From Posed Activation to Gravitational Condensation: A Synthesis of the Einstein-Gap Dynamical Cycle

This deposit presents an updated synthesis of one research cycle of the Geometric Relay Programme within the Entangled Relativity framework. 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. 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. 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 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.