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 predicted here empirical seismic–escape equality 𝑣𝑃−𝑤𝑎𝑣𝑒 𝑟𝑎𝑑 ≈𝑣esc within selfgravitating 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 Universe requiring no dark components or singular origins. On galactic scales the model reproduces the curvature of the radialacceleration relation (RAR) with baryons only and no free parameters (𝑎0≃𝐻0𝑐/2𝜋). On cluster scales the relativistic kinematic term naturally yields the 𝜎𝑣 2 lensing/kinematics scaling—without dark matter. In this framework, the cosmic microwave background (CMB) arises not from a primordial thermal bath but from a near-horizon visibility shell formed by delayed re-emission at the event-horizon boundary, whose statistical inhomogeneity accounts for the small observed anisotropies. Keywords: Discrete Cosmology Model · Group Delay · 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 2 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. 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 assuming every mass’s shape is sustained by countless interactions at finite speeds, creating cumulative delays that shape gravity and redshift—no dark components needed. DCM is 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, 3 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 discrete relativistic delays caused by their rotation, producing stepwise growth (Fig. 1). 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 F of the Supplemental Material. 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 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. 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: 4 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. 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 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. 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. 5 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. 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 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 mechanism, 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 6 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. 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. Figure 4: Longitudinal Doppler and the observer-relative Event Horizon 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 Longitudinal Doppler 𝑧(𝑅) 7 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 flat-spacetime observer looking into Einstein’s stationary gravitational elevator: the elevator need not move, yet the observer perceives a redshift due to time dilation. 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 8 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) 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-ofsight 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 The CMB as Horizon-Shell ReEmission In the DCM framework, radiation originating from beyond the observable horizon experiences cumulative group-delay saturation at z ∗ ≈ 1100. Using the longitudinal Doppler relation 1+z= √(1+β) (1−β) ⁄, this corresponds to an effective propagation velocity β ∗ =0.99999835011,veff=β ∗ c≈299,791.963 k m/s, only ≈ 𝟎.𝟓 𝒌𝒎/𝒔 below the speed of light. This finite delay limit marks the formation of a thin visibility shell where energy is scattered and re-emitted with near-Planck spectral weighting. The observed temperature follows 𝑇obs =𝑇emit (1+𝑧∗ ⁄), yielding Tobs = 2.73 K for Temit ≈ 3000 K. The shell’s near-spherical geometry explains the isotropy of the CMB, while small anisotropies (𝛿𝑇/𝑇≈10−5) arise from inhomogeneities in the outer universe projected onto the horizon screen. This replaces the need for a hot early epoch while preserving the CMB spectrum and polarization features, consistent with a stationary-curvature universe. The CMB’s observed E-mode polarization amplitude (~5µK) and strong E/B asymmetry can be interpreted in DCM as reemission from the same horizon-shell. The shell has a thin radial thickness (Δr ≲ 0.1– 1 Mpc; fiducially Δr≈0.1 Mpc ≃ 100 kpc), which preserves spectral purity and limits line-of-sight damping. 