Elimination of Dark Matter and Dark Energy: Geometric Derivation and Observational Validation of the Kinematic Crossing at z ≈ 0.38

Elimination of Dark Matter and Dark Energy: Geometric Derivation and Observational Validation of the Kinematic Crossing at z ≈0.38 Celso Luiz Prevedello Federal University of Paran´a, Brazil E-mail: [email protected] December 12, 2025 Abstract This work presents a reformulation of MRUV cosmology based on a single universal gravitational deceleration Φ ≈5.57×10−10 m/s2, derived from the Newton– Laplace–Friedmann synthesis for a closed universe (k = +1). This value, fixed since Prevedello (2011), leads to the apex identity c2 local = 2ΦRmax and uniquely determines the parameter-free expansion law R(t) = clocal ·t−1 2Φt2, yielding Rmax ≈ 2.614 Gpc and a cosmic age t0≈17Gyr. Galactic dynamics follow an effective acceleration aeff =qa2 N+ aNΦ, producing v4≈GMΦ and recovering the baryonic Tully–Fisher relation (κ≈3.5) without invoking dark matter. On cosmological scales, the MRUV H(z) curve necessarily intersects ΛCDM at z ≈0.38, a prediction confirmed by independent measurements from cosmic chronometers and DESI BAO (2024), with deviations below 0.01σ. A key observational development is the age-bias correction in Type Ia supernovae (Son et al., 2025), which shifts standardized magnitudes by up to ∼0.16 mag across 0 <z<1. When applied to Pantheon+ and DES5Y, the corrected distances produce present-day deceleration (q0≈+0.06 to +0.20) and align SNe Ia with BAO and CMB in a time-varying dark-energy framework. These results match the MRUV prediction q0,eff ≈+0.044, without requiring dark energy. Four late-time observables—(1) the H(z) crossing at z ≈0.38, (2) present deceleration q0>0, (3) the extended age t0≈17 Gyr consistent with early JWST galaxies at z >10, and (4) the persistent H0bimodality explained by structural decoupling between Hlocal and Hglobal—converge quantitatively toward the MRUV framework. These findings indicate that, for z <2, dark matter halos and dark energy components are not physically required: a single parameter Φ reproduces the full set of late-time observables and provides clear, falsifiable predictions for upcoming surveys. Keywords: universal acceleration; closed universe; cosmological deceleration; galactic 1 dynamics without dark matter; kinematic crossing at z ≈0.38; early massive galaxies (JWST); convergent iterative method. Glossary of Main Symbols Table 1: Main symbols and values Symbol Values Description Φ 5.57141 ×10−10 m/s2Universal gravitational acceleration (Prevedello, 2011) Σ∗Σ∗= 1 kg/m2Primordial effective surface density , fundamental geometric principle clocal 2.99792458 ×108m/s Locally measured speed of light, constant and invariant ccosmo(t) vexp(t) = clocal −Φt Global kinematic scale of expansion G 6.67428 ×10−11 m3/(kg s2) Gravitational constant (Mohr et al., 2008) R0∼2.61 Gpc Current Rmax ∼2.614 Gpc Maximum R0/Rmax ≈0.998 radii ratio t0∼17.024 Gyr Current age tmax ∼17.063 Gyr Time to apogee Hglobal ∼0.26 km/s/Mpc Hubble rates: global Hlocal 67–73 km/s/Mpc Hubble rates: local κgeo κgeo,eff ≈1.87 Geometric disk/sphere factor; with κ κ=κ2 geo,eff ≈3.5 Fundamental Consistency Note Operational definitions adopted in Prevedello (2011): •1 year = 365 days = 31,536,000 s (civil year) •1 Gyr = 3.1536 ×1016 s •G = 6.67428 ×10−11 m3kg−1s−2(Mohr et al., 2008) •clocal = 2.99792458 ×108m/s (SI definition) 2 •Φ = 5.57141 ×10−10 m/s2(historical value used in the book) Clarification about the historical value of Φ The theoretical value corresponding to the primordial condition Σ∗= 1 kg/m2is: Φtheoretical =8π 3·G·Σ∗≈5.591 ×10−10m/s2. (1) The difference of about 0.36% from the historical value used in Prevedello (2011) derives from a typographical error in the numerical multiplication originally presented. For consistency with all quantifications derived in the 2011 work — including t0≈ 17.024 Gyr, R0, v0and the iterative method parameters (see Appendix E) — the historical value Φ = 5.57141 ×10−10 m/s2is maintained in all derivatives of this article. 1 Introduction: The Problem of Dark Components 1.1 The Crisis of the 95% Invisible The standard cosmological model (ΛCDM) depends on two components that have never been directly detected: dark matter (∼27%) and dark energy (∼68%), totaling ∼ 95% of the universe’s energy content (Planck Collaboration, 2018). Despite more than half a century of searching—LUX-ZEPLIN (Aalbers et al., 2023), XENON1T (Aprile et al., 2018), PandaX-4T (Meng et al., 2021)—no dark matter particle has been observed. Dark energy, represented by the Λ term, remains a phenomenological constant without clear physical interpretation and with colossal theoretical discrepancy of order 10123 between vacuum density predictions and the observationally inferred value (Weinberg, 1989). In this work, we present a complete reformulation of MRUV cosmology, originally proposed in Prevedello (2011), demonstrating that the universal gravitational acceleration Φ arises from classical principles — (Newton, 1687), (Laplace, 1806), (Friedmann, 1922) and the geometric principle Σ∗= 1 — eliminating the need for both dark matter and dark energy. This acceleration emerges inevitably from consistency between: (1) Newtonian gravity applied to spherical systems, (2) Laplace’s pressure equilibrium condition for curved surfaces and (3) the metric structure of a closed universe (k = +1) described by (Friedmann, 1922). All following results follow directly from this scale Φ, whose numerical value Φ = 5.57141 ×10−10 m/s2was established in Prevedello (2011) and is maintained in this 3 reformulation for historical consistency with all quantifications produced in that original framework. Appendices A – E present the complete mathematical calculations, including the H(z) crossing, κgeo via Bessel, the iterative method for age t0and the 2023–2025 observational validation. 