The Cosmological Constant Problem in Gauged Constant Vacuum-Mode Cosmology (GCV): Radiative stability, observed-value localization of residual Λ_eff, and positive-Λ realization via a four-step route

This preprint presents a compact synthesis of the Gauged Constant Vacuum-Mode (GCV) route to the cosmological constant problem (CCP). The underlying GCV record has become modular enough that a reader would otherwise need to reconstruct the full claim by moving across separate papers on radiative stability, residual Lambda_eff, UV anchoring, survey-facing interfaces, value selection, and sign selection. The purpose of the present paper is therefore programmatic rather than derivational: it states the three CCP pillars addressed by GCV, explains the four-step route by which they were reached, and clarifies the appropriate completion criterion for a flux-discretuum framework.

The first pillar is radiative stability. On the GR-exact branch of GCV, the strictly spacetime-constant vacuum mode is gauge/flux-controlled and does not reappear as a freely running local cosmological-constant coupling. The local Einstein equations retain GR form with a flux-fixed integration constant Lambda_eff, while local excitations gravitate normally.

The second pillar is observed-value localization. Starting from the radiatively stable GR-exact branch, the numerical-selection, closeout, and UV budget-partition modules compress the residual-value problem from order-unity to near-unity and then to a narrow UV-selected operational window. In the current synthesis, that stage yields a viable UV-pass interval d_pass = [178, 206], aligned with independent geometry statistics near the population median d_proxy ≈ 200.

The third pillar is positive-Lambda realization. Once the magnitude stage is fixed near the observed residual scale, late-time cosmological viability introduces a physical asymmetry between the two signs: Lambda < 0 universes recollapse in finite proper time, while Lambda > 0 universes do not. The surviving near-zero residuals are therefore biased toward the positive branch.

Within the stated GR-exact branch and module assumptions, the synthesis argues that GCV resolves the three classical CCP pillars at programme level up to a narrow UV-selected discretuum rather than a unique continuous endpoint: radiative stability is secured first, the allowed magnitude is localized near the observed scale, and the positive branch is selected by late-time viability.

The paper does not claim a unique UV completion or a purely ultraviolet theorem for the sign. Its role is to provide a compact entry point to the GCV programme and a clear map connecting the companion records.