Discrete Cosmology Model (DCM): Relativistic Group Delays as a Testable Origin of Gravity

1 Discrete Cosmology Model: Relativistic Group Delays as a Testable Origin of Gravity and Redshift Nick Markov Bulgarian Academy of Sciences [email protected] Abstract The Discrete Cosmology Model (DCM) reinterprets gravitation and cosmological expansion as emergent manifestations of discrete, stepwise mass–energy expansion. Each domain expands at its Compton frequency, while finite-speed propagation of phase information introduces small interdomain delays. The ensemble of these delays forms a delay-correlation tensor 𝐺𝜇𝜈 whose second-order coherence reproduces the metric curvature of General Relativity for the single-body case and extends it to multi-body coherence on cosmic scales. Minimization of global delay variance, 𝛿𝑆DCM =0, yields the universal acceleration law 𝑎DCM =1 2 ⁄𝐻0 2𝑅, linking microscopic delay variance to macroscopic curvature through the single parameter 𝐻0. This framework reproduces the observed redshift relation, galactic rotation without dark matter, and the empirical seismic–escape equality 𝑣𝑃−𝑤𝑎𝑣𝑒 𝑟𝑎𝑑 ≈𝑣esc within self-gravitating bodies. DCM preserves Newtonian and relativistic limits for isolated masses while revealing that curvature, inertia, and redshift emerge collectively from synchronized discrete expansion delays in a steady-state Universe requiring no dark components or singular origins. Keywords: Discrete Cosmology Model · Group Delay · Steady-State Universe · Emergent Gravity · Seismic–Escape Correlation · Dark Matter Alternative · Hubble Acceleration · Delay Variance Principle 1 Introduction Contemporary cosmology successfully accounts for many large-scale observations within the ΛCDM framework. Yet its reliance on dark matter, dark energy, and inflation introduces major unexplained components [1], together representing more than 95% of the Universe’s inferred content. While phenomenologically powerful, these constructs remain unobserved and lack a causal–mechanical explanation. The ΛCDM framework is also facing persistent tensions such as the discrepancy in the Hubble constant [2]. Numerous alternatives have been proposed, from stepwise cosmologies [3], MOND [4] and TeVeS [5] to causal set and stochastic spacetime models [6]. These often modify general relativity’s curvature dynamics or introduce additional components, yet lack direct empirical anchoring. 2 The Discrete Cosmology Model (DCM) offers a complementary perspective. Whereas General Relativity (GR) describes how mass–energy curves spacetime, it does not explain why mass has this capacity. DCM addresses this foundational gap by treating mass as a discretely expanding entity, where finite-speed propagation of expansion and rotation gives rise to cumulative relativistic group delays. Gravity and cosmological redshift thus emerge as consequences of interaction delays rather than as axiomatic inputs. This approach retains Einstein’s field equations, but upgrades the stress–energy tensor to include delay terms. In doing so, DCM offers both a physical interpretation of mass–energy and testable predictions across scales: (i) flat galactic rotation curves without dark matter halos, (ii) quadratic suppression of cosmological redshift consistent with the Hubble tension, and (iii) seismic–escape velocity convergence, predicted in advance and validated by Apollo and InSight data. The purpose of this paper is to formalize the delay-based stress–energy tensor and demonstrate its empirical consequences, thereby establishing DCM as a testable, GRcompatible pathway to resolving the puzzles of dark matter and dark energy. 2 Physical Principles 2.1 Discrete Expansion Lag While speculative, this section introduces a discrete-expansion hypothesis providing the minimal kinematic mechanism required for radial group delays in the next sections. The empirical predictions developed later (seismic–escape convergence, rotation curves, and redshift suppression) depend only on the existence of cumulative delays, not on the microphysical details of §2.1. Quantum mechanics does not interpret spin as literal particle rotation, since accounting for the measured magnetic moment this way would require superluminal surface speeds— assuming a fixed mass and radius. Special relativity, however, allows for mass increase at relativistic speeds, which alters the dynamics of rotation and angular momentum. Figure 1: Step expansion of a particle, showing overshoot/undershoot (a schematic illustration of the hypothesis). The Discrete Cosmology Model (DCM) builds on this by proposing that particle mass varies discretely at Compton frequencies, supporting relativistic surface motion and discrete radial growth. This oscillatory mass behavior reconciles the observed magnetic moment with relativistic limits and provides a deterministic physical mechanism for intrinsic spin. DCM hypothesizes that particles overshoot/undershoot space’s expansion due to relativistic discrete delays caused by their rotation, producing stepwise growth (Fig. 1). 