Gravity from a lattice of events.

Before a real-time sum over geometries on a lattice with a physical spacing can claim Einstein gravity as its long-wavelength limit, it has to supply eight things. The paper states each one with its status: a theorem as stated, a theorem that rests on a named model or on the identification κ = 1, or open (its Table 1).

The vertices are the points of a four-dimensional hypercubic lattice, periodic with side N. Kuhn's rule cuts each unit cell into 24 four-simplices, one for each ordering of the four axes: the simplex of the order (a₁, a₂, a₃, a₄) runs from x through x + e_a₁, x + e_a₁ + e_a₂ and x + e_a₁ + e_a₂ + e_a₃ to x + (1,1,1,1). Roček and Williams began linearized Regge calculus on this triangulation. Here it carries a Lorentzian metric.

The motivation for this lattice is a reading of its coordinates. If events are records kept on four counters, each advanced one step at a time, then the orders in which the counters can go from 0000 to 1111 are exactly the simplices of a cell. A one-step edge is a posting, and the sum of a vertex's coordinates is its record time. Each simplex is a chain of four postings, and all 24 share the edge from 0000 to 1111.

A configuration is a real squared length on every edge. The graph never changes, so the path sum runs over lengths, never over triangulations. Every edge joins a vertex x to x + d for one of fifteen classes d in {0,1}⁴ without 0: four postings, six two-step edges, four three-step edges and the main diagonal, the time edge.

In lattice coordinates the flat metric is g_κ = I − κ 11^⊤, so the class d has squared length |d|(1 − κ|d|), where |d| counts its ones. κ is the lattice's one number. For κ above ¼ the metric is Lorentzian: it has exactly one negative eigenvalue, 1 − 4κ, and its eigenvector (1,1,1,1) is timelike. We take that direction as time. It is not the only timelike direction: every direction inside its cone is timelike too, and at κ = 1 that includes the face and body diagonals, with squared lengths −2 and −6.

Follow an event for one tick of record time. Record time counts postings, and position is the part of a step orthogonal to time, a point of the hyperplane where the coordinates sum to zero. A posting on axis a advances record time by one tick and moves the position by σ_a = e_a − ¼(1,1,1,1), whose length is |σ_a| = √(1 − ½ + ¼) = √3/2 (Section 2.2). So in one tick a posting moves by √3/2, in one of four directions. The interval's future cone at one tick is a ball of radius v_c = √(κ − ¼), the cone speed. Every pair in the posting order is future causal exactly when κ ≥ 1, that is, when the ball contains all four postings (Theorem 2.1). At κ = 1 the two speeds are equal and every posting is null. Only at that value does the interval read the posting order on an interval of positive duration (Theorem 2.2).

That the physical interval reads the posting order on at least one interval of positive duration is the paper's identification. The definitions do not force it: every κ ≥ 1 makes the order causal. Every theorem stated at κ = 1 inherits the identification; theorems stated for all κ > ½ do not. It concerns the postings and the graviton's cone. Whether light travels on the same cone is a question about photons, which the paper does not address.

The paper says where its motivation enters: through κ, and otherwise only as the source of one candidate construction of the measure. It also uses the programme’s cost function in three models, each labelled. The rest is Lorentzian Regge calculus on the Kuhn triangulation.

The dynamics is Regge’s action, S(ℓ) = Σ A_h(2π/n_h − θ_σ,h), summed over every pair of a Kuhn simplex σ and one of its ten triangles h: A_h is the area of the timelike triangle, θ_σ,h the dihedral angle of σ at h, and n_h the number of simplices that share h. The path-sum weight is e^iaS, with no Wick rotation. The Lorentzian sign is not inserted separately: within the length family |d|(1 − κ|d|) the causal range forces κ ≥ 1 (Theorem 2.1), and every κ > ¼ makes the metric Lorentzian, with exactly one negative eigenvalue. At the flat κ = 1 lattice the Hessian of one simplex is C/(16√3), with C a 10 × 10 integer matrix printed in Appendix B, when each simplex takes its own flat dihedral angle as its share of 2π at a triangle; with the equal share 2π/n_h the simplex Hessians differ on three pairs of edges by terms that cancel at every triangle, so the lattice operator is the same. Summed over the 24 simplices of a cell with Bloch phases, it gives the operator whose zero modes are the waves below.