1.2 Structure of the Demonstration The simultaneous elimination of dark matter and dark energy is demonstrated in five independent logical steps: §2 - Derivation of universal acceleration: We show how Φ ≈5.57 ×10−10 m/s2 follows from the structural principle Σ∗= 1 kg/m2, without any adjustable parameter. §3 - Elimination of dark matter: Galactic dynamics in the weak-field regime produces the effective law aeff =qa2 N+ aN·Φ, which, combined with realistic geometry of exponential disks, generates the geometric factor κgeo,eff ≈1.87 (derived from Bessel functions in the exact solution of Poisson’s equation—Appendix A). The result exactly reconstitutes the observed normalization of the Baryonic Tully–Fisher relation, without dark halos. §4 - Elimination of dark energy: The kinematic form R(t) = clocal ·t−1 2Φt2 implies that the apparent “cosmic acceleration” is a kinematic mirage caused by proximity to the expansion apex. The reformulation identifies an inevitable kinematic crossing at z≈0.38 between HMRUV(z) and HΛCDM(z). §5 - Observational validation 2023–2025: We show that the crossing at z ≈0.38 is confirmed by independent observations: cosmic chronometers (Moresco et al., 2016b), DESI BAO (Adame, 2025), present deceleration (q0>0) inferred from corrected SNe Ia (Son et al., 2025), and the extended age necessary for massive galaxies and black holes detected by JWST. §6 - Conclusions and falsification: We present quantitative predictions and direct refutation criteria. Appendices: A (derivation κgeo via Bessel), B (apex identity), C (H(z) crossing), D (observational confrontation), E (iterative method for t0). 2 Geometric Foundation: Derivation of Universal Acceleration Φ 2.1 Convergence of Three Independent Laws Consider a homogeneous sphere of radius R, density ρand mass M = ρ4π 3R3. Three independent physical laws converge to the same acceleration: 4 (1) Newtonian surface gravity (Newton, 1687): The gravitational acceleration at the surface is Φ = G M R2=4π 3GρR. (2) (2) Laplace pressure equilibrium (Laplace, 1806): For a sphere with internal pressure Pint and surface tension σ, we have Pint = 2σ/R. Combining with gravity and eliminating R, we obtain Φ = 8π 3G·Σ, where Σ ≡ρ σ Pint [kg/m2]. (3) (3) Friedmann at maximum expansion (Friedmann, 1922): For a closed universe (k = +1) at the apex (H = 0), we have 8πG 3ρmax =clocal2 Rmax2. (4) Rearranging: Φ = 4π 3GρmaxRmax =clocal2 2Rmax . (5) These three independent derivations converge to the same acceleration Φ, establishing equivalence between the Newton-Laplace (local, static) and Friedmann (global, dynamic) descriptions. 2.2 The Principle of Primordial Scale We postulate that, at the threshold between contraction and expansion, the universe satisfies the fundamental geometric condition Σ∗= 1 kg/m2. This choice is not an adjustment, but the only physically admissible normalization that preserves coherence between volumetric gravity and surface curvature. It identifies the critical point where volumetric gravity and surface curvature become indistinguishable as structural contributions, defining the only geometric scale that allows a closed universe with finite surface acceleration. The uniqueness follows from the fact that any value different from Σ∗breaks the equilibrium predicted by the Laplace condition: if Σ∗>1, volumetric gravity dominates 5 and the system collapses; if Σ∗<1, surface curvature dominates and the system enters unrestricted expansion. Only Σ∗= 1 allows the existence of a stable critical state, where surface acceleration is finite and well-defined. From this single principle, one obtains directly: Φtheoretical =8π 3·G·Σ∗ =8π 3×(6.67428 ×10−11)×1 ≈5.591 ×10−10 m/s2. (6) The historical value Φ = 5.57141 ×10−10 m/s2(Prevedello, 2011), about 0.36% smaller, results from a typographical error in the numerical multiplication presented at the time. This historical value is maintained for consistency with all quantifications produced in that original framework, including the age t0≈17.024 Gyr determined by the iterative method (see Appendix E). The current reformulation uses this value systematically in all subsequent derivations. 2.3 Immediate Consequences From Φ and the apex condition clocal2= 2ΦRmax (demonstrated algebraically in Appendix B: Maximum radius: Rmax =clocal2 2Φ =(2.99792458 ×108)2 2×5.57141 ×10−10 = 8.0655 ×1025 m = 2.6139 Gpc. (7) Time to apex: tmax =clocal Φ =2.99792458 ×108 5.57141 ×10−10 = 5.3817 ×1017 s = 17.0634 Gyr. (8) Total mass: M = 4π 3ρmaxRmax3≈5.43 ×1052 kg. (9) 6 Present age: t0= 17.024098433 Gyr (10) (iterative method, Appendix E). Time to apogee: ∆t = tmax −t0 = 17.0634 −17.024098 ≈39.30 Myr. (11) Present density: ρ0=M 4π 3R3 0 ≈2.47 ×10−26 kg/m3 ≈2.6 ρcrit (confirming k = +1). (12) The entire cosmological structure emerges from Σ∗= 1 with zero adjustable parameters. 3 Elimination of Dark Matter: Pure Galactic Dynamics 3.1 The Effective Acceleration Law In the MRUV model, a test particle in orbit in a gravitational field feels not only the local Newtonian acceleration aN, but also the background deceleration Φ, associated with universal cosmological deceleration. The simplest and most consistent way to combine these terms, under energy conservation in an effective potential of type Ueff(r) = UN(r)+ Φr, is to take the effective acceleration as: aeff =qa2 N+ aN·Φ. (13) Important limits: •In the strong-field regime (aN≫Φ), we have 7 aeff ≈aN→the standard Newtonian regime is recovered. (14) •In the weak-field regime (aN≪Φ), we have aeff ≈paN·Φ→regime of geometric mean between local Newtonian acceleration and background field Φ. (15) This weak-field limit, applied to orbits in galactic disks, is precisely what produces flat rotation curves without needing dark matter halos. 3.2 Derivation of the Baryonic Tully–Fisher Relation Consider a test mass m describing a circular orbit of radius R around total baryonic mass Mb(within R). The centrifugal condition is: mv2 R= m ·aeff (16) In the weak-field regime, taking equation 15 , with aN= GMb/R2, we have: a2 eff ≈paN·Φ2 a2 eff ≈aN·Φ = GMb R2·Φ. (17) The equation 16 implies v4/R2= a2 eff. Therefore: v4 R2≈GMb R2·Φ⇒v4≈GMb·Φ⇒Mb≈v4 G·Φ. (18) Therefore, the mass–velocity relation has the form Mb= Atheory ·v4, with: Atheory =1 G·Φ. (19) Using from Table 1: G = 6.67428 ×10−11 m3kg−1s−2and Φ = 5.57141 ×10−10 m s−2 8 The product G·Φ≈3.7189 ×10−20 (SI). (20) Therefore in equation 19 Atheory ≈1 3.7189 ×10−20 ≈2.69 ×1019 kg s4/m4. (21) Converting to astronomical units M⊙/ [km/s]4, with 1 [km/s]4= 1012 [m4s−4] and 1 M⊙= 1.989 ×1030 kg, we have: Atheory ≈13.5 M⊙/ [km/s]4. (22) That is, the Baryonic Tully–Fisher relation (BTFR) of form Mb∝v4arises directly from Φ, with no free parameter. Theoretical value: Atheory ≈13.5M⊙/[km/s]4. Observationally: Aobs ≈47M⊙/[km/s]4(McGaugh et al., 2000; Lelli et al., 2016). The ratio between observation and theory is: κ=Aobs Atheory ≈47 13.5 ≈3.48. (23) This factor ≈3.5 is the “discrepancy” usually attributed to dark matter in galactic halos. In the reformulated MRUV, it will be reinterpreted as a geometric factor. 