Gravity / Newton
A resting mass gives Newton’s law.
Put a smooth body at rest on the lattice and solve for its static field. Outside the body the potential is Newton’s −GM/r, with G = 1/(8πs) fixed by the scale s of the lattice action. The lattice shows only at the body’s own scale, and that correction shrinks as the body grows.
Newton’s law at long wavelengths: provedκ = 1: identified, not derived
- The lattice, three directions
- Newton’s law
- The body
- Static resonance
Live · solved from Appendix B
Lattice ÷ Newton − 1, by direction
- (1,1,−1,−1)
- (3,−1,−1,−1)
- (−1,1,0,0)
- distance in lattice units
Contours of R = χttq/cN, the lattice’s static response over Newton’s kernel, at 1/R = 0.99 (heavy: within 1% of Newton), 0.9, 0.75, 0.5 and 0.25. Red: R is infinite, a static zero mode of the lattice. Magenta: the body’s waves, out to |k| = 2/σ; dashed, the band limit of the computed body.
Solving the lattice equations
6.0 LATTICE UNITS
A smaller body has more short waves, and short waves are where the lattice differs from Newton. A body cannot be smaller than a few lattice units at all: its waves would reach the resonance.
In this computation
- Centre
- –deeper than Newton
- At 3σ
- –from Newton, largest of three directions
- At 60
- –from Newton at 60 lattice units
- Waves
- 0.33the body’s reach in |k|; the nearest resonance is at 0.741
What is live. Your browser builds the κ = 1 operator from the cell Hessian printed in Appendix B, as the program kuhn_bloch.py does, and solves its static response at 38,400 waves, then sums it over a smooth body: a Gaussian of width σ whose spectrum is cut off smoothly below |k| = 0.7, inside the nearest resonance. Newton’s curve is the potential of the same body under Newton’s law.
What it is. The linear static field of the lattice, the kernel of Theorem 5.4. Theorem 4.10 proves the full statement for exact solutions.
With JavaScript on, this figure solves the lattice’s static field for a smooth resting body in your browser and sets it against Newton’s −GM/r. For a body of width 6 lattice units the lattice potential is 3.7% deeper than Newton’s at the centre, within 0.8% of it at three body widths and within 0.2% at 60 lattice units.
The numbers
Newton, to the precision anyone has tested.
Newton’s constant
G = 1/8πs
The only coupling for which every static mode of a point mass has httq = 8πGM, the lattice’s Newton clause.
Earlier paper, Section 3.5
Inverse-square law
3×10−23
The largest relative correction to Newton’s kernel at any wave that tests of the inverse-square law reach, down to 52 μm, for a lattice spacing up to 106 Planck lengths.
Earlier paper, Theorem 5.4
Static resonance
10.3
Lattice spacings: the wavelength of the static resonance along (−1,1,0,0), where cos t* = 9/11 and |k| = 0.866. It is a lattice artefact, and the reason the body has to be smooth.
Earlier paper, Section 4.4 and Appendix B
Nearest resonance
0.741
The smallest |k| at which the static response diverges, along the four postings’ spatial steps, (3,−1,−1,−1) and its images.
Numerical: a scan of 1,500 static directions; along (3,−1,−1,−1) it is 0.7407, as kuhn_bloch.py --static finds
How it works
Why Newton comes out.
1 · The source
A resting body pulls on the time edges.
A body at rest on the lattice is a clock on the time edges, the edges from 0000 to 1111 of each cell, with the point-particle action of a static mass. Its source sits on those edges, and the field equation is s H₀h = −τ: the lattice’s own Hessian, scaled by the action scale s, against the source τ.
Earlier paper, Section 3.5. Reading M as the body’s mass uses the point-particle reading of a clock, which the paper labels a model.
2 · The kernel
Long waves answer exactly as Newton requires.
Wave by wave, the time-time response χtt of the gauge-fixed operator to that source, times q(k) = |k|², tends to the Newton coefficient cN as the wave gets long, in every static direction, with no adjustable factor. The ratio R = χttq/cN is the map above: 1 at the centre, within 1% of 1 inside the heavy contour. Only one coupling makes this hold for every static mode, G = 1/(8πs).
Earlier paper, Section 3.5 and Theorem 5.4: |R − 1| ≤ 2.2 × 1026‖k‖² for 0 < ‖k‖ ≤ 10−10
3 · The potential
Outside a smooth body, the potential is −GM/R.
