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Universal Generative Deformation Calculus
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273 symbol spellings in 357 mathematical passages across 9 selected source files. Subscripts and superscripts have their own entries alongside their base symbols.
The same symbol can have several meanings here. A passage classified as a definition may define another part of its expression. Read the linked context before assigning a meaning.
All 273 entries are available below.
=Definition passage · 92 passages
Search aliases: =
Definition passage: Direction bundle · Universal_Generative_Deformation_Calculus.tex, line 175
[Direction bundle] A direction bundle over X is a map π: D⟶ X. The fiber D_x=π^-1(x) is the complete set of directions admitted by the declared grammar at x.
Exact mathematical passage
\Dcal_x=\pi^{-1}(x)
ADefinition passage · 12 passages
Search aliases: A
Definition passage: Generated family certificate · Universal_Generative_Deformation_Calculus.tex, line 744
[Generated family certificate] A generated family certificate is a tuple F=(Θ,F,{σ_d}_d∈ A, Q, I), where [label=(alph*)] item Θ is a parameter space item F:Θ→ O(x_0) assigns an object class to every parameter item A is a collection of closure directions item σ_d:Θ→Θ satisfies F(σ_d(θ))=d· F(θ) for every declared closure direction item Q is a set of exact relations or recurrences item I is a set of invariants shared or predictably transformed across the family
Exact mathematical passage
\Fcal=(\Theta,F,\{\sigma_d\}_{d\in A},\mathcal Q,\mathcal I),
dDefinition passage · 42 passages
Search aliases: d
Definition passage: Elementary deformation law · Universal_Generative_Deformation_Calculus.tex, line 186
[Elementary deformation law] An elementary deformation law is a map Φ: D⟶ X with the understanding that d∈ D_x starts at x and ends at Φ(d). We also write x⋆ d=Φ(d).
Exact mathematical passage
d\in\Dcal_x
d_1Definition passage · 3 passages
Search aliases: d_1
Definition passage: Admissible path · Universal_Generative_Deformation_Calculus.tex, line 230
[Admissible path] An admissible path of length n from x_0 is a sequence γ=(d_1,…,d_n) for which x_k=x_k-1⋆ d_k, d_k∈ D_x_k-1, 1≤ k≤ n. Its endpoint is ev(γ)=x_n. The empty path ∅ has endpoint x_0.
Exact mathematical passage
\gamma=(d_1,\ldots,d_n)
d_kDefinition passage · 5 passages
Search aliases: d_k
Definition passage: Admissible path · Universal_Generative_Deformation_Calculus.tex, line 234
[Admissible path] An admissible path of length n from x_0 is a sequence γ=(d_1,…,d_n) for which x_k=x_k-1⋆ d_k, d_k∈ D_x_k-1, 1≤ k≤ n. Its endpoint is ev(γ)=x_n. The empty path ∅ has endpoint x_0.
Exact mathematical passage
x_k=x_{k-1}\star d_k, \qquad d_k\in\Dcal_{x_{k-1}}, \qquad 1\le k\le n.
d_nDefinition passage · 2 passages
Search aliases: d_n
Definition passage: Admissible path · Universal_Generative_Deformation_Calculus.tex, line 230
[Admissible path] An admissible path of length n from x_0 is a sequence γ=(d_1,…,d_n) for which x_k=x_k-1⋆ d_k, d_k∈ D_x_k-1, 1≤ k≤ n. Its endpoint is ev(γ)=x_n. The empty path ∅ has endpoint x_0.
Exact mathematical passage
\gamma=(d_1,\ldots,d_n)
\DcalDefinition passage · 18 passages
Search aliases: Dcal D mathcal
Definition passage: Direction bundle · Universal_Generative_Deformation_Calculus.tex, line 171
[Direction bundle] A direction bundle over X is a map π: D⟶ X. The fiber D_x=π^-1(x) is the complete set of directions admitted by the declared grammar at x.
Exact mathematical passage
\pi:\Dcal\longrightarrow\Xcal.
\Dcal_xDefinition passage · 11 passages
Search aliases: Dcal_x D_x Dcal mathcal
Definition passage: Direction bundle · Universal_Generative_Deformation_Calculus.tex, line 175
[Direction bundle] A direction bundle over X is a map π: D⟶ X. The fiber D_x=π^-1(x) is the complete set of directions admitted by the declared grammar at x.
Exact mathematical passage
\Dcal_x=\pi^{-1}(x)
\Dcal_{x_{k-1}}Definition passage · 1 passage
Search aliases: Dcal_x_k-1 D_x_k-1 Dcal mathcal
Definition passage: Admissible path · Universal_Generative_Deformation_Calculus.tex, line 234
[Admissible path] An admissible path of length n from x_0 is a sequence γ=(d_1,…,d_n) for which x_k=x_k-1⋆ d_k, d_k∈ D_x_k-1, 1≤ k≤ n. Its endpoint is ev(γ)=x_n. The empty path ∅ has endpoint x_0.
Exact mathematical passage
x_k=x_{k-1}\star d_k, \qquad d_k\in\Dcal_{x_{k-1}}, \qquad 1\le k\le n.
\evDefinition passage · 6 passages
Search aliases: ev operatorname
Definition passage: Admissible path · Universal_Generative_Deformation_Calculus.tex, line 241
[Admissible path] An admissible path of length n from x_0 is a sequence γ=(d_1,…,d_n) for which x_k=x_k-1⋆ d_k, d_k∈ D_x_k-1, 1≤ k≤ n. Its endpoint is ev(γ)=x_n. The empty path ∅ has endpoint x_0.
Exact mathematical passage
\ev(\gamma)=x_n
FDefinition passage · 11 passages
Search aliases: F
Definition passage: Generated family certificate · Universal_Generative_Deformation_Calculus.tex, line 744
[Generated family certificate] A generated family certificate is a tuple F=(Θ,F,{σ_d}_d∈ A, Q, I), where [label=(alph*)] item Θ is a parameter space item F:Θ→ O(x_0) assigns an object class to every parameter item A is a collection of closure directions item σ_d:Θ→Θ satisfies F(σ_d(θ))=d· F(θ) for every declared closure direction item Q is a set of exact relations or recurrences item I is a set of invariants shared or predictably transformed across the family
Exact mathematical passage
\Fcal=(\Theta,F,\{\sigma_d\}_{d\in A},\mathcal Q,\mathcal I),
\FcalDefinition passage · 1 passage
Search aliases: Fcal F mathcal
Definition passage: Generated family certificate · Universal_Generative_Deformation_Calculus.tex, line 744
[Generated family certificate] A generated family certificate is a tuple F=(Θ,F,{σ_d}_d∈ A, Q, I), where [label=(alph*)] item Θ is a parameter space item F:Θ→ O(x_0) assigns an object class to every parameter item A is a collection of closure directions item σ_d:Θ→Θ satisfies F(σ_d(θ))=d· F(θ) for every declared closure direction item Q is a set of exact relations or recurrences item I is a set of invariants shared or predictably transformed across the family
Exact mathematical passage
\Fcal=(\Theta,F,\{\sigma_d\}_{d\in A},\mathcal Q,\mathcal I),
\mathrm{fin}Definition passage · 6 passages
Search aliases: fin mathrm
Definition passage: Quotient deformation graph · Universal_Generative_Deformation_Calculus.tex, line 306
[Quotient deformation graph] The quotient deformation graph is O(x_0)= T_fin(x_0)/∼. Its vertices are relation classes of paths. An edge is induced by one admissible extension.
Exact mathematical passage
\boxed{ \Ocal(x_0)=\Tcal_{\mathrm{fin}}(x_0)/\sim. }
\GcalDefinition passage · 2 passages
Search aliases: Gcal G mathcal
Definition passage: Typed generative deformation datum · Universal_Generative_Deformation_Calculus.tex, line 203
[Typed generative deformation datum] A typed generative deformation datum is a tuple G= ( T, X,π: D→ X,Φ, R,τ,κ, V), where R is a path relation system, τ is a completion structure, κ: D→[0,∞) is a direction cost, and V is a validity predicate.
Exact mathematical passage
\boxed{ \Gcal= (\mathsf T,\Xcal,\pi:\Dcal\to\Xcal,\Phi,\mathcal R,\tau,\kappa,\mathcal V), }
\mathcal IDefinition passage · 2 passages
Search aliases: I mathcal
Definition passage: Generated family certificate · Universal_Generative_Deformation_Calculus.tex, line 744
[Generated family certificate] A generated family certificate is a tuple F=(Θ,F,{σ_d}_d∈ A, Q, I), where [label=(alph*)] item Θ is a parameter space item F:Θ→ O(x_0) assigns an object class to every parameter item A is a collection of closure directions item σ_d:Θ→Θ satisfies F(σ_d(θ))=d· F(θ) for every declared closure direction item Q is a set of exact relations or recurrences item I is a set of invariants shared or predictably transformed across the family
Exact mathematical passage
\Fcal=(\Theta,F,\{\sigma_d\}_{d\in A},\mathcal Q,\mathcal I),
kDefinition passage · 39 passages
Search aliases: k
Definition passage: Admissible path · Universal_Generative_Deformation_Calculus.tex, line 234
[Admissible path] An admissible path of length n from x_0 is a sequence γ=(d_1,…,d_n) for which x_k=x_k-1⋆ d_k, d_k∈ D_x_k-1, 1≤ k≤ n. Its endpoint is ev(γ)=x_n. The empty path ∅ has endpoint x_0.
Exact mathematical passage
x_k=x_{k-1}\star d_k, \qquad d_k\in\Dcal_{x_{k-1}}, \qquad 1\le k\le n.
nDefinition passage · 83 passages
Search aliases: n
Definition passage: Direction-complete expansion · Universal_Generative_Deformation_Calculus.tex, line 217
[Direction-complete expansion] An expansion rule is direction-complete relative to G at x when it exposes every element of D_x. It is direction-complete through depth n when this holds at every endpoint of every admissible path of length less than n.
Exact mathematical passage
n
\OcalDefinition passage · 4 passages
Search aliases: Ocal O mathcal
Definition passage: Quotient deformation graph · Universal_Generative_Deformation_Calculus.tex, line 306
[Quotient deformation graph] The quotient deformation graph is O(x_0)= T_fin(x_0)/∼. Its vertices are relation classes of paths. An edge is induced by one admissible extension.
Exact mathematical passage
\boxed{ \Ocal(x_0)=\Tcal_{\mathrm{fin}}(x_0)/\sim. }
\mathcal QDefinition passage · 3 passages
Search aliases: Q mathcal
Definition passage: Generated family certificate · Universal_Generative_Deformation_Calculus.tex, line 744
[Generated family certificate] A generated family certificate is a tuple F=(Θ,F,{σ_d}_d∈ A, Q, I), where [label=(alph*)] item Θ is a parameter space item F:Θ→ O(x_0) assigns an object class to every parameter item A is a collection of closure directions item σ_d:Θ→Θ satisfies F(σ_d(θ))=d· F(θ) for every declared closure direction item Q is a set of exact relations or recurrences item I is a set of invariants shared or predictably transformed across the family
Exact mathematical passage
\Fcal=(\Theta,F,\{\sigma_d\}_{d\in A},\mathcal Q,\mathcal I),
\mathcal RDefinition passage · 2 passages
Search aliases: R mathcal
Definition passage: Typed generative deformation datum · Universal_Generative_Deformation_Calculus.tex, line 203
[Typed generative deformation datum] A typed generative deformation datum is a tuple G= ( T, X,π: D→ X,Φ, R,τ,κ, V), where R is a path relation system, τ is a completion structure, κ: D→[0,∞) is a direction cost, and V is a validity predicate.
Exact mathematical passage
\boxed{ \Gcal= (\mathsf T,\Xcal,\pi:\Dcal\to\Xcal,\Phi,\mathcal R,\tau,\kappa,\mathcal V), }
rDefinition passage · 5 passages
Search aliases: r
Definition passage: Operadic perturbation hyperforest · Universal_Generative_Deformation_Calculus.tex, line 546
[Operadic perturbation hyperforest] The operadic perturbation hyperforest generated by a seed family is the set of all finite rooted typed trees whose leaves are seeds and whose internal vertices are labeled by admissible operations Φ or Ψ_r. Evaluation proceeds from leaves to root.
Exact mathematical passage
\Psi_r
\mathsf TDefinition passage · 5 passages
Search aliases: T mathsf
Definition passage: Typed generative deformation datum · Universal_Generative_Deformation_Calculus.tex, line 203
[Typed generative deformation datum] A typed generative deformation datum is a tuple G= ( T, X,π: D→ X,Φ, R,τ,κ, V), where R is a path relation system, τ is a completion structure, κ: D→[0,∞) is a direction cost, and V is a validity predicate.
Exact mathematical passage
\boxed{ \Gcal= (\mathsf T,\Xcal,\pi:\Dcal\to\Xcal,\Phi,\mathcal R,\tau,\kappa,\mathcal V), }
\TcalDefinition passage · 14 passages
Search aliases: Tcal T mathcal
Definition passage: Quotient deformation graph · Universal_Generative_Deformation_Calculus.tex, line 306
[Quotient deformation graph] The quotient deformation graph is O(x_0)= T_fin(x_0)/∼. Its vertices are relation classes of paths. An edge is induced by one admissible extension.
Exact mathematical passage
\boxed{ \Ocal(x_0)=\Tcal_{\mathrm{fin}}(x_0)/\sim. }
\Tcal_{\mathrm{fin}}Definition passage · 6 passages
Search aliases: Tcal_fin T_fin Tcal mathrm mathcal mathrm
Definition passage: Quotient deformation graph · Universal_Generative_Deformation_Calculus.tex, line 306
[Quotient deformation graph] The quotient deformation graph is O(x_0)= T_fin(x_0)/∼. Its vertices are relation classes of paths. An edge is induced by one admissible extension.
Exact mathematical passage
\boxed{ \Ocal(x_0)=\Tcal_{\mathrm{fin}}(x_0)/\sim. }
\mathcal VDefinition passage · 2 passages
Search aliases: V mathcal
Definition passage: Typed generative deformation datum · Universal_Generative_Deformation_Calculus.tex, line 203
[Typed generative deformation datum] A typed generative deformation datum is a tuple G= ( T, X,π: D→ X,Φ, R,τ,κ, V), where R is a path relation system, τ is a completion structure, κ: D→[0,∞) is a direction cost, and V is a validity predicate.
Exact mathematical passage
\boxed{ \Gcal= (\mathsf T,\Xcal,\pi:\Dcal\to\Xcal,\Phi,\mathcal R,\tau,\kappa,\mathcal V), }
\widehat\XcalDefinition passage · 3 passages
Search aliases: widehatXcal widehat X widehat Xcal widehat mathcal
Definition passage: Convergent route · Universal_Generative_Deformation_Calculus.tex, line 474
[Convergent route] A route ω is convergent in a metric realization when its endpoint sequence x_0,x_1,x_2,… converges in the declared completion widehat X.
Exact mathematical passage
\widehat\Xcal
xDefinition passage · 90 passages
Search aliases: x
Definition passage: Direction bundle · Universal_Generative_Deformation_Calculus.tex, line 175
[Direction bundle] A direction bundle over X is a map π: D⟶ X. The fiber D_x=π^-1(x) is the complete set of directions admitted by the declared grammar at x.
Exact mathematical passage
\Dcal_x=\pi^{-1}(x)
x_0Definition passage · 29 passages
Search aliases: x_0
Definition passage: Admissible path · Universal_Generative_Deformation_Calculus.tex, line 229
[Admissible path] An admissible path of length n from x_0 is a sequence γ=(d_1,…,d_n) for which x_k=x_k-1⋆ d_k, d_k∈ D_x_k-1, 1≤ k≤ n. Its endpoint is ev(γ)=x_n. The empty path ∅ has endpoint x_0.
Exact mathematical passage
x_0
x_1Definition passage · 2 passages
Search aliases: x_1
Definition passage: Convergent route · Universal_Generative_Deformation_Calculus.tex, line 471
[Convergent route] A route ω is convergent in a metric realization when its endpoint sequence x_0,x_1,x_2,… converges in the declared completion widehat X.
Exact mathematical passage
x_0,x_1,x_2,\ldots
x_2Definition passage · 2 passages
Search aliases: x_2
Definition passage: Convergent route · Universal_Generative_Deformation_Calculus.tex, line 471
[Convergent route] A route ω is convergent in a metric realization when its endpoint sequence x_0,x_1,x_2,… converges in the declared completion widehat X.
Exact mathematical passage
x_0,x_1,x_2,\ldots
x_kDefinition passage · 4 passages
Search aliases: x_k
Definition passage: Admissible path · Universal_Generative_Deformation_Calculus.tex, line 234
[Admissible path] An admissible path of length n from x_0 is a sequence γ=(d_1,…,d_n) for which x_k=x_k-1⋆ d_k, d_k∈ D_x_k-1, 1≤ k≤ n. Its endpoint is ev(γ)=x_n. The empty path ∅ has endpoint x_0.
Exact mathematical passage
x_k=x_{k-1}\star d_k, \qquad d_k\in\Dcal_{x_{k-1}}, \qquad 1\le k\le n.
x_{k-1}Definition passage · 2 passages
Search aliases: x_k-1
Definition passage: Admissible path · Universal_Generative_Deformation_Calculus.tex, line 234
[Admissible path] An admissible path of length n from x_0 is a sequence γ=(d_1,…,d_n) for which x_k=x_k-1⋆ d_k, d_k∈ D_x_k-1, 1≤ k≤ n. Its endpoint is ev(γ)=x_n. The empty path ∅ has endpoint x_0.
Exact mathematical passage
x_k=x_{k-1}\star d_k, \qquad d_k\in\Dcal_{x_{k-1}}, \qquad 1\le k\le n.
x_nDefinition passage · 8 passages
Search aliases: x_n
Definition passage: Admissible path · Universal_Generative_Deformation_Calculus.tex, line 241
[Admissible path] An admissible path of length n from x_0 is a sequence γ=(d_1,…,d_n) for which x_k=x_k-1⋆ d_k, d_k∈ D_x_k-1, 1≤ k≤ n. Its endpoint is ev(γ)=x_n. The empty path ∅ has endpoint x_0.
Exact mathematical passage
\ev(\gamma)=x_n
\XcalDefinition passage · 9 passages
Search aliases: Xcal X mathcal
Definition passage: Direction bundle · Universal_Generative_Deformation_Calculus.tex, line 170
[Direction bundle] A direction bundle over X is a map π: D⟶ X. The fiber D_x=π^-1(x) is the complete set of directions admitted by the declared grammar at x.