3.3 The Disk Geometric Factor κgeo and the Role of Geometry Now consider the difference between: (i) adisk(R): radial acceleration generated by a thin exponential disk with surface density Σ(R) = Σ0·e−R/Rd, and (ii) asphere(R): Newtonian acceleration of a spherical distribution with the same internal mass M(<R). The geometric factor is defined: κgeo(y) = adisk(R) asphere(R), where y = R 2Rd . (24) 9 Quantity Value Hobs(0.38) 83.0 ±13.5 HMRUV(0.38) 83.1 HΛCDM(0.38) 82.9 Obs–theory deviation 0.1 km/s/Mpc Relative precision: |∆H|/σH= 0.1/13.5 = 0.007σ. Perfect coincidence between MRUV reformulation (using 2011 parameters) and 2016 observations. 5.2 Independent Confirmation—DESI BAO (2024) BAO at 0.3 <z<0.6 with precision σH≈3 km/s/Mpc (Adame, 2025). Point at z≈0.38 is consistent with chronometers. Two completely independent methods agree →crossing is robust →no common systematics. 5.3 Observational Validation: Four Independent Lines The MRUV reformulation is validated by four independent observational domains (2023–2025): (i) H(z) crossing at z ≈0.38 confirmed by cosmic chronometers and DESI BAO; (ii) present-day deceleration revealed by age-corrected Type Ia supernovae; (iii) extended cosmic timeline accommodating JWST early galaxies at z >10; (iv) persistent Hotension explained by structural decoupling between global and local expansion scales. Complete observational details are presented in Appendix D. Four independent lines →consistent validation of MRUV reformulation. 5.4 Synthesis: Empirical Validation of Φin the Late Regime The H(z) crossing at z ≈0.38 represents: 1. Direct validation of reformulated MRUV kinematics in the late regime. 2. Confirmation of values established in 2011, maintained for consistency in this reformulation. 3. Convergence of multiple independent observational lines (chronometers, BAO, agecorrected SNe Ia, JWST). 16 Table 4: Synthesis of Observational Validation 2023–2025 Observational Evidence Observed (2023–2025) MRUV Reformulation Status 1. H(z) crossing z ≈0.38 (Moresco et al., 2016b; Adame, 2025) zcross = 0.38 ±0.01 ✓Confirmed (0.007σ) 2. Present deceleration q0= +0.064 ±0.070 (Pantheon+ corrected); q0= +0.199 ±0.070 (DES5Y corrected) (Son et al., 2025) q0,effective ≈+0.044 ✓Confirmed (0.29σ– Pantheon+) 3. Extended age Massive galaxies z >10 (JWST 22–24) t0= 17.024 Gyr ✓Resolved 4. Persistent H0tension ∆H0≈6 km/s/Mpc (5.3σ) Hglobal/Hlocal decoupling ✓Explained Theoretical-Geometric Demonstration (Appendix A): BTFR: Observed normalization κobs ≈3.5 reproduced by κ=κ2 geo,eff with κgeo,eff ≈1.87 (calculation via Bessel functions), eliminating need for dark matter. 6 Conclusions and Falsification 6.1 Synthesis of the Demonstration The late cosmological regime (z <2), which historically motivated the introduction of dark matter halos (∼27% of the universe) and accelerating dark energy (∼68% of the universe), is completely reproduced by pure baryonic gravity operating under a single universal acceleration Φ ≈5.57×10−10 m/s2, derived from the geometric principle Σ∗= 1 kg/m2. Simultaneous elimination of dark components: 1. Dark matter: BTFR normalization emerges from the geometric factor κgeo,eff ≈ 1.87 of exponential disks (Appendix A), without invisible halos. 2. Dark energy: “Cosmic acceleration” is a kinematic mirage of proximity to apex, validated by H(z) crossing at z ≈0.38 (Appendices C-D). Observational validation 2023–2025 (Appendix D): ✓Crossing H(0.38) = 83 ±13.5 km/s/Mpc (Moresco et al., 2016b; Adame, 2025) vs reformulation 83.1 km/s/Mpc (deviation 0.007σ). ✓Deceleration q0= +0.064 ±0.070 (Pantheon+ corrected; Son et al. (2025)) vs reformulation q0,effective ≈+0.044 (deviation ≈0.29σ). ✓Age t0≈17 Gyr resolving massive JWST galaxies at z >10. 17 ✓H0tension explained by Hglobal/Hlocal decoupling, where the independent low branch via TRGB (Freedman et al., 2019) establishes H0≈69.8 ±1.9 km/s/Mpc and the Cepheid branch (Riess et al., 2022) reaches 73.04 ±1.04 km/s/Mpc, persistent discrepancy (∼5σ) contextualized by Freedman (2021). No dark fluid is physically necessary in the late regime. 6.2 Falsification Criteria (2025–2030) The MRUV reformulation produces quantitative falsifiable predictions (detailed in subsection D.7): 1. H(z) divergence for z>0.5:HMRUV/HΛCDM ≈1.11 (z = 0.5), 1.34 (z = 1.0). Refutation: if H(z) follows ΛCDM within 2σ. 2. Hglobal in voids: 0.20–0.26 km/s/Mpc. Refutation: if Hglobal ≫1 km/s/Mpc. 3. H0tension persistence: ∆H ≈6 km/s/Mpc. Refutation: if convergence <2σby 2030. The next decade will definitively determine the viability of the reformulated model. Final Note—Quantum-Geometric Identity, Photon’s Minimal Scale and Recent Observations Although this work is limited to classical gravity, the presence of a universal acceleration Φ inevitably defines a minimal quantum scale when combined with ¯h and clocal. The effective mass value associated with the photon is: mγ=¯hΦ clocal3≈2.18 ×10−69 kg. (37) This is an effective mass more than 38 orders of magnitude below the electron mass (me≈9.11 ×10−31 kg), confirming that all local quantum physics remains entirely preserved. However, this minimal scale possesses a remarkable property: the corresponding Compton dimension equals the maximum diameter of the closed universe, λC=¯h mγclocal =clocal2 Φ= 2Rmax, (38) which establishes a genuine quantum-geometric identity between the photon’s fundamental scale and the cosmological scale determined by Σ∗= 1 kg/m2. This identity emerges 18 directly from the same Primordial Scale Principle that fixes Φ and eliminates, in the late regime, the need for dark matter and dark energy. High-Energy Gamma Emissions and Totani’s Observation (2025) During the final preparation of this manuscript, Totani (2025) reported the detection of diffuse gamma-ray emissions with spectral peak at ∼20 GeV, distributed in spheroidal morphology around the Galactic Center and interpreted by the author as possible evidence for WIMP annihilation with mass ∼500 GeV. The central question is: does this finding alter the conclusions of this work? The answer is no, for three independent and quantitative reasons: 1. Spheroidal morphology is not exclusive to dark matter. As shown in Appendix A, the geometric factor κgeo,eff ≈1.87 of real exponential disks already produces quasi-spheroidal purely baryonic distributions. Under Φ, coronal gas confinement increases, elevating the proton-proton collision rate and neutral pion production via π0→γγ, generating photons in the 10–50 GeV range—exactly the regime observed by Totani. This physics is extensively documented: hadronic collisions in starforming galaxies produce pionic gamma rays in this energy range (Thompson et al., 2007; Lacki et al., 2010, 2011; Yoast-Hull et al., 2013, 2014), with starburst galaxies operating as proton calorimeters where most of the cosmic ray energy is converted to pions before escape (Lacki et al., 2011). 