From Posed Activation to Geometric Memory: A v1.2 Synthesis of the Einstein-Gap Dynamical Cycle

This deposit presents the v1. 2 synthesis of one research cycle of the Geometric Relay Programme within the Entangled Relativity framework. The entry document for this version is "grpcycleₛynthesisᵥ1₂. pdf", which supersedes the previous "grpcycleₛynthesis. pdf". It links the first Boltzmann-level CLASS implementation of an Einstein-gap relay sector, the derivation of a microphysical activation gate, the ER halo-source addendum, the reduced relay-map condensation programme, the P6 analysis of the closure coefficient, and the corrected geometric derivation of the memory sector. The cycle establishes a connected chain: a microphysically selected fraction source leads to the W2 structure-formation gate; the gate yields a parameter-free bias "b (a) "; the strict relay branch becomes tachyonically unstable; the instability is gravitational in nature and saturates into a finite, scale-locked condensate; the resulting condensate reproduces the dark-matter budget in sign and order of magnitude, without fitted parameters. The exact dark-budget normalisation remains open. Two structural relations previously assumed in the programme are derived in this cycle: the gravitational slip "Psi = Phi + chi" and the dilaton identification "theta² = 1/kappa". The primary observational consequence is a redshift-dependent suppression of the "EG" statistic, with "EGGRP/EGGR ~= 0. 964" at "z = 0. 57", "0. 948" at "z = 0. 30", and "0. 93" as "z -> 0", making low-redshift bins especially discriminant. This updated version corrects and closes the interpretation of the Mori-Zwanzig memory coefficients within the reduced closure. The coefficients are no longer treated as outputs of a second-order "k -> 2k" mode-coupling calculation. Instead, the memory rate "f = 1/4" is derived from the conformal normic coefficient "cₙorm = -1/4" of the four-dimensional Einstein-gap, while the remaining coefficients follow from branch friction "gamma = 1", structural dressing, and "O (s) " matching. This gives " (f, betachi, betaₚi) = (1/4, 5/64, 7/16) " within the reduced closure, reproducing the full reduced self-energy and the position-memory coefficient. A companion stochastic-projection clarification is also included. It shows that the operational memory variable is not the bare kinetic gradient variance "ᵢso": the Gaussian projection of that bilinear variable onto the slow " (chi, pi) " sector vanishes at leading order, and its connected autocorrelation decays too rapidly. The memory register used in the reduced closure should instead be understood as the dressed normic component of the conformal Einstein-gap "Qₘunu". The observational part of this update adds a first parameter-free confrontation of the GRP W2 prediction with current "EG" measurements. Across six heterogeneous published determinations, the GRP W2 curve gives "chi² = 15. 1", compared with "chi² = 18. 9" for GR+LambdaCDM with "Omegaₘ = 0. 3153". This is a modest aggregate improvement, not a detection. Current errors remain at the 15-20% level, and the decisive test is the low-redshift slope of "EG", which awaits Stage-IV precision. The deposit records the remaining open frontiers honestly. The exact dark-budget normalisation still requires the collapsed-geometry relay-gradient correlator "". A first-principles stochastic projection of the dressed normic sector, with explicit noise correlators, remains open. The operator-level derivation of the dressing rule also remains open. The model is therefore not claimed as confirmed; it is presented as a corrected, reproducible, parameter-free dynamical cycle that survives current observational contact and defines concrete falsifiable tests.