A drawing can make a collection of points look spatial before its model has explained any geometry. A more useful starting point is operational: which records can be connected by an elementary change, and what does that change cost? From those relations we can construct neighbours and distances. To obtain physical locality, we must then show that the dynamics actually respects them.
Begin with a relation that can be used
Consider eight distinguishable records labelled 0 through 7. The labels carry no built-in spatial meaning. Declare one reversible operation, P, which advances a label by one modulo eight. Its inverse moves back by one. Two records are neighbours when one application of P or its inverse connects them.
This operation table defines an eight-cycle. A regular octagon or a distorted loop can represent the same adjacency. What matters is which elementary transitions are admitted.
Assign every admitted transition a cost of one step. The distance between two labels is the least number of steps needed to connect them. For this particular cycle,
d(a, b) = min(|a − b|, 8 − |a − b|).
The distance from 0 to 3 is three: the shorter route uses three forward steps. From 0 to 7 it is one, using the inverse operation. From 0 to 4 it is four by either route. These answers follow without measuring the drawing.
Reversibility makes the distance symmetric. Joining a shortest route from a to b to one from b to c provides a route from a to c; the shortest route can only be shorter or equal. This proves the triangle inequality. The relation supplies an intrinsic metric under the unit-cost convention.
Adjacency, embedding and dimension answer different questions
An embedding assigns coordinates to the records for visualization or comparison with a laboratory geometry. It does not automatically turn coordinate distance into operational distance.
For example, draw the cycle as a regular octagon with unit-length sides. Adjacent vertices are one graph step apart, but opposite vertices are four graph steps apart while their straight-line separation is about 2.61. The chord cuts through a part of the drawing on which no elementary transition has been supplied. The two distances measure different permitted journeys.
Nor does the two-dimensional drawing determine the model's physical dimension. A sufficiently large cycle has neighbourhoods whose number of vertices initially grows as 2r + 1 with graph radius r. That growth eventually saturates because the graph is finite. On the eight-cycle, radii zero through four contain 1, 3, 5, 7 and 8 vertices. These local counts differ from the area growth of a two-dimensional lattice.
To discuss an emergent dimension, one needs a family of models, a scale range and a specified measurement—such as neighbourhood growth or a spectral counting law. A finite picture alone cannot identify a unique continuum geometry. This is one reason the geometry section separates observable rank, relations, metric and transport.
A small propagation theorem
Give each record a data register. Suppose the update is synchronous: at round n + 1, the new value at a vertex depends only on the previous values at that vertex and its immediate neighbours. All right-hand inputs are read from round n, so information cannot pass through several vertices within one round.
Compare two initial preparations that differ only at vertex 0. After one round, they can differ only at 0 and its two neighbours. After two rounds, they can differ only within graph distance two of 0. More generally, after n rounds their differences are confined to the radius-n graph ball around 0.
The proof is induction. The initial claim holds at radius zero. If the two preparations agree outside the radius-n ball, then a vertex farther than n + 1 from 0 has itself and every neighbour outside that ball. Its update receives identical inputs in the two preparations, so its next values agree. This establishes the next radius bound.
The result is an upper bound on influence. A particular update may transmit less information or none at all. It establishes neither a universal physical speed nor the dynamics of matter. It shows exactly what a local update rule buys us: a relation between operation count and the region that can depend on a localized preparation.
Continuous quantum dynamics requires a different estimate. The original Lieb–Robinson result for quantum spin systems obtains a finite propagation-velocity bound under its finite-range interaction assumptions. Its suppression outside an effective cone should be distinguished from the exact finite-round support statement proved here.
One shortcut changes both geometry and influence
Add an elementary connection between vertices 0 and 4. Their operational distance immediately falls from four to one. A nearest-neighbour rule on this enlarged graph can now transmit information between them in one round.
Relative to the new graph, the same induction still proves locality. Relative to the old cycle, it appears to permit a remote jump. The apparent contradiction disappears when the operation language is stated. Locality is meaningful only relative to the relations that the complete dynamics can actually use.
This matters for a proposed single source. A controller may visit the records serially while carrying hidden information from one visit to the next. If that information can affect a distant record without passing through the declared neighbourhood structure, the controller supplies a shortcut. Its memory and transitions must therefore enter the locality analysis.
Similarly, converting one graph step per round into a physical velocity requires a length calibration and a measured round duration. Calling these ℓ and Δτ gives a bound ℓ/Δτ. The graph proof does not determine either quantity, and a serial implementation may make the duration depend on system size.
The SMM/DTT research question
For SMM/DTT, an operation-generated graph is a useful intermediate construction. It makes clear how distinguishable records can acquire neighbours without first occupying coordinates. The deeper objective is to explain why the common source selects such relations and preserves them while producing matter and completed observations.
Several tests sharpen that objective. Can the same source prepare the graph and enforce its allowed transport? Do small changes preserve its neighbourhood structure or produce uncontrolled shortcuts? Does increasing the number of records reveal a stable geometric regime? Can the physical clock be measured independently of the graph-step count?
The nested-readout chapter and the open-problems collection connect these questions to the broader programme. A constructed lattice establishes that the mechanism can realize locality. A selection argument must explain why locality follows within the admissible source dynamics.
Which source constraint could generate a neighbourhood relation and prevent hidden transport from bypassing it? I welcome focused collaboration on an explicit model, a stability result or a counterexample that identifies the missing condition.
Sources and further reading
- Spatial emergence: from source relations to geometry — the programme's distinction between locations, adjacency, distance and dynamics.
- From six nested roles to a stable three-dimensional spatial readout — operational neighbourhoods, retained records and restrictions on local transport.
- The finite group velocity of quantum spin systems, Elliott H. Lieb and Derek W. Robinson, Communications in Mathematical Physics 28 (1972), 251–257 — the primary finite-range quantum propagation result used for comparison.