Boundary-Selected Weyl-Squared Gravity and EFT at Conformal Boundaries

Paper I: Boundary-Selected Weyl-Squared Gravity and Effective-Field-Theory at Conformal Boundaries This paper establishes the structural foundation of the framework by proving that the Weyl-squared operator is uniquely compatible with a well-posed variational principle at smooth conformal boundaries in four dimensions. It demonstrates the existence of a strictly hyperbolic radiative attractor that ensures smooth conformal extension across the boundary without requiring additional degrees of freedom. Paper II: Microscopic Accumulation of Geometric Memory via Dimension-8 Operators in Conformal Cosmology This study details the microscopic mechanism of memory accumulation through a dimension-8 operator coupling Weyl curvature to scalar field kinetic terms. It demonstrates how nonlinear structure formation, particularly from supermassive black holes, saturates the memory function to provide the boundary data that sets mass parameters for the subsequent cycle. Paper III: Geometric Memory Transfer and Conformal Transitions in Boundary-Compatible Gravitational EFT It introduces the geometric memory functional ($\alpha_W$), a conformally invariant quantity that captures the accumulated Weyl curvature of a cosmological phase. The work proposes a transition mechanism where inherited geometric radiation drives the Hubble rate above particle mass scales, enabling conformal transitions without assuming particle decay or modifications to the Standard Model. Paper IV: Primordial Cosmological Perturbations in Boundary-Compatible Geometric EFT It derives the primordial scalar and tensor power spectra within the boundary-selected EFT regime. The paper predicts a universal correlated suppression of primordial amplitudes by a factor of $-1.046 \epsilon_{HD}$, while preserving the standard inflationary consistency relations at leading order. Paper V: Ultralight Dilaton Dark Matter from Near-Conformal Dynamics in Geometric EFT It identifies a dark matter candidate emerging from the near-conformal hidden sector responsible for generating the framework's Weyl coefficients. The model predicts an ultralight dilaton with a mass of approximately $10^{-22}$ eV, naturally generated through non-perturbative scale-symmetry breaking, which reproduces fuzzy dark matter phenomenology. Paper VI: Observational Predictions of Geometric Memory: Hubble Tension and Dark Matter Abundance in the REM Framework It closes the five-paper REM series by converting its theoretical structure into quantitative observational predictions. The paper derives from first principles the dynamic vacuum energy sourced by the running Weyl memory function, integrates it against a realistic ΛCDM background using the Hopkins et al. SMBH mass function, and obtains a Hubble tension correction consistent with the observed discrepancy ΔH₀ ≈ 5.8 km/s/Mpc. It simultaneously closes the dark matter chain by treating the hidden-sector intermediate scale M∗ as a prediction rather than an input: requiring the correct dilaton relic abundance with natural initial conditions uniquely fixes M∗ ≈ 5.6 × 10¹³ GeV and a hidden-sector confinement scale Λ_NP ≈ 7.5 TeV. A cross-consistency test confirms that the same memory profile responsible for the Hubble correction saturates within the BBN bound and reproduces the correct dark matter abundance, closing the programme without additional free parameters. Paper VII: Quantum Structure of the Conformal Boundary: Asymptotic Safety, BMS Memory, and Bootstrap Constraints on the REM ProgrammeIt addresses three fundamental structural gaps of the framework by activating the conformal boundary through three complementary quantum and geometric lenses. The paper proves Weyl continuity as a rigorous theorem within Asymptotic Safety, uniquely derives the Markov retention coefficient using Carrollian/BMS symmetry, and bounds the Weyl-Higgs coupling via the conformal bootstrap, unifying these approaches through a central cross-consistency condition. Paper VIII: Flat Space Holography and the Geometric Memory Operator: Fefferman--Graham Expansion, Celestial CFT, and the Computation of δα_inheritIt embeds the REM framework into flat space holography, identifying the geometric memory functional as the expectation value of the integrated news-energy operator of the Celestial CFT. The study utilizes a Fefferman-Graham expansion to determine the exact Weyl vanishing rate (fixing the retention exponent q=3) and calculates the previously undetermined trans-Planckian residual from the conformally soft graviton block in the operator product expansion. Paper IX: Derivation of the Weyl--Higgs Coupling from the Two-Loop Renormalisation Group Flow of the Near-Conformal Hidden SectorIt eliminates the last free parameter of the REM programme by deriving the Weyl-Higgs coupling (κ) from first principles. Using dimensional transmutation and the two-loop beta function in the Banks-Zaks conformal window, the paper shows that a hidden SU(4) gauge sector with 18 Dirac flavors predicts exactly κ ≈ 1.41 × 10⁻⁴, perfectly matching the phenomenological requirements for memory saturation and dark matter without fine-tuning. Paper X: Stochastic Dynamics of the Cosmological State Vector: Invariant Measure, Non-Convergence, and Structural Habitability in the REM FrameworkIt completes the programme by extending the inter-cycle dynamics to the full cosmological state vector, proving it evolves as a contractive Markov chain with a unique invariant probability measure. The paper establishes a Non-Convergence Theorem—showing the universe never statically optimizes due to irreducible quantum and structural noise—and a Structural Habitability Theorem, proving that life-permitting cosmological parameters emerge organically with positive frequency without anthropic selection.