Gravity / Einstein’s equation
The lattice’s field equation is Einstein’s, at every amplitude.
The lattice’s equations act on squared edge lengths, fifteen at each vertex. Sample a smooth curved spacetime on the lattice, its cells nondegenerate and Lorentzian all the way from the flat reference, and expand each edge’s equation in the spacing. The leading term, projected onto the ten components of a metric, is −½ times Einstein’s tensor density, with its full nonlinear dependence on the metric. Your browser checks this on a random spacetime. It is a statement about the field equation, not yet about its solutions: lattice solutions that converge to solutions of the full Einstein equations are open.
The identity: proved and independently verifiedκ = 1: identified (the identity holds for every κ above ½)Nonlinear lattice solutions: open
- The ten components
- Lattice = Einstein
Live · computed in your browser
1, identified
0.80
- Largest
- …component of −½ℰ
- Mismatch
- …largest difference, … of the largest component
- Time
- …for 90 lattice row evaluations
Computing the lattice rows…
The ten components, side by side
What your browser does. It draws a random metric g near the flat reference I − κ11ᵀ, with random first and second derivatives P and Q, and checks, at 21 points of the straight path from the reference to g, that the cells stay nondegenerate and Lorentzian. Each squared edge is the metric at the edge’s midpoint applied to its four-bit edge vector, at lattice spacing s. Each triangle’s area comes from its Gram determinant and each angle from the Gram matrix of its 4-simplex; the deficit around a triangle is 2π minus its angles, and the equation of an edge is the sum over its triangles of the area’s derivative times the deficit. The coefficient of s² comes from spacings 0.02, 0.01 and 0.005. Projected onto the ten metric components it is compared with −½ times the Einstein density computed from g, P and Q. Nothing is taken from the paper’s own programs.
Einstein-equation paper, Sections 2 and 3 and Theorem 3.3; quantum gravity paper, Theorem 4.2
With JavaScript on, your browser checks the identity on a random curved spacetime: the ten metric components of the lattice’s leading field equation equal −½ times the Einstein density, to better than one part in 107.
How it works
From edges to Einstein in four steps.
1 · Edges from a metric
Fifteen squared edges, ten metric numbers.
Each vertex has fifteen edge classes, one for each nonempty string of four bits. A constant metric g gives the edge of class v the squared length vᵀgv: fifteen numbers from ten components. The other five combinations are lattice variables of their own; they have equations too, which a full solution must meet.
For a constant metric the five combinations vanish and the fifteen edges give back the ten numbers exactly. Sample a curved metric at the edge midpoints and the five pick up its derivatives.
Live · sampled in your browser
…
- The five
- …largest of μ₀₁₂, μ₀₁₃, μ₀₂₃, μ₁₂₃ and μ₀₁₂₃
- The ten
- …largest difference between the ten metric combinations and g at the vertex
Einstein-equation paper, Section 2
2 · The lattice equation
Area times deficit, varied edge by edge.
The action is Regge’s: every triangle contributes its area times the deficit angle around it, curvature in lattice form. Its derivative with respect to one squared edge is that edge’s equation. On a smooth metric sampled at spacing s, the equation’s leading term, its coefficient of s², is the lattice’s field equation at long wavelengths.
Einstein-equation paper, Section 2, equation (3)
3 · Linear, then nonlinear
A linear identity and a change of coordinates give every order.
At a constant background the identity is linear in the second derivatives, and the lattice’s hinges pair up to give the Ricci tensor and its trace. A quadratic change of coordinates of a flat metric is still a flat placement of the lattice, so both sides vanish on it; that fixes the dependence on the first derivatives as well. No smallness of the metric’s distance from the reference, or of its derivatives, is needed, as long as the cells stay nondegenerate and Lorentzian.
Einstein-equation paper, Lemmas 3.1 and 3.2, Theorem 3.3
4 · What it does not say
The field equation, not yet its solutions.
