Why does
matter fall?

Water flows through a whirlpool while its shape persists. The lasting thing is a pattern of motion.

RS asks how matter persists through its own internal activity. In its light-pattern account, a material object persists through recurring physical changes and relationships. The dynamics must explain how those changes preserve the object.

Here, recognition means a physical distinction and comparison: a later state can carry information about an earlier one. It does not require a mind watching. And “light pattern” is a proposed microscopic account, not a claim that an atom is a tiny whirlpool of ordinary visible light.

A pendulum swings. An atom oscillates. A clock uses a repeatable change to count duration. To compare two clocks, ask: how many cycles does one complete while the other completes a fixed number?

Distance is also a physical comparison. For example, send a light pulse to a mirror and back. Half the round-trip time, multiplied by the local speed of light, gives the distance in that observer’s measurement.

RS seeks to build these clock and distance relationships from its physical events and their connections. The resulting network of times, lengths and angles is what physicists call geometry. That account must supply physical measurements: knowing which events are connected does not yet say how many metres or seconds separate them.

Suppose two neighbouring states have a positive ratio x. Perfect agreement is x = 1. RS’s canonical comparison rule assigns a number to their mismatch. It gives the same number whether you compare A with B or B with A.

Now connect many comparisons. Changing one location changes the comparisons on its links. Those links connect to other locations. A locally imposed source can therefore require a field extending beyond the source. A field is simply a value at every location.

Here is a small, exact version of that idea. The two ends are held fixed. A source at the centre favours a lower central value; the comparison rule resists abrupt differences. The balanced shape fits these requirements together.

Why does the balance bind things together?

Binding depends on the combined energy of matter and field. The field costs energy to establish, while the matter energy falls. In the weak-field balance, the matter energy falls by twice the field’s positive cost. The combined change is therefore negative.

Try making too little or too much field around a fixed source. At first, the reduction in matter energy wins. Eventually the extra field cost wins. The minimum is the balanced field.

Now place two positive sources near each other. Each responds to the other’s field through the same clock coupling. After subtracting their separate self-energies, the three-dimensional weak-field interaction is negative and proportional to 1/separation. Halve the separation and this interaction energy becomes twice as negative. Its derivative gives attraction.

The energy minimum specifies the static balance and force. How the field forms requires time-dependent equations and boundary conditions. The next chapter shows the motion of a test object once the field is present.

A connected field must satisfy comparisons at every location, including locations with no matter source of their own.

For gravity, the next physical identification is to connect this field to clock and distance measurements. In the weak-field construction, it becomes the gravitational potential: a quantity whose spatial differences determine free-fall acceleration. The comparison rule plus a specified source gives a response law; choosing what physically supplies that source and how the field is measured is also part of the model.

Read the same field as clocks and motion.

To read the example physically, rescale the balanced field and identify it with gravitational potential in the weak-field limit. Lower potential means a slower clock rate relative to the clocks at the ends. This clock reading is an input connecting the comparison model to physical measurements.

Look at the same three central locations. The centre has the lowest value and counts the fewest cycles. On the left, the potential falls as you move right; on the right, it falls as you move left. Free-fall acceleration points toward lower potential, so it points toward the source from both sides. The slope, or change per unit distance, sets the acceleration. A constant clock-rate offset everywhere would produce no slope and no such acceleration.

Now zoom in on either side of the source. Over a small region, the potential has an almost constant slope. We have shown the inward acceleration arrows; next we explain them through the motion of a matter wave. A held clock is a way to measure the rate field, not a separate device pulling on an object.

Phase is position within a wave’s repeating cycle. Think of an angle on a clock face. A matter wave’s phase changes at a rate set by its energy. A slower physical clock rate therefore changes how much phase it accumulates compared with the same matter at a higher rate.

Near Earth, the clock rate increases upward. Over the same reference time, different heights acquire different extra phases. This makes a spatial phase gradient: a tilt in phase across the wave.

In quantum mechanics, a spatial phase gradient carries momentum. This connects timing to motion. With the clock-energy coupling used here, the extra phase gradient gives momentum downward. As time passes, that downward momentum grows: the object accelerates toward lower clock rate.

Clocks are not pulling on the object. The same physical rate enters its matter-wave equation, and that equation changes its motion. A constant rate offset everywhere would shift the phase uniformly, producing no such force.

Why do different masses fall together?

A mass three times larger acquires three times the phase tilt and three times the momentum change. But it also takes three times the momentum change to give it the same change in velocity. The same mass appears in both places and cancels. That is why the two acceleration readings agree.

This cancellation uses a substantive physical condition: the energy coupled to the rate field and the mass governing inertia belong to the same matter dynamics. It would not follow merely from calling an arbitrary source coefficient “mass.”

What changes when you fall with it?

Picture a small room falling with the object. In that room’s frame, the local first-order timing difference disappears. This is another description of the same acceleration. The RS coherent-continuation result identifies this cancellation for a supplied potential.

