Status: CANONICAL
SSZ defines five regimes based on the ratio r/r_s (distance to Schwarzschild radius):
| Regime | r/r_s range | Operative Xi branch | Physical context |
|---|---|---|---|
| very_close | < 1.8 | g2 / inner exponential | At/near horizon |
| blended | 1.8–2.2 | Hermite C² interpolation | Smooth transition |
| photon_sphere | 2.2–3.0 | g1 formula, photon-sphere regime | Photon orbit zone |
| strong | 3.0–10.0 | g1 formula, strong physical regime | Neutron stars, compact objects |
| weak | > 10.0 | Ξ_weak = r_s/(2r) | Solar System, GPS, stars |
The operative formula domain and the physical regime label are not identical. Above r/r_s=2.2, the current calculation branch uses the g1/weak-form expression for Xi, but the physical regime is still "photon_sphere" up to 3.0 and "strong" up to 10.0.
Before using any Ξ formula, determine regime via r/r_s.
The wrong formula in the wrong regime produces incorrect results. This is the most common source of apparent contradictions in SSZ.
def get_regime(r, r_s):
ratio = r / r_s
if ratio < 1.8:
return 'very_close' # g2 / inner exponential
elif ratio <= 2.2:
return 'blended' # Hermite C²
elif ratio <= 3.0:
return 'photon_sphere' # physical regime; operative branch is g1
elif ratio <= 10.0:
return 'strong' # physical regime; operative branch is g1
else:
return 'weak' # Ξ_weakThe transition between Ξ_weak and Ξ_strong is NOT a discontinuity. It uses Hermite C² interpolation:
t = (r/r_s - 1.8) / 0.4 (normalized: 0 at r/r_s=1.8, 1 at r/r_s=2.2)
Ξ_blend = H₅(t) (quintic Hermite polynomial)
This guarantees:
- C⁰ continuity: Ξ matches at both boundaries
- C¹ continuity: dΞ/dr matches at both boundaries
- C² continuity: d²Ξ/dr² matches at both boundaries
Never mix formulas without an explicit blend rule.
For the exact formula-domain table, see regime and formula domain clarification. That file is authoritative for deciding which Xi branch is evaluated.
The transition from weak (g₁) to strong (g₂) is unidirectional:
g₁ → g₂: Irreversible (spacetime regime assignment)
g₂ → g₁: FORBIDDEN (spacetime does not spontaneously de-segment)
This is a fundamental postulate, not a computational convenience. Once a region of spacetime enters the strong-field segmentation regime, it does not spontaneously return to weak-field behavior.
SSZ distinguishes two conceptually different processes that must not be conflated:
Layer 1 — Spacetime regime assignment (irreversible): Once a spatial region acquires g₂ segmentation (e.g., by gravitational collapse), the regime label is permanent. The spacetime itself does not "un-segment." This is what "g₂ → g₁: FORBIDDEN" means.
Layer 2 — Matter/radiation moving through regimes (physical process): Matter and radiation CAN physically move from a g₂ region outward into g₁ space (e.g., during supernova explosion, black hole ringdown, or metric perturbation emission). When this happens, the coherence structure of the segment lattice undergoes an irreversible collapse — the ordered g₂ packing is destroyed as the material expands into the disordered g₁ environment. This process is described in detail in Book Chapter 25.
The g₂ → g₁ coherence collapse (Layer 2) is NOT the reversal of the g₁ → g₂ regime assignment (Layer 1). It is a distinct, entropy-producing physical process analogous to melting: the spacetime region remains g₂, but matter ejected from it loses its coherent segment structure as it enters g₁ space. ΔS_seg > 0 always.
Summary:
- Spacetime regime: g₁ → g₂ irreversible, g₂ → g₁ forbidden
- Matter motion: matter CAN move from g₂ regions to g₁ regions; when it does, coherence collapses irreversibly (ΔS > 0)
Some repositories use r/r_s = 90–110 as boundaries. These are NOT regime boundaries!
They are probe radii (test sampling points) for continuity checks in unified_validation.py. The actual physical regime boundaries are at r/r_s = 1.8 and 2.2.
| Regime | Ξ range | Example objects |
|---|---|---|
| very_close | 0.5–0.802 | Horizon, near-BH |
| blended | 0.22–0.5 | Compact NS, r* zone |
| photon_sphere | 0.13–0.22 | Photon orbits |
| strong | 0.05–0.13 | Outer NS envelope |
| weak | < 0.05 | Everything in Solar System |
- Segment density Ξ(r) — Weak, strong, blend formulas
- Regime vs formula domains — Why formula boundaries ≠ regime boundaries
- Coherence collapse — g₁→g₂ irreversible transition
- Time dilation D(r) — D = 1/(1+Ξ)
- Formula compendium §B.2 — Hermite C² details
- Special values — Ξ(rₛ)=0.802, D(rₛ)=0.555
- Forbidden formulas — Deprecated variants
- Test:
test_regime_definitions.pyin segmented-calculation-suite
© 2025–2026 Carmen N. Wrede, Lino P. Casu