Gravity / Glossary

The words the lattice uses, in plain terms.

Each entry says what the word means on these pages, and where you can see it work. The status words at the end are the ones every page uses to say how far a result has got.

The ledger and the lattice.

Record: An event kept on the ledger. The records are the points of a four-dimensional lattice, one coordinate for each counter. See it work
Counter: One of the four accounts of the ledger. They advance one step at a time, and each step is a posting.
Posting: One step of one counter: the edge from a record to the record one step further along that counter.
Record count, record time: The number of postings so far, t(x) = x₀ + x₁ + x₂ + x₃. It orders the records before any geometry is assigned, and it is the natural clock of the path sum. See it work
Tick: A lattice translation that advances the record count by one.
Kuhn lattice: The lattice with each unit four-cube cut into 24 simplices, one for each order in which the four counters can advance. See it work
Four-simplex: The four-dimensional triangle: five records and the ten edges between them. Each order of the four counters through a cell is a chain of five records, from 0000 to 1111, and those five span one four-simplex; the 24 of them fill the cell. See it work
Time edge: The edge of a cell from 0000 to 1111, along which all four counters advance. Every one of the cell’s 24 four-simplices contains it. See it work
Edge class: The fifteen nonempty strings of four bits, one for each set of counters an edge advances. Every vertex has one edge of each class. See it work
κ (kappa): The one number that fixes the lattice’s flat geometry: the metric I − κ11ᵀ in record coordinates. At κ = 1 every posting is null, on the light cone, and the lattice interval reads the posting order. Read another way, κ is a speed: one posting moves at v with v² = 3/(4κ − 1) in units of light’s, so κ = 1 is a posting at the speed of light and κ ≥ 1 is the causal range. κ = 1 is an identification; the foundation does not derive it. See it work

Geometry and the action.

Squared edge length: The lattice’s variables, one number for each edge. A smooth metric g gives an edge v the squared length vᵀgv; a vertex has fifteen of them for the ten numbers of a metric, so five combinations are lattice variables of their own.
Regge action: The lattice’s action: every triangle contributes its area times the deficit angle around it. Its leading field equation on a smooth metric is Einstein’s. See it work
Hinge charge: What each hinge contributes to a lattice action of Regge’s form: a smooth function C(A, δ) of the hinge’s area and deficit angle that vanishes at zero deficit. Regge’s charge is sAδ. A charge that keeps flat space a vacuum at every stretch of the time diagonal beyond κ = 1 has Regge’s slope, and its long-wave vertices are Einstein–Hilbert’s (the slope paper). See it work
Einstein–Hilbert action: General relativity’s action, (16πG)⁻¹∫√|g| R: the curvature R summed over spacetime with its volume. The lattice’s long-wave quadratic and cubic vertices are compared with its expansion around flat space. See it work
Fierz–Pauli form: The quadratic part of the Einstein–Hilbert action on plane waves around flat space: the linearized Einstein operator, with its polarizations. A charge that keeps flat space a vacuum at every stretch has this long-wave quadratic action, times σ(1)/ℏ and a fixed normalization, −√3π²/2 (the slope paper, Theorem 3.1). See it work
Deficit angle: 2π minus the sum of the angles of the simplices that meet at a triangle. It is zero when the lattice is flat: curvature in lattice form.
Action scale s: The overall factor of the Regge action. Newton’s law on the lattice holds for exactly one coupling, G = 1/(8πs). See it work
Vertex move: The change of squared edge lengths that displacing the vertices of the flat lattice produces. It describes the same flat space, so a solution that is a vertex move is not physical. See it work
Constant strain: A uniform stretch of space and time. In the continuum it is only a change of coordinates, so the one-loop effective action of constant strains depends on the strain only through the volume; on the lattice what is left is the remainder. See it work
Face Laplacian: The operator of the cubic lattice that weighs a cell by 6 and each of its six face neighbours by −1. In the point-source paper the static field equation of the lattice action, on the deformations that carry gravitational waves, is this operator’s Poisson equation. See it work
Green function: The field of one unit at one cell: the solution of the face Laplacian’s equation with that source that decays far away. It is finite at the source, 0.2527… (Watson’s integral), and falls off as 1/(4πr) in every direction. See it work
Light-bending parameter γ: The number that compares the space-space part of a static mass’s field with its time-time part, the part that bends light with the part that gives Newton’s law. General relativity has γ = 1, and the Cassini measurement holds |γ − 1| below 2.3 × 10⁻⁵. On the lattice γ = 1 at long wavelengths, at every κ > ½ (the sourced-Einstein paper, Theorem 4.5). See it work

Waves and time.

