Registered Age and Branch Proto-Geometry: An Intrinsic Age from Accumulated Erasure Memory, Zero on Pure Order and Logarithmic on P2, with Places, Branch Dimension and the Global Age–Geometry Identity — RCD Program, Technical Note TIME-0 (v0.1)

Technical Note TIME-0 of the RCD program (continues REP-1STAT, DOI 10.5281/zenodo.23824079; REP-0, DOI 10.5281/zenodo.23774093; BMR-1, DOI 10.5281/zenodo.23717139; JANO-BR-0, DOI 10.5281/zenodo.23785719; and the Adjudication Record, DOI 10.5281/zenodo.23717352). Question: can a registered age and a proto-geometry arise as two invariants of the same relational-memory matrix M built from erasure branches (BHR/BMR)? Results (exact, no hardware): (i) a registered age τ = Σ_k Tr M_k, with δ_k the load of registered memory present at level k, which is intrinsic, covariant, free of any external stage index, non-decreasing and strict exactly when the level carries distinct erasure branches: chain and antichain have τ ≡ 0 exactly, star, P2 and D2 have τ > 0, and families of equal size have different ages, so τ is not a counter; (ii) closed forms: for P2, Tr M(n) = 49(503n − 1266)/(32448 n²) and τ(n) = (24647/32448)H_n − (10339/5408)H_n⁽²⁾ + 3381/2704 ~ 0.76 log n; for the star, Tr M(n) = 2(n − 1)/n² and τ(n) = 2H_n − 2H_n⁽²⁾ − 1/2 ~ 2 log n: registered age grows without bound and logarithmically; (iii) τ is a function of state on P2 and D2; (iv) a branch proto-geometry: places (classes of elements with the same erasure branch), dim_branch = rank M and a pseudo-distance between erasure roles — P2 has three places and dimension 2, incomparable pairs being farther apart than comparable ones in P2 but not in general; (v) the exact Global Age–Geometry Identity δ_k = (1/2k²) Σi,j d_k(i,j)², while the simple local relation is ruled out; (vi) 168 of the 2,045 closures of the BMR-1 bank are rank-3 candidates, the smallest being chains with a rhombus. Stated limits: nothing here is called time or space; RCD's candidate temporal structure is the pair (≺, τ); rank 3 is not space without unbounded places and extensivity; and reading counterfactual erasures as physical memory remains the hypothesis. Scripts with SHA-256 manifest. English and Spanish versions; the Spanish text prevails.