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Geometry definitions
Definitions for similarity dimension, box counting, generalized dimensions, singularity spectra, Poincaré distance, hyperbolic curvature, Laplace–Beltrami modes, inverse participation ratio, and the $E_{47}$ embedding boundary are consolidated in Fractals and Hyperbolic Geometry.
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Linguistic entities and notation lock
Definitions for $\Sigma,K,\Psi,\Gamma,\Lambda,\Omega,\mathcal I,\mathcal M$, the lexical flow $A_t$, symbol matrix $w$, pre-grammar space $V^*$, and the distinction between historical $J/M$ notation and current Interpretation/Mnemosyne notation are maintained in the full Invariant Grammar monograph.
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Skyrmion constructs added
Definitions and status boundaries for $U$, $L_i$, $B$, $\mathbf n$, $Q$, $\mathcal O_{v_0}$, $M_{ab}(v_0)$, $c_4$, Derrick stationarity, and the RI correspondence are collected on Skyrmions.
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Computational definitions and implementation status
The canonical numerical realizations of $C$, $K$, $E_{47}$, $P_{47}$, the complement $Q$, contraction maps, spectra, residuals, and synthetic reference objects are catalogued in Python and Numerical Validation.
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Entities and Constructs
Canonical definitions for the K47/125 formalism and its invariant grammar.
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$V_2$ is the spin-2 irreducible representation of $SU(2)$, with
$\dim V_2=5.$
The canonical carrier is
$\Sigma:=V_2^{\otimes3},\qquad \dim\Sigma=125.$
$`J_a^{mathrm{tot}}=J_aotimes Iotimes I+Iotimes J_aotimes I+Iotimes Iotimes J_a,$$
for $a\in\{x,y,z\}$.
$C=(J_x^{\mathrm{tot}})^2+(J_y^{\mathrm{tot}})^2+(J_z^{\mathrm{tot}})^2.$ It is Hermitian and acts by $j(j+1)$ on the spin-$j$ sector.
$\Sigma\cong V_0\oplus3V_1\oplus5V_2\oplus4V_3\oplus3V_4\oplus2V_5\oplus V_6.$ The sector dimensions are $(1,9,25,28,27,22,13).$
$K=(C-6I)(C-30I).$ It annihilates exactly the $j=2$ and $j=5$ isotypic sectors.
$`E_{47}:=ker K=W_2oplus W_5cong5V_2oplus2V_5,$$
$\dim E_{47}=47.$