3 Spin, reinterpreted as variable-mass rotation, separates expansion (interaction-heavy) and rotation (minimal expansion and interaction, high magnetic moment) phases at Compton frequencies (Fig. 2). A phase-weighted toy model for the gyromagnetic ratio, based on variable inertia at Compton frequency, is developed in Appendix E of the Supplemental Material. Moreover, this framework offers a new perspective on quantum tunneling: if spin arises from relativistically rotating mass, then mass-energy could transiently exceed classical thresholds, allowing particles to bypass energy or momentum barriers in a manner consistent with tunneling observations. Figure 2: Reinterpretation of particle spin as variablemass rotation: larger size indicating lower mass. Particle size seen from the expanding space perspective (Fig. 1). A schematic illustration of the hypothesis This interpretation may have experimental implications, particularly in systems involving spin-polarized tunneling, anomalous magnetic responses, or timeresolved scattering at Compton-scale intervals. The next sections will focus on gravity where empirical correlations offer substantial evidence supporting DCM. Although the present work focuses on gravitational and cosmological scales, where cumulative group delays dominate, the same principle may manifest at the quantum scale as discrete delays, consistent with the oscillators in hydrodynamic quantum analogs [28-30]; a detailed treatment of spectral structure, however, lies beyond the scope of this paper. 2.2 Expansion Delay and Group Phenomena The hypothesis to test is that gravity arises from cumulative relativistic delays in grouped masses, where inner particles have to perpetually displace outer layers, at finite interaction speeds. Here, gravity is a collective effect, not sourced by single particles. The objective would be to prove that the hypothesized group delay matches the gravitational time dilation from the Schwarzschild metric: 𝒕𝟎=𝒕𝒇 √𝟏−𝒗𝒆𝒔𝒄 𝟐(𝒓) 𝒄𝟐 (1) expressed as a function of the escape velocity 𝑣𝑒𝑠𝑐. The GR-consistent formulation of the delay-based stress–energy tensor is presented in Appendix A, where its decomposition, closure, and conservation properties are derived. Although introduced heuristically, the delay-based metric used in the next sections effectively encodes gravitational and kinematic time dilation, and in Appendix A it is shown to be consistent with Einstein’s field equations. 4 2.3 Seismic–Gravitational Velocity Convergence as Empirical Evidence for Relativistic Group Delay A notable empirical regularity observed in self-gravitating bodies is the convergence between the average seismic wave velocity within the core 𝒗𝒔 and the escape velocity at the surface 𝒗𝒆𝒔𝒄. This convergence appears across a wide range of planetary bodies and moons, and cannot be dismissed as coincidental. It can be formalized as: 𝒗𝒆𝒔𝒄 =𝒗𝒔 (2) where 𝒗𝒔 is the measured or modeled average P-wave or S-wave velocity (depending on the tidal lock), and 𝒗𝒆 is the classical escape velocity, with M the mass and 𝑅 the radius of the body. Appendix A.5 provides the formal calibration, showing that the effective delay modulus links seismic velocity to escape velocity. Empirical data supporting the law Table 1 and Fig. 3 summarize seismic wave speeds and escape velocities for selected planetary bodies and moons, using published interior models. This relationship is not anticipated by standard models of planetary structure, which treat seismic wave propagation and gravitational binding as independent phenomena. However, in the Discrete Cosmology Model, the convergence follows naturally from the assumption that both seismic wave propagation and gravity arise from cumulative interaction delays—specifically, the finite-speed of propagation. Table 1: Empirical data supporting the velocity convergence law backed by Apollo and InSight missions for Moon and Mars [7–8]. Body 𝒗𝒔 km/s Wave 𝒗𝒆𝒔𝒄 km/s Ratio 𝑣𝑠/𝑣𝑒𝑠𝑐 Reference Earth 11.2 P 11.2 1.00 [17] Venus 10.3 P 10.4 1.00 [20] Mars 5.0 P 5.0 1.00 [19] Moon 2.4 S 2.4 1.00 [18] Io 2.2 S 2.4 0.92 Estimated* Asteroids < 0.5 P < 0.1 ≫ 1 (disordered) Estimated* Sun 510 P 618 0.82 [9] * Based on interior modeling Figure 3: Seismic–escape velocity ratio R with source-based 1σ uncertainties. Proxies: Earth—inner-core P-wave speed (PREM); Mars—core P at CMB (InSight); Moon (S-wave)— solid inner core speed (as inferred from Apollo and GRAIL data); Venus*— Perple_X model suite; Sun*—helioseismic sound speed (deep interior). Asterisks (*) indicate model/inversion-based proxies rather than direct core seismology. The ratio of seismic to escape velocity— remains close to unity in all bodies known to be internally differentiated by gravity. In contrast, irregular or non-differentiated planetary cores show either sub-seismic escape velocities or chaotic propagation 5 regimes, consistent with the absence of internal coherence delays. Let’s consider another less accurate but intriguing proxy for Moon: the average radial P-wave speed from center to surface is 7.44 𝑘𝑚/𝑠, and the escape velocity is: 𝒗𝒆𝒔𝒄 =𝟐.𝟑𝟖𝒌𝒎 𝒔 ≈ 𝟕.𝟒𝟒 𝝅=𝟐.