The spacing b of the lattice is physical, and shrinking it at fixed combinatorics would not help: rescaling every squared length rescales the Regge action and leaves its stationary points where they were. So the limit in which Einstein's theory should appear is a long-wavelength limit, with waves spread over many spacings of a lattice that stays put. The first test is the free waves of the flat lattice. Do they travel on the light cone?

They do, in every direction, up to a relative correction of order (k_phb)², where k_ph is the physical wave number (Theorem 5.1, Corollary 5.2). Near the origin of the Brillouin zone there is no slow branch, no fast branch and no extra low-frequency mode. One direction can be solved exactly. Along the paired covector u = (z+t, z+t, z−t, z−t) the wave number is |k| = 2t, the frequency is ω = 2z/√3, and each polarization satisfies a printed polynomial equation (Appendix F). The figure plots both.

At gravitational-wave frequencies the correction is far too small to see. For a spacing up to 10⁶ Planck lengths and frequencies up to 5000 Hz, every non-gauge mode of wavelength at least 2π × 10¹³ spacings, in every direction, has a phase speed within 2 × 10⁻⁵⁰ of light (Corollary 5.3). Along the paired direction the plus mode differs from light by about 7 × 10⁻⁶⁶; whether a bound that small holds in every direction is open. Both numbers lie far below the bound from GW170817. Here “light” means the null cone of the lattice metric: reading GW170817 as a test of it assumes that photons travel on the same cone, which the paper does not derive.

Free waves on the light cone are the first classical test. The second asks more: solutions of the lattice equations with sources must converge to solutions of Einstein’s equations, without an unproved stability constant. Section 4 of the paper proves this at linear order in the field.

The equations are the unconstrained Regge equations: stationarity of the action, with matter, under variations of all fifteen edge classes, at configurations where every simplex is Lorentzian. The posting components of their solutions need not vanish, so the solutions are not configurations of the path sum at κ = 1, which keeps every posting null; the two are related at linear order. They are solved on a torus of side N, periodic in record time as well as in space, with sources whose waves avoid the light cone. The source problem is therefore elliptic in character: radiation and retarded propagation are not part of these theorems. And N → ∞ is a longer box at fixed spacing, so the waves grow longer in lattice units: a long-wavelength limit, not a refinement.

The core is the linear limit (Theorem 4.6). For a stress T conserved on a cone of waves that avoids the light cone, the solution of the gauge-fixed lattice equations tends at long wavelength to the de Donder solution of the linearized Einstein equations:

q_κ(u) h_μν(u) → 16πG (T_μν(0) − ½ g_μνT(0)), G = 1/(8πs).

The stability bound is a closed-form matrix plus any margin, not an assumed constant, and the theorem holds for every κ > ½, so it does not use the identification. The flat-kernel and solution statements below (Theorems 4.2, 4.4 and 4.8 to 4.10, Propositions 4.5 and 4.11) are stated at κ = 1 and inherit it; the closed-torus Gauss law (Theorem 4.7) and the fixed-strength Theorems 4.12 to 4.16 hold for every κ > ½, like Theorem 4.6. Toward the light cone the gauge-fixed solution operator is not uniformly bounded: a bound holds outside the quartic shell |q_κ(u)| ≥ c‖u‖⁴, the exponent 4 is sharp among integer powers, and the zero modes found inside the shell are lattice gauge modes.

On a torus the waves are the points 2πm/N, and the operator has to be invertible at every one of them, apart from the zero modes every flat lattice has: the constant strains, which deform the torus flatly, and the vertex moves, the lattice’s gauge directions. Through the Bloch decomposition this is a statement about one integer polynomial F₁ with 741 terms, the gauge-fixed Bloch determinant: it must not vanish at a nonzero wave of the torus. Theorem 4.2 proves that for every odd side not divisible by 105. Even sides fail, for the reason of section 06: when N is even the axis corner πe_c is a wave of the torus, and the corner mode is a zero mode that is neither a strain nor a vertex move (Theorem 4.4). The exclusion of 105 is a limit of the proof, not a known failure. The proof is 2-adic, with three routes that need N prime to 3, to 5 or to 7, and the odd multiples of 105 are open.

