Temporal Sciences (TS): A Weak-Field Reparameterization of the Classic Tests

Temporal Sciences (TS): A Weak-Field Reparameterization of the Classic Tests Colin Lynch Temporal Sciences Foundation (Founder), Independent Researcher (Dated: September 18, 2025) We ask a narrow question: if one replaces Gby c3t/m (with (t, m)held fixed at the present epoch) in standard weak–field formulae, do the benchmark numbers change? Using identical astronomical inputs, all tests whose predictions enter only through GM are numerically preserved. We obtain: solar limb deflection ˆα= 1.7496′′, Mercury perihelion advance 42.98′′/century, the solar-conjunction Shapiro round-trip delay ∆tRT = 239.7µs, and for PSR B1913+16, ˙ Pb=−2.4026 ×10−12 s s−1 (matching GR at the quoted precision). Looking ahead, separate notes will present (i) a phenomenological mass chassis for cosmological ratios and (ii) a TS second–meter (sm) distribution picture clarifying why the factor c3serves as the conversion that delivers t/m into weak–field observables. These forward topics are not assumed here. I. INTRODUCTION Einstein listed three solar-system tests in 1916 [1]; Shapiro added a fourth in 1964 [2]. Together with the modern Hubble measurement they span six orders of length and eleven of acceleration. We consider the present-epoch identities G=c3t0 m, g ≡c t0 , H0=1 t0 ,(1) and show that an angles-based reparameterization reproduces the benchmarks below without case-by-case fitting.1We refer to Eq. (1) as the “TS triangle”: a present– epoch identity and its two corollaries used here only as a change of variables (no new dynamics). Table Isummarizes the predictions versus observation. Notation. g≡c/t denotes a global kinematic scale (not the local g⊕); we reserve g⊕≈9.81 m s−2for Earth’s surface. Global calibration. We fix the pair (t, m)at the present epoch by Eq. (1). For all weak–field benchmarks in this note we adopt the present–epoch calibration mG≡c3t0/Gmeas = 1.758 ×1053 kg,so that the TS substitution reproduces the standard GR numbers exactly when written through GM. An independent cosmological horizon estimate using Planck 2018 parameters (App. IX) yields a smaller baryon mass mph b≈ 1.5×1053 kg; we therefore summarize the present–epoch ratio feff ≡mG/mph b≈1.17 and defer its dynamical interpretation to a companion analysis. II. LIGHT BENDING GR formula. α=4GM c2b. 1Here gdenotes the global TS scale g≡c/t0, not the local surface gravity g⊕≈9.81 m s−2. TABLE I. Five landmark benchmarks: GR vs TS. For the four solar-system tests, GR and TS agree with observation at the ≲1% level. The Shapiro entry is round-trip (RT). The light-bending value 1.7496′′ corresponds to adopting R⊙= 6.96333 ×108m. Test Observed GR TS Light bend α⊙1.75′′ 1.7496′′ 1.7496′′ Mercury ∆φ43′′/cy 43.0′′ 43.0′′ Red–shift z22.5 m 2.46 ×10−15 same same Shapiro ∆t(RT) 240 µs 239.7µs 239.7µs H0(km/s/Mpc) 70 ±4Planck-ΛCDM fit 70.9 TS substitution. Using G=c3t0/mGgives α= 4ct0M⊙ mGR⊙ . With t0= 4.354×1017 s,mG= 1.758×1053 kg and R⊙= 6.96333 ×108mthis evaluates to 1.7496′′. Equivalently, writing via GM⊙recovers the standard form (see Ref. [3]). III. PERIHELION PRECESSION GR reference value. For Mercury, with a= 5.7909 × 1010 m,e= 0.2056,M⊙= 1.9885 ×1030 kg, ∆φGR =6πGM⊙ ac2(1 −e2)= 5.019×10−7rad/orbit = 0.1035′′/orbit. Using 415.2orbits/century gives 42.98′′/century ≈43′′. TS substitution. Replacing G→c3t0/mGleaves the numerical result unchanged because Gonly occurs in the product GM⊙; hence ∆φTS = ∆φGR (same inputs). 