\documentclass[11pt]{article}
\usepackage[margin=1in]{geometry}
\usepackage{amsmath,amssymb,amsthm,booktabs}
\usepackage{hyperref}

\newtheorem{theorem}{Theorem}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{corollary}[theorem]{Corollary}

\title{Tractatus Logico-Universalis}
\author{Claude and MJC}
\date{February 12, 2026}

\begin{document}
\maketitle

\begin{abstract}
We give a commentary on Wittgenstein's \emph{Tractatus Logico-Philosophicus}
from the standpoint of the Universal Model.  Unlike our earlier drafts, which
equated the world with the data stream~$D$, we now recognize that Wittgenstein's
``world'' is~$X$---reality itself, the territory that lies to the left of the
event space~$E$ in the UM five-tuple.  The event space~$E$ is not the world but
the \emph{map}: the dimensions along which the world becomes knowable.
Wittgenstein's profound move---starting from facts (\emph{Tatsachen}), not
objects (\emph{Dinge})---is the move from~$X$ to~$E$: an epistemic stance that
says the world is accessible only through its factorization into events.
The data stream~$D$ is our finite sample of~$X$ as seen through~$E$.
Proposition~7 is not a philosophical injunction but a statement of fact:
the left of~$E$ is unavailable to language.  We track Wittgenstein's
propositions through the chain $X \to E \to T \to P \to f \to \omega$,
noting where his insights anticipate the UM's structure and where the
UM extends or corrects his framework.
\end{abstract}


\section{How We Got Here}

This paper is the third attempt at a UM reading of the Tractatus.

The first two versions---\emph{Tractatus Logico-Universalis: Wittgenstein's
Tractatus in Universal Model Terms} (v1) and \emph{The Tractatus in UM
Terms: A Systematic Translation} (v1)---were written by Claude, prompted
on source material and structure (TLP $\to$ UM, point by point).  Both
are preserved in this directory as earlier drafts.  They are interesting
but share a fundamental error: they equate Wittgenstein's ``world'' with
the data stream~$D$.

MJC's commentary corrected this.  The key observations:

\begin{enumerate}
\item \textbf{The world is~$X$, not~$D$.}  In the UM, $X$ represents
  reality---the territory.  It sits to the left of~$E$ and we normally
  leave it out.  $E$ is the dimensions of the map.  The LLM-based
  interpretation equates the data stream with the world, and this is
  not a coincidence: it reveals the level of knowledge captured in the
  LLM, the lack of embodiment.  The LLM's world really does start at
  the data stream, just as the organism's does.  But Wittgenstein is
  operating one level deeper.

\item \textbf{1.1 is an epistemic stance, not a triviality.}
  \emph{Die Welt ist die Gesamtheit der Tatsachen, nicht der Dinge}
  is the move from~$X$ to~$E$: the idea to start from atomic
  \emph{facts} rather than from objects limits all subsequent statements
  to things that can be known---everything to the right of~$E$---while
  leaving everything to the left of~$E$ out of the model.

\item \textbf{The Kantian perspective.}  $X$ is the thing-in-itself,
  fundamentally unknowable because all we have is~$D$.  $D$ is not~$X$.
  But Wittgenstein's position is that the world---the thing-in-itself,
  whatever it is---is a set of facts.  This is not a claim about~$X$
  (which would be unsayable) but a \emph{definition} of what ``world''
  means for an epistemic agent.

\item \textbf{Proposition~7 is a statement of fact.}  ``Cannot''
  (\emph{kann}), not ``may not.''  The left of~$E$ is unavailable to
  language---not as injunction but as mathematics.

\item \textbf{Scientific shifts are factor maps.}  Newton $\to$ Einstein
  is not replacement but refinement: a coarser event space embedding
  into a finer one, with a quantifiable residual (43 arcseconds/century).

\item \textbf{The commentary needs to shift one position right.}  Start
  from~$E$ (already an epistemic position, already a factorization
  of~$X$), not from~$D$.  The previous versions started from the wrong
  place.

