Why it is a question
THEORY-205 holds, exactly, that in the slab geometry on a cylinder the uninformative information-bottleneck encoder loses local stability at β = (λ0/λ1)^(2L_B), and that the unstable direction is the boundary weak value of the leading excited transfer-matrix eigenvector. That point is β = 1/ρ_m², with ρ_m the maximal correlation between block and environment. The uninformative encoder stops being globally optimal earlier, at β = 1/s*. Here s* = sup I(U;E)/I(U;V) over encoders U, and s* ≥ ρ_m².
In the circumference-3 critical Ising check the two thresholds agree to six figures. In a nearest-neighbour chain of 3-state units, they differ 27-fold at a buffer of two transfer steps. There an encoder that flags a rare state beats the uninformative one well below the bifurcation.
So the IB’s first transition in a lattice model is the bifurcation only under some further condition, perhaps a symmetry. Whether the first feature the global optimum keeps is still the leading excited mode is open. Until 2026-10-19 the second-model half of this was THEORY-205’s promote_when.
What would answer it
- A proof that s*(V;E) = ρ_m²(V;E) for a stated class of transfer matrices, such as those whose leading excited mode is odd under an involutive symmetry. Or a characterisation of when it fails.
- An exact transfer-matrix computation on a model other than Ising, such as a Potts or non-integrable chain on a cylinder wide enough to separate ∆₁ from ∆₂, that finds the global IB optimum just past 1/s* and says which feature it keeps.
Against THEORY-205’s feature claim: a short-range model with a symmetric transfer matrix whose unstable direction at the bifurcation, in the slab geometry, is not that eigenvector.
What would not answer it: results for planar blocks inside a buffer shell, or for RSMI at a fixed alphabet. They bear on the wider claim THEORY-205 does not make.