Update README.md

README.md

@@ -284,45 +284,32 @@ These objects have a clear mathematical interpretation as symmetric functions of
 
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-# 8. The architecture did not scale monotonically
+# ## 8. Non-Monotonic Scaling with Network Depth
 
-One of the strongest negative results appeared when we increased network depth.
+Increasing network depth did not consistently improve numerical performance or symbolic recoverability in our experiments.
 
-Larger models did not automatically produce better numerical or symbolic behavior.
+In several 32-layer configurations, training entered a collapsed regime in which the network produced nearly constant outputs across substantially different polynomial inputs. One representative failed run exhibited a mean absolute error (MAE) of approximately
 
-Some 32-layer configurations entered a collapsed regime in which the output became almost constant across very different polynomial inputs.
+\[
+\mathrm{MAE}\approx 12.5,
+\]
 
-A representative failure had approximately
+whereas successful runs on real-root instances achieved MAE values near or below 1.
 
-$$
-\mathrm{MAE}\approx12.5,
-$$
-
-compared with values near 1 or below 1 for successful real-root runs.
-
-This happened even when the parameter budget was large.
-
-Later, a particularly clean comparison used approximately the same parameter count:
-
-$$
-24L/600K
-$$
+A particularly informative comparison involved two models with approximately matched parameter budgets of 600K parameters: 
+a 24-layer model and a 32-layer model. Under the tested training configurations, 
+the 24-layer model trained successfully, whereas the 32-layer model collapsed after only a few epochs.
 
-versus
+These observations establish that, for the tested configurations, increasing depth did not yield a monotonic improvement in numerical performance. 
+Instead, the deeper model exhibited a substantially less favorable training outcome despite a comparable parameter budget.
 
-$$
-32L/600K.
-$$
-
-The 24-layer model worked normally, while the 32-layer model collapsed after only a few epochs.
-
-This is an important result because it shows:
-
-$$
-\boxed{\text{more depth does not automatically produce more algebraic reasoning}}
-$$
+One possible explanation is that additional layers exacerbate optimization difficulties associated with the composition of nonlinear transformations, 
+including the propagation of gradients through polynomial activations. 
+However, the experiments reported here do not establish this mechanism, nor do they imply that 24 layers are inherently superior to 32 layers. 
+Matching parameter counts can also require different hidden widths, which may independently affect optimization and representational capacity.
 
-and, in this architecture, depth interacts strongly with optimization stability.
+We therefore treat this result as evidence of a depth-related optimization failure in the tested configurations,
+motivating a more systematic investigation of gradient propagation, activation statistics, and training stability as functions of depth and width.
 
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