9 The angular scale of the first polarization peak instead reflects the transverse coherence on the shell, with characteristic L ⊥∼ 50100 Mpc, giving ℓpeak ≈𝜋 𝑅EH 𝐿⊥≈ 150–300, in agreement with the first E-mode peak measured by Planck. This interpretation naturally explains the observed suppression of primordial B-modes and links CMB anisotropy to interactions between the inner and outer universe. The full mathematical derivation is provided in Appendix C. 2.8 Expansion of a Galactic MultiBody System 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 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 16 particle’s expansion cycle, producing absorption. The coherence limit 𝑣Compton =𝑐=𝑣esc (27) thus, unites photon absorption and gravitational trapping as manifestations of perfect delay synchronization. The coherence is testable via time-resolved photon absorption spectra near Compton wavelengths (e.g., tens of MeV), as outlined in Supplement C. The logic here is consistent with variablemass dynamics in §2.1 (spin reinterpretation) and Appendix H (lepton shells). Supplement C: Quantum Saturation Mechanism At the quantum scale, DCM interprets photon absorption and emission as transient coherence phenomena within discrete expansion shells. A particle’s Compton expansion front propagates at 𝑐, defining a microscopic horizon of synchronization. When a photon field becomes phase-matched to this front, the relative group delay vanishes, Δ𝜏𝛾𝑒 →0, (28) and the photon’s energy merges into the local delay potential Ψ. Here, Ψ represents the particle’s cumulative delay field energy, modulated by Compton cycles. The resulting condition 𝑣Compton =𝑐=𝑣esc, (29) represents quantum saturation—the same coherence limit that defines the macroscopic gravitational horizon. Both processes correspond to zero relative group delay, where propagation and expansion become indistinguishable. Photons are emitted when the local front overshoots equilibrium, absorbed when it re-aligns, and reflected when coherence cannot be established. This framework unifies optical and gravitational interactions under the same delay-variance principle. 6.5 CMB Polarization The Discrete Cosmology Model predicts that the faint polarization of the cosmic microwave background arises not from primordial recombination, but from anisotropic scattering at a thin visibility shell near the event horizon. A thin radial window (Δr ≲ 0.1–1 Mpc; we adopt Δr ≈ 0.1 Mpc ≃ 100 kpc as a fiducial value) preserves spectral purity and limits line-of-sight damping, while the angular scale of the polarization peaks is set by the transverse coherence on the shell (characteristic size L⊥ ∼ 50–100 Mpc), yielding ℓ ≈ πREH/L⊥ ≈ 150– 300, consistent with Planck. This mechanism naturally gives an E-mode amplitude of ~5– 10 μK, with negligible primordial B-modes (lensing only) and a rapid decline of E–B cross-power toward large scales. The measured E-mode amplitude and its angular dependence thus provide a direct test of the horizon-scattering interpretation. 7. Conclusion We have presented the Discrete Cosmology Model (DCM), a framework that complements GR by attributing curvature and time dilation to relativistic group delays 17 in discretely expanding matter. This interpretation upgrades the definition of mass within the stress–energy tensor, providing a causal–mechanical foundation rather than treating mass as an unexplained source term. DCM preserves Einstein’s equations while enriching the source sector with delay terms, ensuring conservation and consistency with established geometry. The resulting framework yields three independent, testable consequences: flat galactic rotation curves, quadratic suppression of cosmological redshift, and seismic–escape velocity convergence. These predictions, especially the seismic relation confirmed by Apollo and InSight missions, distinguish DCM from phenomenological alternatives such as MOND or ΛCDM extensions. By linking microphysical discreteness (Compton-scale oscillations) to macroscopic astrophysical observables, DCM establishes a bridge between foundational physics and cosmology. This causal–mechanical perspective provides a novel, testable approach to the problems of dark matter and dark energy while preserving the structure of General Relativity. DCM unifies rotation curves (RAR curvature from a single 𝐻0-anchored scale), cluster lensing (relativistic 𝜎𝑣2 scaling), and cosmological redshift suppression. The interpretation extends naturally to the CMB, whose near-perfect isotropy arises from horizon-shell re-emission rather than from a primordial thermal epoch. 