Gravity / A massless graviton

In the model, the graviton is massless only if a Ward defect vanishes.

In the continuum a uniform stretch of spacetime is only a change of coordinates, so the one-loop effective action of a constant strain depends on the strain only through the volume. On the lattice the part that does not is the remainder, a mass term for the graviton in the stated one-loop model, and it equals i/2 times the difference of two Ward defects. The lattice’s native graviton fails: its remainder does not vanish. An action from the ledger whose summed Ward defect vanishes is the precise open target. Every candidate computed so far misses it, and many are excluded by proof.

The Ward criterion: provedκ = 1 and the measure: identifiedA massless graviton: open

  • Meets
  • Met at one point, as computed
  • Fails
  • Fails, as computed
  • Fails, verification pending
  • Open

Interactive · from Target 6.8 and Sections 6.3 to 6.7

  1. The zero phase

    (a) fails, proved.

    Its long-wave quadratic vertex vanishes, and the quadratic clause rejects every phase with a zero Hessian. Discussion after Target 6.8

  2. The Regge action, the native path sum

    (a) meets.(b) meets.(c) fails, proved.

    Meets (a) and (b); its remainder is the native one, which does not have ordered limit zero, so the native model graviton is not massless. Computed: 0.75 + 6.71i, 0.62 + 6.93i and 0.45 + 7.03i at ε = 0.1, 0.05 and 0.025. Theorem 3.3; Section 6.3 (numerical)

  3. The Regge action with the derived weight

    (c) fails, as computed.

    The weight removes the doublers, not the remainder: 0.1130 + 24.656i on the eleven non-posting channels. A real normalization of the amplitude changes only the imaginary part, to 0.113 − 0.358i with the modulus normalized or 0.113 + 12.15i with the Born weight normalized, so the remainder is nonzero in each. Its ordered limit still needs a local statement at the origin. Section 6.3 (numerical)

  4. Phases cS + Σh δh³Γh, cubic terms in the deficits

    (c) fails, proved.

    None has an exact vertex symmetry, and at the same slope and weight each has exactly the remainder of cS on every probe. Theorems 6.7 and 6.10

  5. A real weight or density in the measure

    (c) cannot be supplied this way, proved.

    No real weight or density makes a vertex-translation flow a symmetry of the path-sum exponent, so none supplies the symmetry that would give (c). Theorem 6.6

  6. Refinement invariance of the identified lattice

    (c) cannot be supplied this way, proved.

    It would force the remainder to vanish, but the identified lattice does not have it, and at κ = 1 refinement carries the doublers to zero momentum. Theorem 6.9

  7. Area-additive local charges

    (c) fails, as computed.

    Their remainder is that of a one-parameter family in β, and no value of β computed gives zero. Sections 6.6 and 6.7 (iv) (numerical)

  8. An even part homogeneous of degree one in a hinge’s area and its reference area

    (c): it fails the corner condition, proved.

    Under every positive rescaling of both areas it fails the corner condition exactly, and a massless graviton needs that condition: without it the corner logarithm stays (inference). Sections 6.4 and 6.7 (v)

  9. An even part that sees the area only through its ratio to the flat area

    (c) fails, as computed.

    It meets the corner condition with one choice of its slope, but leaves a remainder of modulus above 6 on the probe fE for every profile. Sections 6.6 and 6.7 (v) (numerical)

  10. Local charges within the readings the ledger supplies

    (c) fails, proof complete, verification pending.

    A local charge’s remainder depends only on its slope and even curvature, and the ledger’s readings make the even curvature 0 or proportional to the area. With the Regge slope the first has exactly the Regge charge’s remainder, so it is not massless; the second is not, as computed. Sections 6.6 and 6.7 (vi) (proof complete, verification pending; the second case numerical)

  11. A tuned even term, eight lattice-scale numbers

    (a) meets.(c) met at one point, as computed.(d) not met: the numbers are chosen, not derived.

    On the bare phase, whose doublers are still massless, the remainder can be tuned to zero at least at one point by choosing eight numbers; every such charge meets (a). Nothing in the ledger has yet been shown to supply those numbers, so it is a fit, not a derivation. Discussion after Target 6.8 (numerical)

  12. The ledger’s holonomy phase, a non-local route

    (c) fails, proof complete, verification pending.

    Read through two identifications, it is the Regge phase on every geometry and at every coupling, so it has the native remainder and misses the corner condition. Section 6.7 (proof complete, verification pending)

  13. The action the ledger’s refinement map induces, the other non-local route

    (a) not met: no fixed point with an Einstein–Hilbert long-wave part; proof complete, verification pending.

    At κ = 1 the blocked path sum has no fixed point with an Einstein–Hilbert long-wave part in either order of limits, so this route does not supply the repair. Section 6.7 (proof complete, verification pending)

  14. What remains

    (a) open.(b) open.(c) open.(d) open.

