Cross-Family Universality
Experiment V2.06: Cross-Family Universality
Status: COMPLETE
Goal
Show that V2’s unified capacity definition Q_t = F_timing^(1/(n+2)) gives Gamma* = 1 for ALL detector families, eliminating V1’s spurious family dependence.
The V1 Problem
V1 used C_t = (1/2) log2(F_timing) for all channel families. Since different families have different spectral indices n (so F_timing ~ T^(n+2)), the log2 compresses different power laws differently, leading to family-dependent Gamma*:
| Family | n | V1 alpha | V1 Gamma* | V1 CV(Gamma*) |
|---|---|---|---|---|
| amplitude | 0 | 8.95 | 0.112 | 0.447 |
| UDW | 1 | 5.85 | 0.171 | 0.394 |
| derivative | 2 | 4.07 | 0.246 | 0.387 |
| quadratic | 3 | 3.37 | 0.297 | 0.405 |
V1’s Gamma varies by a factor of 2.7 across families.* The CV of the point-by-point Gamma* is ~40% (not constant within a family either).
The V2 Solution
V2 uses Q_t = F_timing^(1/(n+2)) with the spectral index n appropriate for each detector-field coupling:
| Family | n | V2 alpha | V2 Gamma* | C_Q | CV(Q_t/T) | Max T error |
|---|---|---|---|---|---|---|
| amplitude | 0 | 1.0000 | 1.0000 | 1.710 | ~10^-16 | 3.4 x 10^-5 |
| UDW | 1 | 1.0000 | 1.0000 | 1.858 | ~10^-16 | 2.1 x 10^-5 |
| derivative | 2 | 1.0000 | 1.0000 | 2.155 | ~10^-16 | 9.5 x 10^-7 |
| quadratic | 3 | 1.0000 | 1.0000 | 2.490 | ~10^-16 | 4.1 x 10^-6 |
V2’s Gamma = 1.000000 for ALL families.* Zero spread. The CV of Q_t/T is at machine precision (~10^-16).
Why V1 Was Wrong
The core issue: V1 used log2 to define capacity, but the correct definition uses a fractional power.
For a detector with spectral response F(Omega) ~ Omega^n n_BE(Omega/T):
- F_timing = max[Omega^2 * 4F] ~ T^(n+2)
- V1: C_t = (1/2) log2(F_timing) ~ ((n+2)/2) log2(T) -> nonlinear in kappa
- V2: Q_t = F_timing^(1/(n+2)) ~ T -> linear in kappa
The log2 creates a nonlinear relationship between C_t and kappa, and the nonlinearity depends on n. Hence V1’s Gamma* is family-dependent.
Sigma Independence (Phase 3)
The slope alpha = 1 is independent of switching width sigma for all families:
| Family | sigma = 1 | sigma = 4 | sigma = 16 |
|---|---|---|---|
| amplitude (n=0) | 1.0000 | 1.0000 | 1.0000 |
| UDW (n=1) | 1.0000 | 1.0000 | 1.0000 |
| derivative (n=2) | 1.0000 | 1.0000 | 1.0000 |
| quadratic (n=3) | 1.0000 | 1.0000 | 1.0000 |
C_Q depends on both sigma and n (analytically known), but the slope is always exactly 1.
Physical Interpretation of the Spectral Index
| n | Coupling | Physical detector |
|---|---|---|
| 0 | phi (amplitude) | Scalar field amplitude measurement |
| 1 | linear phi | Standard Unruh-DeWitt point detector |
| 2 | dphi/dtau | Derivative-coupled detector |
| 3 | phi^2 | Quadratic (intensity) coupling |
The spectral index n is determined by the detector-field coupling operator. Different couplings probe different aspects of the Wightman function, but the KMS property ensures that all give the same temperature when the correct capacity observable is used.
Universal Constants
For each family, the capacity observable Q_t = C_Q(n, sigma) * T where:
C_Q(n, sigma) = (4 A c_{n+2})^(1/(n+2))with:
- A = sqrt(pi) sigma / (2 pi) (switching normalization)
- c_m = x_m^m / (e^{x_m} - 1) (from optimizing x^m/(e^x-1))
- x*_m satisfies m(1 - e^{-x}) = x
| m = n+2 | x*_m | c_m |
|---|---|---|
| 2 (n=0) | 1.594 | 0.648 |
| 3 (n=1) | 2.821 | 1.421 |
| 4 (n=2) | 3.921 | 2.411 |
| 5 (n=3) | 4.965 | 3.543 |
What This Proves
-
The slope law is universal across detector families. The choice of detector (UDW, derivative-coupled, etc.) does not affect the extracted temperature, provided one uses Q_t = F_timing^(1/(n+2)).
-
V1’s Gamma family dependence was an artifact.* It arose from using log2 instead of the correct fractional power.
-
The spectral index n is observable. One can determine n from the response function F(Omega) (its power-law behavior at low Omega), then use the correct power 1/(n+2) to extract temperature.
-
C_Q encodes detector properties, not geometry. The proportionality constant C_Q depends on (n, sigma) but NOT on kappa or the spacetime. Combined with V2.05, this means the slope law extracts geometric information (temperature) cleanly separated from detector information (C_Q).
Files
| File | Purpose | Tests |
|---|---|---|
src/cross_family.py | Multi-family analysis, V1 vs V2 comparison | |
tests/test_cross_family.py | Validation tests | 13/13 |
run_experiment.py | Full 3-phase experiment |
Next Steps -> V2.07
V2.07 begins Phase 2 (Metric Recovery): showing that null geodesics are the curves that maximize timing capacity transfer.