\section{The Recursive Kernel}
\label{sec:kernel}

The ordered triad becomes a world-building mechanism only when it closes the
loop from current facts through action to successor facts.

\subsection{From fact cut to successor fact cut}

Let $F_t$ be the declared fact cut available at time $t$. A participant forms
a purpose-bound trust assessment $T_t$, decides whether to cooperate through
$C_t$, and performs an action whose real occurrence is $E_{t,t+1}$. Evidence
from that occurrence is reviewed and admitted into a successor fact cut:

\begin{equation}
F_t
\xrightarrow{\mathrm{assessment}} T_t
\xrightarrow{\mathrm{value,choice}} C_t
\xrightarrow{\mathrm{action}} E_{t,t+1}
\xrightarrow{\mathrm{review,admission}} F_{t+1}.
\label{eq:recursive-kernel}
\end{equation}

$E_{t,t+1}$ is not identical to its record, and its record does not
automatically become accepted fact. The distinction prevents an actor, tool,
or model from defining reality solely through its own report. Admission may
accept, reject, qualify, or leave evidence unresolved. What matters is that
the path remains inspectable enough for later correction.

The kernel has three useful properties.

\paragraph{Persistence.}
Another participant can identify the prior cut, assessment, commitment, and
result without recovering a private conversation.

\paragraph{Correction.}
New consequence can revise trust, expose a bad decision, or invalidate an
object model while preserving what the earlier cut actually contained.

\paragraph{Generation.}
Cooperation does not terminate in agreement. It creates new actions,
artifacts, failures, counterexamples, and commitments that expand the world
available to later participants.

Facts are therefore not merely the beginning of the chain. They are also the
governed return point. Trust is not merely confidence. It is the bridge that
makes a fact cut actionable. Cooperation is not merely social harmony. It is
the generative transition through which the world can change.

\subsection{Cross-session continuation}

Return to the agent task from Section~\ref{sec:problem}. A strong continuation
implements the loop as follows:

\begin{enumerate}[leftmargin=*]
  \item The ending session leaves authoritative source coordinates, changed
        artifacts, check results, the unresolved failure, known omissions, and
        responsibility boundaries. The transcript may be evidence, but is not
        silently promoted into the whole fact world.
  \item A fresh participant checks a bounded completion claim against those
        facts. The result may be warning rather than pass because one failure
        remains unresolved.
  \item The participant sees the value of continuing, the allowed actions,
        the blocked or approval-gated actions, and the party responsible for
        the next decision.
  \item The participant chooses a bounded continuation, performs it, and
        leaves new source changes, observations, receipts, and verification
        results.
  \item Review admits those results into a successor cut without rewriting
        the first session's evidence boundary.
\end{enumerate}

The session can disappear because it is not the identity of the work. The
continuation does not require blind trust in the first agent, because trust is
reconstructed from retained facts. The second participant does not merely
obey a next-step prompt, because value and constraints are inspectable. The
result becomes material for a third participant. Continuity is thus an outcome
of the entire triad, not of storage alone.

\subsection{Why this is a foundation for KFD}

Every later KFD can introduce new objects, procedures, or domain profiles, but
it must re-enter this kernel. A new object needs a stable fact identity. Claims
about it need purpose-bound assessment. Its use with reasoning participants
needs trusted value and visible constraint. Its action produces consequences
that must return through review rather than self-certification.

This dependency explains foundation without requiring the first paper to
catalog every later KFD. The triad is a generative kernel, not a compressed
table of contents.