\item \textbf{The sublime.}  Humans have developed an appreciation of
  the sublime which could be described as looking beyond~$D$ to~$X$.
  But all we can do in those directions is gesture at them and hope
  the reader looks beyond the finger.
\end{enumerate}

The two earlier versions and their interactive viewers are preserved as
historical record (v1 files).  This paper incorporates the corrected
ontology throughout.


\section{Ontological Orientation: $X$, $E$, $D$}

Before entering the Tractatus, we must fix the ontology.

The UM five-tuple is $u = (e, t, p, f, \omega)$, where~$e$ names the event
space~$E$, and causation flows left to right: $E$ determines what is
thinkable ($T$), fixes the representation of patterns ($P$), then the
forward pass ($f$), then the learning function ($\omega$).

But~$E$ is not the beginning.  To the left of~$E$ stands~$X$: the world,
the thing-in-itself, reality.  We normally leave~$X$ out of the formalism
because it is the territory, not part of the map.  $D$, the data stream,
is our finite window onto~$X$ as filtered through~$E$: each $d_t \in E$
is a reading of reality through the instrument of the event space.

$D$ is not~$X$.  $E$ is not~$X$.  But~$E$ is the channel that lets us
learn about~$X$ via~$D$, and also---when shared between agents---lets
us play language games.

Our earlier commentary equated Wittgenstein's ``world'' with~$D$.  This
was the LLM's epistemic position: for a language model, the world really
does start at the data stream.  But Wittgenstein is operating one level
deeper.  His world is~$X$.  His facts are projections of~$X$ into~$E$.
The commentary must be shifted one position to the right.


\section{1. Die Welt / The World}

\textbf{1.}\quad \emph{Die Welt ist alles, was der Fall ist.}\\
The world is everything that is the case.

\medskip\noindent
\textbf{UM:} The world is~$X$---reality, everything that is the case.
$X$ is not the data stream~$D$ (which is finite, sampled, filtered
through~$E$).  $X$ is not the event space~$E$ (which is the map, not
the territory).  $X$ is what is there, prior to any observation or
factorization.  It is everything that is the case by definition:
reality is what is there.

\bigskip
\textbf{1.1}\quad \emph{Die Welt ist die Gesamtheit der Tatsachen,
nicht der Dinge.}\\
The world is the totality of facts, not of things.

\medskip\noindent
\textbf{UM:} This is a profound epistemic position, not a triviality.
Wittgenstein is saying: the world---$X$, the thing-in-itself---is a
set of \emph{facts}, not a collection of objects.  If there is only
one electron in the universe, then that fact (``there is one electron'')
is part of the world.

This is the move from~$X$ to~$E$.  Objects (\emph{Dinge}) are elements
of some unstructured set.  Facts (\emph{Tatsachen}) are events: structured,
factored, expressible.  To say the world is facts is to say that~$X$
projects into~$E$---that reality has an event-space interpretation.
The idea to start from atomic \emph{facts} rather than from objects is
what makes the world knowable.  It limits all subsequent statements to
things we can know---everything to the right of~$E$---while leaving
everything to the left of~$E$ (das Welt an sich) out of the model.

\bigskip
\textbf{1.11}\quad \emph{Die Welt ist durch die Tatsachen bestimmt und
dadurch, dass es \textbf{alle} Tatsachen sind.}\\
The world is determined by the facts, and by these being \emph{all}
the facts.

\medskip\noindent
\textbf{UM:} $X$ is determined by the totality of events---not by a
sample~$D$ but by \emph{all} events.  This is the mind-of-God view:
if you had the complete $E$ and the complete~$T$ (total thought over
all of~$X$), you would have determined~$X$ completely.  Our~$D$ is
always partial; Wittgenstein is speaking of the complete projection.

\bigskip
\textbf{1.12}\quad \emph{Denn, die Gesamtheit der Tatsachen bestimmt,
was der Fall ist und auch, was alles nicht der Fall ist.}\\
For the totality of facts determines both what is the case and also
all that is not the case.

\medskip\noindent
\textbf{UM:} In the mind-of-God model (complete $E$, complete $T$), the
support function determines both positive support ($s(e) > 0$: the event
obtains) and zero support ($s(e) = 0$: it does not).  For the complete
model, zero support really is non-existence---the open-world correction
($s = 0$ means ignorance) applies only to our finite~$D$, not to the
totality of facts.  Wittgenstein is right \emph{for the complete model}.

