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%  The Phase Transition of Language
%  A doctoral thesis (design-science monograph)
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  pdftitle={The Phase Transition of Language},
  pdfauthor={Sean Chatman},
  pdfsubject={The convergence of LSP, MCP, and A2A into a universal law-state runtime},
  pdfkeywords={process mining, conformance checking, LSP, MCP, A2A, agentic AI, OCEL},
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\title{The Phase Transition of Language}
\author{Sean Chatman}
\date{June 2026}

% ============================================================================
\begin{document}
% ============================================================================

\pagenumbering{roman}

% ---------------------------------------------------------------- Title page
\begin{titlepage}
  \centering
  \vspace*{1.2cm}
  {\scshape\large Doctoral Thesis\par}
  \vspace{1.0cm}
  {\Huge\bfseries The Phase Transition of Language\par}
  \vspace{0.9cm}
  {\large\itshape On the Convergence of the Language Server, Model Context,\\
   and Agent-to-Agent Protocols into a Universal Law-State Runtime\par}
  \vspace{0.4cm}
  {\large A Process-Mining Vision for 2030\par}
  \vspace{1.6cm}
  {\large Sean Chatman\par}
  \vspace{0.4cm}
  {\normalsize ChatmanGPT\par}
  \vfill
  {\normalsize Prepared for the consideration of\par}
  \vspace{0.2cm}
  {\large Prof.\ dr.\ ir.\ Wil M.\ P.\ van der Aalst\par}
  \vspace{1.2cm}
  {\normalsize June 2026\par}
  \vspace{0.6cm}
  {\footnotesize Workspace artifact under study: \lspmax{} (CalVer 26.6.18).\par}
\end{titlepage}

% ---------------------------------------------------------------- Abstract
\chapter*{Abstract}
\addcontentsline{toc}{chapter}{Abstract}
\markboth{Abstract}{Abstract}

The Language Server Protocol (LSP) decoupled editors from language intelligence,
collapsing an $M \times N$ integration burden into $M + N$ and turning code
understanding into a reusable service \citep{lsp-redhat-2016,lsp-official}.
A decade later, two further protocols have repeated this move at higher layers:
the Model Context Protocol (MCP), open-sourced by Anthropic in November~2024,
standardised how models reach tools, data, and context \citep{mcp-anthropic-2024};
and the Agent-to-Agent protocol (A2A), published by Google in April~2025,
standardised how autonomous agents discover and coordinate with one another
\citep{a2a-google-2025}. This thesis argues that the fusion of these three
protocols---LSP for meaning, MCP for context, A2A for coordination---constitutes
not an incremental integration but a \emph{phase transition} in machine-mediated
language, analogous to the roughly $1{,}600$--$1{,}700$-fold volume expansion when
water becomes steam at its boiling point \citep{phys-illinois-steam,nationalboard-steam}.
The input that drives the transition is standardisation: a large, discontinuous
investment---a latent heat---absorbed at apparent constant ``temperature'' that,
once paid, expands the addressable surface of language by orders of magnitude.

We develop the argument through the lens of process mining, the discipline
founded by Wil van der Aalst, in which behaviour recorded as event logs is checked
for conformance against a model of how it ought to unfold
\citep{pm-manifesto-2011,vdaalst-book-2016}. We contend that when LSP is
generalised beyond source code to all forms of language---natural language,
legal and regulatory text, mathematics, biological sequence, music, and business
process---each such language acquires an event log and therefore a conformance
surface. Echoing van der Aalst's recent dictum that there is ``no AI without PI''
(process intelligence), in which object-centric process mining is ``the missing
link connecting data and processes'' \citep{noai-nopi-2025}, we argue that there
is no trustworthy conformance for all language without an object-centric event
abstraction \citep{ocel2-2024}.

