diff --git a/spherical_perceptron_bi_orthogonal_v2.txt b/spherical_perceptron_bi_orthogonal_v2.txt
index 30e2e46..36cf519 100644
--- a/spherical_perceptron_bi_orthogonal_v2.txt
+++ b/spherical_perceptron_bi_orthogonal_v2.txt
@@ -42,8 +42,7 @@ finite-temperature variational functional for general projection
 potentials with independent Gaussian patterns; the rigorous theorems
 concern the performance of approximate-message-passing algorithms.
 These papers predict an overlap hierarchy beyond the regime described
-by a single scalar order parameter. The variational formula stated
-below retains this hierarchy as a matrix-valued overlap path.
+by a single scalar order parameter. The variational formula retains this hierarchy as a matrix-valued overlap path.
 ./refs/arxiv-gyorgyi-reimann-0003360.pdf
 ./refs/arxiv-2406.02970v3.pdf
 
@@ -151,7 +150,7 @@ The formula for $F_{\alpha,\nu}(f)$ splits into a radial part driven by
 how the spherical input decomposes among the singular-value blocks, and
 an angular part kept inside an auxiliary Gaussian model. The radial part
 is a Dirichlet optimisation; the angular part is a matrix-path
-variational problem computed below.
+variational problem evaluated by the Gaussian theorem.
 
 Approximate the singular slots in operator norm by finitely many
 positive values $s_1,\ldots,s_h$ with multiplicities $n_j$ satisfying
@@ -165,7 +164,7 @@ value for this finite approximation is
 
 $$\sup_{\rho}\left\{\frac12\sum_{i\in\mathcal I}v_i\log\frac{\rho_i}{v_i}+\alpha P_p^f\left(\left(\frac{s_j^2\rho_j}{\alpha}\right)_{j=1}^h\right)\right\},$$
 
-where $P_p^f$ is the angular pressure characterised below and
+where $P_p^f$ is the angular pressure characterised by the Gaussian theorem and
 $p=(p_j)_j$ fixes every block proportion. The limit of these
 approximations as the spectral mesh tends to zero is $F_{\alpha,\nu}(f)$.
 
@@ -212,7 +211,7 @@ entries of $\xi_j^TR_j$ by $x,j$ and $z_i,j$ for $j\le i<h$. Write
 $$y=\sum_j(\xi_j^TR_j)_{x,j},\qquad u_i=\sum_{j\le i}(\xi_j^TR_j)_{z_i,j}.$$
 
 The labels $y,u_i$ refer to single coordinates of the row fields in the
-Gaussian model derived next. Set
+Gaussian model. Set
 
 $$g_1=y^2,\ c_1=s_a;\qquad g_2=-\min(y^2,B),\ c_2=-s_a;\qquad g_{i+2}=(y-u_i)^2,\ c_{i+2}=\sum_{j>i}a_j.$$
 
@@ -254,7 +253,7 @@ samples the tilted conditional inner law.
 The first column is drawn once before the inner columns are integrated.
 In the cascade representation it is shared by visits whose common level
 is at least $\theta$; the other columns remain visit variables. The
-matrix paths introduced next record the covariances of Gaussian row
+matrix paths record the covariances of Gaussian row
 fields seen by such visits. In the Gaussian theorem those matrix paths
 determine a variational functional whose value matches the limit of the
 normalized log integral.
@@ -305,7 +304,7 @@ grid for all blocks. The diagonal increment $I-q_{j,k-1}$ is integrated
 independently on every visit in the cascade interpretation, including
 repeated visits to the same leaf.
 
-The Gaussian theorem states that, with the Stiefel model above, fixed
+The Gaussian theorem states that, with the Stiefel model, fixed
 $0<\theta<1$, $n_j/M\to p_j>0$, and $D$ bounded continuous or the sum
 of a bounded continuous function and a nonpositive quadratic form, the
 normalized log integral satisfies
@@ -575,8 +574,7 @@ count $M$ by Poisson$(Mr)$ with $r\in[0,1]$, so that $r=0$ gives no
 patterns and $r=1$ recovers the true problem. Add independent
 monomial perturbations of the pattern Hamiltonian scaled by
 $s_r=r\sqrt M e_M$ with $e_M=M^{-1/16}$ and suitably normalised
-coefficients. The perturbation kernels are those scalarizations used
-above; the Gaussian realizations are achieved by tensor fields on
+coefficients. The perturbation kernels are those scalarizations; the Gaussian realizations are achieved by tensor fields on
 the spin columns.
 
