CNRS-P1-Ext: Problem 1 Extension: The β = (3 + √5)/2 Test Case and the Invariance Gap Supplement to the CNRS Problem 1 Working Paper

The Problem 1 working paper 5 establishes explicit (−β) -expansion admissibility theorems for two Pisot numbers, β = 1 + √2 and β = 1 + √3, in the band (2, 3]. Both yield period-2 endpoint orbits with admissibility blocks (01) ^ω and (20) ^ω, motivating the uniform admissibility conjecture (the strengthened periodicity conjecture of 5). The natural next test case is βφ = (3 + √5) /2 ≈2. 618, the Pisot number in (2, 3] whose minimal polynomial is x2 −3x+1 = 0. The table in 5 records that lβ has period 1 and rβ has period 3 under T−β, already differing from the two proved instances (both period 2). We carry out the full orbit computation for βφ and find that the map T−β is not Iβ-invariant at this value: the image of lβφ exits Iβ at the first step. This confirms the invariance gap identified in 5 and shows that the uniform admissibility conjecture cannot be verified for βφ without first resolving the invariance question. We then apply the Pisot conjugate contraction strategy (Route B of 5) concretely to βφ, identifying the conjugate β = (3−√5) /2 ≈ 0. 382 and the contraction rate |β| ≈ 0. 382 < 1. We show that the same structural mechanism operates here as in the two proved instances — the Pisot conjugate norm provides contraction — though the specific numerical rate differs (|β¯φ| ≈0. 382 versus 0. 414 and 0. 732 for the two proved cases). The contraction rate at βφ is in fact the fastest of the three, giving strong heuristic support for the uniform conjecture once invariance is established. Finally we state the revised priority ordering for Problem 1 as a consequence of this computation, and identify the invariance gap as the sole blocking obstacle to completing the admissibility programme.