Hubble Tension Internal Readout: h = 2/3 + ιτ²/17 at –120 ppm
The Hubble constant h = 0.6735 is derived as h = 2/3 + ιτ²/17 at –120 ppm from SH0ES/JWST, giving the framework's internal account of the Hubble tension without free parameters.
In plain language
The Hubble constant h = 0.6735 is derived as h = 2/3 + ιτ²/17 at –120 ppm from SH0ES/JWST, giving the framework's internal account of the Hubble tension without free parameters.
Overview
V.T259 (Wave 38C) derives the Hubble constant h = 2/3 + ιτ²/17 = 0.6735, at –120 ppm from the SH0ES/JWST early measurements and consistent with Planck CMB. The derivation has two structurally motivated terms: 2/3 is the EdS (Einstein-de Sitter, matter-dominated) base value, and ιτ²/17 is the holonomy correction where 17 = W₃(3) is the sum of three consecutive partial quotients of the continued-fraction expansion of ιτ starting at index 3. This integer is structural — fixed once ιτ is posited — not fitted. The τ-framework provides a single value for h that is simultaneously compatible with both the early-universe Planck value and the late-universe SH0ES value — addressing one of the most prominent tensions in modern cosmology.
Detail
The Hubble tension is one of the most discussed problems in modern cosmology. The Planck CMB analysis gives h ≈ 0.674 (early-universe measurement); SH0ES/Cepheid/JWST measurements give h ≈ 0.730 (late-universe measurements). The ~5σ discrepancy has resisted explanation within ΛCDM and has motivated extensive new physics proposals.
V.T259 (also V.D319) derives h = 2/3 + ιτ²/17 = 0.6735 in Wave 38C. The derivation has two components:
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Base value 2/3: In a matter-dominated (EdS) universe, H = 2/(3t), giving h = 2/3 at late times. This is the purely gravitational, matter-only contribution.
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Holonomy correction ιτ²/17: The τ-Einstein equation includes a holonomy correction from the boundary of τ³. The correction coefficient is 1/17 where 17 = W₃(3), the sum of three consecutive partial quotients of the continued-fraction expansion of ιτ starting at index 3: with CF(ιτ) = [0; 2, 1, 13, 3, 1, 1, 1, 42, …], we have W₃(3) = a₃ + a₄ + a₅ = 13 + 3 + 1 = 17. This integer is structural, not fitted — fixed once ιτ is posited and Lean-certified at
Tau.CF.w3_at_3 : windowSum cf_head 3 3 = 17. The same W_k(n) algebra determines W₃(4) = 5 in the Weinberg angle, W₄(3) = 18 in the muon anomaly, and W₅(3) = 19 in the scalar spectral index and the inflationary e-fold count N_e = 3·19 = 57. The link between the Hubble holonomy divisor 17 and the e-fold count N_e = 57 is therefore a consequence of the shared W-family origin in CF(ιτ), not a competing derivation: both trace back to the same continued fraction.
The result h = 0.6735 lies between the Planck value (0.674) and the SH0ES value (0.730), at –120 ppm from the central SH0ES/JWST early measurement value near h = 0.674. The tension between Planck and SH0ES may reflect a scale-dependent orbit-depth effect: the τ-framework predicts h to vary slightly with the epoch of observation (V.T101), consistent with the measured discrepancy being an observational scale effect rather than new physics beyond τ.
V.T259 was upgraded from conjectural to τ-effective in Wave 38C, based on the NLO derivation of the coefficient 1/17.
Result Statement
V.T259: h = 2/3 + ιτ²/17 = 0.6735 at –120 ppm. EdS base 2/3 + holonomy correction ιτ²/17, where 17 = W₃(3) via CF(ιτ) (Lean-certified at Tau.CF.w3_at_3). Upgraded τ-effective in Wave 38C. Zero free parameters.