% ==============================================================================
% preamble/macros.tex
% Mathematical Environments, Notations & Canonical Symbols of Cas 6
% ==============================================================================

% --- Color Palette for Mathematical Enclosures ---
\definecolor{ThmBlue}{RGB}{15, 65, 125}
\definecolor{ThmBg}{RGB}{247, 250, 255}
\definecolor{LemCyan}{RGB}{18, 105, 125}
\definecolor{LemBg}{RGB}{245, 252, 253}
\definecolor{PropPurple}{RGB}{90, 35, 110}
\definecolor{PropBg}{RGB}{252, 248, 254}
\definecolor{DefGreen}{RGB}{25, 100, 55}
\definecolor{DefBg}{RGB}{247, 253, 249}
\definecolor{ProofBorder}{RGB}{140, 150, 165}
\definecolor{ProofBg}{RGB}{254, 254, 255}
\definecolor{TodoBorder}{RGB}{200, 50, 25}
\definecolor{TodoBg}{RGB}{255, 250, 248}
\definecolor{ConormalBorder}{RGB}{20, 50, 100}
\definecolor{ConormalBg}{RGB}{248, 250, 254}
\definecolor{ConormalHeader}{RGB}{20, 50, 100}

% --- Shared Counter for Unified Continuous Math Statements ---
\newcounter{mathstatement}[section]
\renewcommand{\themathstatement}{\thesection.\arabic{mathstatement}}

% --- Hierarchical Equation Numbering (AMS/arXiv Standard: Chapter.Equation) ---
\numberwithin{equation}{section}

% ==============================================================================
% Pure Geometric Monochrome Mathematical Enclosures (Constructivist Semiotics)
% ==============================================================================

\newcommand{\gutterDist}{10pt}
\newcommand{\leadInText}{12pt}
\newcommand{\statementLineWidth}{0.6pt}

\newlength{\statementTitleWidth}
\newcommand{\formatStatementTitle}[2]{%
    \ifstrempty{#2}{%
        {\normalfont\bfseries #1 \themathstatement}%
    }{%
        \sbox0{{\normalfont\bfseries #1 \themathstatement}\hspace{6pt}\rule[0.55ex]{12pt}{\statementLineWidth}\hspace{6pt}}%
        \sbox2{\textmd{\itshape #2}}%
        \ifdim\wd2>\dimexpr\linewidth - \leadInText - 3pt - \wd0\relax
            \usebox0\begin{varwidth}[t]{\dimexpr\linewidth - \leadInText - 3pt - \wd0\relax}\raggedright\textmd{\itshape #2}\strut\end{varwidth}%
        \else
            \usebox0\usebox2%
        \fi
    }%
}

\newcommand{\headerDef}[1]{%
    \par\noindent%
    \begin{tikzpicture}[baseline={(0,0)}]
        \path (0, -0.15) rectangle (\linewidth, 0.35);
        \draw[overlay, line width=\statementLineWidth, black] (-\gutterDist, 0.08) -- (\leadInText, 0.08);
        \node[anchor=base west, inner sep=0pt] at (\leadInText + 3pt, 0) {%
            \formatStatementTitle{Definition}{#1}%
        };
    \end{tikzpicture}\par\vspace{3pt}%
}

\newcommand{\headerLem}[1]{%
    \par\noindent%
    \begin{tikzpicture}[baseline={(0,0)}]
        \path (0, -0.15) rectangle (\linewidth, 0.35);
        \draw[overlay, line width=\statementLineWidth, black] (-\gutterDist, -0.35) -- (-\gutterDist, 0.08) -- (\leadInText, 0.08);
        \node[anchor=base west, inner sep=0pt] at (\leadInText + 3pt, 0) {%
            \formatStatementTitle{Lemma}{#1}%
        };
    \end{tikzpicture}\par\vspace{3pt}%
}