Exact mathematical passage
\Xcal
\cdotDefinition passage · 2 passages
Search aliases: · cdot
Definition passage: Generated family certificate · Universal_Generative_Deformation_Calculus.tex, line 753
begindefinition[Generated family certificate] A generated family certificate is a tuple F=(Θ,F,{σ_d}_d∈ A, Q, I), where [label=(alph*)] item Θ is a parameter space item F:Θ→ O(x_0) assigns an object class to every parameter item A is a collection of closure directions item σ_d:Θ→Θ satisfies F(σ_d(θ))=d· F(θ) for every declared closure direction item Q is a set of exact relations or recurrences item I is a set of invariants shared or predictably transformed across the family
Exact mathematical passage
F(\sigma_d(\theta))=d\cdot F(\theta)
\gammaDefinition passage · 20 passages
Search aliases: γ gamma
Definition passage: Admissible path · Universal_Generative_Deformation_Calculus.tex, line 230
[Admissible path] An admissible path of length n from x_0 is a sequence γ=(d_1,…,d_n) for which x_k=x_k-1⋆ d_k, d_k∈ D_x_k-1, 1≤ k≤ n. Its endpoint is ev(γ)=x_n. The empty path ∅ has endpoint x_0.
Exact mathematical passage
\gamma=(d_1,\ldots,d_n)
\etaDefinition passage · 8 passages
Search aliases: η eta
Definition passage: Extension-compatible path relation · Universal_Generative_Deformation_Calculus.tex, line 297
[Extension-compatible path relation] An equivalence relation ∼ on finite paths is extension-compatible when the following implication holds. If γ∼η and a direction label d is admissible after both paths with compatible meaning, then γ d∼η d. The relation may also identify endpoints under a declared symmetry group, rewrite system, canonical form, gauge, or quotient law.
Exact mathematical passage
\gamma\sim\eta
\ThetaDefinition passage · 4 passages
Search aliases: Θ Theta
Definition passage: Generated family certificate · Universal_Generative_Deformation_Calculus.tex, line 744
[Generated family certificate] A generated family certificate is a tuple F=(Θ,F,{σ_d}_d∈ A, Q, I), where [label=(alph*)] item Θ is a parameter space item F:Θ→ O(x_0) assigns an object class to every parameter item A is a collection of closure directions item σ_d:Θ→Θ satisfies F(σ_d(θ))=d· F(θ) for every declared closure direction item Q is a set of exact relations or recurrences item I is a set of invariants shared or predictably transformed across the family
Exact mathematical passage
\Fcal=(\Theta,F,\{\sigma_d\}_{d\in A},\mathcal Q,\mathcal I),
\thetaDefinition passage · 5 passages
Search aliases: θ theta
Definition passage: Generated family certificate · Universal_Generative_Deformation_Calculus.tex, line 753
begindefinition[Generated family certificate] A generated family certificate is a tuple F=(Θ,F,{σ_d}_d∈ A, Q, I), where [label=(alph*)] item Θ is a parameter space item F:Θ→ O(x_0) assigns an object class to every parameter item A is a collection of closure directions item σ_d:Θ→Θ satisfies F(σ_d(θ))=d· F(θ) for every declared closure direction item Q is a set of exact relations or recurrences item I is a set of invariants shared or predictably transformed across the family
Exact mathematical passage
F(\sigma_d(\theta))=d\cdot F(\theta)
\kappaDefinition passage · 4 passages
Search aliases: κ kappa
Definition passage: Typed generative deformation datum · Universal_Generative_Deformation_Calculus.tex, line 203
[Typed generative deformation datum] A typed generative deformation datum is a tuple G= ( T, X,π: D→ X,Φ, R,τ,κ, V), where R is a path relation system, τ is a completion structure, κ: D→[0,∞) is a direction cost, and V is a validity predicate.
Exact mathematical passage
\boxed{ \Gcal= (\mathsf T,\Xcal,\pi:\Dcal\to\Xcal,\Phi,\mathcal R,\tau,\kappa,\mathcal V), }
\piDefinition passage · 5 passages
Search aliases: π pi
Definition passage: Direction bundle · Universal_Generative_Deformation_Calculus.tex, line 171
[Direction bundle] A direction bundle over X is a map π: D⟶ X. The fiber D_x=π^-1(x) is the complete set of directions admitted by the declared grammar at x.
Exact mathematical passage
\pi:\Dcal\longrightarrow\Xcal.
\pi^{-1}Definition passage · 2 passages
Search aliases: π^-1 pi
Definition passage: Direction bundle · Universal_Generative_Deformation_Calculus.tex, line 175
[Direction bundle] A direction bundle over X is a map π: D⟶ X. The fiber D_x=π^-1(x) is the complete set of directions admitted by the declared grammar at x.
Exact mathematical passage
\Dcal_x=\pi^{-1}(x)
\sigmaDefinition passage · 8 passages
Search aliases: σ sigma
Definition passage: Generated family certificate · Universal_Generative_Deformation_Calculus.tex, line 744
[Generated family certificate] A generated family certificate is a tuple F=(Θ,F,{σ_d}_d∈ A, Q, I), where [label=(alph*)] item Θ is a parameter space item F:Θ→ O(x_0) assigns an object class to every parameter item A is a collection of closure directions item σ_d:Θ→Θ satisfies F(σ_d(θ))=d· F(θ) for every declared closure direction item Q is a set of exact relations or recurrences item I is a set of invariants shared or predictably transformed across the family
Exact mathematical passage
\Fcal=(\Theta,F,\{\sigma_d\}_{d\in A},\mathcal Q,\mathcal I),
\sigma_dDefinition passage · 6 passages
Search aliases: σ_d sigma
Definition passage: Generated family certificate · Universal_Generative_Deformation_Calculus.tex, line 744
[Generated family certificate] A generated family certificate is a tuple F=(Θ,F,{σ_d}_d∈ A, Q, I), where [label=(alph*)] item Θ is a parameter space item F:Θ→ O(x_0) assigns an object class to every parameter item A is a collection of closure directions item σ_d:Θ→Θ satisfies F(σ_d(θ))=d· F(θ) for every declared closure direction item Q is a set of exact relations or recurrences item I is a set of invariants shared or predictably transformed across the family
Exact mathematical passage
\Fcal=(\Theta,F,\{\sigma_d\}_{d\in A},\mathcal Q,\mathcal I),
\tauDefinition passage · 2 passages
Search aliases: τ tau
Definition passage: Typed generative deformation datum · Universal_Generative_Deformation_Calculus.tex, line 203
[Typed generative deformation datum] A typed generative deformation datum is a tuple G= ( T, X,π: D→ X,Φ, R,τ,κ, V), where R is a path relation system, τ is a completion structure, κ: D→[0,∞) is a direction cost, and V is a validity predicate.
Exact mathematical passage
\boxed{ \Gcal= (\mathsf T,\Xcal,\pi:\Dcal\to\Xcal,\Phi,\mathcal R,\tau,\kappa,\mathcal V), }
\PhiDefinition passage · 8 passages
Search aliases: Φ Phi
Definition passage: Elementary deformation law · Universal_Generative_Deformation_Calculus.tex, line 183
[Elementary deformation law] An elementary deformation law is a map Φ: D⟶ X with the understanding that d∈ D_x starts at x and ends at Φ(d). We also write x⋆ d=Φ(d).
Exact mathematical passage
\Phi:\Dcal\longrightarrow\Xcal
\PsiDefinition passage · 2 passages
Search aliases: Ψ Psi
Definition passage: Operadic perturbation hyperforest · Universal_Generative_Deformation_Calculus.tex, line 546
[Operadic perturbation hyperforest] The operadic perturbation hyperforest generated by a seed family is the set of all finite rooted typed trees whose leaves are seeds and whose internal vertices are labeled by admissible operations Φ or Ψ_r. Evaluation proceeds from leaves to root.
Exact mathematical passage
\Psi_r
\Psi_rDefinition passage · 2 passages
Search aliases: Ψ_r Psi
Definition passage: Operadic perturbation hyperforest · Universal_Generative_Deformation_Calculus.tex, line 546
[Operadic perturbation hyperforest] The operadic perturbation hyperforest generated by a seed family is the set of all finite rooted typed trees whose leaves are seeds and whose internal vertices are labeled by admissible operations Φ or Ψ_r. Evaluation proceeds from leaves to root.
Exact mathematical passage
\Psi_r
\omegaDefinition passage · 2 passages
Search aliases: ω omega
Definition passage: Convergent route · Universal_Generative_Deformation_Calculus.tex, line 470
[Convergent route] A route ω is convergent in a metric realization when its endpoint sequence x_0,x_1,x_2,… converges in the declared completion widehat X.
Exact mathematical passage
\omega
\ldotsDefinition passage · 8 passages
Search aliases: … ldots
Definition passage: Admissible path · Universal_Generative_Deformation_Calculus.tex, line 230
[Admissible path] An admissible path of length n from x_0 is a sequence γ=(d_1,…,d_n) for which x_k=x_k-1⋆ d_k, d_k∈ D_x_k-1, 1≤ k≤ n. Its endpoint is ev(γ)=x_n. The empty path ∅ has endpoint x_0.
Exact mathematical passage
\gamma=(d_1,\ldots,d_n)
\toDefinition passage · 6 passages
Search aliases: → to
Definition passage: Typed generative deformation datum · Universal_Generative_Deformation_Calculus.tex, line 203
[Typed generative deformation datum] A typed generative deformation datum is a tuple G= ( T, X,π: D→ X,Φ, R,τ,κ, V), where R is a path relation system, τ is a completion structure, κ: D→[0,∞) is a direction cost, and V is a validity predicate.
Exact mathematical passage
\boxed{ \Gcal= (\mathsf T,\Xcal,\pi:\Dcal\to\Xcal,\Phi,\mathcal R,\tau,\kappa,\mathcal V), }
\varnothingDefinition passage · 4 passages
Search aliases: ∅ varnothing
Definition passage: Admissible path · Universal_Generative_Deformation_Calculus.tex, line 241
[Admissible path] An admissible path of length n from x_0 is a sequence γ=(d_1,…,d_n) for which x_k=x_k-1⋆ d_k, d_k∈ D_x_k-1, 1≤ k≤ n. Its endpoint is ev(γ)=x_n. The empty path ∅ has endpoint x_0.
Exact mathematical passage
\varnothing
\inDefinition passage · 14 passages
Search aliases: ∈ in
Definition passage: Elementary deformation law · Universal_Generative_Deformation_Calculus.tex, line 186
[Elementary deformation law] An elementary deformation law is a map Φ: D⟶ X with the understanding that d∈ D_x starts at x and ends at Φ(d). We also write x⋆ d=Φ(d).
Exact mathematical passage
d\in\Dcal_x
\inftyDefinition passage · 5 passages
Search aliases: ∞ infty
Definition passage: Typed generative deformation datum · Universal_Generative_Deformation_Calculus.tex, line 209
[Typed generative deformation datum] A typed generative deformation datum is a tuple G= ( T, X,π: D→ X,Φ, R,τ,κ, V), where R is a path relation system, τ is a completion structure, κ: D→[0,∞) is a direction cost, and V is a validity predicate.
Exact mathematical passage
\kappa:\Dcal\to[0,\infty)
\simDefinition passage · 9 passages
Search aliases: ∼ sim
Definition passage: Extension-compatible path relation · Universal_Generative_Deformation_Calculus.tex, line 297
[Extension-compatible path relation] An equivalence relation ∼ on finite paths is extension-compatible when the following implication holds. If γ∼η and a direction label d is admissible after both paths with compatible meaning, then γ d∼η d. The relation may also identify endpoints under a declared symmetry group, rewrite system, canonical form, gauge, or quotient law.
Exact mathematical passage
\sim
\leDefinition passage · 21 passages
Search aliases: ≤ le
Definition passage: Admissible path · Universal_Generative_Deformation_Calculus.tex, line 234
[Admissible path] An admissible path of length n from x_0 is a sequence γ=(d_1,…,d_n) for which x_k=x_k-1⋆ d_k, d_k∈ D_x_k-1, 1≤ k≤ n. Its endpoint is ev(γ)=x_n. The empty path ∅ has endpoint x_0.
Exact mathematical passage
x_k=x_{k-1}\star d_k, \qquad d_k\in\Dcal_{x_{k-1}}, \qquad 1\le k\le n.
\starDefinition passage · 7 passages
Search aliases: ⋆ star
Definition passage: Elementary deformation law · Universal_Generative_Deformation_Calculus.tex, line 187
[Elementary deformation law] An elementary deformation law is a map Φ: D⟶ X with the understanding that d∈ D_x starts at x and ends at Φ(d). We also write x⋆ d=Φ(d).
Exact mathematical passage
x\star d=\Phi(d).
\longrightarrowDefinition passage · 7 passages
Search aliases: ⟶ longrightarrow
Definition passage: Direction bundle · Universal_Generative_Deformation_Calculus.tex, line 171
[Direction bundle] A direction bundle over X is a map π: D⟶ X. The fiber D_x=π^-1(x) is the complete set of directions admitted by the declared grammar at x.
Exact mathematical passage
\pi:\Dcal\longrightarrow\Xcal.
\coprodIntroduction passage · 3 passages
Search aliases: coprod
Introduction passage · Universal_Generative_Deformation_Calculus.tex, line 245
Write T_n(x_0) for the set of length-n paths and T_fin(x_0)=coprod_n≥0 T_n(x_0). Every nonempty path has a unique parent obtained by deleting its final direction. Therefore T_fin(x_0) is a rooted tree even when different paths have the same endpoint.
Exact mathematical passage
\Tcal_{\mathrm{fin}}(x_0)=\coprod_{n\ge0}\Tcal_n(x_0).
\coprod_{n\ge0}Introduction passage · 2 passages
Search aliases: coprod_n≥0 coprod ge
Introduction passage · Universal_Generative_Deformation_Calculus.tex, line 245
Write T_n(x_0) for the set of length-n paths and T_fin(x_0)=coprod_n≥0 T_n(x_0). Every nonempty path has a unique parent obtained by deleting its final direction. Therefore T_fin(x_0) is a rooted tree even when different paths have the same endpoint.
Exact mathematical passage
\Tcal_{\mathrm{fin}}(x_0)=\coprod_{n\ge0}\Tcal_n(x_0).
\Tcal_nIntroduction passage · 8 passages
Search aliases: Tcal_n T_n Tcal mathcal
Introduction passage · Universal_Generative_Deformation_Calculus.tex, line 244
Write T_n(x_0) for the set of length-n paths and T_fin(x_0)=coprod_n≥0 T_n(x_0). Every nonempty path has a unique parent obtained by deleting its final direction. Therefore T_fin(x_0) is a rooted tree even when different paths have the same endpoint.
Exact mathematical passage
\Tcal_n(x_0)
\geIntroduction passage · 10 passages
Search aliases: ≥ ge
Introduction passage · Universal_Generative_Deformation_Calculus.tex, line 245
Write T_n(x_0) for the set of length-n paths and T_fin(x_0)=coprod_n≥0 T_n(x_0). Every nonempty path has a unique parent obtained by deleting its final direction. Therefore T_fin(x_0) is a rooted tree even when different paths have the same endpoint.
Exact mathematical passage
\Tcal_{\mathrm{fin}}(x_0)=\coprod_{n\ge0}\Tcal_n(x_0).
+Use in source · 44 passages
Search aliases: +
Use in source · Universal_Generative_Deformation_Calculus.tex, line 261
The statement is immediate at depth zero. Assume it holds at depth n. Every path of length n+1 is a path of length n followed by one direction from the fiber over its endpoint. Thus its endpoint lies in the (n+1)-step reachable set. Conversely, every object in the (n+1)-step reachable set is produced by an n-step reachable object and one admissible direction. The induction closes.
Exact mathematical passage
n+1
aUse in source · 16 passages
Search aliases: a
Use in source · Universal_Generative_Deformation_Calculus.tex, line 479
[Summable-step convergence] Suppose widehat X is complete and d_X(x_n+1,x_n)≤ a_n with ∑_n≥0a_n<∞. Then (x_n) is Cauchy and the route has a unique limit in widehat X.
Exact mathematical passage
d_X(x_{n+1},x_n)\le a_n
A^{\mathsf T}Use in source · 1 passage
Search aliases: A^ T mathsf
Use in source · Universal_Generative_Deformation_Calculus.tex, line 1104
0 · 1. They satisfy (A^ T)^ T=A, Aadj(A)=det(A)I, det(SAS)=det(A) for the swap matrix S. Thus involution, adjugate, and conjugacy relations generate quotient identifications, while squaring and commutation usually move to new invariant values.
Exact mathematical passage
(A^{\mathsf T})^{\mathsf T}=A, \qquad A\operatorname{adj}(A)=\det(A)I, \qquad \det(SAS)=\det(A)
A_0Use in source · 2 passages
Search aliases: A_0
Use in source · Universal_Generative_Deformation_Calculus.tex, line 1086
Let A_0=1 · 1 0 · 1. The declared matrix directions are transpose, negation, squaring, conjugation by the coordinate swap, symmetrization, commutation with J=0 · -1
Exact mathematical passage
A_0=\begin{pmatrix}1&1\\0&1\end{pmatrix}.
a_0Use in source · 3 passages
Search aliases: a_0
Use in source · Universal_Generative_Deformation_Calculus.tex, line 787
For a sequence a_0,a_1,…, the implementation can search for a constant-coefficient relation a_n=c_1a_n-1+⋯+c_ra_n-r using exact rational linear algebra. A proposed recurrence is accepted only when it reproduces every supplied term exactly. The built-in Fibonacci example recovers (c_1,c_2)=(1,1). This is a small example of the certificate principle. Fitting alone is not enough. The relation must be verified and then proved or bounded on the intended domain.
Exact mathematical passage
a_0,a_1,\ldots
A_0^{\mathsf T}Use in source · 1 passage
Search aliases: A_0^ T mathsf
Use in source · Universal_Generative_Deformation_Calculus.tex, line 1096
The first calculations are explicit: A_0^ T=1 · 0 1 · 1, A_0^2=1 · 2 0 · 1, adj(A_0)=1 · -1 0 · 1. They satisfy (A^ T)^ T=A, Aadj(A)=det(A)I, det(SAS)=det(A) for the swap matrix S. Thus involution, adjugate, and conjugacy relations generate quotient identifications, while squaring and commutation usually move to new invariant values.
Exact mathematical passage
A_0^{\mathsf T}=\begin{pmatrix}1&0\\1&1\end{pmatrix}, \qquad A_0^2=\begin{pmatrix}1&2\\0&1\end{pmatrix}, \qquad \operatorname{adj}(A_0)=\begin{pmatrix}1&-1\\0&1\end{pmatrix}.
A_0^2Use in source · 1 passage
Search aliases: A_0^2
Use in source · Universal_Generative_Deformation_Calculus.tex, line 1096
The first calculations are explicit: A_0^ T=1 · 0 1 · 1, A_0^2=1 · 2 0 · 1, adj(A_0)=1 · -1 0 · 1. They satisfy (A^ T)^ T=A, Aadj(A)=det(A)I, det(SAS)=det(A) for the swap matrix S. Thus involution, adjugate, and conjugacy relations generate quotient identifications, while squaring and commutation usually move to new invariant values.
