The Foundation: Distance, Time, Information, and the Prohibition of Singularities
Richard Kent Gates
mail@richardkentgates.com
September 16, 2026
1. The Foundation
Three principles form the foundation of existence:
- Distance necessitates time. You cannot have spatial separation without temporal separation. A signal crossing distance \(\Delta x\) requires time \(\Delta t = \Delta x / c\). Distance without time is operationally undefined.
- Time is information. A clock tick is a state change. Each state change is one bit. The Bekenstein bound gives \(\sim 9\) bits per Planck time. Time elapsed equals information generated.
- Information cannot be lost. Unitarity requires that information is preserved. The Bekenstein bound sets the maximum information content of any region. Singularities violate this by requiring infinite information in zero volume.
2. Distance Necessitates Time
From the metric \(ds^2 = -c^2 dt^2 + dx^2\):
\[\text{Lightlike: } dx = c\,dt \quad\Rightarrow\quad \text{distance REQUIRES time}\]
\[\text{Timelike: } dt > dx/c \quad\Rightarrow\quad \text{time EXCEEDS distance}/c\]
At the Planck scale: \(l_{\text{Pl}} = c \cdot t_{\text{Pl}}\). Distance IS time at the fundamental level.
3. Time Is Information
The Bekenstein bound at the Planck scale:
\[I_{\text{max}} = \frac{2\pi R E}{\hbar c \ln 2} = \frac{2\pi}{\ln 2} \approx 9.06 \text{ bits per Planck time}\]
For the observable universe (holographic bound):
\[I_{\text{univ}} \sim \left(\frac{R}{l_{\text{Pl}}}\right)^2 \sim 10^{122} \text{ bits}\]
4. Information Cannot Be Lost
A singularity (\(r \to 0\), \(\rho \to \infty\)) would require infinite information in zero volume. This violates the Bekenstein bound. Therefore singularities are forbidden.
The scalar field \(G(t)\) enforces this:
\[G \to \gamma^{-1} \text{ as collapse proceeds} \quad\Rightarrow\quad \text{the time gradient weakens} \quad\Rightarrow\quad \text{collapse stops}\]
\[M_{\text{relic}} > 0, \quad R_{\text{relic}} > 0 \quad\Rightarrow\quad \text{distance} > 0, \text{ time} > 0, \text{ information preserved}\]
5. The Information-Distance-Time Triangle
DISTANCE
/ \
/ \
/ MEASURE \
/ \
TIME ———— INFORMATION
Distance requires time (signal propagation). Time is information (state changes = bits). Information requires distance (Bekenstein bound: \(I \sim RE\)). All three are fundamentally coupled.
6. Accelerator Evidence
Accelerators probe the smallest distances, directly testing the distance-time-information coupling:
6.1. Lorentz Factor and Lifetime Dilation
| \(\gamma\) | \(\tau_{\text{lab}}\) (s) | Distance (m) | Information |
|---|
| 1 | \(2.20 \times 10^{-6}\) | 659 | invariant |
| 100 | \(2.20 \times 10^{-4}\) | 65,900 | invariant |
| 10,000 | \(2.20 \times 10^{-2}\) | 6,590,000 | invariant |
| 100,000 | \(2.20 \times 10^{-1}\) | 65,900,000 | invariant |
As \(\gamma\) increases: lifetime and distance scale together, but internal information is invariant. Distance and time are coupled; information is preserved.
6.2. Energy-Time Uncertainty
\[\Delta E \cdot \Delta t \ge \frac{\hbar}{2} \quad\Rightarrow\quad \Delta x = c \cdot \Delta t = \frac{\hbar c}{2 \Delta E}\]
Higher energy probes shorter distances and shorter times. Each measurement carries at least 1 bit. Distance, time, and information are coupled.
6.3. Cross-Sections Carry Field Information
\[\sigma_{\text{eff}} = \sigma_0 \left(1 + c_X G(t)\right)^2\]
The scalar field modifies cross-sections measurably. At \(G = 10^{-3}\): \(\sigma_{\text{eff}}/\sigma_0 = 1.002\). Accelerator data contains information about \(G(t)\).
7. Singularity = Information Catastrophe
A singularity means:
- Distance \(\to 0\) (no space)
- Time \(\to\) undefined (no clocks)
- Information \(\to\) undefined (no states)
This violates the foundation. Therefore singularities are forbidden.
The scalar field \(G(t)\) is the mathematical expression of this prohibition:
- Black holes: \(G \to \gamma^{-1}\) stops collapse, relic mass \(> 0\)
- Cosmology: \(V(\phi) > 0\) always, \(H > 0\) always, no beginning
8. Theory Closure
All 8 checks pass:
(1) Distance requires time (metric constraint)
(2) Time is information (Bekenstein: ~9 bits/Planck time)
(3) Information preserved in black holes (relic mass > 0)
(4) Information preserved in cosmology (no singularity)
(5) Accelerator: lifetime dilation preserves information
(6) Accelerator: energy-time uncertainty couples distance and time
(7) Accelerator: cross-sections carry G(t) information
(8) Singularities forbidden by the foundation