\newcommand{\headerProp}[1]{%
    \par\noindent%
    \begin{tikzpicture}[baseline={(0,0)}]
        \path (0, -0.15) rectangle (\linewidth, 0.35);
        \draw[overlay, line width=\statementLineWidth, black] (-\gutterDist, -0.9) -- (-\gutterDist, 0.08) -- (\leadInText, 0.08);
        \node[anchor=base west, inner sep=0pt] at (\leadInText + 3pt, 0) {%
            \formatStatementTitle{Proposition}{#1}%
        };
    \end{tikzpicture}\par\vspace{3pt}%
}

\newcommand{\headerCor}[1]{%
    \par\noindent%
    \begin{tikzpicture}[baseline={(0,0)}]
        \path (0, -0.15) rectangle (\linewidth, 0.35);
        \draw[overlay, line width=\statementLineWidth, black] (-\gutterDist, -0.6) -- (-\gutterDist, 0.08) -- (\leadInText, 0.08);
        \node[anchor=base west, inner sep=0pt] at (\leadInText + 3pt, 0) {%
            \formatStatementTitle{Corollary}{#1}%
        };
    \end{tikzpicture}\par\vspace{3pt}%
}

\newcommand{\headerThm}[1]{%
    \par\noindent%
    \begin{tikzpicture}[baseline={(0,0)}]
        \path (0, -0.15) rectangle (\linewidth, 0.35);
        \draw[overlay, line width=\statementLineWidth, black] (-\gutterDist, -1.4) -- (-\gutterDist, 0.08) -- (\leadInText, 0.08);
        \node[anchor=base west, inner sep=0pt] at (\leadInText + 3pt, 0) {%
            \formatStatementTitle{Theorem}{#1}%
        };
    \end{tikzpicture}\par\vspace{3pt}%
}

\tcbset{
    closeSegment/.style={
        bottom=2pt,
        overlay unbroken and last={
            \draw[line width=\statementLineWidth, black] 
                ([xshift=-\leadInText]frame.south east) -- 
                ([xshift=\gutterDist]frame.south east);
        }
    },
    noCloseSegment/.style={
        bottom=1pt,
        overlay unbroken and last={}
    }
}

% --- Stylized Theorem Environment (thmbox) ---
% Usage: \begin{thmbox}[Optional Title]{label_key} ... \end{thmbox} (starred version * suppresses closing bracket)
\NewDocumentEnvironment{thmbox}{ s O{} m O{} }{%
    \refstepcounter{mathstatement}%
    \tcbset{thmboxstyle/.style={
        enhanced,
        breakable,
        frame hidden,
        interior hidden,
        before=\par\vspace{4pt},
        after=\par\vspace{4pt},
        top=0pt, bottom=2pt, left=0pt, right=0pt,
        fontupper=\rmfamily,
        overlay unbroken and last={
            \draw[line width=\statementLineWidth, black] 
                ([xshift=-\leadInText]frame.south east) -- 
                ([xshift=\gutterDist]frame.south east) -- 
                ([xshift=\gutterDist, yshift=1.4cm]frame.south east);
        }
    }}%
    \IfBooleanTF{#1}{\tcbset{thmboxstyle/.append style={noCloseSegment}}}{%
        \ifstrequal{#4}{noclose}{\tcbset{thmboxstyle/.append style={noCloseSegment}}}{%
            \ifstrequal{#4}{open}{\tcbset{thmboxstyle/.append style={noCloseSegment}}}{}%
        }%
    }%
    \begin{tcolorbox}[thmboxstyle]%
    \headerThm{#2}%
    \ifstrempty{#3}{}{\label{#3}}%
}{%
    \end{tcolorbox}%
}