For a smooth static body, the exact solutions of the coupled lattice equations have −½htt = −GM/R at proper distance R outside the body, up to an explicit finite-size remainder and in a suitable lattice gauge, with G = 1/(8π) at unit action scale. The limits are taken in order: the coupling goes to zero on each torus first, then the torus grows. On a closed torus a static positive mass alone has no small solution, so the theorem’s body carries a uniform compensation of its mean density.
Earlier paper, Theorems 4.7 and 4.10
4 · The resonance
The body has to be smooth.
Along (−1,1,0,0) the static lattice has a zero mode at cos t* = 9/11, a wavelength of about 10.3 lattice spacings, with its largest component on the time edge, so a resting source is not solvable there. It is a lattice artefact of the kind familiar from finite elements, and every compact lattice body of nonzero mass couples to it. So the theorem covers band truncations of smooth bodies, and its content is the long-wave part.
Earlier paper, Section 4.4, Theorem 4.10 and Proposition 4.11
5 · One loop
The quantum correction carries 41/(10π).
With the lattice’s long-wave rules, the one-loop potential between two masses gains −(41/10π)G²ℏm₁m₂/(r³c³), read through named identifications: the value of general relativity’s effective theory. It is a theorem about the long-wave rules. At κ = 1 the four corner doublers also see a mass’s static field, and whether the full lattice sum keeps this coefficient is open.
Earlier paper, Theorem 7.5, Corollary 7.9 and Proposition 7.6
Light bending
Space curves as much as time: γ = 1.
Newton’s law is the time-time part of a resting mass’s field. General relativity also curves space, hij = (2GM/r)δij, which is what bends starlight and delays radar echoes; the Cassini measurement holds the parameter γ that compares the two parts to |γ − 1| < 2.3 × 10−5. The sourced-Einstein paper derives γ = 1 on the lattice at long wavelengths, at every κ > ½, with no parameter adjusted.
The two parts
γ = 1
The time-time and space-space parts of a static mass’s field are the same function times the metric, gauge invariantly, at the level of the long-wavelength symbol, with no mixed part.
Sourced-Einstein paper, Theorem 4.5
Far away
2GM/r
A band-limited static body has, in de Donder gauge and in the scaling limit, h00 → 2GM/r and hij → (2GM/r)gij: the linearized Schwarzschild far field, with G = 1/(8πs).
Sourced-Einstein paper, Theorems 6.1 and 6.2
Any matter
16πG/q
Sourced by any conserved stress tensor, moving matter included, the long-wavelength field is hμν = (16πG/q)(Tμν − ½gμνT): the linearized Einstein equations, with the coefficient fixed by the action scale alone.
Sourced-Einstein paper, Theorems 4.2 to 4.6
The proviso
One wave
The far-field theorems use smooth, band-limited sources. At one static wave, cos t* = 1 − 6(2κ − 1)/(κ(4κ − 1)(16κ − 5)), the static operator has a zero mode that is not a coordinate change, and an unsmeared point source has no solution there; at κ = 1 that is the resonance above, cos t* = 9/11.
Sourced-Einstein paper, Theorem 5.1
And β = 1. The slope paper goes from the Regge action to its solutions at κ = 1: the lattice’s own long-wave solutions with point-mass jets have γ = β = 1, the values that light bending, the Shapiro delay and the perihelion advance measure, and such a solution exists near a point, the isotropic Schwarzschild metric expanded through second order in the mass. These concern the isotropic point-mass ansatz on a ball of radius 1/8, long-wave and through second order in the amplitude, not exact solutions at finite size.
Slope paper, Section 8, Theorems 8.2 and 8.3
These are linear statements about the flat lattice, proved at every κ > ½. Open, as the paper states them: the far field of a compactly supported body and of an unsmeared point source; a far field for hij that is gauge invariant in position space (Theorem 6.2 is a de Donder statement, and its gauge-invariant form is the symbol-level Theorem 4.5); the nonlinear theory; and the derivation of the lattice itself.
Sourced-Einstein paper, Sections 1, 4 to 6 and 8; Cassini: Bertotti, Iess and Tortora (2003), as the paper cites it
The value of G
Newton’s constant is one number.
With the speed of light and Planck’s constant fixed, Newton’s constant is one pure number. On the ledger that number is Ne, the count of ledger ticks in one Compton period of the electron, and the tree-level coupling is π²ℏc/(ln 2 · me²Ne²). The measured G puts Ne at 9.0156 × 10²² at tree level. Deriving Ne from the ledger is the open target.
