Geometric Phenomenology, Background Budget: A v1.3.1 Synthesis of the Einstein-Gap Dynamical Cycle

This deposit presents the v1. 3. 1 synthesis of one research cycle of the Geometric Relay Programme within the Entangled Relativity framework. The entry document remains grpcycleₛynthesisᵥ1₃. pdf, which supersedes grpcycleₛynthesisᵥ1₂. pdf. The cycle 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 geometric memory sector, the stochastic-projection clarification, the budget refutation, the first parameter-free confrontation with current EG data, and the new companion frontier note on environmental EG signatures. The principal v1. 3 correction is unchanged: the previous v1. 2 claim that the gravitational condensate reproduces the dark-matter budget in sign and order of magnitude is withdrawn. A fully normalised calculation closes the three natural routes negatively. The perturbation gradient energy gives Omegacond ~ 10^-6; the homogeneous background gap is degenerate, giving 0 or 1; and the baryon-forced intermediate branch remains bounded by Omegab, around 0. 02. Therefore Omegaₘ - Omegab is retained as a background input R* rather than derived from the relay. The GRP is therefore framed as a modified-gravity / geometric-phenomenology programme, not as a generator of the dark density. The Einstein-gap governs how structured matter modulates observable gravity, producing signatures in clustering, lensing, mu, Sigma, and EG. The derived slip Psi = Phi + chi and the dilaton identification theta² = 1/kappa stand independently of the budget refutation. The geometric memory sector is retained and strengthened. The memory rate f = 1/4 follows from the conformal normic coefficient cₙorm = -1/4 of the four-dimensional Einstein-gap, while the remaining reduced coefficients follow from branch friction gamma = 1, structural dressing, and O (s) matching. The operational memory variable is the dressed normic component of the conformal Einstein-gap Qₘunu, not the bare kinetic gradient variance. The v1. 3. 1 / Zenodo v4 update adds a companion frontier note developing the post-v1. 3 interpretation of the Einstein-gap as a geometric transfer function. It derives a parameter-free normic delay of approximately 0. 48 e-folds between the W2 structure-formation gate and the clustering response, set by tauₘem = 4 H^-1 with f = 1/4. The companion note also derives, rather than assumes, the environmental gate shift using a peak-background split on the local collapse threshold. The resulting shifts are small and asymmetric, from -0. 078 e-fold in cluster-like environments to +0. 017 e-fold in deep voids. Propagated through the conservative EG route, this gives a derived void-cluster EG contrast of about 0. 73% at z = 0. 2. This replaces the earlier illustrative ~2% environmental benchmark with a bounded sub-percent prediction. The sign is robust: voids are less suppressed than clusters, EGᵛoid > EGᶜluster. However, the amplitude is small and below realistic near-term detectability. The result should therefore be read as a clean, budget-independent theoretical consequence of the transfer-function interpretation, not as an immediate observational discriminator. The first EG confrontation remains a consistency check, not a detection. 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. Current errors remain at the 15-20% level, and the decisive mean-field test is the low-redshift slope of EG. This version records both positive and negative results as first-class outcomes. The dark budget is not derived from the relay and remains a background input. The Einstein-gap survives as a geometric response operator for perturbative observables, with a retained mean EG prediction and a new derived but small environmental EG consequence.