Multimode Memory Flow and the RS Scalar Obstruction: An Erasure-Closed Causal Architecture with Two Exact Memory Modes (BMR-1)

Technical note BMR-1 of the RCD program (Radial Coherential Dynamics), continuing "Relational Memory under Causal Erasure" (10.5281/zenodo.23630978). The previous record found an exact memory-flow law on the star family, which carries a single memory mode. This note asks for the minimal causal architecture able to carry two independent memories and survive its own coarse-graining. Three exact results. (1) Branch–rank lemma: if the N erasure positions produce r distinct branches, rank M_N ≤ r − 1, with equality under affine independence; two modes require at least three causally inequivalent erasures. (2) The P2_n sector (a chain with one pendant element at height ≤ 2) is closed under erasure for all n ≥ 4, has unbounded height and two exact memory modes: exact transfer map R_n, two flow invariants (y, J), closed-form solution, Hermitian memory M_n with rank 2 iff uv ≠ 0, and linear transport of (|u|², Re ūv, Im ūv, |v|²) with spectrum {a², ac, ac, c²}, a = (n−2)/n, c = (n−3)/n; the relative phase is carried as an independent memory component. (3) Step-closure theorem: the support of every Rideout–Sorkin (RS) amplitude is fixed by the local pair (|↓x|, #max ↓x) and always contains the antichain, so no closed family omitting it, P2_n in particular, admits an RS realization; the star is the only RS-supportable family found. An exact classification of RS closure families 5→4 on restricted supports finds only the star line, plus a certified isolated conjugate pair that closes one step and not the next. The obstruction is against the locality of the RS amplitude support, not against scalar amplitudes as such. Includes the erasure automaton of unlabeled posets up to N = 7 (1, 2, 5, 16, 63, 318, 2045 classes) and all scripts with a SHA-256 manifest. Exact results are stated as such; the enumeration of isolated points on large supports is not certified. English and Spanish versions; the Spanish text prevails. Developed at the Panel Cerezo: ChatGPT (formulation of BMR-1, branch–rank lemma, Hermitian correction), Claude (automaton, P2_n, step-closure theorem, classification, verification and drafting), under the direction and adjudication of the principal.