A floor prevents that freely falling motion by pushing upward on you. That supporting push is what you feel as weight. Try finding the falling frame below, then add a tidal difference that cannot be removed across the whole room.

At free fall, all three marks line up if the field is uniform over this small region. Add a tidal difference: the centre can still be in free fall, but one common acceleration cannot erase the remaining variation across the whole object.

This is the difference between free fall here and curvature across a region. Two neighbouring free objects can change their separation even while neither feels a supporting push.

We have connected energy balance, clock rates and motion in a weak static field. Full gravity also includes distances, directions, pressure and propagating disturbances. These must belong to the same coupled description.

RS’s comparison approach asks both questions using one mathematical rule. You have already used a simple example: the chain’s total cost. Test a small change in its shape, and the change in cost tells you whether it is balanced.

For moving systems, the rule assigns a number to a whole possible history. That number is called the action. Its equations come from testing tiny changes to the history and finding where the score has zero initial slope for each allowed change. This is called stationarity.

Now ask two questions of that same action. Change the matter while keeping geometry fixed: this gives the matter equation. Change the geometry while keeping matter fixed: this gives the matter’s gravitational source.

Why change geometry? Because the same physical pattern is compared differently if you change lengths, angles or clock rates. The change in its action tells you how strongly it responds. This includes directional flow and pressure, not only “how much stuff is here.”

Stretch the setting. Keep the pattern difference.

Imagine a field whose value changes from one side of a small region to the other. Keep that difference fixed. Stretch the region horizontally: the same change now happens over a longer distance, so the horizontal gradient becomes smaller. The region also becomes larger. Both effects enter the matter action.

For this particular action, doubling the horizontal length halves the score when the field varies horizontally. When the field varies vertically, the horizontal stretch leaves that gradient unchanged and doubles the score through the larger volume. The response remembers direction. That is why a gravitational source carries more information than a single amount of mass.

Clocks, distances and angles belong to one geometry. The same field that affects timing must also give consistent spatial measurements. Matter and geometry are then solved together; changing the matter changes the problem the geometry must satisfy.

Once a gravitational field is established, Newton’s law describes its slow-motion, weak-field effects. For an isolated spherical source in that limit, the field spreads through three-dimensional space. Double the distance from its centre and the same total flux is spread over four times the area: the acceleration is one quarter as large.

An apple falls almost straight down because it has little sideways speed. The Moon is continually falling too, but its sideways motion carries it around Earth. A space station and its occupants fall together, so the occupants do not press on a supporting floor.

Einstein’s geometric description makes the larger connection: freely moving matter follows the locally straight paths of spacetime, while relative free fall reveals curvature. RS asks how the patterns and comparisons beneath that description can produce it. The sourced-field construction and the coupled example above are concrete parts of that work.

Everything so far has been classical: a field settles into balance, and matter moves through it. A quantum theory of gravity asks for more. It does not pick one geometry. It adds up every geometry the lattice can carry, each with a phase, and the classical field has to come out of that sum when waves are long.

In RS, events are records kept on a ledger. Four counters advance one step at a time, and each step is called a posting. The records form a four-dimensional lattice, and the candidate quantum theory is a sum over the geometries of that lattice in real time, weighted by Regge’s action, the form general relativity takes on a lattice. Quantum Gravity from the Recognition Ledger asks whether that sum is a quantum theory of gravity. A yes needs four things.

The shortest wave and the number κ

The flat lattice has one adjustable number, κ. It sets the length of every edge, and a posting, one counter’s single step, has squared length 1 − κ. RS identifies κ = 1. It is the only value at which the lattice’s interval reads the order of the postings over some stretch of time, and there each posting is null: it moves exactly at the speed of the light cone. In one tick a posting moves its position by √3/2, and the cone’s speed is √(κ − ¼), so κ = 1 says that the two speeds agree.

The paper finds that the same value does something else. On a lattice the shortest wave along an axis flips sign at every step. Usually the lattice pushes back on it, so exciting it costs energy. The push is κ − 1 times a fixed amount, and at κ = 1 it vanishes. The lattice then carries four extra massless waves, one for each axis, at the edge of its range of wavelengths. They are called doublers, and lattice models of quarks and electrons have them too.

Across every κ above ½, massless corner waves occur only at κ = 1 (Theorem 2.1). So deriving κ = 1 and deriving the doublers are the same problem. The foundation as stated does not decide κ: a world at κ = 1 and a world at κ = 3/2 satisfy the same premises (Theorem 2.3). Try the second one above. Its order of postings is still causal, but its interval no longer reads it, and its corner waves have a restoring push.

A cost of curvature gives the doublers a gap

The doublers spoil the first quantum correction. A calculation like this needs a regulator to stay finite, and the regulator must be removed at the end. As it is removed, one response of the graviton at zero momentum grows like the logarithm of one over the regulator, without limit, in the stated model of the one-loop diagrams. That was the result of the earlier paper on this lattice. The growth is plain in that paper’s model of the four corners: once the regulator is small, each further factor of ten by which it shrinks adds the same amount. Give the corner waves a gap, and the growth stops.