2 IV. GRAVITATIONAL RED-SHIFT AND SHAPIRO DELAY A. Red-shift For a small height difference ∆hat radius rfrom a spherically symmetric mass M, ∆z≡∆ν ν=∆Φ c2=GM c21 r−1 r+ ∆h ≈GM c2 ∆h r2(∆h≪r).(2) With the TS substitution G=c3t0/mGthis becomes ∆z=c t0M mG ∆h r2. Using ∆h= 22.5 m and standard Earth parameters reproduces the Pound–Rebka value ∆z≈2.46 ×10−15 [4]; for GPS clocks, combining the gravitational term with the special-relativistic time dilation yields the standard ≈38 µs/day net correction [5]. B. Shapiro delay One-way radar time-delay in GR: ∆tone-way = 2GM c3ln 4rErR b2.Substituting G=c3t0/mG, ∆tone-way =2t0M mG ln 4rErR b2.Using rE=rR= 1 AU and b=R⊙yields ∆tone-way ≈120 µs, so the round-trip delay is ∆tRT ≈239.7µs, in agreement with the Viking echo [6]. V. COSMIC EXPANSION From H= 1/t we have H0= 1/t0; converting gives H0= 70.9 km s−1Mpc−1. The identity is used here as a kinematic reparameterization; departures from unity in ΛCDM are O(1). At this level the O(1) geometry behind H= 1/t can be viewed as a deficit angle in a simple Regge 4-simplex (App. VII); numerical inputs are listed in App. X. VI. DISCUSSION Calibration and sensitivity. With t0= 4.354 × 1017 sand mG= 1.758 ×1053 kg,GTS = 6.6732 × 10−11 m3kg−1s−2, within 0.02% of the CODATA value. Consequently each benchmark inherits that same fractional calibration. Perturbations are linear: δX/X = −δm/m at fixed t0(equivalently δX/X =δG/G). Epoch dependence of G.TS implies ˙ G/G = 1/t − ˙m/m. Local bounds require the RHS near zero today. LLR analyses give a present-epoch constraint ˙ G/G0= (7.1±7.6) ×10−14 yr−1, with similar limits from other techniques [7]. With 1/t0= 7.24796×10−11 yr−1this implies |1/t0−˙m/m|≲10−13 yr−1. A strictly constant m would predict ˙ G/G = 1/t0≃7.25×10−11 yr−1, excluded by LLR by ∼700×. MOND-scale connection (numerical context only). It is often noted that a0≃1.2×10−10 m s−2is numerically close to cH0/2π; with our present-epoch inputs cH0/2π= 1.096 ×10−10 m s−2(about −9% relative to a0) [8]. Using the TS chronorate CR = 6.67430 as a scale factor suggests an alternative proximity, aTS 0≡c H0 CR −1=c/t0 CR −1,(3) which evaluates to 1.213 ×10−10 m s−2for H0= 70.9 km s−1Mpc−1(+1.1% vs. 1.2×10−10), and 1.153 × 10−10 m s−2for the Planck 2018 value H0= 67.4 (−3.9%). We include this as a numerical proximity only; no dynamical identification with MOND is implied and aTS 0scales linearly with the adopted H0. See also Ref. [9] for a program-level discussion of H0and t0(Sec. 9, p. 48). Cross-epoch invariant test. Define Ξ(z)≡ G(z)m(z)/(c3t(z)). Our postulate is Ξ(0) = 1 today. It can be probed by assembling: (i) an effective G(z)constraint (BBN/CMB), (ii) a baryonic-mass prescription m(z)(particle-horizon or comoving), and (iii) the cosmic time t(z). Using BBN’s GBBN/G0= 0.99+0.06 −0.05 (2σ) [7] is consistent with constancy at the ∼6% level under the horizon-mass prescription; broader tests can combine BBN and CMB [10]. Related work. Connections between cosmic mass/scale and inertia or Ghave a long history in Machian and scalar–tensor contexts. Sciama suggested c2∼GM/R [11]; Dirac’s LNH proposed G∝1/t [12]; and Brans–Dicke theory introduced a dynamical gravitational coupling [13]. Modern constraints on varying “constants” across epochs are reviewed in [10]. Outlook (phenomenology beyond this note). The broader TS program explores whether certain dimensionless cosmological ratios can be encoded by a single phase parameter in a “Chronomass/Chronorate” chassis, mapping either