Book reference: Ch 3, Ch 4, Appendix B.7.1
Test files: test_intersection.py, test_ssz_physics.py
Primary sources: 06_STRONG_FIELD/GR_SSZ_INTERSECTION_PHI_DISCRETIZATION.md, really-full-output.md
The symbol r* is used in two related but distinct comparisons. Do not collapse them into one number.
Both comparisons solve the same invariant equation:
D_SSZ(x) = D_GR(x), x = r/r_s
D_GR(x) = sqrt(1 - 1/x)
D_SSZ(x) = 1/(1 + Xi(x))
The value of r*/r_s depends on which SSZ Xi form is being compared with GR.
| Context | Xi form | r*/r_s | Xi(r*) | D*(=D_GR) | Use |
|---|---|---|---|---|---|
| Decay / global comparison | Xi_A(x)=1-exp(-phi/x) |
1.594811 | 0.637439 | 0.610710 | segcalc constants, global D comparison |
| Saturation / local metric-pure comparison | Xi_B(x)=1-exp(-phi*x) |
1.386562 | 0.893914 | 0.528007 | metric-pure/local saturation tests |
Both values are mass-independent because the equation contains only x=r/r_s.
r* is not the solution of Xi_weak = Xi_strong in the outer domain. With Xi_weak=1/(2x), the weak branch is a first-order GR-matching proxy; it is not the same object as the GR time-dilation comparison above.
Therefore the phrase "universal intersection" must always say which comparison is meant:
r*/r_s = 1.594811for the decay/globalD_SSZ = D_GRcomparison.r*/r_s = 1.386562for the saturation/localD_SSZ = D_GRcomparison.
The two exponential forms agree at the Schwarzschild radius:
Xi_A(1) = Xi_B(1) = 1 - exp(-phi) = 0.801711847
D(r_s) = 1/(1 + Xi(r_s)) = 0.555027710
This is why both notations can appear in papers without contradicting the finite-horizon result.
Both intersection values lie in the phi bracket:
1 < r*/r_s < phi = 1.618033988...
This bracket is the invariant statement. The exact numerical value inside the bracket depends on the selected Xi comparison.
r* is not itself the hard code boundary of the operational blend. The canonical computational blend remains:
very_close: x < 1.8
blended: 1.8 <= x <= 2.2
outer: x > 2.2
Physical regimes then classify the outer side further into photon-sphere, strong, and weak contexts. See regime definitions and regime/formula domain clarification.
- GR/SSZ Intersection and Phi Discretization — detailed two-form derivation
- Regime Definitions — hard boundaries and physical regimes
- Special Values — Xi(r_s), D(r_s), D*
- Confusion Prevention — false-alarm patterns