# PRE-REGISTRATION: BIG-SPARC Blind Test of the Frozen Galaxy Closure Laws

**Frozen**: 2026-06-11
**Purpose**: convert the framework's strongest empirical result (SPARC/THINGS rotation-curve
performance) into an overfitting-proof prediction. The coefficients below are FROZEN as of this
date. When the BIG-SPARC catalog (~4000 galaxies, Lelli et al., expected 2026-2027) is released,
these laws will be applied with ZERO re-fitting and the results published regardless of outcome.

## 1. Frozen theorem anchor

- g_A = c*H0/(2*pi) with Planck 2018 H0 = 67.36 km/s/Mpc  =>  g_A = 1.0416e-10 m/s^2
- BE-RAR interpolating function: nu(y) = 1 / (1 - exp(-sqrt(y))), y = g_bar / g_dag(r)
- Screening: Poisson limit (ell_screen = c/(sqrt(3) H0) >> r_galaxy)

## 2. Frozen closure laws (trained on SPARC 171 + THINGS overlap, 2026-03-28/29 campaign)

### 2a. Transition-state bridge (linear; primary low-flexibility law)
```
lambda_gal = clip(-0.18 - 0.08 * z(log_r_eff_kpc), -0.5, 0.5)

T_gal = mean_z[ +log_compact
                +log_g_ratio
                -geom_log_hi_to_star
                -geom_type_scaled
                +env_neighbor_tidal_proxy ]

q_gal = clip(0.097 + 0.047 * T_gal, 0.0, 0.35)
w_gal = clip(0.184 + 0.106 * T_gal, 0.0, 0.85)

g_dag(r) = g_A * [1 + lambda_gal * F_mix(r) + q_gal * tanh(log(r / r_eff))]
F_mix(r) = (1 - w_gal) * F_sph(r) + w_gal * F_axis(r)
```

### 2b. Staged-activation refinement (secondary)
```
q_gal = clip(0.097 + 0.047*T_gal + 0.005*relu(T_gal + 0.600), 0.0, 0.35)
w_gal = clip(0.184 + 0.106*T_gal - 0.040*relu(T_gal - 0.200), 0.0, 0.85)
```

### 2c. Support-phase boundary-state law (strongest structured bridge)
```
k_phase = tanh((c0 - C_support) / w_phase)
p0 = -1.00, p_src = 0.00, p_obs = 4.00, s_proj = 0.15, c0 = 0.80, w_phase = 0.12
```

### 2d. Stellar-vs-gas balance term (second-order; conservative variant)
```
q_local -> q_local + a_bal * z(stellar_minus_gas) * (1 - p_proj)^gamma
feature = rms_stellar_minus_gas, gamma = 2.00, a_bal = 0.080
```

z(.) denotes the standard-score transform with mean/std computed ON THE TRAINING SAMPLE
(SPARC 171) and FROZEN — new galaxies are standardized against the training statistics, never
re-standardized on the blind sample. Exact implementations: `analysis_v2/transition_state_predictive_bridge.py`,
`analysis_v2/transition_staged_activation_bridge.py`, `analysis_v2/first_principles_support_phase_bridge.py`,
`analysis_v2/support_phase_visibility_balance_probe.py`.

## 3. Training-set reference metrics (for honest comparison)

| Law | SPARC median chi2/dof | THINGS blind | THINGS overlap |
|-----|----------------------|--------------|----------------|
| g_A + BE-RAR alone (no closure) | 1.951 | — | — |
| Transition bridge (2a) | 1.6118 | 0.3984 | 1.4378 |
| Staged (2b) | 1.6144 | 0.4022 | 1.4329 |
| Support-phase (2c) | 1.5807 | 0.3477 | 0.6281 |
| MOND-simple reference | 1.855 | — | — |

## 4. Blind protocol (binding)

1. **No re-fitting** of any coefficient in §2 on BIG-SPARC data, in whole or in part.
2. Per-galaxy nuisances identical to training: Y_disk in [0.2, 0.8], distance factor in
   [0.7, 1.3]. The SAME nuisance freedom is granted to all comparison models.
3. Comparison models: (a) MOND-simple with a0 = 1.2e-10 m/s^2 fixed; (b) NFW with standard
   c-M priors; both with identical nuisances.
4. Primary endpoints: median chi2/dof over the blind sample; per-galaxy win rate vs MOND-simple
   (Wilcoxon signed-rank p); both for law 2a (primary) and 2c (secondary).
5. Pre-stated interpretation:
   - Frozen laws within 0.1 of their SPARC training medians on the blind sample => the closure
     layer captures real galaxy structure (strong confirmation).
   - Blind medians degrade by >0.4 => the closure layer was overfitting; only the theorem
     anchor + BE-RAR survive as claims.
   - Intermediate => partial transfer; report the gradient honestly.
6. Results are published regardless of outcome, alongside this document and its hash.

## 5. Tamper evidence

SHA-256 of this file (computed over the file with this field blank, recorded in
`PREREGISTRATION_BIG_SPARC_2026_06_11.sha256` beside it). Recommended additional timestamping:
git commit + Zenodo deposit of this file pair.