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 hydrodynamic quantum analogs; a detailed treatment of spectral structure, however, lies beyond the scope of this paper. Concluding Highlights • Seismic–gravitational law: Average seismic velocities converge with escape velocities across self-gravitating bodies, revealing a new empirical regularity. • Delay-based mechanism: Gravity and cosmological redshift arise from cumulative relativistic group delays in discretely expanding matter. • Flat rotation curves: Galactic dynamics are explained by combined gravitational and kinematic delays, without invoking dark matter. • Hubble tension: Quadratic redshift suppression near the cosmic horizon naturally accounts for the observed discrepancy in H₀. • Falsifiability: Predictions can be tested with Artemis lunar seismology, galaxy rotation spectroscopy, and local-group redshift surveys. 8 Future Work • Test the seismic wave correlation for other bodies with gravity shaped cores. • Future DCM tests may explore stellar bodies, predicting the Sun’s P-wave velocity (~510 km/s) aligns with its 18 escape velocity (618 km/s, ratio ~0.82) via radial projection, testable with advanced helioseismology. • Confirm Moon’s circular S-wave propagation with Artemis [7]. • Upscale Q-Drive at low temperatures [16]. Acknowledgements The author thanks colleagues and computational tools for feedback and editing support. References 1. Sahni, V., Dark matter and dark energy: The mystery deepens, Nat. Phys. 2, 635– 637, (2006). 2. Planck Collaboration, Planck 2018 results. VI. Cosmological parameters. 3. Richard Lieu, Are dark matter and dark energy omnipresent?, Class. Quantum Grav. 2025. doi: 10.1088/13616382/adbed1. 4. Milgrom, M., A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis, Astrophys. J. 270, 365–370 (1983),. doi: 10.1086/161130. 5. Bekenstein, J. D., Relativistic gravitation theory for the MOND paradigm. 6. Oppenheim J, et al. A postquantum theory of classical gravity? Nature Communications 14, 43348 2023. doi: 10.1038/s41467-023-43348-2. 7. Garcia R.F, et al. Lunar seismology: A data and instrumentation review. Icarus 332, 66–78 2019. 8. Irving J.C.E, et al. First observations of core-transiting seismic phases on Mars. Proc. Natl. Acad. Sci. USA 120(18), e2217090120 2023. doi: 10.1073/pnas.2217090120. 9. Hartlep T, Mansour N.N. Acoustic wave propagation in the Sun. Center for Turbulence Research, Annual Research Briefs 2005. 10. Riess, A. G. et al., The expansion of the Universe is faster than expected. 11. Courteau S. The structure of spiral galaxies: I. Near-infrared properties of bulges and disks. ApJ Supplement Series 103, 363 1996. doi: 10.1086/192284. 12. Courteau S. Optical rotation curves and linewidths for Tully-Fisher applications. AJ 114, 2402 1997. doi: 10.1086/118656. 13. Sofue Y. Rotation curves of spiral galaxies. PASJ, 1–10 2014. doi: 10.1093/pasj/psu070. 14. Sofue Y. Rotation curve of the Milky Way and the Galactic constants. PASJ, 1–10 2017. doi: 10.1093/pasj/psw103. 15. Lynden-Bell, D. et al., Spectroscopy and photometry of elliptical galaxies. V. Galaxy streaming toward the new supergalactic center. 16. Markov N. A quantum propulsion method. In: Proceedings of the International Maritime Association of the Mediterranean (IMAM2019), Varna, September 2019. 19 17. Dziewonski, Adam M.; Anderson, Don L. 1981. "Preliminary reference Earth model". Physics of the Earth and Planetary Interiors. 25 (4): 297–356. Bibcode:1981PEPI...25..297D. doi: 10.1016/0031-9201(81)90046-7. ISSN 0031-9201. 18. Weber, Renee C.; Lin, Pei-Ying; Garnero, Edward J.; Williams, Quentin; Lognonné, Philippe (2011-01-21). "Seismic Detection of the Lunar Core". Science. 331 (6015): 309–312. Bibcode:2011Sci...331..309W. doi: 10.1126/science.1199375. ISSN 00368075. PMID 21212323. S2CID 206530647. 