What changes
The present curvature response depends causally on past curvature through a finite retarded kernel.
Independent theoretical-physics programme
Causal Memory Gravity
Gravity with a finite causal memory.
CMG adds a retarded curvature-memory channel to Einstein gravity. A Dynamic Planck Network is the proposed microscopic origin; its continuum matching remains conditional.
Public edition I–XIII · 1 October 2026
What is strongest
A precisely stated near-GR linear closure and its fixed-background DESI DR1 benchmark.
What remains open
Action-to-closure matching, full Boltzmann likelihoods, nonlinear screening and microscopic calibration.
Locked gravitational notation
The gravitational framework.
The action-level coupling and the DESI-facing closure amplitude are not identified without a derivation.
Nonlocal action
Effective sector
$\begin{aligned}S&\supset-\dfrac{\tilde\alpha}{2}\int d^4x\sqrt{-g}\\[-2pt]&\qquad\times R(\Box+\beta_{\rm rate}^2)^{-1}_{\rm ret}R\end{aligned}$
$(\Box+\beta_{\rm rate}^2)U=-R$
α̃ ∈ (−1,0), with ηact ≡ −α̃ ∈ (0,1).Source records CMG I ↗ Microscopic Foundations ↗
Fixed-background no-slip closure
DESI-facing
$\mu(k,a)=1-\dfrac{\eta_{\rm cl}a^2\beta_{\rm IR}^2}{k^2+a^2\beta_{\rm IR}^2}$
$\gamma(k,a)=1$
βIR ≡ βrate/c is a dimensional bridge, not an amplitude-matching proof.Source records CMG VII ↗
Symbols and assumptions
- $\tilde\alpha,\;\eta_{\rm act}$
- Action coupling; $\eta_{\rm act}=-\tilde\alpha\in(0,1)$.
- $\eta_{\rm cl}$
- Fixed-background closure amplitude; $0\le\eta_{\rm cl}\le1$. No action-to-closure amplitude matching is assumed.
- $\beta_{\rm rate},\;\beta_{\rm IR}$
- Memory rate and inverse-length scale, related by $\beta_{\rm IR}=\beta_{\rm rate}/c$.
- $k,\;a,\;\mu,\;\gamma$
- Comoving wave number, scale factor, Poisson modifier and slip parameter. The displayed no-slip closure fixes $\gamma=1$.
- $R,\;\Box,\;U$
- Ricci scalar, covariant wave operator and auxiliary field with retarded boundary conditions.
Observational baseline
The DESI DR1 benchmark.
DESI permits a near-GR sector; it does not select CMG.
0.087σ from the GR value Ag = 1, using the full 6 × 6 covariance and exact ΛCDM growth baseline.Source records CMG VII ↗
Representative large-βratio range; at βratio = 30 the physical interval remains unconstrained.Source records CMG VII ↗
Substantial late-time suppression would have to arise from the separate, still-uncalibrated nonlinear response sector.Source records CMG VII ↗
Claim boundary
The DESI limits constrain ηcl in a fixed-background phenomenological closure. They are not limits on ηact from the nonlocal action.
Research corpus
Publications.
Titles and dates from public DOI metadata, checked . Version numbers are not supplied in these metadata; the linked Zenodo records contain the available versions and files. Research summaries reflect the 1 October 2026 edition.
15 publications
Handbook + Synthesis#
Causal Memory Gravity: A Handbook and Synthesis Article for Understanding the Theory
Recommended entry point explaining the CMG framework, notation, assumptions, active corpus structure, and claim boundaries.
$\displaystyle \eta_{\rm act}=-\tilde\alpha,$ $\displaystyle \eta_{\rm cl}\;{\rm distinct}$
DOI: 10.5281/zenodo.19204265 ↗
Microscopic Foundations#
Causal Memory Gravity Microscopic Foundations, Continuum Limit, and Mean-Field Validation
Foundation-level validation connecting DPN dynamics, continuum CMG equations, screening notation, and controlled mean-field approximations.
$\displaystyle \Gamma_{\rm eff}\uparrow\Rightarrow\beta_\mu\downarrow\Rightarrow\mu\to1$
DOI: 10.5281/zenodo.18776920 ↗
CMG I#
Causal Memory Gravity I: Cosmological Baseline and Reduced Almost-Λ Dynamics
Core cosmological baseline: nonlocal action, stability window, GR limits, reduced late-time dynamics, and near-GR perturbation interface.
