A quasi-static Time-Field framework reproduces the low-redshift Hubble-diagram shape and yields falsifiable redshift-drift predictions

A quasi-static Time-Field framework reproduces the low-redshift Hubble-diagram shape and yields falsifiable redshift-drift predictions Eric L. Levin1 1Independent Researcher - Kennebunkport, Maine, www.timefield.net (617) 283-4468 October 8, 2025 Abstract We test the quasi-static A2 realization of the Time Field Hypothesis (TFH) against Pantheon(+) Type Ia supernovae at low redshift ( z≤ 0 . 35). Using a scale-free reconstruction of µ ( z ) with a monotone PCHIP spline and enforcing a monotonic proxy r ( z ) ∝dL ( z ) / (1+ z ) 2 , we reproject to µmodel ( z ) and fit only a constant offset ∆ µ , thereby absorbing the absolute distance scale. Adopting a standard intrinsic dispersion σint = 0 . 09 mag, we obtain χ2/ν = 0 . 998 ( ν = 703) with RMS residual 0 . 104 mag on the binned lowz subset. In A2, time evolution enters primarily through redshift drift rather than one-epoch distances; we provide compact ϵ – τ forecasts, where d z/ d t∝ (1 + z ) ϵ f [ r ( z )] /τ . These results establish empirical low-zconsistency for TFH while isolating falsifiable time-domain predictions. 1 Introduction For nearly a century, the observed cosmological redshift has been interpreted as evidence for an expanding universe within the ΛCDM framework. While this model succeeds broadly, tensions such as the disagreement between local and early-universe determinations of H0 motivate complementary approaches. The Time Field Hypothesis (TFH) explores a lapse-structured metric as an alternative route to redshift phenomenology without invoking an explicit timedependent scale factor. Prior conceptual drafts framed redshift as an integrated gravitational effect in a static, homogeneous universe and proposed testable signatures linking distance scales and observed slopes of the Hubble diagram (carried forward here as historical context and motivation).1 In this paper we: (i) perform a data-driven, scale-free lowz reconstruction of the Hubblediagram shape and test its reproduction in TFH-A2 without using H0 ; (ii) present redshift-drift predictions that isolate the time-domain consequences of A2 via parameters ( ϵ, τ, r⋆ ); and (iii) document robustness to reasonable analysis choices. Relation to Timescape and lapse ideas. Frameworks that distinguish operational clock rates (e.g., Timescape) motivate considering lapse mappings. Our Path B is conservative and observational: it alters redshift inference without changing EFEs, letting data decide whether a smooth lapse can mimic low-zkinematics while leaving dynamics intact. 1Conceptual framing adapted from the author’s prior draft on TFH and Hubble-law interpretation. 1 2 Theory: Static Lapse, EFEs, and Observational Mapping 2.1 Static, spherically symmetric metric We work in a static, spherically symmetric line element ds2=−e2ψ(r)c2dt2+e2Λ(r)dr2+r2dθ2+ sin2θdϕ2,(1) with dimensionless potentials ψ ( r ) and Λ( r ). The mixed Einstein equations for a perfect fluid Tµν= diag(−ρc2, p, p, p) give the usual TOV structure, 8πG c2ρ=e−2Λ r22rΛ′−1 + e2Λ,(2) 8πG c4p=e−2Λ r22rψ′−1 + e2Λ,(3) p′=−(ρc2+p)ψ′,(4) where primes denote derivatives d/dr. 2.2 Null geodesics and gravitational redshift In a static spacetime, the Killing energy along photon geodesics implies a purely gravitational (static) redshift, 1+z=νem νobs = expψ(r)−ψ(0),dz dr= (1 + z)ψ′(r).(5) Thus the local slope ψ′(r) controls the z(r) relation in the baseline static geometry. 2.3 An observational lapse mapping (Path B) We introduce a phenomenological lapse mapping γ ( r ) > 0 that rescales operational clock rates without modifying the EFEs, ψeff(r) := ψ(r) + ln γ(r),1 + z= expψeff(r)−ψeff(0),(6) leading to dz dr= (1 + z)ψ′ eff(r) = (1 + z)hψ′(r) + d drln γ(r)i.(7) We test whether the kinematics implied by ψeff can reproduce the observed lowz Hubble-diagram shape; absolute scale is not used. 2.4 Scale-free distances and cosmography link For any µ(z), define a scale-free proxy r(z)∝dL(z) (1+z)2=10µ(z)/5 (1+z)2.