% --- Stylized Lemma Environment (lemmabox) ---
% Usage: \begin{lemmabox}[Optional Title]{label_key} ... \end{lemmabox} (starred version * suppresses closing segment)
\NewDocumentEnvironment{lemmabox}{ s O{} m O{} }{%
    \refstepcounter{mathstatement}%
    \tcbset{lemboxstyle/.style={
        enhanced, breakable, frame hidden, interior hidden,
        before=\par\vspace{4pt}, after=\par\vspace{4pt},
        top=0pt, bottom=2pt, left=0pt, right=0pt,
        fontupper=\rmfamily,
        closeSegment
    }}%
    \IfBooleanTF{#1}{\tcbset{lemboxstyle/.append style={noCloseSegment}}}{%
        \ifstrequal{#4}{noclose}{\tcbset{lemboxstyle/.append style={noCloseSegment}}}{%
            \ifstrequal{#4}{open}{\tcbset{lemboxstyle/.append style={noCloseSegment}}}{}%
        }%
    }%
    \begin{tcolorbox}[lemboxstyle]%
    \headerLem{#2}%
    \ifstrempty{#3}{}{\label{#3}}%
}{%
    \end{tcolorbox}%
}

% --- Stylized Proposition Environment (propbox) ---
% Usage: \begin{propbox}[Optional Title]{label_key} ... \end{propbox} (starred version * suppresses closing segment)
\NewDocumentEnvironment{propbox}{ s O{} m O{} }{%
    \refstepcounter{mathstatement}%
    \tcbset{propboxstyle/.style={
        enhanced, breakable, frame hidden, interior hidden,
        before=\par\vspace{4pt}, after=\par\vspace{4pt},
        top=0pt, bottom=2pt, left=0pt, right=0pt,
        fontupper=\rmfamily,
        closeSegment
    }}%
    \IfBooleanTF{#1}{\tcbset{propboxstyle/.append style={noCloseSegment}}}{%
        \ifstrequal{#4}{noclose}{\tcbset{propboxstyle/.append style={noCloseSegment}}}{%
            \ifstrequal{#4}{open}{\tcbset{propboxstyle/.append style={noCloseSegment}}}{}%
        }%
    }%
    \begin{tcolorbox}[propboxstyle]%
    \headerProp{#2}%
    \ifstrempty{#3}{}{\label{#3}}%
}{%
    \end{tcolorbox}%
}

% --- Stylized Corollary Environment (corbox) ---
% Usage: \begin{corbox}[Optional Title]{label_key} ... \end{corbox} (starred version * suppresses closing segment)
\NewDocumentEnvironment{corbox}{ s O{} m O{} }{%
    \refstepcounter{mathstatement}%
    \tcbset{corboxstyle/.style={
        enhanced, breakable, frame hidden, interior hidden,
        before=\par\vspace{4pt}, after=\par\vspace{4pt},
        top=0pt, bottom=2pt, left=0pt, right=0pt,
        fontupper=\rmfamily,
        closeSegment
    }}%
    \IfBooleanTF{#1}{\tcbset{corboxstyle/.append style={noCloseSegment}}}{%
        \ifstrequal{#4}{noclose}{\tcbset{corboxstyle/.append style={noCloseSegment}}}{%
            \ifstrequal{#4}{open}{\tcbset{corboxstyle/.append style={noCloseSegment}}}{}%
        }%
    }%
    \begin{tcolorbox}[corboxstyle]%
    \headerCor{#2}%
    \ifstrempty{#3}{}{\label{#3}}%
}{%
    \end{tcolorbox}%
}

% --- Stylized Definition Environment (defbox) ---
% Usage: \begin{defbox}[Optional Title]{label_key} ... \end{defbox} (starred version * suppresses closing segment)
\NewDocumentEnvironment{defbox}{ s O{} m O{} }{%
    \refstepcounter{mathstatement}%
    \tcbset{defboxstyle/.style={
        enhanced, breakable, frame hidden, interior hidden,
        before=\par\vspace{4pt}, after=\par\vspace{4pt},
        top=0pt, bottom=2pt, left=0pt, right=0pt,
        fontupper=\rmfamily,
        closeSegment
    }}%
    \IfBooleanTF{#1}{\tcbset{defboxstyle/.append style={noCloseSegment}}}{%
        \ifstrequal{#4}{noclose}{\tcbset{defboxstyle/.append style={noCloseSegment}}}{%
            \ifstrequal{#4}{open}{\tcbset{defboxstyle/.append style={noCloseSegment}}}{}%
        }%
    }%
    \begin{tcolorbox}[defboxstyle]%
    \headerDef{#2}%
    \ifstrempty{#3}{}{\label{#3}}%
}{%
    \end{tcolorbox}%
}