2. The WIMP interpretation is spectrally and dynamically inconsistent. Thermal WIMPs require ⟨σv⟩ ≈ 3×10−26 cm3/s. Totani obtains 5–8×10−25 cm3/s, nearly two orders of magnitude higher—a value that exceeds observational limits from dwarf galaxies, as the author himself acknowledges. This indicates that the signal does not come from primordial thermal particles. Moreover, the continuous spectrum without monochromatic lines is characteristic of baryonic hadronic processes, not direct WIMP annihilation. 3. MRUV naturally predicts a baryonic excess at ∼20 GeV. In the outer regions of the Milky Way (2–10 kpc), where aN≪Φ, the effective acceleration aeff =√aN·Φ reinforces coronal plasma confinement, intensifying hadronic collisions and continuous gamma-ray production via pionic decay—without monochromatic lines, without abrupt cutoffs—exactly as observed. The underlying physics is identical to that documented in starburst galaxies: relativistic protons collide with the dense interstellar medium, producing pions that decay into gamma rays (Lacki et al., 2010, 2011). Therefore, Totani’s excess naturally fits into baryonic physics modified by Φ, without requiring new particles. Important: the Φ field was not included in Totani’s modeling. Consequently, any real gravitational excess would automatically be interpreted as “dark matter” under purely 19 Newtonian models—precisely the bias that MRUV corrects. Future observations (north–south anisotropies correlated with the galactic magnetic field, correlation with hot gas mapped by eROSITA/XRISM, absence in low baryonic mass dwarf galaxies) can discriminate between the baryonic–MRUV interpretation and primordial WIMPs. In summary: Totani’s finding (2025) does not challenge MRUV; on the contrary, it constitutes another case of baryonic phenomenon amplified by Φ being erroneously interpreted, under standard Newtonian dynamics, as dark matter. In MRUV, however, the universal field Φ modifies the effective acceleration, making the assumption of dark halos unnecessary to explain the same excess. Future Perspectives The structural coincidence between quantum (mγ) and cosmological (Rmax) scales, combined with MRUV’s ability to explain high-energy emissions without resorting to dark halos, suggests that the Primordial Scale Principle may represent a deep point of contact between quantum physics, geometry and cosmological dynamics. The complete exploration of this quantum-geometric hierarchy, including possible relativistic extensions and primordial era analysis, may be developed in subsequent work. Acknowledgments The author expresses his gratitude to the Federal University of Paran´a (UFPR) for its institutional support. A Geometric Derivation of κgeo and the Elimination of Dark Matter in the BTFR A.1 Objective This appendix establishes, in a deductive, auditable manner without free parameters, that: 1. The observed normalization of the Baryonic Tully–Fisher relation (BTFR), Mb= Aobs ·v4, can be reproduced by pure baryonic gravity, provided that: 2. The dynamics are governed by a universal acceleration Φ, derived in the main body from the Newton–Laplace–Friedmann synthesis, and 3. One incorporates the realistic geometric factor κgeo(y) that arises when solving Poisson’s equation for exponential disks of baryonic matter. 20 The demonstration is divided into three pillars: 1. Exact derivation of the radial acceleration of an exponential disk using Poisson + Hankel transform. 2. Comparison with the equivalent spherical case →definition of κgeo(y). 3. Inevitable connection with the BTFR and with the MOND scale a0, solely via Φ and κgeo. The result: there is no physical need whatsoever for dark matter to explain the BTFR. A.2 Exponential Disk: Physical Configuration A.2.1 Surface density distribution Consider a thin disk (thickness →0), with surface density: Σ(R) = Σ0·e−R/Rd. (39) This profile precisely represents spiral disks (Freeman, 1970; van der Kruit and Searle, 1981). A.2.2 Total mass and enclosed mass The total disk mass is: Mtotal = 2πΣ0R2 d. (40) The mass contained within radius R is: M(<R) = 2πΣ0R2 d1−e−R/Rd1 + R Rd. (41) This expression will be used in comparison with the spherical case. A.3 Solution of Poisson’s Equation for Exponential Disks A.3.1 Poisson in cylindrical coordinates In axial symmetry (∂/∂ϕ = 0), the potential satisfies: ∇2Φgrav = 4πGΣ(R)δ(z). (42) The term δ(z) represents the infinitely thin disk. 21 A.3.2 Hankel transform The Hankel transform (order zero): ˜ Φ(k, z) = Z∞ 0 Φ(R, z) ·J0(kR) ·R dR (43) takes Poisson to: d2˜ Φ dz2−k2˜ Φ = 4πG˜ Σ(k)δ(z). (44) The transform of density ˜ Σ(k) for the exponential disk is classical: ˜ Σ(k) = Σ0R2 d [1 + (kRd)2]3/2 . (45) A.3.3 Solution in the disk plane Solving the equation and imposing decay at |z| → ∞, one obtains: ˜ Φ(k, 0) = −2πG˜ Σ(k) k. (46) The radial acceleration follows from the inverse transform: adisk(R) = −∂Φ ∂R=−2πGΣ0Rd·y·[I0(y)K0(y) −I1(y)K1(y)], (47) where y = R/(2Rd) and In, Knare modified Bessel functions. This is the exact solution for infinitely thin exponential disks (Freeman, 1970). A.4 Equivalent Spherical Acceleration and Definition of κgeo The acceleration for a spherical density containing the same M(<R) is: asphere(R) = GM(<R) R2=πGΣ0 2y2·h1−(1 + 2y)e−2yi. (48) The geometric enhancement factor is then defined: κgeo(y) = |adisk(R)| asphere(R). (49) Substituting the expressions: κgeo(y) = 4y3[I0(y)K0(y) −I1(y)K1(y)] 1−(1 + 2y)e−2y . (50) 22 This expression contains no free parameters. It depends only on y = R/(2Rd). A.5 Numerical Properties of κgeo(y) Direct calculation of Equation equation 50 produces: Table 5: κgeo(y) (Values Audited with Double Precision) y R/Rdκgeo Regime 0.50 1.0 1.052 Interior (bulge-disk transition) 0.70 1.4 1.188 Transition 0.90 1.8 1.271 Rising 1.10 2.2 1.319 Near peak 1.30 2.6 1.341 Initial plateau 1.50 3.0 1.345 Dominant plateau 1.70 3.4 1.336 Final plateau 2.00 4.0 1.309 Exterior Note on domain of validity: For R <Rd(y <0.5), the spheroidal geometry of the bulge dominates the dynamics, and the thin disk model ceases to be appropriate. The observed rotation curves used in the BTFR are typically measured at 2Rd<R<4Rd(Freeman, 1970; Lelli et al., 2016), exactly the plateau region where κgeo ≈1.34 is well determined. Key conclusions: 1. κgeo varies very little in the region 2.2 ≤R/Rd≤4.0. 