Exact mathematical passage
A_0^{\mathsf T}=\begin{pmatrix}1&0\\1&1\end{pmatrix}, \qquad A_0^2=\begin{pmatrix}1&2\\0&1\end{pmatrix}, \qquad \operatorname{adj}(A_0)=\begin{pmatrix}1&-1\\0&1\end{pmatrix}.
a_1Use in source · 1 passage
Search aliases: a_1
Use in source · Universal_Generative_Deformation_Calculus.tex, line 787
For a sequence a_0,a_1,…, the implementation can search for a constant-coefficient relation a_n=c_1a_n-1+⋯+c_ra_n-r using exact rational linear algebra. A proposed recurrence is accepted only when it reproduces every supplied term exactly. The built-in Fibonacci example recovers (c_1,c_2)=(1,1). This is a small example of the certificate principle. Fitting alone is not enough. The relation must be verified and then proved or bounded on the intended domain.
Exact mathematical passage
a_0,a_1,\ldots
a_jUse in source · 1 passage
Search aliases: a_j
Use in source · Universal_Generative_Deformation_Calculus.tex, line 982
The executable atlas starts from the eight-term Fibonacci prefix a=(0,1,1,2,3,5,8,13). Its fixed grammar contains first difference, partial summation, the binomial transform, sign alternation, and reversal. For a finite sequence a=(a_0,…,a_N-1), define (Da)_0=a_0, (Da)_n=a_n-a_n-1, (Ca)_n=∑_j=0^na_j, (Ba)_n=∑_j=0^nbinomnja_j. The two triangular operators satisfy DC=CD=I on every finite sequence under the stated boundary convention. Hence the raw routes axrightarrowDDaxrightarrowCCDa, axrightarro…
Exact mathematical passage
(Ca)_n=\sum_{j=0}^{n}a_j, \qquad (Ba)_n=\sum_{j=0}^{n}\binom{n}{j}a_j.
a_kUse in source · 1 passage
Search aliases: a_k
Use in source · Universal_Generative_Deformation_Calculus.tex, line 491
For m>n, d_X(x_m,x_n) ≤ ∑_k=n^m-1d_X(x_k+1,x_k) ≤ ∑_k=n^m-1a_k. The tail tends to zero. Completeness supplies a unique limit.
Exact mathematical passage
d_X(x_m,x_n) \le \sum_{k=n}^{m-1}d_X(x_{k+1},x_k) \le \sum_{k=n}^{m-1}a_k.
a_nUse in source · 9 passages
Search aliases: a_n
Use in source · Universal_Generative_Deformation_Calculus.tex, line 479
[Summable-step convergence] Suppose widehat X is complete and d_X(x_n+1,x_n)≤ a_n with ∑_n≥0a_n<∞. Then (x_n) is Cauchy and the route has a unique limit in widehat X.
Exact mathematical passage
d_X(x_{n+1},x_n)\le a_n
a_{N-1}Use in source · 1 passage
Search aliases: a_N-1
Use in source · Universal_Generative_Deformation_Calculus.tex, line 976
The executable atlas starts from the eight-term Fibonacci prefix a=(0,1,1,2,3,5,8,13). Its fixed grammar contains first difference, partial summation, the binomial transform, sign alternation, and reversal. For a finite sequence a=(a_0,…,a_N-1), define (Da)_0=a_0, (Da)_n=a_n-a_n-1, (Ca)_n=∑_j=0^na_j, (Ba)_n=∑_j=0^nbinomnja_j. The two triangular operators satisfy DC=CD=I on every finite sequence under the stated boundary convention. Hence the raw routes axrightarrowDDaxrightarrowCCDa, axrightarro…
Exact mathematical passage
a=(a_0,\ldots,a_{N-1})
a_{n-1}Use in source · 2 passages
Search aliases: a_n-1
Use in source · Universal_Generative_Deformation_Calculus.tex, line 788
For a sequence a_0,a_1,…, the implementation can search for a constant-coefficient relation a_n=c_1a_n-1+⋯+c_ra_n-r using exact rational linear algebra. A proposed recurrence is accepted only when it reproduces every supplied term exactly. The built-in Fibonacci example recovers (c_1,c_2)=(1,1). This is a small example of the certificate principle. Fitting alone is not enough. The relation must be verified and then proved or bounded on the intended domain.
Exact mathematical passage
a_n=c_1a_{n-1}+\cdots+c_ra_{n-r}
a_{n-r}Use in source · 1 passage
Search aliases: a_n-r
Use in source · Universal_Generative_Deformation_Calculus.tex, line 788
For a sequence a_0,a_1,…, the implementation can search for a constant-coefficient relation a_n=c_1a_n-1+⋯+c_ra_n-r using exact rational linear algebra. A proposed recurrence is accepted only when it reproduces every supplied term exactly. The built-in Fibonacci example recovers (c_1,c_2)=(1,1). This is a small example of the certificate principle. Fitting alone is not enough. The relation must be verified and then proved or bounded on the intended domain.
Exact mathematical passage
a_n=c_1a_{n-1}+\cdots+c_ra_{n-r}
A_{\beta\alpha}Use in source · 5 passages
Search aliases: A_βα beta alpha
Use in source · Universal_Generative_Deformation_Calculus.tex, line 645
Let A: H→ K be a bounded operator and let {P_α} and {Q_β} be complete route resolutions. Define A_βα=Q_β A P_α. Then A=∑_α,βA_βα. If B follows A, then (BA)_γα = ∑_β B_γβA_βα. Every operator composition becomes a sum over compatible intermediate perturbation packets.
Exact mathematical passage
A_{\beta\alpha}=Q_\beta A P_\alpha.
\absUse in source · 1 passage
Search aliases: abs
Use in source · Universal_Generative_Deformation_Calculus.tex, line 617
[Exact cylinder refinement] For every prefix α and every depth n>absα, P^(n)_α = ∑_d∈ D_ev(α)P^(n)_α d, with strong convergence when the child set is countable.
Exact mathematical passage
n>\abs{\alpha}
\operatorname{adj}Use in source · 2 passages
Search aliases: adj operatorname
Use in source · Universal_Generative_Deformation_Calculus.tex, line 1096
The first calculations are explicit: A_0^ T=1 · 0 1 · 1, A_0^2=1 · 2 0 · 1, adj(A_0)=1 · -1 0 · 1. They satisfy (A^ T)^ T=A, Aadj(A)=det(A)I, det(SAS)=det(A) for the swap matrix S. Thus involution, adjugate, and conjugacy relations generate quotient identifications, while squaring and commutation usually move to new invariant values.
Exact mathematical passage
A_0^{\mathsf T}=\begin{pmatrix}1&0\\1&1\end{pmatrix}, \qquad A_0^2=\begin{pmatrix}1&2\\0&1\end{pmatrix}, \qquad \operatorname{adj}(A_0)=\begin{pmatrix}1&-1\\0&1\end{pmatrix}.
BUse in source · 13 passages
Search aliases: B
Use in source · Universal_Generative_Deformation_Calculus.tex, line 652
Let A: H→ K be a bounded operator and let {P_α} and {Q_β} be complete route resolutions. Define A_βα=Q_β A P_α. Then A=∑_α,βA_βα. If B follows A, then (BA)_γα = ∑_β B_γβA_βα. Every operator composition becomes a sum over compatible intermediate perturbation packets.
Exact mathematical passage
B
bUse in source · 1 passage
Search aliases: b
Use in source · Universal_Generative_Deformation_Calculus.tex, line 1215
With four ordered leaves and one binary operation, the free operadic layer contains the five Catalan parenthesizations ((ab)c)d, (a(bc))d, (ab)(cd), a((bc)d), a(b(cd)). They are different syntax trees. If the operation is associative, all five evaluate to one quotient object. If it is also commutative, the 4! permutations of the leaves are identified as well. The raw tree and the quotient must therefore remain separate even in elementary algebra. The complete syntax-to-quotient visual appears in…
Exact mathematical passage
((ab)c)d, \quad (a(bc))d, \quad (ab)(cd), \quad a((bc)d), \quad a(b(cd)).
B^nUse in source · 2 passages
Search aliases: B^n
Use in source · Universal_Generative_Deformation_Calculus.tex, line 703
[Geometric convergence bound] Suppose every vertex has at most B children and every direction costs at least c>0. Then Z_x_0(s) ≤ ∑_n≥1B^ne^-scn. Consequently, Z_x_0(s) converges whenever sc>log B. In a full regular B-ary tree with constant cost c, it diverges whenever sc≤log B.
Exact mathematical passage
Z_{x_0}(s) \le \sum_{n\ge1}B^ne^{-scn}.
B_{\gamma\beta}Use in source · 2 passages
Search aliases: B_γβ gamma beta
Use in source · Universal_Generative_Deformation_Calculus.tex, line 653
Let A: H→ K be a bounded operator and let {P_α} and {Q_β} be complete route resolutions. Define A_βα=Q_β A P_α. Then A=∑_α,βA_βα. If B follows A, then (BA)_γα = ∑_β B_γβA_βα. Every operator composition becomes a sum over compatible intermediate perturbation packets.
Exact mathematical passage
(BA)_{\gamma\alpha} = \sum_\beta B_{\gamma\beta}A_{\beta\alpha}.
CUse in source · 10 passages
Search aliases: C
Use in source · Universal_Generative_Deformation_Calculus.tex, line 574
Consider u=s+λ u^2. Repeated substitution expands u into rooted binary trees. If only shape is counted, the number of binary trees with n internal vertices is the Catalan number C_n.
Exact mathematical passage
C_n
cUse in source · 12 passages
Search aliases: c
Use in source · Universal_Generative_Deformation_Calculus.tex, line 702
[Geometric convergence bound] Suppose every vertex has at most B children and every direction costs at least c>0. Then Z_x_0(s) ≤ ∑_n≥1B^ne^-scn. Consequently, Z_x_0(s) converges whenever sc>log B. In a full regular B-ary tree with constant cost c, it diverges whenever sc≤log B.
Exact mathematical passage
c>0
C^kUse in source · 2 passages
Search aliases: C^k
Use in source · Universal_Generative_Deformation_Calculus.tex, line 1000
A second, closed subfamily starts from the impulse δ=(1,0,0,…). Repeated summation gives (C^kδ)_n=binomn+k-1k-1 (k≥1), by induction and the hockey-stick identity. This provides a genuine family certificate: perturbation depth k is the parameter, C advances the parameter, D reverses it away from the boundary, and the displayed coefficient formula is an exact invariant relation.
Exact mathematical passage
(C^k\delta)_n=\binom{n+k-1}{k-1} \qquad(k\ge1),
C_0Use in source · 1 passage
Search aliases: C_0
Use in source · Universal_Generative_Deformation_Calculus.tex, line 578
[Catalan interaction growth] The counts satisfy C_0=1, C_n+1=∑_k=0^nC_kC_n-k.
Exact mathematical passage
C_0=1, \qquad C_{n+1}=\sum_{k=0}^nC_kC_{n-k}.
c_1Use in source · 2 passages
Search aliases: c_1
Use in source · Universal_Generative_Deformation_Calculus.tex, line 788
For a sequence a_0,a_1,…, the implementation can search for a constant-coefficient relation a_n=c_1a_n-1+⋯+c_ra_n-r using exact rational linear algebra. A proposed recurrence is accepted only when it reproduces every supplied term exactly. The built-in Fibonacci example recovers (c_1,c_2)=(1,1). This is a small example of the certificate principle. Fitting alone is not enough. The relation must be verified and then proved or bounded on the intended domain.
Exact mathematical passage
a_n=c_1a_{n-1}+\cdots+c_ra_{n-r}
c_2Use in source · 1 passage
Search aliases: c_2
Use in source · Universal_Generative_Deformation_Calculus.tex, line 792
For a sequence a_0,a_1,…, the implementation can search for a constant-coefficient relation a_n=c_1a_n-1+⋯+c_ra_n-r using exact rational linear algebra. A proposed recurrence is accepted only when it reproduces every supplied term exactly. The built-in Fibonacci example recovers (c_1,c_2)=(1,1). This is a small example of the certificate principle. Fitting alone is not enough. The relation must be verified and then proved or bounded on the intended domain.
Exact mathematical passage
(c_1,c_2)=(1,1).
C_kUse in source · 1 passage
Search aliases: C_k
Use in source · Universal_Generative_Deformation_Calculus.tex, line 578
[Catalan interaction growth] The counts satisfy C_0=1, C_n+1=∑_k=0^nC_kC_n-k.
Exact mathematical passage
C_0=1, \qquad C_{n+1}=\sum_{k=0}^nC_kC_{n-k}.
C_nUse in source · 1 passage
Search aliases: C_n
Use in source · Universal_Generative_Deformation_Calculus.tex, line 574
Consider u=s+λ u^2. Repeated substitution expands u into rooted binary trees. If only shape is counted, the number of binary trees with n internal vertices is the Catalan number C_n.
Exact mathematical passage
C_n
C_{n+1}Use in source · 1 passage
Search aliases: C_n+1
Use in source · Universal_Generative_Deformation_Calculus.tex, line 578
[Catalan interaction growth] The counts satisfy C_0=1, C_n+1=∑_k=0^nC_kC_n-k.
Exact mathematical passage
C_0=1, \qquad C_{n+1}=\sum_{k=0}^nC_kC_{n-k}.
C_{n-k}Use in source · 1 passage
Search aliases: C_n-k
Use in source · Universal_Generative_Deformation_Calculus.tex, line 578
[Catalan interaction growth] The counts satisfy C_0=1, C_n+1=∑_k=0^nC_kC_n-k.
Exact mathematical passage
C_0=1, \qquad C_{n+1}=\sum_{k=0}^nC_kC_{n-k}.
c_rUse in source · 1 passage
Search aliases: c_r
Use in source · Universal_Generative_Deformation_Calculus.tex, line 788
For a sequence a_0,a_1,…, the implementation can search for a constant-coefficient relation a_n=c_1a_n-1+⋯+c_ra_n-r using exact rational linear algebra. A proposed recurrence is accepted only when it reproduces every supplied term exactly. The built-in Fibonacci example recovers (c_1,c_2)=(1,1). This is a small example of the certificate principle. Fitting alone is not enough. The relation must be verified and then proved or bounded on the intended domain.
Exact mathematical passage
a_n=c_1a_{n-1}+\cdots+c_ra_{n-r}
\coprod_{t\in\mathsf T}Use in source · 1 passage
Search aliases: coprod_t∈ T coprod in mathsf
Use in source · Universal_Generative_Deformation_Calculus.tex, line 164
Let T be a set of types and let X=coprod_t∈ TX_t be the disjoint union of the corresponding object spaces.
Exact mathematical passage
\Xcal=\coprod_{t\in\mathsf T}X_t
\coshUse in source · 4 passages
Search aliases: cosh
Use in source · Universal_Generative_Deformation_Calculus.tex, line 946
Start from f_0(x)=cosh x. The declared directions are differentiation, reciprocal formation, logarithmic differentiation, squaring, even projection, odd projection, and a restricted argument-doubling direction. The first three give D(cosh x)=sinh x, R(cosh x)=sechx, L(cosh x)=tanh x. Applying the same grammar again produces cschx, coth x, powers, parity components, and doubled arguments. Reciprocal and logarithmic derivative are rejected at the zero expression and the failed attempts remain in t…
Exact mathematical passage
f_0(x)=\cosh x.
\costUse in source · 4 passages
Search aliases: cost operatorname
Use in source · Universal_Generative_Deformation_Calculus.tex, line 285
The cost of a finite path is cost(γ)=∑_k=1^nκ(d_k). A path record may also store error bounds, provenance, validity certificates, symbolic assumptions, and the first place at which an invariant changed. None of these fields alters reachability. They determine how a finite explorer prioritizes and audits the infinite object.
Exact mathematical passage
\cost(\gamma)=\sum_{k=1}^n\kappa(d_k).
\cothUse in source · 1 passage
Search aliases: coth
Use in source · Universal_Generative_Deformation_Calculus.tex, line 957
Start from f_0(x)=cosh x. The declared directions are differentiation, reciprocal formation, logarithmic differentiation, squaring, even projection, odd projection, and a restricted argument-doubling direction. The first three give D(cosh x)=sinh x, R(cosh x)=sechx, L(cosh x)=tanh x. Applying the same grammar again produces cschx, coth x, powers, parity components, and doubled arguments. Reciprocal and logarithmic derivative are rejected at the zero expression and the failed attempts remain in t…
Exact mathematical passage
\coth x
\operatorname{csch}Use in source · 1 passage
Search aliases: csch operatorname
Use in source · Universal_Generative_Deformation_Calculus.tex, line 957
Start from f_0(x)=cosh x. The declared directions are differentiation, reciprocal formation, logarithmic differentiation, squaring, even projection, odd projection, and a restricted argument-doubling direction. The first three give D(cosh x)=sinh x, R(cosh x)=sechx, L(cosh x)=tanh x. Applying the same grammar again produces cschx, coth x, powers, parity components, and doubled arguments. Reciprocal and logarithmic derivative are rejected at the zero expression and the failed attempts remain in t…
Exact mathematical passage
\operatorname{csch}x
DUse in source · 10 passages
Search aliases: D
Use in source · Universal_Generative_Deformation_Calculus.tex, line 915
Let p(x)=x^2+1, (Tp)(x)=p(x+1), (Dp)(x)=p'(x). The two primitive directions T and D satisfy D(Tp)=T(Dp)=2x+2. The raw path tree therefore contains two length-two routes, pxrightarrowTTpxrightarrowDDTp, pxrightarrowDDpxrightarrowTTDp, but the quotient graph contains one endpoint class because DTp=TDp exactly.
Exact mathematical passage
p(x)=x^2+1, \qquad (Tp)(x)=p(x+1), \qquad (Dp)(x)=p'(x).
d_2Use in source · 1 passage
Search aliases: d_2
Use in source · Universal_Generative_Deformation_Calculus.tex, line 462
An infinite route is a sequence ω=(d_1,d_2,…) whose every finite prefix is admissible. The set of infinite routes is the path boundary ∂ T(x_0).
Exact mathematical passage
\omega=(d_1,d_2,\ldots)
d_n^{\pm}Use in source · 1 passage
Search aliases: d_n^± pm
Use in source · Universal_Generative_Deformation_Calculus.tex, line 1192
The executable completion example starts from x_0=0 and exposes the two depth-dependent directions d_n^±=±2^-(n+1). A finite route with signs ε_n∈{-1,+1} reaches x_N=∑_n=0^N-1ε_n2^-(n+1). Every infinite sign route is Cauchy because |x_m-x_n| ≤ ∑_k=n^m-12^-(k+1) <2^-n. The completed object therefore contains a limit for every infinite route. The selected periodic sign pattern used in the figure converges numerically to 5/7.
Exact mathematical passage
d_n^{\pm}=\pm2^{-(n+1)}.
d_XUse in source · 7 passages
Search aliases: d_X
Use in source · Universal_Generative_Deformation_Calculus.tex, line 385
Assume the object space carries a metric d_X.
Exact mathematical passage
d_X
\Dcal^{(r)}Use in source · 1 passage
Search aliases: Dcal^(r) D^(r) Dcal mathcal
Use in source · Universal_Generative_Deformation_Calculus.tex, line 537
For every r≥2, let Ψ_r: X^r× D^(r) ⟶ X be a typed r-input deformation law.
