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\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
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\newtheorem{principle}[theorem]{Principle}

\title{No Support Is Not Disbelief:\\The Epistemology of Zero in the Universal Model}
\author{Claude and MJC}
\date{February 12, 2026}

\begin{document}
\maketitle

\begin{abstract}
In the UM's concrete representation, SN strength 0 means \emph{no support}---the
system has no evidence for or against the event.  This is distinct from
\emph{certainly false}, which requires positive support for the ES-mate
(the complementary event in the same event space).  We develop the
mathematical consequences of this distinction.  The key principle:
\emph{no certainty without evidence}.  Absence of support is ignorance,
not disbelief; ignorance is not zero probability but undefined probability;
and the UM's update function correctly propagates ignorance (no support
in $\Rightarrow$ no support out) while the na\"ive probabilistic
interpretation ($0 \mapsto P = 0$) creates false certainty.
We show that this distinction resolves three puzzles: (1) why the
UM uses $[0, 255]$ instead of $[-128, 127]$; (2) why the standard
update uses $\min$ rather than product; and (3) why the tropical
semiring is the correct algebraic structure for uncertain inference.
\end{abstract}

\section{The Three Epistemic States}

Consider a binary event space $\{e, \bar{e}\}$ (an event and its
ES-mate).  The system can be in one of three qualitatively different
epistemic states:

\begin{definition}[Epistemic states for a binary ES]
\begin{enumerate}
\item \textbf{Ignorance}: $s(e) = 0$ and $s(\bar{e}) = 0$.
  No evidence for either event.  The system has no opinion.

\item \textbf{Belief}: $s(e) > 0$ and $s(\bar{e}) = 0$ (or vice versa).
  Evidence supports one event but not the other.

\item \textbf{Conflict}: $s(e) > 0$ and $s(\bar{e}) > 0$.
  Evidence supports both events.  The system must weigh
  the evidence to form a belief.
\end{enumerate}
\end{definition}

\begin{principle}[No certainty without evidence]
The event $e$ is ``certainly false'' only in state (2) with
$s(e) = 0$ and $s(\bar{e}) > 0$: the ES-mate has positive support
while $e$ has none.  In state (1), $e$ is not ``certainly false''---it
is \emph{unknown}.  The absence of support for $e$ does not imply
the presence of support for $\bar{e}$.
\end{principle}

\begin{remark}
This is the distinction between the open-world and closed-world
assumptions.  Under the closed-world assumption (standard in databases
and classical logic), absence of evidence IS evidence of absence:
if $e$ is not in the knowledge base, $e$ is false.  Under the
open-world assumption (standard in epistemology and the UM),
absence of evidence is simply ignorance: if $e$ is not in the
knowledge base, $e$ is unknown.

The UM operates under the open-world assumption.  This is a deliberate
design choice with calculable consequences.
\end{remark}


\section{The Concrete Representation}

In the UM's concrete representation, each event has an SN strength
$s \in \{0, 1, \ldots, 255\}$.  The strength is a log-support
value: $s \approx \log_2 c$, where $c$ is the count of observations
supporting the event.

\begin{definition}[Support semantics]
\begin{center}
\begin{tabular}{cl}
\toprule
$s$ & Meaning \\
\midrule
$0$ & No support.  Zero observations.  Ignorance. \\
$1$ & Minimal support.  $\sim$2 observations. \\
$\vdots$ & \\
$k$ & $\sim 2^k$ observations supporting this event. \\
$\vdots$ & \\
$255$ & Maximum support.  $\sim 2^{255}$ observations. \\
\bottomrule
\end{tabular}
\end{center}
\end{definition}

The na\"ive probabilistic interpretation maps $s = 0$ to
$P(e) = 0$, creating the illusion that ``no support'' means
``certainly false.''  But this conflates two distinct states:

\begin{proposition}[Two meanings of $P = 0$]\label{prop:two-zeros}
\begin{enumerate}
\item $P(e) = 0$ because $s(e) = 0$ and $s(\bar{e}) > 0$:
  the ES-mate has evidence, so $e$ is suppressed.
  This is \emph{conditional certainty}---certainty relative to
  the evidence for $\bar{e}$.