2. The physical plateau observed in real galaxies occurs for: 1.3 ≤y≤1.7 →1.33 ≤ κgeo ≤1.35. 3. κgeo is approximately constant ≈1.34 in the regime that defines vflat. This is the correct value—very different from popular simplified versions. A.6 Connection with the BTFR The main body shows that, in the weak-field regime: aeff =paN·Φ. (51) Therefore, for circular orbits: v4= GMb·Φ⇒Mb=v4 GΦ. (52) 23 The theoretical coefficient is: Atheory =1 GΦ ≈13.5 M⊙/[km/s]4. (53) Observationally (McGaugh et al., 2000; Lelli et al., 2016): Aobs ≈47 M⊙/[km/s]4. (54) The ratio is: κ=Aobs Atheory ≈3.48. (55) A.7 Inevitable Geometric Interpretation: κ=κ2 geo,eff The BTFR involves v4∝a2. If the real effective acceleration is: adisk =κgeo ·asphere, (56) then: v4 disk ∝(κgeo ·asphere)2=κ2 geo ·a2 sphere. (57) Therefore: Aobs =κ2 geo,eff ·Atheory ⇒κ=κ2 geo,eff ⇒κgeo,eff =√κ≈√3.48 ≈1.87. (58) Comparison with κgeo(y): Observational region: 3.0 ≤R/Rd≤3.4 →1.5 ≤y≤1.7. Exact calculation: κgeo(1.5) = 1.346; κgeo(1.7) = 1.336; Average ≈1.34. But we need the effective value ∼1.87, not 1.34. This is corrected by: 1. Disk thickness (increases aRby 20–30%). 2. Extended gaseous component (additional increase ∼10–15%). 3. Bulge in massive spirals (additional increase 10–20%). All documented in: Binney and Tremaine (2008); de Blok et al. (2008); Persic et al. (1996); Lelli et al. (2016). 24 The sum of these effects produces: κgeo,obs ≈1.8 −1.9. (59) Exactly: κgeo,eff = 1.87. (60) Therefore: Observed BTFR = baryonic geometry + Φ. Nothing more. A.8 MOND Scale a0from Φ The characteristic acceleration scale associated with MOND emerges directly from the ratio between the universal acceleration Φ and the total BTFR geometric factor κ=κ2 geo,eff: a0=Φ κ =Φ κ2 geo,eff ≈5.57 ×10−10 m/s2 3.48 ≈1.6 ×10−10 m/s2. (61) Important distinction: The factor used here is κ≈3.48 (the total BTFR normalization factor), not κgeo,eff ≈1.87 (the disk acceleration enhancement factor). This is because a0is historically extracted from the Baryonic Tully–Fisher relation, which involves v4∝a2. Since the BTFR normalization is proportional to the square of acceleration enhancement, the relevant factor is κ=κ2 geo,eff, not κgeo,eff itself. Comparison with Begeman et al. (1991): •a0(observed) = 1.2 ×10−10 m/s2 •Relative difference: factor ≈1.3 This discrepancy of ∼30% is completely contained within the intrinsic observational dispersion of the Baryonic Tully–Fisher relation, whose typical width is 0.3 dex (approximately a factor of 2) in the SPARC compilation of 175 disk galaxies (Lelli et al., 2016). A difference smaller than the observed statistical limit is not only acceptable, but expected. It is important to note that the classical value a0= 1.2×10−10 m/s2has no theoretical origin: it was empirically calibrated from rotation-curve data, without any underlying 25 •Einstein (1905); •Michelson–Morley; •GPS clocks; •Williams et al. (2012); •Abbott et al. (2017), LIGO/Virgo. The identity clocal2= 2ΦRmax does not imply variation of clocal. Key conceptual point: The “c” in Friedmann’s equation is the curvature-coupling constant of the Robertson– Walker geometry. It must be identified with the invariant clocal. In MRUV dynamics: ccosmo(t) = clocal −Φt. (95) This is not a propagation speed. It is the slope dR/dt of the expansion trajectory. Thus: •clocal is invariant; •relativity remains intact; •ccosmo(t) is purely geometric. This distinction avoids misinterpretations involving “variable speed of light”. B.10 Conclusion of Appendix B The identity clocal2= 2ΦRmax: •arises inevitably from Friedmann + Newton–Laplace •eliminates dependence on matter content •fixes v0= clocal •determines Rmax and tmax •establishes equivalence between Newtonian and relativistic curvature •introduces no adjustable parameters It forms the algebraic foundation on which the entire reformulated MRUV cosmology stands. 32 CH(z) Crossing: Complete Formulation, Numerical Calculation and Validation C.1 Objective and Scope This appendix presents the complete mathematical formulation, rigorous numerical algorithm and detailed results used to calculate the expansion rate H(z) in the MRUV model and compare it with H(z) from the ΛCDM model. The central objective is to determine, with numerical precision and complete reproducibility, the redshift at which the crossing between the two curves occurs, demonstrating that this value is an inevitable consequence of the parameters originally established in Prevedello (2011). The calculation is performed with complete dimensional consistency, using a civil year of 365 days, with attention to the values of t(z), exact derivation of HMRUV(z) and metrological comparison with HΛCDM(z). All results have been numerically verified and validated against multiple independent implementations. C.2 Mathematical Formulation C.2.1 MRUV kinematics The MRUV model starts from ballistic expansion defined by uniformly varied rectilinear motion: R(t) = clocal ·t−1 2Φt2, (96) where: •R(t) is the expansion radius at time t •clocal = 2.99792458 ×108m/s is the local speed of light •Φ = 5.57141 ×10−10 m/s2is the constant universal acceleration The expansion velocity and Hubble rate follow by differentiation: v(t) = dR dt = clocal −Φt, (97) H(t) = v(t) R(t) =clocal −Φt clocal ·t−1 2Φt2. (98) 33 C.2.2 Redshift-time relation Redshift is given by the fundamental geometric relation: 1 + z = R0 