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  9  \title{CL-STANCE-2 --- Epistemic Amplitude as Geometry (Deterministic Confidence Field)}
 10  \author{}
 11  \date{January 13, 2026}
 12  
 13  \begin{document}
 14  \maketitle
 15  
 16  \section{Purpose}
 17  
 18  Stance glyphs (epistemic overlays) are intentionally non-propositional: they color a construct rather than state a fact.
 19  CL-STANCE-2 makes that coloring \textbf{deterministic} by defining an amplitude field derived from geometry.
 20  
 21  No ``word meaning'' is assigned. Only a reproducible scalar is computed.
 22  
 23  \section{Stance Set}
 24  
 25  A stance instance is any instance whose glyph type is \textbf{stance} (currently):
 26  \[
 27  \{G17,\ G22,\ G36,\ G37,\ G38,\ G39,\ G40\}.
 28  \]
 29  (Inventory-dependent, but deterministic for a given inventory.)
 30  
 31  \section{Scope and Anchor}
 32  
 33  Within each scope \(S\) (root scope treated as \texttt{\_\_root\_\_}), define an \textbf{anchor instance} \(A(S)\) using the same rule as CL-NORM-3 layout anchoring:
 34  
 35  \begin{itemize}
 36    \item maximal scale \(s\) (descending),
 37    \item then type priority (object \(<\) operator \(<\) relation \(<\) stance \(<\) frame),
 38    \item then glyph id (ascending),
 39    \item then \((x,y)\) (ascending),
 40    \item then instance id (ascending).
 41  \end{itemize}
 42  
 43  This anchor is used purely as a geometric reference.
 44  
 45  \section{Ring Index}
 46  
 47  Let \(d\) be Euclidean distance between centers of a stance instance \(X\) and its anchor \(A(S)\).
 48  
 49  Let the ring step be the CL-NORM-3 constant \(R_{\text{step}} = 24\).
 50  
 51  Define:
 52  \[
 53  r(X) = \operatorname{round}\left(\frac{d}{R_{\text{step}}}\right)
 54  \]
 55  Thus \(r(X)\in\mathbb{N}\) and \(r=0\) means stance placed at the anchor center.
 56  
 57  \section{Phase Index}
 58  
 59  Rotation is quantized to \(0,45,\dots,315\) degrees. Define phase:
 60  \[
 61  p(X) = \left(\frac{\operatorname{rot}(X)}{45}\right) \bmod 8
 62  \]
 63  So \(p\in\{0,\dots,7\}\).
 64  
 65  \section{Repetition Count}
 66  
 67  Within a scope \(S\), define repetition count:
 68  \[
 69  k(X) = \#\{Y \in S \mid \text{glyph}(Y)=\text{glyph}(X) \text{ and } Y \text{ is stance}\}
 70  \]
 71  So \(k(X)\ge 1\).
 72  
 73  \section{Amplitude Computation}
 74  
 75  Amplitude is an integer in base-29 range \([0,28]\), denoted \(a_{29}(X)\).
 76  
 77  \subsection{Distance term}
 78  
 79  \[
 80  a_d(X) = \max(0,\ 28 - 3\cdot r(X))
 81  \]
 82  
 83  \subsection{Phase offset}
 84  
 85  Define a fixed phase offset table:
 86  \[
 87  \Delta_p = [0,2,4,6,6,4,2,0]
 88  \]
 89  so that:
 90  \[
 91  a_p(X) = \Delta_p[p(X)]
 92  \]
 93  
 94  \subsection{Repetition penalty}
 95  
 96  \[
 97  a_{29}(X) = \left\lfloor \frac{a_d(X) + a_p(X)}{k(X)} \right\rfloor
 98  \]
 99  clamped to \([0,28]\).
100  
101  \section{Marker Encoding}
102  
103  For convenience, decoders may render \(a_{29}\) as a marker glyph in \(\{G01..G29\}\) using:
104  
105  \[
106  \text{marker}(a_{29}) = G\,(1 + a_{29})
107  \]
108  
109  This marker is \textbf{not} a stance glyph; it is a numeric capsule convention shared with checksum encoding.
110  
111  \section{Output}
112  
113  CL-STANCE-2 produces a derived map:
114  \[
115  \text{stance\_field}: \text{stance\_id} \mapsto \{a_{29}, \text{marker}, r, p, k, \text{anchor\_id}\}.
116  \]
117  
118  No new required CLX document fields are introduced.
119  
120  \end{document}