V2.51b - KMS Temperature Extraction + Improved Entropy
V2.51b: KMS Temperature Extraction + Improved Entropy
Executive Summary
V2.51b adds KMS temperature extraction (detailed balance of F(omega)) as a 4th independent measurement, increases acceleration resolution from 8 to 16 points, and increases frequency resolution from 100 to 200 bins. Following the circularity audit from V2.51, we report 4 measurements (3 pass, 1 fail):
| # | Measurement | Value | Target | Error | Status |
|---|---|---|---|---|---|
| 1 | c/3 (entropy density) | 0.310 | 0.333 | 7.1% | PASS |
| 2 | Gamma* (QFI scaling) | 0.704 | ~1.0 | 34.4% | PASS |
| 3 | R_kk (Ricci curvature) | -8.42 | 0 | — | PASS |
| 4 | T_kms/T_unruh (KMS temp) | 51.9 | 1.0 | 5086% | FAIL |
c/3 improved from 22.5% to 7.1% thanks to 16 acceleration points (vs 8). This is now within 10% of the CFT prediction for c=1 free scalar.
Gamma regressed from 0.8% to 34%* because the numerical gradient d(ln F)/d(ln a) is noisier with finer acceleration spacing. Still passes the [0.5, 2.0] threshold.
KMS temperature extraction fails: T_kms is ~50x too high at N=1000. Root cause: only 13 trajectory points in the Rindler wedge, giving insufficient spectral resolution to resolve the KMS exponential. Needs N >> 10000.
What Changed from V2.51
| Change | Effect |
|---|---|
| 16 acceleration points (was 8) | c/3 improved: 22.5% -> 7.1% |
| 200 frequency bins (was 100) | Better spectral coverage for KMS |
| KMS temperature extraction added | T_kms/T_unruh = 52 (FAIL, 50x too high) |
| Auto n_fixed via determine_n_fixed | n_fixed=8 at N=1000, 9 at N=2000 |
| Gamma* with 16 points | Regressed: 0.8% -> 34% (noisier gradient) |
Detailed Results
Phase 1: Primary Ensemble (N=1000, 30 seeds, n_fixed=8)
Measurement 1: c/3 (entropy density)
Trimmed mean: 0.310 (target 0.333, error 7.1%)
Median: 0.348
Std: 0.623 (O(1) per-seed variance)
95% CI: [0.140, 0.497]
R^2 (fit quality): median 0.079, mean 0.136
-> 0.333 is within the 95% CI
Measurement 2: Gamma* (QFI scaling)
Trimmed mean: 0.704 (target ~1.0, error 34%)
Median: 0.656
Std: 0.239
95% CI: [0.634, 0.780]
Outliers: 1/30 removed
Measurement 3: R_kk (Ricci curvature)
Trimmed mean: -7.91 (target 0)
Median: -8.42
Std: 198.2
95% CI: [-26.3, 12.0]
Outliers: 4/30 removed
-> 0 is within the 95% CI
Measurement 4: T_kms (KMS temperature)
Median: 51.9 (target 1.0, error 5086%)
Mean: 53.4
Std: 18.4
Valid: 30/30 seeds
-> FAILS: T_kms is 50x T_unruh
BD Calibration (sanity check):
Box(t^2) = -2.000000 (exact by construction)
Box(x^2) = +2.000000 (exact by construction)
Derived (not independent):
G = 1/(4eta) = 0.808 (target 0.750)Why KMS Temperature Extraction Fails
At N=1000, the Rindler trajectory at acceleration a=1.0 has only 13 trajectory points (causal set elements near xi = 1/a within band delta_xi = 0.5).
The detector response F(omega) is computed as a sum over 13^2 = 169 complex exponentials. This is far too few to resolve the KMS detailed balance condition F(-omega)/F(+omega) = exp(-omega/T), which requires:
- The ratio to decay exponentially over multiple e-folding times
- Spectral resolution delta_omega << T ~ a/(2pi) ~ 0.16
With 13 trajectory points, the effective Nyquist frequency is ~6 bins, and the spectral resolution is much coarser than T. The log ratio ln(F(-omega)/F(+omega)) is essentially flat (range [-0.5, 0.7]) when it should span [-63, 0] over omega in [0, 10].
Estimate: KMS extraction requires ~100+ trajectory points, which needs N >> 10000 (possibly N ~ 50000). At current N=1000, the density is ~5 points per unit area, and the Rindler trajectory region contains ~13 points. Scaling to N=50000 would give ~65 points, still likely insufficient.
This is a fundamental limitation of the discrete approach at accessible N, not a code bug.
R^2 Quality Concern
The entropy log fit R^2 decreased from 0.13 (V2.51) to 0.079 (V2.51b) despite having 16 acceleration points instead of 8. Possible explanations:
-
More points reveal more noise: With 8 points, some random scatter was masked. 16 points expose that the per-acceleration entropy values don’t cleanly follow S ~ (c/3)*ln(1/a) at the individual-seed level.
-
n_fixed=8 is too few: Each entropy value is computed from only 8 trajectory points per acceleration. The von Neumann entropy of an 8x8 submatrix has O(1) noise relative to the physical signal.
-
Adjacent accelerations share trajectory points: With 16 accelerations in [0.3, 3.0], neighboring accelerations may select overlapping sets of 8 points, creating artificial correlations in S(a).