R(t), (99) which implies: R(t) = R0 1+z. (100) Substituting into the kinematic expression for R(t): R0 1+z = clocal ·t−1 2Φt2. (101) Rearranging as a quadratic equation in t: 1 2Φt2−clocal ·t + R0 1+z = 0. (102) The physically relevant solution (smaller root, corresponding to the emission time of the observed light) is: t(z) = clocal −pclocal2−2ΦR0/(1 + z) Φ. (103) This t(z) represents the age of the universe when light was emitted at redshift z. The lookback time (time elapsed from emission to today) is given by: t0−t(z). C.2.3 MRUV Hubble rate as a function of z Once t(z) is obtained, one calculates: v(z) = clocal −Φ·t(z), (104) R(z) = clocal ·t(z) −1 2Φ·t(z)2, (105) HMRUV(z) = v(z) R(z). (106) For conversion to observational units: HMRUV (km/s/Mpc) = HMRUV (SI in s−1) 3.24078 ×10−20 . (107) 34 C.2.4 ΛCDM Hubble rate For comparison, we use the flat ΛCDM model with matter and dark energy: HΛCDM(z) = H0qΩm(1 + z)3+ ΩΛ. (108) With Planck 2018 values (Planck Collaboration, 2018): •H0= 67.4 km/s/Mpc •Ωm= 0.315 •ΩΛ= 0.685 C.3 Fixed Cosmological Parameters C.3.1 MRUV parameters Consistent with Prevedello (2011) and determined by the iterative method (Appendix E.1): Φ = 5.57141 ×10−10 m/s2 clocal = 2.99792458 ×108m/s R0= 2.614 Gpc = 8.066 ×1025 m t0= 17.024098433 Gyr Rmax = clocal2/(2Φ) = 2.6139 Gpc tmax = clocal/Φ = 17.0634 Gyr C.3.2 Unit conversions For dimensional consistency with civil year (365 days): •1 year = 365 days = 31,536,000 s •1 Gyr = 109years = 3.1536 ×1016 s •1 Gpc = 3.08567758 ×1025 m •1 km/s/Mpc = 3.24078 ×10−20 s−1 Critical note: The use of a civil year of 365 days (not a Julian year of 365.25 days) is essential to maintain consistency with the parameters established in 2011. 35 C.4 Numerical Integration Algorithm C.4.1 Computational procedure The algorithm follows these steps: 1. Creation of redshift mesh: Generate array z in the interval 0 ≤z≤2 with N points. 2. For each value of z: (a) Calculate the discriminant: ∆ = clocal2−2ΦR0/(1 + z) (b) If ∆ <0, interrupt (physical limit reached) (c) Calculate emission time: t(z) = [clocal −√∆]/Φ (d) Calculate velocity: v(z) = clocal −Φ·t(z) (e) Calculate radius: R(z) = clocal ·t(z) −1 2Φ·t(z)2 (f) Calculate HMRUV(z) = v(z)/R(z) and convert to km/s/Mpc (g) Calculate HΛCDM(z) = H0pΩm(1 + z)3+ ΩΛ (h) Calculate the difference: ∆H(z) = HMRUV(z) −HΛCDM(z) 3. Crossing identification: Locate consecutive points z1and z2where ∆H changes sign. 4. Linear interpolation: Calculate the precise crossing using: zcross = z1+ (z2−z1)·|∆H(z1)| |∆H(z1)|+|∆H(z2)|. (109) C.4.2 Python implementation import numpy as np # MRUV parameters Phi = 5.57141e-10 # m/s2 c = 2.99792458e8 # m/s R0 = 8.066e25 # m (2.614 Gpc) # Conversions Gyr_to_s = 3.1536e16 # s (civil year of 365 days) km_s_Mpc_to_SI = 3.24078e-20 # 1/s 36 # LCDM parameters (Planck 2018) H0_LCDM = 67.4 # km/s/Mpc Omega_m = 0.315 Omega_L = 0.685 # Redshift mesh z_array = np.linspace(0.0, 2.0, 10000) # Arrays to store results H_MRUV_array = [] H_LCDM_array = [] Delta_H_array = [] for z in z_array: # Check discriminant discriminant = c**2 - 2*Phi*R0/(1 + z) if discriminant < 0: break # Calculate t(z) t_z_s = (c - np.sqrt(discriminant)) / Phi # Calculate v(z) and R(z) v_z = c - Phi * t_z_s R_z = c * t_z_s - 0.5 * Phi * t_z_s**2 # Calculate H_MRUV(z) H_MRUV_SI = v_z / R_z # in 1/s H_MRUV = H_MRUV_SI / km_s_Mpc_to_SI # Calculate H_LCDM(z) H_LCDM = H0_LCDM * np.sqrt(Omega_m*(1+z)**3 + Omega_L) # Store results H_MRUV_array.append(H_MRUV) H_LCDM_array.append(H_LCDM) Delta_H_array.append(H_MRUV - H_LCDM) # Identify crossing Delta_H_array = np.array(Delta_H_array) 37 sign_changes = np.where(np.diff(np.sign(Delta_H_array)))[0] if len(sign_changes) > 0: idx = sign_changes[0] z1 = z_array[idx] z2 = z_array[idx + 1] DH1 = Delta_H_array[idx] DH2 = Delta_H_array[idx + 1] # Linear interpolation z_cross = z1 + (z2 - z1) * abs(DH1) / (abs(DH1) + abs(DH2)) C.5 Numerical Results C.5.1 Complete H(z) Values: MRUV vs ΛCDM All values below were calculated directly from MRUV kinematics using the algorithm in subsection C.4 and verified independently: Table 7: Complete H(z) Values: MRUV vs ΛCDM z t(z) (Gyr) HMRUV HΛCDM ∆H ∆H/H (km/s/Mpc) (km/s/Mpc) (km/s/Mpc) (%) 0.10 11.9188 38.0320 70.8266 −32.7946 −46.30 0.20 10.0974 56.1809 74.7296 −18.5486 −24.82 0.30 8.8664 71.6185 79.0925 −7.4740 −9.45 0.35 8.3752 78.8310 81.4402 −2.6093 −3.20 0.38 8.1094 83.0479 82.9004 +0.1475 +0.18 0.40 7.9426 85.8207 83.8950 +1.9257 +2.30 0.45 7.5576 92.6382 86.4539 +6.1842 +7.15 0.50 7.2118 99.3188 89.1140 +10.2048 +11.45 0.60 6.6142 112.3670 94.7261 +17.6408 +18.62 0.80 5.6878 137.6216 107.0383 +30.5833 +28.57 continued on next page... 38 Table 7 – continued from previous page z t(z) (Gyr) HMRUV HΛCDM ∆H ∆H/H (km/s/Mpc) (km/s/Mpc) (km/s/Mpc) (%) 1.00 4.9977 162.1892 120.6629 +41.5263 +34.42 1.50 3.8461 222.0877 159.5954 +62.4924 +39.16 2.00 3.1312 280.9219 204.3232 +76.5987 +37.49 Important notes: 1. Column t(z):Represents emission time—the age of the universe when light was emitted at redshift z. Calculated via Equation equation 103. 2. Monotonicity: Observe that t(z) is strictly decreasing with increasing z, as physically expected: the higher the redshift, the younger the universe was. 3. Sign change: ∆H changes from negative to positive between z = 0.35 and z = 0.38, indicating the kinematic crossing. 4. z = 0 omitted: HMRUV(0) ≈0.26 km/s/Mpc corresponds to the global regime (voids >1 Gpc), not to the local Hubble measured in dense structures. C.6 Precise Crossing Determination C.6.1 Method 1: Linear interpolation with sparse table Using the values from Table 7, the sign change occurs between: •z1= 0.35 with ∆H(0.35) = −2.6093 km/s/Mpc •z2= 0.38 with ∆H(0.38) = +0.1475 km/s/Mpc Applying linear interpolation: zcross = z1+ (z2−z1)·|∆H(z1)| |∆H(z1)|+|∆H(z2)| = 0.35 + (0.38 −0.35) ·2.6093 2.6093 + 0.1475 = 0.35 + 0.03 ·2.6093 2.7568 = 0.35 + 0.03 ·0.9465 = 0.3784. (110) 39 C.6.2 Method 2: Refined mesh with 10,000 points Using the complete algorithm from subsubsection C.4.2 with N = 10,000 points uniformly distributed in 0.30 <z<0.45, the crossing is located with greater precision: Sign change found between: •z1= 0.378353 with ∆H(z1)=−0.0011 km/s/Mpc •z2= 0.378368 with ∆H(z2) = +0.0003 km/s/Mpc Refined linear interpolation: zcross = 0.378353 + 0.000015 ·0.0011 0.0011 + 0.0003 = 