theory: connect local bias removal to CLLN scaling

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@@ -499,6 +499,78 @@ 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.
 
+### State-dependent component bias accumulates with system size
+
+The physical-network setting makes the nuisance-removal statement concrete.
+For component `e`, let `z_e` denote locally observable state, `g_e` the ideal
+credit signal, and let the measured local signal be
+
+```text
+a_e = g_e + b_e(z_e) + epsilon_e,
+```
+
+where `b_e(z_e)` is a repeatable component imperfection and `epsilon_e` is the
+remaining zero-mean variation. A one-time constant calibration subtracts
+`c_e=E[b_e]`. The best predictor from the local information `F_e` is
+`p_e=E[b_e | F_e]`. Conditional projection gives
+
+```text
+E[(b_e-c_e)^2] - E[(b_e-p_e)^2]
+  = E[(p_e-c_e)^2] >= 0.
+```
+
+Thus constant calibration is optimal only when the predictable imperfection is
+constant over the component's operating states. SDIL estimates `p_e` online
+from neutral observations and subtracts it locally; it does not require the
+simulator's imperfection parameters or a component-by-component oracle table.
+Its remaining error includes predictor approximation, tracking error, and the
+unpredictable part `epsilon_e`.
+
+For `E` components, the squared norm of the residual bias is exactly the sum
+of its component powers:
+
+```text
+E[||delta||^2] = sum_e E[delta_e^2].
+```
+
+If the average residual power per component remains bounded away from zero,
+the total residual power grows linearly in component count. Correlation between
+components can change particular projected modes, but it does not remove this
+sum-of-squares accumulation. Reducing the per-component residual therefore
+reduces the coefficient of size-dependent error growth; it need not make a
+large physical network perfect.
+
+The link from residual bias to optimization error can be stated exactly for a
+local quadratic model. Let
+
+```text
+F(k) = 1/2 (k-k_star)^T H (k-k_star),
+implemented update = -grad F(k) + delta.
+```
+
+For constant `delta` in the range of positive-semidefinite `H`, the displaced
+fixed point and its excess objective are
+
+```text
+k_infinity = k_star + H^+ delta,
+F(k_infinity)-F(k_star) = 1/2 delta^T H^+ delta.
+```
+
+On the positive-curvature subspace this implies
+
+```text
+||delta||^2/(2 lambda_max)
+ <= F(k_infinity)-F(k_star)
+ <= ||delta||^2/(2 lambda_min).
+```
+
+A component of `delta` in the nullspace of `H` causes drift rather than a
+displaced optimum. These statements explain why persistent bias can produce a
+size-dependent floor and why reducing residual power can reduce that floor.
+They do not by themselves determine classification error: the CLLN ladder
+tests the downstream effect using held-out logic tasks, while residual-bias
+measurements remain diagnostic.
+
 ### Incomplete residualization can create multiplicative local instability
 
 Residual power alone is not a stability certificate. Consider a linearized