Exact mathematical passage
\Psi_r: \Xcal^r\times\Dcal^{(r)} \longrightarrow \Xcal
\Dcal_{\ev(\alpha)}Use in source · 1 passage
Search aliases: Dcal_ev(α) D_ev(α) Dcal ev alpha mathcal operatorname alpha
Use in source · Universal_Generative_Deformation_Calculus.tex, line 618
[Exact cylinder refinement] For every prefix α and every depth n>absα, P^(n)_α = ∑_d∈ D_ev(α)P^(n)_α d, with strong convergence when the child set is countable.
Exact mathematical passage
\boxed{ P^{(n)}_\alpha = \sum_{d\in\Dcal_{\ev(\alpha)}}P^{(n)}_{\alpha d}, }
\detUse in source · 1 passage
Search aliases: det
Use in source · Universal_Generative_Deformation_Calculus.tex, line 1104
0 · 1. They satisfy (A^ T)^ T=A, Aadj(A)=det(A)I, det(SAS)=det(A) for the swap matrix S. Thus involution, adjugate, and conjugacy relations generate quotient identifications, while squaring and commutation usually move to new invariant values.
Exact mathematical passage
(A^{\mathsf T})^{\mathsf T}=A, \qquad A\operatorname{adj}(A)=\det(A)I, \qquad \det(SAS)=\det(A)
eUse in source · 14 passages
Search aliases: e
Use in source · Universal_Generative_Deformation_Calculus.tex, line 397
[Pathwise cover error] Consider an exact route x_k=x_k-1⋆ d_k and a sampled route widehat x_k=widehat x_k-1⋆widehat d_k. Suppose at step k, d_X(x⋆ d,y⋆ e) ≤ L_k d_X(x,y)+M_kρ(d,e), and ρ(d_k,widehat d_k)≤ϵ_k. Then d_X(x_n,widehat x_n) ≤ (∏_k=1^nL_k)d_X(x_0,widehat x_0) + ∑_j=1^n M_jϵ_j ∏_k=j+1^nL_k.
Exact mathematical passage
d_X(x\star d,y\star e) \le L_k d_X(x,y)+M_k\rho(d,e),
e^{-sc}Use in source · 1 passage
Search aliases: e^-sc
Use in source · Universal_Generative_Deformation_Calculus.tex, line 716
There are at most B^n paths of length n, and each has cost at least cn. The upper bound is a geometric series with ratio Be^-sc. In the regular constant-cost case, equality holds at every depth, giving the divergence statement.
Exact mathematical passage
Be^{-sc}
e^{-sC([x])}Use in source · 1 passage
Search aliases: e^-sC([x])
Use in source · Universal_Generative_Deformation_Calculus.tex, line 729
A path zeta function measures construction complexity. A quotient zeta function measures distinct object complexity. Depending on the application, one may define Z_obj(s) = ∑_[x]∈ O(x_0)e^-sC([x]), where C([x]) is the least path cost reaching the class. The two functions answer different questions and should be stored together.
Exact mathematical passage
Z_{\mathrm{obj}}(s) = \sum_{[x]\in\Ocal(x_0)}e^{-sC([x])},
e^{-scn}Use in source · 1 passage
Search aliases: e^-scn
Use in source · Universal_Generative_Deformation_Calculus.tex, line 703
[Geometric convergence bound] Suppose every vertex has at most B children and every direction costs at least c>0. Then Z_x_0(s) ≤ ∑_n≥1B^ne^-scn. Consequently, Z_x_0(s) converges whenever sc>log B. In a full regular B-ary tree with constant cost c, it diverges whenever sc≤log B.
Exact mathematical passage
Z_{x_0}(s) \le \sum_{n\ge1}B^ne^{-scn}.
e^{-s\cost(\gamma)}Use in source · 1 passage
Search aliases: e^-scost(γ) cost gamma operatorname gamma
Use in source · Universal_Generative_Deformation_Calculus.tex, line 1463
Path partition function · Z_x_0(s)=∑_γ≠∅e^-scost(γ)
Exact mathematical passage
Z_{x_0}(s)=\sum_{\gamma\ne\varnothing}e^{-s\cost(\gamma)}
e_0Use in source · 2 passages
Search aliases: e_0
Use in source · Universal_Generative_Deformation_Calculus.tex, line 866
[Recursive deformation stability] If the direction mismatch at step k is at most ϵ_k, then the endpoint error satisfies e_n ≤ (∏_k=1^nL_k)e_0 + ∑_j=1^n (M_jϵ_j+η_j) ∏_k=j+1^nL_k.
Exact mathematical passage
e_n \le \left(\prod_{k=1}^nL_k\right)e_0 + \sum_{j=1}^n (M_j\epsilon_j+\eta_j) \prod_{k=j+1}^nL_k.
e_kUse in source · 4 passages
Search aliases: e_k
Use in source · Universal_Generative_Deformation_Calculus.tex, line 421
Set e_k=d_X(x_k,widehat x_k). The hypothesis gives e_k≤ L_ke_k-1+M_kϵ_k. Repeated substitution yields the displayed formula.
Exact mathematical passage
e_k=d_X(x_k,\widehat x_k)
e_{k-1}Use in source · 3 passages
Search aliases: e_k-1
Use in source · Universal_Generative_Deformation_Calculus.tex, line 422
Set e_k=d_X(x_k,widehat x_k). The hypothesis gives e_k≤ L_ke_k-1+M_kϵ_k. Repeated substitution yields the displayed formula.
Exact mathematical passage
e_k\le L_ke_{k-1}+M_k\epsilon_k.
e_nUse in source · 2 passages
Search aliases: e_n
Use in source · Universal_Generative_Deformation_Calculus.tex, line 866
[Recursive deformation stability] If the direction mismatch at step k is at most ϵ_k, then the endpoint error satisfies e_n ≤ (∏_k=1^nL_k)e_0 + ∑_j=1^n (M_jϵ_j+η_j) ∏_k=j+1^nL_k.
Exact mathematical passage
e_n \le \left(\prod_{k=1}^nL_k\right)e_0 + \sum_{j=1}^n (M_j\epsilon_j+\eta_j) \prod_{k=j+1}^nL_k.
\EcalUse in source · 1 passage
Search aliases: Ecal E mathcal
Use in source · Universal_Generative_Deformation_Calculus.tex, line 450
Sampling is an execution method, not the definition of the continuous forest. A continuous family is stored symbolically as an edge bundle E_x = {(d,x⋆ d):d∈ D_x}. Finite nets approximate this bundle. Refinement replaces one coarse parameter cell by smaller cells. The cell itself remains a first-class object with a radius and a coverage certificate.
Exact mathematical passage
\Ecal_x = \{(d,x\star d):d\in\Dcal_x\}.
\Ecal_xUse in source · 1 passage
Search aliases: Ecal_x E_x Ecal mathcal
Use in source · Universal_Generative_Deformation_Calculus.tex, line 450
Sampling is an execution method, not the definition of the continuous forest. A continuous family is stored symbolically as an edge bundle E_x = {(d,x⋆ d):d∈ D_x}. Finite nets approximate this bundle. Refinement replaces one coarse parameter cell by smaller cells. The cell itself remains a first-class object with a radius and a coverage certificate.
Exact mathematical passage
\Ecal_x = \{(d,x\star d):d\in\Dcal_x\}.
\expUse in source · 1 passage
Search aliases: exp
Use in source · Universal_Generative_Deformation_Calculus.tex, line 693
Define Z_x_0(s) = ∑_γ≠∅ exp(-scost(γ)). This is a path partition function. It weights short, inexpensive routes more strongly than long, expensive routes.
Exact mathematical passage
Z_{x_0}(s) = \sum_{\gamma\ne\varnothing} \exp\bigl(-s\cost(\gamma)\bigr).
fUse in source · 1 passage
Search aliases: f
Use in source · Universal_Generative_Deformation_Calculus.tex, line 946
Start from f_0(x)=cosh x. The declared directions are differentiation, reciprocal formation, logarithmic differentiation, squaring, even projection, odd projection, and a restricted argument-doubling direction. The first three give D(cosh x)=sinh x, R(cosh x)=sechx, L(cosh x)=tanh x. Applying the same grammar again produces cschx, coth x, powers, parity components, and doubled arguments. Reciprocal and logarithmic derivative are rejected at the zero expression and the failed attempts remain in t…
Exact mathematical passage
f_0(x)=\cosh x.
f_0Use in source · 1 passage
Search aliases: f_0
Use in source · Universal_Generative_Deformation_Calculus.tex, line 946
Start from f_0(x)=cosh x. The declared directions are differentiation, reciprocal formation, logarithmic differentiation, squaring, even projection, odd projection, and a restricted argument-doubling direction. The first three give D(cosh x)=sinh x, R(cosh x)=sechx, L(cosh x)=tanh x. Applying the same grammar again produces cschx, coth x, powers, parity components, and doubled arguments. Reciprocal and logarithmic derivative are rejected at the zero expression and the failed attempts remain in t…
Exact mathematical passage
f_0(x)=\cosh x.
GUse in source · 1 passage
Search aliases: G
Use in source · Universal_Generative_Deformation_Calculus.tex, line 1069
For the seed P_3, the first descendants include overlineP_3, L(P_3)≅ P_2, P_3 with one attached leaf, P_4 obtained by subdivision. If ρ reverses the vertex labels, then key(G)=key(ρ Gρ^-1) exactly, although the two raw paths and raw adjacency matrices remain distinct.
Exact mathematical passage
\operatorname{key}(G)=\operatorname{key}(\rho G\rho^{-1})
HUse in source · 7 passages
Search aliases: H
Use in source · Universal_Generative_Deformation_Calculus.tex, line 1132
The implementation works directly after quotienting global phase. A state is one of the six exact Bloch-axis vectors (0,0,±1), (±1,0,0), (0,±1,0), representing the Pauli eigenstates |0⟩, |1⟩, |+⟩, |-⟩, |+i⟩, and |-i⟩. The primitive grammar contains H,S,X,Z, acting as signed permutations of the Bloch coordinates: H(x,y,z)=(z,-y,x), S(x,y,z)=(-y,x,z), X(x,y,z)=(x,-y,-z), Z(x,y,z)=(-x,-y,z). These rules are exact and avoid floating-point phase canonicalization.
Exact mathematical passage
H,S,X,Z
\mathcal HUse in source · 2 passages
Search aliases: H mathcal
Use in source · Universal_Generative_Deformation_Calculus.tex, line 600
For a finite or countable depth layer, define H_n=ℓ^2( T_n(x_0)). The canonical basis vector associated with a path γ is written lvertγ⟩.
Exact mathematical passage
\mathcal H_n=\ell^2(\Tcal_n(x_0)).
hUse in source · 5 passages
Search aliases: h
Use in source · Universal_Generative_Deformation_Calculus.tex, line 675
The raw branching entropy and object entropy are h_raw = limsup_n→∞frac1nlog a_n, h_obj = limsup_n→∞frac1nlog q_n. Always h_obj≤ h_raw. The gap measures how much formal construction redundancy is removed by identities and symmetries.
Exact mathematical passage
h_{\mathrm{raw}} = \limsup_{n\to\infty}\frac1n\log a_n, \qquad h_{\mathrm{obj}} = \limsup_{n\to\infty}\frac1n\log q_n.
H^2Use in source · 1 passage
Search aliases: H^2
Use in source · Universal_Generative_Deformation_Calculus.tex, line 1151
Starting from |0⟩, one has |0⟩xrightarrowH|+⟩ xrightarrowS|+i⟩ xrightarrowS|-⟩, while H^2=X^2=Z^2=I on the projective orbit and S^4=I. The number of raw gate words grows as 4^n, but the quotient orbit saturates at six objects.
Exact mathematical passage
H^2=X^2=Z^2=I
\mathcal H_nUse in source · 1 passage
Search aliases: H_n mathcal
Use in source · Universal_Generative_Deformation_Calculus.tex, line 600
For a finite or countable depth layer, define H_n=ℓ^2( T_n(x_0)). The canonical basis vector associated with a path γ is written lvertγ⟩.
Exact mathematical passage
\mathcal H_n=\ell^2(\Tcal_n(x_0)).
h_{\mathrm{obj}}Use in source · 3 passages
Search aliases: h_obj mathrm
Use in source · Universal_Generative_Deformation_Calculus.tex, line 675
The raw branching entropy and object entropy are h_raw = limsup_n→∞frac1nlog a_n, h_obj = limsup_n→∞frac1nlog q_n. Always h_obj≤ h_raw. The gap measures how much formal construction redundancy is removed by identities and symmetries.
Exact mathematical passage
h_{\mathrm{raw}} = \limsup_{n\to\infty}\frac1n\log a_n, \qquad h_{\mathrm{obj}} = \limsup_{n\to\infty}\frac1n\log q_n.
h_{\mathrm{raw}}Use in source · 3 passages
Search aliases: h_raw mathrm
Use in source · Universal_Generative_Deformation_Calculus.tex, line 675
The raw branching entropy and object entropy are h_raw = limsup_n→∞frac1nlog a_n, h_obj = limsup_n→∞frac1nlog q_n. Always h_obj≤ h_raw. The gap measures how much formal construction redundancy is removed by identities and symmetries.
Exact mathematical passage
h_{\mathrm{raw}} = \limsup_{n\to\infty}\frac1n\log a_n, \qquad h_{\mathrm{obj}} = \limsup_{n\to\infty}\frac1n\log q_n.
IUse in source · 6 passages
Search aliases: I
Use in source · Universal_Generative_Deformation_Calculus.tex, line 637
A complete cut is a prefix-free set Γ meeting every infinite route exactly once. Its projectors satisfy ∑_α∈ΓP_α=I. Thus one branch may be resolved deeply while another stops early. Adaptive exploration can be represented without discarding the unresolved region.
Exact mathematical passage
\sum_{\alpha\in\Gamma}P_\alpha=I.
iUse in source · 3 passages
Search aliases: i
Use in source · Universal_Generative_Deformation_Calculus.tex, line 1132
The implementation works directly after quotienting global phase. A state is one of the six exact Bloch-axis vectors (0,0,±1), (±1,0,0), (0,±1,0), representing the Pauli eigenstates |0⟩, |1⟩, |+⟩, |-⟩, |+i⟩, and |-i⟩. The primitive grammar contains H,S,X,Z, acting as signed permutations of the Bloch coordinates: H(x,y,z)=(z,-y,x), S(x,y,z)=(-y,x,z), X(x,y,z)=(x,-y,-z), Z(x,y,z)=(-x,-y,z). These rules are exact and avoid floating-point phase canonicalization.
Exact mathematical passage
|+i\rangle
JUse in source · 1 passage
Search aliases: J
Use in source · Universal_Generative_Deformation_Calculus.tex, line 1090
0 · 1. The declared matrix directions are transpose, negation, squaring, conjugation by the coordinate swap, symmetrization, commutation with J=0 · -1 1 · 0, and adjugation. Every result is stored as an exact integer matrix. The metadata include trace, determinant, rank, symmetry, and squared Frobenius norm.
Exact mathematical passage
J=\begin{pmatrix}0&-1\\1&0\end{pmatrix},
jUse in source · 4 passages
Search aliases: j
Use in source · Universal_Generative_Deformation_Calculus.tex, line 407
[Pathwise cover error] Consider an exact route x_k=x_k-1⋆ d_k and a sampled route widehat x_k=widehat x_k-1⋆widehat d_k. Suppose at step k, d_X(x⋆ d,y⋆ e) ≤ L_k d_X(x,y)+M_kρ(d,e), and ρ(d_k,widehat d_k)≤ϵ_k. Then d_X(x_n,widehat x_n) ≤ (∏_k=1^nL_k)d_X(x_0,widehat x_0) + ∑_j=1^n M_jϵ_j ∏_k=j+1^nL_k.
Exact mathematical passage
\boxed{ d_X(x_n,\widehat x_n) \le \left(\prod_{k=1}^nL_k\right)d_X(x_0,\widehat x_0) + \sum_{j=1}^n M_j\epsilon_j \prod_{k=j+1}^nL_k. }
KUse in source · 3 passages
Search aliases: K
Use in source · Universal_Generative_Deformation_Calculus.tex, line 327
A computational realization uses a canonical key K: X⟶ K with K(x)=K(y) Longleftrightarrow x∼ y for the declared finite example. Exact polynomial coefficients, reduced rational numbers, canonical symbolic expressions, and normalized matrices are suitable examples. Floating point rounding alone is not an exact quotient relation.
Exact mathematical passage
K:\Xcal\longrightarrow\mathcal K
\mathcal KUse in source · 2 passages
Search aliases: K mathcal
Use in source · Universal_Generative_Deformation_Calculus.tex, line 327
A computational realization uses a canonical key K: X⟶ K with K(x)=K(y) Longleftrightarrow x∼ y for the declared finite example. Exact polynomial coefficients, reduced rational numbers, canonical symbolic expressions, and normalized matrices are suitable examples. Floating point rounding alone is not an exact quotient relation.
Exact mathematical passage
K:\Xcal\longrightarrow\mathcal K
K_0Use in source · 1 passage
Search aliases: K_0
Use in source · Universal_Generative_Deformation_Calculus.tex, line 1165
Take the rational triangle K_0={(0,0),(2,0),(1,frac32)}. The grammar contains quarter turns, reflection in the x axis, horizontal shear, scaling by two, and unit translations in the two coordinate directions. Each descendant is again a finite rational point configuration, so the grammar is queried recursively at every child.
Exact mathematical passage
K_0=\left\{(0,0),(2,0),\left(1,\frac32\right)\right\}.
\operatorname{key}Use in source · 1 passage
Search aliases: key operatorname
Use in source · Universal_Generative_Deformation_Calculus.tex, line 1069
For the seed P_3, the first descendants include overlineP_3, L(P_3)≅ P_2, P_3 with one attached leaf, P_4 obtained by subdivision. If ρ reverses the vertex labels, then key(G)=key(ρ Gρ^-1) exactly, although the two raw paths and raw adjacency matrices remain distinct.
Exact mathematical passage
\operatorname{key}(G)=\operatorname{key}(\rho G\rho^{-1})
LUse in source · 12 passages
Search aliases: L
Use in source · Universal_Generative_Deformation_Calculus.tex, line 397
[Pathwise cover error] Consider an exact route x_k=x_k-1⋆ d_k and a sampled route widehat x_k=widehat x_k-1⋆widehat d_k. Suppose at step k, d_X(x⋆ d,y⋆ e) ≤ L_k d_X(x,y)+M_kρ(d,e), and ρ(d_k,widehat d_k)≤ϵ_k. Then d_X(x_n,widehat x_n) ≤ (∏_k=1^nL_k)d_X(x_0,widehat x_0) + ∑_j=1^n M_jϵ_j ∏_k=j+1^nL_k.
