GEP Analysis Hub
This hub is the central index for all GEP empirical analysis and interactive tools. Each property contains real data from billion-digit precision computation on the Lumiea Systems R810 cluster.
Kurtosis Convergence
Isotropy Test
Helix Winding Numbers
All 11 Constants Live
7 Q Values Interactive
10 Window Sizes
Black Holes • Cosmology
ERPC Hardware • Nexus
Falsification Tracker
The Guided Entropy Principle (GEP) is a unified framework derived from a single entropy evolution equation: ΔS = D·C·R·(1 + αE − β‖∇S‖) with β > 0 and Smin > 0. This hub presents empirical evidence from Tsallis q-sweep analysis at billion-digit precision on the Lumiea Systems R810 cluster, consistent with the GEP prediction that mathematical constants share common underlying entropy surface structure.
Gary W. Floyd — Lumiea Systems Research Division / ThunderStruck Service LLC, New Caney, Texas — Published as Open Defensive Prior Art under Creative Commons. Full paper series on Academia.edu.
Q-Sweep Findings: Overview
Eleven mathematical constants of entirely different mathematical origins — transcendental, algebraic, and of unknown irrationality status — produce Tsallis entropy kurtosis values differing by less than 0.2 percent at window sizes 50 to 100. This convergence is consistent across all seven q values tested.
This is the kurtosis convergence signal. If the digit sequences were independent random processes conditioned only on normality, the expected coefficient of variation would be substantially larger. The observed convergence is consistent with the GEP entropy-time surface prediction that mathematical constants share a common underlying generative structure.
Two independent signatures were found: kurtosis convergence and helix winding number convergence. The critical control analysis — applying the same method to genuinely random digit sequences — has not yet been performed and is the necessary next step.
Spanning transcendental numbers (pi, e, lemniscate), algebraic irrationals (sqrt2, sqrt3, phi), logarithmic constants (ln2), special function constants (Catalan, Euler-Mascheroni gamma, Gauss), and combinatorial sequences (Fibonacci).
Hardware: Lumiea Systems R810 server cluster (Dell PowerEdge R810, K80 GPUs), approximately 72 hours compute time.
Precision: Billion-digit precision per constant. Digit sources: y-cruncher precision computation output files.
Method: Sliding window Tsallis entropy gradient distribution; kurtosis, skewness, KS significance recorded per (constant, q, window) combination.
Verification: Complete dataset embedded as 648 KB JSON in the interactive explorer. All results independently verifiable in browser at darkt22002.github.io/gep-tsallis-explorer
The Convergence Signal
Eleven constants, sorted by kurtosis. Range: 0.0178. Mean: 2.4030. CV: 0.194%.
| Constant | Kurtosis (q=1.0, W=100) | Dev from Mean | Dev % |
|---|---|---|---|
| MEAN | — | — | — |
| STDEV | — | — | — |
| CV | — | — | — |
Convergence tightness (CV) is lowest at W=50 to 100 and increases at larger windows as the distribution approaches the asymptotic floor (~2.229).
| Window | Mean Kurtosis | CV (%) | Notes |
|---|
Q-Dependence of the Signal
As q decreases from 1.0 toward 0, the Tsallis entropy becomes increasingly sensitive to low-probability events. The mean kurtosis amplifies dramatically. At q=0.1, mean kurtosis reaches 530, a 177-fold elevation above the Gaussian baseline. All 11 constants show this elevation simultaneously, with CV remaining below 0.2%.
| q Value | Mean K at W=100 | CV (%) | Elevation vs Gaussian (3.0) | Regime |
|---|
The kurtosis magnitude changes dramatically with q, but the convergence tightness does not. The CV stays below 0.2% across all q values at W=100. This means the signal is not a q-artifact — it is present in the digit structure regardless of which entropy weighting regime is applied.
At q=0.1, the Tsallis entropy is highly sensitive to rare digit combinations. The fact that all 11 constants produce the same rare-event structure (same kurtosis, CV < 0.14%) is the most striking version of the convergence signal.
