Universal Generative Deformation Calculus

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1 \begin{longtable}{L{0.17\textwidth}L{0.29\textwidth}L{0.43\textwidth}}
2 \caption{Worked finite realizations in the visual-atlas edition.}\label{tab:worked-realizations}\\
3 \toprule
4 Domain & Seed and directions & What the quotient, certificate, or completion reveals \\
5 \midrule
6 \endfirsthead
7 \toprule
8 Domain & Seed and directions & What the quotient, certificate, or completion reveals \\
9 \midrule
10 \endhead
11 Integer walk & $0$ with $+1$ and $-1$ & $2^n$ routes collapse to $n+1$ endpoints, while exact binomial route multiplicities remain available. \\
12 Polynomials & $x^2+1$ with differentiation, translation, reflection, difference, and multiplication & Commuting and involutive operator routes become exact quotient relations between polynomial endpoints. \\
13 Analytic functions & $\cosh x$ with derivative, reciprocal, logarithmic derivative, parity, square, and doubling & Hyperbolic families, poles, singular descendants, and invalid operations coexist in one audited graph. \\
14 Sequences & A Fibonacci prefix and the impulse $\delta_0$ with difference, summation, binomial transform, sign alternation, and reversal & Exact transform relations generate structured integer families rather than one undifferentiated list of examples. \\
15 Gaussian laws & $N(0,1)$ with shift, scale, reflection, convolution, centering, and standardization & The relation-reduced atlas lives in exact $(\mu,\sigma^2)$ coordinates and exposes state-dependent canonicalization. \\
16 Matrices & $\left(\begin{smallmatrix}1&1\\0&1\end{smallmatrix}\right)$ with transpose, square, adjugate, conjugation, symmetrization, and commutator & Exact trace, determinant, rank, and symmetry metadata distinguish invariant-preserving routes from genuine deformation. \\
17 Affine shapes & A rational triangle with rotations, reflections, shear, scale, and translations & Concrete descendant geometry remains recursively deformable, with exact coordinate identities and finite-order symmetries. \\
18 Graphs & $P_3$ with complement, line graph, subdivision, leaf addition, and relabeling & Different labeled adjacency matrices collapse to canonical unlabeled graph-isomorphism classes. \\
19 Quantum states & The projective seed $|0\rangle$ with exact Bloch actions of $H,S,X,Z$ & Exponentially many gate words saturate at six Pauli eigenstates after global phase has been quotiented out. \\
20 Congruence lifts & Roots of $x^2=1$ over $3^m$ and $x^2=0$ over $2^m$ & Simple roots lift uniquely, while singular roots acquire state-dependent branching and dead ends. \\
21 Completion and products & Summable signed dyadic increments and multi-input products & Infinite routes receive limits only after completion, while Catalan syntax trees collapse only after associativity is imposed. \\
22 \bottomrule
23 \end{longtable}

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