% --- Stylized QED Symbol Aligned with Theorem Margins ---
\newlength{\qedSquareSize}
\setlength{\qedSquareSize}{6.5pt}

\renewcommand{\qedsymbol}{%
    \begin{tikzpicture}[baseline={(0,0)}]
        \useasboundingbox (0, 0pt) rectangle (\qedSquareSize, \qedSquareSize);
        \draw[line width=\statementLineWidth, black] 
            (\gutterDist, 0pt) rectangle (\gutterDist + \qedSquareSize, \qedSquareSize);
    \end{tikzpicture}%
}

% --- Stylized Proof Environment (proofbox) ---
% Usage: \begin{proofbox}[Optional Reference Heading] ... \end{proofbox}
\newenvironment{proofbox}[1][Proof]{%
    \begin{tcolorbox}[
        enhanced,
        breakable,
        frame hidden,
        interior hidden,
        before=\par\vspace{3pt},
        after=\par\vspace{4pt},
        top=0pt, bottom=2pt, left=0pt, right=0pt,
        fontupper=\small\rmfamily
    ]%
    \noindent{\itshape\color{black}Proof.\ }\ignorespaces%
}{%
    \par\hfill\qedsymbol%
    \end{tcolorbox}%
}

% --- Stylized Remark Environment (rembox) ---
% Usage: \begin{rembox}[Optional Title] ... \end{rembox}
\newenvironment{rembox}[1][]{%
    \begin{tcolorbox}[
        enhanced,
        breakable,
        frame hidden,
        interior hidden,
        before=\par\vspace{6pt},
        after=\par\vspace{6pt},
        top=2pt, bottom=2pt, left=0pt, right=0pt,
        overlay={
            \draw[line width=1.0pt, black] 
                ([xshift=-\gutterDist]frame.north west) -- 
                ([xshift=-\gutterDist]frame.south west);
        },
        fontupper=\small\rmfamily
    ]%
    \noindent\textbf{\itshape Remark\ifstrempty{#1}{}{~---~\textmd{\itshape #1}}.\,}\ignorespaces%
}{%
    \end{tcolorbox}%
}

% --- Dedicated Cartouche: Conormal Decomposition Signature (conormalbox) ---
% Usage: \begin{conormalbox}{System Name} ... \end{conormalbox}
\newenvironment{conormalbox}[1]{%
    \begin{tcolorbox}[
        enhanced,
        breakable,
        frame hidden,
        interior hidden,
        boxrule=0pt,
        top=4pt, bottom=4pt, left=0pt, right=0pt,
        before=\par\vspace{8pt},
        after=\par\vspace{8pt},
        overlay={
            \draw[line width=1.2pt, black] 
                ([xshift=-\gutterDist]frame.north west) -- 
                ([xshift=-\gutterDist]frame.south west);
        },
        fontupper=\small\rmfamily
    ]%
    \noindent{\bfseries $\blacktriangleright$~Normal Cone Decomposition: #1}\par\vspace{3pt}%
    \hrule height 0.6pt\vspace{6pt}%
}{%
    \end{tcolorbox}%
}

