% Options for packages loaded elsewhere
\PassOptionsToPackage{unicode}{hyperref}
\PassOptionsToPackage{hyphens}{url}
%
\documentclass[
  11pt,
]{article}
\usepackage{amsmath,amssymb}
\usepackage{iftex}
\ifPDFTeX
  \usepackage[T1]{fontenc}
  \usepackage[utf8]{inputenc}
  \usepackage{textcomp} % provide euro and other symbols
\else % if luatex or xetex
  \usepackage{unicode-math} % this also loads fontspec
  \defaultfontfeatures{Scale=MatchLowercase}
  \defaultfontfeatures[\rmfamily]{Ligatures=TeX,Scale=1}
\fi
\usepackage{lmodern}
\ifPDFTeX\else
  % xetex/luatex font selection
  \setmainfont[]{DejaVu Serif}
  \setmonofont[]{DejaVu Sans Mono}
\fi
% Use upquote if available, for straight quotes in verbatim environments
\IfFileExists{upquote.sty}{\usepackage{upquote}}{}
\IfFileExists{microtype.sty}{% use microtype if available
  \usepackage[]{microtype}
  \UseMicrotypeSet[protrusion]{basicmath} % disable protrusion for tt fonts
}{}
\makeatletter
\@ifundefined{KOMAClassName}{% if non-KOMA class
  \IfFileExists{parskip.sty}{%
    \usepackage{parskip}
  }{% else
    \setlength{\parindent}{0pt}
    \setlength{\parskip}{6pt plus 2pt minus 1pt}}
}{% if KOMA class
  \KOMAoptions{parskip=half}}
\makeatother
\usepackage{xcolor}
\usepackage[margin=22mm]{geometry}
\usepackage{longtable,booktabs,array}
\usepackage{calc} % for calculating minipage widths
% Correct order of tables after \paragraph or \subparagraph
\usepackage{etoolbox}
\makeatletter
\patchcmd\longtable{\par}{\if@noskipsec\mbox{}\fi\par}{}{}
\makeatother
% Allow footnotes in longtable head/foot
\IfFileExists{footnotehyper.sty}{\usepackage{footnotehyper}}{\usepackage{footnote}}
\makesavenoteenv{longtable}
\setlength{\emergencystretch}{3em} % prevent overfull lines
\providecommand{\tightlist}{%
  \setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}
\setcounter{secnumdepth}{-\maxdimen} % remove section numbering
\ifLuaTeX
\usepackage[bidi=basic]{babel}
\else
\usepackage[bidi=default]{babel}
\fi
\babelprovide[main,import]{american}
\ifPDFTeX
\else
\babelfont{rm}[]{DejaVu Serif}
\fi
% get rid of language-specific shorthands (see #6817):
\let\LanguageShortHands\languageshorthands
\def\languageshorthands#1{}
\ifLuaTeX
  \usepackage{selnolig}  % disable illegal ligatures
\fi
\usepackage{bookmark}
\IfFileExists{xurl.sty}{\usepackage{xurl}}{} % add URL line breaks if available
\urlstyle{same}
\hypersetup{
  pdftitle={KURPISHEV LOGIC 2: Final Monograph of the Doctrine},
  pdfauthor={Ivan Borisovich Kurpishev - Independent Researcher, Kaliningrad - me@kurpishev.ru},
  pdflang={en-US},
  hidelinks,
  pdfcreator={LaTeX via pandoc}}

\title{KURPISHEV LOGIC 2: Final Monograph of the Doctrine}
\usepackage{etoolbox}
\makeatletter
\providecommand{\subtitle}[1]{% add subtitle to \maketitle
  \apptocmd{\@title}{\par {\large #1 \par}}{}{}
}
\makeatother
\subtitle{Volume V. Kurpishev Chemistry and Special FOS Reductions:
C@C\_chem, Rep\_chem, Evidence-D, Reactions, RBD Graphs and Biosystems}
\author{Ivan Borisovich Kurpishev - Independent Researcher, Kaliningrad
- me@kurpishev.ru}
\date{2026 - Assembly point:
KLT-DOCTRINE-FINAL-MONOGRAPH-VOLUME-V-CHEMISTRY-EN-v8.0}

\begin{document}
\maketitle

\section{Editorial passport of Volume
V}\label{editorial-passport-of-volume-v}

\textbf{Control point.}
\texttt{KLT-DOCTRINE-FINAL-MONOGRAPH-VOLUME-V-CHEMISTRY-EN-v8.0}.