𝟑𝟕 𝒌𝒎 𝒔 (3) While for Earth, 𝒗𝒆𝒔𝒄 ≈10.5 𝑘𝑚 𝑠= 𝒗𝑷−𝒘𝒂𝒗𝒆 the P-wave average speed from center to surface. The 𝝅-factor implies that the interactions propagate radially for the tidallyfree and circularly for the tidally-locked satellites. This could be the reason why Pwave and S-wave correlations distinguish tidally-free from tidally-locked bodies. The core data proxy (Table 1) is more reliable than the proxy based on the average planetary radial P-wave speeds (e.g. Eq. 3) likely due to the homogenous core composition in comparison to mantle and crust. Besides,the seismic data from Mars and Venus is less reliable than the data from Earth and Moon. The Sun’s acoustic P-wave average speed (510 km/s, [9]) deviates from 𝒗𝒆𝒔𝒄 (618 km/s, ratio ~0.82) likely due to the enhanced propagation of the P-wave by nuclear reaction accelerating particles to hundreds of km/s. The 𝒗𝒆𝒔𝒄 =𝒗𝒔 relationship drives a similarity of time dilations providing direct, empirical support for the DCM’s interpretation of gravity as a relativistic group-delay phenomenon. It also offers a new predictive diagnostic: bodies exhibiting ratio ≈ 1 can be inferred to have gravityshaped cores, even if their internal structures are otherwise poorly constrained. Interpretation within the DCM framework In DCM, mass is defined dynamically: as a delay in the local expansion of discrete spacetime elements due to relativistic coupling with nearby mass. Gravity emerges as a macroscopic consequence of this group delay, and its cumulative effect manifests in the form of an escape-velocity-quantified spacetime curvature. Simultaneously, the ability of a medium to transmit internal stresses (measured as seismic wave speed) is constrained by the same delay mechanisms— namely, the propagation time of interactions across the body's interior. Thus, the observed convergence between and signals a relativistic limit on internal signal coherence. This suggests that seismic and gravitational metrics are not independent but both emerge from the same delay-governed structure of matter. Choice of wave type: For tidally-free bodies P-wave speeds are used as the radial interaction proxy; for tidally-locked bodies we use S-wave speeds as a circumferential proxy. This is a DCM hypothesis and a direct test: it should be supported by anisotropy patterns; we do not assume it proven. The seismic wave average speeds and ranges in Fig. 3 are taken from the references. Escape velocity: 𝒗𝒆𝒔𝒄 =√2𝐺𝑀 𝑅 ⁄ with modern GM and mean radius R; uncertainties are small vs seismic ones. 6 We treat the seismic–escape convergence as an empirical regularity predicted by DCM’s group-delay mechanism. It is not assumed as proof of the mechanism; rather, it constitutes a falsifiable signature: gravity-shaped cores should satisfy once uncertainties are propagated. We pre-specify the proxy choice (P for tidally free; S-wave proxy for tidally locked) and provide a prospective target list; deviations outside the stated band would falsify this claim. 2.4 Cosmological Redshift as Expansion Delay While the seismic correlation provides a compelling local verification of the model's reinterpretation of gravity, the same principles can be extended to cosmological scales, where the cumulative effect of discrete delays manifests as redshift. For systems of grouped masses, gravity can be analyzed from two complementary observational perspectives: that of an insider within the gravitational system, and that of an outsider observing from a distant, noninertial frame. Drawing on the elevator analogy, the flat-spacetime insider experiences a longitudinal Doppler effect, consistent with local free-fall conditions. In contrast, the distant observer at the "top" perceives a consistent with Eq. 1 transverse Doppler effect, reflecting time dilation across the gravitational field. In the standard cosmological model, the redshift of light from distant galaxies is attributed to the stretching of space itself—a Doppler-like effect due to metric expansion. Within the Discrete Cosmology Model, the cosmological redshift is reinterpreted as a cumulative gravitational time delay experienced by photons traversing an expanding vacuum. Unlike tired-light hypotheses [21] that invoke path-length photon fatigue, DCM explains redshift as an observer-relative time-dilation effect from cumulative interaction delays, thereby preserving image coherence [22] and supernova time dilation [23] while simultaneously constraining local seismology and galactic dynamics within a single, testable framework. This delay is observer-relative: the farther we look, the more delayed the expansion of matter appears to us. Light emitted from such regions originates from a slower-clock domain relative to the observer’s frame, resulting in a lower observed frequency— i.e., a redshift. Importantly, this redshift emerges without the need for recessional velocity or expanding metric. It is the gravitational analog of the longitudinal Doppler effect (Fig. 7) seen by a flatspacetime observer looking into Einstein’s stationary gravitational elevator: the elevator need not move, yet the observer perceives a redshift due to time dilation. Figure 4: Longitudinal Doppler and the observer-relative Event Horizon Longitudinal Doppler 𝑧(𝑅) 7 This reinterpretation also provides a new derivation for the Hubble law: cosmological redshift results from gravitational delays, not metric expansion, scaling with distance R: 𝑣(𝑅) ~ √𝜌𝑅 (4) derived from the escape velocity formula rewritten in density (𝜌)terms: 𝑣=√2𝐺𝑀 𝑅 ⁄=√8 3 ⁄𝜋𝐺𝜌 𝑅 (5) The gravitational delay acts as if the universe is expanding in appearance, but not in spacetime itself, distinguishing DCM from tired light or earlier non-metric models. Table 2: Density vs. cosmic mean Scale Density vs. Mean Evidence <10 Mpc Overdense 2MASS, SDSS ~50 Mpc Possibly overdense Laniakea 100–300 Mpc Conflicting Mixed claims >300 Mpc Cosmic mean Planck CMB According to Eq. 4, 18% local overdensity may explain the Hubble tension [10] of 8% – 9%. Table 2 points to potential sources of overdensity that may affect the relationship in Eq. 4. 