19. Zheng, Yingcai; Nimmo, Francis; Lay, Thorne 2015. "Seismological implications of a lithospheric low seismic velocity zone in Mars". Physics of the Earth and Planetary Interiors. 240: 132– 141. Bibcode:2015PEPI..240..132Z. doi: 10.1016/j.pepi.2014.10.004. ISSN 00319201. 20. Dumoulin T. et al. Tidal constraints on the interior of Venus. J. Geophys. Res. Planets 121, 1727–1743 2016. doi: 10.1002/2016JE005159. 21. Zwicky, F. “On the Red Shift of Spectral Lines,” Proc. Natl. Acad. Sci. 15 (1929). 22. Lubin, L. M., & Sandage, A. “The Tolman Surface Brightness Test… IV,” AJ 122, 1084 (2001). 23.Blondin, S. et al. “Time Dilation in Type Ia Supernova Spectra at High Redshift,” The Astrophysical Journal, Volume 682, Issue 2, pp. 724-736 (2008) 24. Kocevski D.D., Ebeling H. 2006. On the origin of the Local Group’s peculiar velocity. Astrophys. J. 645, 1043–1053. doi: 10.1086/504360 25. Lavaux G., Hudson M.J. 2011. The 2M++ galaxy redshift catalogue. Mon. Not. R. Astron. Soc. 416, 2840–2858. doi: 10.1111/j.1365-2966.2011.19233.x 26. Courtois H.M., Pomarède D., Tully R.B., Hoffman Y., Courtois D. 2013. Cosmography of the Local Universe. Astron. J. 146, 69. doi: 10.1088/00046256/146/3/69 27. Tully R.B., Courtois H., Hoffman Y., Pomarède D. 2014. The Laniakea supercluster of galaxies. Nature 513, 71– 73. doi: 10.1038/nature13674 28. 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 20 APPENDIX A: GRCompatible Stress-Energy 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 finite speed 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 ⟨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) 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) 21 Ψkin(𝑟)=∫ 𝑣𝑐2(𝑠) 𝑠 𝑟 𝑟0 𝑑𝑠, 𝑄𝜈=𝜌𝑏 ∇𝜈(𝜒𝑘Ψkin). (A10) 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. 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) 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. 22 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 scalartensor 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: 23 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 algorithm here 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. 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. APPENDIX C: Cosmic Background and HorizonShell Polarization C.1. Origin of the Background Field In the Discrete Cosmology Model (DCM), the cosmic microwave background (CMB) is not interpreted as relic radiation from a primordial hot epoch, but as a stationary reemission phenomenon arising at the visibility 24 shell near the cosmological event horizon. Radiation from the outer Universe—where delay gradients exceed the local groupvelocity limit—becomes trapped, scattered, and thermally re-equilibrated within a narrow layer of high delay compression. The Plancklike spectrum thus emerges from the cumulative redshift and finite-speed propagation of interactions, rather than from early-Universe thermalization. C.2. Radiative Transfer Through the Horizon Shell Let 𝑊(𝑟) denote the local delay-weighting kernel along the radial coordinate 𝑟, normalized such that ∫𝑊 𝑅EH 0(𝑟) 𝑑𝑟=1, (C1) where 𝑅EH is the effective event-horizon radius. The observed intensity 𝐼obs(𝜈) is given by the convolution 𝐼obs(𝜈)=∫𝑊 𝑅EH 0(𝑟) 𝐼emit (𝜈 1+𝑧(𝑟))𝑑𝑟 1+𝑧(𝑟), (C2) where 𝐼emit is the local emissivity and 𝑧(𝑟) is the cumulative redshift obtained from Eq. (8) in the main text. A sharply peaked 𝑊(𝑟) near 𝑟≈𝑅EH corresponds to a “thin-shell’’ visibility function, producing a nearly Planckian spectrum when the shell thickness Δ𝑟/𝑅EH ≲10−3. C.3. Visibility Shell Thickness and Delay Saturation The finite‐delay saturation defining the visibility shell occurs at the characteristic redshift 𝑧∗≈1100, where the cumulative group‐delay reaches its relativistic limit. Using the longitudinal Doppler relation 1+𝑧=√1+𝛽 1−𝛽, (C3) the corresponding effective propagation velocity is 𝛽∗=0.99999835011, 𝑣eff =𝛽∗𝑐 ≈299,791.963 km/s, which is only ≈0.5 km/s below the speed of light. This defines the asymptotic delay boundary beyond which further redshift accumulation is negligible, marking the onset of photon reprocessing and isotropization