\textbf{Volume title.} \emph{Volume V. Kurpishev Chemistry and Special
FOS Reductions: C@C\_chem, Rep\_chem, Evidence-D, Reactions, RBD Graphs
and Biosystems}.

\textbf{Author.} Ivan Borisovich Kurpishev, Independent Researcher,
Kaliningrad, \texttt{me@kurpishev.ru}.

\textbf{Status.} Volume V is the English fundamental edition of the
chemical branch of the Doctrine. It follows Volume IV, where the
physical reduction of FOS was fixed, and transfers the common core

\[
C@C\to Rep(R,I,U;D)\to \lambda\to Status\to RBD
\]

into the chemical domain.

\textbf{Editorial law.} This volume is not a chemistry cheat sheet and
not a compressed reference table. It is a fundamental doctrinal volume:
chemistry is treated as a reper-packet theory of substances, reactions,
conditions, sufficient foundations and special reductions of FOS.

\textbf{Boundary of claims.} Classical chemistry, atomic structure, the
periodic system, valency, chemical bonding, pH, solubility,
thermodynamics, kinetics, spectroscopy and analytical chemistry are not
claimed as authorial discoveries. The authorial contribution consists in
placing these layers into a single KLT/RBD normalization: substance,
formula, reaction, ionic graph, material, solubility, pH,
thermodynamics, kinetics, spectroscopy and analysis become a chain of
verifiable Reper nodes.

\section{Abstract}\label{abstract}

Volume V introduces \textbf{Kurpishev Chemistry} as a reper-packet
reduction of Fundamental Support Connectivity. A chemical world is not
defined as a flat list of substances and not as a collection of reaction
equations. It is a special domain

\[
\mathcal C_K=\Theta_{chem}(FOS;D_{reaction}).
\]

The minimal chemical unit is event@state:

\[
C@C_{chem}=(e_{chem},s_{chem}),
\]

where \(e_{chem}\) is a chemical event - bond formation, bond breaking,
electron transfer, protonation, precipitation, phase change, catalytic
act or spectral registration - and \(s_{chem}\) is a state -
composition, concentration, phase, temperature, pressure, pH, charge,
solubility, material, medium and instrumental situation.

The chemical Reper has the form

\[
Rep_{chem}(X)=(R_X,I_X,U_X;D_X),
\]

where \(R_X\) is an established chemical fact, \(I_X\) is a formula,
valency, charge, configuration or identifier, \(U_X\) is the field of
possible compounds, reactions and conditions, and \(D_X\) is the
sufficient foundation: table, experiment, procedure, medium, source,
mass-charge balance, thermodynamic or kinetic condition, or spectral
confirmation.

The main thesis is

\[
Truth_{chem}(X)
\Longleftrightarrow
Dom_{chem}(X)\land D_X\land cr(U_X,I_X;R_X,D_X)=-1.
\]

If the domain, medium, foundation, mass-charge balance, phase, pH,
temperature, source or method is missing, the chemical node does not
receive truth-status. It receives \texttt{gap}, \texttt{candidate},
\texttt{hypothesis} or \texttt{needs-Evidence-D}.

\section{Authorial constructions fixed in Volume
V}\label{authorial-constructions-fixed-in-volume-v}

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  Kurpishev Chemistry as a reduction of FOS.
\item
  Chemical object as \(C@C_{chem}\).
\item
  Chemical Reper \(Rep_{chem}(X)=(R_X,I_X,U_X;D_X)\).
\item
  Evidence-D as sufficient chemical foundation.
\item
  Reaction as Reper transition.
\item
  Ionic graph as the proof form of a reaction in solution.
\item
  Mass and charge balance as a minimal Dom/D admission condition.
\item
  Chemical CGI as an index of proof-chain hole density.
\item
  KLT-RBD-CHEM as a graph of substances, formulas, reactions,
  conditions, sources and statuses.
\item
  Special FOS reductions: DNA, biosystems, materials, culture and art.
\end{enumerate}

\section{Source frame and prior-art
boundary}\label{source-frame-and-prior-art-boundary}

The internal basis of Volume V is formed by Monograph 5.0, the Master2
editorial branch, the FOS journal article, PILOT-01, KLT-RBD-CHEM v0.1,
the KLT-RBD-CHEM publication/site bundle v1.2, the restored chemistry
branch of the website and the preceding volumes of the final monograph.

