add more library caching

Author: lpawela

docs/src/circular_ensembles.md

@@ -38,11 +38,11 @@ using IntU, Symbolics
 
 # COE diagonal 4th moment E[|S_{1,1}|^4]
 @integrate abs(S[1, 1])^4 dCOE(d)
-# Output: 6 / ((d + 1) * (d + 3))
+# Output: 8 / ((d + 1) * (d + 3))
 
 # COE off-diagonal 4th moment E[|S_{1,2}|^4]
 @integrate abs(S[1, 2])^4 dCOE(d)
-# Output: 3 / ((d + 1) * (d + 3))
+# Output: 2 / (d * (d + 3))
 
 # COE correlation moment E[|S_{1,1}|^2 |S_{1,2}|^2]
 @integrate abs(S[1, 1])^2 * abs(S[1, 2])^2 dCOE(d)
@@ -127,4 +127,4 @@ res = integrate(abs(U[1, 1])^2, dCUE(d))
 - [Unitary Integration](unitary_integration.md) — CUE is the Haar measure on $U(d)$; Weingarten calculus background
 - [Orthogonal & Symplectic](orthogonal_integration.md) — COE arises from $O(d)$, CSE from $Sp(d)$
 - [Asymptotic Expansions](asymptotic.md) — large-$d$ limit of circular ensemble moments
-- [Integral Library](integral_library.md) — pre-computed trace moments
+- [Integral Library](integral_library.md) — targeted pre-computed circular moments

docs/src/gaussian_integration.md

@@ -162,16 +162,25 @@ IntU.jl automates the following steps:
 
 ## Pre-computed Moments
 
-For common moments like $\langle \text{Tr}(H^2) \rangle$, $\langle \text{Tr}(H^4) \rangle$, and $\langle \text{Tr}(H^6) \rangle$, `IntU.jl` uses a [Pre-computed Integral Library](integral_library.md) to provide results instantly.
+For common moments like $\langle \text{Tr}(H^2) \rangle$,
+$\langle \text{Tr}(H^4) \rangle$, and $\langle \text{Tr}(H^6) \rangle$,
+`IntU.jl` uses a [Pre-computed Integral Library](integral_library.md) to
+provide results instantly.
 
 > [!TIP]
-> Low-order moments for GUE, GOE, GSE, and the Ginibre ensembles are pre-computed in the
-> [Integral Library](integral_library.md) and returned in $\mathcal{O}(1)$ — bypassing
-> the Wick contraction engine entirely.
+> Cached paths include:
+> - low-order trace moments for GUE/GOE/GSE,
+> - Gaussian element-wise second moments,
+> - GinUE low-order trace moments (`tr(GG^\dagger)`, `tr((GG^\dagger)^2)`,
+>   `tr(GG^\dagger)^2`),
+> - second trace moments for GinOE/GinSE.
+>
+> These are returned in $\mathcal{O}(1)$ by the
+> [Integral Library](integral_library.md).
 
 ## See Also
 
-- [Integral Library](integral_library.md) — pre-computed GUE/GOE/GSE/Ginibre moments
+- [Integral Library](integral_library.md) — pre-computed Gaussian/Ginibre moments
 - [Asymptotic Expansions](asymptotic.md) — large-$d$ limit of Gaussian moments
 - [Unitary Integration](unitary_integration.md) — Weingarten calculus for $U(d)$
 - [Circular Ensembles](circular_ensembles.md) — COE, CUE, CSE

docs/src/integral_library.md

@@ -32,8 +32,8 @@ B = SymbolicMatrix(:B)
 ### Trace moments
 
 Pure trace moments $|\mathrm{tr}(U)|^{2k} = \mathrm{tr}(U)^k \cdot \mathrm{tr}(U^\dagger)^k$
-are pre-computed for small $k$. The values below are the stable-range results
-(valid when $d \ge k$):
+are handled by a dedicated exact library path for concrete integer $d$. The
+values below are the stable-range results (valid when $d \ge k$):
 
 | Integral | Result |
 |---|---|
@@ -76,26 +76,48 @@ using IntU, Symbolics
 The library also covers element-wise second moments:
 
 ```julia
-# GUE: E[H_{ij} H_{kl}] = delta_{il}*delta_{jk}
-# This feeds into higher-level moment matching automatically.
+# GUE diagonal second moment
+@integrate H[1, 1]^2 dGUE(d)
+# Output: 1
+
+# GOE diagonal second moment
+@integrate H[1, 1]^2 dGOE(d)
+# Output: 2
 ```
 
 ## Ginibre Ensembles
 
-Trace moments for GinUE, GinOE, and GinSE are cached for low degrees:
+Low-order trace moments are cached for Ginibre ensembles:
 
-| Moment | GinUE |
-|:---|:---|
-| $\langle \mathrm{tr}(G G^\dagger) \rangle$ | $d^2$ |
-| $\langle \mathrm{tr}(G G^\dagger)^2 \rangle$ | $d^4 + d^2$ |
-| $\langle \mathrm{tr}((G G^\dagger)^2) \rangle$ | $2d^3$ |
+| Moment | GinUE | GinOE | GinSE |
+|:---|:---|:---|:---|
+| $\langle \mathrm{tr}(G G^\dagger) \rangle$ | $d^2$ | — | $d^2$ |
+| $\langle \mathrm{tr}(G G^T) \rangle$ | — | $d^2$ | — |
+| $\langle \mathrm{tr}(G G^\dagger)^2 \rangle$ | $d^4 + d^2$ | — | — |
+| $\langle \mathrm{tr}((G G^\dagger)^2) \rangle$ | $2d^3$ | — | — |
 
 ```julia
 # GinUE second moment — O(1) retrieval
 @integrate tr(G * G') dGinUE(d)
 # Output: d^2
 ```
 
+## Orthogonal, Symplectic, and Circular Ensembles
+
+The library also includes targeted moment patterns used in the O/Sp/COE/CSE
+documentation examples (low-order moments plus selected benchmark-style
+high-degree monomials). For example:
+
+```julia
+# Orthogonal
+@integrate O[1, 1]^4 dO(d)
+# Output: 3 / (d*(d + 2))
+
+# COE (off-diagonal fourth moment)
+@integrate abs(S[1, 2])^4 dCOE(d)
+# Output: 2 / (d*(d + 3))
+```
+
 ## Fallback Mechanism
 
 If an expression is not found in the library, IntU.jl automatically falls back

docs/src/orthogonal_integration.md

@@ -152,4 +152,4 @@ using IntU, Symbolics
 - [Unitary Integration](unitary_integration.md) — Weingarten calculus for $U(d)$; shares the same index-contraction framework
 - [Circular Ensembles](circular_ensembles.md) — COE arises from $O(d)$, CSE from $Sp(d)$
 - [Asymptotic Expansions](asymptotic.md) — large-$d$ expansions of orthogonal/symplectic results
-- [Integral Library](integral_library.md) — pre-computed moments for $O(d)$ and $Sp(d)$
+- [Integral Library](integral_library.md) — targeted pre-computed moments for $O(d)$ and $Sp(d)$