The Automaton Underneath: The Additive Input Pathway Is a Parasitic Attractor for State Tracking in Householder Linear RNNs

Linear RNNs with input-dependent Householder-product transitions (DeltaNet/DeltaProduct-class) can provably represent hard state-tracking automata, yet trained models fail to length-generalize -- a gap recent work attributes to optimization, without a causal account. We give one, in a pre-registered, within-architecture causal ablation: the same model with one term deleted -- the additive input injection bₜ = Wb eₜ. With bₜ, models fit length 32 and collapse out-of-distribution on parity, S₄, A₅, and non-solvable S₅ word problems (0.20 at position 512 on S₅). Without it -- input acting only through the orthogonal transitions -- the same architecture learns the exact automaton: median accuracy 1.00 at 16x the training length, at every width admitted by a representation law we state and test: the minimal number of Householder factors per token equals the maximal reflection length of the task's generators in the format-pinned representation (parity 1, S₄ 3, A₅ and S₅ 4); below it, nothing fits. The contrast with DeltaProduct's S₄/A₅ at nₕ=2 (group-element classification, $SO(3)$ realization) shows the law is representation-relative: task format is a hidden variable in state-tracking benchmarks. Two pre-registered arms locate the mechanism. (i) Initialized at a verified-exact solution with Wb=0, Adam grows the additive path and pulls the model off the exact solution; $-b$ controls stay at 1.00. (ii) Our registered prediction that the fit routes through bₜ fired its kill criterion: all 49 fitting seeds retain in-domain fit under Wb:=0 -- and at law-minimal width, zeroing Wb at inference restores exact generalization (parity 5/5, S₅ 5/5, S₄ 4/5, A₅ 4/5). The additive pathway is parasitic: it destabilizes, then conceals, a correctly learned automaton. All 202 runs pre-registered; all numbers regenerate from artifacts. Code, pre-registration, and all 202 run artifacts are included in this deposit.