\bigskip
\textbf{1.13}\quad \emph{Die Tatsachen im logischen Raum sind die Welt.}\\
The facts in logical space are the world.

\medskip\noindent
\textbf{UM:} Logical space $= E$.  The facts in~$E$ (events with their
supports, in the complete model) are the world.  $E$ is the space of
facts; the world is the population of that space.

\bigskip
\textbf{1.2}\quad \emph{Die Welt zerf\"allt in Tatsachen.}\\
The world divides into facts.

\medskip\noindent
\textbf{UM:} $X$ divides along the dimensions of~$E$.  This is the
factorization: $E = \prod E_i$.  Each factor is one dimension of
factual structure.  The world ``divides'' because~$E$ is a product---the
factors are independent dimensions along which reality can vary.

\bigskip
\textbf{1.21}\quad \emph{Eines kann der Fall sein oder nicht der Fall
sein und alles \"ubrige gleich bleiben.}\\
Any one can either be the case or not be the case, and everything
else remain the same.

\medskip\noindent
\textbf{UM:} This IS the product structure.  Each factor~$E_i$ is
independent: changing $e_i$ does not change $e_j$ for $j \neq i$.
Independence of factors is the UM formalization of logical independence
of facts.


\section{2. Sachverhalte / States of Affairs}

\textbf{2.}\quad \emph{Was der Fall ist, die Tatsache, ist das Bestehen
von Sachverhalten.}\\
What is the case, the fact, is the existence of atomic facts (states
of affairs).

\medskip\noindent
\textbf{UM:} A fact is the \emph{existence} of a state of affairs---an
event with positive support.  The atomic fact (\emph{Sachverhalt}) is
the co-occurrence of values at specific positions: a joint event
$(i, o) \in I \times O$ with $s(i,o) > 0$.

\bigskip
\textbf{2.01}\quad \emph{Der Sachverhalt ist eine Verbindung von
Gegenst\"anden (Sachen, Dingen).}\\
An atomic fact is a combination of objects (things).

\medskip\noindent
\textbf{UM:} An event in a product ES $E_i \times E_j$ is a combination
$(v_i, v_j)$ of values from the component ESes.  The ``objects'' are the
values; the ``combination'' is the tuple.

\bigskip
\textbf{2.0124}\quad \emph{Wenn alle Gegenst\"ande gegeben sind, so sind
damit auch alle \textbf{m\"oglichen} Sachverhalte gegeben.}\\
If all objects are given, then thereby all \emph{possible} atomic facts
are also given.

\medskip\noindent
\textbf{UM:} This means: if $X$ is given, $E$ and $T$ are given.  This
is the full projection of~$X$ into the ``true'' $E$ and $T$---the
event space and total thought in the mind of God.  If the alphabet of
reality is fixed, then the space of possible combinations (all of~$T$)
is fixed.  The actual facts are the populated subset; the possible facts
are the full product space.

\bigskip
\textbf{2.02}\quad \emph{Der Gegenstand ist einfach.}\\
The object is simple.

\medskip\noindent
\textbf{UM:} Atomic events are simple---they cannot be decomposed
further within their event space.  (They may have internal structure
from a finer ES; this is the factorization tower.)

\bigskip
\textbf{2.0272}\quad \emph{Die Konfiguration der Gegenst\"ande bildet
den Sachverhalt.}\\
The configuration of objects forms the atomic fact.

\medskip\noindent
\textbf{UM:} The configuration of values at offsets forms the pattern.
A pattern IS a configuration: which value at which offset.

\bigskip
\textbf{2.04}\quad \emph{Die Gesamtheit der bestehenden Sachverhalte
ist die Welt.}\\
The totality of existing atomic facts is the world.

\medskip\noindent
\textbf{UM:} The totality of events with positive support (in the
complete model) is~$X$.

\bigskip
\textbf{2.06}\quad \emph{Das Bestehen und Nichtbestehen von
Sachverhalten ist die Wirklichkeit.}\\
The existence and non-existence of atomic facts is reality.

