Operational BHR/BMR Bridge: Heritage/Memory Bifurcation as Two Readouts of One Two-Qubit State, with Embedding in the JANO-1 Circuit — RCD Program, Technical Note JANO-BR-0 (v0.1)

Technical Note JANO-BR-0 of the RCD program (continues “La doble herencia de la coherencia”, DOI 10.5281/zenodo.23520439; the Adjudication Record “Relational Memory under Causal Erasure”, DOI 10.5281/zenodo.23717352; BMR-1, DOI 10.5281/zenodo.23717139; and REP-0, DOI 10.5281/zenodo.23774093). Question: can the heritage/memory bifurcation of BHR-BMR become a physical readout of one and the same circuit? Results (exact, no hardware): in BHR-3 the mixed relational-memory state ρ_mix(r) has rank 2 whenever Ω₃ > 0, so a single qubit c purifies it, |Ψ(r)⟩ = √λ₊|0⟩|ψ₊⟩ + √λ₋|1⟩|ψ₋⟩, with a three-gate {Ry, CNOT} construction and a closed map r ↦ (θ_c, η); tracing out c returns ρ_mix (R⊥), while measuring c in the HJW basis leaves the system exactly in the coherent-heritage state ρ_coh with probability p_coh = Tr D_coh/Tr D_mix and in the residual-memory state |−⟩ (independent of r) with probability p_M; the three readouts are tied by the exact identity V_mix = p_coh·V_coh − p_M (at r = 1: 145/149 = 147/149 − 2/149). Verified against the original JANO-1 code: the five-qubit JANO-1 circuit with α = α_BR(r) = arcsin √λ₋(r) is literally this purification, and its frozen observables reduce to P_rec = λ₊ and R∥ = R⊥ = √(2λ₊λ₋) = √S_lin(ρ_mix) — a spectral readout of the memory — while the HJW readout is the complementary face that separates heritage from memory. Stated limit: the construction reproduces ρ_mix(r) by design and is not a falsifiable prediction of RCD; hardware stays closed. Scripts with SHA-256 manifest. English and Spanish versions; the Spanish text prevails.