within the horizon shell. The shell’s finite thickness therefore reflects the transition between nearly luminal propagation and complete delay saturation. The shell’s finite radial thickness (𝛥𝑟≲ 0.1 𝑀𝑝𝑐) defines the visibility window, while its transverse coherence (L ⊥ ≈ 50−100 ) sets the angular polarization scale (ℓ≈200− 300). For 𝑘≈0.08−0.10, the effective coherence scale is Δ𝑟≲ 𝑐 k𝐻0(1+𝑧∗) ≈50−100 Mpc, (C4) corresponding to the first E‐mode multipole ℓ≈𝜋𝑅EH/Δ𝑟≈200−300. This range reproduces the angular polarization peak observed by Planck and WMAP, linking the measured CMB coherence to the finite‐delay structure of the event‐horizon shell. C.4. Spectral Purity and Energy Balance Because each photon experiences cumulative redshift delay rather than scattering in an 25 expanding medium, energy conservation holds locally. The equilibrium spectrum approaches 𝐼𝜈=2ℎ𝜈3 𝑐2[exp( ℎ𝜈 𝑘𝐵𝑇obs)−1]−1, (C5) where 𝑇obs =𝑇emit/(1+𝑧⋆) and 𝑧⋆≃1100 corresponds to the mean horizon redshift. The observed monopole temperature 𝑇obs ≈ 2.73 K is reproduced for 𝑇emit ≈3000 K— matching the effective radiative temperature of a moderately dense extragalactic medium beyond the horizon. Ongoing energy flux from the outer Universe sustains the blackbody spectrum. C.5. Liouville Suppression and Coherence The Liouville invariance of phase-space density implies that the redshifted field maintains spectral coherence. However, intensity fluctuations from the outer Universe are attenuated by a factor ∼(1+𝑧)−4, ensuring angular uniformity at the level of 𝛿𝑇/𝑇∼10−5. This suppression replaces the isotropizing role of inflation in standard cosmology, providing a causal explanation for the uniform background. C.6. Relation to the Hubble Tension The observed quadratic suppression in 𝑧(𝑟), introduced by the factor (1−𝑘𝑟/𝑅EH), causes the local Hubble constant to appear ∼10 % higher than the global value. The same 𝑘 parameter that defines the redshiftpinched shell also governs the apparent anisotropy level of the CMB through the delay-weighting gradient ∂𝑊/∂𝑟. C.7. Inhomogeneity and Temporal Compression Neither the outer Universe nor the eventhorizon shell is perfectly homogeneous. Local curvature and density variations beyond 𝑅EH produce anisotropic illumination patterns projected inward onto the shell. The shell itself acts as a causal-compression interface: the extreme time dilation near the horizon reduces temporal variation and averages small-scale inhomogeneities, producing a comparatively uniform reemission pattern. Formally, the residual anisotropy of the reprocessed field may be expressed as 𝛿𝑇 𝑇∝∂𝑊(𝑧) ∂Ω , (C6) where 𝑊(𝑧) is the local delay-weighting kernel across solid angle Ω. This dual-layer non-uniformity—inhomogeneous outer illumination and compressed local emission—naturally yields the observed angular fluctuation amplitude of order 10−5. C.8. Polarization Signatures Radiation reprocessed in the visibility shell acquires linear polarization through anisotropic scattering within the delayweighted medium. The characteristic angular coherence of polarization arises from the transverse correlation length 𝐿⊥ on the visibility shell. Here 𝐿⊥∼50–100 Mpc represents the projected physical separation on the horizon surface at z ∗ ≈ 1100, corresponding to an angular scale of 0.7–1.3° or multipoles ℓpeak ∼150–300. The geometry of the horizon shell supports only curl-free (E-mode) polarization patterns, 32 • Third channel participation. The second resonance might activate an electromagnetic self-interaction channel (radiative term) coherently with the longitudinal/transverse pair. Its inclusion at resonance order 𝑁=2 could boost the scale to 𝒪(103) while keeping the muon’s 𝛿 mechanism intact (H.2), i.e., 𝑚𝜏/𝑚𝑒 ≈ (3/2∝−1) 𝑅(𝑁= 2,𝛼) + 𝛿𝜏, (H12) 𝛿𝜏 from the same (𝑓,χ,𝑔𝑒,η) law with Nscaling. What we don’t do: We do not present a number for 𝑚𝜏/𝑚𝑒 from a simplistic channel sum. Instead, we elevate tau to a target for a forthcoming microderivation that uses the same machinery as H.2 (phase-weighted inertia) but extended to compound closures and/or a third channel. Falsifiable forecast (band, not a point): Once a specific compound-closure