algorithm: calibrate feedback in representable subspace

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@@ -76,6 +76,62 @@ The important dependence is `O(sigma^2)`, with a dimension-dependent constant.
 Reducing K removes variance but not this bias; increasing width/depth without a
 sigma audit is therefore unsafe.
 
+### Perturbing the representable convolutional subspace
+
+The failed ResNet-20 run exposed a separate projection problem.  Its
+translation-shared feedback field for example `i`, channel `c`, and spatial
+site `s` is
+
+```text
+p_ics = b_ic + t_ics v_ic,       t_ics = tanh(h_ics).
+```
+
+The legacy estimator perturbs every `h_ics` independently, constructs a full
+unit target, and only then averages its delta-rule update into the shared
+`b/v` parameterization.  At K1, almost all sampled energy can lie outside this
+two-dimensional per-channel subspace.  SDIL-v2 instead draws independent
+Rademacher coefficients `z^b_ic,z^v_ic` and intervenes along
+
+```text
+xi_ics = (z^b_ic + t_ics z^v_ic) / sqrt(2).
+```
+
+Let `q_ics=-dL/dh_ics`, let `S` be the number of spatial sites, and let `D` be
+the centered directional loss derivative.  Then
+
+```text
+qhat^b_ic = -sqrt(2) D z^b_ic / S,
+qhat^v_ic = -sqrt(2) D z^v_ic / S
+```
+
+are unbiased for the two negative-gradient moments
+
+```text
+qbar^b_ic = mean_s q_ics,
+qbar^v_ic = mean_s (q_ics t_ics).
+```
+
+All other examples, channels, and layers contribute zero-mean cross terms.
+For BatchNorm, `D` is the derivative of the summed minibatch loss, so the same
+identity targets each example's contribution including the true cross-example
+Jacobian coupling.
+
+This does not mistake moments for expansion coefficients.  The predicted
+moments of `p=b+t v` are
+
+```text
+pbar^b = b + mean(t) v,
+pbar^v = mean(t) b + mean(t^2) v.
+```
+
+Consequently updating the two vectorizer pathways with
+`qhat^b-pbar^b` and `qhat^v-pbar^v` is exactly the expected gradient of the
+original full-field squared prediction error.  It uses the same two causal
+loss queries per direction as unit perturbation, but avoids estimating
+unrepresentable spatial components.  It remains a variance reduction, not a
+guarantee of useful feedback or stable end-to-end optimization; those require
+a separately frozen empirical gate.
+
 ## 2. What is sufficient for a descent step
 
 Flatten all forward parameters into `theta`, let `p = grad L(theta)`, and let