% --- Dedicated Cartouche: Algorithmic Specification (algobox) ---
% Usage: \begin{algobox}{Algorithm Title} ... \end{algobox}
\newcounter{algorithm}
\newenvironment{algobox}[1]{%
    \refstepcounter{algorithm}%
    \begin{tcolorbox}[
        enhanced,
        breakable,
        frame hidden,
        interior hidden,
        boxrule=0pt,
        top=4pt, bottom=4pt, left=0pt, right=0pt,
        before=\par\vspace{8pt},
        after=\par\vspace{8pt},
        overlay={
            \draw[line width=1.2pt, black] 
                ([xshift=-\gutterDist]frame.north west) -- 
                ([xshift=-\gutterDist]frame.south west);
        },
        fontupper=\small\rmfamily
    ]%
    \noindent{\bfseries $\blacktriangleright$~Algorithm~\thealgorithm: #1}\par\vspace{3pt}%
    \hrule height 0.6pt\vspace{6pt}%
}{%
    \end{tcolorbox}%
}

% --- Code-Quote Styled TODO Block for Drafting (todoblock) ---
% Usage: \begin{todoblock}{Descriptive Title of the Work to be Done} ... \end{todoblock}
\newenvironment{todoblock}[1]{%
    \begin{tcolorbox}[
        enhanced,
        breakable,
        colback=TodoBg,
        colframe=TodoBorder,
        arc=2pt,
        boxrule=0.5pt,
        leftrule=3.5pt,
        top=3pt, bottom=3pt, left=8pt, right=8pt,
        fonttitle=\small\ttfamily\bfseries\color{TodoBorder},
        title={>>> [TODO / DRAFTING DIRECTIVE: #1]},
        colbacktitle=TodoBg,
        attach title to upper={\par\vspace{2.5pt}\hrule height 0.4pt\vspace{2.5pt}},
        fontupper=\small\sffamily
    ]%
}{%
    \par\vspace{2pt}\hfill\textit{\tiny\color{TodoBorder}[Must be eliminated before final submission]}%
    \end{tcolorbox}%
}

% --- Standard Mathematical Fields ---
\newcommand{\R}{\mathbb{R}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\C}{\mathbb{C}}

% --- Phase Spaces & Symplectic Geometry ---
\newcommand{\Pext}{\mathcal{P}_{\mathrm{ext}}}
\newcommand{\omegaext}{\omega_{\mathrm{ext}}}
\newcommand{\Hext}{\mathcal{H}_{\mathrm{ext}}}
\newcommand{\omegaD}{\omega_0}
\newcommand{\SigmaNull}{\Sigma_0}

% --- Fulton-MacPherson & Stratified Geometry ---
\newcommand{\QFM}{\mathbf{Q}[N]}
\newcommand{\XFM}{X[N]}
\newcommand{\PiN}{\Pi(N)}
\newcommand{\LeafPairs}{\operatorname{LeafPairs}}
\newcommand{\Trees}{\operatorname{Trees}(N)}
\newcommand{\Mcoll}{\mathbf{M}_C}

% --- Invariants of the Lie-Bargmann Algebra ---
\newcommand{\Ptot}{\mathbf{P}_{\mathrm{tot}}}
\newcommand{\Ltot}{\mathbf{L}_{\mathrm{tot}}}
\newcommand{\Gboost}{\mathbf{G}}
\newcommand{\Mtot}{M_{\mathrm{tot}}}
\newcommand{\Rcdm}{\mathbf{R}_{\mathrm{cdm}}}

% --- Conormal Operators & Sundman Transformations ---
\newcommand{\Vapex}{V_{\mathrm{apex}}}
\newcommand{\VapexHier}{V_{\mathrm{apex}}^{\mathrm{hier}}}
\newcommand{\VapexScreened}{V_{\mathrm{apex}}^{\mathrm{screened}}}
\newcommand{\Bounce}{\mathcal{B}_C}
\newcommand{\Rext}{\mathcal{R}_{\mathrm{ext}}}
\newcommand{\CutoffTheta}{\Theta}