The lattice also has fewer zero modes than general relativity on the same torus. Time is periodic, so every null integer wave, such as m = (1, 1, 1, 0), carries two transverse polarizations that solve the linearized Einstein equations. On the lattice, on the sides of Theorem 4.2, those waves carry no zero mode beyond the strains and vertex moves. Along the null lines of Proposition 4.5, the waves s(1, 1, 1, 0) and their permutations, F₁ vanishes at the origin to order 8, against order 4 along an axis line. The paper reads this as the two transverse eigenvalues being of order |k|⁴ there rather than |k|² (an inference): at such a null torus wave only the lattice correction, of order (2π|m|/N)⁴ rather than (2π|m|/N)², keeps the operator invertible. These small divisors get smaller as N grows, as for time-periodic wave equations, and the paper takes them to be why the constants of the solution theorems depend on N (also an inference).

A fixed source cannot simply be solved, and two proved obstructions say why. On a closed torus the equations with a static positive mass have no small solution: the flat lattice pairs with the mass at first order with a fixed sign, while the left side of the equations is of second order (Theorem 4.7). A static positive mass must be compensated, the discrete form of ∫ ρ = 0 on a three-torus (an inference). And a source whose linear response has nonzero self-energy needs compensating stresses at second order, a discrete von Laue condition; such sources are dense among the compatible ones, though no physically selected source has been shown to be one.

So the solution theorems carry three pieces of matter, fixed before any solution is sought: the resting body, clocks on the time edges with the point-particle action; a uniform compensation of its mean density, a modelling choice; and a medium, a spring on the proper lengths of the timelike edges with the flat lattice as its rest lengths. The medium’s first variation at the flat lattice vanishes, so the linear field is the body’s own. The medium holds the equations along the constant strains and vertex moves, which vanish at first order and first act at second order in the coupling. It is a model, with no degrees of freedom of its own.

With that matter, and the medium’s amplitude large enough for the side and the body, on every odd side not divisible by 105 the full coupled equations on every edge have exact solutions for small coupling σ, in the Lorentzian chamber and locally unique (Theorem 4.8). The limits are then taken in a fixed order.

The order matters. A source of fixed strength on a growing torus would need a stability bound uniform in N, and at κ = 1 none holds over all wavelengths, because of the static resonance that section 05 shows: a static wave within 2π/N of it is nearly a zero mode on every large torus, and there are flat lattices arbitrarily close to the κ = 1 lattice with exact non-gauge zero modes (Proposition 4.11). A medium of fixed amplitude cannot remove them, and one strong enough to do so would act as a mass for the long-wave field. For a static, band-limited source at a fixed small strength α, what holds uniformly is the long-wave part. The band equations have exactly one small solution on each large torus, with constants that do not depend on N; divided by α, it lies within Cα of the linearized Einstein field at every edge once N is large, up to constant strains and vertex moves. The gauge directions are controlled, and the band field has an explicit second-order limit, for pure gravity with a fixed external source and for the band equations only (Theorems 4.12 to 4.16). Imposing the remaining equations at fixed strength is open.

So the content of the weak-field limits is linear. The nonlinear solutions add existence on each torus, not nonlinear content in the limit, and nonlinear Einstein solutions are not reached. Section 09 says what would have to replace the medium.

The source law is the other half of the classical test. A resting body is a clock on the time edges, with the point-particle action of a static mass. With the action scale s, the linear field equation is s H₀h = −τ for an edge source τ, and the lattice Newton clause, h_tt q(u) = 8πGM for every static mode of the field of a point mass, holds for exactly one coupling: G = 1/(8πs) (Section 3.5). Reading G as Newton’s constant uses that action scale; reading M as the body’s mass uses the point-particle reading of a clock, which the paper labels a model.

Theorem 5.4 makes the static kernel precise. The time-time response χ_tt of the gauge-fixed κ = 1 operator to a time-edge source satisfies |χ_tt(k) q(k)/c_N − 1| ≤ 2.2 × 10²⁶ ‖k‖² for 0 < ‖k‖ ≤ 10⁻¹⁰, where c_N = 8πG·2M_p for a source of one-site proper mass M_p. The figure computes the ratio itself, from the operator of Appendix B, out to the lattice scale.

For a body of finite size, Theorem 4.10 turns the kernel into the potential, in the ordered limit of section 04. For a smooth static density of compact support, the exact coupled solutions have −½h_tt = −GM/R at proper distance R outside the body, up to an explicit finite-size remainder, in a suitable lattice gauge, with G = 1/(8π) at unit action scale. The lattice body is a band truncation of a smooth body. A body supported on finitely many lattice sites is not covered, because at κ = 1 every compact lattice body of nonzero mass couples to the static resonance shown in the figure.

