GRUT Phase IV: Fundamental Closure of Phase III

GRUT Phase IV: Fundamental Closure of Phase III (Action-Principle Landscape, Strong-Field Lapse, and Effective Lapse Closure) & GRUT-RAI v2.0 DescriptionThe Grand Responsive Universe Theory (GRUT) introduces a viscoelastic, memory-lagging metric to spacetime. This record contains the Phase IV manuscript and the accompanying GRUT-RAI v2.0 computational physics codebase. Phase IV is presented as the foundational closure pass that follows the completed Phase I–III effective framework. Where Phase III established the effective covariant closure of the strong-field sector, including the constrained endpoint law and mixed-viscoelastic interior classification, Phase IV deepens the derivational and structural basis of that closure. Phase IV delivers four principal advances. First, it maps the action-principle landscape of the GRUT memory sector and formalizes the local–conservative–first-order trilemma: the GRUT relaxation law τ Φ˙+Φ=X\tau \,\dot{\Phi} + \Phi = XτΦ˙+Φ=X cannot be obtained from a theory that is simultaneously local, conservative, and first-order. This classifies the viable foundational routes and establishes the nonlocal retarded formulation (Route C) as the strongest structural parent of the current memory law, while also sharply defining the strengths and limitations of the doubled-field dissipative route (Route B). Second, it clarifies the ontological status of the scalar memory field Φ\PhiΦ. Under the exponential kernel, Φ\PhiΦ is established not as a fundamental particle-like degree of freedom, but as an exact local Markovian reduction of a deeper nonlocal retarded history. This elevates the GRUT memory sector from a constitutive-effective closure toward a more sharply defined foundational structure. Third, it strengthens the strong-field collapse sector. Phase IV proves a self-healing property of the GRUT endpoint: at equilibrium, the source term driving the lapse-sector correction vanishes identically because Mdrive=agravM_{drive}=a_{grav}Mdrive=agrav. As a result, the constrained endpoint law Req/rs=1/3R_{eq}/r_s = 1/3Req/rs=1/3 and the Phase III force-balance structure remain preserved. The remaining strong-field uncertainty is localized not to the existence of the endpoint, but to the sub-horizon metric-to-lapse mapping. Fourth, it resolves the last major heuristic in the strong-field Route C chain by elevating the central lapse value from a heuristic estimate to a constitutive-derived central proxy. The exact algebraic barrier-to-gravity ratio at the endpoint is Φbarrier/Φgrav=11+βQ,\Phi_{barrier}/\Phi_{grav} = \frac{1}{1+\beta_Q},Φbarrier/Φgrav=1+βQ1, which for the canon choice βQ=2\beta_Q=2βQ=2 yields 1/31/31/3. This confirms the prior central value numerically while showing that the dependence is controlled by the memory barrier stiffness exponent βQ\beta_QβQ rather than fundamentally by αvac\alpha_{vac}αvac. The equality with αvac=1/3\alpha_{vac}=1/3αvac=1/3 in the canon case is therefore a parametric coincidence, not a first-principles identity. The accompanying GRUT-RAI v2.0 Python framework verifies the mathematical bounds, tensor constructions, perturbative checks, and strong-field scans used in the manuscript. At the final Phase IV pass, the repository reports: 1,222 passed 3 skipped 5 warnings (pre-existing codebase notices) 0 failures with zero regressions from Phase III. This upload is presented as a foundational closure pass, not as a final first-principles completion of all GRUT sectors. Remaining unresolved issues are sharply localized, including the sub-horizon interior metric/lapse mapping, deeper first-principles closure of the memory sector, and later frontier extensions beyond the core Phase IV program.