rule is chosen, it must: • reduce to H.1 for 𝑁=1 (muon), • keep the same 𝛿-law modulo an explicit 𝑁-scaling, and • hit 𝑚𝜏/𝑚𝑒 within a narrow, parameter-free leading band, with δτ\delta_\tauδτ fixed by the same micro-correction structure (no refit). (Editorial note to reviewers: this section explicitly acknowledges the current limitation and sets a testable program rather than retrofitting numbers.) We interpret the tau as a second-order resonance requiring compound closure; we outline constraints and a falsifiable program but defer a numeric derivation to future work. H.4 — Shell-Layered Resonance and Lepton Stability In DCM, the electron’s effective envelope is set by one reduced Compton wavelength, 𝜆𝐶    =ℏ 𝑚𝑒𝑐, (H13) since each Compton cycle can expand the interaction field by at most 𝑐Δ𝑡=𝜆𝐶    . We interpret this envelope as the cumulative extent of the particle’s delay field. Shell-layering mechanism Higher resonances (muon, tau) arise as coherent overshoots in which additional Compton-scale shells are stacked. Each shell corresponds to the activation of an additional delay channel: • Electron (𝑵=𝟎): baseline shell. • Muon (𝑵=𝟏): longitudinal + transverse channels close coherently, yielding one extra shell. • Tau (N = 2): requires compounded closure with a radiative selfinteraction channel, stacking yet another shell. Thus, the effective envelope grows as 𝑅𝑁≈(𝑁+1) 𝜆𝐶    , (H14) with each shell individually constrained by the light-speed limit. The observed lepton 33 generations correspond to the first three such closures. Mass ratios and shells The muon ratio follows from the two-channel closure (H.1): 𝑚μ 𝑚𝑒 ≈ 3/2 α−1+δ, (H15) δ≈1.21. The tau can then be interpreted as the secondorder closure (𝑁=2) requiring compounded shells. A general ansatz consistent with H.1–H.3 is: 𝑚𝑁 𝑚𝑒 ≈ 3/2 α−1 𝒬(𝑁,α), (H16) with 𝒬(𝑁=1)=1 (muon) and 𝒬(𝑁= 2)∼2α−1/3 (tau), giving the observed ∼ 3477. Deriving 𝒬(𝑁,α) from microphysics remains a task for future work, but the shell framework provides a natural scaffold. Lifetimes from coherence decay Each additional shell increases phase complexity and reduces stability. We model the coherence lifetime as τ𝑁 ∼ τ0 (𝑁+1)2, τ0≈10−21 s, (H17) the Compton timescale. This scaling yields (Fig. H.1): • 𝑁=0 (electron): τ0→ ∞. • 𝑁=1 (muon): τ1∼2 μs. • 𝑁=2 (tau): τ2∼0.3 ps. These values are consistent with observed lepton lifetimes (𝜇: 2.2 𝜇𝑠; 𝜏: 0.29 𝑝𝑠), supporting the dual criterion of phase closure + coherence threshold. Figure H.1: Lepton shells Testable predictions • Finite spectrum: No higher leptons exist beyond tau, as additional shells collapse before forming physical states. • Envelope effects: Precision scattering near Compton scales could reveal layered structures in effective charge distributions. • Anomalous g-factors: Muon and tau 𝑔−2 should show deviations consistent with altered delay-field envelopes. • Scaling consistency: The same shelllayering logic underpins both microscopic lepton structure and macroscopic plasma outflows (Appendix I). Experimental tests could include scattering at energy scales near the electron Compton wavelength (tens of MeV), where multi-shell structures might leave measurable deviations in effective charge distributions. Alternatively, high-precision muon and tau 34 𝑔−2 experiments could reveal anomalous contributions from layered delay envelopes. APPENDIX I — Expansion-Driven Winds and Jets I.1 Solar Wind in the DCM Framework Observations reveal two distinct regimes of the solar wind: the fast polar streams (~600–800 km/s) from coronal holes and the more variable slow streams (~300–450 km/s) from the heliosphere current sheet and streamer belt. Within the Discrete Cosmology Model (DCM), both are understood through the same core mechanism: inner layers undergoing discrete expansion displace and delay the outer plasma layers. This displacement sets an effective basal pressure and modifies the expansion factor κ in the corona. Slow Wind (Main DCM Application) The slow wind shows enhanced FIP bias, higher charge states, and strong variability with longitude and solar cycle—signatures of delayed outer-layer release. In DCM, the