A Lorentzian symbol vanishes on a cone, and a cone is unbounded. On a lattice that cone runs out to the edge of the Brillouin zone, so the operator can vanish far from the origin. At κ = 1 the gauge-fixed operator has an exact zero mode at each of the four axis corners πe_c, the waves with one coordinate π and the others 0. A Euclidean lattice operator that is positive away from the origin has no such zeros. Lattice fermions have doublers for a different reason; these belong to real-time discretizations, and at κ = 1 they sit exactly at the axis corners because the postings are null.

At the corner πe_c the gauge-fixed operator A = B + ΓΓ^† has a one-dimensional kernel. It is spanned by v_c(d) = 2^|d|−3 on the seven classes with d_c = 0, and 0 on the other eight (Proposition 6.1). The Bloch matrix and the vertex-move columns annihilate v_c separately, so it is a zero mode for every gauge weight, not an artefact of the gauge term, and it is not a vertex move.

Near the corner the Schur complement of A onto the doubler's direction, the doubler's effective operator, is (√3/640) h^⊤Kh at leading order, where h is the offset from the corner and K has one negative and three positive eigenvalues, with det K = −1013375 = −55²·335 (Proposition 6.2). Each doubler is massless and propagates on a cone of its own. The two kinds of cone point different ways: record time is timelike for the graviton and spacelike for the doubler, and no direction is timelike or null for both (Theorem 6.15). The figure shows this on one plane of waves.

The doublers feel a constant background directly, without derivatives. A constant strain H moves the zero at corner c at first order by δ_c(H), the pairing of v_c with the derivative of the Bloch matrix along H. The shift vanishes for every strain of the irreducible representation E and for the conformal strain. For the T₂ probe it is ∓√3/32, and the four shifts always sum to −396/(1152√3) times tr H (Proposition 6.3), so the doublers' squared mass is linear in the background strain. The trace has a direct meaning: H_aa is the change of the squared length of the posting along axis a, so tr H is the total change of the four postings’ squared lengths. A massless particle that couples to a constant background without derivatives gives, in a pair loop at zero external momentum in four dimensions, a logarithm of the infrared cutoff. In the ordered limit the only infrared cutoff on the lattice is the regulator ε. The figure puts each background the paper names on the lattice.

Nothing in the construction gives the doublers a mass. A flat background that keeps every posting null shifts the four zeros, at first order, by amounts that sum to zero, so it cannot lift all four, and any nonzero shift of that kind makes at least one doubler tachyonic. A strain that moves all four the same way changes the posting lengths: raising κ above 1, the strain −11^⊤, gives all four a mass, and it leaves κ = 1 (Theorem 6.10). The doublers belong to the null postings.

A massless graviton needs more than the right tree-level propagator. At one loop its self-energy must not develop a mass, a width or an anisotropy that survives the removal of the regulator. That is also where an observable effect could hide. At tree level the lattice’s correction is suppressed by (k_phb)² and lies far below the GW170817 bound (section 03), but a one-loop mass, width or shift of the cone would be suppressed only by the loop factor (an inference). Read as a loop effect, a cone shift δ = 8π(ℓ_P/b)² d (an identification) would contradict GW170817 at a spacing up to two Planck lengths for every |d| ≥ 10⁻¹⁴, when the other terms are negligible (Section 5.4); no theorem yet computes d.

The regulator ε is the damping of the path sum’s Gaussian-order construction, a regulator model: the real-time weight carries a factor e^−ε|x|²/2, and its normalized moments converge to (ε − iaA_R)⁻¹, the damped inverse of the flat Hessian A_R on the retained edges R (Theorem 3.2). The paper asks this in a stated diagram model: the graviton and scalar sunset and seagull diagrams at zero external momentum, with the Feynman-type gauge term at unit weight, no ghost line, the Feynman sign of iε and one massless scalar for matter (Section 6.1). Two traceless spatial probes, one in each of the representations E and T₂ of the axis permutations, give the same value for any covariant self-energy once each is divided by its DeWitt norm, so the real part of their difference, Δ_N(ε), measures an anisotropy. The limits are taken in a fixed order.

