A causal nonlocal infrared modification of General Relativity in which spacetime retains finite memory of past curvature perturbations. The current public baseline is a near-GR fixed-background closure in the DESI DR1 linear-growth window; the action-level coupling, nonlinear screening, and companion DPN sectors are kept distinct.
A plain-language overview of what CMG proposes and where it currently stands.
Modern cosmology is remarkably successful, yet persistent tensions remain in structure-growth measurements, galaxy dynamics, and cosmic distance determinations. CMG treats these as tests of a controlled nonlocal gravitational response, not as solved anomalies.
Causal Memory Gravity (CMG) adds a finite retarded curvature-memory channel to the local Einstein response. The Dynamic Planck Network (DPN) is the proposed microscopic substrate, but generic convergence from directed rewiring graphs to the retarded Lorentzian kernel remains conditional.
The locked linear closure is infrared and late-time in character. Local recovery is specified through an environment-dependent nonlinear response map at the EFT/PM-N-body level; compatibility with laboratory, Solar-System, binary-pulsar, lensing, and nonlinear-structure data still requires implementation and calibration.
In practice, the minimal public gravitational closure predicts subtle changes in linear structure growth through \(\mu(k,a)\) with \(\gamma=1\). Galactic SEM--RAR response, CMB likelihoods, and nonlinear screening are separate test layers, not completed replacements for dark matter or \(\Lambda\)CDM.
CMG is an active research program. The baseline gravitational sector is constrained to be near-GR by DESI DR1. Companion papers explore DPN matter, gauge, chirality, FRG, and measurement sectors as theoretical extensions with explicitly stated claim boundaries.
Standard General Relativity contains no explicit finite-memory curvature kernel. CMG adds a retarded nonlocal response in which the present gravitational field depends on past curvature. The locked phenomenological closure is intended for the late-time infrared regime.
Spacetime does not forget its curvature history instantaneously. Instead, a retarded Green's function with a finite memory scale \(\beta_{\rm rate}\) introduces a delayed gravitational response. This is the single physical idea at the core of CMG.
The proposed DPN bridge uses the conditional spectral matching \(\beta_{\rm rate}^2 = K_\theta \lambda_2/V_{\rm graph}\). It identifies the slow graph mode, but its normalization and SI calibration are not yet a parameter-free microscopic prediction.
The dimensionless coupling is restricted to \(-1 < \tilde{\alpha} < 0\), defining the tree-level stability window of the linearized effective-scalar truncation, not full nonlinear or UV stability. The local screening map algebraically restores GR as the screened inverse-length scale tends to zero, but its astrophysical calibration remains open.
The minimal memory-based gravitational closure. Linear cosmology follows from \(\mu(k,a)\) and \(\gamma=1\); nonlinear screening enters through a separate effective response map.
$$ S_{\text{eff}} \supset -\frac{\tilde{\alpha}}{2} \int d^4x \sqrt{-g}\; R\, (\Box + \beta_{\rm rate}^2)_{\text{ret}}^{-1}\, R $$
Auxiliary field convention: \((\Box+\beta_{\rm rate}^2)U = -R\). The retarded propagator preserves causality.
Linearized effective-scalar window: \(-1 < \tilde{\alpha} < 0\). The action-level amplitude is \(\eta_{\rm act}\equiv-\tilde{\alpha}\in(0,1)\); it is not globally identified with the DESI closure amplitude \(\eta_{\rm cl}\).
Action-level relaxation rate \(\beta_{\rm rate}\). The dimensional inverse length is \(\beta_{\rm rate}/c\); using it operationally as \(\beta_{\rm IR}\) does not supply the missing action-to-perturbation amplitude matching.
Only the retarded Green's function is used. The retarded prescription is adopted as a causal boundary condition on the effective equations, physically motivated by the underlying Dynamic Planck Network — a directed graph with asymmetric causal transfer rules. A generic proof that the graph dynamics converges to the retarded continuum kernel remains open.
The source of memory is the spacetime curvature scalar, not matter density directly.
