# Structured-shift retention: exact coverage, explicit cost

**Status:** executed fresh-seed linear regression experiments and standard robust linear-algebra derivations. All uncertainty operators are known inputs. This is not a method for predicting unknown capability shifts and is not a claim of novel robust optimization or demonstrated RSI.

## Model and exact certificate

Let an update \(d\in\mathbb R^p\) change retained responses by

\[
B_\theta d,\qquad B_\theta=B_0+\sum_{\ell=1}^m\theta_\ell B_\ell,
\qquad \|\theta\|_2\le\rho.
\]

The baseline and all \(B_\ell\in\mathbb R^{r\times p}\) are supplied to the controller. Define \(M(d)=[B_1d\ \cdots\ B_md]\).

**Proposition 1 — Exact nominal-null uncertainty cost.** If \(B_0d=0\), then

\[
\sup_{\|\theta\|_2\le\rho}\|B_\theta d\|_2
=\rho\|M(d)\|_{\rm op}.
\]

**Proof.** The left side equals \(\sup\|M(d)\theta\|\). The induced operator norm gives the upper bound; a top right singular vector multiplied by \(\rho\) attains it. For general \(d\), the triangle inequality gives the valid but potentially conservative bound \(\|B_0d\|+\rho\|M(d)\|_{\rm op}\). Equality of this sum is not claimed when \(B_0d\ne0\).

**Proposition 2 — Uniform zero preservation.** For \(\rho>0\), an update preserves every operator in the uncertainty ball exactly if and only if

\[
d\in\ker\mathcal S,\qquad
\mathcal S=\operatorname{stack}(B_0,B_1,\ldots,B_m).
\]

**Proof.** Membership makes every summand zero. Conversely, the zero parameter gives \(B_0d=0\), and parameters \(\pm\rho e_\ell\) then force \(B_\ell d=0\) separately.

If \(\mathcal S\) has rank \(R\), its nullity is \(p-R\). If its columns are full spark, every nonzero support circuit has at least \(R+1\) coordinates, and any selected \(R+1\) columns supply a circuit when \(p\ge R+1\). **Proof:** every set of at most \(R\) columns is independent; \(R+1\) columns are dependent, with no coefficient zero by the same independence. This generic arity statement does not apply to arbitrary rank-deficient column subsets. If \(R=p\), no nonzero update exists; the numeric \(R+1\) arity exceeds the available dictionary and must not be interpreted as an existing packet.

For a nominal-null direction \(d_0\), choosing

\[
\alpha=\min\left\{1,\frac{\tau}{\|B_0d_0\|+\rho\|M(d_0)\|_{\rm op}}\right\},
\qquad d=\alpha d_0
\]

certifies retained-response change at most \(\tau\) throughout the entire modeled ball. The zero-denominator case uses \(\alpha=1\). This is a standard robust radial restriction, not an invented optimizer. It permits bounded change rather than strict invariance.

## Frozen experiment

The source and protocol were frozen before outcomes. Sixteen seeds cover every combination of \(p\in\{64,128\}\), \(n\in\{96,256\}\), retained rank \(r\in\{2,4\}\), and two or four uncertainty components: 256 main tasks. A further 16 rank-exhaustion tasks use \(p=64,r=4,m=16\). All supplied operators have orthonormal rows; ridge regularization is fixed at 0.01, \(\rho=0.25\), and \(\tau=0.03\).

Targets are a fixed equal-norm mixture of a common-kernel signal and a conflicting supplied-operator rowspace signal, with observation noise standard deviation 0.1. Training uses \(n\) Gaussian inputs and evaluation uses 1,024 fresh inputs. This deliberately exposes the learning cost of preserving conflicting retained responses. It does not resemble an unrestricted real-world capability suite. No validation-based regularization repair was performed after seeing results.

Methods are nominal-null ridge, a dense robust-stack projector ridge solve, an optimized mathematically identical robust-null-basis ridge solve, the certified scalar-budget nominal update, and no change. Factorization and solve time are charged independently to each method; shared synthetic data generation is excluded.

For each update, the study computes the **uniform analytical certificate**, evaluates its top singular-vector adversary, and checks 64 independently sampled in-ball shifts. Random-shift success is a diagnostic; the guarantee comes from the proved bound. Separate stress tests use radius \(2\rho\) and a new operator direction outside the modeled span.

