Memory Localization and Transport: A Finite Core Theorem for Erasure Memory, the Origin of the Logarithmic Registered Age, All-N Transport under Coarse-Graining on P2 and D2, and the Binomial Transport Locus — RCD Program, Technical Note MEM-0 (v0.1)

Technical Note MEM-0 of the RCD program (continues TIME-0, DOI 10.5281/zenodo.23850769; REP-1STAT, DOI 10.5281/zenodo.23824079; JANO-BR-0, DOI 10.5281/zenodo.23785719; BMR-1, DOI 10.5281/zenodo.23717139). Question: where is the memory registered by the relational-memory matrix M of BHR/BMR, and is its distribution consistent under the erasure channel that defines it? Results (exact in rationals, no hardware): (i) Finite Core Theorem — if N − k elements share one erasure branch (bulk) and k are exceptional (core), then m_bulk = ‖Δ‖²/N², m_r = ‖d_r − Δ/N‖² and Σ m_i = Q − ‖Δ‖²/N exactly, hence δ_N = Tr M_N = Q/N − ‖Δ‖²/N²; verified on all forms of P2, D2 and star with constants fixed for all n ≥ 4, yielding the closed form of D2; (ii) localization — the normalized memory of the growing homogeneous bulk is O(1/N) and the core's tends to 1, so τ_N = Q·H_N − ‖Δ‖²·H_N^{(2)} + const ~ Q log N: TIME-0's logarithm is the imprint of a finite core of exceptional erasure in a growing universe; (iii) transport — with the uniform BMR channel and raw mass, the L1 error between the transported and the native memory measure is rational in N for all N ≥ 5 on P2 (structural proof for the pendant family) and on D2, and tends to zero with lim N·E_N = 986915592/2078356921 ≈ 0.4749 (P2) and 119913192/196616173 ≈ 0.6099 (D2), with element-by-element agreement at leading order; star is an exact control; (iv) binomial transport locus — with free weights, A(w) = 4(w₀² + w₁²)|w₀² − 2w₁² − w₂²|/S², and on w₀² = 2w₁² + w₂² the conditions A = 0, asymptotic core memory (1/4, 1/2, 1/4) and E_N = B/N² with B > 0 (α = 2 exactly) are equivalent; (v) all coefficients (Q, ‖Δ‖², A, α, ν^∞) are species-dependent and are not RCD numbers while the dynamics does not select the amplitudes. Also recorded: the BHR/BMR amplitudes coincide term by term with classical sequential growth of causal sets (Rideout–Sorkin 1999), PF-0 is the first step of the factorial tower t_k = t₀ r^k/k! rather than transitive percolation, and the structural lessons of the closed diligences NUM-0, GEO-0, SIG-0 and VOL-0. Stated limits: ν is not a volume; E_N → 0 is a mathematical consistency, not a physical law; nothing here is time or space, and the order remains an input. Scripts with SHA-256 manifest. English and Spanish versions; the Spanish text prevails.