R12 Minimax Causal Broadcast Subspace No-Go

Status: THEOREM-REJECTED AS A REASONING MECHANISM BEFORE CODE. MCBS may be useful as a read-only causal-mediation diagnostic, but it cannot identify a reusable workspace or authorize training a writer/updater in its subspace.

1. Proposed object

For latent contrast delta_i, downstream consumer f_ic, a frozen residual metric G, and rank-k basis B, define

P_B = B (B^T G B)^(-1) B^T G
Delta_ic   = f_ic(h_i + delta_i) - f_ic(h_i)
Delta^B_ic = f_ic(h_i + P_B delta_i) - f_ic(h_i)

The proposed objective minimizes the worst-consumer normalized error between Delta^B_ic and Delta_ic. The metric must be frozen; otherwise the meaning of the complement changes under residual rescaling.

2. Exact collapse theorem

For affine consumers f_c(h) = A_c h + b_c, projection preservation and complement ablation are the same condition:

Delta^B_c = Delta_c
iff A_c (I - P_B) delta = 0
iff Delta^perp_c = 0.

Let D = span{delta_i} and N = D intersect (intersection_c kernel(A_c)). The minimum exact dimension is

k_min = dim(D) - dim(N).

Therefore MCBS recovers only the observable quotient of the chosen consumers. That quotient need not contain task state, an update law, or information useful to a new consumer.

3. Motor-bundle counterexample

Let the true state be (a, b) and every fitted consumer depend only on a XOR b. A one-dimensional parity coordinate preserves every fitted causal effect, while complement ablation removes every fitted effect. It nevertheless cannot answer a held-out query for a. More generally, a finite set of consumers can be served by a post-hoc answer bundle (g_1(s), ..., g_m(s)).

MCBS cannot distinguish that bundle from a reusable state. Adding more fitted consumers only enlarges the answer table unless the test also requires closure under unseen state updates and unseen consumers revealed after the subspace is frozen.

4. Prior-art boundary

The ingredients already overlap established methods:

projection-based causal subspaces and donor swaps: Distributed Alignment Search, https://proceedings.mlr.press/v236/geiger24a.html;
counterfactual intervention training: Interchange Intervention Training, https://proceedings.mlr.press/v162/geiger22a.html;
causal activation localization: ROME causal tracing, https://proceedings.neurips.cc/paper_files/paper/2022/hash/6f1d43d5a82a37e89b0665b33bf3a182-Abstract-Conference.html;
worst-group shared subspaces: Fair PCA, https://proceedings.neurips.cc/paper_files/paper/2019/file/2201611d7a08ffda97e3e8c6b667a1bc-Paper.pdf.

The defensible synthesis is a vocabulary-free, consumer-supervised, minimax causal-localization diagnostic. It is not a new reasoning primitive.

5. Decision and successor boundary

No Shohin writer, updater, SFT, or coordinate swap is authorized from MCBS. Any successor must separate reusable state from a finite answer table by freezing the representation before revealing both:

held-out consumer functions;
held-out state-update operators;
a committed output-code permutation;
source-deleted multi-step reuse.

It must pass a workspace-positive synthetic model and reject a dimension- matched motor-only model under the same compute and scorer. Readout accuracy, projection preservation, and complement ablation alone are insufficient.