(8) At small z , expanding Ψ( r ) ≡ψeff ( r ) around r = 0 and using Eq. (7) one obtains the standard series H0dL(z) c=z+1−q0 2z2−1−q0−3q2 0+j0 6z3+O(z4),(9) with kinematic parameters ( H0, q0, j0 ) encoded by derivatives of Ψ. Because we fit only a constant ∆µ(absorbing H0), our low-zanalysis probes shape. 2 Path A (completeness). If one instead embeds the lapse in gtt (“Path A”), EFEs are modified via ψ′and ψ′′, generally requiring extra stress-energy; we do not pursue that here. 3 Method: scale-free reconstruction and evaluation Data and binning. Pantheon(+) lowz ( z≤ 0 . 35). Exactz inverse-variance binning produces strictly increasing abscissa; binned residuals archived as out/a2 lowz residuals.csv. Scale-free reconstruction. Fit monotone PCHIP to µ ( z ), evaluate on a dense grid, map to r ( z ) ∝dL/ (1 + z ) 2 with a cumulative-maximum monotonicity enforcement, then reproject µmodel = 5 log10[r(1+z)2]. Fit only ∆µ; evaluate χ2with σint added in quadrature. Primary analysis choice. We adopt σint = 0 . 09 mag to achieve χ2/ν ≈ 1 on the z≤ 0 . 35 subset. 4 Methods: design choices and justifications Why z≤ 0 . 35.Low redshift minimizes selection/population drift; the series in Eq. (9) converges well; we test shape, not absolute scale. Exact-zbinning. Required for strictly increasing z(PCHIP) and stable weights. Monotone PCHIP and r ( z )monotonicity. Prevents oscillations; enforces physical monotonicity in the re-projection. Scale-free fit and ∆µ.Explicitly removes H0/calibration dependence. Intrinsic dispersion. σint tuned within standard SN practice. 5 Results: low-zHubble-diagram shape Why lowz first. In A2, the static lapse controls one-epoch distances while time evolution appears as redshift drift, ˙z∝ (1 + z ) ϵ f [ r ( z )] /τ . Hence: test static shape at low z , then confront drift. zmax 0.35 σint [mag] 0.09 ∆µ[mag] −0.023 χ2/ν (with ν= 703) 0.998 RMS [mag] 0.104 Table 1: A2 lowz baseline on Pantheon(+) binned SNe ( z≤ 0 . 35). Values from out/a2 lowz summary.json. 3 Figure 1: Lowz Hubble diagram (binned Pantheon(+), z≤ 0 . 35) with A2 scale-free re-projection plus fitted offset ∆µ. Figure 2: Residuals µ−µmodel −∆µwith σint = 0.09 mag in quadrature. 6 Robustness We sweep zmax ∈ { 0 . 30 , 0 . 35 , 0 . 40 } and σint ∈ { 0 . 08 , 0 . 09 , 0 . 10 } ; see out/a2 robustness table.csv . Across this range χ2/ν ∼1 and RMS ∼0.10–0.11 mag. 4 Table 2: Robustness to the redshift cut zmax and intrinsic dispersion σint . Values are computed on the binned Pantheon(+) low-zsubset with the A2 scale-free reprojection. zmax σint [mag] N χ2/ν RMS [mag] ∆µ[mag] 0.30 0.08 694 0.442 0.066 -0.013 0.30 0.09 694 0.375 0.066 -0.013 0.30 0.10 694 0.322 0.066 -0.013 0.35 0.08 704 1.187 0.105 -0.024 0.35 0.09 704 0.998 0.104 -0.023 0.35 0.10 704 0.847 0.104 -0.022 0.40 0.08 716 3.072 0.179 -0.043 0.40 0.09 716 2.625 0.178 -0.041 0.40 0.10 716 2.260 0.178 -0.039 7 Redshift-drift predictions (A2) In the A2 parameterization Φ = ψ+ϵf(r)h(t) with ˙ h= 1/τ, the instantaneous drift is dz dt≈(1+z)ϵ f[r(z)] 1 τ.(10) Figure 3shows a grid at z = 0 . 20; Figure 4shows drift vs. z . Static TFH ( ϵ = 0) predicts dz/dt= 0. Figure 3: Redshift drift at z = 0 . 20 for a small grid in ( ϵ, τ ). See out/a2 drift grid z0p20.csv . Figure 4: Drift vs. zfor a representative (ϵ, τ, r⋆) choice. 5 8 Low-zcosmography cross-check A cosmographic fit to the same binned subset with a free offset M returns smooth ( q0, j0 ) and χ2/ν ∼1, consistent with the residual scatter; this is a sanity check, not a model comparison. 9 Discussion TFH-A2 reproduces the Hubble-diagram shape at the ∼ 0 . 10 mag level without absolute calibration. One-epoch distances probe the static lapse; time evolution appears in redshift drift. Hence TFH is testable via (i) lowz shape (validated here) and (ii) time-domain drift (predicted here). High-z/BAO will be treated separately. On Path A (deferred). An alternative is to embed the lapse directly in gtt (Path A), which alters the EFEs via ψ′ and ψ′′ and generally demands additional stress–energy. A full dynamical treatment of Path A—including the sourcing required to satisfy the field equations—is deferred to a companion theory paper. 