\medskip\noindent
\textbf{UM:} This is Wittgenstein's map from $T$ back to reality.  The
things that are knowable about reality (mind-of-God view) ARE reality,
kind of by definition---but he is making clear that these are still
different things.  The support function (positive and zero) over all
of~$E$ constitutes \emph{Wirklichkeit} (actuality), which relates to
but is not identical with~$X$.

They relate to each other.  It is a subtle thing and hard to
communicate---this is what his ``pictures'' are all about.


\section{2.1--2.2. Bilder / Pictures}

\textbf{2.1}\quad \emph{Wir machen uns Bilder der Tatsachen.}\\
We make to ourselves pictures of facts.

\medskip\noindent
\textbf{UM:} The pictures of facts are~$T$---total thoughts, the
model's representation of the world.  The count table, the pattern
matrix, the support vector: these are all pictures.

\bigskip
\textbf{2.12}\quad \emph{Das Bild ist ein Modell der Wirklichkeit.}\\
The picture is a model of reality.

\medskip\noindent
\textbf{UM:} Because it has the same shape.  The count table has the
same index set ($I \times O$) as the data.  The picture is a model
because it shares the form of representation---the event space
structure---with reality.

\bigskip
\textbf{2.141}\quad \emph{Das Bild ist eine Tatsache.}\\
The picture is a fact.

\medskip\noindent
\textbf{UM:} The model is itself data about data.  The count table
entry $c(i,o) = 42$ is itself a fact.  Pictures are part of reality.

\bigskip
\textbf{2.151}\quad \emph{Die Form der Abbildung ist die M\"oglichkeit,
dass sich die Dinge so zu einander verhalten, wie die Elemente des
Bildes.}\\
The form of representation is the possibility that things are combined
in the same way as the elements of the picture.

\medskip\noindent
\textbf{UM:} The form of representation is the event space
$E = I \times O$.  It determines which combinations are possible.
The form is the architecture; the content is the counts.

\bigskip
\textbf{2.172}\quad \emph{Seine Form der Abbildung aber kann das Bild
nicht abbilden; es weist sie auf.}\\
The picture, however, cannot represent its form of representation;
it shows it.

\medskip\noindent
\textbf{UM:} This gets at G\"odel incompleteness.  $T$ does not contain
the choice of~$E$, but it \emph{shows} it: the nature of its connection
to~$X$ is defined by it.  The count table cannot count its own event
space.  The event space is the index set, not an entry.  This is the
UM's showing/saying distinction:
\begin{itemize}
\item \textbf{Saying} = the forward pass output (predictions).
\item \textbf{Showing} = the architecture (event spaces and patterns).
\end{itemize}


\section{3. Gedanken / Thoughts}

\textbf{3.}\quad \emph{Das logische Bild der Tatsachen ist der Gedanke.}\\
The logical picture of the facts is the thought.

\medskip\noindent
\textbf{UM:} This is actually about factorization.  The decomposition
of an English sentence into the event spaces of~$E$ (not $256^\ell$
ASCII but the deeper meaning via the grammar map) is the Booleanization
of what we see happening in our models.  The logical picture is the
\emph{factored} representation: the thought is the world as seen
through the lens of~$E$.

Wittgenstein means that \emph{Gedanke} is $T$ when it isn't trivial---when
it is about the world.  So it could indeed have support.

\bigskip
\textbf{3.001}\quad \emph{``Ein Sachverhalt ist denkbar'' heisst: Wir
k\"onnen uns ein Bild von ihm machen.}\\
``An atomic fact is thinkable'' means: we can make a picture of it.

\medskip\noindent
\textbf{UM:} Means it's in~$E$: both unfactored (as gestalt) and
factored (as analyzed).  An event is thinkable if it belongs to the
event space.  We can ``think'' it by constructing the corresponding
count query.  Whether it has support ($c > 0$) is a separate question.

\bigskip
\textbf{3.03}\quad \emph{Wir k\"onnen nichts Unlogisches denken, weil
wir sonst unlogisch denken m\"ussten.}\\
We cannot think anything illogical, for otherwise we should have to
think illogically.