2.5 Redshift as cumulative gravitational delay: a minimal derivation We model the observable redshift as arising from cumulative time dilation along the photon path through an interaction-limited, discretely expanding medium. In the weakfield, stationary limit we use an effective isotropic metric ds2=−e2Φ𝑒𝑓𝑓 𝑐2𝑐2𝑑𝑡2 (6) +e−2Φ𝑒𝑓𝑓 𝑐2(𝑑𝑟2+𝑟2dΩ2), with the path-averaged potential governing clock rates of the medium. For null geodesics the frequency shift between emission at r and observation at 0 is, to leading order, 1+𝑧≃𝑒𝑥𝑝(Φ𝑒𝑓𝑓(0)−Φ𝑒𝑓𝑓(𝑟) 𝑐2) (7) ≃1+Φ𝑒𝑓𝑓(0)−Φ𝑒𝑓𝑓(𝑟) 𝑐2 We decompose Φ𝑒𝑓𝑓 = Φ𝑔+ Φ𝑘 into (i) a gravitational delay term Φ𝑔 determined by the mass distribution along the line of sight and (ii) a kinematic delay term Φ𝑘 accounting for the finitespeed support of expanding multibody systems (see §2.7). For cosmological sightlines we approximate Φ𝑔 by a slowly varying function of proper distance r and expand to quadratic order in r/REH (REH an effective event-horizon scale, Fig. 7): 𝑧(𝑟)≃(𝐻0 𝑐)𝑟(1−𝑘 𝑟 𝑅𝐸𝐻), 0 ≤ r ≲ REH, (8) 8 where 𝑘 is a dimensionless coefficient aggregating the cumulative delay relative to the linear Hubble law. This form is dimensionally consistent, reduces to Hubble’s law at small r, and yields a suppression Δz/zlin ≃ 𝑘 at r ≃ REH. Fits to present SN Ia+BAO reconstructions suggest 𝑘 ≈ 0.08–0.10 if the entire tension is attributed to delay. The cosmological closure of the delay tensor leading to this quadratic redshift suppression is given in Appendix A.4. 2.6 Interpreting 𝒌 from the line-of-sight potential Let the line-of-sight effective potential be Φ𝑒𝑓𝑓(𝑟)=1 𝑐∫𝑎∥(𝑠)𝑑𝑠 𝑟 0, where 𝑎∥ encodes the retarded interaction coupling. In the weak-field limit the fractional frequency shift accumulates as 𝑧(𝑟)≃1 𝑐2∫𝜕Φ𝑒𝑓𝑓(𝑠) 𝜕𝑠 𝑑𝑠 𝑟 0=1 𝑐2Φ𝑒𝑓𝑓(𝑟) (9) Assuming a smoothly saturating potential Φ𝑒𝑓𝑓(𝑟)≃A r −𝐵𝑟2 𝑅𝐸𝐻 (10) with A≃𝐻0𝑐 and B≃k𝐻0𝑐 , we recover the quadratic parameterization above. The single dimensionless parameter 𝑘 is the (rescaled) ratio of the horizon-scale contribution to the linear Hubble term. In data applications can be inferred by a one-parameter regression of H(z) or DL(z) against ΛCDM baselines. 2.7 Expansion of a Galactic MultiBody System 2.7.1 Metric ansatz and lensing check We introduce an effective stationary, spherically-symmetric metric for the exterior of a disk-dominated system: ds2=−e2Φ𝑒𝑓𝑓(𝑟) 𝑐2𝑐2𝑑𝑡2 (11) +e−2Φ𝑒𝑓𝑓(𝑟) 𝑐2(𝑑𝑟2+𝑟2dΩ2), with Φ𝑒𝑓𝑓(𝑟) = Φ𝑔(𝑟) + Φ𝑘(𝑟). The gravitational term Φ𝑔(𝑟) reduces to the Newtonian potential of the observed baryons in the weak-field limit. The kinematic term Φ𝑘(𝑟) encodes the finite-speed support of the multi-body expansion and, for approximately flat rotation curves 𝑣φ(r)≃ 𝑣c, takes the isothermal form Φ𝑘(𝑟)=−1 2𝑣𝑐2ln(r/𝑟0). (12) A formal derivation of the effective delay density and the conservation check for axisymmetric disks is presented in Appendix A.3. Time-dilation then reads TTD ≃ 1 + [Φg(r) + Φk(r)]/c², recovering eq. (11) at leading order. Lensing follows from Φ and Ψ which coincide in this isotropic ansatz; the deflection angle is α(b)=4 𝑐2∫∇⊥Φ𝑒𝑓𝑓 𝑑𝑧 . (13) For 𝛷𝑘 above one obtains the standard singular isothermal sphere result α ≃ 4π (vc²/c²), i.e., the Einstein radius θE ≃ 4π (σv²/c²)(Dls/Ds), matching strongand weaklensing phenomenology that scales with 9 velocity dispersion—without invoking additional matter. Just as discrete phase dynamics govern quantum forces, similar principles apply to large-scale systems. The interplay between gravitational and kinematic delays becomes essential in explaining the curvature of multibody structures like galaxies. Sections 2.2 and 2.3 examined the spacetime curvature arising from relativistic delay within a single gravitationally bound body, such as a planet. In this context, gravitational time dilation emerges from the finite speed at which electromagnetic interactions propagate through mass. This time lag can be empirically estimated using seismic wave velocities, as demonstrated with data from Earth and Mars. The same principle extends to multi-body systems, where each constituent contributes to a cumulative expansion-related delay. In this case, we need to change the observer’s perspective to an “outsider” in relation to the observed galaxies, witnessing a transverse Doppler effect per the elevator analogy. However, in such systems, a second source of relativistic lag must be considered. The expanding volume of a multi-body system— such as a galaxy—cannot be sustained without the rotational motion of its members. Absent this kinematic support, the system would collapse under its own inertia. This introduces an additional expansion (transverse Doppler) delay component: kinematic time dilation, arising from orbital motion. To fully describe the curvature in a multi-body system, both gravitational (Eq. 1, written