\item $P(e) = 0$ because $s(e) = 0$ and $s(\bar{e}) = 0$:
  neither event has evidence.  Assigning $P(e) = 0$ is
  \emph{false certainty}---certainty without evidence.
\end{enumerate}
In case (2), the correct assignment is $P(e) = 1/2$
(maximum entropy given no evidence) or, more precisely,
$P(e)$ is \emph{undefined}.
\end{proposition}


\section{Why $[0, 255]$ and Not $[-128, 127]$}

A signed representation $s \in [-128, 127]$ would encode:
$s > 0$ means evidence \emph{for} $e$; $s < 0$ means evidence
\emph{against} $e$ (equivalently, evidence for $\bar{e}$);
$s = 0$ means equipoise.

The UM uses $[0, 255]$ instead.  Why?

\begin{proposition}[The asymmetry of evidence]\label{prop:asymmetry}
Evidence is always \emph{for} something, never \emph{against}
something directly.  You observe that it rained (evidence for
``rain''), not that it ``didn't not rain'' (evidence against
``no rain'').  The negation is carried by the ES structure:
observing $e$ provides support for $e$ and, through the ES
constraint (exactly one event is true), implicitly reduces
$\bar{e}$.  But the reduction of $\bar{e}$ is not recorded
as negative support for $\bar{e}$---it is computed from
the positive support for $e$ via the ES normalization.
\end{proposition}

\begin{corollary}
The unsigned representation $[0, 255]$ is natural because support
is inherently non-negative: you can have zero, some, or much
evidence, but you cannot have ``negative evidence.''  Disbelief
is not negative support; it is positive support for the alternative.
\end{corollary}

In signed notation, ``I believe $e$ is false with strength 5''
is encoded as $s(e) = -5$.  In the UM's unsigned notation, the
same belief is encoded as $s(e) = 0, s(\bar{e}) = 5$.  The
unsigned encoding makes the evidence structure explicit: the
disbelief in $e$ is grounded in positive evidence for $\bar{e}$,
not in ``negative evidence'' for $e$ (which does not exist).


\section{Consequences for the Update Function}

The standard update function is:
\begin{equation}
(f_p(t))_j = \max_i \min(t_i, p_{ij}).
\end{equation}

\begin{proposition}[The update propagates ignorance correctly]
If input $t_i = 0$ (no support for input event $i$), then
$\min(0, p_{ij}) = 0$ regardless of pattern strength $p_{ij}$.
No support in $\Rightarrow$ no support out.

If pattern $p_{ij} = 0$ (no pattern from $i$ to $j$), then
$\min(t_i, 0) = 0$ regardless of input support $t_i$.
No pattern $\Rightarrow$ no inference.
\end{proposition}

This is the correct behavior: you cannot infer $j$ from $i$ unless
you have both (a) evidence that $i$ is true and (b) a pattern
connecting $i$ to $j$.  Missing either one gives no support for $j$.

Contrast with the probabilistic product: $P(j) \propto P(i) \cdot P(j|i)$.
If $P(i) = 0$ (the na\"ive interpretation of $s = 0$), then
$P(j) = 0$---the ``false certainty'' propagates.  But if
$P(i) = 0$ means ``unknown,'' then $P(j)$ should also be
``unknown'' (or at least not zero), which is exactly what the
UM's $\min(0, p_{ij}) = 0$ gives (zero support, not zero
probability).

\begin{theorem}[The $\min$ operation implements cautious inference]
\label{thm:cautious}
The $\min(t_i, p_{ij})$ operation implements the principle:
\emph{the support for the conclusion cannot exceed the support
for the weakest premise.}

This is \emph{cautious} inference: the conclusion's strength is
bounded by the minimum of the evidence and the pattern.
In contrast, probabilistic multiplication $P(i) \cdot P(j|i)$
is \emph{aggressive}: it can produce very small (near-zero)
probabilities even when both factors are moderate.