At identified κ = 1 the outer limit does not exist: Δ(ε) grows without bound. The logarithm comes from the four doublers. In the language of Feynman integrals, each axis corner is a pinch: the doubler's eigenvalue vanishes quadratically there, so its gradient is zero: no direction increases it at first order, and no deformation of the integration contour avoids the iε prescription. At zero external momentum both lines of the sunset carry the same wave, so each corner contributes a double pole. Every other degenerate zero except the origin has a direction along which all its vanishing eigenvalues increase at first order (Theorem 6.7), so the contour can be moved off it and its contribution is finite (Proposition 6.9). At the origin the cubic vertex vanishes on constant strains, so that contribution is finite too, and nothing cancels the corners' logarithm. That reading is a summary; the proof is Propositions 6.6 and 6.9 with Theorems 6.7 and 6.8. Two further facts are exact: the scalar’s operator does not vanish at the axis corners, where its symbol is 8√3/3, and a ghost loop, were one added, could not cancel a corner logarithm either, because the vertex-move Gram determinant is nonzero away from k = 0.

Near each corner the calculation reduces to one integral (Propositions 6.5 and 6.6). Over the Lorentzian form Q(y) = −y₀² + y₁² + y₂² + y₃²,

I(ε) = ∫ e^−|y|² (Q(y) + iε)⁻² d⁴y = −π² ∫₀^∞ e^−ετ τ(1 + iτ)(1 + τ²)^−3/2 dτ.

The real part tends to −π². The imaginary part has the integrand e^−εττ²(1 + τ²)^−3/2, which falls like 1/τ until the regulator cuts it off near τ = 1/ε. Plotted against log τ it is a plateau, so once ε is small each factor of ten in ε adds the same area, ln 10. The four corners together give the doubler term D(ε) = (5/(44√335 π²)) log(1/ε) + O(1), about 6.3 × 10⁻⁴ log(1/ε), and Δ(ε) − D(ε) has a finite limit.

The logarithm needs the doublers to be exactly massless. Give each corner line a constant dressing z, so that (r + iε)⁻¹ becomes (r + z + iε)⁻¹, and the doubler term has a finite limit for every z ≠ 0, massive, tachyonic or complex (Proposition 6.13): the thin curves follow D(ε) until ε falls below about |z|, and then stop. At κ = 1 nothing in the construction supplies such a gap (section 06). Whether the doublers’ own one-loop self-energy does is open. It depends on momentum, power counting suggests that it grows like log(1/ε) itself, and the statement for a constant z does not cover a dressing of that kind (Section 6.6). Near the corners it is as large as the doublers’ tree-level operator at the regulators reached, so a finite anisotropy obtained that way would not be a massless graviton with small corrections (an inference).

The logarithm is also slow. The paper’s computed anisotropy at ε = 0.0125 is 0.1541 ± 0.0008, of which the logarithmic term accounts for 0.0028 (numerical). The logarithm overtakes a finite part of that size only when log(1/ε) exceeds about 240, that is, for ε below about 10⁻¹⁰⁴, so at every regulator a computation can reach, the anisotropy is dominated by its finite part. The divergence is a property of the limit, and Theorem 6.8 proves it.

The statement holds in the stated diagram model at identified κ = 1, and the proof's computations are exact. It is a failure of one quantum limit, not of the classical results: none of this touches the recovery of linearized Einstein gravity and Newton's law.

The long-wave graviton has a zero-momentum question of its own. In the continuum a constant strain is a linear change of coordinates, so the one-loop effective action of constant strains can depend on it only through the volume: a cosmological term and nothing else. Anything else, the remainder R, is a mass term for the transverse-traceless graviton at zero momentum. On the lattice a linear map must also keep each null posting on its own axis, and only ±1 does (Theorem 6.17); the paper finds no other symmetry that would force R to vanish (inference).

No gauge fixing, ghost or covariant measure factor changes R, wave by wave at ε = 0 (Theorem 6.17), and one inequality on one explicit zone integral would prove it nonzero (Proposition 6.18). That inequality is not proved. Numerically R is about +7i in the graviton sector, and none of the measure weights examined cancels it (Proposition 6.20). What an imaginary R means depends on whether the real-time path sum has a unitary evolution in record time, which is not established: with one, R would be a width, which observed gravitational waves exclude given the identification of the lattice spacing; without one, the path sum at identified κ = 1 would not be a consistent quantum theory in record time.

General relativity treated as an effective field theory predicts a definite quantum correction to the potential between two masses (Donoghue; Bjerrum-Bohr, Donoghue and Holstein):

V(r) = −(Gm₁m₂/r)(1 + 3G(m₁ + m₂)/(rc²)) − (41/(10π)) G²ℏm₁m₂/(r³c³).