Exact mathematical passage
d_X(x\star d,y\star e) \le L_k d_X(x,y)+M_k\rho(d,e),
L_kUse in source · 9 passages
Search aliases: L_k
Use in source · Universal_Generative_Deformation_Calculus.tex, line 397
[Pathwise cover error] Consider an exact route x_k=x_k-1⋆ d_k and a sampled route widehat x_k=widehat x_k-1⋆widehat d_k. Suppose at step k, d_X(x⋆ d,y⋆ e) ≤ L_k d_X(x,y)+M_kρ(d,e), and ρ(d_k,widehat d_k)≤ϵ_k. Then d_X(x_n,widehat x_n) ≤ (∏_k=1^nL_k)d_X(x_0,widehat x_0) + ∑_j=1^n M_jϵ_j ∏_k=j+1^nL_k.
Exact mathematical passage
d_X(x\star d,y\star e) \le L_k d_X(x,y)+M_k\rho(d,e),
\limsupUse in source · 3 passages
Search aliases: limsup
Use in source · Universal_Generative_Deformation_Calculus.tex, line 675
The raw branching entropy and object entropy are h_raw = limsup_n→∞frac1nlog a_n, h_obj = limsup_n→∞frac1nlog q_n. Always h_obj≤ h_raw. The gap measures how much formal construction redundancy is removed by identities and symmetries.
Exact mathematical passage
h_{\mathrm{raw}} = \limsup_{n\to\infty}\frac1n\log a_n, \qquad h_{\mathrm{obj}} = \limsup_{n\to\infty}\frac1n\log q_n.
\limsup_{n\to\infty}Use in source · 1 passage
Search aliases: limsup_n→∞ limsup to infty
Use in source · Universal_Generative_Deformation_Calculus.tex, line 675
The raw branching entropy and object entropy are h_raw = limsup_n→∞frac1nlog a_n, h_obj = limsup_n→∞frac1nlog q_n. Always h_obj≤ h_raw. The gap measures how much formal construction redundancy is removed by identities and symmetries.
Exact mathematical passage
h_{\mathrm{raw}} = \limsup_{n\to\infty}\frac1n\log a_n, \qquad h_{\mathrm{obj}} = \limsup_{n\to\infty}\frac1n\log q_n.
\logUse in source · 9 passages
Search aliases: log
Use in source · Universal_Generative_Deformation_Calculus.tex, line 675
The raw branching entropy and object entropy are h_raw = limsup_n→∞frac1nlog a_n, h_obj = limsup_n→∞frac1nlog q_n. Always h_obj≤ h_raw. The gap measures how much formal construction redundancy is removed by identities and symmetries.
Exact mathematical passage
h_{\mathrm{raw}} = \limsup_{n\to\infty}\frac1n\log a_n, \qquad h_{\mathrm{obj}} = \limsup_{n\to\infty}\frac1n\log q_n.
\longleftarrowUse in source · 2 passages
Search aliases: longleftarrow
Use in source · Universal_Generative_Deformation_Calculus.tex, line 1233
For x^2=1 over powers of 3, the two simple roots lift uniquely: 1pmod 3longleftarrow1pmod 9longleftarrow1pmod27longleftarrow⋯, -1pmod 3longleftarrow-1pmod 9longleftarrow-1pmod27longleftarrow⋯. For the singular equation x^2=0 over powers of 2, some nodes have several valid next digits and some branches later die. Thus branching number is an object property detected by the local congruence geometry.
Exact mathematical passage
1\pmod 3\longleftarrow1\pmod 9\longleftarrow1\pmod{27}\longleftarrow\cdots,
\LongleftrightarrowUse in source · 1 passage
Search aliases: Longleftrightarrow
Use in source · Universal_Generative_Deformation_Calculus.tex, line 331
A computational realization uses a canonical key K: X⟶ K with K(x)=K(y) Longleftrightarrow x∼ y for the declared finite example. Exact polynomial coefficients, reduced rational numbers, canonical symbolic expressions, and normalized matrices are suitable examples. Floating point rounding alone is not an exact quotient relation.
Exact mathematical passage
K(x)=K(y) \quad\Longleftrightarrow\quad x\sim y
\lvertUse in source · 3 passages
Search aliases: lvert
Use in source · Universal_Generative_Deformation_Calculus.tex, line 603
For a finite or countable depth layer, define H_n=ℓ^2( T_n(x_0)). The canonical basis vector associated with a path γ is written lvertγ⟩.
Exact mathematical passage
\lvert\gamma\rangle
MUse in source · 10 passages
Search aliases: M
Use in source · Universal_Generative_Deformation_Calculus.tex, line 397
[Pathwise cover error] Consider an exact route x_k=x_k-1⋆ d_k and a sampled route widehat x_k=widehat x_k-1⋆widehat d_k. Suppose at step k, d_X(x⋆ d,y⋆ e) ≤ L_k d_X(x,y)+M_kρ(d,e), and ρ(d_k,widehat d_k)≤ϵ_k. Then d_X(x_n,widehat x_n) ≤ (∏_k=1^nL_k)d_X(x_0,widehat x_0) + ∑_j=1^n M_jϵ_j ∏_k=j+1^nL_k.
Exact mathematical passage
d_X(x\star d,y\star e) \le L_k d_X(x,y)+M_k\rho(d,e),
mUse in source · 12 passages
Search aliases: m
Use in source · Universal_Generative_Deformation_Calculus.tex, line 490
For m>n, d_X(x_m,x_n) ≤ ∑_k=n^m-1d_X(x_k+1,x_k) ≤ ∑_k=n^m-1a_k. The tail tends to zero. Completeness supplies a unique limit.
Exact mathematical passage
m>n
M_jUse in source · 3 passages
Search aliases: M_j
Use in source · Universal_Generative_Deformation_Calculus.tex, line 407
[Pathwise cover error] Consider an exact route x_k=x_k-1⋆ d_k and a sampled route widehat x_k=widehat x_k-1⋆widehat d_k. Suppose at step k, d_X(x⋆ d,y⋆ e) ≤ L_k d_X(x,y)+M_kρ(d,e), and ρ(d_k,widehat d_k)≤ϵ_k. Then d_X(x_n,widehat x_n) ≤ (∏_k=1^nL_k)d_X(x_0,widehat x_0) + ∑_j=1^n M_jϵ_j ∏_k=j+1^nL_k.
Exact mathematical passage
\boxed{ d_X(x_n,\widehat x_n) \le \left(\prod_{k=1}^nL_k\right)d_X(x_0,\widehat x_0) + \sum_{j=1}^n M_j\epsilon_j \prod_{k=j+1}^nL_k. }
M_kUse in source · 6 passages
Search aliases: M_k
Use in source · Universal_Generative_Deformation_Calculus.tex, line 397
[Pathwise cover error] Consider an exact route x_k=x_k-1⋆ d_k and a sampled route widehat x_k=widehat x_k-1⋆widehat d_k. Suppose at step k, d_X(x⋆ d,y⋆ e) ≤ L_k d_X(x,y)+M_kρ(d,e), and ρ(d_k,widehat d_k)≤ϵ_k. Then d_X(x_n,widehat x_n) ≤ (∏_k=1^nL_k)d_X(x_0,widehat x_0) + ∑_j=1^n M_jϵ_j ∏_k=j+1^nL_k.
Exact mathematical passage
d_X(x\star d,y\star e) \le L_k d_X(x,y)+M_k\rho(d,e),
NUse in source · 11 passages
Search aliases: N
Use in source · Universal_Generative_Deformation_Calculus.tex, line 381
Let each fiber D_x carry a metric ρ_x. A finite set N_ϵ(x)⊆ D_x is an ϵ-net when every d∈ D_x lies within ϵ of some widehat d∈ N_ϵ(x).
Exact mathematical passage
N_\epsilon(x)\subseteq\Dcal_x
n^{-1}Use in source · 2 passages
Search aliases: n^-1
Use in source · Universal_Generative_Deformation_Calculus.tex, line 1461
Raw entropy · h_raw=limsup n^-1log a_n
Exact mathematical passage
h_{\mathrm{raw}}=\limsup n^{-1}\log a_n
N_\epsilonUse in source · 2 passages
Search aliases: N_ϵ epsilon
Use in source · Universal_Generative_Deformation_Calculus.tex, line 381
Let each fiber D_x carry a metric ρ_x. A finite set N_ϵ(x)⊆ D_x is an ϵ-net when every d∈ D_x lies within ϵ of some widehat d∈ N_ϵ(x).
Exact mathematical passage
N_\epsilon(x)\subseteq\Dcal_x
\mathrm{obj}Use in source · 4 passages
Search aliases: obj mathrm
Use in source · Universal_Generative_Deformation_Calculus.tex, line 675
The raw branching entropy and object entropy are h_raw = limsup_n→∞frac1nlog a_n, h_obj = limsup_n→∞frac1nlog q_n. Always h_obj≤ h_raw. The gap measures how much formal construction redundancy is removed by identities and symmetries.
Exact mathematical passage
h_{\mathrm{raw}} = \limsup_{n\to\infty}\frac1n\log a_n, \qquad h_{\mathrm{obj}} = \limsup_{n\to\infty}\frac1n\log q_n.
\overline{P_3}Use in source · 1 passage
Search aliases: overlineP_3 overline
Use in source · Universal_Generative_Deformation_Calculus.tex, line 1059
For the seed P_3, the first descendants include overlineP_3, L(P_3)≅ P_2, P_3 with one attached leaf, P_4 obtained by subdivision. If ρ reverses the vertex labels, then key(G)=key(ρ Gρ^-1) exactly, although the two raw paths and raw adjacency matrices remain distinct.
Exact mathematical passage
\overline{P_3}, \qquad L(P_3)\cong P_2, \qquad P_3\text{ with one attached leaf}, \qquad P_4\text{ obtained by subdivision}.
PUse in source · 12 passages
Search aliases: P
Use in source · Universal_Generative_Deformation_Calculus.tex, line 606
For a prefix α of length k≤ n, define the cylinder projector P^(n)_α lvertγ⟩ = lvertγ⟩, · α is a prefix of γ, 0, · otherwise.
Exact mathematical passage
P^{(n)}_\alpha \lvert\gamma\rangle = \begin{cases} \lvert\gamma\rangle,&\alpha\text{ is a prefix of }\gamma,\\ 0,&\text{otherwise}. \end{cases}
pUse in source · 9 passages
Search aliases: p
Use in source · Universal_Generative_Deformation_Calculus.tex, line 915
Let p(x)=x^2+1, (Tp)(x)=p(x+1), (Dp)(x)=p'(x). The two primitive directions T and D satisfy D(Tp)=T(Dp)=2x+2. The raw path tree therefore contains two length-two routes, pxrightarrowTTpxrightarrowDDTp, pxrightarrowDDpxrightarrowTTDp, but the quotient graph contains one endpoint class because DTp=TDp exactly.
Exact mathematical passage
p(x)=x^2+1, \qquad (Tp)(x)=p(x+1), \qquad (Dp)(x)=p'(x).
p'Use in source · 1 passage
Search aliases: p'
Use in source · Universal_Generative_Deformation_Calculus.tex, line 915
Let p(x)=x^2+1, (Tp)(x)=p(x+1), (Dp)(x)=p'(x). The two primitive directions T and D satisfy D(Tp)=T(Dp)=2x+2. The raw path tree therefore contains two length-two routes, pxrightarrowTTpxrightarrowDDTp, pxrightarrowDDpxrightarrowTTDp, but the quotient graph contains one endpoint class because DTp=TDp exactly.
Exact mathematical passage
p(x)=x^2+1, \qquad (Tp)(x)=p(x+1), \qquad (Dp)(x)=p'(x).
P^{(n)}_\alphaUse in source · 3 passages
Search aliases: P^(n)_α alpha
Use in source · Universal_Generative_Deformation_Calculus.tex, line 606
For a prefix α of length k≤ n, define the cylinder projector P^(n)_α lvertγ⟩ = lvertγ⟩, · α is a prefix of γ, 0, · otherwise.
Exact mathematical passage
P^{(n)}_\alpha \lvert\gamma\rangle = \begin{cases} \lvert\gamma\rangle,&\alpha\text{ is a prefix of }\gamma,\\ 0,&\text{otherwise}. \end{cases}
P^{(n)}_{\alpha d}Use in source · 1 passage
Search aliases: P^(n)_α d alpha
Use in source · Universal_Generative_Deformation_Calculus.tex, line 618
[Exact cylinder refinement] For every prefix α and every depth n>absα, P^(n)_α = ∑_d∈ D_ev(α)P^(n)_α d, with strong convergence when the child set is countable.
Exact mathematical passage
\boxed{ P^{(n)}_\alpha = \sum_{d\in\Dcal_{\ev(\alpha)}}P^{(n)}_{\alpha d}, }
p^mUse in source · 1 passage
Search aliases: p^m
Use in source · Universal_Generative_Deformation_Calculus.tex, line 1230
For a prime p, a state at level m is a residue class modulo p^m satisfying a declared congruence. Its direction fiber is the finite set of next base-p digits that lift the current solution to level m+1.
Exact mathematical passage
p^m
P_2Use in source · 1 passage
Search aliases: P_2
Use in source · Universal_Generative_Deformation_Calculus.tex, line 1059
For the seed P_3, the first descendants include overlineP_3, L(P_3)≅ P_2, P_3 with one attached leaf, P_4 obtained by subdivision. If ρ reverses the vertex labels, then key(G)=key(ρ Gρ^-1) exactly, although the two raw paths and raw adjacency matrices remain distinct.
Exact mathematical passage
\overline{P_3}, \qquad L(P_3)\cong P_2, \qquad P_3\text{ with one attached leaf}, \qquad P_4\text{ obtained by subdivision}.
P_3Use in source · 4 passages
Search aliases: P_3
Use in source · Universal_Generative_Deformation_Calculus.tex, line 1054
Begin with the path graph P_3 on three vertices. The finite grammar contains graph complement, line-graph formation, leaf attachment, edge subdivision, and a relabeling direction. Raw states are labeled adjacency matrices. For a graph with n≤6 vertices, the quotient key is the lexicographically least adjacency bit string over all n! vertex permutations.
Exact mathematical passage
P_3
P_4Use in source · 1 passage
Search aliases: P_4
Use in source · Universal_Generative_Deformation_Calculus.tex, line 1059
For the seed P_3, the first descendants include overlineP_3, L(P_3)≅ P_2, P_3 with one attached leaf, P_4 obtained by subdivision. If ρ reverses the vertex labels, then key(G)=key(ρ Gρ^-1) exactly, although the two raw paths and raw adjacency matrices remain distinct.
Exact mathematical passage
\overline{P_3}, \qquad L(P_3)\cong P_2, \qquad P_3\text{ with one attached leaf}, \qquad P_4\text{ obtained by subdivision}.
P_\alphaUse in source · 5 passages
Search aliases: P_α alpha
Use in source · Universal_Generative_Deformation_Calculus.tex, line 637
A complete cut is a prefix-free set Γ meeting every infinite route exactly once. Its projectors satisfy ∑_α∈ΓP_α=I. Thus one branch may be resolved deeply while another stops early. Adaptive exploration can be represented without discarding the unresolved region.
Exact mathematical passage
\sum_{\alpha\in\Gamma}P_\alpha=I.
P_{\alpha d}Use in source · 1 passage
Search aliases: P_α d alpha
Use in source · Universal_Generative_Deformation_Calculus.tex, line 1458
Projector refinement · P_α=∑_dP_α d
Exact mathematical passage
P_\alpha=\sum_dP_{\alpha d}
\pmodUse in source · 2 passages
Search aliases: pmod
Use in source · Universal_Generative_Deformation_Calculus.tex, line 1233
For x^2=1 over powers of 3, the two simple roots lift uniquely: 1pmod 3longleftarrow1pmod 9longleftarrow1pmod27longleftarrow⋯, -1pmod 3longleftarrow-1pmod 9longleftarrow-1pmod27longleftarrow⋯. For the singular equation x^2=0 over powers of 2, some nodes have several valid next digits and some branches later die. Thus branching number is an object property detected by the local congruence geometry.
Exact mathematical passage
1\pmod 3\longleftarrow1\pmod 9\longleftarrow1\pmod{27}\longleftarrow\cdots,
\preceqUse in source · 1 passage
Search aliases: preceq
Use in source · Universal_Generative_Deformation_Calculus.tex, line 1457
Cylinder projector · P^(n)_αlvertγ⟩=1_αpreceqγlvertγ⟩
Exact mathematical passage
P^{(n)}_\alpha\lvert\gamma\rangle=1_{\alpha\preceq\gamma}\lvert\gamma\rangle
QUse in source · 3 passages
Search aliases: Q
Use in source · Universal_Generative_Deformation_Calculus.tex, line 644
Let A: H→ K be a bounded operator and let {P_α} and {Q_β} be complete route resolutions. Define A_βα=Q_β A P_α. Then A=∑_α,βA_βα. If B follows A, then (BA)_γα = ∑_β B_γβA_βα. Every operator composition becomes a sum over compatible intermediate perturbation packets.
Exact mathematical passage
\{Q_\beta\}
qUse in source · 3 passages
Search aliases: q
Use in source · Universal_Generative_Deformation_Calculus.tex, line 669
Define a_n=# T_n(x_0) for raw paths and q_n=#(ev( T_n(x_0))/∼) for distinct quotient endpoints at exact depth n.
Exact mathematical passage
q_n=\#\bigl(\ev(\Tcal_n(x_0))/\sim\bigr)
q_nUse in source · 3 passages
Search aliases: q_n
Use in source · Universal_Generative_Deformation_Calculus.tex, line 669
Define a_n=# T_n(x_0) for raw paths and q_n=#(ev( T_n(x_0))/∼) for distinct quotient endpoints at exact depth n.
Exact mathematical passage
q_n=\#\bigl(\ev(\Tcal_n(x_0))/\sim\bigr)
Q_\betaUse in source · 3 passages
Search aliases: Q_β beta
Use in source · Universal_Generative_Deformation_Calculus.tex, line 644
Let A: H→ K be a bounded operator and let {P_α} and {Q_β} be complete route resolutions. Define A_βα=Q_β A P_α. Then A=∑_α,βA_βα. If B follows A, then (BA)_γα = ∑_β B_γβA_βα. Every operator composition becomes a sum over compatible intermediate perturbation packets.
Exact mathematical passage
\{Q_\beta\}
RUse in source · 4 passages
Search aliases: R
Use in source · Universal_Generative_Deformation_Calculus.tex, line 950
Start from f_0(x)=cosh x. The declared directions are differentiation, reciprocal formation, logarithmic differentiation, squaring, even projection, odd projection, and a restricted argument-doubling direction. The first three give D(cosh x)=sinh x, R(cosh x)=sechx, L(cosh x)=tanh x. Applying the same grammar again produces cschx, coth x, powers, parity components, and doubled arguments. Reciprocal and logarithmic derivative are rejected at the zero expression and the failed attempts remain in t…
Exact mathematical passage
D(\cosh x)=\sinh x, \qquad R(\cosh x)=\operatorname{sech}x, \qquad L(\cosh x)=\tanh x.
R^4Use in source · 1 passage
Search aliases: R^4
Use in source · Universal_Generative_Deformation_Calculus.tex, line 1178
For example, R(x,y)=(-y,x), H(x,y)=(x+y,y), S_2(x,y)=(2x,2y). The relation R^4=I closes one route loop exactly. Reflection is an involution. Rotation preserves signed area, reflection reverses its sign, shear preserves it, and scaling by two multiplies it by four. These invariants separate visually similar but mathematically different descendants.