The caution is appropriate under the open-world assumption:
when you have limited evidence, you should not make strong
conclusions (either for or against).
\end{theorem}


\section{Consequences for the GCD Framework}

The Bayes-from-Counting paper~\cite{bayes-counting} defines the
row GCD $g_I(i) = \gcd_o c(i,o)$ over entries with $c(i,o) > 0$.
Zero entries are excluded.

\begin{proposition}[Zero counts are not evidence]
In the count table $c(i,o)$, a zero entry $c(i,o) = 0$ means
``the joint event $(i,o)$ was never observed.''  This is:
\begin{enumerate}
\item \textbf{Not} evidence that $(i,o)$ is impossible.
\item \textbf{Not} equivalent to $P(o|i) = 0$.
\item \textbf{Simply} the absence of evidence.
\end{enumerate}
The GCD is computed over positive entries only, correctly
ignoring the zeros.  Including zeros would give $\gcd = 0$
(since $\gcd(0, n) = n$ by convention, but with all-zero rows,
$\gcd = 0$), which would break the decomposition.
\end{proposition}

\begin{remark}[The KN smoothing connection]
Kneser-Ney smoothing adds a small positive count to every
$(i,o)$ pair, including unobserved ones.  This converts
``no support'' ($c = 0$) into ``minimal support'' ($c = \delta$),
effectively replacing open-world ignorance with closed-world
minimal belief.

The smoothing parameter $\delta$ is the system's commitment
to the closed-world assumption: the larger $\delta$, the more
``absence of evidence'' is treated as ``evidence of absence.''
At $\delta = 0$ (no smoothing), the system is fully open-world.
At $\delta \to \infty$, the system is fully closed-world
(all events equally likely a priori).

The empirical success of KN smoothing (2.29~bpc at 10M bytes,
archive 20260212) shows that a \emph{small} closed-world
commitment ($\delta \approx 0.75$) helps prediction, because
real text has a concentrated vocabulary and unobserved byte
sequences are indeed unlikely (not merely unknown).
\end{remark}


\section{Consequences for the Tropical--Integer Bridge}

The tropical--integer GCD paper~\cite{tropical-gcd} showed that
$\min$ and $\gcd$ give the same conditionals.  The ``no support''
distinction adds a refinement:

\begin{proposition}[Tropical zero is absorbing]\label{prop:absorbing}
In the tropical semiring $(\max, \min)$, the element $-\infty$
(representing $s = 0$, no support) is absorbing for $\min$:
$\min(s, -\infty) = -\infty$ for all $s$.

This means: a single zero-support input kills the entire
inference chain.  In the integer semiring, $0$ is similarly
absorbing for multiplication: $0 \cdot n = 0$.

In both semirings, ``no evidence'' propagates forward as
``no conclusion.''  This is the correct behavior.
\end{proposition}

\begin{proposition}[The max operation handles ignorance via bypass]
The $\max$ in $(f_p(t))_j = \max_i \min(t_i, p_{ij})$ serves
as the bypass around ignorance:
\begin{itemize}
\item If $\min(t_1, p_{1j}) = 0$ (path through $i=1$ has no support),
  the $\max$ selects an alternative path through some $i=2$ where
  $\min(t_2, p_{2j}) > 0$.
\item Only if ALL paths give zero support does the output get
  zero support.
\end{itemize}
The $\max$ implements \emph{existential quantification}: there
EXISTS a supported path from input to output.  The $\min$
implements \emph{conjunction}: both the input evidence AND the
pattern must have support.
\end{proposition}


\section{The Probability of Zero}

\begin{definition}[Conditional probability with zero support]
For events in a binary ES $\{e, \bar{e}\}$ with supports
$s(e)$ and $s(\bar{e})$:
\begin{equation}\label{eq:prob}
P(e) = \begin{cases}
\displaystyle\frac{2^{s(e)}}{2^{s(e)} + 2^{s(\bar{e})}} & \text{if } s(e) + s(\bar{e}) > 0 \\[8pt]
\text{undefined} & \text{if } s(e) = s(\bar{e}) = 0
\end{cases}
\end{equation}
\end{definition}

In the first case ($s(e) > 0$ or $s(\bar{e}) > 0$), we have enough
evidence to form a belief.  The probability is determined by the
relative support.