The classical chemical background is a table-reference corpus: atomic
structure, the periodic system, chemical bonding, classes of substances,
solubility, acidity, electrochemistry, organic classes, laboratory
features and analytical reactions. This background is used as a source
of domains and Evidence-D, but it is not merged with the authorial
contribution.

The mathematical-logical base remains unchanged:

\[
x\mapsto C@C_x\mapsto Rep_x(R,I,U;D)\mapsto \lambda_x\mapsto Status_x\mapsto RBD_x.
\]

\section{Formal dictionary of
notation}\label{formal-dictionary-of-notation}

\begin{longtable}[]{@{}
  >{\raggedright\arraybackslash}p{(\columnwidth - 2\tabcolsep) * \real{0.5000}}
  >{\raggedright\arraybackslash}p{(\columnwidth - 2\tabcolsep) * \real{0.5000}}@{}}
\toprule\noalign{}
\begin{minipage}[b]{\linewidth}\raggedright
Symbol
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Meaning
\end{minipage} \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
\(C@C_{chem}\) & chemical event@state \\
\(e_{chem}\) & chemical event \\
\(s_{chem}\) & chemical state \\
\(Rep_{chem}\) & chemical Reper quadruple \\
\(R_X\) & established chemical fact \\
\(I_X\) & formula, name, valency, charge, configuration, identifier \\
\(U_X\) & field of possible compounds, reactions, phases and
conditions \\
\(D_X\) & sufficient chemical foundation \\
\(D_{reaction}\) & reaction foundation: medium, temperature, pH,
catalyst, procedure \\
Evidence-D & documentary and experimental support of a chemical node \\
\(\lambda_{chem}\) & projective-harmonic coordination indicator of the
Reper quadruple \\
\(\delta_{truth}\) & truth defect \(|\lambda+1|\) \\
\(CGI_{chem}\) & chemical gap index \\
\(RBD_{chem}\) & database of chemical Reper nodes and links \\
\(W_{chem}\) & chemical world as a FOS reduction \\
\end{longtable}

\section{Part I. Chemistry as a Reper
domain}\label{part-i.-chemistry-as-a-reper-domain}

\section{1. Chemical object}\label{chemical-object}

\subsection{1.1 Substance as event@state}\label{substance-as-eventstate}

A classical substance is often described by a formula: \(H_2O\),
\(NaCl\), \(CO_2\), \(Fe_2O_3\). In the KLT/RBD frame the formula is not
enough. The same formula may have different states: gas, liquid,
solution, crystal, hydrate, ion form, complex, radical or intermediate
product.

Therefore we introduce

\[
Substance_X=C@C_{chem}(X)=(e_X,s_X).
\]

The event \(e_X\) fixes the act of appearance or registration of the
substance. The state \(s_X\) fixes phase, medium, concentration,
temperature, pressure, pH, charge and observation procedure.

\textbf{Definition 1.1.} A chemical object is a pair

\[
X_{chem}=(e_X,s_X),
\]

where \(e_X\) is a chemical event and \(s_X\) is a chemical state. The
object is admissible only if \(Dom_{chem}(X)\) is specified.

\textbf{Example 1.2.} The notation \(HCl\) without state is an
incomplete chemical flag. In the gas phase it denotes hydrogen chloride.
In water it denotes hydrochloric acid with ionization into \(H_3O^+\)
and \(Cl^-\). Hence

\[
HCl(g)\neq HCl(aq),
\]

and the two nodes receive different \(C@C_{chem}\) records, different
Evidence-D and different RBD links.

\subsection{1.2 Formula as invariant but not as complete
object}\label{formula-as-invariant-but-not-as-complete-object}

The formula enters the component \(I_X\):

\[
I_X=\text{formula/name/charge/configuration}.
\]

But a chemical fact is not exhausted by a formula. A formula without
state is an invariant label, not a Reper node. Truth-status requires

\[
(R_X,I_X,U_X;D_X).
\]

\textbf{Proposition 1.3.} A chemical formula is a necessary but
insufficient part of a chemical Reper.

\section{2. Chemical Reper}\label{chemical-reper}

\subsection{2.1 Definition}\label{definition}

\textbf{Definition 2.1.} The chemical Reper of an object \(X\) is the
quadruple

\[
Rep_{chem}(X)=(R_X,I_X,U_X;D_X).
\]

Its components are

\[
R_X=\text{established substance, reaction, measurement or property},
\]

\[
I_X=\text{formula, name, valency, charge, configuration, index},
\]

\[
U_X=\text{field of possible compounds, reactions, phases, pH and media},
\]

\[
D_X=\text{table, experiment, balance, procedure, source, instrument, medium}.
\]

\subsection{2.2 The fourth point
Evidence-D}\label{the-fourth-point-evidence-d}

The component \(D_X\) is not externally appended to chemistry. It is the
fourth point without which the triple \(R_X,I_X,U_X\) remains
incomplete.