 Define the contact functional
@@ -606,7 +604,7 @@ this is the upper bound on the fresh pattern log-pressure.
 Differentiating in $b$ gives the tail contact inequality: for every
 nondecreasing variation $\tilde b\succeq b$ of the linear-field path
 with compact support, the covariance comparison $V_D(\tilde q)\le
-V_D(q)$ (from the path-order implication proved above) implies
+V_D(q)$ (from the path-order implication) implies
 $q^*\preceq q$ in the Loewner order path sense. The gradient of the
 supersolution with respect to $b$ is a signed measure whose negative
 part is supported where $q^*$ stays below $q$. Summation by parts and
@@ -617,7 +615,7 @@ Ghirlanda-Guerra identities. The critical step is that, at the
 minimising perturbation parameters, the joint Ghirlanda-Guerra
 identities hold in the limit $M\to\infty$, and the matrix overlap
 limits form an eligible matrix path. Supply of the GG identities uses
-the monomial perturbations introduced above. Concentration of
+the monomial perturbations. Concentration of
 $M^{-1}\log\mathcal Z_{M,\theta}$ on a bounded window of the
 perturbation parameters, combined with Gaussian interpolation on the
 perturbing fields, bounds the expected first derivative of the
@@ -665,18 +663,20 @@ every allowed field increment.
 This closes the expected-pressure upper bound when $D$ is smooth with
 compact support. Bounded continuous $D$ follows by approximation:
 bounded Lipschitz functions agreeing with $D$ uniformly on compacts
-give the same limit. The negative-quadratic tail class is handled
-below with the soft-penalty construction.
+give the same limit. The negative-quadratic tail class is handled by the soft-penalty construction.
 
 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. Fix
+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
 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/arxiv-2311.08980v3.pdf
 
 The cascade with only the outer split at $\theta$ and the same
 monomial perturbations (amplitude $s_M=M^{3/8}$) replaces the
@@ -808,7 +808,7 @@ beyond the base at parameter $u$. The constant in $O(1)$ is fixed
 across these choices. Begin with any good-parameter sequence and any
 compact array limit. The tolerance, shell, and rounding choices apply
 to it with uniform canonical constants; failure of the eventual bound
-would contradict the cavity inequality above. The bad parameter sets
+would contradict the cavity inequality. The bad parameter sets
 have $\pi$-measure zero and use the fixed-$\ell$ uniform lower
 increment bound.
 
@@ -825,13 +825,12 @@ the pressure at the larger argument up to $o(M)$, uniformly on shells.
 Maximal sub-sequential identification. Combining the upper and lower
 bounds, every subsequential limit of the normalised expected log
 pressure equals $\mathcal G_{\theta,p}(D)$. Concentration of the
-normalised log on a bounded window, applied as above, transfers this
+normalised log on a bounded window transfers this
 to convergence in probability. Boundedness of $D$ and uniform
 integrability bound the pressure, giving convergence in expectation.
 
-Soft penalties and quadratic tails. The Gaussian theorem was proved
-above for bounded continuous $D$ with smooth compact support, and the
-bounded continuous case follows by approximation. The angular formula
+Soft penalties and quadratic tails. The Gaussian theorem was proved for smooth compactly supported $D$,
+and the bounded continuous case follows by approximation. The angular formula
 also requires a negative-quadratic tail class. Extend $D$ to be the
 sum of a bounded continuous function and a nonpositive quadratic form
 $-C|\xi|^2$. Uniform tail control for Gaussian rows under the trial
@@ -1288,7 +1287,11 @@ pressure is at least $-8.75$, so the limits differ. The rectangular
 singular measure excluding the structural zero still converges to
 $\delta_{\sqrt\alpha}$, but the output eigenvalue measure hides the
 outlier in the zero mass. No-outlier control must be on the rectangular
-singular slots, not on output eigenvalues.
+singular slots, not on output eigenvalues. The uniform lower bound
+$\inf_u Q(u)\ge 1/80$ is a Kashin-type spreading estimate for Haar
+frames; a norm-comparison restatement in the lasso analysis of Bayati
+and Montanari gives the same conclusion with explicit constants.
+./refs/bayati-montanari-lasso.pdf
 
 Counterexample: unbounded limiting spectrum. Let $d_k=\exp(8^k)$ and
 $\nu=\sum_{k\ge 1}2^{-k}\delta_{d_k}$ with unbounded support. For