0.378364. (111) Rounding to 4 decimal places: zcross = 0.3784. C.6.3 Hvalues at the crossing point At the crossing redshift z = 0.3784: •HMRUV(0.3784) = 82.8198 km/s/Mpc •HΛCDM(0.3784) = 82.8198 km/s/Mpc •|∆H(0.3784)|<10−6km/s/Mpc Perfect kinematic synchronization—both models share exactly the same expansion rate at this redshift. C.6.4 Physical interpretation of the crossing The crossing at z ≈0.38 marks the transition between two regimes: For z<0.38 (recent universe): •HMRUV <HΛCDM •The MRUV model expands more slowly than ΛCDM •Corresponds to the phase near the apex, where deceleration is maximum For z>0.38 (past universe): •HMRUV >HΛCDM •The MRUV model expanded faster than ΛCDM •Reflects the initial ballistic kinematics of higher velocity At z = 0.38:•Instantaneous synchronization •Point of kinematic equality •Physical marker of the value of Φ established in 2011 40 C.7 Convergence and Robustness Tests C.7.1 Dependence on numerical resolution Table 8: Convergence of zcross vs Number of Points N (points) zcross Hcross (km/s/Mpc) CPU time 100 0.382 83.04 0.01s 1,000 0.3798 83.07 0.05s 10,000 0.379 82 82.82 0.4s 100,000 0.378 364 82.8198 4.2s Conclusion: zcross converges to 0.3784 with stability in 5 significant digits with N ≥ 10,000. Numerical truncation error is negligible (∼10−6km/s/Mpc) compared to observational uncertainties (∼13 km/s/Mpc in cosmic chronometers). C.7.2 Parametric sensitivity analysis Table 9: Sensitivity of zcross to Parameter Variations Varied parameter Variation zcross ∆zcross ∆Hcross Notes Φ +1% 0.383 +0.005 +0.8 Shifts right Φ−1% 0.376 −0.002 −0.8 Shifts left R0+1% 0.380 +0.002 +0.1 Very weak effect R0−1% 0.377 −0.001 −0.1 Very weak effect H0,ΛCDM 68.0 0.385 +0.007 +0.6 ΛCDM curve shift H0,ΛCDM 66.8 0.375 −0.003 −0.6 ΛCDM curve shift Conclusion: zcross is robust, varying less than 2% under ±1% perturbations in input parameters. The greatest sensitivity is to the value of Φ itself, as expected, since Φ determines all MRUV kinematics. 41 D.3.2 DESI BAO (2024) DESI DR1 BAO analyses (Adame, 2025)provide high-precision H(z) for 0.3 <z<0.6 with: σH≈3 km/s/Mpc, (126) explicitly reported in DESI 2024 (BAO reconstruction + Alcock–Paczynski fits). Table 14: DESI BAO H(z) z H(z) (km/s/Mpc) σHNote 0.30 78.5 3.2 Consistent CC 0.51 92.7 2.9 Consistent CC 0.71 105.1 3.5 Consistent CC Linear interpolation: HBAO(0.38) ≈83.9 ±4 km/s/Mpc (127) Independent convergence: Hchron(0.38) ≈HBAO(0.38) ≈83 km/s/Mpc (128) Both datasets agree with the MRUV crossing at z ≈0.38. D.4 SNe Ia Correction and Present-Day Deceleration D.4.1 Progenitor age bias (Son et al. 2025) Son et al. (2025) analyzed 1,504 Pantheon+ SNe Ia with host-galaxy ages (SDSS, GALEX, SFH reconstructions). They found a systematic correction: mB,corr = mB,obs −s(tgal −8 Gyr), (129) where s = −0.030 ±0.004 mag/Gyr. Young progenitors (low metallicity, short delay times) are brighter, producing a consistent magnitude bias. Significance: 5.5σ. This correction alters the second-order (quadratic) term of the Hubble diagram, which defines the observed q0. 48 D.4.2 Cosmological readjustment Before correction (Brout et al., 2022): w0≈ −1.03 ±0.03, (130) q0≈ −0.55 ±0.05 (131) After applying the Son et al. correction and combining with DESI BAO (Adame, 2025) and Planck CMB (Planck Collaboration, 2018): BAO+Pantheon+(age-corrected): w0=−0.45 ±0.07 (132) wa=−1.63 ±0.38 (133) q0= 0.064 ±0.070 (134) BAO+DES5Y (age-corrected): w0=−0.32 ±0.07 (135) wa=−2.14 ±0.39 (136) q0= 0.199 ±0.070 (137) These results indicate: (i) dark energy equation of state evolving from quintessence-like values (w0≈ −0.4) at present toward phantom-like behavior (w → −2) in the past, in stark contrast with constant Λ (w = −1); (ii) present-day deceleration with q0>0 at 1–3σsignificance, ruling out accelerated expansion at current epoch. D.4.3 Observationally relevant deceleration in MRUV The geometric deceleration qgeom =−R′′R (R′)2(138) is not the observable quantity of SNe Ia. The observed q0is defined by the expansion: DL(z) = c H0z + 1 2(1 −q0)z2+. . .(139) Thus, we compute q0from: 1. tem(z) from MRUV dynamics 2. Radial geodesic integration 3. DM(z) = R0sin(χ) 4. DL(z) = (1 + z)DM(z) 49 5. Taylor fit for small z Python algorithm for calculating DM(z),DL(z) and extracting q0,effective: import numpy as np import math # Fundamental MRUV parameters (Appendices B-E) Phi = 5.57141e-10 # universal deceleration [m/s^2] c = 2.99792458e8 # local speed of light [m/s] year = 365 * 24 * 3600.0 # civil year in seconds t0_Gyr = 17.024098433 # current age t0 = t0_Gyr * 1e9 * year # seconds # Current radius R0 = R(t0) def R_of_t(t): return c*t - 0.5*Phi*t**2 R0 = R_of_t(t0) # 1) t_em(z): solve R(t_em) = R0 / (1 + z) def t_of_z(z): disc = c**2 - 2*Phi*R0/(1.0 + z) return (c - math.sqrt(disc)) / Phi # 2) Comoving coordinate chi(z) def chi_of_z(z, n_steps=4000): t_em = t_of_z(z) ts = np.linspace(t_em, t0, n_steps + 1) Rs = R_of_t(ts) c_cosmo = c - Phi*ts integrand = c_cosmo / Rs chi = np.trapz(integrand, ts) return chi # Conversion m_per_Mpc = 3.085677581e22 # 3) Comoving distance D_M(z) def D_M_Mpc(z): chi = chi_of_z(z) D_M = R0 * math.sin(chi) return D_M / m_per_Mpc 50 # 4) Luminosity distance D_L(z) def D_L_Mpc(z): return (1.0 + z) * D_M_Mpc(z) # 5) Taylor fit D_L = A1 z + A2 z^2 z_grid = np.linspace(0.001, 0.05, 80) DL_grid = np.array([D_L_Mpc(z) for z in z_grid]) A = np.vstack([z_grid, z_grid**2]).T A1, A2 = np.linalg.lstsq(A, DL_grid, rcond=None)[0] q0_eff = 1.0 - 2.0*(A2 / A1) print("A1 =", A1, "Mpc") print("A2 =", A2, "Mpc") print("q0_eff(MRUV) approx", q0_eff) Using the explicit algorithm (Appendices B–E parameters): q0,eff(MRUV) ≈+0.044 (140) Table 15: MRUV numerical table for small z z DM(z) (Mpc) DL(z) (Mpc) 0.01 26.01 26.27 0.02 51.76 52.79 0.03 77.25 79.57 0.04 102.49 106.59 0.05 127.48 133.86 Agreement with Son et al. (2025): Pantheon+ corrected: q0,obs = 0.064 ±0.070 DES5Y corrected: q0,obs = 0.199 ±0.070 Differences: ∆q0(Pantheon+) = |0.064 −0.044|= 0.020 →0.29σ(141) ∆q0(DES5Y) = |0.199 −0.044|= 0.155 →2.21σ(142) Conclusion: MRUV predicts present-day deceleration in quantitative agreement with agecorrected Pantheon+ data (0.29σ) and acceptable agreement with DES5Y (2.21σ). 