From Comparison Budget to Physical Constants
By John Gavel
Over the past year, I’ve been working on a relational model of physical dynamics that doesn’t begin with spacetime, coordinates, or geometry. Instead, it starts from something far more primitive: binary relational states and the rules that govern how they can change. Recently I've been trying to understand why Lorentz-like structures keep appearing in my framework long before anything resembling distance or velocity is introduced.
To get there, I had to look closely at the update process itself.. the way the substrate decides which sites flip, which remain stable, and how tension between them is resolved. What I eventually realized is that every valid update requires the substrate to make a small number of irreversible binary decisions. These decisions are not optional; they are logically required for the update to be both charge-neutral and tension-descending.
The Three Irreducible Decisions
When I analyzed the update rules in detail, I found that every admissible update must satisfy three independent logical conditions:
- Local tension descent: a site may flip only if more than half of its neighboring relations are in disagreement.
- Charge neutrality: the total contribution of 'flipped' sites must sum to zero.
- Edge disjointness: no edge may have both endpoints flipped simultaneously.
Each of these conditions requires at least one irreversible binary comparison. They act on different variables and different scopes with one local(internal), one global(external), one topological.. and because of that, they cannot be merged or compressed into fewer primitive decisions. The substrate must evaluate all three.
This leads to a simple but powerful conclusion:
\[ \Delta B_{\min} = 3 \]
Here \(\Delta B_{\min}\) is the minimal number of irreversible binary comparisons required to make one closed, admissible update decision. It is the smallest non-negotiable “budget” of logical work the substrate must perform to produce a physically valid update.
Why This Matters
Once I accepted that \(\Delta B_{\min} = 3\) is fundamental, a set of deeper connections emerged. These weren’t forced; they fell out naturally from the structure of the model.
Ground Energy
I define a primitive time step \(\tau_0\). If each closed update requires \(\Delta B_{\min}\) irreversible comparisons, then the ground energy becomes:
\[ E_0 = \frac{\Delta B_{\min}}{\tau_0} \] \]
Action Quantum
The action quantum turns out to be three times the comparison budget:
\[ \hbar = 3\,\Delta B_{\min} \]
Helix Extent
What the Helix Extent Really Is
When I talk about the “helix extent,” I’m referring to something that emerges only after the relational substrate has been running(relating) long enough for stable, repeating update cycles to form. These cycles aren’t geometric at first, they are purely temporal patterns in how the substrate flips and resolves tension. But once those patterns stabilize, they begin to trace out a repeating structure that has both a temporal period and a spatial footprint.
The temporal part is the three-step cycle I call the helix period:
\[ 3\tau_0 \]
This is the smallest closed loop of causal activity the substrate can sustain. Every three primitive ticks, the relational pattern returns to a recognizable state. That’s the “helix” part, a repeating twist in the update dynamics.
But the moment you have a repeating temporal cycle, you also get a repeating spatial footprint. Each cycle propagates influence outward through the relational network, and the amount of outward reach per cycle becomes a well-defined quantity. That quantity is what I call the helix extent:
\[ \ell_h \]
It’s the emergent spatial distance associated with one full three-step helix cycle. It’s not put in by hand. It comes from the structure of the relational network itself.
Where the Formula Comes From
The helix extent depends on several discrete and geometric parameters of the substrate. These parameters describe how many relations exist, how they branch, how they route, and how much coherence is required for a comparison to count as a real event. When you combine all of those constraints, the helix extent becomes a function:
\[ \ell_h = \mathcal{G}(K, H, \pi_2, g(\tau), \lambda_p) \]
This isn’t an arbitrary function. Each symbol represents a structural feature of the relational substrate:
- K — the branching factor of the relational network. It tells you how many neighbors each site can influence per tick.
- H — the total number of reachable sites within one routing shell. It’s a measure of how “wide” the influence spreads.
- \(\pi_2\) — a discrete routing constant that determines how many distinct two-step paths exist between sites. It encodes the substrate’s second-order connectivity.
- g(\tau) — the temporal routing polynomial. This describes how influence propagates outward as a function of time steps. In my work, it takes the form: \[ g(\tau) = 10\tau^2 + 2 \] which is a closed expression derived from the substrate’s update geometry.
- \(\lambda_p\) — the primitive coherence extent. This is the minimum temporal separation required for a comparison to be meaningful. It’s the smallest “chunk” of coherent relational activity.
When you combine all of these, you get a single emergent quantity: the spatial extent of one closed helix cycle. The helix extent exists because the substrate’s update rules naturally produce repeating cycles, and those cycles have a measurable spatial footprint.
Why It Matters
The helix extent is the bridge between the relational substrate and the geometric notion of distance. Before this quantity appears, nothing in the model resembles space. Everything is temporal and relational. But once the helix extent is defined, you can finally talk about how far influence travels per unit time.
And that’s exactly where the speed of light comes from in my framework:
\[ c = \frac{3E_0}{\hbar}\,\mathcal{G}(\dots) \]
The speed of light isn’t a primitive constant. It’s the ratio between the helix extent and the helix period, scaled by the ground energy and the action quantum. In other words, it’s an emergent property of how the substrate routes influence through its relational network.
This is why I keep saying that Lorentz-like behavior shows up early in my work. The algebra isn’t coming from geometry it’s coming from the structure of the update cycles themselves. Once the helix extent exists, the familiar constants start to fall into place.
The characteristic spatial extent of a closed relational cycle, what I call the helix extent, can be expressed as a function of several discrete and geometric parameters:
\[ \ell_h = \mathcal{G}(K, H, \pi_2, g(\tau), \lambda_p) \]
Speed of Light
Finally, the effective speed of light in this framework emerges from the same ingredients:
\[ c = \frac{3E_0}{\hbar}\,\mathcal{G}(\dots) \]
This is the part that surprised me. The constants we normally treat as fundamental — \(E_0\), \(\hbar\), \(\ell_h\), and even \(c\) — begin to look like summaries of a deeper comparison structure. They arise not from geometry or spacetime, but from the logical requirements the substrate must satisfy to produce a valid update.
Why Lorentz Appears Early
This perspective explains why Lorentz-like algebra shows up in my work before anything spatial is introduced. If the underlying dynamics are governed by ratios of flow participation and minimal comparison budgets, then the hyperbolic structure of Lorentz transformations may be a reflection of those ratios. Not of motion through space.
In other words, Lorentz symmetry might be telling us something about the structure of relational decision-making at the substrate level, long before spacetime emerges.
This is still speculative, but the connections are too consistent to ignore. The more I explore this comparison-based foundation, the more it seems that many familiar physical constants may be emergent consequences of a deeper, simpler logic.
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