In chemistry this is critical. A reactional conclusion cannot be
assigned truth-status if it does not state:

\begin{itemize}
\tightlist
\item
  the medium of the reaction;
\item
  the phase of substances;
\item
  whether mass and charge are balanced;
\item
  whether a catalyst is present;
\item
  whether temperature and pressure are admissible;
\item
  how the product is confirmed;
\item
  whether there is spectral, tabular or laboratory support.
\end{itemize}

Thus

\[
(R_X,I_X,U_X)\not\Rightarrow Truth_{chem}(X).
\]

Only

\[
(R_X,I_X,U_X;D_X)\land Dom_{chem}(X)
\Rightarrow
\text{admission to }\lambda\text{-verification}.
\]

\section{3. Chemical truth}\label{chemical-truth}

\subsection{3.1 Lambda closure}\label{lambda-closure}

For a chemical Reper the projective-harmonic indicator is written

\[
\lambda_{chem}(X)=\frac{(U_X-R_X)(I_X-D_X)}{(U_X-D_X)(I_X-R_X)}.
\]

The truth defect is

\[
\delta_{truth}^{chem}(X)=|\lambda_{chem}(X)+1|.
\]

\textbf{Definition 3.1.} A chemical node has truth-status only when

\[
Dom_{chem}(X)\land D_X\land \lambda_{chem}(X)=-1.
\]

The formula is not a numeric fetish. It is a structural requirement:
reality, invariant, universe of possibilities and sufficient foundation
must form a harmonic Reper configuration.

\subsection{3.2 Statuses}\label{statuses}

\begin{longtable}[]{@{}
  >{\raggedright\arraybackslash}p{(\columnwidth - 2\tabcolsep) * \real{0.5000}}
  >{\raggedright\arraybackslash}p{(\columnwidth - 2\tabcolsep) * \real{0.5000}}@{}}
\toprule\noalign{}
\begin{minipage}[b]{\linewidth}\raggedright
Status
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Meaning
\end{minipage} \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
\texttt{truth-status} & domain, foundation, balance and harmonic closure
are present \\
\texttt{gap-domain} & the chemical domain is missing or ambiguous \\
\texttt{gap-foundation} & sufficient foundation is missing \\
\texttt{needs-balance} & mass or charge is not balanced \\
\texttt{needs-medium} & medium, phase, pH or temperature is
unspecified \\
\texttt{candidate} & formula or reaction is plausible but not
authorized \\
\texttt{hypothesis} & the node is an open chemical conjecture \\
\texttt{needs-Evidence-D} & documentation or experiment is required \\
\end{longtable}

\section{Part II. Substance, reaction and ionic
graph}\label{part-ii.-substance-reaction-and-ionic-graph}

\section{4. Substance node}\label{substance-node}

A substance node in KLT-RBD-CHEM is not only a name. It is a record

\[
Node_{sub}(X)=
(id_X,formula_X,name_X,phase_X,charge_X,medium_X,D_X,Status_X).
\]

The same formula may give several nodes:

\begin{longtable}[]{@{}lll@{}}
\toprule\noalign{}
Formula & Node & Difference \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
\(H_2O\) & liquid water & phase and standard conditions \\
\(H_2O\) & ice & solid phase and lattice state \\
\(H_2O\) & vapor & gas phase and temperature-pressure domain \\
\(H_2O\) & solvent node & medium and solvation role \\
\(H_2O\) & product node & reaction output with evidence \\
\end{longtable}

\textbf{Definition 4.1.} A substance node is complete when formula,
state, domain and sufficient foundation are present.

\section{5. Reaction as Reper
transition}\label{reaction-as-reper-transition}

A classical reaction equation

\[
A+B\to C+D
\]

is insufficient in the KLT/RBD frame. A reaction must be treated as a
transition of Reper states:

\[
Rep_{chem}(A,B;s_0)\xrightarrow{\Delta,\Xi,\Upsilon;D_{reaction}}Rep_{chem}(C,D;s_1).
\]

The operator \(\Delta\) fixes the initiating act: contact, heating,
irradiation, mixing, catalyst insertion, voltage or pressure. The
operator \(\Xi\) fixes the process: bond rearrangement, electron
transfer, diffusion, nucleation, phase shift. The operator \(\Upsilon\)
fixes the product-state turn: precipitation, gas release, complex
formation, color change, spectral peak or stabilized material.

