diff --git a/spherical_perceptron_bi_orthogonal_v2.txt b/spherical_perceptron_bi_orthogonal_v2.txt
index 36cf519..115c99e 100644
--- a/spherical_perceptron_bi_orthogonal_v2.txt
+++ b/spherical_perceptron_bi_orthogonal_v2.txt
@@ -82,8 +82,13 @@ volume. A random subdivision of Haar subspaces proves this comparison
 by transferring weighted volume, even when the activation concentrates
 that weight. The one-sided projection tests therefore recover the
 angular pressure without requiring concavity of the activation.
-Gaussian orthogonalisation supplies the underlying Haar frames, as in
-the classical constructions of Mezzadri.
+Gaussian orthogonalisation supplies the underlying Haar frames. The
+constructions of Mezzadri are the standard reference; the abstract
+states the scope plainly: "random unitary matrices from the classical
+compact groups $U(N)$, $O(N)$ and $USp(2N)$ with probability
+distributions given by the respective invariant measures" and the
+algorithm is "straightforward to implement using standard linear
+algebra packages." Only the $O(N)$ case is needed.
 ./refs/arxiv-mezzadri-0609050.pdf
 
 The remaining Gaussian problem involves several Stiefel columns, with
@@ -103,14 +108,14 @@ retaining the prescribed sharing of one coordinate.
 ./refs/arxiv-2010.09114v2.pdf
 ./refs/arxiv-1906.08471v2.pdf
 ./refs/mourrat-panchenko-2020.pdf
-./refs/aizenman-et-al-2004-cond-mat-0403240.pdf
+./refs/aizenman-sims-starr-2003-cond-mat-0306386.pdf
 
 The matrix overlap argument draws on Panchenko's vector-spin
 synchronisation theory and its enriched formulation in Mourrat's
 upper-bound paper. Closely related matrix spherical entropy formulas
 appear in Ko's paper on the Crisanti-Sommers formula for vector spins,
 and the scalar spherical cascade transform is computed by Mourrat. The
-entropy here includes the change of active spin space at the sharing
+entropy includes the change of active spin space at the sharing
 level, which the scalar formula does not cover.
 ./refs/arxiv-panchenko-vector.pdf
 ./refs/arxiv-1911.04355.pdf
@@ -562,7 +567,7 @@ Gaussian functional $\mathcal G_{\theta,p}$. The proof produces an
 expected pressure upper bound that matches $\mathcal G_{\theta,p}$ at
 this trial, uniformly in $q$; taking the infimum over $q$ gives the
 theorem. The comparison follows Mourrat's supersolution method for
-mean-field spin glasses with vector spins, adapted here to the
+mean-field spin glasses with vector spins, adapted to the
 nonlinear pattern Hamiltonian.
 ./refs/arxiv-2004.01679v8.pdf
 ./refs/arxiv-2010.09114v2.pdf
@@ -667,15 +672,19 @@ give the same limit. The negative-quadratic tail class is handled by the soft-pe
 
 Lower bound: cavity with two restored patterns. The proof follows the
 extended variational principle of Aizenman, Sims, and Starr, adapting
-the Aizenman-Sims-Starr cavity to the Stiefel-cascade model. A related
-critical-point method for vector spin glasses with nonconvex
-interactions is due to Chen and Mourrat. Fix
+the Aizenman-Sims-Starr cavity to the Stiefel-cascade model. The
+published framing is explicit about scope: "a broader variational
+principle, in which the lower bound, as well as the actual value, are
+expressed through an optimization procedure for which
+ultrametric/hierarchical structures form only a subset of the
+variational class." A related critical-point method for vector spin
+glasses with nonconvex interactions is due to Chen and Mourrat. Fix
 smooth compactly supported $D$. Replace the pattern count $M$ by
 $M+\ell$ for a fixed cavity size $\ell$ large compared to $1$ and tiny
 compared to $M$. Compare the enlarged pressure with the base $M$
 pressure on the same disorder, using two restored patterns per
 increment.
-./refs/aizenman-et-al-2004-cond-mat-0403240.pdf
+./refs/aizenman-sims-starr-2003-cond-mat-0306386.pdf
 ./refs/arxiv-2311.08980v3.pdf
 
 The cascade with only the outer split at $\theta$ and the same
@@ -991,9 +1000,9 @@ initial integrals of $B^*$ dominate those of $A$ everywhere, with equal
 totals.
 
 The allocation step is the finite-weight Hardy-Littlewood-Polya
-convex-order principle. For the original rearrangement background see
+convex-order principle. For the underlying rearrangement theory see
 Hardy, Littlewood, and Polya's monograph; the specific weight-preserving
-allocation proof is given here.
+allocation proof is given.
 
 Consider allocation tables $\lambda_{ij}\ge 0$ with column sums $p_j$
 and row sums $p_i$. The entry $\lambda_{ij}$ is the dimension