We instantiate the argument in \lspmax, a law-state runtime projected through LSP
whose \code{ConformanceVector} keeps \admitted, \refused, and \unknownstat{} law
axes strictly distinct, whose diagnostics are emitted as object-centric events
(OCEL~2.0), and whose claims are graded only by bounded status \citep{lspmax}.
Using design-science methodology \citep{hevner2004,peffers2007}, we present the
artifact, a fusion architecture, and a 2030 roadmap. Throughout, we adopt the
artifact's own epistemic discipline: we grade each claim as \admitted, \refused,
or \unknownstat; we mark every projection as a \forecast; and we decline the
language of triumph.

\vfill
\noindent\textbf{Keywords:} process mining; conformance checking; object-centric
event logs; Language Server Protocol; Model Context Protocol; Agent-to-Agent
protocol; agentic AI; law-state runtime; protocol convergence.

% ---------------------------------------------------------- Epistemic note
\chapter*{A Note on Method and Language}
\addcontentsline{toc}{chapter}{A Note on Method and Language}
\markboth{A Note on Method and Language}{A Note on Method and Language}

This thesis studies an artifact, \lspmax, whose constitution forbids
\emph{victory language}---unbounded declarations such as ``solved'' or
``g{u}aranteed''---and instead requires every claim to carry a \emph{bounded
status} and, where admission is asserted, a \emph{receipt}: a content-addressed
record binding a digest, a boundary, and a negative-control result
\citep{lspmax}. It would be inconsistent to describe such an artifact in the
register it refuses. We therefore adopt its discipline as our own. Findings are
graded \admitted{} (evidence present and checked), \refused{} (evidence present,
contrary), or \unknownstat{} (admissibility not yet determinable); intermediate
grades \candidate, \partialstat, and \openstat{} are used where appropriate. The
three states never collapse into one another: an \unknownstat{} is never quietly
promoted to \admitted. Numerical predictions about 2030 are commercial or
analyst forecasts, not measurements, and are marked \forecast. Quotations from
primary specifications are reproduced verbatim and cited to the revision in force
at the time of writing; the reader is cautioned that living specifications
(MCP, A2A, and LSP~3.18 among them) continue to evolve.

% ---------------------------------------------------------------- Contents
\cleardoublepage
\tableofcontents
\listoffigures
\listoftables

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\pagenumbering{arabic}
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%
%  Chapters are maintained as separate files under chapters/ and included
%  below. Each \input is uncommented as the chapter is written, so that the
%  document always compiles with whatever content currently exists.
%
\input{chapters/01-introduction}            % Ch 1: Introduction; thesis statement; RQs
\input{chapters/02-background}              % Ch 2: Process mining; LSP; MCP; A2A
\input{chapters/02b-recent-advances}        %        (closing section of Ch 2: recent arXiv 2025-2026)
\input{chapters/02a-math-foundations}       % Ch 3: Mathematical foundations (algebra/logic/analysis/geometry)
\input{chapters/03-conceptual-framework}    % Ch 4: Language as process; phase transition
\input{chapters/04-methodology}             % Ch 5: Design science; bounded-status epistemics
\input{chapters/05-artifact}                % Ch 6: The lsp-max artifact
\input{chapters/06-fusion}                  % Ch 7: The LSP x MCP x A2A fusion architecture
\input{chapters/07-all-language}            % Ch 8: Conformance checking for all language
\input{chapters/08-vision-2030}             % Ch 9: The phase transition realised; roadmap
\input{chapters/09-discussion}              % Ch 10: RQs revisited; threats to validity
\input{chapters/10-conclusion}              % Ch 11: Contributions; future work
% NOTE: chapter numbers follow \input order, not file names.
%
% ============================================================================

% ---------------------------------------------------------------- References
\cleardoublepage
\printbibliography[heading=bibintoc,title={References}]

% ---------------------------------------------------------------- Appendices
\appendix
\input{chapters/A-method-catalog}           % App A: the max/* method catalogue
\input{chapters/B-law-table}                % App B: laws and bounded-status vocabulary
\input{chapters/C-timeline}                 % App C: protocol timeline 2016--2030

\end{document}