\textbf{Definition 5.1.} A reaction is a Reper transition if it contains
initial state, process state, product state and sufficient reaction
foundation.

\subsection{5.1 Balance admission}\label{balance-admission}

The minimal admission test is

\[
Balance_{mass}(r)=1\land Balance_{charge}(r)=1.
\]

If either condition is absent, the reaction does not enter
truth-verification. It receives \texttt{needs-balance}.

\textbf{Example 5.2.} The ionic equation

\[
Ag^+(aq)+Cl^-(aq)\to AgCl(s)
\]

is admissible only with medium, ionic species, precipitate state and
source/procedure of recognition. The equation is not merely symbolic; it
is a Reper transition from solution-state ions to a solid precipitate
node.

\section{6. Ionic graph}\label{ionic-graph}

For solution chemistry the natural proof object is not only a formula
but an ionic graph:

\[
G_{ion}=(V_{ion},E_{trans},D_{medium}).
\]

Here \(V_{ion}\) is the set of ionic nodes, \(E_{trans}\) is the set of
transfer, precipitation, protonation, complexation and redox edges, and
\(D_{medium}\) is the medium foundation.

\textbf{Definition 6.1.} An ionic graph is admissible if every edge has
medium, charge, phase and Evidence-D.

The ionic graph prevents a common error: a formula equation may look
balanced but may be chemically false in the given medium. KLT-RBD-CHEM
therefore checks the graph, not only the string.

\section{7. pH, solubility and phase
gates}\label{ph-solubility-and-phase-gates}

\subsection{7.1 pH gate}\label{ph-gate}

Many chemical statements are true only in a pH domain. We write

\[
Dom_{pH}(X)=[pH_{min},pH_{max}].
\]

A node without \(Dom_{pH}\) is incomplete when acidity or alkalinity
controls the result.

\textbf{Proposition 7.1.} If a reaction depends on protonation,
hydrolysis, precipitation or acid-base equilibrium, then pH is part of
\(D_{reaction}\).

\subsection{7.2 Solubility gate}\label{solubility-gate}

Solubility is an admission condition for solution reactions. Let

\[
Sol(X;T,pH,medium)\in\{soluble,slightly\ soluble,insoluble,conditional\}.
\]

If solubility is unknown, the product-state cannot be fixed.

\subsection{7.3 Phase gate}\label{phase-gate}

The phase gate is written

\[
Phase(X)\in\{s,l,g,aq,crystal,amorphous,complex,intermediate\}.
\]

A chemical equation without phase information is a formula-chain
candidate, not yet a verified Reper transition.

\section{8. Thermodynamics and
kinetics}\label{thermodynamics-and-kinetics}

Thermodynamics and kinetics are not interchangeable. Thermodynamics
answers whether a transition is energetically admissible. Kinetics
answers whether and how fast it occurs in the stated domain.

Thus a reaction node must distinguish

\[
D_{therm}=\{\Delta G,\Delta H,\Delta S,K,T,p\},
\]

from

\[
D_{kin}=\{rate,mechanism,activation,order,catalyst,medium\}.
\]

\textbf{Principle 8.1.} A thermodynamic possibility does not imply
kinetic realization. A kinetic observation does not by itself identify
the complete thermodynamic Reper.

This principle blocks the replacement of proof by plausibility. If a
reaction is only favorable but no mechanism or evidence is known, its
status remains conditional.

\section{9. Spectroscopy and analytical
chemistry}\label{spectroscopy-and-analytical-chemistry}

Spectral and analytical evidence form a privileged subclass of
Evidence-D:

\[
D_{spectral}=\{IR,UV,MS,NMR,Raman,XRD,chromatography,colorimetric,qualitative\ test\}.
\]

\textbf{Definition 9.1.} A spectral Reper is a quadruple

\[
Rep_{spec}(X)=(Signal_X,Assignment_X,Alt_X;D_{instrument}).
\]

Here \(Signal_X\) is the observed signal, \(Assignment_X\) is the
proposed chemical assignment, \(Alt_X\) is the universe of alternative
assignments, and \(D_{instrument}\) is the instrument/procedure
foundation.