The last term comes from the non-analytic log q² part of the one-loop amplitude. Gravity’s one-loop ultraviolet divergences are local and do not touch it, and the coefficient does not depend on how gravity is completed at short distances, provided the light particles are those of general relativity: the graviton and the two masses. That proviso is where a lattice with doublers has to be careful.

Section 7 of the paper assembles the one-loop amplitude from rules taken from the lattice. Each line and vertex but two is the long-wavelength limit of a lattice object. The regulated two-point function of the Regge operator with a de Donder gauge term tends, off the light cone, to the de Donder graviton line. The simplicial scalar tends to the Klein–Gordon line, and its first variation in the edge lengths to the stress-tensor coupling. The three-graviton vertex is the third variation of the Regge action, with exact momentum conservation, and its long-wave limit is the Einstein–Hilbert vertex with the same G as the propagator. Two ingredients are imposed rather than derived. The first is the Feynman iη on every line: the lattice regulator’s own prescription puts strain poles on both sides of the real axis, and the original reading, which used it with a Born cube and no subtraction, fails (Theorem 7.1). The second is the ghost vertex, which the path-sum measure does not supply. The scalar action and the gauge readout are model choices. So the long-wave rules are the Einstein–Hilbert rules in harmonic gauge, written in record coordinates, with G = 1/(8πs), and the loop integrals are continuum integrals.

With these rules the paper computes the one-loop amplitude for a static momentum transfer Q between two masses, class by class, with the iterated tree exchange, the Born term, subtracted. Read at small transfer, the coefficient of log |Q|² is 123u, with u = √3 μ₁μ₂/(960π²as²) for masses μ₁, μ₂ in lattice units (Theorem 7.5). Through the identifications of Corollary 7.9 that is the 41/(10π) above. The figure shows how the total is assembled.

The value depends on how a potential is read from an amplitude, and the paper states its reading: the Born reading of BDH, with the regulator’s logarithm subtracted in a form that does not change when the regulator is rescaled, and the coefficient read at small momentum transfer. The natural first readings are false, which is why the reading matters.

The coefficient is a theorem about the long-wave rules, not about the lattice path sum at κ = 1, and the reason is the proviso above: the lattice’s light particles are not only those of general relativity. The four doublers of section 06 see the static field of a mass. In de Donder gauge the long-wave field of a resting mass M couples to the doubler pair at every axis corner without derivatives, with the same strength at all four (Proposition 7.6):

δ_c(h) = −33M/(64√3 s q(u)) + o(q(u)⁻¹), c = 0, 1, 2, 3.

The reason is in the trace: the field of a mass changes the squared lengths of the null postings. In a local model of one corner, the massless pair loop carries iπ² log t² at small external momentum t off the doubler’s cone, while a gapped pair carries no logarithm (Proposition 7.7). If near each corner the lattice integrand follows that model, and the rest of the zone does not cancel it, the static amplitude gains a term in log q²/q⁴. That term is more singular at small transfer than the log q² behind the r⁻³ term, and with it the amplitude has no long-wave log coefficient at all (Theorem 7.8): the lattice form of requirement 7 would fail at κ = 1. Both conditions are open. In the local model a gap removes the logarithm, but at κ = 1 nothing in the construction supplies one, and moving κ above 1 gives all four doublers a mass at first order but abandons the identification (Theorem 6.10).

Three links between the long-wave amplitude and the path sum are open: the regulator limits of the lattice amplitude and its uniform remainder against the long-wave amplitude, which at κ = 1 contain the doubler corners; the identification of the amplitude with the moments of the path sum, including its ghost factor; and the graviton self-energy at zero momentum, whose ordered anisotropy diverges (section 07).

At identified κ = 1 the classical side succeeds at linear order and the quantum side fails, and both outcomes trace to the null postings. Classically they give the lattice its causal structure and leave its long-wave sector that of linearized gravity. Quantum mechanically they leave the four corner doublers massless.

What the results do not contain is as definite as what they do: κ = 1 as a consequence of the definitions, a nonzero long-wave remainder, any repair of the graviton sector, a derived measure, nonlinear Einstein solutions in the limit, matter whose equation of state needs no identification, and a quantum prediction at identified κ = 1. The one-loop coefficient 41/(10π) concerns the long-wave rules, not the lattice path sum.