Appendix AU - Collapse Ratios
# Appendix AU — Collapse Ratios
v 1.1 · 2025-07-12 · supersedes v 1.0 (2025-06-10)
Idea. Once the quartet α β γ λ is locked, many wildly different
observables collapse onto dimension-less numbers ≈ 1.
A single large deviation would falsify PBG.
All raw formulas are boxed in Foundational Derivations;
strong-sector and gauge data come from
[[Appendix BA – Quartic Bag & Neutron β-Decay]][[Appendix BB – Spinor-Gauge Sector]][[Appendix BC – One-Loop World-Sheet & g-2]].
Family A — Electromagnetic fingerprints (stable)
| Observable |
Pivot | Ratio |
Status |
|---|---|---|---|
| Bohr speed |
0.9998 | ✓ | |
| Fine-split energy |
SM QED | 1.003 | ✓ |
| CODATA-23 | 0.9994 | ✓ |
Family B — Strong / nuclear sector (provisional)
| Indicator | Definition | Value | Δ from 1 |
|---|---|---|---|
| Proton collapse | 0.97 | −3 % | |
| Neutron β-index | 1.03 | +3 % | |
| Δ–N splitting | 0.92 | −8 % |
Matches within ≤ 8 % after λ bag re-minimisation; two-loop shell expected to tighten.
Family C — Electroweak / radiative (provisional)
| Indicator | Definition | Value | Δ from 1 |
|---|---|---|---|
| Anomaly ratio | 0.996 | −0.4 % | |
| Fermi anchoring | 1.01 | +1 % | |
| Weak-mass split | 0.96 | −4 % |
Two-loop world-sheet + SU(2) renormalisation should reduce these spreads to < 1 %.
Family D — Cosmic / Yukawa length (stable)
| Observable | Ratio |
Status |
|---|---|---|
| Hubble radius |
1.00 ± 0.03 | ✓ |
| 0.98 | ✓ |
Legend
- ✓ — agreement within current experimental ±1 % (or quoted error bars).
- Provisional — relies on appendices BA–BC; expect ≤5 % shifts when two-loop and SU(2) work finalised.
Road map
| Next action | Targets |
|---|---|
| Two-loop shell & spin-torsion | Family B Δ–N and β-index |
| Two-loop world-sheet | Family C anomaly ratio |
| Non-Abelian anomaly (SU 2) | Family C Fermi anchoring & weak-split |
A single ratio drifting beyond ±10 % after final corrections will flag an inconsistency alert for the entire framework.