in mass terms) and kinematic contributions must be combined, as formalized in the following equation for the total time dilation factor (TTD): 𝑻𝑻𝑫 (𝒓) = √𝟏−𝟐 𝑮𝑴(𝒓) 𝒓𝒄𝟐−𝒗𝟐(𝒓) 𝒄𝟐 (14) Here, 𝑴(𝒓) is the galactic mass enclosed within radius 𝒓, and 𝒗 is the tangential velocity of stars rotating around the galactic center. The first term corresponds to classical gravitational time dilation (apparent motion) as in general relativity, while the second term accounts for kinematic time dilation arising from orbital (genuine) motion. This combined factor serves as an effective metric-like approximation describing the delay-based curvature of the galactic expansion profile. Fig. 5 presents an indicative correlation to illustrated the proposed physical principle. (A practical prediction algorithm, including the baryonic baseline, self-consistent delay iteration, and ensemble band construction, is detailed in Appendix A.7, including an example plot for UGC 14). Fig. 5 relies on observational data for the orbital velocities [11–12] and the galactic mass distribution [13–14] substituted in Eq. 14. Fig. 5 illustrates how at the periphery of a large spiral galaxy the kinematic component becomes dominant. The lag introduced by high tangential velocities contributes to the overall delay of expansion, leading to the observed flat rotation curves without invoking additional dark matter. In this model, the mass inferred from rotational curves corresponds not to hidden matter but 16 Proc. Natl. Acad. Sci. 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Couder, Y., & Fort, E. (2006). Single-Particle Diffraction and Interference at a Macroscopic Scale. Physical Review Letters, 97(15), 154101. https://doi.org/10.1103/PhysRevLett.97. 154101 29. Bush, J. W. M. (2015). Pilot-Wave Hydrodynamics. Annual Review of Fluid Mechanics, 47, 269–292. https://doi.org/10.1146/annurev-fluid010814-014506 30. Fort, E., Eddi, A., Boudaoud, A., Moukhtar, J., & Couder, Y. (2010). Path-memory induced quantization of classical orbits. Proceedings of the National Academy of Sciences USA, 107(41), 17515–17520. https://doi.org/10.1073/pnas.100738610 7 APPENDIX A: GR-Compatible StressEnergy Tensor for the Discrete Cosmology Model (DCM) The Discrete Cosmology Model (DCM) complements General Relativity (GR) by providing a causal-mechanical foundation for the stress-energy tensor, interpreting mass and curvature as emergent from discrete interaction delays. We maintain Einstein’s field equations, , (A1) but upgrade the source term to include delay effects. A.1 Two-Scale Link: Discrete to Continuum Let θ = ωCτ be the fast Compton phase, with ωC the Compton frequency. The microscopic tensor τµνdisc(x,θ) encodes phase-dependent mass ma(θ) and delay stresses Dµν(x,θ) from finitespeed interactions (e.g., electromagnetic stresses, see Appendix A of the supplemental material). Under scale separation ε = (𝑡𝑠𝑦𝑠 −1 𝜔𝐶 ⁄) ≪ 1 (e.g., ε ∼ 10-20 for planetary cores, ≪ 10-30 for galaxies), the macroscopic tensor is: , (A2) Ensuring . Empirically, the seismic–escape velocity convergence (Table 1, §2.3) calibrates the delay scalar as 18 ⟨Wcore⟩ ≃ ve2/c2, linking discrete dynamics to macroscopic curvature. A.2 Exchange form and total conservation The stress-energy tensor is: 𝑇𝜇𝜈 DCM =𝑇𝜇𝜈 (bar)+Δ𝑇𝜇𝜈 (delay), (A3) where the baryon tensor is: 𝑇𝜇𝜈 (bar)=(𝜌𝑏+𝑝𝑏/𝑐2)𝑢𝜇𝑢𝜈+𝑝𝑏𝑔𝜇𝜈 + 𝑞µ𝑢𝜈+𝜋𝜇𝜈, (A4) And the delay tensor is: Δ𝑇𝜇𝜈 (delay)=𝜌𝑑𝑐2𝑢𝜇𝑢𝜈+𝑝𝑑ℎ𝜇𝜈 +𝜋𝜇𝜈 (𝑑), (A5) with uµ the 4-velocity, 𝑢𝜇𝑢𝜇=−1, ℎ𝜇𝜈 = 𝑔𝜇𝜈 +𝑢𝜇𝑢𝜈, 𝑞𝜇𝑢𝜇=0, 𝜋𝜇 𝜇=𝜋𝜇 (𝑑)𝜇= 0, (A6) 𝜋𝜇𝜈𝑢𝜈=𝜋𝜇𝜈 (𝑑)𝑢𝜈=0. (A7) Figure A.1: Baryons 𝑇𝜇𝜈 (bar)and delay sector 𝑇𝜇𝜈 (delay)exchange fourforce 𝑄ν. The exchanges cancel in the divergence, ensuring ∇𝜇𝑇𝜇𝜈 DCM =0 while allowing finite-speed delay stresses to support rotation and redshift effects. We allow exchange via a 4-force density 𝑄𝜈(Fig.A.1): ∇𝜇𝑇𝜇𝜈 (bar)=−𝑄𝜈, ∇𝜇Δ𝑇𝜇𝜈 (delay)=+𝑄𝜈 ⇒ ∇𝜇𝑇𝜇𝜈 DCM =0. (A8) The weak-field closure used in disks is: 𝑊=𝜒𝑔Φbar 𝑐2+𝜒𝑘𝑣2 𝑐2, (A9) Ψkin(𝑟)=∫ 𝑣𝑐2(𝑠) 𝑠 𝑟 𝑟0 𝑑𝑠, 𝑄𝜈=𝜌𝑏 ∇𝜈(𝜒𝑘Ψkin). (A10) The exchange represents the finite-speed “delay stress” needed to support rotation; 𝜒𝑔,𝜒𝑘∼𝑂(1) and are calibrated empirically, not universal constants. At the microscopic level, 𝑄ν represents momentum transfer from finite-speed Compton-scale expansion (𝜀=𝜆𝐶/𝐿≪1); the macroscopic exchange law (A.6) is the ensemble average over these discrete delays. Thus, 𝜒𝑔 and 𝜒𝑘 do not introduce new universal constants, but instead reflect observational uncertainties (e.g. mass-tolight ratios and baryonic profile scatter) when coarse-grained over galactic or planetary scales. A.3 Stationary, axisymmetric disks: iterative closure Projecting ∇𝜇𝑇𝜇𝜈 (bar)=−𝑄ν radially for a cold disk: 𝑣𝑐 2 𝑟=𝜕𝑟Φbar +𝑄𝑟 𝜌𝑏=𝜕𝑟(Φbar +𝜒𝑘Ψkin)≡ 𝜕𝑟Φeff. (A11) 19 To avoid circularity, we solve selfconsistently: 1. Init 𝑣𝑐(0): baryons only, (𝑣𝑐(0))2/𝑟= 𝜕𝑟Φbar. 2. Update Ψkin (𝑛)(𝑟)= ∫(𝑣𝑐(𝑛−1)(𝑠))2 𝑟 𝑟0/𝑠 𝑑𝑠. 3. Effective Φeff (𝑛)=Φbar +𝜒𝑘Ψkin (𝑛) . 4. Velocity (𝑣𝑐(𝑛))2/𝑟=𝜕𝑟Φeff (𝑛). 