Gravity / The quantum question

On the lattice’s own clock, some waves grow every tick.

A sum over the lattice’s geometries in real time is a quantum theory only if it defines an evolution, and the ledger has a clock: record time, the number of postings so far. Each tick multiplies a wave by a fixed factor, its root w. A one-particle quantum theory with real frequencies needs a positive norm that every tick keeps, so every physical root must lie on the unit circle. At the identified κ = 1 one physical solution of the graviton field grows by √2 every tick, so no such norm exists. An evolution that keeps one is open.

The growing solutions: provedκ = 1: identifiedA norm the tick keeps: open

  • Massless scalar of the one-loop model
  • Graviton field: the growing solution

Live · roots solved in your browser

With JavaScript on, this figure solves the record-time roots live. At κ = 1 the graviton field has a solution that grows by √2 per tick; the massless scalar of the one-loop model has four roots off the unit circle at every κ above ½, the largest of size 2.653 at κ = 1.

The growing solution

√2 per tick, exactly.

At κ = 1 the linearized Regge equation, the lattice’s equation for a weak gravitational field, has a solution that is the same at every point of a slice of constant record count, so it fits a torus of any size, and is multiplied by √2 at every tick. Four of its fifteen edge components are nonzero, with exact values in √2.

  1. 1000
  2. 0100
  3. 1100
  4. 0010
  5. 1010
  6. 0110
  7. 1110
  8. 0001
  9. 1001
  10. 0101
  11. 1101
  12. 0011
  13. 1011
  14. 0111
  15. 1111
The change in the squared length of each of the fifteen edges at a vertex of record count 0, one edge per string of four bits (which counters the edge advances; ink up, red down). Four are nonzero: 4 + 2√2 on 0010, −2√2 on 1110, −4 − 2√2 on 0001 and 2√2 on 1101. At record count t every value is multiplied by √2t. In the paper’s notation hd(x) = √2t(x) + |d| ud, with u = 2 + 2√2, −1, −(2 + 2√2) and 1 on those four edges.

Live · checked in your browser

Your browser builds the Bloch matrix B of the flat κ = 1 Hessian from Appendix B of the earlier paper, at the phases (w, w, w, w), and applies it to the paper’s polarization u. The paper proves more than the last check shows: the field is no vertex move of any shape, at every spatial period.

Quantum gravity paper, Theorem 8.1 and Appendix A.3; record-time paper, Theorem 3.2

  1. Why no norm survives

    A kept norm would equal twice itself.

    Suppose a size n(h) of solutions were positive on every physical solution, every one that is not a vertex move, scaled as n(ch) = |c|² n(h), and unchanged by the tick. The tick sends this solution h to √2 h, so n(h) = n(√2 h) = 2 n(h), which forces n(h) = 0. So no such norm exists at any spatial period. The same holds for the solutions the native path sum keeps, with every posting held fixed, for every phase whose Gaussian field equation is a nonzero multiple of Regge’s, and at κ = 3/2 with the factor (1 + √10)/3.

    Quantum gravity paper, Theorem 8.1 and Proposition 8.2; record-time paper, Theorems 3.4 and 3.5

  2. The doublers

    Record time keeps the four doublers.

    Ordering the path sum by record time does not remove the massless corner doublers. Near each axis corner, for every time direction of the long-wave graviton’s cone, some spatial direction carries two real record-time frequencies and other directions none: the doubler propagates in record time with a mixed spectrum.

    The four doublers, solved live

    Quantum gravity paper, Theorem 3.2

A weight is a damping

Damping of fourth order vanishes in the continuum limit.

A real weight in the sum over geometries enters the record-time equation as a damping, and with the weight derived from the ledger’s cost of curvature the lattice equation is not record-time stable once it has a physical wave away from the zero wave. A weaker test asks only that every family of waves whose roots approach the circle as the spacing shrinks grow by a vanishing amount per unit of physical time. Damping of fourth order in the wave number, the order of the derived weight, passes; damping of second order, the order of the action, fails.

  • Fourth order, m = 2
  • Second order, m = 1

Live · exact, the model of Section 5.1

With JavaScript on, this figure computes the damped model’s growing root at every spacing. With damping of fourth order the growth per unit time falls like c b²q³/2; with damping of second order it levels off at about 0.455 q for c = 1.

  1. The derived weight

    With the derived weight, the roots are stable in the continuum limit.

    At every fixed nonzero strength of the weight built from the ledger’s cost of curvature, every family of physical waves away from the zero wave stays off the circle, and near the zero wave each physical root grows per tick by at most a constant times the cube of its spatial momentum. So the weighted Gaussian equation is stable in the continuum limit, though not on the lattice. This is a statement about roots, not a norm-preserving evolution: no positive physical Hilbert space is constructed. It holds at Gaussian order and rests on κ = 1, the weight’s two identifications and its channel counts. That the lattice-scale modes off the circle leave the continuum theory alone is an inference; for specified two-point observables the record-time paper proves an estimate to that effect, whose independent verification is not yet finished.

    Record-time paper, Section 6; quantum gravity paper, Section 8.1

  2. Without a weight

    The bare doublers fail even the weaker test.

