Book reference: Appendix B.7 (Special Values section)
Test file: test_intersection.py, test_horizon_finite.py
| Quantity | SSZ Value | GR Value | Significance |
|---|---|---|---|
| Xi(r_s) | 0.80171 | (diverges) | Saturation: 1 - exp(-phi) |
| D(r_s) | 0.55503 | 0 | FINITE! (SSZ core result) |
| s(r_s) | 1.80171 | infinity | Finite scaling factor |
| z(r_s) | 0.80171 | infinity | Finite redshift |
D(r_s) = 0.555 is the single most important number in SSZ.
GR predicts D = 0 (singularity). SSZ predicts D = 0.555 (finite, testable).
| Context | r*/r_s | Xi(r*) | D* = D_GR(r*) | Derivation |
|---|---|---|---|---|
Decay/global form 1-exp(-φ/x) |
1.594811 | 0.637439 | 0.610710 | D_SSZ = D_GR |
Saturation/local form 1-exp(-φx) |
1.386562 | 0.893914 | 0.528007 | D_SSZ = D_GR |
The invariant fact is mass-independence and the phi-bracket 1 < r*/r_s < φ. The numerical value depends on which Xi form is used.
| Quantity | Value | Formula |
|---|---|---|
| phi | 1.618034 | (1+sqrt(5))/2 |
| phi^2 | 2.618034 | phi + 1 |
| 1/phi | 0.618034 | phi - 1 |
| phi/2 | 0.809017 | coupling half-ratio |
| 1 - exp(-phi) | 0.80171 | Xi at r_s |
| exp(-phi) | 0.19829 | complement of Xi(r_s) |
| 1/(phi^(2*pi)) | ~0.00730 | 1/N_0 factor for alpha |
| Quantity | Value | Notes |
|---|---|---|
| alpha_measured | 1/137.036 = 7.2974e-3 | CODATA 2018 |
| alpha_SSZ | 1/(phi^(2*pi) * N_0) | N_0 = 4 segments/wavelength |
| alpha_SSZ numerical | ~1/137.08 | within 0.03% of measured |
| Deviation | 0.03% | residual discussed in Ch 5 |
| k | r/r_s = phi^k | D_SSZ | D_GR | D_GR algebraic |
|---|---|---|---|---|
| 0 | 1.000 | 0.555 | 0 | - |
| 1 | 1.618 (phi) | 0.686 | 0.786 | sqrt(1-1/phi) = 0.786 |
| 2 | 2.618 (phi^2) | 0.770 | 0.882 | sqrt(1-1/phi^2) |
| 3 | 4.236 (phi^3) | 0.839 | 0.929 | - |
Algebraic identity: D_GR(phi * r_s) = sqrt(1/phi^2) = 1/phi = 0.618 (Fibonacci!)
Wait, corrected: D_GR(phi * r_s) = sqrt(1 - r_s/(phi*r_s)) = sqrt(1 - 1/phi) = sqrt(0.382) = 0.618. Yes, this equals 1/phi.
For any three radii r_A, r_B, r_C:
I_ABC = [D(r_A)/D(r_B)] * [D(r_B)/D(r_C)] * [D(r_C)/D(r_A)] = 1
This is a topological invariant — path-independent, holds for any choice of radii.
| Transition | r/r_s | Notes |
|---|---|---|
| very_close / blend lower | 1.8 | Xi switches from strong to blend |
| blend upper / photon_sphere | 2.2 | Hermite interpolation ends |
| photon sphere (GR) | 1.5 | r = 3/2 * r_s |
| ISCO (GR, Schwarzschild) | 3.0 | r = 3 * r_s |
Xi_max = Xi(r_s) = 0.80171 [saturation of segment density]
D_min = D(r_s) = 0.55503 [minimum time dilation, finite]
s_max = s(r_s) = 1.80171 [maximum scaling factor, finite]
All these are finite — no infinities in SSZ at any physically accessible radius.
- Intersection Invariance — two r*/r_s comparisons and phi bracket
- phi-Lattice Discretization — phi^k table
- Holonomy Invariants — I_ABC proof
- Structural Constants — phi, alpha
- Singularities Resolved — why D(r_s) != 0