Gravity / The lattice
Four counters, one step at a time, build a lattice of events.
Events are records kept on a ledger. Four counters advance one step at a time: each step is a posting, each state of the four counters is a record, and the number of steps taken is record time. From 0000 to 1111 the counters can advance in 4! = 24 orders, and those orders are the 24 four-simplices that fill one cell of the lattice.
- Histories through the cell
- Yours
- The time edge, shared by all 24
Live · exact combinatorics
The 24 orders 0 found · 0 histories
Each order lists the counters in the order they advance, numbered 1 to 4 from the left of 0000. The six in each column start with the same counter.
Advance one counter
0000
What is exact. The 16 records of one cell, its 32 postings and its 24 orders. Rows are record time; positions within a row are a drawing. A line grows heavier as more histories use it: a posting out of 0000 or into 1111 lies on 6 of the 24 orders, a posting in the middle of the cell on 2, so the lines at the bottom and top carry three times the traffic.
What a simplex is. The order (a₁, a₂, a₃, a₄) is the simplex from x through x + ea₁, x + ea₁ + ea₂ and x + ea₁ + ea₂ + ea₃ to x + (1,1,1,1). Its ten edges are four postings, three two-step edges, two three-step edges and the time edge from 0000 to 1111, which every simplex of the cell shares.
With JavaScript on, histories run through the cell live and you can step one yourself. From 0000 to 1111 the four counters can advance in 4! = 24 orders.
The cell in four dimensions
Twenty-four simplices fill one cell.
The cell is the four-dimensional cube between the records 0000 and 1111. Each of the 24 orders of the four counters is a chain of five records from 0000 to 1111; those five are the corners of one four-simplex, and the 24 simplices fill the cell. Each has ten edges: its four postings, the time edge from 0000 to 1111, which all 24 share, and five more. The drawing projects the cube from four dimensions onto the screen and turns it; the turning is only a view.
- The order’s four postings
- The simplex’s other edges
- The time edge, shared by all 24
Live · drawn in your browser
- Order
- …counters in the order they advance,
- Records
- …
The order you built above, in four dimensions
Build an order with “Advance one counter” in the first figure, and it shows here.
Of the cube’s 384 symmetries, 24 orders of the axes times 16 choices of which to reflect, exactly 48 send every one of these simplices onto another: those that reflect no axis or all four. They keep the record clock or reverse it, and the reversal, 1111 − v, is the lattice’s time reversal, an exact symmetry of the whole theory; reflecting a single axis is neither a symmetry of the cut nor a time reversal.
Earlier paper, Section 2; the symmetries: finite-path-sum paper, Theorem 2.9
The one number
At κ = 1 the light cone passes through the postings.
In one tick of record time a posting moves √3/2 in space, toward one of the four corners of a regular tetrahedron. The interval’s light cone reaches √(κ − ¼) in the same tick. At κ = 1 the two are equal, so every posting is null. That value is identified, not derived.
- The light cone, one tick later
- The four postings
Live · exact closed forms
1.00
- Cone
- 0.866√(κ − ¼), per tick
- Posting
- 0.866√3/2, per tick
- Length²
- 0of a posting, 1 − κ
- Speed
- 1.000of a posting, in units of light’s: v² = 3/(4κ − 1), so κ is a speed
- Time diagonal
- 3.464the proper time of four postings, one on each counter, √(16κ − 4): κ stretches time against unit spatial steps
The open diamond, (2,2,2,−1)/5, lies one tick after the event and inside the cone, so it is future timelike; no chain of postings reaches it, because it needs a negative step on 0001. The posting order is a polyhedral cone and the interval’s cone is round, so no value of κ reads the order on every pair (earlier paper, Theorem 2.2).
Two worlds. The foundation’s premises constrain κ only through causality, which needs the cone to reach at least √3/2 per tick. The realization at κ = 1 and the one at κ = 3/2, with the cone at √5/2, meet every premise with the same model and the same proof. The premise that picks κ = 1 is one more: that some nonzero history of postings is not strictly timelike. Reading proper time as a count of comparisons between different postings gives it, and that reading is an identification.