Modified gravity slip functions for Boltzmann solver interface (CLASS / Hi-CLASS):
$$ \mu(k,a) = 1 - \frac{\eta_{\rm cl}\, a^2 \beta_{\text{IR}}^2}{k^2 + a^2\beta_{\text{IR}}^2}, \quad \gamma(k,a) = 1 $$
Notation bridge:
\(\eta_{\rm act}\equiv-\tilde{\alpha}\in(0,1)\), \(0\le\eta_{\rm cl}\le1\), \(\beta_{\rm IR}\equiv\beta_{\rm rate}/c\) as a dimensional bridge
GR recovery limits:
Action level: \(\tilde{\alpha}\to0\), equivalently \(\eta_{\rm act}\to0\), or \(\beta_{\rm rate}\to\infty\)
Fixed-background closure: \(\eta_{\rm cl}\to0\) or \(\beta_{\rm IR}\to0\)
These are distinct limits and must not be conflated.
The screening ansatz is constructed to recover local GR in dense, nonlinear environments through two effective inputs:
(A) Channel saturation: \(\mathcal{N}(x)/N_0\) driven by local density.
(B) Tidal geometric degradation: \(\sqrt{s_{ij} s^{ij}}\) driven by local shear.
Dense or tidally distorted environments increase \(\Gamma_{\rm eff}(x)\). The local Poisson inverse-length scale then decreases as \(\beta_\mu(x)=\beta_{0,{\rm IR}}\Gamma_0/\Gamma_{\rm eff}(x)\), so \(\Gamma_{\rm eff}\to\infty\Rightarrow\beta_\mu\to0\Rightarrow\mu(k,a,x)\to1\). This implication belongs to the screening ansatz and is not the homogeneous large-\(\beta_{\rm IR}\) limit.
The screening mechanism is specified as an EFT / PM-N-body closure ansatz. Its nonlinear calibration requires PM/N-body or dynamical DPN simulations; no production-level \(S_8\) solution is claimed here.
Current public status after the DESI DR1 full-shape-profiled update and the companion DPN sector revisions. The page separates baseline gravitational claims from companion-sector results.
The linearized effective-scalar truncation is stable at tree level for \(-1<\tilde{\alpha}<0\), equivalently \(0<\eta_{\rm act}<1\). This does not establish full nonlinear, retarded, or ultraviolet stability. For \(\beta_{\rm rate}=H_0\) and the stated retarded initial data, the controlled near-GR branch gives the conditional bound \(\eta_{\rm act}\lesssim0.04\).
In the microfoundation program, the action rate is conditionally matched to the DPN Laplacian gap through \(\beta_{\rm rate}^2 = K_\theta \lambda_2/V_{\rm graph}\). This identifies a slow graph mode; continuum convergence, normalization, and SI calibration remain open.
In the relational GKSL matter closure, total fermion number is exactly conserved: \([Q,H]=0\) and \([Q,A_e]=0\). The lightest state in the \(Q=1\) sector is therefore stable against spontaneous decay by charge superselection and spectral ordering.
The proposed local-loss-of-criticality mechanism is a two-dimensional feedback model \((n_q,\rho)\) with threshold \(J_{c,q}\). Within the model, \(J
The revised DESI DR1 ShapeFit benchmark uses the full 6×6 compressed-growth covariance and the exact flat-\(\Lambda\)CDM linear growth ODE rather than the older \(\Omega_m^{0.55}\) template. It gives \(A_g = 0.9962 \pm 0.0437\), \(\chi^2_{\rm CMG}=5.480\), \(\chi^2_{\Lambda\rm CDM}=5.487\), and \(\Delta\chi^2\equiv\chi^2_{\rm CMG}-\chi^2_{\Lambda\rm CDM}\simeq-0.0075\). With one extra parameter, \(\Delta{\rm AIC}\simeq+1.99\) and \(\Delta{\rm BIC}\simeq+1.78\); linear CMG is allowed but not preferred.
The compressed fit constrains \(\eta_{\rm cl}S_{\rm DESI}(\beta_{\rm IR})=1-A_g\), not a unique \((\eta_{\rm cl},\beta_{\rm IR})\) point.