$\displaystyle \mu(k,a)=1-\eta_{\rm cl}a^2\beta_{\rm IR}^2/(k^2+a^2\beta_{\rm IR}^2)$
DOI: 10.5281/zenodo.18457675 ↗
CMG II#
Causal Memory Gravity II: Unified Cosmology and Galactic Dynamics with a DPN-Saturated SEM-RAR Response
Galactic SEM--RAR response channel with SPARC/RAR calibration and explicit limits of dark-matter replacement claims.
$\displaystyle g_{\rm obs}=g_b/(1-e^{-\sqrt{g_b/g_\dagger}})$
DOI: 10.5281/zenodo.18175576 ↗
CMG III#
Causal Memory Gravity III: From Planck Network to Cosmic Memory - A First-Principles Derivation of Causal Memory Gravity
Conditional DPN-to-continuum bridge and spectral matching of the slow graph mode to the action-level memory rate.
$\displaystyle \beta_{\rm rate}^2=C_V(K_\theta/I_\theta)\lambda_2(L_{\rm el})$
DOI: 10.5281/zenodo.18189822 ↗
CMG V#
Causal Memory Gravity V: Continuum Limit and Emergent Quantum Dynamics from Discrete Planck Networks
Conditional continuum and emergent quantum/gauge constructions; the same Zenodo record includes the SU(2) frame-gap supplement and Maxwell/Plaquette addendum.
$\displaystyle \hbar_{\rm eff}=K_\theta\Delta t$
- SU(2) frame-gap supplementIncluded in the CMG V record
- Maxwell / Plaquette addendumIncluded in the CMG V record
CMG VI#
Causal Memory Gravity VI: Microscopic Origin of Inertia and Dark Energy — Controlled Mean-Field Baseline with Reduced-DPN Fluctuation Validation
Mean-field and reduced-DPN fluctuation validation for inertia and dark-energy-like behavior; includes the Thermal Criticality / Spectral Matching / RG Closure addendum.
$\displaystyle \sigma_{\bar\rho}\propto N^{-0.516}$
- Thermal Criticality / Spectral Matching / RG Closure addendumIncluded in the CMG VI record
CMG VII#
Causal Memory Gravity VII: DESI Compressed Benchmark and Full-Shape Nuisance-Profile Constraint
DESI DR1 compressed and full-shape profiled benchmark; near-GR in the linear sector and not preferred over LambdaCDM.
$\displaystyle A_g=0.9962\pm0.0437$
DOI: 10.5281/zenodo.18670252 ↗
CMG VIII#
Causal Memory Gravity VIII: Emergent Fermionic Matter Sector in a Discrete Planck Network
Emergent fermionic excitations, charge-sector structure, and relational open-system dynamics on the Dynamic Planck Network.
$\displaystyle \partial_t\rho=-i[H,\rho]+\mathcal D[\rho]$
DOI: 10.5281/zenodo.18675763 ↗
CMG IX#
Causal Memory Gravity IX: UV Completion via Functional Renormalization Group (FRG/EAA)
FRG/EAA embedding. Printed minimal-flow exponents are a historical superseded-sign benchmark; the corrected positive-mass flow remains open.
$\displaystyle \partial_t\Gamma_k=\frac12{\rm STr}[(\Gamma_k^{(2)}+R_k)^{-1}\partial_tR_k]$
DOI: 10.5281/zenodo.19119918 ↗
CMG X#
Causal Memory Gravity X: Emergent Fermion Sector, Asymptotic Unitarity, Relational GKSL Dynamics, and CLASS Interface
Extended fermion-sector construction linking relational GKSL dynamics, asymptotic unitarity, and the cosmological solver interface.
$\displaystyle K^{\rm GKSL}_\beta(\tau)=\beta^2e^{-\beta\tau}$
Companion record: CMG X Supplement — Bell / SU(2)
DOI: 10.5281/zenodo.19120016 ↗CMG X Supplement#
Causal Memory Gravity X Supplement: Bell Correlations, SU(2)-Covariant Frame Transport, and Memory-Damped Relational Entanglement
Supplement on Bell correlations, SU(2)-covariant frame transport, and memory-damped relational entanglement in the CMG/DPN matter-sector framework.
