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\begin{document}

\title{BLLM: Beyond the LLM with Deep Human-Mind-AI}

\author{Deep Bhattacharjee\,\orcidlink{0000-0003-0466-750X}}
\email{itsdeep@live.com}
\thanks{$*$\,Corresponding author. Second address: \nolinkurl{d.bhattacharjee@erl-forschung.de}.}
\affiliation{Formerly, Electro-Gravitational Space Propulsion Laboratory (EGSPL), Bhubaneswar, Odisha 751030, India}
\affiliation{Formerly, University of Calcutta, Senate House, 87/1 College Street, Kolkata 700073, West Bengal, India}

\author{Ushashi Bhattacharya\,\orcidlink{0009-0002-2254-3914}}
\email{rai.bhattacharya.in@gmail.com}
\affiliation{Formerly, National Taiwan University, Taipei 106319, Taiwan}

\date{October 19, 2026}

\begin{abstract}
\itshape
A system that improves itself keeps every change that passes a check, so a check that makes errors lets mistakes pile up. We compute the least checking that keeps such a system sound. It is an entropy of the costs of the changes, and an endless chain of improvements stays sound at finite cost exactly when that entropy is finite. A stack of checked layers costs more than its layers would cost alone. We also compute how many guesses a proposer needs before a right one appears: human authors come close to this least number, and language models need $1.5$ to $4.3$ times as many. Anaxia, an engine built on these laws, rebuilt the published 2016 counterexample to Steinberg's conjecture from the rules of a colouring problem alone. The engine is a work in progress, and its rules and search are being revised and extended for the open problems it has not yet settled.
\par\medskip\upshape\noindent\textit{2020 Mathematics Subject Classification.} 68T05; 68T20; 94A17; 60J80; 05C15.
\end{abstract}

\keywords{recursive self-improvement; verification; entropy; automated discovery}

\maketitle
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\tableofcontents

\input{sections/sec01_intro}
\input{sections/sec02_history}
\input{sections/sec02b_related}
\input{sections/sec03_model}
\input{sections/sec04_entropy}
\input{sections/sec05_huffman}
\input{sections/sec06_renyi}
\input{sections/sec07_chains}
\input{sections/sec08_singularity}
\input{sections/sec08b_online}
\input{sections/sec08c_agents}
\input{sections/sec08d_gap}
\input{sections/sec09_branching}
\input{sections/sec09b_towers}
\input{sections/sec12_network}
\input{sections/sec10c_benchmark}
\input{sections/sec10d_experiments}
\input{sections/sec10g_construct}
\input{sections/sec10f_hodge}
\input{sections/sec11_status}

\appendix
\input{sections/appA_derivations}
\input{sections/appB_checks}
\input{sections/appC_hardware}
\input{sections/appD_sources}
\input{sections/sec08e_brains}
\input{sections/sec10_scaling}
\input{sections/sec10b_cases}

\input{sections/declarations}

\input{sections/bibliography}

\end{document}