\textbf{Theorem 9.2.} A spectral signal alone does not authorize a
chemical identity unless the alternative assignments and instrumental
domain are stated.

\textbf{Proof.} The signal is the \(R\) component. The assignment is the
\(I\) component. Alternative assignments form \(U\). The instrumental
protocol and reference library form \(D\). Without \(U\) and \(D\), the
Reper is incomplete. Therefore truth-status is unavailable. \(\square\)

\section{Part III. KLT-RBD-CHEM}\label{part-iii.-klt-rbd-chem}

\section{10. Database schema}\label{database-schema}

KLT-RBD-CHEM is the graph form of Volume V. Its minimal schema is

\[
RBD_{chem}=(Substances,Reactions,Conditions,Evidence,Statuses,Edges).
\]

A substance table stores formula, names, state, charge, phase, hazard
domain, source and status. A reaction table stores reactants, products,
coefficients, conditions, medium, balance checks, evidence and status.
The edge layer links substances to reactions, reactions to evidence,
evidence to sources and statuses to audit rules.

\subsection{10.1 Minimal tables}\label{minimal-tables}

\begin{longtable}[]{@{}ll@{}}
\toprule\noalign{}
Table & Function \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
\texttt{chem\_substance} & formula, state, phase, charge, identifiers \\
\texttt{chem\_reaction} & reactants, products, coefficients,
conditions \\
\texttt{chem\_condition} & pH, T, pressure, solvent, catalyst, medium \\
\texttt{chem\_evidence} & tables, experiments, spectra, methods,
sources \\
\texttt{chem\_edge} & typed graph links \\
\texttt{chem\_status} & truth, gap, candidate, hypothesis, needs-D \\
\end{longtable}

\section{11. Chemical CGI}\label{chemical-cgi}

The chemical gap index measures the hole-density of a chemical chain:

\[
CGI_{chem}(r)=\frac{h_{Dom}+h_D+h_{balance}+h_{medium}+h_{phase}+h_{source}}{n_{claims}+\epsilon}.
\]

Here the numerator records missing domain, missing foundation, missing
balance, missing medium, missing phase and missing source. The
denominator normalizes by the number of chemical claims.

\textbf{Definition 11.1.} A chain is chemically stable if

\[
CGI_{chem}(r)<1
\]

and every critical node has Evidence-D. A chain with \(CGI_{chem}\ge 1\)
requires repair before publication or database ingestion.

\section{12. Import and audit route}\label{import-and-audit-route}

Every chemical text, table or laboratory note is imported by the route

\[
Text\to Extract\to Normalize\to C@C_{chem}\to Rep_{chem}\to Audit\to RBD_{chem}.
\]

The audit stage performs the following checks:

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  domain check;
\item
  sufficient foundation check;
\item
  formula normalization;
\item
  mass balance;
\item
  charge balance;
\item
  phase/medium check;
\item
  pH and solubility gate;
\item
  evidence and source check;
\item
  lambda-harmonic consistency;
\item
  status assignment.
\end{enumerate}

\section{Part IV. Special FOS
reductions}\label{part-iv.-special-fos-reductions}

\section{13. DNA as chemical-informational
Reper}\label{dna-as-chemical-informational-reper}

DNA is not reduced to a mere chemical formula and not detached from
chemistry as pure information. In the KLT/RBD frame it is a
chemical-informational Reper:

\[
Rep_{DNA}=(R_{mol},I_{seq},U_{bio};D_{lab}).
\]

Here \(R_{mol}\) is the molecular realization, \(I_{seq}\) is the
sequence/invariant, \(U_{bio}\) is the field of possible biological
expressions and mutations, and \(D_{lab}\) is laboratory and sequencing
evidence.

\textbf{Proposition 13.1.} DNA is a special FOS reduction because it
simultaneously holds material bonding, sequence invariance, biological
possibility and evidence foundation.

\section{14. Biosystems}\label{biosystems}

A biosystem is a layered chemical Reper:

\[
BioSys=(Chem,Cell,Organism,Environment;D_{bio}).
\]

The chemical layer alone does not define the organism, but no biological
statement is evidence-stable without chemical carriers, conditions and
measurement domain.

KLT-RBD-CHEM therefore treats metabolism, enzymatic catalysis, membrane
transport and signaling as Reper transitions with chemical, biological
and environmental conditions.