The identity is about the equation’s leading term, not about lattice solutions. For a static worldline matter model, lattice solutions and the continuum equation agree through second order in the source. At third order the gravitational source matches Einstein’s cubic coefficient (proof complete, verification pending); the full cubic limit and nonlinear lattice solutions converging to solutions of the full Einstein equations are open. And at finite spacing one first-order static field admits no second-order correction on the thirty edges at a record: its second-order vertex-move defect is 756√3.
Compute the defect below · The static exterior, on the horizons page
Einstein-equation paper, Theorem 5.1, Section 6, Section 7 and Theorem 7.1
Which actions are Einstein’s
If flat space stays a vacuum at every stretch, the slope is Regge’s.
Regge’s action gives each hinge its area times its deficit angle. Any smooth charge C(A, δ) of a hinge’s area and deficit that vanishes at zero deficit makes a lattice action of the same form, and most of them do not have Einstein’s long-wave limit. The slope paper asks that flat space stay a solution when the lattice’s time diagonal is uniformly stretched, at every stretch beyond κ = 1. Then the charge’s slope must be Regge’s, σ(1)A, at every area A ≥ ½, and its long-wave quadratic and cubic vertices around the flat lattice are σ(1)/ℏ times Einstein–Hilbert’s, both from one ½∫√|g| R, whatever the charge does at higher order in δ. Newton’s constant stays free.
- How far from a vacuum
- A vacuum
Live · Proposition 4.1, from the areas of Table 2
1.500
- Class areas
- –A0 to A3 at this κ
- First
- 0σ(A2)/A2 − σ(A3)/A3, over σ(1)
- Second
- 0A0σ(A0) − A1σ(A1) + ¼A3σ(A3), over σ(1)
- Across κ
- –
What is computed. On the flat lattice of parameter κ the hinges fall into four classes, with areas A0² = (2κ − 1)/4, A1² = (3κ − 1)/2, A2² = 3(4κ − 1)/4 and A3² = 4κ − 1; at κ = 1 they are ½, 1, 3/2 and √3. Because the charge vanishes at zero deficit, the first variation of its action at a flat lattice reads only its slope at these four areas, and the lattice is a vacuum exactly when both conditions above vanish. The bump slopes are the paper’s own examples: Regge’s charge plus b(A) sin(kδ)/k, which reproduces every whole posting and adds the bump b to the slope. With the bump at 5/2 the slope is Regge’s below area 2, so every classical long-wave clause holds, and still flat space is not a vacuum at every stretch (Theorem 6.4). With the bump at ½ flat space is a vacuum at every stretch beyond a threshold above κ = 1, yet the long-wave quadratic vertex is not Regge’s: the threshold κ = 1 is sharp (Theorem 6.2). With the bump between 2 and 3 it is a vacuum on (½, 5/4], which does not force Regge’s slope (Proposition 6.3). The charge sAδ + A²(1 − cos kδ) has Regge’s slope and an even part k²A² that is not Regge’s: flat space is a vacuum at every stretch, since the condition reads the slope alone, and the even part does not touch the long-wave vertices (Proposition 7.2).
Slope paper, Table 2, Proposition 4.1, Theorems 6.2 and 6.4, Propositions 6.3 and 7.2
With JavaScript on, this figure checks Proposition 4.1 live for seven hinge charges as the time diagonal is stretched. Regge’s slope keeps flat space a vacuum at every stretch, with or without an even part; a slope with a bump at area 5/2, ½, or between 2 and 3 keeps it a vacuum only on some stretches; σ = A² and σ = √A on none.
The slope
σ(1)A
Regge’s, at every area A ≥ ½, for any C³ hinge charge with C(A, 0) = 0 that keeps flat space a vacuum at every stretch beyond κ = 1.
Slope paper, Theorem 3.1(i)
Quadratic vertex
Fierz–Pauli
The long-wave quadratic action on plane waves around flat κ = 1 is σ(1)/ℏ times −(√3π²/2) times the Fierz–Pauli form, the linearized Einstein operator.
Slope paper, Theorem 3.1(ii)
Cubic vertex
π²/2
The cubic vertex is the same factor times (π²/2) times the Einstein–Hilbert cubic vertex: both from one ½∫√|g| R in lattice units.