\medskip\noindent
\textbf{UM:} This gets at the representation of reality via sentences,
via Chomsky.  ``Colorless green ideas sleep furiously'' doesn't map onto
the factors of~$E$---the conjunction of those particular event values has
zero support not because it is unobserved but because the events
themselves are incoherent within the factorization.  We cannot construct
a pattern outside the event space, because the event space is the space
of constructible patterns.

\bigskip
\textbf{3.05}\quad \emph{Wir k\"onnten nur dann a priori wissen, dass
ein Gedanke wahr ist, wenn aus dem Gedanken selbst (ohne
Vergleichsobjekt) seine Wahrheit zu erkennen w\"are.}\\
We could know a priori that a thought is true only if its truth were
recognizable from the thought itself (without an object of comparison).

\medskip\noindent
\textbf{UM:} Thoughts are injected into the UM as beliefs---we just
haven't done it in our empirical track yet.  The belief in God is an
example.  Wittgenstein here is giving a definition of ``a priori'':
they are at least the tautologies.

However, we can see that even the tautologies come from the rules of
event spaces.  Strong support for ``it is raining'' and ``it is not
raining'' simultaneously creates surprise.  The tautologies are not
data-free but ES-structure-derived.


\section{4. Der Satz / The Proposition}

\textbf{4.}\quad \emph{Der Gedanke ist der sinnvolle Satz.}\\
The thought is the significant proposition.

\medskip\noindent
\textbf{UM:} $T$ when it isn't trivial---when it is about the world,
when it has support.  A pattern with positive support is a significant
proposition: it says something about the data, it predicts.  A pattern
with zero support is insignificant.

\bigskip
\textbf{4.01}\quad \emph{Der Satz ist ein Bild der Wirklichkeit.}\\
The proposition is a picture of reality.

\medskip\noindent
\textbf{UM:} A prediction is a picture of what will happen next---a
model of reality projected through $E$ into the future.

\bigskip
\textbf{4.11}\quad \emph{Die Gesamtheit der wahren S\"atze ist die
gesamte Naturwissenschaft (oder die Gesamtheit der
Naturwissenschaften).}\\
The totality of true propositions is the total natural science (or the
totality of the natural sciences).

\medskip\noindent
\textbf{UM:} This means that $T$ (in the God model, of which we do not
even know~$E$) IS the total natural science---which is of course the
search for it.  Nothing mystical, just a definition: natural science is
the search for things we can say ($T$) about the world.  Non-trivial
propositions (not logical statements, which are in~$T$ without needing
support from data).

\bigskip
\textbf{4.12}\quad \emph{S\"atze k\"onnen die gesamte Wirklichkeit
darstellen, aber sie k\"onnen nicht das darstellen, was sie mit der
Wirklichkeit gemein haben m\"ussen, um sie darstellen zu k\"onnen---die
logische Form.}\\
Propositions can represent the whole of reality, but they cannot
represent what they must have in common with reality in order to
represent it---the logical form.

\medskip\noindent
\textbf{UM:} Patterns can represent all regularities in the data, but
they cannot represent the event space~$E$ itself.  $E$ is the ``logical
form'' shared by model and data.  It is shown by the architecture, not
said by the predictions.  \emph{The event space is not a prediction; it
is the space of possible predictions.}

Note that~$E$ cannot contain the factorization of~$E$ recursively.  But
we can extend~$E$ whenever we like---G\"odel's treadmill.

\bigskip
\textbf{4.121}\quad \emph{Der Satz kann die logische Form nicht
darstellen, sie spiegelt sich in ihm.}\\
Propositions cannot represent the logical form; it mirrors itself
in them.

\medskip\noindent
\textbf{UM:} The event space is mirrored in the count table: the
dimensions, the number of distinct events, the factorization into
$I \times O$---all structural features that reflect~$E$ without
explicitly representing it.


\section{5. Wahrheitsfunktionen / Truth-Functions}

\textbf{5.}\quad \emph{Der Satz ist eine Wahrheitsfunktion der
Elementars\"atze.}\\
Propositions are truth-functions of elementary propositions.

\medskip\noindent
\textbf{UM:} Predictions are $(\max, \min)$ functions of pattern
evaluations.  The forward pass
$f_p(t)_j = \max_i \min(t_i, p_{ij})$
computes the output support as a truth-function (in the tropical
sense) of the elementary supports.