G in SI units is one number: provedThe phase premise: identifiedNe: open
The one number that fixes G
Live · Section 5 of the Newton’s constant paper
–
At this Ne
- G
- –m³ kg⁻¹ s⁻², tree level
- ÷ measured
- –against 6.67430 × 10⁻¹¹
- αG
- –Gme²/(ℏc)
Readings, not predictions
- Record length
- –m, λrec = h/(mecNe)
- Tick
- –s, λrec/c
- Spacing
- –m, b = (2/√3)λrec
What is live. The relation of Section 5, G = π²ℏc/(ln 2 · me²Ne²), with h and c exact, me = 9.1093837139 × 10⁻³¹ kg and, for comparison, the CODATA G = 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻². This is the tree-level coupling, G = 1/(8πs) in the ledger’s units. A laboratory measures Z times it, with a positive factor Z that the quantum path sum fixes and that is not yet computed, so the measured G fixes Ne/√Z = 9.01558(10) × 10²², the window (9.0154, 9.0158) × 10²² at two standard uncertainties.
What it rests on. The phase premise, an identification: a face of area λrec² holds one bit, so a full turn around it halves the weight; then the Planck length is λrec/(2√ln 2) ≈ 0.6006 λrec. The spacing reading also takes κ = 1 and light crossing one record length per record rank, which gives b ≈ 1.92 Planck lengths. The record length, tick and spacing shown are readings of the measured value at Z = 1, used nowhere as inputs. The present premises leave Ne free.
Newton’s constant paper, Sections 2 to 5 and Target 5.1
A second road to the same number. In the ledger’s own units the measured vacuum energy is also one number, and with H0 and ΩΛ measured it fixes G as well: a derivation of the vacuum energy would also predict G. The energy of empty space
Cosmological-constant paper, Section 6
The quantum correction
At a millimetre, the one-loop correction changes Newton’s law by 3 parts in 10⁶⁴.
The lattice’s long-wave rules give the one-loop correction to the potential between two masses, −(41/10π)G²ℏm₁m₂/(r³c³) in an infrared-subtracted reading: the coefficient general relativity’s effective theory gives. Relative to Newton’s term it is (41/10π)(ℓP/r)², of order one only at the Planck length. Whether the lattice path sum itself has this coefficient is open: at κ = 1 its doublers couple to a mass’s static field and could add a more singular term.
- (41/10π)(ℓP/r)²
Live · the coefficient of Theorem 7.5
1.0×10−3 m
- Planck
- 6.2×1031Planck lengths
- Correction
- 3.4×10−64of Newton’s potential
What it rests on. The coefficient is a theorem about the long-wave rules, which are the Einstein–Hilbert rules in lattice coordinates, read through named identifications; it is not yet a property of the lattice path sum. And a test will not come from it: a one-loop correction of relative size ξ(ℓP/r)² stays below the sensitivity of short-range tests of gravity at every distance they reach, for every |ξ| up to 10⁴⁰.
Earlier paper, Theorem 7.5 and Corollary 7.9; quantum gravity paper, Section 8.3
With JavaScript on, choose a distance: the one-loop correction is (41/10π)(ℓP/r)² of Newton’s potential, about 3 × 10−64 at a millimetre.
Check it
Run the program.
The same operator, rebuilt from Appendix B in Python. It checks the static normalization and the resonance printed in Appendix B, and with --static writes the ratio along the three directions drawn above.
kuhn_bloch.py Python 3 with numpy
sha256 b7ca85f6ff5e5aa92b37d46589658f1f10f4b40effa93c3dd7d66d8ca6483059
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 earlier paper is Linearized Einstein gravity from Lorentzian Regge calculus on the Kuhn lattice; the sourced-Einstein paper is Sourced Einstein equations from the recognition lattice, October 2026; the quantum gravity paper is Quantum Gravity from the Recognition Ledger, version 1.2; the Newton’s constant paper is Newton’s Constant and the Recognition Ledger, version 1.2; the cosmological-constant paper is The Cosmological Constant on the Recognition Lattice, version 1.2.
Words on this page, in plain terms: action scale s · Ne · phase premise · γ · κ (kappa).
The gravity portal · Einstein’s equation from the lattice · One unit on the lattice · Next: the four doublers · Newton in the full account · The earlier paper (PDF) · Glossary · Search every page