## Results

|Method|Fresh-input test NMSE [pooled 95% bootstrap CI]|Solver time, ms|Mean uniform retained-response bound|Tasks certified within τ, %|
|---|---:|---:|---:|---:|
|nominal_null_ridge|0.41771 [0.37361, 0.46446]|0.349|0.125|0.0|
|robust_projector_ridge|1.22017 [1.13942, 1.30966]|0.614|1.02e-16|100.0|
|optimized_robust_basis_ridge|1.22017 [1.13942, 1.30966]|0.379|8.81e-17|100.0|
|scalar_budget_nominal|0.68764 [0.67971, 0.69680]|0.427|0.03|100.0|
|no_change|1.00106 [1.00088, 1.00126]|0.001|0|100.0|

Nominal preservation failed all structured uncertainty certificates. Strict common-kernel updates repaired that failure within the declared model but sacrificed predictive quality; with a fixed small ridge penalty, pooled NMSE exceeded the no-change baseline. This is not hidden as a successful learning result. Some better-sampled configurations improved over no change, but the full grid is reported below. The scalar-budget method offered a stronger pooled quality/retention compromise while allowing the declared nonzero tolerance.

The optimized robust-basis control matched dense-projector updates to maximum absolute coefficient error **2.445e-14**, and had identical quality. Thus the projector formulation is not credited with an efficiency improvement over standard constrained least squares. All scalar-budget updates exceeded \(\tau\) under their \(2\rho\) adversary; common-kernel updates remained invariant because they annihilate every supplied direction at any coefficient radius. In rank-exhaustion cases, robust-stack learning collapsed exactly to no change, as the rank argument predicts.

|Configuration|Nominal-null NMSE|Robust-basis NMSE|Scalar-budget NMSE|
|---|---:|---:|---:|
|p64_n96_r2_m2|0.5452|1.2152|0.6983|
|p64_n96_r2_m4|0.3172|1.0676|0.6819|
|p64_n96_r4_m2|0.4923|0.9954|0.6832|
|p64_n96_r4_m4|0.3214|0.9504|0.6659|
|p64_n256_r2_m2|0.2411|0.6325|0.6787|
|p64_n256_r2_m4|0.1375|0.6295|0.6608|
|p64_n256_r4_m2|0.2272|0.6363|0.6537|
|p64_n256_r4_m4|0.1592|0.5867|0.6388|
|p128_n96_r2_m2|0.8935|2.0993|0.7912|
|p128_n96_r2_m4|0.7479|2.0723|0.7275|
|p128_n96_r4_m2|0.9824|2.5272|0.7601|
|p128_n96_r4_m4|0.6498|2.4736|0.7059|
|p128_n256_r2_m2|0.2300|0.9730|0.6819|
|p128_n256_r2_m4|0.1271|0.9536|0.6656|
|p128_n256_r4_m2|0.3470|0.8647|0.6662|
|p128_n256_r4_m4|0.2645|0.8455|0.6425|
|p64_n96_r4_m16|0.3122|1.0006|0.7857|

## Coverage limitations and composition counterexample

Coverage of a known operator span is essential. If an unknown residual operator \(E\) with \(\|E\|_{\rm op}\le\eta\) is also allowed, the certificate becomes

\[
\|B_0d\|+\rho\|M(d)\|_{\rm op}+\eta\|d\|.
\]

For nominal-null updates the worst case is exactly \(\rho\|M(d)\|_{\rm op}+\eta\|d\|\): choose the structured top singular direction, then choose a rank-one \(E\) mapping \(d\) into the same output direction. Therefore common-kernel preservation does not protect nonzero updates from arbitrary unmodeled directions. An empirically justified bound on \(\eta\) is required; it is not supplied by anchor agreement alone. This is an immediate robust-optimization consequence, not a new general safety theorem.

Single-step shifts also need not cover their compositions. In dimension three take \(B_0=e_1^T\), \(A_1=e_1e_2^T\), \(A_2=e_2e_3^T\), and \(d=e_3\). Then \(B_0d=B_0A_1d=B_0A_2d=0\), but \(B_0A_1A_2d=1\), while the reverse composition gives zero. Closing the observation rowspace under these operators raises its rank from two to three and removes the update. This is the established observability/invariant-subspace issue: single-step tests cannot silently justify arbitrary ordered compositions.

## Scientific boundary and reproduction

The result supplies a falsifiable repair to a specified nominal-retention failure: exact robustness for a known finite structured uncertainty class, with measured loss of learning freedom and an optimized equal-solution baseline. It does not establish unknown-shift generalization, autonomous algorithm invention, or worldwide novelty. Robust least squares with uncertain data is established prior art: El Ghaoui and Lebret, *Robust Solutions to Least-Squares Problems with Uncertain Data*, SIAM J. Matrix Analysis and Applications 18(4), 1035–1064 (1997), [DOI 10.1137/S0895479896298130](https://doi.org/10.1137/S0895479896298130). Observation-kernel invariance is standard control theory; see [Stanford EE363 observability notes](https://ee363.stanford.edu/lectures/011_observ.pdf).

Run `OPENBLAS_NUM_THREADS=1 OMP_NUM_THREADS=1 python research/robust_shift/robust_shift.py`, then `python research/robust_shift/report.py`. NumPy is the only dependency. The package includes the source-hashed protocol, all per-task updates and adversaries, CSV rows, full configuration summaries, and five numerical checks. Mathematical proofs are stated above; independent external verification has not occurred.