# SFT Constants and Thresholds Ledger

This ledger collects the project’s threshold inequalities and what they imply.

| Threshold | Name | Meaning | Implication | Where |
|---|---|---|---|---|
| LCM < π | Lift-closeness margin | degree cannot change under perturbation | LDW stability | `universality/UNIV_PASS_4_ROBUSTNESS.md` |
| HMR < P/2 | Holonomy margin | holonomy class cannot change | HW stability | `universality/UNIV_PASS_4_ROBUSTNESS.md` |
| MBC holds | Monotone branch condition | branch-certified covering | anti-collapse bounds valid | `breakthrough/BREAKTHROUGH_PASS_7.md` |
| 0<a≤∂λF≤b | Bounded distortion | density sandwich + TV/FUD controls | quantitative uniformity bounds | `breakthrough/BREAKTHROUGH_PASS_1.md`, `breakthrough/BREAKTHROUGH_PASS_3.md` |
| 2FDR > SHC | Mixing certificate | diffusion dominates shear | exponential L2 decay to Haar target | `breakthrough/BREAKTHROUGH_PASS_5.md`, `universality/UNIV_PASS_7_MIXING_UNIVERSALITY.md` |
| RNF nonresonant | Stroboscopic sweep | increments rationally independent | full torus equidistribution (conditional) | `breakthrough/BREAKTHROUGH_PASS_2.md`, `breakthrough/BREAKTHROUGH_PASS_6.md` |


## Rate table (quantitative convergence)

| Regime | Quantity | Bound | Notes |
|---|---|---|---|
| DSD (pure diffusion) | FUD mode n | \(|\widehat{p}(n,s)|\le |\widehat{p}(n,0)| e^{-\kappa (2\pi n/P)^2 s}\) | slowest mode \(|n|=1\) dominates |
| DSD | FUD_N | \(\mathrm{FUD}_N(s)\le \mathrm{FUD}_N(0) e^{-\kappa (2\pi/P)^2 s}\) | conservative (uses n=1) |
| SADM + MXC | L2 deviation | \(\|p-1/P\|_{L^2}(s)\le \|p-1/P\|_{L^2}(0) e^{-\frac{(2\mathrm{FDR}-\mathrm{SHC})}{2}s}\) | \(\mathrm{FDR}=\kappa(2\pi/P)^2\) |
| SOL + diffusion on free coord | coupled L2 | same as DSD on free coordinate | target is coupled Haar on CST |