$\displaystyle P_{L/R}=(1\mp\gamma^5)/2$
Supplement to CMG X
DOI: 10.5281/zenodo.19163848 ↗CMG XI#
Causal Memory Gravity XI: Directed Bi-Orthogonal Continuum Limit, Emergent Chirality, and Infrared Pseudo-Unitarity
Directed-network continuum limit and conditional chirality theorem using bi-orthogonal modes and infrared pseudo-unitarity.
$\displaystyle P_{L/R}=(1\mp\gamma^5)/2$
DOI: 10.5281/zenodo.19185031 ↗
CMG XII#
Causal Memory Gravity XII: Local Loss of Criticality as a Measurement Mechanism and Stable Fermionic Outcomes
Proposed local-loss-of-criticality measurement model, with stable fermionic outcomes protected by charge-sector structure within the stated closure.
$\displaystyle J=J_{c,q}$
DOI: 10.5281/zenodo.19368389 ↗
CMG XIII#
Causal Memory Gravity XIII: Conditional Microscopic Gauge Dynamics and the Conditional Birth Selector (Parts I and II)
Conditional constructions and bounded negative results for the microscopic gauge provider, scale admission, coefficient-selection law and physical event clock. The frozen corpus is closed; the 7/8 selector ledger and open row 5 are retained.
$\displaystyle \lambda_2(L_{\rm el})\neq\Delta_{\rm transfer}\neq m_{\rm corr}\neq\Delta_{\rm YM}$
- Part I: Conditional Microscopic Gauge Dynamics on Causal Planck Networks — Construction, Falsification Protocol, and Three Negative ResultsIncluded in the CMG XIII record
- Part II: A Conditional Birth Selector, an Open Measure Law, and a Fourth Clock-Anchor Negative ResultIncluded in the CMG XIII record
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Research status
Research status and boundaries.
Controlled results, numerical benchmarks, conditional constructions and open questions. Expand an entry for its scope and source records.
ControlledLinear scalar truncation
Tree-level window −1 < α̃ < 0, equivalently 0 < ηact < 1. This is not a proof of nonlinear or ultraviolet stability.Source records CMG I ↗ Microscopic Foundations ↗
BenchmarkDESI DR1 linear growth
Ag = 0.9962 ± 0.0437, only 0.087σ from GR. CMG is allowed in this sector, but not preferred over ΛCDM.Source records CMG VII ↗
ConditionalDPN → continuum bridge
The spectral relation identifies a slow graph mode with the memory rate under explicit coarse-graining assumptions; normalization and SI calibration remain open.Source records CMG III ↗ Microscopic Foundations ↗
OpenAction-level cosmological closure
The general-gauge SymPy route and radiation cross-check pass, and the prototype provider has passed its internal gate. Direct integration replaces the rejected averaged-fluid route; the start-time criterion fails, the admissible Λ root is non-unique, and production hardening plus a separately frozen likelihood remain open.Technical status Theory map & claim boundaries ↗
BenchmarkRegular SU(2) control
The finite-volume V4 control is reproducible, but it is not a physical DPN port. The elastic graph gap, transfer gap, correlation mass and continuum Yang–Mills gap remain distinct: λ₂(Lel) ≠ Δtransfer ≠ mcorr ≠ ΔYM.Technical status Theory map & claim boundaries ↗
OpenMicroscopic gauge provider and event clock
CMG XIII supplies a conditional gauge construction on fixed oriented complexes and a conditional birth selector with seven of eight inputs. The recovered historical generator derives no gauge provider (3 of 14 fields present, one contradicted), the SU(2) decay estimates fail scale admission, the effective-coefficient law (row 5) is open and a fixed three-volume is not a clock. The frozen corpus was closed on 16 September 2026; no confinement or continuum Yang–Mills gap is claimed.Source records CMG XIII ↗
Companion sectorsGalaxies · Gauge · Matter · Measurement
Galaxies
g† = 1.1735 × 10⁻¹⁰ m s⁻² is an empirical SEM–RAR fit. Its DPN interpretation is not yet a microscopic prediction.Source records CMG II ↗
Gauge sector
Recovered faces verify B₁B₂ = 0 and the discrete Hodge structure. cγ = 0.7380 is a static zero-momentum coefficient; the registered status is D2_CONTRACT_FAILED, while its depth ladder supports a finite-time-transient diagnosis only. A microscopic provider and two-species common cone remain open; the CMG XIII audit finds 3 of 14 provider fields in the recovered generator and derives none.Source records CMG V / Maxwell addendum ↗ CMG XIII ↗
Matter sector
Exact charge conservation holds in the stated GKSL closure; emergent chirality remains conditional on explicit spectral assumptions.Source records CMG VIII ↗ CMG X ↗ CMG XI ↗
Measurement
The motif calculation supplies the necessary positive Q = 3 sign; physical stability still requires the density threshold and consistently extracted gaps. No proton identification follows from the sign test.Source records CMG XII ↗
Open calculationsAction-level closure · Nonlinear response · Joint likelihood · Microscopic scaling
Action-level provider
Use the passed general-gauge and radiation prototype as the starting point; resolve the start-time plateau, choose a physical Λ branch, harden the production provider and obtain independent tensor-tool confirmation before opening a likelihood.
