Exploring the Geometry of Temporal Flow
By John GavelMy work in temporal flow physics has led me down a fascinating path: studying the geometry of the substructure of space and time. What I’ve found challenges the way we usually think about geometry and relational dynamics.
From my perspective, space emerges from time, and time emerges from flow units — fundamental relational points. These points are never neutral; each is either \(F^+\) or \(F^-\), but never 0, and never both simultaneously.
Each point exists along a one-dimensional manifold and has exactly two neighbors, but it can only relate to one neighbor at a time. A point expresses its difference from its neighbor in one of two ways:
- staying the same, or
- flipping relative to its paired neighbor.
Sometimes, however, both neighbors are occupied, leaving the point temporarily ignored in that “tick” of the system. This does not alter the preserved difference at the point itself, but it does change how the geometry expresses the relational dynamics mathematically.
At this stage, the system is direct, not statistical. But as these differences accumulate and propagate, we can use mathematics to quantify and express the resulting gaps. This is where three equations come into play.
Introducing the Three Equations
Total Capacity
\[ N^2 \]
This represents the full relational capacity of \(N\) points, including self-relations. It is the maximum number of relational slots the system can support.
Relational Load
\[ N(N-1) \]
This represents the total number of distinct pairwise relations excluding self-relations. It is the minimum number of interactions required to fully resolve a system of \(N\) points.
Gap Factor
\[ v = \frac{\sqrt{N}+1}{2} \]
This is the geometric gap scaler. It measures how unresolved relational load must be distributed geometrically when temporal constraints prevent full closure.
Why \(N(N-1)\) Appears at All
I didn’t arrive at \(N(N-1)\) by searching for a known combinatorial formula. I ran into it accidentally while trying to understand why certain numbers kept appearing as hard limits in the geometry.
At first, I thought I was encountering kissing numbers. Twelve kept showing up. So did 132. These felt geometric — almost forced — as if the system refused to organize unless those thresholds were met.
But stepping back revealed something important:
- These numbers were not counting neighbors.
- They were counting interactions.
That is exactly what \(N(N-1)\) measures.
What the Equation Is Actually Measuring
The expression \(N(N-1)\) counts the minimum number of distinct relational interactions required to resolve \(N\) points without self-reference.
Each point must differentiate itself from every other point. That means:
- no point can be defined in isolation,
- no relation is optional,
- stability requires mutual constraint resolution.
\(N(N-1)\) is not extra structure. It is the baseline relational obligation a system must satisfy before geometry can stabilize.
TABLE 1 — Relational Load \(N(N-1)\), \(N = 1\) to \(12\)
| N | Expression | Geometric / Physical Role |
|---|---|---|
| 1 | \( N(N-1) = 0 \) | Single point, no relations |
| 2 | \( N(N-1) = 2 \) | First binary interaction |
| 3 | \( N(N-1) = 6 \) | Triangular closure |
| 4 | \( N(N-1) = 12 \) | First 3D relational shell |
| 5 | \( N(N-1) = 20 \) | Curvature begins to matter |
| 6 | \( N(N-1) = 30 \) | Hexagonal efficiency |
| 7 | \( N(N-1) = 42 \) | Prime break in symmetry |
| 8 | \( N(N-1) = 56 \) | Cubic expansion pressure |
| 9 | \( N(N-1) = 72 \) | Square doubling resonance |
| 10 | \( N(N-1) = 90 \) | Transitional shell |
| 11 | \( N(N-1) = 110 \) | High relational strain |
| 12 | \( N(N-1) = 132 \) | Full 12-node closure shell |
This table shows that the numbers I initially mistook for geometric packing limits were actually minimum interaction thresholds. Geometry stabilizes only once these interaction counts are met.
The Emergence of the Gap
Subtracting load from capacity gives:
\[ \text{Gap} = N^2 - N(N-1) \]
TABLE 2 — Capacity vs Load vs Gap
| N | Expression | Geometric / Physical Role |
|---|---|---|
| 1 | \(N^2 = 1,\; N(N-1)=0,\; \text{Gap}=1\) | Self-count gap: the diagonal/isolated contribution when comparing full grid to pairwise links |
| 2 | \(N^2 = 4,\; N(N-1)=2,\; \text{Gap}=2\) | Pairwise deficit: the number of diagonal/self elements absent in the pairwise graph |
| 3 | \(N^2 = 9,\; N(N-1)=6,\; \text{Gap}=3\) | Gap \(=N\): counts local diagonal terms; interpretable as local/vertex self-contributions |
| 4 | \(N^2 = 16,\; N(N-1)=12,\; \text{Gap}=4\) | Geometric gap between square lattice and pairwise links; scales linearly with \(N\) |
| 5 | \(N^2 = 25,\; N(N-1)=20,\; \text{Gap}=5\) | Represents diagonal/self elements removed when forming pairwise-only relations |
| 6 | \(N^2 = 36,\; N(N-1)=30,\; \text{Gap}=6\) | Linear gap \(=N\): useful as a simple measure of 'missing' self-connections |
The gap grows linearly while relational demand grows quadratically. This is the first clear signal that geometry must absorb unresolved relational load.