5. Iterate to |𝑣𝑐(𝑛)−𝑣𝑐(𝑛−1)|/|𝑣𝑐(𝑛−1)|< 𝛿 (e.g., 10−3). As a toy example, for an exponential disk with Σ𝑏(𝑟)=Σ0𝑒−𝑟/𝑅𝑑, with Σ0=108 𝑀⊙/ kpc2 and 𝑅𝑑=3 kpc, the iteration converges after four steps to 𝑣𝑐≈150 km/s at 𝑟≈ 10 𝑘𝑝𝑐. This demonstrates that the selfconsistent closure reproduces flat rotation without nulling the baryonic potential Φbar. This convergent closure yields flat outer segments without nulling Φbar; for an exponential disk it asymptotes to an isothermal-like tail. A.4 Cosmology (FRW): isotropy and continuity On FRW 𝑢𝜇=(1,0,0,0)0), shear-free) require π𝜇𝜈 (𝑑)=0: Δ𝑇𝜇𝜈 (delay)=𝜌𝑑𝑐2𝑢𝜇𝑢𝜈+𝑝𝑑ℎ𝜇𝜈. (A12) With 𝐶≡−𝑢𝜈𝑄𝜈, 𝜌𝑏 󰇗 +3𝐻(𝜌𝑏+𝑝𝑏/𝑐2)=−𝐶, (A13) 𝜌𝑑 󰇗 +3𝐻(𝜌𝑑+𝑝𝑑/𝑐2)=+𝐶, (A14) and (to preserve isotropy) take 𝑄𝜈=𝐶 𝑢𝜈 (energy exchange only). There are two closures: • Conservative 𝐶=0, 𝑤𝑑≃−1+ 𝑂(𝜀). • Algebraic 𝜌𝑑=3𝜀𝐻2/(8𝜋𝐺) with 𝜀≃0.08−0.10 (quadratic redshift suppression used in §2.5). The seismic law 𝑣𝑠2≃𝑣𝑒𝑠𝑐 2≃𝑐2𝑊 provides an independent calibration of the delay scalar, reinforcing that 𝜀≪1 bridges microlevel discreteness and macro-scale observables in both planetary interiors and cosmological expansion. A.5 Two-scale kernel and isothermal tail We define a minimal two-scale kernel acting on 𝑣2: 𝐾(𝑟,𝑟′)=𝜒𝑔 𝛿(𝑟−𝑟′) 𝑟′+𝜒𝑘 Θ(𝑟−𝑟′) 𝑟𝑟′, (A15) 𝑔del(𝑟)=∫𝐾(𝑟,𝑟′) 𝑣2(𝑟′) 𝑟′ 𝑑𝑟′. (A16) This produces Ψkin ∼ln𝑟 over flat segments and ρ𝑑(𝑟)=1 4𝜋𝐺𝑟2𝑑 𝑑𝑟[𝑟𝑣𝑐2]∝𝑟−2, (A17) i.e. an isothermal-like envelope without dark halos. A.6 Comparison to Other Theories Unlike MOND, which introduces an empirical acceleration scale, DCM derives flat rotation curves from kinematic delays without ad hoc parameters. Unlike scalar- 20 tensor theories (e.g., TeVeS), DCM’s delay scalar W is empirically calibrated by seismic data (Table 1), grounding it in observable phenomena As implemented in Appendix A.7: baryonic band from SMD-F/SMD-S; self-consistent iteration in 𝑣𝑐 and 𝑣𝑒; ensemble band cross {Υ∗,𝜒𝑔,𝜒𝑘}∼𝑂(1)reflecting observational uncertainties (not a MOND-like universal parameter). A.7 Prediction Algorithm for Galactic Rotation Curves The delay-based stress–energy formulation can be operationalized into a reproducible algorithm for predicting galaxy rotation curves from photometric mass maps: 1. Baryonic baseline: Surface brightness profiles 𝑆𝑏(𝑅) are converted to stellar surface densities using catalog 𝑀/𝐿. Two limiting cases are considered: 2. Initial velocities: An initial 𝑣𝑐(𝑅) is formed by combining baryonic components. 3. Delay kernel: The two-scale delay operator (Appendix A.3) is applied to 𝑣2, yielding an effective delay acceleration field 𝑔del(𝑅). 4. Iteration: 𝑣𝑐2=𝑅(𝑔bar +𝑔del) is updated iteratively until convergence of both vcv_cvc and the associated escape velocity 𝒗𝒆𝒔𝒄. 5. Ensemble band: Parameters (𝐿1/ℎ,𝐿2/ ℎ,χ1,χ2) are scanned within order-unity ranges. Models within 10% of the best RMSE relative to observed 𝑣obs are retained, defining a predictive band. This procedure produces a family of rotation curves consistent with the observed flat outer profile without invoking dark matter halos. Figure A.2 illustrates the method for galaxy U14, showing the baryonic band [11-12], the DCM band, and the observed velocities [1314]. Figure A.2: DCM prediction for rotational velocities (UGC 14) The proposed here algorithm is not a fit in the MOND sense (no free universal 𝑎0) but a self-consistent closure of the delay tensor with empirical baryons. APPENDIX B: Observer‑Local Factors and Horizon Relay Lemma B.1 No-local-cap lemma Let (1+𝑧𝑜𝑏𝑠)=𝐶𝑙𝑜𝑐 ·(1+𝑧𝑝𝑎𝑡ℎ) with constant 𝐶𝑙𝑜𝑐 >0. If 𝑙𝑖𝑚𝑟→𝑅𝐸𝐻(1+ 𝑧𝑝𝑎𝑡ℎ)=∞ (hard horizon), then 𝑙𝑖𝑚𝑟→𝑅𝐸𝐻(1+𝑧𝑜𝑏𝑠)=∞ and 𝑑𝑧𝑜𝑏𝑠/𝑑𝑟= 𝐶𝑙𝑜𝑐 ·𝑑𝑧𝑝𝑎𝑡ℎ/𝑑𝑟. Thus a constant local factor cannot produce a finite 𝑧𝑚𝑎𝑥 nor enforce 𝑑𝑧/𝑑𝑟→0. B.2 Relay (penetration) clarification Photons originating beyond Earth’s horizon do not arrive at Earth in finite observer time. 21 An observer who relocates outward can receive those photons at their new location because their personal horizon moves; information can then be relayed back to Earth via new local emission, but the original wavefronts have not crossed Earth’s horizon. Hence observer‑relative horizons are consistent: visibility differs by location without contradiction. DCM: SUPLEMENTAL MATERIAL This file contains speculative extensions of the Discrete Cosmology Model (electrostatics [C], particle creation [D], a gyromagnetic ratio from phase-weighted inertia [toy model - E], and a CMB-like monopole [F]) that were deliberately excluded from the main article to preserve focus on the gravitational and cosmological tests. The Supplemental Material also provides two additional exploratory extensions of the Discrete Cosmology Model (DCM). Appendix H develops a shell-layered resonance framework for leptons, interpreting the electron, muon, and tau as successive Compton-scale closures. This yields a natural explanation for lepton mass ratios and lifetimes, linking the muon’s mass to 3/2𝛼−1 and reproducing the observed lifetime hierarchy through coherence decay. Appendix I applies the same discrete expansion principle to astrophysical plasmas, where inner-layer expansion displaces collisionless outer layers. The resulting “expansion thrust” offers a scaling law for stellar and planetary winds (𝑣𝑜𝑢𝑡 ≈ k𝑣𝑒𝑠𝑐) and relativistic jets, consistent with observed outflow velocities across multiple systems. Together, these appendices illustrate the broader unifying potential of DCM, extending its reach from particle microphysics to astrophysical outflows, while remaining complementary to the core cosmological claims of the main manuscript. APPENDIX C: Discrete delays at Compton scales This