\section{15. Materials and technology}\label{materials-and-technology}

A material is not only composition. It is a stateful chemical object:

\[
Material_X=(Composition_X,Structure_X,Process_X,Use_X;D_X).
\]

The same composition can produce different materials under different
synthesis, thermal, mechanical and surface conditions. Therefore a
material node requires process Evidence-D.

\textbf{Example 15.1.} Iron oxide can be an analytical precipitate, a
pigment, a corrosion product or a structural phase. The formula alone
does not decide the Reper.

\section{16. Culture and art}\label{culture-and-art}

Chemical Evidence-D enters culture and art through pigments, binders,
corrosion, restoration, dating, conservation and material analysis. But
a cultural object is not reduced to chemical composition.

We write

\[
Rep_{art}=(R_{object},I_{form},U_{interpret};D_{material}+D_{cultural}).
\]

The chemical layer supplies \(D_{material}\). The cultural layer
supplies \(D_{cultural}\). The two foundations must not be confused.

\section{Part V. Theorems and proof
protocol}\label{part-v.-theorems-and-proof-protocol}

\section{17. Central theorem of chemical
reduction}\label{central-theorem-of-chemical-reduction}

\textbf{Theorem 17.1.} Let \(FOS\) be Fundamental Support Connectivity,
let \(\Theta_{chem}\) be an admissible chemical reducing functor, and
let \(D_{reaction}\) be a sufficient foundation of the chemical domain.
Then the chemical world

\[
W_{chem}=\Theta_{chem}(FOS;D_{reaction})
\]

is a Reper-realized chemical world if and only if for every chemical
object \(X\in W_{chem}\) there exist

\[
C@C_{chem}(X),\quad Rep_{chem}(X)=(R_X,I_X,U_X;D_X),
\]

and

\[
Dom_{chem}(X)\land D_X\land cr(U_X,I_X;R_X,D_X)=-1.
\]

\textbf{Proof.} Necessity follows because a chemical object without
event@state is only a name or a formula, not a realized object. The
Reper quadruple adds real content, invariant, field of possibilities and
sufficient foundation. Without \(D_X\) a chemical chain cannot
distinguish an admissible product from a formal symbol. Sufficiency
follows because \(C@C_{chem}\), \(Rep_{chem}\), domain and harmonic
closure give the object local fixation, possible transitions and
verifiable status. Therefore \(\Theta_{chem}\) produces not a random
list of substances, but a compatible system of chemical Reper
assemblies. \(\square\)

\section{18. Chemical gap theorem}\label{chemical-gap-theorem}

\textbf{Theorem 18.1.} Any chemical formula-chain in which at least one
component is absent - \(Dom_{chem}\), \(D_{reaction}\), mass balance,
charge balance or Evidence-D - does not receive truth-status and is
assigned gap-status.

\textbf{Proof.} A chemical formula-chain is a special case of general
formula-chain audit. General audit requires a domain and sufficient
foundation. In chemistry the domain is realized through medium, phase,
temperature, pH, charge, mass, source and procedure. The absence of any
mandatory component makes the chain underdetermined. \(\square\)

\section{19. Reaction as Reper-transition
theorem}\label{reaction-as-reper-transition-theorem}

\textbf{Theorem 19.1.} A reaction \(A+B\to C+D\) is not a simple arrow
between formulas but a Reper transition between states if and only if
the operator triple

\[
\Delta_{reaction},\quad \Xi_{process},\quad \Upsilon_{product}
\]

and the sufficient foundation of the transition are specified.

\textbf{Proof.} A reaction has beginning, process and product-state. The
beginning is fixed by \(\Delta_{reaction}\). The change of composition
and bonds is fixed by \(\Xi_{process}\). Translation into a new product
state is fixed by \(\Upsilon_{product}\). Without one of these
components the reaction either does not start, does not have a process,
or does not fix the product-state. \(\square\)

\section{20. Proof protocol of Volume
V}\label{proof-protocol-of-volume-v}

Every chemical statement in Volume V must follow the route

\[
Claim_{chem}\to Dom?\to D?\to Balance?\to Evidence?\to \lambda?\to Status.
\]

If the route is not completed, the statement remains in one of the
working statuses: \texttt{gap}, \texttt{candidate}, \texttt{hypothesis},
\texttt{needs-source}, \texttt{needs-balance}, \texttt{needs-lab-D}.

