Review methods for FA operator dynamics

1 files changed, 27 insertions, 1 deletions

diff --git a/notes/18_contribution_roadmap.md b/notes/18_contribution_roadmap.md
index aa26caf..89b578b 100644
--- a/notes/18_contribution_roadmap.md
+++ b/notes/18_contribution_roadmap.md
@@ -230,7 +230,33 @@ This separation avoids overclaiming while still giving a coherent theory chain.
 
 ## Next Priority
 
-The weakest current visual was the phase-transition contribution; the dense
+The weakest current theory gap is not the static capacity law anymore. It is
+the time-dependent operator dynamics:
+
+```text
+e_0 is solved by the actual-FA initial moment theorem
+e_t for t > 0 still needs alignment-gain dynamics
+```
+
+The literature review in `notes/30_literature_methods_for_fa_operator_dynamics.md`
+suggests the most principled route is an FA tangent hierarchy:
+
+```text
+K_FA,t = J_t J_tilde_t^T
+dK_FA,t/dt = dJ_t/dt J_tilde_t^T + J_t dJ_tilde_t^T/dt
+```
+
+with a short-time approximation:
+
+```text
+K_FA,t ≈ K_FA,0 + t dot_K_FA,0
+```
+
+This formalizes the successful time-varying operator estimator. A lower-risk
+full-time theorem can be developed in deep linear FA or in a two-layer
+teacher-student order-parameter setting.
+
+The weakest current visual was the old phase-transition contribution; the dense
 long-training sweep now shows that the correct phenomenon is a soft ramp, not a
 hard kink.