expansion parameter 𝜅 and the basal pressure 𝑝0 are regulated by group delays between the deeper expanding layers and the corona. This leads naturally to asymptotic speeds of 300– 450 km/s. Simple slow wind scaling relation For the slow wind, a minimal DCM scaling law is 𝑣slow ≈ 𝜅exp 𝑣esc(𝑟eff), (I1) where 𝑟eff is the effective depth at which expansion delays couple to the outer plasma. This links the observed wind speed directly to the escape-velocity profile (Figure I.2) and provides a quantitative handle for comparison with data. Predictions: • Correlation between κ, charge states (O⁷⁺/O⁶⁺, Fe), and slow-wind velocity near streamer cusps. • Clear rotational modulation (27-day periodicities and harmonics) from group-delay phasing. • Solar-cycle dependence: stronger DCM signatures during maximum, when streamer belts dominate. • Latitudinal tapering: DCM effects weaken toward high latitudes where fast wind dominates. These predictions can be tested directly with Parker Solar Probe, Solar Orbiter, and archival Ulysses data. Fast Wind (Speculative Extension) The fast wind is treated here as a secondary hypothesis. While MHD/Alfvénic models provide a well-developed explanation, DCM suggests that fast wind signatures may reflect deeper effective coupling depths (reff ∼ 0.2−0.3 with κexp ≈ 0.5−0.8. A 35 distinctive prediction is that such flows should exhibit weaker first-ionization potential (FIP) bias than slow wind, due to their origin in deeper, less fractionated layers. The fast wind is conventionally explained by open-field MHD processes and Alfvénic turbulence. Within DCM, an additional lens emerges: if magnetic flux tubes are anchored in deeper layers where 𝐯esc(𝒓)≳ 𝟕𝟓𝟎 𝒌𝒎/𝒔 (fig. I.1), the displacement of plasma can channel stored potential energy into the corona more directly. This explains why the fast wind shows near-photospheric composition and low FIP bias: the outer layers contribute minimally, while deeper displacements dominate. Figure I.1: Escape velocity as a function of solar radius based on interior density estimates. In the DCM framework, the terminal speed of the slow wind corresponds to vesc(reff) scaled by κexp. For example, faster outflows (750–800 km/s) correspond to deeper couplings near 0.2−0.6R ⊙ . Thus, observed solar wind velocities can be interpreted as signatures of specific expansion-delay depths inside the Sun. Still, the Parker wind framework already explains fast-wind speeds for coronal temperatures of 1–2 MK, so DCM’s role here remains a constrained hypothesis. The primary, testable application remains the slow wind. Planetary Polar Winds (Analogy) The same displacement logic applies in planetary contexts: • Earth: the polar wind, where ionospheric plasma escapes along open geomagnetic field lines, is driven by pressure differences between deep thermospheric/ionospheric layers and the outer magnetosphere. • Mars and Venus: weaker magnetospheres still allow solar EUV-heated lower layers to displace ions outward, producing persistent polar outflows. These planetary winds mirror the Sun’s polar outflow: in all cases, inner layers expand and displace outer plasma along open field lines. Relativistic Winds and Jets At astrophysical extremes (pulsars, AGN, GRBs), the same principle scales upward: relativistic winds and jets may be understood as the most efficient form of inner-layer displacement, where rapid rotation and expansion expel outer plasma at near-light speeds. In this sense, the slow solar wind is the most testable local case, while relativistic jets are the ultimate manifestation of the same mechanism. 36 Falsifiability DCM predicts measurable correlations in the slow wind: κ–composition–velocity links, rotational modulation, and solar-cycle trends. If these signatures fail to appear, the DCM contribution must be rejected. Conversely, confirmation would connect the Sun’s slow wind, planetary polar winds, and astrophysical relativistic outflows under a single displacement principle. This duality highlights DCM’s unifying principle: discrete Compton-scale delays manifest as both microscopic particle structure and macroscopic astrophysical winds and jets.