\section{Appendix A. Insertion map for the final
monograph}\label{appendix-a.-insertion-map-for-the-final-monograph}

\begin{longtable}[]{@{}llll@{}}
\toprule\noalign{}
insert\_id & Place & Title & Status \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
V-CH-01 & Volume V, Part I & Chemistry as Reper domain & ready \\
V-CH-02 & Volume V, Part II & Substance and reaction & ready \\
V-CH-03 & Volume V, Part III & KLT-RBD-CHEM and Evidence-D & ready \\
V-CH-04 & Volume V, Part IV & DNA, biosystems, materials & ready \\
V-CH-05 & Volume V, appendix & Tables and proof-status & ready \\
\end{longtable}

\section{Appendix B. Formula index}\label{appendix-b.-formula-index}

\begin{longtable}[]{@{}
  >{\raggedright\arraybackslash}p{(\columnwidth - 2\tabcolsep) * \real{0.5000}}
  >{\raggedright\arraybackslash}p{(\columnwidth - 2\tabcolsep) * \real{0.5000}}@{}}
\toprule\noalign{}
\begin{minipage}[b]{\linewidth}\raggedright
Formula
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Function
\end{minipage} \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
\(\mathcal C_K=\Theta_{chem}(FOS;D_{reaction})\) & chemistry as FOS
reduction \\
\(C@C_{chem}=(e_{chem},s_{chem})\) & minimal chemical unit \\
\(Rep_{chem}(X)=(R_X,I_X,U_X;D_X)\) & chemical Reper \\
\(Truth_{chem}\Longleftrightarrow Dom\land D\land cr=-1\) &
truth-status \\
\(G_{ion}=(V_{ion},E_{trans},D_{medium})\) & ionic graph \\
\(CGI_{chem}\) & chemical gap index \\
\(Rep_{DNA}\) & DNA as chemical-informational Reper \\
\end{longtable}

\section{Appendix C. Index of authorial
concepts}\label{appendix-c.-index-of-authorial-concepts}

\begin{itemize}
\tightlist
\item
  Kurpishev Chemistry.
\item
  \(C@C_{chem}\).
\item
  \(Rep_{chem}\).
\item
  Evidence-D.
\item
  Chemical CGI.
\item
  Ionic Reper graph.
\item
  Reaction as Reper transition.
\item
  Chemical gap audit.
\item
  KLT-RBD-CHEM.
\item
  Special FOS reductions.
\end{itemize}

\section{Bibliography and source
supports}\label{bibliography-and-source-supports}

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  Kurpishev I. B. \emph{Monograph 5.0: Kurpishev Logic. Non-associative
  packet Reper logic, NAPG 3.0, V}P physics, anthropology of turn and
  KLT/RBD applications*. Kaliningrad, 2026.
\item
  Kurpishev I. B. \emph{PILOT-01: Reper-projective architecture of
  formula chains}. 2026.
\item
  Kurpishev I. B. \emph{Kurpishev FOS, Desargues-Kurpishev Theorem,
  truth harmony and extended PN.2}. 2026.
\item
  Kurpishev I. B. \emph{KLT-RBD-CHEM STRUCTURE v0.1}. 2026.
\item
  Kurpishev I. B. \emph{KLT-RBD-CHEM SITE/MONOGRAPH/PUBLICATION BUNDLE
  v1.2}. 2026.
\item
  Stakheev A. Yu. \emph{All Chemistry in 50 Tables}. Moscow, 1998.
\item
  Ponarin Ya. P. \emph{Affine and Projective Geometry}. Moscow, 2009.
\item
  Bourbaki N. \emph{The Architecture of Mathematics}. Mathematical
  Enlightenment, issue 5, 1960.
\item
  Rashevsky P. K. \emph{Riemannian Geometry and Tensor Analysis}.
  Moscow, 1967.
\item
  Arnold V. I. \emph{Geometry of Complex Numbers, Quaternions and
  Spins}. Moscow, 2002.
\end{enumerate}

\section{Conclusion}\label{conclusion}

Volume V fixes Kurpishev Chemistry as an autonomous layer of the
Doctrine. Its central result is the transfer of chemistry from the mode
of lists of substances and reactions into the mode of a proof-connected
Reper graph. In this frame a substance receives event@state, a formula
enters the invariant component, a reaction becomes a Reper transition,
Evidence-D becomes the fourth point of sufficient foundation, and
truth-status is assigned only after verification of domain, foundation,
balance, source and harmonic closure.

Next control point:

\texttt{KLT-DOCTRINE-FINAL-MONOGRAPH-VOLUME-VI-KLT-RBD-SOFTWARE-LEGAL-RU-v8.1}.

\end{document}
