CTD → DESI-PV Preregistered Forecast

# CTD → DESI-PV Preregistered Forecast **Locked v3.1 (DESI-only scoring; trend-fit amplitudes)** **Timestamp:** 2nd January 2026 **Scope.** Forecast, not a fit to DESI. Defines shell-by-shell expectations for dipole `A1`, quadrupole `A2`, axis behaviour, and wobble handling. **Pass/fail applies to DESI PV only.** CF4 and Pantheon+ are anchors (not scored). --- ## 1) Frozen setup (apply exactly) - **Mask:** Galactic latitude `|b| ≥ 15°` - **Shells (Mpc):** `30–60`, `60–90`, `90–120`, `120–150`, `150–200` - **Midpoints (Mpc):** `r_mid = 45, 75, 105, 135, 175` - **Frame:** CMB (`V_cmb`) - **No DESI PV catalogue consulted prior to this timestamp.** --- ## 2) Frozen parameters (inputs) - **Dipole amplitude law:** `A1(r) = K1 / r`, with locked `K1 = 310.07 ± 15.29` km s⁻¹. - **Quadrupole ratio law:** `r21(r) = A2/A1 = K_r21 / r`, with locked `K_r21 = 19.033 ± 0.923` Mpc. - **Global dipole-axis prior (outside wobble shell):** `RA = 187.84°`, `Dec = −54.65°`. --- ## 3) Wobble handling (locked) A localized **wobble** is expected in `30–60` Mpc (peak `~52–54` Mpc) with: - larger angular scatter of the dipole, - accompanying quadrupole excess, - **snap-back:** for `r ≥ 60` Mpc the axis realigns toward the global prior. > **Policy:** The `30–60` Mpc shell is **qualitative only and unscored**. The statement to check is simply: **“substantial rotation present in 30–60 Mpc.”** --- ## 4) Numerical expectations (for plotting/inspection) ### 4.1 Dipole `A1` (km s⁻¹ Mpc⁻¹), using `A1 = K1 / r_mid` | Shell | r_mid | A1 (central) | A1 (±1σ from K1) | |---:|---:|---:|---:| | 30–60 | 45 | 6.89 | ±0.34 | | 60–90 | 75 | 4.13 | ±0.20 | | 90–120 | 105 | 2.95 | ±0.15 | | 120–150| 135 | 2.30 | ±0.11 | | 150–200| 175 | 1.77 | ±0.09 | **Bias envelope (expectation for shell-mean `H0`):** typical `±½·A1`; worst-case `±A1`. ### 4.2 Quadrupole `A2` (km s⁻¹ Mpc⁻¹), via `r21 = K_r21 / r_mid` and `A2 = r21·A1` | r_mid | r21 (central) | r21 (±1σ) | A2 (central) | A2 (±1σ, propagated) | |---:|---:|---:|---:|---:| | 45 | 0.423 | ±0.021 | 2.914 | ±0.202 | | 75 | 0.254 | ±0.012 | 1.049 | ±0.073 | | 105 | 0.181 | ±0.009 | 0.535 | ±0.037 | | 135 | 0.141 | ±0.007 | 0.324 | ±0.022 | | 175 | 0.109 | ±0.005 | 0.193 | ±0.013 | *Bands above reflect model inputs only; **scoring** uses combined (model ⊕ measurement) 1σ where relevant.* --- ## 5) DESI-only pass/fail criteria (trend-fit amplitudes) **Scored shells are the four outside the wobble:** `60–90`, `90–120`, `120–150`, `150–200`. ### 5.1 GOLD (stringent) 1) **Global dipole amplitude check (trend-fit).** Fit `A1` vs `1/r` across the 4 scored shells (weighted by DESI errors). Require: - fitted `K1̂` within **1σ** of locked `K1 = 310.07 ± 15.29` km s⁻¹, **and** - fit quality `R² ≥ 0.95` *(or equivalently: in log–log, slope within `−1 ± 0.15`)* 2) **Global quadrupole ratio check (trend-fit).** Fit `r21 = A2/A1` vs `1/r` across the scored shells. Require: - fitted `K_r21̂` within **1σ** of locked `K_r21 = 19.033 ± 0.923` Mpc, **and** - fit quality `R² ≥ 0.90`. 3) **Directions (per scored shell).** Best-fit dipole axis within **10°** of `(187.84°, −54.65°)` in each scored shell, **and** a visible snap-back beyond the wobble shell toward the locked global axis. 4) **Shape/anisotropy (global).** Significant `Δχ²` over isotropy using the scored shells. **GOLD overall:** items (1)–(2) must both be satisfied; item (3) must hold in **≥3 of 4** scored shells; item (4) must hold. --- ### 5.2 SILVER (fair; DESI-realistic) 1) **Global dipole amplitude (trend-fit):** `K1̂` within **1.5σ** of locked `K1` **or** `R² ≥ 0.90`. 2) **Global quadrupole ratio (trend-fit):** `K_r21̂` within **1.5σ** of locked `K_r21` **or** `R² ≥ 0.85`. 3) **Directions (per scored shell):** within **20°** of `(187.84°, −54.65°)` in **≥2 of 4** scored shells. 4) **Endpoint coherence:** if only 2 shells pass item (3), both **endpoints** (`60–90` and `150–200`) must pass, **and** the middle shells follow the `1/r` trends for `A1` and `r21` within **1.5σ combined**. 5) **Global shape:** satisfy **2 of 3**: (`A1` slope near `−1`; `r21` slope near `1/r`; significant `Δχ²`). --- ## 6) Declaration This document and all locked values above were fixed and time-stamped **1 January 2026**, prior to viewing any DESI PV catalogue. The `30–60` Mpc shell is **qualitative only** (substantial rotation expected) and **unscored**.