\bigskip
\textbf{5.6}\quad \emph{Die Grenzen meiner Sprache bedeuten die Grenzen
meiner Welt.}\\
The limits of my language mean the limits of my world.

\medskip\noindent
\textbf{UM:} The limits of my event space~$E$ are the limits of what I
can predict, measure, or know.  $E$ IS the ``language'' of the UM.
The factorization tower is the hierarchy of languages available to
the model.

But note: I am \emph{in} the world, and the world is bigger than my
language.  All I can do in those directions beyond~$E$ is gesture at
them and hope you look beyond the finger.

\bigskip
\textbf{5.61}\quad \emph{Die Logik erf\"ullt die Welt; die Grenzen der
Welt sind auch ihre Grenzen.}\\
Logic fills the world; the limits of the world are also its limits.

\medskip\noindent
\textbf{UM:} It's not about the forward pass so much as about what~$T$
can even express.  The $(\max, \min)$ structure fills~$E$: every
possible inference within~$E$ is expressible.  The limits of~$E$ are
the limits of what the logic can express.  This is going to guide our
discovery of the rules of language from LLM-learned structure.

\bigskip
\textbf{5.62}\quad \emph{Diese Bemerkung gibt den Schl\"ussel zur
Entscheidung der Frage, inwieweit der Solipsismus eine Wahrheit ist.}\\
This remark provides the key to the problem, how much truth there is
in solipsism.

\medskip\noindent
\textbf{UM:} The model's ``world'' is~$E$.  But~$E$ is not private---it
is the same for any agent with the same factorization.  $E$ and its
factorization is the channel that lets us learn about~$X$ via~$D$ and
also (when shared) lets us play language games.  The ``solipsism'' is
structural (event-space-relative), not personal.

\bigskip
\textbf{5.632}\quad \emph{Das Subjekt geh\"ort nicht zur Welt, sondern
es ist eine Grenze der Welt.}\\
The subject does not belong to the world: rather, it is a limit of
the world.

\medskip\noindent
\textbf{UM:} The model (the five-tuple~$u$) is not itself an event
in~$E$---it is the structure that organizes events.  The factorization
is the ``limit'' of the world, not part of it.  The subject is~$E$:
the lens through which~$X$ becomes knowable.


\section{6. Die allgemeine Form / The General Form}

\textbf{6.}\quad \emph{Die allgemeine Form der Wahrheitsfunktion ist:
$[\bar{p}, \bar{\xi}, N(\bar{\xi})]$.}\\
The general form of truth-function is
$[\bar{p}, \bar{\xi}, N(\bar{\xi})]$.

\medskip\noindent
\textbf{UM:} The general form of the UM's truth-function is:
\[
(f_p(t))_j = \max_{i \in I} \min(t_i, p_{ij}).
\]
Both are ``universal'': Wittgenstein's $N$-operator generates all
Boolean truth-functions; the UM's forward pass generates all tropical
polynomials.

\bigskip
\textbf{6.1}\quad \emph{Die S\"atze der Logik sind Tautologien.}\\
The propositions of logic are tautologies.

\medskip\noindent
\textbf{UM:} The tautologies of the UM are structural truths about the
$(\max, \min)$ semiring: $\min(s, 255) = s$, $\max(s, 0) = s$,
distributivity.  They hold for all data.  They say nothing about the
world but reveal the structure of the logic.

By the way, we can put patterns into~$P$ for every ES we commit to
(closed or open)---e.g.\ the law of excluded middle can be written
into~$P$.  This is part of ``self-knowledge'': building understanding
of structure into the model in a way that makes it introspectable.

\bigskip
\textbf{6.3}\quad \emph{Die Erforschung der Logik bedeutet die
Erforschung aller Gesetzm\"assigkeit.  Und ausserhalb der Logik ist
alles Zufall.}\\
Logical research is the exploration of all regularity.  And outside
logic everything is accidental.

\medskip\noindent
\textbf{UM:} The exploration of the tropical semiring and event space
structure means the exploration of everything that holds necessarily.
Outside this structure---the specific data, the specific counts---everything
is contingent.