Forward-Reverse Isotropy Test
The GEP entropy-time surface predicts that temporal direction should be detectable in digit streams of mathematical constants, because the constants are projections of the surface. Forward (normal) and reversed digit streams were compared at q=1.0. All 11 constants have reverse data available for this comparison.
| Window | Forward Mean K | Reverse Mean K | Difference | Asymmetry % |
|---|
Forward digit streams produce consistently higher kurtosis than reversed streams at all window sizes tested. The asymmetry is largest at small windows (25% at W=50) and decreases at larger windows as both converge to the asymptotic floor near 2.229.
This asymmetry is consistent across all 11 constants simultaneously. The forward-reverse difference is not noise: it is a systematic directional signal present in every constant analyzed.
Critical caveat: This asymmetry could reflect number-theoretic properties of the digit sequences themselves rather than a physical directional preference. The control analysis — applying the same test to genuinely random sequences — is the necessary falsification test.
Helix Winding Number
The helix winding number measures the spiral advance per unit window in a complex-plane representation of the digit trajectory. Each digit is mapped to a phase angle in the complex plane, and the winding number quantifies how many turns the trajectory makes per window as the digit sequence advances.
For the reversed digit streams at q=1.0, all 11 constants were analyzed. The winding numbers converge to approximately -0.504, with inter-constant variation below 0.009%.
| Constant | Winding Number | Dev from Mean | Dev (ppm) |
|---|---|---|---|
| MEAN | — | — | — |
| RANGE | — | — | — |
| CV | — | — | — |
The winding number converges to approximately -0.50419, close to -1/2. In the GEP entropy-time framework, -1/2 corresponds to a half-period phase advance per window, consistent with a surface topology that is periodic in the imaginary direction with period equal to twice the window size.
This is a second independent convergence signature, entirely separate from the kurtosis convergence. Eleven unrelated constants produce essentially identical spiral structure in their reversed digit representation. The probability of this occurring by chance in independent random sequences is effectively zero if the sequences are truly independent.
The critical question is whether random digit-normal sequences also produce winding numbers near -0.504. This is the control test that determines whether the helix convergence reflects the constants or the method.
Falsification Conditions
The most important test is also the simplest: apply the identical Tsallis q-sweep analysis to genuinely random digit sequences of equivalent length. If random sequences produce CV < 0.2% kurtosis convergence at windows 50-100, the finding is a method artifact, not a property of the constants. This test must be performed before strong conclusions can be drawn.
Conditions That Would Falsify the GEP Interpretation
| # | Falsification Condition | Status |
|---|---|---|
| 1 | Random digit sequences produce CV < 0.2% kurtosis convergence at the same window sizes | NOT YET TESTED |
| 2 | Any mathematical constant at >1B digit precision fails to converge within the observed band | OPEN |
| 3 | Digit normality of irrational numbers is shown to necessarily produce the observed kurtosis values | OPEN |
| 4 | The helix winding number -0.504 is a trivial consequence of base-10 representation or window size | NOT YET TESTED |
| 5 | Random sequences show the same 25% forward-reverse kurtosis asymmetry | NOT YET TESTED |
| 6 | The convergence disappears at trillion-digit precision, revealing it as a finite-precision artifact | OPEN |
Conditions That Would Strengthen the GEP Interpretation
| # | Supporting Evidence | Status |
|---|---|---|
| 1 | Random sequences do NOT produce CV < 0.2% convergence at these window sizes | NEEDED |
| 2 | Additional constants not in the original run show kurtosis within the convergence band | OPEN |
| 3 | Winding number -0.504 confirmed for constants beyond the original 11 | OPEN |
| 4 | Peak convergence window scales predictably with digit precision (scale invariance) | OPEN |
The finding is empirically real: the kurtosis convergence and helix winding number convergence are present in the data and verifiable in the interactive explorer. The finding is consistent with the GEP entropy-time surface prediction. It is not yet confirmed as a property of mathematical constants rather than of the analysis method. The control analysis on random sequences is the next required step.