theory: bound both sides of residual stability

1 files changed, 22 insertions, 5 deletions

diff --git a/THEORY.md b/THEORY.md
index 8299c75..f659d0f 100644
--- a/THEORY.md
+++ b/THEORY.md
@@ -539,11 +539,20 @@ joint eigenmode let `k=d c-lambda`, where `d` and `c>=0` are eigenvalues of
 
 The determinant is `mu`. Evaluating its characteristic polynomial at one
 gives `p(1)=-eta k`; therefore every `k>0` produces a real eigenvalue above
-one. Weight decay stabilizes the mode only if it makes the effective `k`
-nonpositive, with the usual additional discrete-time step-size constraints.
-Consequently a small aggregate residual RMS can coexist with an unstable
-worst-direction coefficient. Deep state dependence can amplify the same mode
-across layers before a slow neutral predictor catches it.
+one. The full second-order Jury conditions give the strict stability window
+
+```text
+|mu| < 1,
+-2 (1 + mu) / eta < k < 0.
+```
+
+The upper boundary excludes positive Hebbian feedback; the lower boundary
+excludes an over-strong negative mode that alternates and grows. Weight decay
+or predictor overestimation must therefore control both sign and magnitude,
+not merely make `k` nonpositive. Consequently a small aggregate residual RMS
+can coexist with an unstable worst-direction coefficient. Deep state
+dependence can amplify the same mode across layers or increase the local
+covariance until an initially stable negative `k` crosses the lower boundary.
 
 This calculation explains the frozen MT-1 failure boundary without changing
 its result. The training-only audit finds the first active nonfinite tensors
@@ -554,6 +563,14 @@ traffic setting. A scalable version needs an operator-level dissipativity or
 gain-control mechanism; the pretraining residual-RMS threshold is
 insufficient by itself.
 
+The subsequent frozen S0 training-prefix screen tests a fixed one-sided
+predictor margin and finds no eligible value in `{0.001,0.003,0.01,0.03}`.
+Every candidate has nonpositive measured neutral residual-soma slope, yet the
+smaller margins overflow BatchNorm state and the larger margins show enormous
+transient loss and parameter growth. This is consistent with the two-sided
+condition above: sign control is necessary but does not bound the evolving
+gain `d c` or nonnormal deep-network transients.
+
 ### Intermittent feedback tracking can hide behind a final cosine
 
 An idealized mirror event every `k` task updates uses