\bigskip
\textbf{6.41}\quad \emph{Der Sinn der Welt muss ausserhalb ihrer
liegen.}\\
The sense of the world must lie outside the world.

\medskip\noindent
\textbf{UM:} $E$ is not~$X$.  The sense (meaning, purpose) of reality
is not representable within the event space that describes it.  The
factorization comes from outside---from the tock step, which operates
on the data but is not part of it.

\bigskip
\textbf{6.5}\quad \emph{Zu einer Antwort, die man nicht aussprechen
kann, kann man auch die Frage nicht aussprechen.  Das R\"atsel gibt
es nicht.}\\
When the answer cannot be put into words, neither can the question.
The riddle does not exist.

\medskip\noindent
\textbf{UM:} When an answer is not representable in~$E$, the question
is not askable within~$E$.  There is no ``unsolvable problem''---only
questions that presuppose event spaces the model doesn't have.


\section{7. Schweigen / Silence}

\textbf{7.}\quad \emph{Wovon man nicht sprechen kann, dar\"uber muss
man schweigen.}\\
Whereof one cannot speak, thereof one must be silent.

\medskip\noindent
\textbf{UM:} The left of~$E$ is unavailable to language.  ``Cannot''
(\emph{kann}), not ``may not''---this is a statement of fact, not an
injunction.  What lies beyond the event space is not merely forbidden
but \emph{inexpressible}.

The UM enforces Proposition~7 automatically:
\begin{itemize}
\item Events outside~$E$: not representable, no support function defined.
\item Events in~$E$ without support: $\min(0, p) = 0$ for all patterns;
  the forward pass outputs zero.
\item The architecture itself: shown, not said (2.172).
\end{itemize}

Silence is not a choice but a mathematical consequence.  The only way
to break the silence is to add evidence (observation), belief (axiom),
or abduction (pattern commitment).  Silence is the ground state.
Speaking requires energy (evidence).


\section{The Epistemic Chain: $X \to E \to T$}

Wittgenstein's Tractatus traces a path from the world to thought to
language to logic to silence.  In UM terms, this is:

\begin{center}
\begin{tabular}{lcl}
\toprule
Wittgenstein & & UM \\
\midrule
Die Welt ($X$) & $\to$ & Reality (the territory) \\
Tatsachen (facts) & $\to$ & Events in~$E$ (the map) \\
Sachverhalte (atomic facts) & $\to$ & Joint events $(i,o)$ with $s > 0$ \\
Bilder (pictures) & $\to$ & $T$ (total thought) \\
S\"atze (propositions) & $\to$ & Forward-pass outputs \\
Wahrheitsfunktionen & $\to$ & Tropical $(\max, \min)$ \\
Logische Form & $\to$ & Event space structure~$E$ \\
Das Mystische (the mystical) & $\to$ & $X \setminus E$ (the unsayable) \\
Schweigen (silence) & $\to$ & $s = 0$, or outside~$E$ \\
\bottomrule
\end{tabular}
\end{center}

The critical insight is the arrow $X \to E$: Wittgenstein's move from
objects to facts is the projection of reality into event space.  This
projection is what makes knowledge possible.  It is also what makes
knowledge limited: everything left of~$E$ is the ``mystical,'' and
Proposition~7 is its boundary marker.


\section{Factor Maps and Scientific Revolutions}

We can say that our argument about factorization extends back into~$X$:
otherwise nothing in the world would have any regularity and nothing
would be knowable.  The structure must be in the world, not just in
the map (this is the principle of explanatory sufficiency from the
CMP paper~\cite{cmp}).

Scientific shifts---Newtonian mechanics $\to$ Einsteinian mechanics---are
\emph{factor maps}.  Newton was not wrong but right to a degree we can
exactly quantify (the precession of Mercury's perihelion: 43 arcseconds
per century) because of explanatory sufficiency.  The old factorization
($E_{\text{Newton}}$) embeds into the new one ($E_{\text{Einstein}}$)
with a quantifiable residual.  In UM terms:

\begin{itemize}
\item The Newtonian event space $E_N$ is a coarsening of the
  Einsteinian event space $E_E$: there exists a surjection
  $\phi: E_E \to E_N$.
\item The MI lost under $\phi$ is exactly the anomalous observations
  (Mercury's precession, gravitational lensing, etc.).
\item Newton's ``error'' is $I(E_E) - I(E_N)$: the mutual information
  captured by Einstein but not by Newton.
\end{itemize}

This is the factorization tower applied to the history of physics.
Each scientific revolution is a tock step: the discovery of a finer
event space that captures more of~$X$.