A doubler differs from the harmless lattice motions in one checkable way: it bends the lattice. To first order its deficit angles, the lattice’s measure of curvature, do not all vanish, while those of every flat change do (Theorem 5.3). So a weight that charges for curvature can see the doublers without touching flat space. Under two identifications that the paper names, the ledger supplies one: its recognition cost J applied to the curvature around each hinge, with no coefficient left to choose once a count of channels at each hinge is fixed (Theorem 5.4).

The weight is zero on flat changes, falls off with the fourth power of the wave number on long waves, so it leaves Einstein’s limit alone, and is positive on all four doublers (Theorem 5.5). With it, the real parts of the graviton’s one-loop responses in the same model have finite limits, the one that diverged included (Theorem 7.4). Lattice models of quarks handle their doublers with a term of the same structure, found by Kenneth Wilson; the likeness is one of structure only, since the fields here are bosonic and the sum runs in real time. Here the term comes from the ledger’s cost, under named identifications, rather than being chosen.

Does the sum evolve in time?

A quantum theory also has to evolve, and its evolution has to be unitary: it must keep the total probability fixed. On the lattice the clock is record time, the count of postings, and the test is whether some positive measure of a wave’s size, a norm, stays the same from tick to tick. The sum without the weight fails it at κ = 1. Its linearized equation has a solution, not a mere move of the vertices, that grows by a factor √2 at every tick, so no such norm exists (Theorem 8.1).

The weight enters the evolution as a damping, so at Gaussian order the weighted sum is not stable on the lattice either. The long waves are another matter. Hold the weight’s strength, its size relative to the action, fixed. Then, as the lattice spacing shrinks relative to the wavelength, every family of physical modes that approaches a steady oscillation grows by an amount per unit of physical time that goes to zero, and no family of lattice-scale modes, the doublers included, approaches a steady oscillation away from zero wave number. Among the modes that approach a steady oscillation, the limit sees only the graviton’s branch. In that sense the record-time roots of the weighted sum, at its Gaussian order, are stable in the continuum limit, given the weight’s two identifications and an inference that the lattice-scale modes, which may grow from tick to tick, do not affect the continuum theory (Section 8.1). That is a statement about the roots of the equation, not an evolution that keeps a norm: the paper does not construct one, and it remains open. If the spacing itself shrinks, the strength changes with it, and that case is not proved.

The open question: is the graviton massless?

Gapping the doublers does not settle the graviton itself. At zero momentum the one-loop correction, in the stated diagram model, leaves a remainder that is not a function of the volume, and whenever it is nonzero it acts as a mass term. The paper proves an exact test. The remainder vanishes if the one-loop determinant does not change under volume-preserving linear moves of the lattice’s vertices, and in general it equals i/2 times the difference of two such defects (Theorems 6.2 and 6.4).

Every candidate computed to convergence leaves the remainder nonzero (numerical). No real weight or density can supply the missing symmetry, and adding terms of third or higher order in the deficit angles to the action leaves the remainder as it was, at the same slope and weight (Theorems 6.6, 6.7 and 6.10). Zero has been reached only by choosing lattice-scale numbers by hand (numerical), which is a fit, not a derivation. Without the weight, the model’s graviton is not massless at one loop (Theorem 3.3); with it, the computed remainder is still nonzero. Read through named identifications, the computed remainders amount to a one-loop mass scale of the model near 10²⁸ eV, about fifty-one orders of magnitude above the measured bound (numerical): a diagnostic of the path sum as it stands, not a prediction. An action derived from the ledger whose remainder vanishes is the paper’s precise open target. Neither of the two routes that are not local supplies it at κ = 1 (awaiting verification): a phase built from the holonomy of postings is the Regge phase itself, and the action that the ledger’s refinement map induces has no fixed point with Einstein’s long-wave form.

Einstein’s equations at long wavelengths

The classical side is where the lattice is strongest. With a resting body and a model medium, exact lattice solutions have linearized Einstein gravity as their long-wave limit, and a compact body gets Newton’s potential with G = 1/(8πs), where s is the scale of the action. With matter that conserves postings in place of the medium, the agreement extends to second order in the field for a static body at small strength, under named identifications and two model assumptions. Read as a statement about continuum fields it holds at first order and at every nonzero wave of the second order; the uniform part is balanced by a constant that is left free rather than checked. Every long-wave vertex of the action is Einstein’s: on a smooth metric sampled at the edges, the metric part of the lattice field equations is the full Einstein density, nonlinear terms included, for every κ above ½ (Theorem 4.2). That is a statement about the field equations, not about their solutions: what is missing is lattice solutions at finite strength. One obstacle is known for vacuum: the lattice’s vertex moves stop being a symmetry at second order in the field, one order in the derivatives past Einstein’s equations (Theorem 4.3), so exact static vacuum solutions of all the lattice equations meet an obstruction there (Section 4).