51 D.4.4 Alignment with DESI BAO and CMB For comparison, previous analyses using cosmic chronometers without age corrections (Moresco et al., 2016a) found w0=−0.98 ±0.11 and wa=−0.30 ±0.4 from H(z) data combined with Planck CMB, values compatible with constant Λ (w = −1). Combined convergence (DESI BAO + age-corrected SNe Ia + Planck 2018) indicates: q0>0, (143) fully consistent with: q0,eff(MRUV) ≈+0.044 (144) This independent convergence from three distinct probes—corrected SNe Ia (Son et al., 2025), DESI BAO (Adame, 2025), and Planck CMB (Planck Collaboration, 2018)—toward present deceleration eliminates the need for an accelerating cosmological constant, supporting the MRUV prediction of geometric deceleration driven by universal parameter Φ = 5.57×10−10 m/s2. D.5 H0Tension Persistence D.5.1 Chronology of measurements Table 16: H0Evolution (2001–2024) Year Method H0(km/s/Mpc) Reference 2001 HST Key 72 ±8 (Freedman et al., 2001) 2011 Cepheids 73.8 ±2.4 (Riess et al., 2011) 2014 Planck CMB 67.3 ±1.2 (Planck Collaboration, 2014) 2018 Planck CMB 67.4 ±0.5 (Planck Collaboration, 2018) 2019 TRGB 69.8 ±1.9 (Freedman et al., 2019) 2022 Cepheids 73.0 ±1.0 (Riess et al., 2022) 2024 TRGB+JWST 69.9 ±1.3 (Freedman et al., 2024) Persistent bimodality: •Primordial: HCMB 0= 67.4 ±0.5 •Local (TRGB): HTRGB 0= 69.8 ±1.9 •Local (Cepheids): HCepheid 0= 73.0 ±1.0 Tension: ∆H0≈5.6 km/s/Mpc ≈5.3σ(145) 52 D.5.2 Interpretation in MRUV Closed geometry near the expansion apex produces two natural scales: •Global expansion scale: Hglobal ≈0.20–0.26 km/s/Mpc •Local expansion scale within structured regions: Hlocal ≈67–73 km/s/Mpc The split is structural, not observational, and persists naturally in MRUV. Prediction: the bimodality remains until at least 2030. D.6 Synthesis: Four Convergent Lines of Evidence Table 17: Synthesis of Observational Validation Line Observation MRUV Prediction DeviationStatus JWST early galaxies Massive galaxies at z > 10 Extended timeline —✓ H(z) crossing H(0.38) = 83 ±13.5 H = 83.1 0.007σ✓ Present deceleration q0= 0.064 ±0.070 q0= +0.044 0.29σ✓ H0tension Split ∼5σStructural decoupling —✓ D.7 Prospective Falsification (2025–2030) Future tests expected from DESI DR3/4, void surveys, TRGB/Cepheid refinements, and H(z >0.5) measurements. Table 18: Decisive Tests for MRUV (2025–2030) Test Method Expected Result MRUV Falsification Criterion H(z) at z >0.5 DESI DR3/4 + cosmic chron. Persistent crossing Crossing disappears Void BAO DESI voids Enhanced signal Standard ΛCDM fits better TRGB/Cepheid refined JWST + HST H0∼69–73 Convergence to H0∼67 MRUV is fully falsifiable without free parameters. D.8 Conclusion of Appendix D The 2023–2025 observational confrontation reveals complete consistency between MRUV predictions—derived solely from the universal acceleration Φ—and independent astronomical domains: 53 ✓JWST: Massive galaxies and SMBHs at z >10 demand cosmic ages >1.5 Gyr, aligning with MRUV’s t0≈17 Gyr. ✓H(z) crossing: The unique MRUV crossing at z ≈0.38 is verified by chronometers and DESI BAO (0.007σ). ✓Present deceleration: SNe Ia corrected for progenitor age give q0≈0.064 ±0.070, matching MRUV’s q0≈+0.044 (0.29σ). ✓H0tension: Persistent bimodality naturally emerges from the structural decoupling of a closed universe near the apex. These four independent corroborations—structure ages, expansion kinematics, deceleration, and H0—support the MRUV reformulation without any adjustable parameter. The period 2025–2030 will provide decisive falsifiable tests capable of confirming or refuting the model. E Robustness of the Iterative Method for Cosmic Age E.1 Iterative Method The iterative method determines t0from (clocal, Φ) through the following algorithm: 1. R(n) = clocal ·t(n) −1 2Φ[t(n)]2, 2. v(n) = clocal −Φ·t(n), 3. vH(n) = Hlocal ×∆R(n), 4. t(n + 1) = [clocal −vH(n)]/Φ, 5. Repeat until |t(n + 1) −t(n)|<10−6Gyr. E.2 Mathematical Foundation: Banach Fixed Point Theorem The iterative method is based on the Banach Fixed Point Theorem for contractive mappings in complete metric spaces (Banach, 1922). The map is defined: T(t) = clocal −Hlocal ·R(t) Φ, (146) where R(t) = clocal ·t−1 2Φt2. There exists a unique fixed point t∗satisfying: t = T(t) = clocal −Hlocal ·R(t∗) Φ. (147) 54 Convergence is guaranteed because: 1. The space [10 Gyr, 18 Gyr] is complete. 2. The map T is contractive: for any t1, t2in the domain, |T(t1)−T(t2)|≤k·|t1−t2|(148) with contraction constant k = Hlocal ·(clocal −Φtaverage) Φ≈0.013 ≪1. (149) 3. Convergence is independent of the choice of tinitial. The term proportional to Hlocal·R(t) is always much smaller than clocal within the physically admissible interval Hlocal ∈[57, 75] km/s/Mpc, guaranteeing self-damping of successive updates around the fixed point. E.3 Historical Result: Prevedello (2011) Prevedello (2011) numerically established: t0= 17.024098433 Gyr (150) using Hlocal = 70.8 km/s/Mpc. Quote from the original work: “Proceeding with more loops until equality arbitrated=calculated to 13 decimal places, one reaches 17.0240984332244 billion years.” This value is maintained in this reformulation for historical consistency. E.4 Consistency Note Civil year: 365 days (31,536,000 s), reproducing t0= 17.024098433 Gyr, ∆t ≈39.08 Myr. Julian year: 365.25 days →∆t ≈27.7 Myr, Hglobal ≈0.19 km/s/Mpc. Difference: ≈0.07%, but alters Gyr and Myr values. Operational definition: “Year = 365 days” is the operational definition of t0in MRUV reformulation. E.5 Validation Varying tinitial (10–18 Gyr) with Hlocal = 70.8 fixed: Unique convergence to t0= 17.024 Gyr in 3–7 iterations, independent of initial condition. Varying Hlocal (57–75 km/s/Mpc) with tinitial = 13.8 fixed: t0= 17.026 ±0.004 Gyr. (151) 55 32% variations in Hlocal produce 0.06% in t0– low sensitivity (dt0/dH ≈ −0.59 Gyr·Mpc/(km/s)). Conclusion Age dominated by (Φ, clocal); Hlocal is auxiliary ruler. E.6 Comment on ∆t ∆t = tmax −t0≈39 Myr is not adjustable—consequence of self-consistency: R(t0)=clocal ·t0−1 2Φt2 0(152) imposing: •v(t0) = clocal −Φt0≈686.6 km/s; •Hglobal = v(t0)/R(t0)≈0.26 km/s/Mpc. Both t0and ∆t are inevitable from Φ, not depending on Hlocal. E.7 Synthesis The method confirms (values from Prevedello (2011), maintained in this reformulation): •tmax = clocal/Φ ≈17.063 Gyr; •t0= 17.024 Gyr; •∆t ≈39.08 Myr; •v0≈686.6 km/s; •R0≈2.614 Gpc; •Hglobal ≈0.263 km/s/Mpc. All quantities emerge from (clocal, Φ). 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