    A phase with an even part outside the ledger’s present readings, or one whose slope map or even kernel is not that of any local charge, for instance one that couples different hinges. If no phase the ledger supplies can meet the target, proving that is a result about the programme. Sections 6.7 and 10

The four conditions of Target 6.8

(a) The classical limit
The phase is three times differentiable and reproduces every posting record; the flat lattice is a stationary point; its flat Hessian vanishes on every direction that leaves all deficits unchanged to first order; and its long-wave quadratic and cubic vertices are one positive multiple of Regge’s. A condition stated without Regge gives all of these: a record-compatible charge with σ(1) > 0 that keeps flat space a vacuum at every stretch of the time diagonal beyond κ = 1 meets every classical clause, with c = σ(1)/ℏ; the converse fails. That condition fixes the slope only; the even part, which the remainder sees, stays free. The slope, live
(b) Regular determinants
Regular one-loop determinants on every torus at every regulator ε > 0.
(c) A vanishing Ward defect
The summed Ward defect has ordered limit zero, the torus growing first and then the regulator removed, on every transverse-traceless probe. By Theorem 6.4 that is exactly a zero-momentum remainder with ordered limit zero.
(d) Derived coefficients
Every coefficient derived from the ledger, so that no number is tuned by hand.

How to read it. A mark is the paper’s statement about that candidate and that condition, with its status; an empty cell is a condition the paper does not decide for that candidate. The quadratic truncation of the Regge action is not in the table because it is not a phase of hinge charges, so it is not a candidate at all. Every statement here concerns the stated diagram model, a gauge term of unit weight with no ghost line and the Feynman iε; a ghost or gauge measure derived from the ledger could change it.

Quantum gravity paper, Target 6.8 and Sections 3, 6 and 10; the vacuum condition: slope paper, Section 6, Theorem 6.4 and Proposition 7.2

The computed remainders

The target is the origin; every remainder computed sits far from it.

A massless graviton of the model needs the remainder to have ordered limit zero on both kinds of transverse-traceless probe. The paper’s computed values, drawn in the complex plane: the native path sum on the probe Htt at three regulators, its finite part on fE, and the derived weight on Htt, fE and fT2.

  • Native path sum
  • With the derived weight

Drawn from the paper’s computed values (numerical)

What is drawn. The values of the paper’s Section 6.3, with the Feynman regulator on the eleven non-posting channels. For the native path sum the real part carries a small corner logarithm, so it has no limit as ε falls; its finite part is 0.27 ± 0.05, and the imaginary part on Htt is about 7.0. With the derived weight, at ε = 0 on midpoint grids, the two probes add to the Htt value. These are values of the model determinant, not of a full gauge, ghost and density amplitude.

Quantum gravity paper, Section 6.3 and Theorem 3.3

The computed remainders: native, 0.75 + 6.71i, 0.62 + 6.93i and 0.45 + 7.03i at ε = 0.1, 0.05 and 0.025, and about 0.31 + 5.67i on fE; with the derived weight, 0.1130 + 24.656i on Htt, 0.0467 + 17.608i on fE and 0.0663 + 7.048i on fT2. The target is zero.

How it works

From a stretch of spacetime to a test for any action.

  1. 1 · The remainder

    A constant strain should only change the volume.

    A constant strain, a uniform stretch of space and time, is in the continuum 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 the part that does not is the zero-momentum remainder: a mass term for the graviton in the model’s one-loop effective action.

    Quantum gravity paper, Sections 3 and 6.1

  2. 2 · The criterion

    An exact symmetry would remove it.

    For every one-loop determinant the remainder equals i/2 times the second-order Ward defect minus the first-order one, and summed over waves for the lattice path sum. If the determinant is unchanged by volume-preserving linear moves of the vertices, the remainder vanishes; near the flat lattice such an invariance is local covariance. A real phase without a regulator gives a purely imaginary remainder, and a real normalization of the determinant changes only its imaginary part.

    Quantum gravity paper, Definition 6.1 and Theorems 6.2 to 6.5

  3. 3 · The native graviton

    The lattice’s own graviton is not massless.

    The native remainder of the model determinant does not have ordered limit zero; computed, its real part carries a small logarithm from the four doublers, and at ε = 0.1 both Ward identities fail, the first-order one as well. A weight from the ledger’s cost of curvature gaps the doublers but not the remainder.

    The four doublers, live

    Quantum gravity paper, Theorem 3.3 and Section 6.3

  4. 4 · The target

    An action from the ledger’s hinge records, with four properties.

    A phase built from the ledger’s hinge records with (a) the classical limit of the Regge action, (b) regular one-loop determinants on every torus at every ε > 0, (c) a summed Ward defect whose ordered limit is zero on every transverse-traceless probe, and (d) every coefficient derived from the ledger. Condition (d) is not decoration: without it a tuned counterterm could meet the target, which is a choice, not a derivation. Given such an action and the measure that gaps the doublers, the model’s graviton is massless at one loop if the remainder also vanishes for the combined path sum, weight included; reading that as a massless pole of a propagating graviton also needs an evolution in record time with a norm the tick keeps.

    Record time, live · The action, on the status ring

    Quantum gravity paper, Target 6.8 and Section 10, items 1 and 2

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 quantum gravity paper is Quantum Gravity from the Recognition Ledger, version 1.2.

Further reading: 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: remainder · Ward defect · constant strain · ordered limit · transverse-traceless probe · corner condition · slope and even curvature · one-loop diagram model · curvature weight.

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