Clarify soft erosion scaling claim

2 files changed, 100 insertions, 0 deletions

diff --git a/notes/18_contribution_roadmap.md b/notes/18_contribution_roadmap.md
index 4805a19..be4d775 100644
--- a/notes/18_contribution_roadmap.md
+++ b/notes/18_contribution_roadmap.md
@@ -71,6 +71,17 @@ So the FA/BP gap need not wait until redundant directions are fully exhausted.
 Redundancy suppresses the expected task-relevant alignment burden, but does not
 make it exactly zero.
 
+The scaling distinction is:
+
+```text
+k fixed, P grows        -> k/P -> 0, erosion vanishes
+k proportional to P     -> k/P stays constant, relative erosion persists
+```
+
+For equal-width MLPs, both parameter count and feedback matrix dimensions scale
+like `Θ(L n^2)`, so a fixed per-matrix alignment burden can remain a constant
+fraction of capacity under width scaling.
+
 Hard functional rank loss remains a limiting sanity check:
 
 ```text
diff --git a/notes/23_soft_capacity_theory_revision.md b/notes/23_soft_capacity_theory_revision.md
index bc5ff3c..c64c36c 100644
--- a/notes/23_soft_capacity_theory_revision.md
+++ b/notes/23_soft_capacity_theory_revision.md
@@ -189,3 +189,92 @@ Scaling Law and Soft Capacity Erosion
 
 Hard exhaustion should appear as a corollary or limiting case, not the main
 claim.
+
+## Selling Point
+
+The natural intuition is:
+
+```text
+if the model is sufficiently overparameterized, it should absorb the random
+feedback-alignment burden and behave like BP.
+```
+
+Existing FA/DFA work often supports the softer intuition that the network can
+learn to make random feedback useful, and overparameterized two-layer analyses
+show convergence in settings where random feedback is used. Our result should
+not deny those convergence results. Instead, it sharpens the question:
+
+```text
+does overparameterization make the alignment burden exactly free?
+```
+
+The soft-erosion answer is no in general.
+
+Overparameterization dilutes the burden only through the ratio `k/P`:
+
+```text
+E[K_k | J] = (1 - k/P) K_BP
+```
+
+So if `k` is fixed and `P -> infinity`, then:
+
+```text
+k/P -> 0
+```
+
+and the soft capacity loss vanishes.
+
+But FA does not usually keep `k` fixed while scaling the model. In an equal-width
+MLP, parameter dimension scales like:
+
+```text
+P = Θ(L n^2)
+```
+
+The feedback-alignment matrices also scale with the layer matrix dimensions:
+
+```text
+D_l = n_l n_{l+1}
+sum_l D_l = Θ(L n^2)
+```
+
+If the effective alignment burden is proportional to these feedback matrix
+dimensions, then:
+
+```text
+k = α P
+```
+
+for some effective burden fraction `α`. Then:
+
+```text
+k/P = α
+E[K_k | J] = (1 - α) K_BP
+```
+
+The relative tangent-capacity reduction does not vanish just because width grows.
+
+This gives the clean scaling message:
+
+```text
+large models can make the absolute performance strong;
+but if the feedback-alignment burden scales with model size, the relative
+capacity erosion can remain nonzero.
+```
+
+This also explains why empirical gaps can look small in large models:
+
+1. BP and FA losses may both be very low, so the absolute task metric gap is
+   visually small.
+2. The residual or test metric can be insensitive once both methods are in a
+   high-performance regime.
+3. The correct theoretical object is the fractional erosion `k/P` or the
+   operator reduction `K_FA` versus `K_BP`, not only the final raw loss gap.
+
+Therefore, the main claim should be:
+
+```text
+capacity does not make feedback alignment free; it only controls the erosion
+rate. The gap vanishes only under scalings where the effective alignment burden
+grows sublinearly relative to parameter dimension.
+```