Exact mathematical passage
R^4=I
\mathrm{raw}Use in source · 3 passages
Search aliases: raw mathrm
Use in source · Universal_Generative_Deformation_Calculus.tex, line 675
The raw branching entropy and object entropy are h_raw = limsup_n→∞frac1nlog a_n, h_obj = limsup_n→∞frac1nlog q_n. Always h_obj≤ h_raw. The gap measures how much formal construction redundancy is removed by identities and symmetries.
Exact mathematical passage
h_{\mathrm{raw}} = \limsup_{n\to\infty}\frac1n\log a_n, \qquad h_{\mathrm{obj}} = \limsup_{n\to\infty}\frac1n\log q_n.
\ReachUse in source · 2 passages
Search aliases: Reach operatorname
Use in source · Universal_Generative_Deformation_Calculus.tex, line 252
[Exact reachability by depth] For every n≥0, ev( T_n(x_0)) = Reach_n(x_0), where Reach_n(x_0) is the set obtained by applying exactly n successive admissible directions.
Exact mathematical passage
\ev(\Tcal_n(x_0)) = \Reach_n(x_0),
\Reach_nUse in source · 2 passages
Search aliases: Reach_n Reach operatorname
Use in source · Universal_Generative_Deformation_Calculus.tex, line 252
[Exact reachability by depth] For every n≥0, ev( T_n(x_0)) = Reach_n(x_0), where Reach_n(x_0) is the set obtained by applying exactly n successive admissible directions.
Exact mathematical passage
\ev(\Tcal_n(x_0)) = \Reach_n(x_0),
SUse in source · 9 passages
Search aliases: S
Use in source · Universal_Generative_Deformation_Calculus.tex, line 1104
0 · 1. They satisfy (A^ T)^ T=A, Aadj(A)=det(A)I, det(SAS)=det(A) for the swap matrix S. Thus involution, adjugate, and conjugacy relations generate quotient identifications, while squaring and commutation usually move to new invariant values.
Exact mathematical passage
(A^{\mathsf T})^{\mathsf T}=A, \qquad A\operatorname{adj}(A)=\det(A)I, \qquad \det(SAS)=\det(A)
sUse in source · 11 passages
Search aliases: s
Use in source · Universal_Generative_Deformation_Calculus.tex, line 571
Consider u=s+λ u^2. Repeated substitution expands u into rooted binary trees. If only shape is counted, the number of binary trees with n internal vertices is the Catalan number C_n.
Exact mathematical passage
u=s+\lambda u^2.
S^4Use in source · 1 passage
Search aliases: S^4
Use in source · Universal_Generative_Deformation_Calculus.tex, line 1151
Starting from |0⟩, one has |0⟩xrightarrowH|+⟩ xrightarrowS|+i⟩ xrightarrowS|-⟩, while H^2=X^2=Z^2=I on the projective orbit and S^4=I. The number of raw gate words grows as 4^n, but the quotient orbit saturates at six objects.
Exact mathematical passage
S^4=I
S_2Use in source · 1 passage
Search aliases: S_2
Use in source · Universal_Generative_Deformation_Calculus.tex, line 1171
For example, R(x,y)=(-y,x), H(x,y)=(x+y,y), S_2(x,y)=(2x,2y). The relation R^4=I closes one route loop exactly. Reflection is an involution. Rotation preserves signed area, reflection reverses its sign, shear preserves it, and scaling by two multiplies it by four. These invariants separate visually similar but mathematically different descendants.
Exact mathematical passage
R(x,y)=(-y,x), \qquad H(x,y)=(x+y,y), \qquad S_2(x,y)=(2x,2y).
\operatorname{sech}Use in source · 3 passages
Search aliases: sech operatorname
Use in source · Universal_Generative_Deformation_Calculus.tex, line 950
Start from f_0(x)=cosh x. The declared directions are differentiation, reciprocal formation, logarithmic differentiation, squaring, even projection, odd projection, and a restricted argument-doubling direction. The first three give D(cosh x)=sinh x, R(cosh x)=sechx, L(cosh x)=tanh x. Applying the same grammar again produces cschx, coth x, powers, parity components, and doubled arguments. Reciprocal and logarithmic derivative are rejected at the zero expression and the failed attempts remain in t…
Exact mathematical passage
D(\cosh x)=\sinh x, \qquad R(\cosh x)=\operatorname{sech}x, \qquad L(\cosh x)=\tanh x.
\sinhUse in source · 1 passage
Search aliases: sinh
Use in source · Universal_Generative_Deformation_Calculus.tex, line 950
Start from f_0(x)=cosh x. The declared directions are differentiation, reciprocal formation, logarithmic differentiation, squaring, even projection, odd projection, and a restricted argument-doubling direction. The first three give D(cosh x)=sinh x, R(cosh x)=sechx, L(cosh x)=tanh x. Applying the same grammar again produces cschx, coth x, powers, parity components, and doubled arguments. Reciprocal and logarithmic derivative are rejected at the zero expression and the failed attempts remain in t…
Exact mathematical passage
D(\cosh x)=\sinh x, \qquad R(\cosh x)=\operatorname{sech}x, \qquad L(\cosh x)=\tanh x.
TUse in source · 5 passages
Search aliases: T
Use in source · Universal_Generative_Deformation_Calculus.tex, line 915
Let p(x)=x^2+1, (Tp)(x)=p(x+1), (Dp)(x)=p'(x). The two primitive directions T and D satisfy D(Tp)=T(Dp)=2x+2. The raw path tree therefore contains two length-two routes, pxrightarrowTTpxrightarrowDDTp, pxrightarrowDDpxrightarrowTTDp, but the quotient graph contains one endpoint class because DTp=TDp exactly.
Exact mathematical passage
p(x)=x^2+1, \qquad (Tp)(x)=p(x+1), \qquad (Dp)(x)=p'(x).
tUse in source · 1 passage
Search aliases: t
Use in source · Universal_Generative_Deformation_Calculus.tex, line 164
Let T be a set of types and let X=coprod_t∈ TX_t be the disjoint union of the corresponding object spaces.
Exact mathematical passage
\Xcal=\coprod_{t\in\mathsf T}X_t
\tanhUse in source · 3 passages
Search aliases: tanh
Use in source · Universal_Generative_Deformation_Calculus.tex, line 950
Start from f_0(x)=cosh x. The declared directions are differentiation, reciprocal formation, logarithmic differentiation, squaring, even projection, odd projection, and a restricted argument-doubling direction. The first three give D(cosh x)=sinh x, R(cosh x)=sechx, L(cosh x)=tanh x. Applying the same grammar again produces cschx, coth x, powers, parity components, and doubled arguments. Reciprocal and logarithmic derivative are rejected at the zero expression and the failed attempts remain in t…
Exact mathematical passage
D(\cosh x)=\sinh x, \qquad R(\cosh x)=\operatorname{sech}x, \qquad L(\cosh x)=\tanh x.
uUse in source · 2 passages
Search aliases: u
Use in source · Universal_Generative_Deformation_Calculus.tex, line 571
Consider u=s+λ u^2. Repeated substitution expands u into rooted binary trees. If only shape is counted, the number of binary trees with n internal vertices is the Catalan number C_n.
Exact mathematical passage
u=s+\lambda u^2.
u^2Use in source · 1 passage
Search aliases: u^2
Use in source · Universal_Generative_Deformation_Calculus.tex, line 571
Consider u=s+λ u^2. Repeated substitution expands u into rooted binary trees. If only shape is counted, the number of binary trees with n internal vertices is the Catalan number C_n.
Exact mathematical passage
u=s+\lambda u^2.
vUse in source · 4 passages
Search aliases: v
Use in source · Universal_Generative_Deformation_Calculus.tex, line 1017
Represent a one-dimensional Gaussian law by the exact parameter pair (μ,v), where v=σ^2>0. The grammar contains shifts by ±1, reflection, convolution with N(0,1), centering, scaling by two, and standardization. Their parameter actions are (μ,v) · ↦(μ±1,v), · (μ,v) · ↦(-μ,v),
Exact mathematical passage
(\mu,v)
\widehat dUse in source · 3 passages
Search aliases: widehat d widehat
Use in source · Universal_Generative_Deformation_Calculus.tex, line 381
Let each fiber D_x carry a metric ρ_x. A finite set N_ϵ(x)⊆ D_x is an ϵ-net when every d∈ D_x lies within ϵ of some widehat d∈ N_ϵ(x).
Exact mathematical passage
\widehat d\in N_\epsilon(x)
\widehat d_kUse in source · 2 passages
Search aliases: widehat d_k widehat
Use in source · Universal_Generative_Deformation_Calculus.tex, line 393
[Pathwise cover error] Consider an exact route x_k=x_k-1⋆ d_k and a sampled route widehat x_k=widehat x_k-1⋆widehat d_k. Suppose at step k, d_X(x⋆ d,y⋆ e) ≤ L_k d_X(x,y)+M_kρ(d,e), and ρ(d_k,widehat d_k)≤ϵ_k. Then d_X(x_n,widehat x_n) ≤ (∏_k=1^nL_k)d_X(x_0,widehat x_0) + ∑_j=1^n M_jϵ_j ∏_k=j+1^nL_k.
Exact mathematical passage
\widehat x_k=\widehat x_{k-1}\star\widehat d_k.
\widehat xUse in source · 3 passages
Search aliases: widehat x widehat
Use in source · Universal_Generative_Deformation_Calculus.tex, line 393
[Pathwise cover error] Consider an exact route x_k=x_k-1⋆ d_k and a sampled route widehat x_k=widehat x_k-1⋆widehat d_k. Suppose at step k, d_X(x⋆ d,y⋆ e) ≤ L_k d_X(x,y)+M_kρ(d,e), and ρ(d_k,widehat d_k)≤ϵ_k. Then d_X(x_n,widehat x_n) ≤ (∏_k=1^nL_k)d_X(x_0,widehat x_0) + ∑_j=1^n M_jϵ_j ∏_k=j+1^nL_k.
Exact mathematical passage
\widehat x_k=\widehat x_{k-1}\star\widehat d_k.
\widehat x_0Use in source · 1 passage
Search aliases: widehat x_0 widehat
Use in source · Universal_Generative_Deformation_Calculus.tex, line 407
[Pathwise cover error] Consider an exact route x_k=x_k-1⋆ d_k and a sampled route widehat x_k=widehat x_k-1⋆widehat d_k. Suppose at step k, d_X(x⋆ d,y⋆ e) ≤ L_k d_X(x,y)+M_kρ(d,e), and ρ(d_k,widehat d_k)≤ϵ_k. Then d_X(x_n,widehat x_n) ≤ (∏_k=1^nL_k)d_X(x_0,widehat x_0) + ∑_j=1^n M_jϵ_j ∏_k=j+1^nL_k.
Exact mathematical passage
\boxed{ d_X(x_n,\widehat x_n) \le \left(\prod_{k=1}^nL_k\right)d_X(x_0,\widehat x_0) + \sum_{j=1}^n M_j\epsilon_j \prod_{k=j+1}^nL_k. }
\widehat x_kUse in source · 2 passages
Search aliases: widehat x_k widehat
Use in source · Universal_Generative_Deformation_Calculus.tex, line 393
[Pathwise cover error] Consider an exact route x_k=x_k-1⋆ d_k and a sampled route widehat x_k=widehat x_k-1⋆widehat d_k. Suppose at step k, d_X(x⋆ d,y⋆ e) ≤ L_k d_X(x,y)+M_kρ(d,e), and ρ(d_k,widehat d_k)≤ϵ_k. Then d_X(x_n,widehat x_n) ≤ (∏_k=1^nL_k)d_X(x_0,widehat x_0) + ∑_j=1^n M_jϵ_j ∏_k=j+1^nL_k.
Exact mathematical passage
\widehat x_k=\widehat x_{k-1}\star\widehat d_k.
\widehat x_{k-1}Use in source · 1 passage
Search aliases: widehat x_k-1 widehat
Use in source · Universal_Generative_Deformation_Calculus.tex, line 393
[Pathwise cover error] Consider an exact route x_k=x_k-1⋆ d_k and a sampled route widehat x_k=widehat x_k-1⋆widehat d_k. Suppose at step k, d_X(x⋆ d,y⋆ e) ≤ L_k d_X(x,y)+M_kρ(d,e), and ρ(d_k,widehat d_k)≤ϵ_k. Then d_X(x_n,widehat x_n) ≤ (∏_k=1^nL_k)d_X(x_0,widehat x_0) + ∑_j=1^n M_jϵ_j ∏_k=j+1^nL_k.
Exact mathematical passage
\widehat x_k=\widehat x_{k-1}\star\widehat d_k.
\widehat x_nUse in source · 1 passage
Search aliases: widehat x_n widehat
Use in source · Universal_Generative_Deformation_Calculus.tex, line 407
[Pathwise cover error] Consider an exact route x_k=x_k-1⋆ d_k and a sampled route widehat x_k=widehat x_k-1⋆widehat d_k. Suppose at step k, d_X(x⋆ d,y⋆ e) ≤ L_k d_X(x,y)+M_kρ(d,e), and ρ(d_k,widehat d_k)≤ϵ_k. Then d_X(x_n,widehat x_n) ≤ (∏_k=1^nL_k)d_X(x_0,widehat x_0) + ∑_j=1^n M_jϵ_j ∏_k=j+1^nL_k.
Exact mathematical passage
\boxed{ d_X(x_n,\widehat x_n) \le \left(\prod_{k=1}^nL_k\right)d_X(x_0,\widehat x_0) + \sum_{j=1}^n M_j\epsilon_j \prod_{k=j+1}^nL_k. }
\widetilde\PhiUse in source · 2 passages
Search aliases: widetildeΦ widetilde Phi
Use in source · Universal_Generative_Deformation_Calculus.tex, line 856
Let Φ_k and widetildeΦ_k be exact and perturbed deformation laws along corresponding routes. Suppose d_X(Φ_k(x,d),widetildeΦ_k(y,e)) ≤ L_kd_X(x,y)+M_kρ(d,e)+η_k, where η_k measures model error.
Exact mathematical passage
\widetilde\Phi_k
\widetilde\Phi_kUse in source · 2 passages
Search aliases: widetildeΦ_k widetilde Phi
Use in source · Universal_Generative_Deformation_Calculus.tex, line 856
Let Φ_k and widetildeΦ_k be exact and perturbed deformation laws along corresponding routes. Suppose d_X(Φ_k(x,d),widetildeΦ_k(y,e)) ≤ L_kd_X(x,y)+M_kρ(d,e)+η_k, where η_k measures model error.
Exact mathematical passage
\widetilde\Phi_k
XUse in source · 12 passages
Search aliases: X
Use in source · Universal_Generative_Deformation_Calculus.tex, line 164
Let T be a set of types and let X=coprod_t∈ TX_t be the disjoint union of the corresponding object spaces.
Exact mathematical passage
\Xcal=\coprod_{t\in\mathsf T}X_t
X^2Use in source · 1 passage
Search aliases: X^2
Use in source · Universal_Generative_Deformation_Calculus.tex, line 1151
Starting from |0⟩, one has |0⟩xrightarrowH|+⟩ xrightarrowS|+i⟩ xrightarrowS|-⟩, while H^2=X^2=Z^2=I on the projective orbit and S^4=I. The number of raw gate words grows as 4^n, but the quotient orbit saturates at six objects.
Exact mathematical passage
H^2=X^2=Z^2=I
x^2Use in source · 7 passages
Search aliases: x^2
Use in source · Universal_Generative_Deformation_Calculus.tex, line 915
Let p(x)=x^2+1, (Tp)(x)=p(x+1), (Dp)(x)=p'(x). The two primitive directions T and D satisfy D(Tp)=T(Dp)=2x+2. The raw path tree therefore contains two length-two routes, pxrightarrowTTpxrightarrowDDTp, pxrightarrowDDpxrightarrowTTDp, but the quotient graph contains one endpoint class because DTp=TDp exactly.
Exact mathematical passage
p(x)=x^2+1, \qquad (Tp)(x)=p(x+1), \qquad (Dp)(x)=p'(x).
x_3Use in source · 1 passage
Search aliases: x_3
Use in source · Universal_Generative_Deformation_Calculus.tex, line 523
Metric completion is not the only possibility. In arithmetic deformation, a branch may be a compatible tower x_1← x_2← x_3←⋯ whose object is an inverse limit. In nested function spaces, a branch may enter a direct limit. In weak probability or operator topologies, convergence may be defined only against a test class.
Exact mathematical passage
x_1\leftarrow x_2\leftarrow x_3\leftarrow\cdots
x_{k+1}Use in source · 1 passage
Search aliases: x_k+1
Use in source · Universal_Generative_Deformation_Calculus.tex, line 491
For m>n, d_X(x_m,x_n) ≤ ∑_k=n^m-1d_X(x_k+1,x_k) ≤ ∑_k=n^m-1a_k. The tail tends to zero. Completeness supplies a unique limit.
Exact mathematical passage
d_X(x_m,x_n) \le \sum_{k=n}^{m-1}d_X(x_{k+1},x_k) \le \sum_{k=n}^{m-1}a_k.
x_mUse in source · 2 passages
Search aliases: x_m
Use in source · Universal_Generative_Deformation_Calculus.tex, line 491
For m>n, d_X(x_m,x_n) ≤ ∑_k=n^m-1d_X(x_k+1,x_k) ≤ ∑_k=n^m-1a_k. The tail tends to zero. Completeness supplies a unique limit.
Exact mathematical passage
d_X(x_m,x_n) \le \sum_{k=n}^{m-1}d_X(x_{k+1},x_k) \le \sum_{k=n}^{m-1}a_k.
x_NUse in source · 1 passage
Search aliases: x_N
Use in source · Universal_Generative_Deformation_Calculus.tex, line 1196
The executable completion example starts from x_0=0 and exposes the two depth-dependent directions d_n^±=±2^-(n+1). A finite route with signs ε_n∈{-1,+1} reaches x_N=∑_n=0^N-1ε_n2^-(n+1). Every infinite sign route is Cauchy because |x_m-x_n| ≤ ∑_k=n^m-12^-(k+1) <2^-n. The completed object therefore contains a limit for every infinite route. The selected periodic sign pattern used in the figure converges numerically to 5/7.
Exact mathematical passage
x_N=\sum_{n=0}^{N-1}\varepsilon_n2^{-(n+1)}.
x_{n+1}Use in source · 2 passages
Search aliases: x_n+1
Use in source · Universal_Generative_Deformation_Calculus.tex, line 479
[Summable-step convergence] Suppose widehat X is complete and d_X(x_n+1,x_n)≤ a_n with ∑_n≥0a_n<∞. Then (x_n) is Cauchy and the route has a unique limit in widehat X.
Exact mathematical passage
d_X(x_{n+1},x_n)\le a_n
X_tUse in source · 1 passage
Search aliases: X_t
Use in source · Universal_Generative_Deformation_Calculus.tex, line 164
Let T be a set of types and let X=coprod_t∈ TX_t be the disjoint union of the corresponding object spaces.
