theory: expose intermittent feedback tracking lag

1 files changed, 35 insertions, 0 deletions

diff --git a/THEORY.md b/THEORY.md
index c64eb46..d82e171 100644
--- a/THEORY.md
+++ b/THEORY.md
@@ -470,6 +470,41 @@ The relevant timescale is therefore the product of learning rate and number of
 neutral updates. Too few neutral updates leave traffic; task-period updates
 converge to the wrong coefficient regardless of speed.
 
+### Intermittent feedback tracking can hide behind a final cosine
+
+An idealized mirror event every `k` task updates uses
+
+```text
+Q_m^+ = Q_m + eta_M (W_m - Q_m).
+```
+
+Let `E_m=Q_m-W_m` immediately before that event and let
+`U_m=W_(m+1)-W_m` be the sum of the following `k` forward-weight changes.
+The tracking error obeys the exact recurrence
+
+```text
+E_(m+1) = (1 - eta_M) E_m - U_m.
+```
+
+If every task update has norm at most `d`, then `||U_m|| <= k d` and
+
+```text
+limsup_m ||E_m|| <= k d / eta_M.
+```
+
+The bound is attained in the scalar or constant-collinear case. Thus reducing
+the mirror cadence or increasing its learning rate only controls lag relative
+to how fast the task weights move; a high feedback cosine is not automatic.
+For the frozen RRM setting `k=16, eta_M=0.1`, the ideal worst-case lag scale is
+`160` single-task-update norms before stochastic finite-probe error is added.
+
+Conversely, after the task updates stop (`U_m=0`), the endpoint error decays as
+`(1-eta_M)^m`. A learning-rate drop or quiet tail can therefore produce an
+excellent final cosine even when the feedback was badly out of date during
+the loss-producing part of training. Endpoint alignment must be reported with
+the alignment trajectory and task loss. This is a standard tracking fact used
+to audit the mirror baseline, not an SDIL novelty claim.
+
 ## 4. Hardware-independent complexity
 
 Let `H` be the number of hidden layers, `K` the perturbation directions per