A separate fixed-background DESI DR1 packaged full-shape test profiles the nuisance sector for each fixed CMG point. With \(\beta_{\rm ratio}=\beta_{\rm IR}/(H_0/c)\), the one-sided 95% full-shape profiled limits are \(\eta_{{\rm cl},95}^{\rm FS}=(0.2544,0.0680,0.0316,0.0212,0.0153)\), while the joint ShapeFit-vector limits are \(\eta_{{\rm cl},95}^{\rm joint}=(0.2269,0.05764,0.02860,0.02006,0.01483)\), for \(\beta_{\rm ratio}=(100,300,1000,3000,10^4)\). At \(\beta_{\rm ratio}=30\), \(0\le\eta_{\rm cl}\le1\) is unconstrained. The profile reference is \(\chi^2_{\rm GR,prof}=339.560305\), with maximum shallow improvement \(0.0501\).
For DESI-sensitive large \(\beta_{\rm ratio}\), the fixed-background linear amplitude is restricted to \(\eta_{\rm cl}\lesssim0.015\)–\(0.032\) at one-sided 95%. These limits are fixed-background, linear, and first-order in the ShapeFit response; this is not yet a full self-consistent CLASS/Hi-CLASS CMG cosmology and does not validate nonlinear screening or a nonlinear \(S_8\) solution.
The action-derived FLRW background was evaluated with the DESI DR2 BAO operator at fixed standard cosmological parameters. The GR reference is \(\chi^2_{\rm BAO,GR}=22.998238\) with \(r_d^{\rm GR}=147.274006\,\mathrm{Mpc}\); the lowest sampled grid value occurs at \(\eta_{\rm act}=0.1\), \(\beta_{\rm rate}/H_0=10^5\), with \(\Delta\chi^2_{\rm BAO}=-5.415287\).
This is a fixed-parameter sparse-grid interface test, not a profiled BAO constraint, preference, or evidence measure. It must not be combined with the DESI DR1 \(\eta_{\rm cl}\) limits.
On one exported DPN sample with \(|V|=2000\), \(|E|=9203\), and \(|F|=4326\), the incidence operators satisfy \(B_1B_2=0\) exactly, gauge invariance holds to machine precision, and the discrete Hodge decomposition is verified. Plaquette curvature \(\Theta_p\) is carried by the coexact sector, with type-asymmetry \(\mathcal R_\Theta=1.333\).
This supports the structural Maxwell realization, but the physical calibration of \(\alpha_{\rm em}\) remains proxy-level.
The minimal quadratic local model does not generically stabilize \(Q=3\) composites because the sign condition \(u_3-3u_1>0\) fails. Adding the cooperative memory term \(\Delta H_{\rm coop}=-\chi\rho(\hat N_D^2-\hat N_D)\) gives \(u_3-3u_1=6\chi>0\). On the exported DPN ensemble, all 1159 proton-like \(K_4\) motifs satisfy \(u_3-3u_1>0\) under the stated proxy identification.
This is a real-DPN motif test, not yet a calibrated hadron-mass derivation.
A production \(S_8\) claim requires PM/N-body or dynamical DPN calibration of the nonlinear response sector. The current DESI results are linear or fixed-background linear tests and do not validate nonlinear screening.
The profiled full-shape calculation inserts a fixed-background linear growth provider into the public DESI packaged likelihood. A complete CMG CLASS/Hi-CLASS implementation with background, perturbations, lensing, CMB, and covariance-level likelihoods remains the decisive cosmology calculation.
The ladder-ring directed transfer construction yields the chiral projectors \(P_L=(1-\gamma^5)/2\), \(P_R=(1+\gamma^5)/2\) under explicit assumptions: spectral isolation of a slow quartet, weak directed scaling, and tensor-product doublet structure. It is a conditional structural theorem, not yet a universal DPN classification.
The DPN Maxwell structure is numerically verified at the oriented 2-complex level, but the physical photon normalization, Villain defect renormalization, and independent extraction of \(\alpha_{\rm em}\) remain open.
The current observational message is deliberately narrow: DESI allows a near-GR linear CMG sector, but does not prefer it and does not establish a nonlinear \(S_8\) solution.
The target signature is a scale- and redshift-dependent suppression in \(f\sigma_8(k,z)\). Current DESI DR1 compressed data give \(A_g\approx1\); future full-shape and weak-lensing releases must test whether any residual CMG response survives beyond nuisance profiling.