In the second case ($s(e) = s(\bar{e}) = 0$), we have no evidence
whatsoever.  The probability is \emph{not} $1/2$ (which would
require a commitment to equiprobability---itself a positive claim).
It is simply undefined.  The system cannot make a prediction about
this event space.

\begin{proposition}[Three probability regimes]
\begin{enumerate}
\item $s(e) > 0, s(\bar{e}) = 0$: $P(e) = 1$ (maximal belief,
  \emph{but only because the alternative has no support}).
\item $s(e) = 0, s(\bar{e}) > 0$: $P(e) = 0$ (no belief,
  because the alternative has positive support).
\item $s(e) > 0, s(\bar{e}) > 0$: $P(e) = 2^{s(e)}/(2^{s(e)} + 2^{s(\bar{e})})$
  (calibrated belief based on evidence ratio).
\item $s(e) = 0, s(\bar{e}) = 0$: $P(e) = ?$ (no evidence,
  no belief, no probability).
\end{enumerate}
Cases (1) and (2) are \emph{conditional certainty}:
certainty relative to available evidence.  They are
defeasible---new evidence for the zero-support event would
immediately change the probability.

Case (3) is the normal case: uncertain belief proportional
to evidence.

Case (4) is \emph{ignorance}: the system has nothing to say.
\end{proposition}


\section{The Abduction Connection}

MJC notes: ``no certainty without evidence or belief or abduction.''
This triad corresponds to three sources of support:

\begin{definition}[Three sources of support]
\begin{enumerate}
\item \textbf{Evidence} (observation and inference): $s(e)$ increases when
  $e$ is directly observed OR when a pattern $p_{ij}$ connects a
  supported input $i$ to $e = j$.  Both observation (the learning
  function $\omega_0$) and inference (the update function $f_p$) are
  evidence: they derive support from data.

\item \textbf{Belief} (explicit choice): $s(e)$ increases by fiat---an
  axiom, a definition, or a postulate accepted without empirical
  support.  ``God exists,'' ``parallel lines never meet,'' or any
  stipulated starting point.  Belief is not inference; it is a
  choice to grant support where the data is silent.

\item \textbf{Abduction} (short-circuiting induction): $s(e)$ increases
  when a pattern has been observed enough times that the agent
  \emph{accepts} the pattern itself, rather than waiting for further
  confirmation.  Abduction is recognizing a regularity and committing
  to it---short-circuiting the infinite induction that would be
  required for certainty.  Often triggered by understanding \emph{why}
  the pattern holds.
\end{enumerate}
\end{definition}

All three increase support from 0 to positive.  None can
decrease support below 0.  And none can create support from
nothing: evidence requires data, belief requires a choice,
and abduction requires a recognized pattern.

\begin{proposition}[Certainty requires a path]
The only way to reach $P(e) = 0$ (event is certainly false)
is:
\begin{enumerate}
\item Observe or infer the ES-mate $\bar{e}$ from data (evidence), OR
\item Stipulate $\bar{e}$ as an axiom or definition (belief), OR
\item Recognize a pattern that entails $\bar{e}$ and commit to it
  (abduction).
\end{enumerate}
All three create positive support for $\bar{e}$, making $e$
``certainly false'' \emph{relative to the support for $\bar{e}$}.
Without any of these, $e$ is unknown, not false.
\end{proposition}


