ctt progress 2026-07-02 workspace/paper

workspace/paper/notes/theory_ctt.md

@@ -70,13 +70,14 @@ approximation error.
 
 ### 5. CTT Support Regret Bound
 
-If CTT returns `K` transported proposals whose minimum distance to target
-positive support is at most `epsilon_T`, and utility is locally Lipschitz in
-tangent code with constant `L_U`, then the proposal support regret to the best
-positive tangent in the target chart is bounded by:
+If source retrieval radius is `r`, positive tangent sets satisfy
+`d_H(P_s,P_t) <= L ||z_s-z_t||`, CTT transport error is at most `epsilon_T`,
+and utility is locally Lipschitz in tangent code with constant `L_U`, then the
+proposal support regret to the best positive tangent in the target chart is
+bounded by:
 
 ```text
-SupportRegret <= L_U * epsilon_T + epsilon_label + epsilon_model
+SupportRegret <= L_U * (L * r + epsilon_T) + epsilon_label + epsilon_model
 ```
 
 The bound is a proxy-support statement. It becomes an outcome claim only when

workspace/paper/sections/theory.tex

@@ -24,8 +24,10 @@ In this draft we implement residual and gated residual forms:
 \[
   \hat\xi_t = \xi_s^+ + \Delta_{\phi}(z_s,z_t,\xi_s^+),
   \qquad
-  \hat\xi_t = \xi_s^+ + g_{\phi}(z_s,z_t,\xi_s^+)
-  \odot \Delta_{\phi}(z_s,z_t,\xi_s^+).
+  \hat\xi_t =
+  g_{\phi}(z_s,z_t,\xi_s^+)\odot\xi_s^+
+  +(1-g_{\phi}(z_s,z_t,\xi_s^+))\odot
+  \Delta_{\phi}(z_s,z_t,\xi_s^+).
 \]
 \end{definition}
 
@@ -76,11 +78,12 @@ $\epsilon_{\phi}$ is approximation error.
 \end{proposition}
 
 \begin{proposition}[Proxy support regret bound]
-If CTT produces a proposal whose distance to target positive support is at most
-$\epsilon_T$ and utility is locally Lipschitz in tangent code with constant
-$L_U$, then proxy support regret to the nearest target positive is bounded by
+If source retrieval radius is $r=\|z_s-z_t\|$, the positive tangent bundle is
+$L$-smooth in Hausdorff distance, CTT transport error is at most $\epsilon_T$,
+and utility is locally Lipschitz in tangent code with constant $L_U$, then proxy
+support regret to the nearest target positive is bounded by
 \[
-  L_U\epsilon_T+\epsilon_{\mathrm{label}}+\epsilon_{\mathrm{model}}.
+  L_U(Lr+\epsilon_T)+\epsilon_{\mathrm{label}}+\epsilon_{\mathrm{model}}.
 \]
 \end{proposition}