appendix consolidates the micro‑foundations previously in §2.8. In DCM, electrostatic attraction/repulsion emerges from complementary/synchronous phases of discrete particle expansion at Compton rates (Table 3). Here we add a concrete calculation linking the Bohr radius to discrete steps: Let the reduced Compton wavelength of the electron be λC = ħ/(me c). The Bohr radius is a₀ = ħ²/(me e²/(4πϵ₀)) = ħ/(me c α) = λC / α, with α ≈ 1/137 the fine‑structure constant. Thus a₀ ≈ 137 × λC. If the elementary radial expansion step is λC, then the ground‑state electron–proton separation corresponds to ~137 steps. This matches the empirical scale a₀ without invoking continuous charge distributions. The DCM interpretation is that out‑of‑phase Compton dynamics minimize interaction lag at ~137 steps, reproducing the Bohr radius scale. 22 Table 3: Electrostatic interactions in DCM Pair Relative Compton Phase DCM Prediction Experimental Support e⁻– e⁺ Out-ofphase Strong attraction Positronium observed e⁻–p Compton mismatch Moderate attraction Hydrogen forms e⁺–p Compton mismatch Moderate attraction Scattering matches e⁻–p e⁻– e⁻ Same phase Repulsion Seen in scattering e⁺– e⁺ Same phase Repulsion Confirmed Worked numerical values (SI): λC ≈ 3.8616×10⁻¹³ m; a₀ ≈ 5.2918×10⁻¹¹ m; ratio a₀/λC ≈ 137.036. Interestingly, recent hydrodynamic quantum analogs [28-30] reproduce many of the interference and quantization patterns expected from discrete group delays, suggesting that the same causal mechanism underpinning DCM may extend naturally into the quantum domain. Connection to Hydrodynamic Quantum Analogs (HQAs). Hydrodynamic quantum analogs (HQAs) such as walking-droplet experiments on vibrating baths, have successfully reproduced many features of quantum mechanics, including diffraction, interference, and quantized orbital states. These systems behave as oscillating point sources radiating waves into a surrounding medium, with trajectories guided by their self-generated pilot-wave field. The Discrete Cosmology Model (DCM) interprets this not as an analogy but as a physical mechanism: discrete Compton-scale delays of a particle act as oscillatory monopole sources in spacetime, producing wave fields that manifest as the quantum wave function. Unlike HQAs, which remain laboratory analogs, DCM extends the same causal principle to real particles (Fig. 2), providing a unifying interpretation across scales. For example, the Bohr radius emerging from a ~137-fold Compton mismatch exemplifies how discrete delays give quantized radii, while interference and tunneling arise naturally from overlapping delay fields. Thus, HQAs can be seen as experimental analogues validating the plausibility of DCM’s causal, testable framework. Limitations: while the present argument fixes the fundamental scale (e.g., the Bohr radius), it does not reproduce the full hydrogen spectrum. In principle, the level spacings could be obtained by quantizing allowed phase trajectories, analogous to hydrodynamic quantum analogs where orbital quantization emerges from pathmemory dynamics. A fuller treatment of spectral structure remains beyond the scope of this supplement. APPENDIX D: Particle Creation via Rotating Voids (sketch) We summarize the rotating‑void creation mechanism: a rapidly rotating low‑density 23 gap (e.g., photon‑borne disturbance) can, via relativistic time‑dilation, inhibit local vacuum expansion, allowing a nearby mass fluctuation to nucleate a new particle in discrete steps. The new particle inherits the phase of the source mass; antiparticles) emerge with a π phase shift. Annihilation releases the rotating gap. Testability: look for phase‑synchronized birth events near nuclei; search for transient, step‑wise growth signatures in ultrafast pump–probe experiments; test protonpositron beams for attraction, filtering magnetic effects; measure tunneling rates in STM/quantum wells under varying fields (predicts mass fluctuations) [16]. These are high‑risk, high‑reward tests intentionally segregated from the gravitational/ cosmological core of the manuscript to avoid overreach. APPENDIX E: Gyromagnetic Ratio from Phase-Weighted Inertia (Toy Model) Assume charge circulates during a fraction f of a Compton cycle with rotating mass mrot and radius r. For a ring (κ =1), 𝜇≈𝑞 2𝑓𝑟2𝜔,𝑆≈𝑓𝑚𝑟𝑜𝑡𝑟2𝜔 => 𝜇 𝑆≈𝑞 2𝑚𝑟𝑜𝑡 (E1) Identifying 𝜇≈𝑔 𝑞 2𝑚0S yields 𝑔 ≈ 𝑚0/𝑚𝑟𝑜𝑡. If 𝑚𝑟𝑜𝑡 = 𝑚0/2, then 𝑔 ≈2 without superluminal rotation. Small phase asymmetries and EM self-interactions yield a natural 𝑔−2 correction dependent on the delay scalar 𝑊. APPENDIX F: CMB-like Background Without a Primordial Thermal Bath (DCM) This appendix formalizes how a CMB-like monopole can arise in DCM without a primordial thermal bath. Two ingredients are required: (i) a global redshift–distance map z(r) that exhibits a near-horizon pinch, and (ii) a physically narrow line-of-sight visibility (weight) 𝑊(𝑧) peaked near a finite 𝑧⋆≈103. The mapping alone is insufficient; spectral purity follows only if the effective kernel is sharply confined. F.1 Global mapping and horizons We use an observer-normalized map with 𝑧(0)=0 and 𝑑𝑧/𝑑𝑟|0=𝐻0/𝑐. Two horizon choices are useful. Hard horizon (infinite ceiling): choose an effective speed 𝛽(𝑟)=𝑣𝑒𝑓𝑓(𝑟)/𝑐→1 as 𝑟→𝑅𝐸𝐻, then 1+ 𝑧=√(1+𝛽)/(1−𝛽)→∞. Soft horizon (finite ceiling): impose z(r)→z_max<∞ with 𝑑𝑧/𝑑𝑟→0 as 𝑟→𝑅𝐸𝐻. A convenient family is 𝑧(𝑟)=𝑧𝑚𝑎𝑥 ·𝑡𝑎𝑛ℎ([𝐻0𝑟/(𝑐𝑧𝑚𝑎𝑥)]· [1−(𝑟/𝑅𝐸𝐻)𝑝]−𝛾) with 𝑝,𝛾>0. Both recover 𝑧≈(𝐻0/𝑐)𝑟 at small 𝑟. F.2 Radiative transfer and the role of 𝑾(𝒛) For a statistically homogeneous medium the observed specific intensity at 𝜈0 is 𝐼𝜈0= ∫𝑑𝑧·[𝑑𝑟/𝑑𝑧]·𝑊(𝑧,𝜈0)·𝐵((1+𝑧)𝜈0, 𝑇𝑒𝑚𝑖𝑡(𝑧))/(1+𝑧)3 (C1) where 𝐵 is the Planck function and 𝑊 encodes emissivity, geometry, opacity, and transport (visibility). 