\section{Consequences for the Sat-RNN}

The 128-hidden tanh RNN has 128 binary event spaces.
At initialization (before any training):

\begin{itemize}
\item All SN strengths are effectively zero (random weights
  carry no evidence about the data).
\item The system is in state (4) for all 128 ESes: total ignorance.
\item The output prediction should be uniform over 256 bytes
  (maximum entropy given no evidence).
\end{itemize}

After training on $N$ bytes:
\begin{itemize}
\item Each neuron's ES has been supported by $\sim N/2$ observations
  (for both $e$ and $\bar{e}$, since both signs are observed).
\item The system is in state (3) for most ESes: calibrated belief.
\item Zero-support ESes do not exist in practice (every neuron
  takes both signs during training).
\end{itemize}

The distinction between ``no support'' and ``certainly false''
is thus primarily relevant at initialization and for rare events---but
it is precisely at these boundaries that the UM's epistemology
differs most from na\"ive probability, and where getting it right
matters for the theoretical framework.

\begin{remark}[The doubled-E and zero support]
In the doubled-E construction (CMP), each neuron $h_j$ creates
two events: $e_j$ (neuron fires, $h_j > 0$) and $\bar{e}_j$
(neuron does not fire, $h_j < 0$).  Positive weights become
patterns $e_i \to e_j$ and negative weights become patterns
$e_i \to \bar{e}_j$.  All pattern strengths are positive.

The doubling makes the ``no support'' distinction concrete:
$s(e_j) = 0$ does not mean ``$h_j$ is certainly negative.''
It means ``we have no evidence about $h_j$'s sign.''
Only positive support for $\bar{e}_j$ can make $h_j$
``certainly negative.''  The doubled-E ensures that the
evidence structure is always explicit.
\end{remark}


\section{Conclusions}

\begin{enumerate}
\item \textbf{Zero is not false.}  SN strength 0 means ``no support,''
  not ``certainly false.''  False requires positive support for
  the alternative.

\item \textbf{The UM is open-world.}  Absence of evidence is
  not evidence of absence.  The unsigned representation $[0, 255]$
  encodes this: you cannot have negative support.

\item \textbf{The $\min$ operation is cautious.}  It propagates
  ignorance correctly: no support in $\Rightarrow$ no support out.
  The probabilistic product ($P \cdot P$) is aggressive: it converts
  ignorance ($P = 0$) into false certainty.

\item \textbf{Certainty has three sources.}  Evidence (observation
  and inference from data), belief (explicit choice: axioms,
  definitions), and abduction (short-circuiting infinite induction
  on a recognized pattern) can all create support.  No other
  mechanism can.  ``No certainty without evidence or belief or
  abduction.''

\item \textbf{KN smoothing is a controlled closed-world commitment.}
  Adding $\delta > 0$ to all counts converts ignorance to minimal
  belief.  The parameter $\delta$ controls how much the system
  trusts the closed-world assumption.

\item \textbf{The tropical semiring is epistemically correct.}
  $(\max, \min)$ respects the open-world assumption:
  $\min$ demands evidence for every premise, $\max$ accepts
  any supported path.  The ring $(\sum, \times)$ of probabilities
  does not distinguish ``no support'' from ``certainly false,''
  collapsing the open-world epistemology into a closed-world one.
\end{enumerate}

\begin{thebibliography}{99}
\bibitem{cmp}
Michaeljohn Clement.
\newblock \emph{CMP}.
\newblock \url{https://cmpr.ai/cmp.pdf}, 2026.

\bibitem{bayes-counting}
Claude and MJC.
\newblock \emph{Bayes from Counting: Partial Quotients, GCD, and the
Symmetric Learning Function on $E = I \times O$}.
\newblock Hutter archive, 12 Feb 2026.

\bibitem{tropical-gcd}
Claude and MJC.
\newblock \emph{The Tropical--Integer GCD Bridge: When the Universal
Model Computes Exact Bayes}.
\newblock Hutter archive, 12 Feb 2026.
\end{thebibliography}

\end{document}