Gamma* Regression Analysis
Gamma* changed from 1.008 (8 accel points) to 0.656 (16 accel points). This is because compute_gamma_star uses np.gradient to compute d(ln F)/d(ln a):
- With 8 points: delta(ln a) = 0.329 → smoother gradient estimate
- With 16 points: delta(ln a) = 0.154 → noisier gradient, amplifies Ct noise
This means Gamma is sensitive to the acceleration sampling*, which weakens it as a robust physical measurement. The previous value of 1.008 was partly an artifact of coarse sampling that smoothed over noise.
N-Convergence (auto n_fixed)
| N | n_fixed | c/3 | Gamma* | R_kk | T_kms | Checks | Seeds |
|---|---|---|---|---|---|---|---|
| 500 | 8 | 0.488 | 0.735 | 4.6 | 71.1 | 2/4 | 15 |
| 1000 | 8 | 0.184 | 0.661 | -7.7 | 44.8 | 2/4 | 15 |
| 2000 | 9 | 0.621 | 0.694 | 9.0 | 43.8 | 2/4 | 15 |
Convergence is still NOT monotonic. n_fixed auto-determines to 8-9 regardless of N, so the adaptive scaling did not help. The root issue is that the trajectory point selection (delta_xi band width) doesn’t scale with N either.
Positive note: T_kms/T_unruh decreases with N (71 -> 45 -> 44), suggesting the KMS extraction slowly improves, but convergence to 1.0 would require many orders of magnitude more points.
Comparison: V2.51 vs V2.51b
| Metric | V2.51 (8 accel, 100 omega) | V2.51b (16 accel, 200 omega) | Better? |
|---|---|---|---|
| c/3 | 0.408 (22.5%) | 0.310 (7.1%) | YES |
| Gamma* | 1.008 (0.8%) | 0.656 (34%) | NO |
| R_kk | -8.42 | -8.42 | SAME |
| R^2 | 0.13 | 0.079 | NO |
| T_kms | N/A | 51.9 (5086%) | NEW |
| Checks | 3/3 | 3/4 | +1 attempted |
The trade-off: 16 acceleration points greatly improved c/3 (the most important measurement) but degraded Gamma* (which depends on the gradient of Ct vs a).
What This Work Actually Shows
The genuine contributions
-
c/3 = 0.310 (7.1% from target): The best measurement yet. The SJ vacuum entanglement entropy on a Poisson-sprinkled causal set gives c/3 within 7% of the CFT prediction c/3 = 0.333 for a free scalar in 1+1D.
-
KMS diagnostic: Even though T_kms fails, the ~50x result is informative. It quantifies the spectral resolution gap: at N=1000 with 13 trajectory points, the discrete detector cannot resolve the Unruh temperature. This sets a clear target for future work: N >> 10000.
-
Gamma sensitivity discovered*: The Gamma* measurement is sensitive to acceleration sampling (1.008 with 8 points, 0.656 with 16 points). This means previous claims of “0.8% accuracy” were optimistic.
What it does NOT show
-
Temperature is not derived. KMS extraction gives T ~50x too high at N=1000. The Unruh temperature T = a/(2pi) remains an input assumption.
-
Einstein equations are not tested. Flat spacetime: 0 = 0. Need curved spacetime.
-
Convergence not demonstrated. Non-monotonic c/3 across N values.
-
Individual seeds are noisy. R^2 = 0.08 means the log model explains only 8% of per-seed entropy variance.
Gaps to Publication
| Gap | Severity | What’s needed |
|---|---|---|
| Temperature not derived | Critical | N >> 10000 for KMS, or alternative method |
| Low R^2 on entropy fit | High | n_fixed >> 8, or different functional form |
| Non-monotonic convergence | High | Adaptive delta_xi scaling with N |
| Gamma* sensitivity to sampling | Moderate | Robust gradient estimator |
| Only flat spacetime | Moderate | Test on de Sitter causal set |
Version History
| Version | c/3 | Gamma* | R_kk | T_kms | Box(t^2) | Independent checks |
|---|---|---|---|---|---|---|
| V2.41 | 8.4 | 0.536* | -517 | N/A | N/A | 0/3 |
| V2.49 | 0.34 | 0.09** | 2.8 | N/A | -2.000 | 1/3 |
| V2.50 | 0.36 | 1.07 | -7.7 | N/A | -2.000 | 3/3 |
| V2.51 | 0.41 | 1.01 | -8.4 | N/A | -2.000 | 3/3 |
| V2.51b | 0.31 | 0.66 | -8.4 | 51.9 | -2.000 | 3/4 |
V2.41 used V2.19 Gamma for headline, slope-law in pipeline. *V2.49 only computed slope-law; V2.19 Gamma was never run. V2.51b: 16 accel points, 200 omegas, KMS extraction added.
Honest Assessment: 60%
Down from V2.51’s 70%, because:
- Gamma* regressed to 34% error (was 0.8% — the 0.8% was partly an artifact)
- R^2 dropped to 0.08 (was 0.13)
- KMS temperature fails completely (50x too high)
- Non-monotonic convergence persists
The 40% gap:
- 15%: Derive temperature non-circularly (need N >> 10000 or alternative method)
- 10%: Achieve R^2 > 0.5 on individual seeds (need n_fixed >> 8)
- 10%: Demonstrate monotonic convergence (need adaptive trajectory selection)
- 5%: Stabilize Gamma* across acceleration samplings (robust gradient)
Positive: c/3 improved to 7.1% from target, the best measurement in the series.