Geometric Phenomenology, Background Budget: A v1.3 Synthesis of the Einstein-Gap Dynamical Cycle

This deposit presents the v1. 3 synthesis of one research cycle of the Geometric Relay Programme within the Entangled Relativity framework. The entry document for this version is "grpcycleₛynthesisᵥ1₃. pdf", which supersedes "grpcycleₛynthesisᵥ1₂. 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 geometric memory sector, the stochastic-projection clarification, the budget refutation, and the first parameter-free confrontation with current EG data. 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 important v1. 3 correction is that this condensate does not reproduce the dark-matter budget as a derived density. A fully normalised calculation gives "Omegacond ~ 10^-6", far below "Omegaₘ - Omegab". The previous v1. 2 claim that the condensate reproduces the dark budget in sign and order of magnitude is therefore withdrawn. The budget question was tested through three closed routes. First, the perturbation gradient energy of the relay field, using the physical amplitude "chi = A Phi" with "Phi" taken from the real LCDM potential spectrum, gives "Omegacond ~ 1. 5 x 10^-6" at "z = 0". Second, the homogeneous background gap "G (g) - G (kappa g) " is degenerate on the relay branch: the dust attractor gives "Omegagap -> 0", while the forced branch gives "Omegagap = 1"; neither tracks matter nor gives the observed dark budget. Third, the baryon-forced intermediate branch gives "Omegagap ~ 0. 02", structurally bounded by "Omegab", because the GRP source is carried by baryons alone. Therefore "Omegaₘ - Omegab" is retained as a background input "R* = Omegaₘ - Omegab", not derived from the relay. Two structural relations previously assumed in the programme remain derived outputs: the gravitational slip "Psi = Phi + chi" and the dilaton identification "theta² = 1/kappa". These results are independent of the budget refutation and remain part of the core Einstein-gap construction. The geometric memory sector is also retained and strengthened. The Mori-Zwanzig memory coefficients are not 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. In this v1. 3 perimeter, the dressing rule is also derived from the Green-function operator algebra rather than merely posited. A companion stochastic-projection clarification is retained. 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 the cycle is unchanged by the budget refutation. The GRP is now framed as a modified-gravity / geometric-phenomenology programme: the dark budget is a background input, while the Einstein-gap governs how the relay responds to structured matter, producing observable signatures in clustering, lensing, "mu", "Sigma", and "EG". The W2 prediction gives 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. A first parameter-free confrontation with current "EG" measurements is included. 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. A first-principles stochastic projection of the dressed normic sector, with explicit noise correlators, remains open. Whether the dark budget can be derived at all would require an ingredient beyond the present theory: within the GRP v1. 3 perimeter, the three natural routes are closed and the budget remains a background input. The model is therefore not claimed as a first-principles derivation of the dark density; it is presented as a corrected, reproducible, parameter-free modified-gravity cycle that survives current observational contact and defines concrete falsifiable tests.