From a3f6c103678e0dcae3a682945c784a3f88d5039f Mon Sep 17 00:00:00 2001 From: YurenHao0426 Date: Fri, 5 Jun 2026 15:57:01 -0500 Subject: Review methods for FA operator dynamics --- notes/18_contribution_roadmap.md | 28 +++++++++++++++++++++++++++- 1 file changed, 27 insertions(+), 1 deletion(-) (limited to 'notes/18_contribution_roadmap.md') 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. -- cgit v1.2.3