The updated DESI packaged-likelihood test shows that fixed-reference improvements can disappear or reverse after nuisance refitting. The physically relevant constraint is therefore the profiled \(\chi^2_{\rm prof}(\eta_{\rm cl},\beta_{\rm ratio})\), not an unprofiled visual improvement.
The next decisive astrophysical test is not another constant-amplitude fit. It is a nonlinear response calculation where void/web cells, filaments, and halo/tidal regions receive different effective memory activation.
The minimal linear closure keeps \(\gamma=1\) and is near-GR at early times. A full TT/TE/EE and CMB-lensing likelihood still requires a self-consistent CMG Boltzmann implementation, not only a post-processed growth provider.
The supported statement is: \(A_g=0.9962\pm0.0437\) is a near-GR compressed benchmark; the profiled full-shape large-\(\beta_{\rm ratio}\) constraint is \(\eta_{\rm cl}\lesssim0.015\)--\(0.032\) at one-sided 95%; nonlinear screening remains uncalibrated.
The present linear benchmark uses the minimal \(\mu(k,a)\) closure. A fully self-consistent CLASS / Hi-CLASS CMG cosmology remains the required next implementation step.
The fixed-background closure recovers GR as eta_cl → 0 or beta_IR → 0; this block is a linear interface, not an action-derived perturbation system or nonlinear screening implementation.
Current, claim-controlled answers after the DESI full-shape-profiled and DPN companion-sector updates.
Not as a completed claim. CMG modifies the gravitational sector and provides a candidate route to galactic phenomenology through memory-modified gravity. Whether this can replace particle dark matter in clusters, nonlinear structure formation, and the full CMB likelihood remains open.
No. The action-level memory correction decouples for \(\tilde{\alpha}\to0\), equivalently \(\eta_{\rm act}\to0\), or \(\beta_{\rm rate}\to\infty\). In the fixed-background closure, GR is recovered for \(\eta_{\rm cl}\to0\) or \(\beta_{\rm IR}\to0\).
No. The revised DESI DR1 compressed benchmark gives \(A_g=0.9962\pm0.0437\), centered on GR. The profiled full-shape test restricts large-\(\beta_{\rm ratio}\) linear amplitudes to \(\eta_{\rm cl}\lesssim0.015\)–\(0.032\) at one-sided 95%. A linear-only \(S_8\) solution is therefore not claimed.
It showed that an apparent fixed-reference improvement is not a profile-likelihood improvement. The nuisance sector must be refit for every fixed \((\eta_{\rm cl},\beta_{\rm ratio})\) point. After profiling, DESI-sensitive large inverse lengths allow only a small linear amplitude, approximately \(\eta_{\rm cl}\lesssim0.015\)–\(0.032\) at one-sided 95%.
The linearized effective-scalar truncation is stable at tree level inside \(-1<\tilde{\alpha}<0\), equivalently \(0<\eta_{\rm act}<1\). This is not full nonlinear or ultraviolet stability. For \(\beta_{\rm rate}=H_0\) and the stated retarded initial data, the controlled near-GR branch gives \(\eta_{\rm act}\lesssim0.04\).
The structural sector is now numerically supported on one exported DPN 2-complex: \(|V|=2000\), \(|E|=9203\), \(|F|=4326\), \(B_1B_2=0\), Hodge decomposition verified, and \(\mathcal R_\Theta=1.333\). This is not yet a calibrated derivation of the physical electromagnetic coupling \(\alpha_{\rm em}\).
The relational GKSL closure gives exact fermion-number superselection, so the lightest \(Q=1\) mode is stable. Measurement is modeled as local loss of criticality: below \(J_{c,q}\) a mode remains coherent and near-critical; above \(J_{c,q}\) it flows to a dissipative attractor. The \(Q=3\) composite stabilization result is numerically supported on 1159 real DPN \(K_4\) motifs under the stated proxy identification.
It is derived under explicit structural assumptions. In the ladder-ring directed-transfer construction, the slow-sector projectors converge to \(P_L=(1-\gamma^5)/2\) and \(P_R=(1+\gamma^5)/2\), assuming spectral isolation, weak directed scaling, and a tensor-product doublet structure. A universal classification over generic irregular DPN graph families remains open.