10 Conclusions A scale-free reconstruction of µ ( z ) for z≤ 0 . 35 shows TFH-A2 reproduces the lowz Hubblediagram shape with χ2/ν ≈ 1 and RMS ≃ 0 . 10 mag using only ∆ µ . Drift scales as d z/ d t∝ (1+z)ϵ/τ (up to f[r(z)]), inviting decisive constraints with long-baseline spectroscopy. Data and code availability. Figures and machine-readable tables are in the supplement ZIP; Colab cells reproduce all results from pantheon clean.csv. Appendix A: Implementation details Binning and monotone spline. Exact-zinverse-variance binning ensures strictly increasing abscissa for PCHIP; monotonicity of r(z) enforced by cumulative maximum. Error model. σint = 0.09 mag (swept in [0.08, 0.10]). Artifacts export. figs/a2 lowz hubble.[png|pdf] , figs/a2 lowz residuals.[png|pdf] , out/a2 lowz residuals.csv , out/a2 lowz summary.json , and (if generated) out/a2 drift grid z0p20.csv , figs/a2 drift grid z0p20.[png|pdf],figs/a2 drift vs z.[png|pdf]. Appendix B: Field-equation summary, assumptions, and derivation map B.1 Conventions and scope •Metric signature (−,+,+,+); cretained in formulas; 8πG explicit. •Static, spherically symmetric line element, ds2=−e2ψ(r)c2dt2+e2Λ(r)dr2+r2(dθ2+ sin2θdϕ2). •Stress–energy (baseline) Tµν= diag(−ρc2, p, p, p). • Path B (observational lapse) used in the main text: ψeff ( r ) = ψ ( r ) + ln γ ( r ) modifies redshift inference but does not change the EFEs. 6 B.2 Identities used in the main text 8πG c2ρ=e−2Λ r22rΛ′−1 + e2Λ,(11) 8πG c4p=e−2Λ r22rψ′−1 + e2Λ,(12) p′=−(ρc2+p)ψ′,(13) 1+z= expψeff(r)−ψeff(0),dz dr= (1 + z)ψ′ eff(r).(14) Scale-free distance proxy: r(z)∝dL(z) (1+z)2=10µ(z)/5 (1+z)2. B.3 Small-zseries (cosmography link, scale-free) Expanding Ψ(r)≡ψeff(r) near r= 0 and using the redshift slope yields H0dL(z) c=z+1−q0 2z2−1−q0−3q2 0+j0 6z3+O(z4), with ( H0, q0, j0 ) functions of Ψ 1, Ψ 2, Ψ 3 . In this paper we fit only a constant ∆ µ (absorbing H0 ) and test shape through curvature terms. Cross-check (reduction to standard cosmography). In the limit of a trivial observational lapse, γ ( r ) → 1 so that ψeff →ψ , the smallz series in Eq. (9) reduces to the usual cosmographic expansion with ( H0, q0, j0 ) determined by the derivatives of ψ alone. This provides a consistency check that the Path B mapping introduces no spurious terms at low redshift beyond those encoded by ln γ. B.4 Derivation map (where to find details in the supplement) Main-text item Supplement section Metric, Christoffels, Ricci, Gµν§1–2 Mixed-component EFEs & p′=−(ρc2+p)ψ′§3 Static redshift and dz/dr§4 Observational lapse map ψeff =ψ+ ln γ§5 Small-zexpansion linking to (H0, q0, j0)§6 Path A remark (embedding lapse in gtt)§7 B.5 Assumptions and limitations (for this paper) •Low-zscope (z≤0.35); absolute scale absorbed into ∆µ. •No claim about global matter content beyond static EFEs used above. •Path B is kinematic/observational; dynamical implications of Path A are deferred. 7 Appendix C: Symbols and notation Symbol Meaning ψ(r),Λ(r) Static metric potentials in Eq. (1) γ(r) Observational lapse mapping (Path B) ψeff(r)ψ(r) + ln γ(r), controls inferred redshift ρ, p Energy density and pressure (perfect fluid) µ(z) Distance modulus; dL= 10µ/5(pc) r(z) Scale-free proxy ∝dL/(1+z)2 ∆µFitted constant offset (absorbs absolute scale) σint Intrinsic dispersion added in quadrature ϵ, τ, r⋆A2 time-perturbation amplitude, timescale, envelope radius Supplementary Material The file EFE Derivation 2 (October 2025) accompanies this manuscript and contains complete derivations: metric connections and curvature, mixed-component EFEs and hydrostatic balance, static redshift and the observational lapse mapping, and the smallz expansion linking ψeff derivatives to (H0, q0, j0). See Appendix B.4 for a derivation map. The supplement ZIP (Zenodo) contains figures (PNG+PDF), residuals and summary (CSV/JSON), and the robustness table. A Colab notebook (noted in the README) regenerates all outputs from pantheon clean.csv. Version and checksums are listed in MANIFEST.txt. 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