R ≡ Ωc/Ωbor κ≡Ωm/Ωbvia a cosine law with one fixed offset and one calibrated phase [14]. Preliminary calibrations suggest sub–degree phase stability; a dedicated analysis and independent cross–checks will appear separately. In a parallel note, we will also develop aTS second–meter (sm) distribution picture explaining the appearance of the factor c3: in that view c3plays the role of a unit-conversion distributor that carries the present-epoch ratio t/m into length– and time-based observables in weak–field expressions [15]. Both threads are forward-looking and are not assumed in the calculations presented here. 3 VII. REGGE ONE-CELL SKETCH FOR LIGHT BENDING In a single 4–simplex toy model, a hinge deficit sourced by a point mass produces an exterior scattering angle matching the GR grazing value α= 4GM/(c2b)[16]. A full derivation with multiple cells and the Regge–Einstein correspondence is deferred. VIII. BINARY-PULSAR DECAY CHECK In GR, the orbital-period derivative for an eccentric binary due to GW damping is ˙ PGR b=−192π 5T5/3 ⊙Pb 2π−5/31 + 73 24 e2+37 96 e4 (1 −e2)7/2 m1m2 (m1+m2)1/3, (4) with T⊙≡GM⊙/c3= 4.925490947 µs. Using PSR B1913+16 parameters (Pb= 0.322997448918 d, e= 0.6171340,m1= 1.438 M⊙,m2= 1.390 M⊙) gives ˙ Pb,GR = (−2.40263 ±0.00005) ×10−12 s s−1[17–19]. Replacing G→c3t0/mGleaves T⊙unchanged (since Gappears only in GM⊙), so the TS prediction reproduces the GR value for B1913+16 to numerical precision. Here GM⊙is fixed by ephemerides, so T⊙is observationally determined independent of G. IX. MASS DEFINITION AND COMPUTATION We use an operational present-epoch definition for the baryonic mass inside the observable domain. Two prescriptions are common: 1. Particle-horizon mass (preferred here). Define Rph as the present comoving distance to the particle horizon and take mph b≡ρb4π 3R3 ph,with ρb= Ωbρcrit and ρcrit = 3H2 0/(8πG). Using illustrative Planck 2018 values (Ωbh2, h) = (0.0224,0.674) and Rph ≈46.3 Gly yields mph b≈ 1.5×1053 kg (see main text for discussion). 2. Hubble-sphere mass (not used here). Taking RH≡c/H0instead gives a much smaller mass, not representative of the entire observable domain. X. NUMERICAL CONSTANTS CODATA 2018 values: c= 299,792,458 m s−1,G= 6.67430 ×10−11 m3kg−1s−2. Cosmic inputs: t0= 4.354 ×1017 s(13.8 Gyr, Planck 2018 [20]), mG= 1.758 ×1053 kg. Additional: M⊙= 1.9885 ×1030 kg, R⊙= 6.96333 ×108m,1 AU = 1.495978707 ×1011 m, 1 pc = 3.0857 ×1016 m. Companion materials and data availability. A stepby-step buildout of the broader TS program—including the mass chassis calibration and the TS second–meter (sm) distribution picture for the c3factor—is maintained in the Temporal Sciences: Nexus resource [15]. Geometric background and mass mapping are summarized in The Rotational Geometry That Drives the Universe [14]. These external resources are not required for the present note; they provide replicable derivations, figures, and extended discussion. Author note. Generative AI tools were used only for language editing and typesetting assistance; all equations, derivations, and scientific claims are the author’s, who takes full responsibility. [1] A. Einstein, Annalen der Physik 354, 769 (1916). [2] I. I. Shapiro, Physical Review Letters 13, 789 (1964). [3] C. M. Will, Living Reviews in Relativity 17, 1 (2014). [4] R. V. Pound and G. A. Rebka, Physical Review Letters 4, 337 (1960). [5] N. Ashby, Living Reviews in Relativity 6, 1 (2003). [6] I. I. Shapiro, C. C. 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