Brillouin zone: The waves of the lattice, k in [−π, π]⁴. Its sixteen corners are the waves with every coordinate 0 or π. See it work
Light cone: The waves that travel at light’s speed. Long lattice waves travel on it in every direction, with corrections of order (kb)² for a spacing b. See it work
Doubler: A massless lattice wave at an axis corner of the Brillouin zone, on a cone of its own. At κ = 1 there are four, and in the whole family κ > ½ they occur only there. See it work
Record-time root: The factor w by which a wave is multiplied at each tick. A norm that the tick keeps needs every physical root on the unit circle. See it work

The quantum theory.

Path sum: The sum over the lattice’s geometries in real time, each weighted by a phase from its Regge action: the candidate quantum theory of gravity. With a stated regulator it is a finite integral, so its expectations are defined off a discrete set of phase scales (the finite-path-sum paper); the regulator is not derived. See it work
Phase premise: The identification that reads a face of area λ_rec² as holding one bit: a full turn around it, weighed in imaginary time, halves the weight. It fixes the product of the phase and action scales, and with it the Planck length in record lengths. See it work
N_e: The number of ledger ticks in one Compton period of the electron. In SI units the tree-level coupling is G = π²ℏc/(ln 2 m_e² N_e²); deriving N_e is open. See it work
One-loop diagram model: The stated model in which the one-loop statements hold: a gauge term of unit weight, no ghost line, the Feynman iε. A ghost or gauge measure derived from the ledger could change it. See it work
Ordered limit: The one-loop limit taken in a fixed order: first the torus grows at a fixed regulator ε > 0, then the regulator is removed. See it work
Curvature weight: A measure weight built from the ledger’s cost of curvature, under two named identifications and a channel count. It gaps the four doublers and vanishes to fourth order in the wave number on long waves. See it work
Remainder: The part of the one-loop response to a constant strain that is not a function of the volume: a mass term for the graviton in the model’s effective action. See it work
Transverse-traceless probe: A strain H of the flat lattice that is symmetric, zero on the postings and traceless for the metric. These strains make a five-dimensional space that carries both irreducible representations of the axis permutations, and the computed probes f_E and f_T2 are one in each. A massless graviton needs the remainder to have ordered limit zero on every such probe, one by one: on their sum H_tt alone, the two could cancel without either vanishing. See it work
Corner condition: A condition on an even term w added to the Regge charge: at unit action scale, w′(½) − w′(3/2) = −1/(3√3). It holds exactly when the derivative along the strain f_T2 of the phase’s quadratic form on the doubler vanishes at all four axis corners, on every torus of even side. Without it, the corner logarithm stays and the graviton is not massless (an inference). See it work
Slope and even curvature: For a local hinge charge C(A, δ) that vanishes at zero deficit, two functions of the hinge’s area: the slope σ(A), its first derivative in the deficit δ at δ = 0, and the even curvature e(A), its second. The one-loop remainder, on every probe, with every weight and at every regulator, depends on the charge only through these two (proof complete, verification pending). Within the readings of a charge that the ledger supplies, e is 0 or βA. The slope paper fixes the slope from a vacuum condition: if flat space stays a vacuum at every stretch of the time diagonal beyond κ = 1, the slope is Regge’s, σ(1)A, at every area A ≥ ½, and the even curvature, which stays free, drops out of the long-wave vertices. The slope, live See it work
Ward defect: How far the one-loop determinant fails, at second and at first order, to be unchanged by volume-preserving linear moves of the vertices. The remainder is i/2 times the second defect minus the first. For a classical field the static-exteriors paper uses the same word one order up: the second-order Ward defect pairs a field’s third variation with a move of one vertex, and for the static cubic field at a record it is 756√3. See it work, one loop One order up

Status words.

Every result on these pages carries one of these words, used as the papers use them.

Proved: Proved as stated in the paper named, with the premises listed there.
Independently verified: A proof that has also been checked independently.
Proof complete, verification pending: A complete proof whose independent verification is not yet finished. It is never listed as proved.
Identification: A physical premise that the definitions do not force, such as κ = 1 or the phase premise.
Model: An object specified for study, such as the one-loop diagram model or a matter carrier.
Inference: A consequence stated without a completed proof.
Numerical: Evidence from a computation at stated inputs; it supports a statement only there.
Open: A precise statement that is neither proved nor refuted.

Sources: the status words as defined in the quantum gravity paper (Table 1), the Einstein-equation paper (Section 1) and the static-exteriors paper (Section 1); the phase premise as stated in the Newton’s constant paper (Section 3); the face Laplacian and the Green function as in the point-source paper (Sections 2 to 4); γ as in the sourced-Einstein paper (Theorem 4.5); the path sum’s finiteness and κ as a speed as in the finite-path-sum paper (Table 1 and Theorem 2.11); and the slope from a vacuum as in the slope paper (Theorem 3.1).

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