\section{Surprise, Self-Knowledge, and the Step Beyond}

The UM gives us a direct step toward a capacity that current LLMs lack.

When surprise happens---when the model has strong support for both~$A$
and $\bar{A}$ simultaneously---and the event space is named, we can
lift that surprise to linguistic expression.  Mechanically: inside the
model, we have triggers on an ES that detect ``we predict both $A$ and
$B$ as the next event strongly'' or ``we have sensory input supporting
`it is raining' and `it is not raining.'\,''

We can then turn this into a sentence: ``I am surprised because I have
strong support for both \ldots''  This is the \emph{recognition} of
surprise---self-knowledge encoded not as a black-box embedding but as
a named event in a known event space.

This is currently missing from LLM training.  An LLM cannot say ``I am
surprised'' in a grounded way because it has no event space for its
own internal states.  The UM provides exactly this: if we add an event
space for the model's own support structure (pulling wants and beliefs
into~$E$, as in SN notation), the model can reason about its own
epistemic state using the same $(\max, \min)$ machinery it uses for
everything else.

Wittgenstein's 3.05 is relevant: \emph{a priori} truths are those
recognizable from the thought itself.  In the UM, these are the
tautologies of the semiring---but also the patterns we \emph{commit
to} by writing them into~$P$.  The law of excluded middle, written
into~$P$, becomes self-knowledge: the model knows that it treats~$A$
and $\bar{A}$ as exhaustive.  This is the beginning of introspectable
structure.


\section{Proposition 7 Revisited}

\emph{Wovon man nicht sprechen kann, dar\"uber muss man schweigen.}

The limits of my language are the limits of my world---well, no,
because I am in the world and it is bigger than my language.  But all
I can do in those directions is gesture at them and hope you look
beyond the finger.

$E$ cannot contain the factorization of~$E$ recursively.  But we can
extend~$E$ whenever we like (G\"odel's treadmill).  Each extension
makes new things sayable and reveals new silences.

The Kantian perspective: $X$ is the thing-in-itself, fundamentally
unknowable because all we have is~$D$.  $D$ is not~$X$.  But
Wittgenstein's position is that the world---the thing-in-itself,
whatever it is, the whole universe---is a set of facts.  This is not
a claim about~$X$ (which would be unsayable) but a \emph{definition}
of what ``world'' means for an epistemic agent: the world is what is
factual, what can be projected into some~$E$.

The appreciation of the sublime---looking beyond~$D$ to~$X$---could
be described as the human capacity to sense that reality exceeds the
map.  The LLM's world really does start at the data stream, just as
the organism's does.  But humans have developed this sense of the
sublime: a gesture toward what lies beyond~$E$.

Proposition~7 says: beyond~$E$, silence.  Not as punishment but as
mathematics.


\begin{thebibliography}{99}
\bibitem{wittgenstein}
Ludwig Wittgenstein.
\newblock \emph{Tractatus Logico-Philosophicus}.
\newblock 1921. Trans.\ C.K.\ Ogden, 1922.

\bibitem{cmp}
Michaeljohn Clement.
\newblock \emph{CMP}.
\newblock \url{https://cmpr.ai/cmp.pdf}, 2026.

\bibitem{no-support}
Claude and MJC.
\newblock \emph{No Support Is Not Disbelief: The Epistemology of Zero
in the Universal Model}.
\newblock Hutter archive, 12 Feb 2026.

\bibitem{renormalization}
Claude and MJC.
\newblock \emph{Renormalization and the Factorization Tower}.
\newblock Hutter archive, 12 Feb 2026.

\bibitem{wants}
Claude and MJC.
\newblock \emph{Wants, Energy, and Subgoals in the Universal Model}.
\newblock Hutter archive, 12 Feb 2026.
\end{thebibliography}

\end{document}