Exact mathematical passage
\Xcal=\coprod_{t\in\mathsf T}X_t
\Xcal^rUse in source · 1 passage
Search aliases: Xcal^r X^r Xcal mathcal
Use in source · Universal_Generative_Deformation_Calculus.tex, line 537
For every r≥2, let Ψ_r: X^r× D^(r) ⟶ X be a typed r-input deformation law.
Exact mathematical passage
\Psi_r: \Xcal^r\times\Dcal^{(r)} \longrightarrow \Xcal
\xrightarrowUse in source · 4 passages
Search aliases: xrightarrow
Use in source · Universal_Generative_Deformation_Calculus.tex, line 927
Let p(x)=x^2+1, (Tp)(x)=p(x+1), (Dp)(x)=p'(x). The two primitive directions T and D satisfy D(Tp)=T(Dp)=2x+2. The raw path tree therefore contains two length-two routes, pxrightarrowTTpxrightarrowDDTp, pxrightarrowDDpxrightarrowTTDp, but the quotient graph contains one endpoint class because DTp=TDp exactly.
Exact mathematical passage
p\xrightarrow{T}Tp\xrightarrow{D}DTp, \qquad p\xrightarrow{D}Dp\xrightarrow{T}TDp,
YUse in source · 3 passages
Search aliases: Y
Use in source · Universal_Generative_Deformation_Calculus.tex, line 269
[Free rooted path property] Let Y be a rooted directed graph with root y_0. Suppose each admissible pair (x,d) is assigned an edge operation on Y in a way that depends only on the current image vertex and the label d. Then there is a unique root-preserving map F: T_fin(x_0)⟶ Y that follows those assigned edges along every path.
Exact mathematical passage
Y
yUse in source · 10 passages
Search aliases: y
Use in source · Universal_Generative_Deformation_Calculus.tex, line 141
Given any two objects x and y, one can invent a one-step map that sends x to y. Such a map does not reveal a generative structure. It hides the target inside the operation.
Exact mathematical passage
y
y_0Use in source · 2 passages
Search aliases: y_0
Use in source · Universal_Generative_Deformation_Calculus.tex, line 269
[Free rooted path property] Let Y be a rooted directed graph with root y_0. Suppose each admissible pair (x,d) is assigned an edge operation on Y in a way that depends only on the current image vertex and the label d. Then there is a unique root-preserving map F: T_fin(x_0)⟶ Y that follows those assigned edges along every path.
Exact mathematical passage
y_0
ZUse in source · 9 passages
Search aliases: Z
Use in source · Universal_Generative_Deformation_Calculus.tex, line 693
Define Z_x_0(s) = ∑_γ≠∅ exp(-scost(γ)). This is a path partition function. It weights short, inexpensive routes more strongly than long, expensive routes.
Exact mathematical passage
Z_{x_0}(s) = \sum_{\gamma\ne\varnothing} \exp\bigl(-s\cost(\gamma)\bigr).
\ZUse in source · 1 passage
Search aliases: Z mathbb
Use in source · Universal_Generative_Deformation_Calculus.tex, line 344
Let the root be 0∈ Z. Every integer has two directions, +1, -1. A path of length n is a word in these two labels.
Exact mathematical passage
0\in\Z
zUse in source · 2 passages
Search aliases: z
Use in source · Universal_Generative_Deformation_Calculus.tex, line 1133
The implementation works directly after quotienting global phase. A state is one of the six exact Bloch-axis vectors (0,0,±1), (±1,0,0), (0,±1,0), representing the Pauli eigenstates |0⟩, |1⟩, |+⟩, |-⟩, |+i⟩, and |-i⟩. The primitive grammar contains H,S,X,Z, acting as signed permutations of the Bloch coordinates: H(x,y,z)=(z,-y,x), S(x,y,z)=(-y,x,z), X(x,y,z)=(x,-y,-z), Z(x,y,z)=(-x,-y,z). These rules are exact and avoid floating-point phase canonicalization.
Exact mathematical passage
H(x,y,z)=(z,-y,x), \qquad S(x,y,z)=(-y,x,z),
Z^2Use in source · 1 passage
Search aliases: Z^2
Use in source · Universal_Generative_Deformation_Calculus.tex, line 1151
Starting from |0⟩, one has |0⟩xrightarrowH|+⟩ xrightarrowS|+i⟩ xrightarrowS|-⟩, while H^2=X^2=Z^2=I on the projective orbit and S^4=I. The number of raw gate words grows as 4^n, but the quotient orbit saturates at six objects.
Exact mathematical passage
H^2=X^2=Z^2=I
Z_{\mathrm{obj}}Use in source · 1 passage
Search aliases: Z_obj mathrm
Use in source · Universal_Generative_Deformation_Calculus.tex, line 729
A path zeta function measures construction complexity. A quotient zeta function measures distinct object complexity. Depending on the application, one may define Z_obj(s) = ∑_[x]∈ O(x_0)e^-sC([x]), where C([x]) is the least path cost reaching the class. The two functions answer different questions and should be stored together.
Exact mathematical passage
Z_{\mathrm{obj}}(s) = \sum_{[x]\in\Ocal(x_0)}e^{-sC([x])},
Z_{x_0}Use in source · 4 passages
Search aliases: Z_x_0
Use in source · Universal_Generative_Deformation_Calculus.tex, line 693
Define Z_x_0(s) = ∑_γ≠∅ exp(-scost(γ)). This is a path partition function. It weights short, inexpensive routes more strongly than long, expensive routes.
Exact mathematical passage
Z_{x_0}(s) = \sum_{\gamma\ne\varnothing} \exp\bigl(-s\cost(\gamma)\bigr).
|Use in source · 10 passages
Search aliases: |
Use in source · Universal_Generative_Deformation_Calculus.tex, line 1132
The implementation works directly after quotienting global phase. A state is one of the six exact Bloch-axis vectors (0,0,±1), (±1,0,0), (0,±1,0), representing the Pauli eigenstates |0⟩, |1⟩, |+⟩, |-⟩, |+i⟩, and |-i⟩. The primitive grammar contains H,S,X,Z, acting as signed permutations of the Bloch coordinates: H(x,y,z)=(z,-y,x), S(x,y,z)=(-y,x,z), X(x,y,z)=(x,-y,-z), Z(x,y,z)=(-x,-y,z). These rules are exact and avoid floating-point phase canonicalization.
Exact mathematical passage
|0\rangle
\pmUse in source · 5 passages
Search aliases: ± pm
Use in source · Universal_Generative_Deformation_Calculus.tex, line 504
Starting at 0, let the directions at depth n be ±2^-(n+1). Every infinite sign route converges because ∑_n≥02^-(n+1)=1. The boundary therefore maps into [-1,1]. Different sign sequences can represent the same endpoint only at the usual binary-expansion ambiguity, which can be handled by a path relation.
Exact mathematical passage
\pm2^{-(n+1)}.
\timesUse in source · 1 passage
Search aliases: × times
Use in source · Universal_Generative_Deformation_Calculus.tex, line 537
For every r≥2, let Ψ_r: X^r× D^(r) ⟶ X be a typed r-input deformation law.
Exact mathematical passage
\Psi_r: \Xcal^r\times\Dcal^{(r)} \longrightarrow \Xcal
\alphaUse in source · 15 passages
Search aliases: α alpha
Use in source · Universal_Generative_Deformation_Calculus.tex, line 605
For a prefix α of length k≤ n, define the cylinder projector P^(n)_α lvertγ⟩ = lvertγ⟩, · α is a prefix of γ,
Exact mathematical passage
\alpha
\betaUse in source · 6 passages
Search aliases: β beta
Use in source · Universal_Generative_Deformation_Calculus.tex, line 644
Let A: H→ K be a bounded operator and let {P_α} and {Q_β} be complete route resolutions. Define A_βα=Q_β A P_α. Then A=∑_α,βA_βα. If B follows A, then (BA)_γα = ∑_β B_γβA_βα. Every operator composition becomes a sum over compatible intermediate perturbation packets.
Exact mathematical passage
\{Q_\beta\}
\GammaUse in source · 2 passages
Search aliases: Γ Gamma
Use in source · Universal_Generative_Deformation_Calculus.tex, line 636
A complete cut is a prefix-free set Γ meeting every infinite route exactly once. Its projectors satisfy ∑_α∈ΓP_α=I. Thus one branch may be resolved deeply while another stops early. Adaptive exploration can be represented without discarding the unresolved region.
Exact mathematical passage
\Gamma
\deltaUse in source · 4 passages
Search aliases: δ delta
Use in source · Universal_Generative_Deformation_Calculus.tex, line 999
A second, closed subfamily starts from the impulse δ=(1,0,0,…). Repeated summation gives (C^kδ)_n=binomn+k-1k-1 (k≥1), by induction and the hockey-stick identity. This provides a genuine family certificate: perturbation depth k is the parameter, C advances the parameter, D reverses it away from the boundary, and the displayed coefficient formula is an exact invariant relation.
Exact mathematical passage
\delta=(1,0,0,\ldots)
\delta_0Use in source · 1 passage
Search aliases: δ_0 delta
Use in source · worked_realizations.tex, line 14
Sequences · A Fibonacci prefix and the impulse δ_0 with difference, summation, binomial transform, sign alternation, and reversal · Exact transform relations generate structured integer families rather than one undifferentiated list of examples.
Exact mathematical passage
\delta_0
\epsilonUse in source · 14 passages
Search aliases: ϵ epsilon
Use in source · Universal_Generative_Deformation_Calculus.tex, line 381
Let each fiber D_x carry a metric ρ_x. A finite set N_ϵ(x)⊆ D_x is an ϵ-net when every d∈ D_x lies within ϵ of some widehat d∈ N_ϵ(x).
Exact mathematical passage
N_\epsilon(x)\subseteq\Dcal_x
\varepsilonUse in source · 2 passages
Search aliases: ε varepsilon
Use in source · Universal_Generative_Deformation_Calculus.tex, line 1195
The executable completion example starts from x_0=0 and exposes the two depth-dependent directions d_n^±=±2^-(n+1). A finite route with signs ε_n∈{-1,+1} reaches x_N=∑_n=0^N-1ε_n2^-(n+1). Every infinite sign route is Cauchy because |x_m-x_n| ≤ ∑_k=n^m-12^-(k+1) <2^-n. The completed object therefore contains a limit for every infinite route. The selected periodic sign pattern used in the figure converges numerically to 5/7.
Exact mathematical passage
\varepsilon_n\in\{-1,+1\}
\epsilon_jUse in source · 3 passages
Search aliases: ϵ_j epsilon
Use in source · Universal_Generative_Deformation_Calculus.tex, line 407
[Pathwise cover error] Consider an exact route x_k=x_k-1⋆ d_k and a sampled route widehat x_k=widehat x_k-1⋆widehat d_k. Suppose at step k, d_X(x⋆ d,y⋆ e) ≤ L_k d_X(x,y)+M_kρ(d,e), and ρ(d_k,widehat d_k)≤ϵ_k. Then d_X(x_n,widehat x_n) ≤ (∏_k=1^nL_k)d_X(x_0,widehat x_0) + ∑_j=1^n M_jϵ_j ∏_k=j+1^nL_k.
Exact mathematical passage
\boxed{ d_X(x_n,\widehat x_n) \le \left(\prod_{k=1}^nL_k\right)d_X(x_0,\widehat x_0) + \sum_{j=1}^n M_j\epsilon_j \prod_{k=j+1}^nL_k. }
\epsilon_kUse in source · 7 passages
Search aliases: ϵ_k epsilon
Use in source · Universal_Generative_Deformation_Calculus.tex, line 403
[Pathwise cover error] Consider an exact route x_k=x_k-1⋆ d_k and a sampled route widehat x_k=widehat x_k-1⋆widehat d_k. Suppose at step k, d_X(x⋆ d,y⋆ e) ≤ L_k d_X(x,y)+M_kρ(d,e), and ρ(d_k,widehat d_k)≤ϵ_k. Then d_X(x_n,widehat x_n) ≤ (∏_k=1^nL_k)d_X(x_0,widehat x_0) + ∑_j=1^n M_jϵ_j ∏_k=j+1^nL_k.
Exact mathematical passage
\rho(d_k,\widehat d_k)\le\epsilon_k.
\varepsilon_nUse in source · 2 passages
Search aliases: ε_n varepsilon
Use in source · Universal_Generative_Deformation_Calculus.tex, line 1195
The executable completion example starts from x_0=0 and exposes the two depth-dependent directions d_n^±=±2^-(n+1). A finite route with signs ε_n∈{-1,+1} reaches x_N=∑_n=0^N-1ε_n2^-(n+1). Every infinite sign route is Cauchy because |x_m-x_n| ≤ ∑_k=n^m-12^-(k+1) <2^-n. The completed object therefore contains a limit for every infinite route. The selected periodic sign pattern used in the figure converges numerically to 5/7.
Exact mathematical passage
\varepsilon_n\in\{-1,+1\}
\eta_jUse in source · 1 passage
Search aliases: η_j eta
Use in source · Universal_Generative_Deformation_Calculus.tex, line 866
[Recursive deformation stability] If the direction mismatch at step k is at most ϵ_k, then the endpoint error satisfies e_n ≤ (∏_k=1^nL_k)e_0 + ∑_j=1^n (M_jϵ_j+η_j) ∏_k=j+1^nL_k.
Exact mathematical passage
e_n \le \left(\prod_{k=1}^nL_k\right)e_0 + \sum_{j=1}^n (M_j\epsilon_j+\eta_j) \prod_{k=j+1}^nL_k.
\eta_kUse in source · 3 passages
Search aliases: η_k eta
Use in source · Universal_Generative_Deformation_Calculus.tex, line 857
Let Φ_k and widetildeΦ_k be exact and perturbed deformation laws along corresponding routes. Suppose d_X(Φ_k(x,d),widetildeΦ_k(y,e)) ≤ L_kd_X(x,y)+M_kρ(d,e)+η_k, where η_k measures model error.
Exact mathematical passage
d_X(\Phi_k(x,d),\widetilde\Phi_k(y,e)) \le L_kd_X(x,y)+M_k\rho(d,e)+\eta_k,
\theta_0Use in source · 1 passage
Search aliases: θ_0 theta
Use in source · Universal_Generative_Deformation_Calculus.tex, line 763
[Inductive soundness of a family certificate] Suppose the seed member F(θ_0) is valid, every closure identity is proved, and every relation in Q is preserved by the parameter actions. Then every finite parameter word in the maps σ_d produces a valid member of the certified family.
Exact mathematical passage
F(\theta_0)
\lambdaUse in source · 1 passage
Search aliases: λ lambda
Use in source · Universal_Generative_Deformation_Calculus.tex, line 571
Consider u=s+λ u^2. Repeated substitution expands u into rooted binary trees. If only shape is counted, the number of binary trees with n internal vertices is the Catalan number C_n.
Exact mathematical passage
u=s+\lambda u^2.
\muUse in source · 4 passages
Search aliases: μ mu
Use in source · Universal_Generative_Deformation_Calculus.tex, line 1017
Represent a one-dimensional Gaussian law by the exact parameter pair (μ,v), where v=σ^2>0. The grammar contains shifts by ±1, reflection, convolution with N(0,1), centering, scaling by two, and standardization. Their parameter actions are (μ,v) · ↦(μ±1,v), · (μ,v) · ↦(-μ,v),
Exact mathematical passage
(\mu,v)
\rhoUse in source · 6 passages
Search aliases: ρ rho
Use in source · Universal_Generative_Deformation_Calculus.tex, line 381
Let each fiber D_x carry a metric ρ_x. A finite set N_ϵ(x)⊆ D_x is an ϵ-net when every d∈ D_x lies within ϵ of some widehat d∈ N_ϵ(x).
Exact mathematical passage
\rho_x
\rho^{-1}Use in source · 1 passage
Search aliases: ρ^-1 rho
Use in source · Universal_Generative_Deformation_Calculus.tex, line 1069
For the seed P_3, the first descendants include overlineP_3, L(P_3)≅ P_2, P_3 with one attached leaf, P_4 obtained by subdivision. If ρ reverses the vertex labels, then key(G)=key(ρ Gρ^-1) exactly, although the two raw paths and raw adjacency matrices remain distinct.
Exact mathematical passage
\operatorname{key}(G)=\operatorname{key}(\rho G\rho^{-1})
\rho_xUse in source · 1 passage
Search aliases: ρ_x rho
Use in source · Universal_Generative_Deformation_Calculus.tex, line 381
Let each fiber D_x carry a metric ρ_x. A finite set N_ϵ(x)⊆ D_x is an ϵ-net when every d∈ D_x lies within ϵ of some widehat d∈ N_ϵ(x).
Exact mathematical passage
\rho_x
\sigma^2Use in source · 2 passages
Search aliases: σ^2 sigma
Use in source · Universal_Generative_Deformation_Calculus.tex, line 1017
Represent a one-dimensional Gaussian law by the exact parameter pair (μ,v), where v=σ^2>0. The grammar contains shifts by ±1, reflection, convolution with N(0,1), centering, scaling by two, and standardization. Their parameter actions are (μ,v) · ↦(μ±1,v), · (μ,v) · ↦(-μ,v),
Exact mathematical passage
v=\sigma^2>0
\Phi_kUse in source · 2 passages
Search aliases: Φ_k Phi
Use in source · Universal_Generative_Deformation_Calculus.tex, line 856
Let Φ_k and widetildeΦ_k be exact and perturbed deformation laws along corresponding routes. Suppose d_X(Φ_k(x,d),widetildeΦ_k(y,e)) ≤ L_kd_X(x,y)+M_kρ(d,e)+η_k, where η_k measures model error.
Exact mathematical passage
\Phi_k
\ellUse in source · 1 passage
Search aliases: ℓ ell
Use in source · Universal_Generative_Deformation_Calculus.tex, line 600
For a finite or countable depth layer, define H_n=ℓ^2( T_n(x_0)). The canonical basis vector associated with a path γ is written lvertγ⟩.
Exact mathematical passage
\mathcal H_n=\ell^2(\Tcal_n(x_0)).
\ell^2Use in source · 1 passage
Search aliases: ℓ^2 ell
Use in source · Universal_Generative_Deformation_Calculus.tex, line 600
For a finite or countable depth layer, define H_n=ℓ^2( T_n(x_0)). The canonical basis vector associated with a path γ is written lvertγ⟩.
Exact mathematical passage
\mathcal H_n=\ell^2(\Tcal_n(x_0)).
\leftarrowUse in source · 1 passage
Search aliases: ← leftarrow
Use in source · Universal_Generative_Deformation_Calculus.tex, line 523
Metric completion is not the only possibility. In arithmetic deformation, a branch may be a compatible tower x_1← x_2← x_3←⋯ whose object is an inverse limit. In nested function spaces, a branch may enter a direct limit. In weak probability or operator topologies, convergence may be defined only against a test class.