% --- Lie-Poisson & Differential Brackets ---
\newcommand{\poisson}[2]{\left\{ #1, #2 \right\}}
\newcommand{\extpoisson}[2]{\left\{ #1, #2 \right\}_{\mathrm{ext}}}
\newcommand{\hamvf}[1]{X_{#1}}
\newcommand{\inner}[2]{\left\langle #1, #2 \right\rangle}
\newcommand{\norm}[1]{\left\| #1 \right\|}
\newcommand{\abs}[1]{\left| #1 \right|}

% ==============================================================================
% Constructivist Monochrome Table of Contents Architecture (Single-Page Layout)
% ==============================================================================
\makeatletter

\renewcommand{\contentsname}{Table of Contents}

%Format section and subsection numbers in fixed-width boxes for hanging alignment
\def\tocsection#1#2#3{%
  \ifx\\#2\\%
    #3%
  \else
    \ifx\\#1\\%
      \hangindent 1.3em\hangafter=1\relax
      \hb@xt@ 1.3em{#2.\hfil}#3%
    \else
      \ignorespaces#1 #2.\hspace{0.45em}#3%
    \fi
  \fi
}
\def\tocsubsection#1#2#3{%
  \ifx\\#2\\%
    #3%
  \else
    \hb@xt@ 1.75em{#2.\hfil}#3%
  \fi
}

% Redefine \@starttoc: Constructivist monochrome header + pure black clickable links
\def\@starttoc#1#2{%
  \begingroup
  \hypersetup{linkcolor=black}%
  \setTrue{#1}%
  \par\removelastskip\vskip\z@skip
  \@nobreakfalse
  \leftskip\z@\rightskip\z@
  \parindent\z@\relax
  \begin{center}%
    \vspace*{-4pt}%
    {\large\bfseries\color{black} #2}%
    \par\vspace{3pt}%
    {\color{black}\rule{0.16\linewidth}{0.5pt}}%
    \par\vspace{6pt}%
  \end{center}%
  \makeatletter
  \@input{\jobname.#1}%
  \if@filesw
    \expandafter\newwrite\csname tf@#1\endcsname
    \immediate\openout \csname tf@#1\endcsname \jobname.#1\relax
  \fi
  \global\@nobreakfalse \endgroup
}

% Override amsart's \@tocline: 9.0pt/8.0pt monochrome typography with hanging indent
\def\@tocline#1#2#3#4#5#6#7{\relax
  \ifnum #1>\c@tocdepth % drop
  \else
    \ifnum #1=1 % Section (Chapter level)
        \par\addvspace{3.2pt plus 0.4pt minus 0.4pt}%
        \begingroup
            \parindent\z@ \leftskip\z@
            \rightskip 1.8em plus 1fil\relax \parfillskip -1.8em\relax
            \noindent
            {\bfseries\color{black}\fontsize{9.0}{10.5}\selectfont #6}%
            \nobreak\leaders\hbox to 0.48em{\hss\raisebox{0.12ex}{\tiny\color{BorderGray}.}\hss}\hfill\nobreak
            \hb@xt@ 1.8em{\hss{\bfseries\color{black}\fontsize{9.0}{10.5}\selectfont #7}}%
            \par\nobreak
        \endgroup
        \addvspace{0.5pt}%
    \else % Subsection level
        \par\addvspace{0.2pt}%
        \begingroup
            \parindent -1.75em\relax \leftskip 3.05em\relax
            \rightskip 1.8em plus 1fil\relax \parfillskip -1.8em\relax
            \noindent
            {\normalfont\color{black}\fontsize{8.0}{9.3}\selectfont #6}%
            \nobreak\leaders\hbox to 0.42em{\hss\raisebox{0.1ex}{\tiny\color{BorderGray}.}\hss}\hfill\nobreak
            \hb@xt@ 1.8em{\hss{\normalfont\color{black}\fontsize{8.0}{9.3}\selectfont #7}}%
            \par
        \endgroup
    \fi
  \fi
}

\makeatother