The Third Equation: Measuring the Real Gap
TABLE 3 — Example: \(N = 2\)
| Quantity | Expression | Value |
|---|---|---|
| Relational Load | \(N(N-1)\) | 2 |
| Total Capacity | \(N^2\) | 4 |
| Capacity Utilization | \(\frac{N(N-1)}{N^2}\) | 0.5 |
| Gap Size | \(N^2 - N(N-1)\) | 2 |
| Gap Factor | \(\frac{\sqrt{N}+1}{2}\) | 1.207 |
Only half of the system’s relational capacity can be realized. The gap factor \(v\) quantifies the geometric cost of resolving even a single binary distinction under temporal constraints.
Gap Factor \(v\) for \(N = 1\) to \(12\)
| N | Expression | Value |
|---|---|---|
| 1 | \( v = (\sqrt{N}+1)/2 \) | 1.000 |
| 2 | \( v = (\sqrt{N}+1)/2 \) | 1.207 |
| 3 | \( v = (\sqrt{N}+1)/2 \) | 1.366 |
| 4 | \( v = (\sqrt{N}+1)/2 \) | 1.500 |
| 5 | \( v = (\sqrt{N}+1)/2 \) | 1.618 |
| 6 | \( v = (\sqrt{N}+1)/2 \) | 1.724 |
| 7 | \( v = (\sqrt{N}+1)/2 \) | 1.822 |
| 8 | \( v = (\sqrt{N}+1)/2 \) | 1.914 |
| 9 | \( v = (\sqrt{N}+1)/2 \) | 2.000 |
| 10 | \( v = (\sqrt{N}+1)/2 \) | 2.081 |
| 11 | \( v = (\sqrt{N}+1)/2 \) | 2.158 |
| 12 | \( v = (\sqrt{N}+1)/2 \) | 2.232 |
So, in my work Space is not fundamental and Geometry is the residue of unresolved, time-ordered relations.
Connecting the Dots: How the Constants Emerged
As I continued exploring finite-N geometries, I noticed the same numbers appearing repeatedly in different forms. Each time I ran simulations or examined structural limits, constants like phi, sqrt(5), and fractions of small integers kept resurfacing. Eventually, it became clear: these were not coincidences, but projections of the same underlying relational constraints expressed in different domains.
To summarize these connections, here is a table showing how each constant relates back to the base equations and what aspect of the geometry it governs:
| Quantity | Expression | Domain | Geometric / Physical Role |
|---|---|---|---|
| tau | \( \tau = \frac{H}{K^2 + (H/2)} \) | Temporal | Effective relational throughput; how fast interactions can propagate under capacity and load limits |
| w | \( w = \frac{4}{3 \sqrt{5}} \) | Angular | Rotational stability for icosahedral adjacency; limits angular motion to preserve relational order |
| O | \( O = \frac{5 \pi}{4} \) | Phase | Phase offset due to pentagonal frustration; unavoidable misalignment in Euclidean embedding |
| m | \( m = \frac{3}{5^3} \) | Volume | Packing density of pentagonal structures; volumetric cost of maintaining 3D order |
| a | \( a = \phi + \frac{1}{26} \) | Curvature | Vertex curvature with finite-N correction; real-world adjustment of ideal phi geometry |
| S | \( S = \sqrt{5} \times 1.01 \) | Duality | Scaling factor for dual structures; introduces slack to allow unresolved gaps to persist |
Each of these constants is a lens on the same underlying principle: the difference between relational capacity and minimum interaction load. Whether expressed as a temporal rate, angular constraint, phase offset, volumetric scale, vertex curvature, or duality factor, they all arise from the same relational system governed by N*(N-1), N^2, and the gap factor v = (sqrt(N)+1)/2.
This is the “punchline” of the geometry: one relational constraint, six manifestations, all revealed by studying the fundamental equations of temporal flow.
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