Register with Residual Capacity: An Informative Reading Need Not Consume the Carrier's Quantum Capacity — The A_η Family in the Tracial Sector, Exact Non-Markovianity, Finite-History Reversal, Four Separate Persistences with Flagged Entanglement Surv

Technical Note I-C of the RCD program, Act I (continues I-B1, DOI 10.5281/zenodo.23990720; I-B0, DOI 10.5281/zenodo.23965760; and I-A0/I-A1, DOI 10.5281/zenodo.23961107). I-B1 left the boundary that every realization is a sample entering conditionally; I-C asks whether an event can be registered without consuming the coherence needed for the next one, leave an inheritable trace, and take part in an inheritance network. In the tracial sector ρ = P/r (r = 2, 3, 4; no hardware; theorems with numerical control, 92 checks with declared tolerances, all passed), results: (I) Register-with-Residual-Capacity Theorem. A rank-one reading on the whole system admits only nuclear instruments (Haapasalo–Pellonpää) and hence breaks entanglement (Horodecki–Shor–Ruskai); the family A_η(Q) = (1−η)P + ηrQ with Lüders operation L_Q = aP + bQ gives, for 0 < η < 1, an informative reading (I_η > 0, with I₁ = ln r − H_r + 1) together with a non-entanglement-breaking depolarizing channel, p_η = 1 − b²/(r+1) > 1/(r+1), whose posterior state (1−η)P/r + ηQ is the I-B1 segment produced by a mechanism. Exact non-Markovianity on labels, k(Q₃|Q₁,Q₂) − k(Q₃|Q₂) = η²√(1−η²) in r = 2. Finite-history reversal theorem: w_n(Q₁…Q_n) = w_n(Q_n…Q₁) for every n, so no arrow comes from label statistics. Four separate persistences: the classical archive; the averaged residual, decaying as ηp_ηⁿ and never zero; entanglement with discarded registers, dying at n_EB = ⌈ln(1/(r+1))/ln p_η⌉; and entanglement with kept registers, with Schmidt rank r along every trajectory and the exact bound λ_min ≥ [(1−η)/(1+(r−1)η)]ⁿ/r — flagged-survival corollary: with the classical register kept the negativity is an integral of branch negativities and stays positive for every finite n; the classical register is the flag that preserves the resolution of entangled branches, not a source of entanglement. (II) Inheritance network, ratified conditionally under a declared connection scheme H_con (carrier registers): acyclicity by strict inclusion, invariance of density, state, registers and predecessors under all compatible linear extensions and relabeling, rhombus with incomparability sustained by commutation and no-signaling, and the exact counterexample 13/8, 5/4, 1/16 showing that a product label does not imply independence (local statistics factorize iff the posterior is a product). Orientation: the boundary source dies (P/r has a preimage; in a qubit L_{Q⊥}L_Q = √(1−η²)P), the contraction source belongs to the average, and accumulation remains declared. Locality no-go: the Kraus operators span M_r, the commutant is trivial and Fix(Φ_η) = ℂP, even for product labels on ℂ²⊗ℂ²: locality cannot be extracted from the algebra of the global instrument. Diagnostics: Rideout–Sorkin Bell causality transcribed to the label law fails; the prefixed rhombus network is not sprinkling-like. Not claimed: time, space, gravity, irreversibility, Bell causality, asymptotic quantum capacity, or the derivation of η, accumulation, connections or factorization; the event debt (No Free Actualization) remains open. Scripts with SHA-256 manifest; run incidents documented. English and Spanish versions; the Spanish text prevails.