Slope paper, Theorem 3.1(iii) and Remark 3.2
Newton’s constant
free
The factor σ(1) is not fixed, nor its sign; read as Newton’s, G = 1/(8πσ(1)), which for Regge’s charge is G = 1/(8πs) (an inference).
Slope paper, Remark 3.2 and Section 9
Two cautions, as the paper states them. First, given C(A, 0) = 0 the vacuum condition is equivalent to Regge’s slope on the areas it reads (Theorem 5.1): the result recasts Regge’s slope as the statement that flat space stays a vacuum at every time-stretch, and does not derive Einstein’s equations from nothing. Second, it covers the long-wave plane-wave vertices at second and third order around flat space, not the full nonlinear theory, strong fields or other backgrounds. The charge’s even part stays free and drops out of these vertices (Proposition 7.2). Whether the ledger requires the premise is open, and κ = 1, edge lengths as a metric and time along the record diagonal are identifications.
Slope paper, Sections 1, 2.5, 5, 7 and 9
One order past Einstein
A static field the next order cannot correct.
At κ = 1 the static field of the Einstein-equation paper’s Proposition 7.2, the cubic potential Φ = ξ₁³ − 3ξ₁ξ₂² read at the edge midpoints with a fixed correction on ten edge classes, solves every first-order Regge equation. Moving one record y is a zero mode of the flat lattice: the flat Hessian annihilates it. But the field’s second-order rows on the thirty edges at y, paired with a move u of y, give 756√3 (u₀ + u₁ − u₂ − u₃). A second-order correction would have to cancel that pairing, and every correction pairs with the move to zero, so none exists there: the lattice’s vertex moves stop being a symmetry at second order in the field, one order in the derivatives past Einstein’s equations.
- Second-order rows at the thirty edges
- Each times the move of y
Live · computed in your browser
Move y by
- Defect
- …
- First order
- …largest of the thirty first-order rows: zero up to rounding (flat lattice …)
- Correction
- …a random correction’s second-order rows paired with the move:
- Time
- …for the thirty rows, seven times each
Computing the thirty rows…
What your browser does. It builds the squared edges ℓ₀ + t h around the record y = (2, 1, −1, 0): ℓ₀ = n − n² for an edge that advances n counters, the flat lattice at κ = 1, and h the field (31) of the paper. It computes the thirty rows at y as the figure at the top of this page does, from triangle areas, the angles of each 4-simplex and Schläfli’s identity, and takes their first and second derivatives in t from symmetric differences at t = 2×10⁻⁴, 10⁻⁴ and 0.5×10⁻⁴ with two Richardson steps. Moving y by u changes an edge that leaves it by −2dᵀgu and one that arrives by +2dᵀgu, with g = I − 11ᵀ. The correction gives every edge an arbitrary value. Nothing is taken from the paper’s programs; the check covers the thirty rows at y, and the paper proves the first-order equations at every record.
What it does not say. The obstruction is exact for this first-order field and these local rows. A different first-order field, a restricted equation or a modified action are left open, and so are nonlinear lattice solutions.
Einstein-equation paper, Section 7.2, Theorem 7.1, Proposition 7.2 and equation (31) (status: theorem, independently verified); quantum gravity paper, Theorem 4.3
With JavaScript on, your browser computes the thirty Regge rows at the record y for the static field of Proposition 7.2: the first-order rows vanish, and the second-order rows paired with a move of y along one counter give 756√3 ≈ 1309.43.
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 Einstein-equation paper is Einstein’s Metric Field Equation at Every Order from the Recognition Lattice; the quantum gravity paper is Quantum Gravity from the Recognition Ledger, version 1.2.
Words on this page, in plain terms: Regge action · hinge charge · Einstein–Hilbert action · Fierz–Pauli form · slope and even curvature · squared edge length · deficit angle · edge class · four-simplex · vertex move · κ (kappa).
The gravity portal · Gravitational waves on the lattice · Next: Newton’s law from the lattice · Glossary · Search every page