Quantum gravity paper, Section 2 and Theorem 2.3; earlier paper, Theorem 2.2; the posting’s speed: finite-path-sum paper, Theorem 2.11; the time diagonal: slope paper, Section 2.2
| Steps |d| | Classes | Squared length | Character |
|---|---|---|---|
| 1 | 4 postings | 0 | null |
| 2 | 6 | −2 | timelike |
| 3 | 4 | −6 | timelike |
| 4 | 1, the time edge | −12 | timelike |
What is drawn. The event is at the centre; the cyan ball is the interval’s future cone one tick later, and each short cyan mark shows where the cone crosses the ray to a posting. The figure turns so that you can see it in three dimensions; distances are to scale, in lattice units.
The numbers
One cell, counted.
Orders in one cell
24
The orders in which four counters can step from 0000 to 1111, 4! of them: the four-simplices of Kuhn’s triangulation of the cell.
Earlier paper, Section 2
Edge classes
15
Every edge joins a record x to x + d for one of the fifteen nonzero d in {0,1}⁴: four postings, six two-step edges, four three-step edges and the time edge.
Earlier paper, Section 2
One posting, one tick
√3/2
How far a posting moves in space: σa = ea − ¼(1,1,1,1) has length √(1 − ½ + ¼).
Earlier paper, Section 2.2
The identified value
κ = 1
Where the light cone’s reach √(κ − ¼) equals the posting’s √3/2, so every posting is null.
Earlier paper, Theorems 2.1 and 2.2
How it works
From records to a light cone.
1 · The cell
Kuhn’s rule cuts each cell into 24 simplices.
The records are the points of a four-dimensional hypercubic lattice, periodic with side N. Kuhn’s rule cuts each unit cell into 24 four-simplices, one for each ordering of the four axes, and all 24 share the edge from 0000 to 1111. Roček and Williams began linearized Regge calculus on this triangulation; here it carries a Lorentzian metric. The graph never changes: what varies is the squared length of every edge.
Earlier paper, Sections 2 and 3.1
2 · The metric
One number fixes the flat geometry.
In lattice coordinates the flat metric is gκ = I − κ11⊤, so the class d has squared length |d|(1 − κ|d|), where |d| counts its ones. For κ above ¼ the metric is Lorentzian: it has exactly one negative eigenvalue, 1 − 4κ, and its eigenvector (1,1,1,1), the direction of record time, is timelike.
Earlier paper, Section 2
3 · The cone
Every pair in the posting order is causal exactly when κ ≥ 1.
A posting reaches √3/2 in one tick and the cone reaches √(κ − ¼), so the cone contains all four postings exactly when κ ≥ 1, and every pair in the posting order is then future causal. At κ = 1 the two reaches are equal and every posting is null. Only at that value does the interval read the posting order on an interval of positive duration.
Earlier paper, Theorems 2.1 and 2.2
4 · The identification
κ = 1 is identified, not derived.
The definitions do not force it: every κ ≥ 1 makes the order causal, and the foundation as stated admits a world at κ = 1 and a world at κ = 3/2 alike. The weakest premise that, with causality, gives κ = 1 says that some nonzero history of postings is not strictly timelike. In the whole family κ > ½ the four massless corner doublers occur exactly at κ = 1, so a derivation of κ = 1 would derive the doublers too. Every theorem stated at κ = 1 inherits the identification; theorems stated for all κ > ½ do not.
Quantum gravity paper, Theorems 2.1 to 2.3
Sources: the slope paper is Regge’s Slope and the Long-Wave Einstein–Hilbert Vertices from Flat-Space Stationarity on the Recognition Lattice, 4 November 2026; the finite-path-sum paper is A finite gravitational path sum on the recognition lattice, October 2026; the earlier paper is Linearized Einstein gravity from Lorentzian Regge calculus on the Kuhn lattice; the quantum gravity paper is Quantum Gravity from the Recognition Ledger, version 1.2.
Words on this page, in plain terms: record · counter · posting · tick · record time · four-simplex · time edge · Kuhn lattice · κ (kappa).
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