Nonlinear calibration
Embed the response in PM/N-body or dynamic DPN simulations; calibrate βμ(x) and Γeff(x) against halo observables.
Joint likelihood
Compute CMB TT/TE/EE, lensing, DESI growth and BAO with a self-consistent action-level Boltzmann implementation.
Microscopic provider and scaling
Derive a generator-native DPN provider before physical gauge sampling and close birth-selector row 5 (the coefficient law left open in CMG XIII), then test ξphys, continuum convergence and retarded-kernel scaling on large dynamic ensembles.
Programme-status infographicView the full research overview
Open the full programme-status infographic ↗
Interactive linear response
Explore the linear response.
Large k recovers μ → 1. At long wavelengths, the no-slip closure approaches μ → 1 − ηcl.
Explorer range: 0–0.3.
30–10,000; the slider uses a logarithmic scale.
0.0001–5. Click the curve or use this field to select a point.
Selected pointμ(k,1) = 0.954123k = 0.1 h Mpc⁻¹; βIR = 0.333564 h Mpc⁻¹Long-wavelength limit: μ → 0.950
Pedagogical shape explorer at a = 1; H₀/c = 3.33564 × 10⁻⁴ h Mpc⁻¹. It is not a likelihood fit and does not implement the separate action-level perturbation candidate.Source record CMG VII ↗
Direct answers
Frequently asked questions.
What is Causal Memory Gravity?+
CMG is an active theoretical-gravity research programme built around the retarded infrared action term S ⊃ −(α̃/2) R(□ + β²)⁻¹ret R. A Dynamic Planck Network is the proposed microscopic origin, but continuum matching and the action-level cosmological closure remain conditional.Source records CMG I ↗
What does finite causal memory mean?+
Finite causal memory means the auxiliary field obeys (□ + β²)U = −R with retarded boundary conditions, so U(x) depends on curvature in the causal past and never on future data. The parameter β sets the memory-decay scale.Source records CMG I ↗
How does CMG recover General Relativity?+
At the action level, GR is recovered for α̃ → 0 or βrate → ∞. In the separate phenomenological closure, μ(k,a) → 1 for ηcl → 0 or βIR → 0. These are distinct limits because ηact and ηcl have not been identified.Source records CMG I ↗ CMG VII ↗
Do DESI DR1 data favor CMG over General Relativity?+
No. The fixed-background benchmark gives Ag = 0.9962 ± 0.0437, only 0.087σ from the GR value Ag = 1. It permits a near-GR closure and does not select CMG over ΛCDM.Source records CMG VII ↗
Does CMG replace dark matter?+
Not as a completed claim. The SEM–RAR sector reproduces an empirical galactic response, but cluster dynamics, nonlinear structure formation and the full CMB likelihood remain open tests.Source records CMG II ↗
Has CMG solved the S₈ tension?+
No. The DESI-compatible linear closure is near GR. Any material late-time suppression would require a separately calibrated nonlinear screening sector.Source records CMG VII ↗
Are ηact and ηcl the same parameter?+
No identification is currently justified. ηact belongs to the nonlocal action; ηcl is the amplitude of the fixed-background no-slip closure used in the DESI-facing test.Source records CMG I ↗ CMG VII ↗
Is the reported photon speed physical?+
The value 0.7380 is a static zero-momentum coefficient on recovered faces, not cγ/cDPN. The preregistered D2 contract failed; its depth ladder supports a finite-time-transient diagnosis only. A physical answer requires raw-export dispersion, a microscopic provider, dynamic weights and two species compared with one physical clock.Source records CMG V / Maxwell addendum ↗