# T_H0_ENVELOPE — Pre-registered analysis recipe

**Test:** Hubble-tension envelope test. DCT predicts a deterministic
ratio between the cosmic-mean (Einstein-frame) and high-information-
density (physical-frame) determinations of H₀, sourced by the
conformal-frame mapping `H_phys / H_E = 1 / √P₀ = 1.0830`.

## Null hypothesis (ΛCDM)

`H₀` is a single universal Hubble flow value. The "tension" between
SH0ES (high-density, Cepheid + SNe) and Planck (cosmic-mean, CMB) is
either statistical (low probability) or due to unmodeled systematic
in one of the two methods.

## Alternative hypothesis (DCT)

The conformal-frame mapping `H_phys = H_E / √P₀` predicts a
*deterministic ratio*:
- Einstein-frame methods (CMB, BAO+BBN, large-scale low-density
  averages) → `H_E ≈ 67.4 km/s/Mpc`
- Physical-frame methods (Cepheid SH0ES, TRGB, megamaser — high
  information density at the source) → `H_phys ≈ 73.06 km/s/Mpc`
- Ratio `H_phys / H_E = 1 / √P₀ = 1 / √0.851 = 1.0830`

Zero free parameters: `P₀ = 0.851` is the corpus-canonical condensate
amplitude (DCT-FND-V2 / published Foundation deposit).

## Public dataset

Two-method comparison:
- Planck PR3 (Aghanim+ 2020): H₀ = 67.40 ± 0.50 km/s/Mpc
- SH0ES (Riess+ 2022): H₀ = 73.04 ± 1.04 km/s/Mpc
- Cross-checks: ACT DR4 (Aiola 2020), TRGB (Freedman 2021), Mira
  variables (Huang 2020), DESI BAO+BBN (DESI 2024), Wong 2020 H0LiCOW
  (Wong+ 2020), Pesce 2020 megamaser, etc.

## Statistical method

Two-method:
1. Compute observed ratio `R_obs = H_SH0ES / H_Planck = 73.04/67.40 =
   1.0837 ± 0.0173` (errors propagated in quadrature).
2. Compare to DCT prediction `R_DCT = 1 / √0.851 = 1.0830`.
3. σ-distance from DCT: `(R_obs − R_DCT) / σ_R_obs ≈ 0.04σ`.

Multi-method: 17 published H₀ measurements with method-specific
information-density-rank assignment (acknowledged post-hoc; see
`exploratory/h0_info_density_rigorous.py` for the full pre-registration-
required treatment).

## Decision rule

- |R_obs − R_DCT| within 1σ → **PASS** (DCT prediction matches)
- |R_obs − R_DCT| within 1–3σ → **SOFT PASS**
- |R_obs − R_DCT| > 5σ → **MISS**

## Falsifier signature

A measured ratio outside DCT's `1.0830 ± 0.005` band (allowing for
P₀ uncertainty) at >5σ would falsify DCT's conformal-frame H₀
mechanism.

## Important caveats

- **H₀ ratio vs photon BAO.** Operational \(H_{\rm phys} = H_E/\sqrt{P_0}\)
  is a matter-clock target on \(\tilde g = Pg\). It must **not** be
  reverse-engineered into the standard photon comoving \(\chi(z)\) used in
  BAO — homogeneous \(P(t)\) cancels from radial nulls (DCT-BAO-01; May
  2026 geometry revision). The legacy large DESI \(\Delta\chi^2\) came from
  an obsolete \(D_M\) rescaling map, not from a null-cone theorem.
- This test reports the two-method \(H_0\) ratio on its own; the BAO
  geometry caveat is recorded explicitly in the result JSON / Markdown.
- The **17-method information-density slope** has a post-hoc rank
  assignment in its present form. Pre-registration of the rank is
  required to promote that test to genuinely pre-dictive.

## Cross-validation

Independent: TRGB (Freedman 2021, 69.8 ± 1.7) is consistent with DCT
prediction at intermediate ranks; Pesce 2020 megamaser (73.9 ± 3.0)
is consistent at the high-density extreme.