The Inevitable Constant: Why c Is the Pulse of the Lattice
John Gavel
From Rules to Geometry
This post marks a deliberate departure from rule-seeking approaches to fundamental physics (such as Wolfram-style cellular automata) and enters the domain of geometric derivation.
Rules are guesses.
Geometry is necessity.
In the Unified Lattice framework, physical constants are not inputs to be tuned or measured after the fact. They are structural consequences of how closure, adjacency, and recursion work in a discrete topology. The speed of light is not assumed, postulated, or imposed as a limit. It emerges—inevitably—from the way the lattice fails to close.
What follows is not a model layered on physics, but a derivation from the lattice itself.
1. The Frame Topology (Before Units, Before Physics)
Everything begins with a frame: a discrete closure structure with two distinct but inseparable modes.
Given a frame number \( N_d \) at depth \( d \):
- Max Frame (Full Closure)
\[ M_f(d) = N_d^2 \] - Process Frame (Near Closure)
\[ P_f(d) = (N_d - 1)^2 \] - Recursive Generator (Next Frame)
\[ N_{d+1} = N_d (N_d - 1) \]
This is not arbitrary. It is the minimal topology that distinguishes:
- area vs boundary
- completion vs process
- closure vs propagation
The lattice does not grow by addition.
It grows by boundary multiplication.
2. Why This Works Starting at \( N = 1 \)
The structure is valid from the very first nontrivial frame.
\( N = 1 \)
- \( M_f = 1^2 = 1 \)
- \( P_f = 0^2 = 0 \)
This is pure closure with no interior — no propagation possible.
\( N = 2 \)
- \( M_f = 4 \)
- \( P_f = 1 \)
This is the first appearance of an interior defect — the ghost of adjacency.
\( N = 4 = 2^2 \) (The Seed)
This is the first self-closing prime square. From here onward, the recursion becomes coherent and self-similar across depths.
From this point forward:
- full closure scales as \( N^2 \)
- process closure lags as \( (N-1)^2 \)
- recursion advances by \( N(N-1) \)
3. The Invariant Gap (The Ghost in the Frame)
Normalize the process frame so it completes in the same basis as the max frame:
\[ P_f^{(\text{norm})} = (N-1)^2 \cdot \frac{N}{N-1} = N(N-1) \]Now compute the difference:
\[ \Delta = M_f - P_f^{(\text{norm})} \] \[ \Delta = N^2 - N(N-1) = \boxed{N} \]The lattice always misses closure by one generator per cycle.
That missed unit is not noise.
It is not error.
It is structure.
4. Locking the Frame to Reality: The Planck Foundation
Now—and only now—do we introduce physical units.
We do not approximate with Planck units.
We use them as definitions.
-
One frame unit (one tick):
\[ t_P = 5.391 \times 10^{-44}\ \text{s} \] -
One causal pixel:
\[ \ell_P = 1.616 \times 10^{-35}\ \text{m} \]
By definition:
\[ \ell_P = c \cdot t_P \]One lattice tick in time corresponds to one pixel of causal distance.
5. Calculating Effective Speed from the Topology
Over one full max-frame closure:
-
Total Time
\[ T = M_f \cdot t_P = N^2 t_P \] -
Total Distance Advanced (by the gap)
\[ D = \Delta \cdot \ell_P = N \ell_P \]
Velocity:
\[ v = \frac{D}{T} = \frac{N \ell_P}{N^2 t_P} = \frac{\ell_P}{N t_P} \]Substitute \( \ell_P = c t_P \):
\[ v = \frac{c}{N} \]6. Renormalization: Why c Survives Every Scale
At the microscopic frame level, propagation scales as \( 1/N \).
Physical reality, however, is an average over vast numbers of closures.
Under renormalization:
- all \( N \)-dependence cancels
- only the ratio \( \ell_P / t_P \) survives
Conclusion: c Is the Lattice Gap
In rule-based approaches, the speed of light is guessed and preserved by symmetry.
In the Unified Lattice, it is forced.
The speed of light is not a limit imposed on the universe.
It is the rate at which the lattice’s irreducible gap propagates relative to its
closure time.
The universe is not constrained by c.
The universe is the light-speed propagation of its own internal non-closure.
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