    Without a weight, near the corner πe0 the κ = 1 equation has an exact family of physical solutions on a decaying branch, each multiplied by at most e−3/4 per unit physical time however small the spacing. That rate does not vanish as the spacing shrinks, so the bare equation is not stable in the continuum limit. Its proof is complete; its independent verification is not yet finished.

    Record-time paper, Theorem 5.4

Four tests

Two of the four are not yet earned.

A quantum theory of gravity built from the ledger has to meet four demands: a lattice, action and measure that are derived; finiteness, beginning at one loop; a massless graviton; and Einstein gravity at long wavelengths. Here is where each stands, with what it rests on.

Einstein at long wavelengths proved

Holds at linear and second order.

Linearized Einstein gravity and Newton’s law with G = 1/(8πs) at κ = 1; the second order with worldline matter, under identifications of the matter law; every long-wave vertex of the action is Einstein’s, as field equations on sampled jets. Open: nonlinear lattice solutions.

Quantum gravity paper, Section 4 and Theorem 4.2

Finite proved

Real parts finite at one loop.

With the derived weight, the real parts of the one-loop model’s graviton responses have finite ordered limits, where the native sum diverges like log(1/ε). The imaginary part: proof complete, verification pending. Open: the full path sum and higher loops. A finite limit is not a massless graviton.

Quantum gravity paper, Theorems 3.1 and 7.4, Proposition 7.5

Derived not yet

κ and the measure are identified.

κ = 1 is an identification: the foundation admits a world at κ = 1 and one at κ = 3/2. The measure is the ledger’s under two identifications and a channel count. An action whose one-loop remainder vanishes has not been found.

Quantum gravity paper, Theorem 2.3 and Sections 5 and 6

A massless graviton open

The precise open target.

In the one-loop model the native graviton is not massless, and every candidate computed to convergence leaves a nonzero remainder. The target it has to meet is stated exactly below.

Quantum gravity paper, Theorem 3.3 and Target 6.8

Every one-loop statement on this page concerns a stated diagram model: a gauge term of unit weight, no ghost line, the Feynman iε. A ghost or gauge measure derived from the ledger could change it. An evolution in record time with a norm the tick keeps, the top of this page, is open as well.

The sum itself

With a stated regulator, the path sum is a finite integral.

A sum over Lorentzian geometries needs a space of geometries, an action, a measure and a reason the sum exists. The finite-path-sum paper gives all four on the fixed Kuhn graph and states the status of each: proved, conditional (proved from a premise that is not derived), a model (a stated construction), or open.

1 · The geometries

Proved An open cone of admissible geometries for κ ≠ 1, with a null stratum at κ = 1 and an explicit ball around the flat lattice; every simplex has signature (3, 1); κ is the posting speed, and κ ≥ 1 is causal.

Model or conditional The fixed Kuhn graph, real squared lengths, time on the body diagonal; one diagonal per cell, given a clock without folds.

Open The value of κ.

2 · The action

Proved The Regge action, smooth and homogeneous on the cone, with its first variation by Schläfli’s identity.

Conditional The Lorentzian sign and κ ≥ 1, given name-blindness and a causal reading of the cell.

Open The native phase scale; the continued action.

3 · The measure

Proved Finite, nonzero, a density for every admissible family, with no Faddeev–Popov factor; the family, the radius and the weight change it.

Model The ledger chart, the account family, the box radius and a flat bare weight.

Open Whether the bare weight is flat; refinement.

4 · Finiteness

Proved A finite integral at a stated regulator, with a normalization entire in the phase scale, so expectations are defined off a discrete set; finite with nonzero normalization at κ = 1.

Model The regulator, a bound on squared lengths or on the ledger box.

Open The Abel limit of the uncut sum; reflection positivity; a derived cutoff.

The regulator is stated, not derived. Without it the sum is not absolutely convergent, and whether its Abel regularization converges is open.

Finite-path-sum paper, Section 1 and Table 1

The massless graviton

A Ward identity decides whether the graviton stays massless.

In the stated one-loop model the zero-momentum remainder, a mass term for the graviton, equals i/2 times the difference of two Ward defects. The native graviton’s remainder does not vanish, and an action from the ledger whose summed Ward defect vanishes is the precise open target. The criterion, the four conditions of Target 6.8 and every candidate the paper has tested against them have a page of their own.

A massless graviton: the open target

Sources: the quantum gravity paper is Quantum Gravity from the Recognition Ledger, version 1.2; the finite-path-sum paper is A finite gravitational path sum on the recognition lattice, October 2026; the record-time paper is Record-Time Root Stability of the Recognition Path Sum; the earlier paper is Linearized Einstein gravity from Lorentzian Regge calculus on the Kuhn lattice.

Further reading on the massless-graviton target: The Native Lattice Graviton Is Not Massless: Ward Criteria and the Limits of Local Regge Repairs at κ = 1 and Non-Local Recognition Phases and the Massless-Graviton Target, both preliminary preprints of 1 November 2026.

Words on this page, in plain terms: tick · record time · record-time root · doubler · path sum · curvature weight · one-loop diagram model.

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