24 A near-Planck spectrum requires 𝑊(𝑧) to be sharply peaked near a single z⋆ so that 𝑇𝑒𝑚𝑖𝑡(𝑧)/(1+𝑧) is effectively constant across the kernel. F.3 Width requirement (mapping alone is not enough) A pinch in 𝑧(𝑟) means 𝑑𝑧/𝑑𝑟→0 at the horizon, but the integrand carries the Liouville factor (1+𝑧)−3 and the Wien tail of 𝐵, which suppress high‑z contributions. To emulate a delta‑shell, the effective kernel 𝐾(𝑧)≡[𝑑𝑟/𝑑𝑧]·𝑊/(1+𝑧)3 must be narrow. For a representative hard‑horizon calibration (𝛼≡𝑅𝐸𝐻/(𝑐/𝐻0)=1.8,𝑝= 3,𝛾=0.4), we find 𝑑𝑟/𝑑𝑧|𝑧⋆≈1100 ≈ 0.027 𝑀𝑝𝑐. Thus 𝛥𝑧≈3−10 implies 𝛥𝑟≈ 0.08−0.27 𝑀𝑝𝑐 (80−270 𝑘𝑝𝑐) for the contributing shell; ppm‑level spectral purity likely demands even thinner shells. F.4 Near‑horizon kinematics at 𝒛⋆≈ 𝟏𝟏𝟎𝟎 Using the longitudinal Doppler relation 1+ 𝑧=√(1+𝛽)/(1−𝛽) gives, 𝑓𝑜𝑟 𝑧⋆≈ 1100, 𝛽⋆≈0.99999835011 and 𝑣𝑒𝑓𝑓 ≈ 0.99999835011 𝑐≈299,791.963 𝑘𝑚/𝑠− only ≈0.495 𝑘𝑚/𝑠 below 𝑐. This quantifies how close to the horizon the contributing shell sits if 𝑧⋆ is fixed. F.5 Horizon‑sourced visibility shell (transport‑limited layer) In the hard‑horizon map, direct high‑z contributions are Liouville‑suppressed. If the discrete‑delay medium enforces strong transport limits near an effective horizon, inbound ultra‑redshifted radiation from the outer universe can be interrupted (e.g., scattered, decohered) and re‑emitted within a thin layer at 𝑧⋆. This produces a sharply peaked visibility 𝑊(𝑧)=𝑒−𝜏(𝑧)·𝑑𝜏/𝑑𝑧, acting as a last‑interaction surface. Provided 𝑇𝑒𝑚𝑖𝑡(𝑧)/(1+𝑧) is ~constant across the narrow kernel, the monopole approaches a single Planck curve at 𝑇𝑜𝑏𝑠 =𝑇𝑒𝑚𝑖𝑡/(1+ 𝑧⋆). Spectral‑distortion limits then translate into a shell thickness constraint 𝛥𝑟≲102 kpc at 𝑧⋆. F.6 Energy‑budget constraint (conservative check) Let 𝑢𝐶𝑀𝐵 denote the observed monopole energy density. A horizon‑sourced shell must satisfy an energy balance: the net inbound radiative power intercepted and reprocessed by the layer (integrated over solid angle and weighted by 𝑊) must reproduce 𝑢𝐶𝑀𝐵 after redshifting. This condition sets a lower bound on the product of the near‑horizon optical‑depth rise and the pre‑shell radiation field. Failure to meet this bound falsifies the mechanism irrespective of spectral fits. F.7 Testable implications • Spectral purity: 𝑅𝑀𝑆 fractional deviation < 10−4 across 30−600 𝐺𝐻𝑧 requires 𝛥𝑟≲ 102 kpc at 𝑧⋆. • Polarization: a scattering shell implies specific E‑mode features akin to a thin visibility spike; absence beyond limits disfavors the model. F.8 Summary DCM can, in principle, reproduce a CMB‑like monopole without a primordial thermal bath if (i) 𝑧(𝑟) exhibits a soft‑ or hard‑horizon pinch and (ii) a physical, thin, horizon‑adjacent visibility shell peaks 𝑊(𝑧) near 𝑧⋆≈103. Local blueshifts are small 25 multiplicative calibrations and cannot set 𝑧⋆ or create the pinch. • Anisotropy: any coupling of 𝑊(𝑧) to large‑scale structure induces tiny but testable departures from isotropy. • Redshift drift: soft‑horizon variants predict distinct ż(𝑧) at 𝑧~1−3; measuring the sign/magnitude discriminates microphysics. APPENDIX H: Resonant Overshoot Hypothesis for Lepton Masses In the Discrete Cosmology Model (DCM), particle mass fluctuates discretely at Compton frequencies. While the electron represents the stable baseline of this cycle, heavier leptons may be understood as resonant overshoot states that occur when discrete delays accumulate coherently. H.1 A Toy Derivation of the Muon/Electron Mass Ratio In the Discrete Cosmology Model (DCM), mass oscillates at Compton frequency. The electron corresponds to the stable baseline, while the muon arises as the first coherent resonant overshoot state. Two delay channels contribute per Compton cycle: • a longitudinal (gravitational-like) channel, linear in delay quanta, • a transverse (kinematic) channel, weighted by ½β² as in the weak-field expansion of γ. DCM predicts that the first overshoot occurs when these channels close coherently. The effective muon/electron ratio follows from combining them: (𝑚𝜇/𝑚𝑒)≈𝛼−1+½𝛼−1 =1.5𝛼−1 (H1) Numerically, with 𝛼−1 =137.036: (3/2)𝛼−1 =205.554 (H2) The observed value is: 𝑚𝜇/𝑚𝑒=206.768 (H3) leaving a small residual correction 𝛿=(𝑚𝜇/𝑚𝑒)−(3/2)𝛼−1 ≈1.214 (H4) Thus the leading term explains 99.4% of the ratio without free parameters, while the residual is plausibly due to micro-level inertia effects. H.2 Relating the Correction δ to PhaseWeighted Inertia Appendix E proposed a phase-weighted inertia toy model: during each Compton cycle, a fraction f of the mass rotates (m_rot) and the rest expands (m_exp). The cycleaveraged inertia is 𝑚 =𝑓𝑚𝑟𝑜𝑡+(1−𝑓)𝑚𝑒𝑥𝑝 (H5) For the electron baseline, the rotating mass satisfies 𝑚𝑟𝑜𝑡 (𝑒)≈𝑚0/𝑔𝑒, with 𝑔𝑒≃2(1+ 𝛼/2𝜋). Define the rotational share: 𝜒=𝑚𝑟𝑜𝑡 (𝑒)/𝑚(𝑒) (H6) At the first resonance, let the rotating mass be amplified by a factor η: