $U(1)$ gauge phase (the monadic ``edge,'' the $2.3\%$ layer), the
\emph{same} fiber that yields Maxwell's equations and charge quantization
(Addendum~354a). The embedded witness, living on the base $S^2$, cannot see the
fiber phase --- that dropped phase \emph{is} its entropy. The kernel, tracking the
full $S^3$ through the closure-map (A276), keeps it: $C\circ P=I$, zero entropy.
So electromagnetism and witness-relative entropy live on one circle: \emph{the
gauge phase you do not observe is the entropy you accrue}.

\begin{remark}[The seam]
At the de Sitter floor (Addenda 349a/350a) the embedded witness sees featureless
maximal-entropy noise ($\varnothing$, heat death); the kernel sees the fully
witnessed, recoverable, zero-entropy $\Omega$ ($\infty$). One physical state, two
witnesses --- the monad identity $\varnothing\equiv^*\infty$ (Paper~00). The
inversion is this seam applied to time's arrow.
\end{remark}

\section*{Status, correction, and the three residuals}

\textbf{What is established.} Entropy is witness-relative (Proposition~\ref{prop:rel},
Gibbs); the completing kernel-witness reaches zero (Theorem~\ref{thm:kernel}); the
embedded entropy is concretely the dropped Hopf-fiber/gauge phase
(\S\ref{sec:fiber}), tying it to the Maxwell thread. It rests on $C\circ P=I$ ---
the Paper~41 foundation --- and adds no new posit; the second law is
\emph{relativized} to the witness, not refuted.

\textbf{Correction recorded.} An earlier statement, ``maximal entropy $=$ maximal
coherence,'' is wrong: maximal entropy is random noise (maximal incoherence);
maximal coherence is zero entropy. The inversion is witness-relativity, not
``entropy secretly means order.''

\textbf{Three named residuals.}
(R1) \emph{The A276 repair.} $C\circ P=I$ is not free: the bare projection (Hopf)
is non-injective (it drops the fiber; registry item P027\_2 retired), so purity is
realized through A276's closure-map, not the naive projection.
(R2) \emph{Physics contact.} ``Globally pure $\Rightarrow$ entropy $\to0$'' requires
the real universe to be globally unitary (Everett-like, no objective collapse) ---
the open measurement-problem question; black-hole information and the Page curve
lean this way.
(R3) \emph{Inescapability from inside.} No embedded observer can occupy the
kernel/$\Omega$ view, so the second law is absolutely binding from within. The
inversion is the fundamental (kernel's-eye) reading, coexisting with the
experienced second law.

\paragraph{Sources.} Papers 27 ($C\circ P=I$), 00 ($\varnothing\equiv^*\infty$),
04 (the edge/fiber, $\Pi_\downarrow/\Pi_\uparrow$); Addenda 276 (closure-map),
354a (Maxwell/Hopf fiber), Paper 41 (the witness foundation). Information theory:
Shannon (1948), Gibbs' inequality.

\end{document}