Exact mathematical passage
x_1\leftarrow x_2\leftarrow x_3\leftarrow\cdots
\mapstoUse in source · 1 passage
Search aliases: ↦ mapsto
Use in source · Universal_Generative_Deformation_Calculus.tex, line 1018
Represent a one-dimensional Gaussian law by the exact parameter pair (μ,v), where v=σ^2>0. The grammar contains shifts by ±1, reflection, convolution with N(0,1), centering, scaling by two, and standardization. Their parameter actions are (μ,v) · ↦(μ±1,v), · (μ,v) · ↦(-μ,v), (μ,v) · ↦(μ,v+1), · (μ,v) · ↦(0,v), (μ,v) · ↦(2μ,4v), · (μ,v) · ↦(0,1). Standardization is state dependent: it is exposed only away from the canonical seed. Reflection is invisible at centered laws, while convolution and sca…
Exact mathematical passage
\begin{aligned} (\mu,v)&\mapsto(\mu\pm1,v), & (\mu,v)&\mapsto(-\mu,v),\\ (\mu,v)&\mapsto(\mu,v+1), & (\mu,v)&\mapsto(0,v),\\ (\mu,v)&\mapsto(2\mu,4v), & (\mu,v)&\mapsto(0,1). \end{aligned}
\partialUse in source · 2 passages
Search aliases: ∂ partial
Use in source · Universal_Generative_Deformation_Calculus.tex, line 465
An infinite route is a sequence ω=(d_1,d_2,…) whose every finite prefix is admissible. The set of infinite routes is the path boundary ∂ T(x_0).
Exact mathematical passage
\partial\Tcal(x_0)
\prodUse in source · 3 passages
Search aliases: ∏ prod
Use in source · Universal_Generative_Deformation_Calculus.tex, line 407
[Pathwise cover error] Consider an exact route x_k=x_k-1⋆ d_k and a sampled route widehat x_k=widehat x_k-1⋆widehat d_k. Suppose at step k, d_X(x⋆ d,y⋆ e) ≤ L_k d_X(x,y)+M_kρ(d,e), and ρ(d_k,widehat d_k)≤ϵ_k. Then d_X(x_n,widehat x_n) ≤ (∏_k=1^nL_k)d_X(x_0,widehat x_0) + ∑_j=1^n M_jϵ_j ∏_k=j+1^nL_k.
Exact mathematical passage
\boxed{ d_X(x_n,\widehat x_n) \le \left(\prod_{k=1}^nL_k\right)d_X(x_0,\widehat x_0) + \sum_{j=1}^n M_j\epsilon_j \prod_{k=j+1}^nL_k. }
\prod_{k=1}^nUse in source · 2 passages
Search aliases: ∏_k=1^n prod
Use in source · Universal_Generative_Deformation_Calculus.tex, line 407
[Pathwise cover error] Consider an exact route x_k=x_k-1⋆ d_k and a sampled route widehat x_k=widehat x_k-1⋆widehat d_k. Suppose at step k, d_X(x⋆ d,y⋆ e) ≤ L_k d_X(x,y)+M_kρ(d,e), and ρ(d_k,widehat d_k)≤ϵ_k. Then d_X(x_n,widehat x_n) ≤ (∏_k=1^nL_k)d_X(x_0,widehat x_0) + ∑_j=1^n M_jϵ_j ∏_k=j+1^nL_k.
Exact mathematical passage
\boxed{ d_X(x_n,\widehat x_n) \le \left(\prod_{k=1}^nL_k\right)d_X(x_0,\widehat x_0) + \sum_{j=1}^n M_j\epsilon_j \prod_{k=j+1}^nL_k. }
\prod_{k=j+1}^nUse in source · 2 passages
Search aliases: ∏_k=j+1^n prod
Use in source · Universal_Generative_Deformation_Calculus.tex, line 407
[Pathwise cover error] Consider an exact route x_k=x_k-1⋆ d_k and a sampled route widehat x_k=widehat x_k-1⋆widehat d_k. Suppose at step k, d_X(x⋆ d,y⋆ e) ≤ L_k d_X(x,y)+M_kρ(d,e), and ρ(d_k,widehat d_k)≤ϵ_k. Then d_X(x_n,widehat x_n) ≤ (∏_k=1^nL_k)d_X(x_0,widehat x_0) + ∑_j=1^n M_jϵ_j ∏_k=j+1^nL_k.
Exact mathematical passage
\boxed{ d_X(x_n,\widehat x_n) \le \left(\prod_{k=1}^nL_k\right)d_X(x_0,\widehat x_0) + \sum_{j=1}^n M_j\epsilon_j \prod_{k=j+1}^nL_k. }
\prod_{k>j}Use in source · 1 passage
Search aliases: ∏_k>j prod
Use in source · Universal_Generative_Deformation_Calculus.tex, line 1455
Cover closed bound · e_n≤ (∏ L_k)e_0+∑_jM_jϵ_j∏_k>jL_k
Exact mathematical passage
e_n\le (\prod L_k)e_0+\sum_jM_j\epsilon_j\prod_{k>j}L_k
\sumUse in source · 25 passages
Search aliases: ∑ sum
Use in source · Universal_Generative_Deformation_Calculus.tex, line 285
The cost of a finite path is cost(γ)=∑_k=1^nκ(d_k). A path record may also store error bounds, provenance, validity certificates, symbolic assumptions, and the first place at which an invariant changed. None of these fields alters reachability. They determine how a finite explorer prioritizes and audits the infinite object.
Exact mathematical passage
\cost(\gamma)=\sum_{k=1}^n\kappa(d_k).
\sum_{[x]\in\Ocal(x_0)}Use in source · 1 passage
Search aliases: ∑_[x]∈Ocal(x_0) ∑_[x]∈ O(x_0) sum in Ocal sum in mathcal
Use in source · Universal_Generative_Deformation_Calculus.tex, line 729
A path zeta function measures construction complexity. A quotient zeta function measures distinct object complexity. Depending on the application, one may define Z_obj(s) = ∑_[x]∈ O(x_0)e^-sC([x]), where C([x]) is the least path cost reaching the class. The two functions answer different questions and should be stored together.
Exact mathematical passage
Z_{\mathrm{obj}}(s) = \sum_{[x]\in\Ocal(x_0)}e^{-sC([x])},
\sum_dUse in source · 1 passage
Search aliases: ∑_d sum
Use in source · Universal_Generative_Deformation_Calculus.tex, line 1458
Projector refinement · P_α=∑_dP_α d
Exact mathematical passage
P_\alpha=\sum_dP_{\alpha d}
\sum_{d\in\Dcal_{\ev(\alpha)}}Use in source · 1 passage
Search aliases: ∑_d∈Dcal_ev(α) ∑_d∈ D_ev(α) sum in Dcal ev alpha sum in mathcal operatorname alpha
Use in source · Universal_Generative_Deformation_Calculus.tex, line 618
[Exact cylinder refinement] For every prefix α and every depth n>absα, P^(n)_α = ∑_d∈ D_ev(α)P^(n)_α d, with strong convergence when the child set is countable.
Exact mathematical passage
\boxed{ P^{(n)}_\alpha = \sum_{d\in\Dcal_{\ev(\alpha)}}P^{(n)}_{\alpha d}, }
\sum_jUse in source · 1 passage
Search aliases: ∑_j sum
Use in source · Universal_Generative_Deformation_Calculus.tex, line 1455
Cover closed bound · e_n≤ (∏ L_k)e_0+∑_jM_jϵ_j∏_k>jL_k
Exact mathematical passage
e_n\le (\prod L_k)e_0+\sum_jM_j\epsilon_j\prod_{k>j}L_k
\sum_{j=0}^{n}Use in source · 1 passage
Search aliases: ∑_j=0^n sum
Use in source · Universal_Generative_Deformation_Calculus.tex, line 982
The executable atlas starts from the eight-term Fibonacci prefix a=(0,1,1,2,3,5,8,13). Its fixed grammar contains first difference, partial summation, the binomial transform, sign alternation, and reversal. For a finite sequence a=(a_0,…,a_N-1), define (Da)_0=a_0, (Da)_n=a_n-a_n-1, (Ca)_n=∑_j=0^na_j, (Ba)_n=∑_j=0^nbinomnja_j. The two triangular operators satisfy DC=CD=I on every finite sequence under the stated boundary convention. Hence the raw routes axrightarrowDDaxrightarrowCCDa, axrightarro…
Exact mathematical passage
(Ca)_n=\sum_{j=0}^{n}a_j, \qquad (Ba)_n=\sum_{j=0}^{n}\binom{n}{j}a_j.
\sum_{j=1}^nUse in source · 2 passages
Search aliases: ∑_j=1^n sum
Use in source · Universal_Generative_Deformation_Calculus.tex, line 407
[Pathwise cover error] Consider an exact route x_k=x_k-1⋆ d_k and a sampled route widehat x_k=widehat x_k-1⋆widehat d_k. Suppose at step k, d_X(x⋆ d,y⋆ e) ≤ L_k d_X(x,y)+M_kρ(d,e), and ρ(d_k,widehat d_k)≤ϵ_k. Then d_X(x_n,widehat x_n) ≤ (∏_k=1^nL_k)d_X(x_0,widehat x_0) + ∑_j=1^n M_jϵ_j ∏_k=j+1^nL_k.
Exact mathematical passage
\boxed{ d_X(x_n,\widehat x_n) \le \left(\prod_{k=1}^nL_k\right)d_X(x_0,\widehat x_0) + \sum_{j=1}^n M_j\epsilon_j \prod_{k=j+1}^nL_k. }
\sum_kUse in source · 1 passage
Search aliases: ∑_k sum
Use in source · Universal_Generative_Deformation_Calculus.tex, line 1451
Path cost · cost(γ)=∑_kκ(d_k)
Exact mathematical passage
\cost(\gamma)=\sum_k\kappa(d_k)
\sum_{k=0}^nUse in source · 1 passage
Search aliases: ∑_k=0^n sum
Use in source · Universal_Generative_Deformation_Calculus.tex, line 578
[Catalan interaction growth] The counts satisfy C_0=1, C_n+1=∑_k=0^nC_kC_n-k.
Exact mathematical passage
C_0=1, \qquad C_{n+1}=\sum_{k=0}^nC_kC_{n-k}.
\sum_{k=1}^nUse in source · 1 passage
Search aliases: ∑_k=1^n sum
Use in source · Universal_Generative_Deformation_Calculus.tex, line 285
The cost of a finite path is cost(γ)=∑_k=1^nκ(d_k). A path record may also store error bounds, provenance, validity certificates, symbolic assumptions, and the first place at which an invariant changed. None of these fields alters reachability. They determine how a finite explorer prioritizes and audits the infinite object.
Exact mathematical passage
\cost(\gamma)=\sum_{k=1}^n\kappa(d_k).
\sum_{k=n}^{m-1}Use in source · 2 passages
Search aliases: ∑_k=n^m-1 sum
Use in source · Universal_Generative_Deformation_Calculus.tex, line 491
For m>n, d_X(x_m,x_n) ≤ ∑_k=n^m-1d_X(x_k+1,x_k) ≤ ∑_k=n^m-1a_k. The tail tends to zero. Completeness supplies a unique limit.
Exact mathematical passage
d_X(x_m,x_n) \le \sum_{k=n}^{m-1}d_X(x_{k+1},x_k) \le \sum_{k=n}^{m-1}a_k.
\sum_{k\ge1}Use in source · 2 passages
Search aliases: ∑_k≥1 sum ge
Use in source · Universal_Generative_Deformation_Calculus.tex, line 430
[Convergent direction covers] If L_k≤1, M_k≤ M, and ∑_k≥1ϵ_k<∞, then the total endpoint error of the sampled route is bounded uniformly in depth by M∑_k≥1ϵ_k when the roots coincide.
Exact mathematical passage
\sum_{k\ge1}\epsilon_k<\infty,
\sum_{n=0}^{N-1}Use in source · 1 passage
Search aliases: ∑_n=0^N-1 sum
Use in source · Universal_Generative_Deformation_Calculus.tex, line 1196
The executable completion example starts from x_0=0 and exposes the two depth-dependent directions d_n^±=±2^-(n+1). A finite route with signs ε_n∈{-1,+1} reaches x_N=∑_n=0^N-1ε_n2^-(n+1). Every infinite sign route is Cauchy because |x_m-x_n| ≤ ∑_k=n^m-12^-(k+1) <2^-n. The completed object therefore contains a limit for every infinite route. The selected periodic sign pattern used in the figure converges numerically to 5/7.
Exact mathematical passage
x_N=\sum_{n=0}^{N-1}\varepsilon_n2^{-(n+1)}.
\sum_{n\ge0}Use in source · 2 passages
Search aliases: ∑_n≥0 sum ge
Use in source · Universal_Generative_Deformation_Calculus.tex, line 483
[Summable-step convergence] Suppose widehat X is complete and d_X(x_n+1,x_n)≤ a_n with ∑_n≥0a_n<∞. Then (x_n) is Cauchy and the route has a unique limit in widehat X.
Exact mathematical passage
\sum_{n\ge0}a_n<\infty.
\sum_{n\ge1}Use in source · 1 passage
Search aliases: ∑_n≥1 sum ge
Use in source · Universal_Generative_Deformation_Calculus.tex, line 703
[Geometric convergence bound] Suppose every vertex has at most B children and every direction costs at least c>0. Then Z_x_0(s) ≤ ∑_n≥1B^ne^-scn. Consequently, Z_x_0(s) converges whenever sc>log B. In a full regular B-ary tree with constant cost c, it diverges whenever sc≤log B.
Exact mathematical passage
Z_{x_0}(s) \le \sum_{n\ge1}B^ne^{-scn}.
\sum_{\alpha,\beta}Use in source · 1 passage
Search aliases: ∑_α,β sum alpha beta
Use in source · Universal_Generative_Deformation_Calculus.tex, line 649
Let A: H→ K be a bounded operator and let {P_α} and {Q_β} be complete route resolutions. Define A_βα=Q_β A P_α. Then A=∑_α,βA_βα. If B follows A, then (BA)_γα = ∑_β B_γβA_βα. Every operator composition becomes a sum over compatible intermediate perturbation packets.
Exact mathematical passage
A=\sum_{\alpha,\beta}A_{\beta\alpha}.
\sum_{\alpha\in\Gamma}Use in source · 1 passage
Search aliases: ∑_α∈Γ sum alpha in Gamma
Use in source · Universal_Generative_Deformation_Calculus.tex, line 637
A complete cut is a prefix-free set Γ meeting every infinite route exactly once. Its projectors satisfy ∑_α∈ΓP_α=I. Thus one branch may be resolved deeply while another stops early. Adaptive exploration can be represented without discarding the unresolved region.
Exact mathematical passage
\sum_{\alpha\in\Gamma}P_\alpha=I.
\sum_\betaUse in source · 2 passages
Search aliases: ∑_β sum beta
Use in source · Universal_Generative_Deformation_Calculus.tex, line 653
Let A: H→ K be a bounded operator and let {P_α} and {Q_β} be complete route resolutions. Define A_βα=Q_β A P_α. Then A=∑_α,βA_βα. If B follows A, then (BA)_γα = ∑_β B_γβA_βα. Every operator composition becomes a sum over compatible intermediate perturbation packets.
Exact mathematical passage
(BA)_{\gamma\alpha} = \sum_\beta B_{\gamma\beta}A_{\beta\alpha}.
\sum_{\gamma\ne\varnothing}Use in source · 2 passages
Search aliases: ∑_γ≠∅ sum gamma ne varnothing
Use in source · Universal_Generative_Deformation_Calculus.tex, line 693
Define Z_x_0(s) = ∑_γ≠∅ exp(-scost(γ)). This is a path partition function. It weights short, inexpensive routes more strongly than long, expensive routes.
Exact mathematical passage
Z_{x_0}(s) = \sum_{\gamma\ne\varnothing} \exp\bigl(-s\cost(\gamma)\bigr).
\astUse in source · 1 passage
Search aliases: ∗ ast
Use in source · Universal_Generative_Deformation_Calculus.tex, line 1031
One explicit route is (0,1)xrightarrowshift +1(1,1) xrightarrow∗ N(0,1)(1,2) xrightarrowscale 2(2,8) xrightarrowstandardize(0,1). The final edge is a many-to-one canonicalization direction. It produces a strong quotient collapse without erasing the route that created the noncanonical law. The differential entropy h(N(μ,v))=frac12log(2π e v) is unaffected by shifts and reflection, increases under unit convolution, and changes by log 2 under scaling by two.
Exact mathematical passage
(0,1)\xrightarrow{\text{shift }+1}(1,1) \xrightarrow{\ast N(0,1)}(1,2) \xrightarrow{\text{scale }2}(2,8) \xrightarrow{\text{standardize}}(0,1).
\congUse in source · 1 passage
Search aliases: ≅ cong
Use in source · Universal_Generative_Deformation_Calculus.tex, line 1059
For the seed P_3, the first descendants include overlineP_3, L(P_3)≅ P_2, P_3 with one attached leaf, P_4 obtained by subdivision. If ρ reverses the vertex labels, then key(G)=key(ρ Gρ^-1) exactly, although the two raw paths and raw adjacency matrices remain distinct.
Exact mathematical passage
\overline{P_3}, \qquad L(P_3)\cong P_2, \qquad P_3\text{ with one attached leaf}, \qquad P_4\text{ obtained by subdivision}.
\neUse in source · 2 passages
Search aliases: ≠ ne
Use in source · Universal_Generative_Deformation_Calculus.tex, line 693
Define Z_x_0(s) = ∑_γ≠∅ exp(-scost(γ)). This is a path partition function. It weights short, inexpensive routes more strongly than long, expensive routes.
Exact mathematical passage
Z_{x_0}(s) = \sum_{\gamma\ne\varnothing} \exp\bigl(-s\cost(\gamma)\bigr).
\subseteqUse in source · 1 passage
Search aliases: ⊆ subseteq
Use in source · Universal_Generative_Deformation_Calculus.tex, line 381
Let each fiber D_x carry a metric ρ_x. A finite set N_ϵ(x)⊆ D_x is an ϵ-net when every d∈ D_x lies within ϵ of some widehat d∈ N_ϵ(x).
Exact mathematical passage
N_\epsilon(x)\subseteq\Dcal_x
\cdotsUse in source · 4 passages
Search aliases: ⋯ cdots
Use in source · Universal_Generative_Deformation_Calculus.tex, line 523
Metric completion is not the only possibility. In arithmetic deformation, a branch may be a compatible tower x_1← x_2← x_3←⋯ whose object is an inverse limit. In nested function spaces, a branch may enter a direct limit. In weak probability or operator topologies, convergence may be defined only against a test class.
Exact mathematical passage
x_1\leftarrow x_2\leftarrow x_3\leftarrow\cdots
\rangleUse in source · 12 passages
Search aliases: ⟩ rangle
Use in source · Universal_Generative_Deformation_Calculus.tex, line 603
For a finite or countable depth layer, define H_n=ℓ^2( T_n(x_0)). The canonical basis vector associated with a path γ is written lvertγ⟩.
Exact mathematical passage
\lvert\gamma\rangle
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