Gravity / Where it stands

Ten parts, and what would settle each.

The quantum gravity paper divides its programme into ten parts and gives each a status: what is proved and under which premises, what is open, and what would settle it. Einstein gravity at long wavelengths holds at linear and second order, and finiteness holds for the real parts of the one-loop responses in the stated diagram model. A derived lattice, action and measure, and a massless graviton, are not yet earned.

Einstein at long wavelengths, finite real parts: provedκ = 1 and the measure’s two premises: identifiedThe action and a massless graviton: open

With JavaScript on, the ten parts sit on a ring with the links between them, and you can rearrange them in the order in which they decide the outcome. Here every part’s status is listed in turn.

Part 1 · Section 2

The lattice and κ

Which flat lattice the theory lives on. One number, κ, fixes its geometry, and the value κ = 1 is identified, not derived.

Proved: In the family of flat metrics I − κ11ᵀ with κ > ½, massless corner doublers occur exactly at κ = 1, and then only at the four axis corners. The same value is equivalent to the interval reading the posting order on an interval of positive duration, to κ being the least causal value, and to some history of record cells being null. Two worlds, at κ = 1 and at κ = 3/2, satisfy the same premises of the foundation as stated (Theorems 2.1 to 2.3).
Identified: κ = 1: the physical interval reads the posting order on at least one interval of positive duration.
Open: A premise of the ledger, stated without κ, that forces its value; and the derivation of the carrier itself, records on four counters with the 24 arrival orders as the Kuhn simplices, from the ledger.
What would settle it: A theorem deriving that premise from the ledger. A premise that forces κ = 1 forces the doublers, so it settles this part only together with a measure that gaps them.

Waits for part 3, the measure.

Move κ and see the two worlds · The four doublers

Quantum gravity paper, Table 1 and the box of Section 2; Section 10, item 3

Part 2 · Section 6

The action

The rule that weighs each geometry of the lattice. Whether the model’s graviton comes out massless depends on it, and every candidate computed to convergence fails that test.

Proved: For every one-loop determinant, the zero-momentum transverse-traceless remainder equals i/2 times the difference of the second- and first-order defects of a Ward identity, and it vanishes when the determinant obeys that identity (Theorems 6.2 and 6.4). No real weight or density makes a vertex-translation flow a symmetry of the path-sum exponent (Theorem 6.6). No phase cS + Σ_h δ_h³Γ_h has an exact vertex symmetry, and at the same slope and weight such a phase has exactly the remainder of cS (Theorems 6.7 and 6.10). Refinement invariance 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).
Computed: The remainder of the model determinant is nonzero for the native path sum, for the derived weight in each of its normalizations, and for the other candidates computed; curvature-squared terms of mixed sign are not resolved (numerical, Sections 6.3 and 6.6).
Proof complete, verification pending: The remainder of every local hinge charge depends only on its slope and even curvature; the readings of a hinge charge that the ledger supplies make the even curvature 0 or proportional to the area, and neither gives a massless graviton in the model. Neither non-local route examined supplies the repair at κ = 1: the ledger’s holonomy phase is the Regge phase, and the blocked path sum has no fixed point with an Einstein–Hilbert long-wave part (Sections 6.6 and 6.7).
Open, and decisive: An action derived from the ledger, every coefficient included, with the classical limit of the Regge action, whose model determinant has a summed Ward defect with ordered limit zero on every transverse-traceless probe (Target 6.8).
What would settle it: A phase built from the ledger’s hinge records that meets Target 6.8, with the doublers gapped by the measure; or a proof that no phase the ledger can supply meets it.

Waits for part 3, the measure. Constrains, and is constrained by, part 7, unitarity. Parts 5, 6 and 9 wait for it.

Since this paper. The slope paper (4 November 2026) gives a condition stated without Regge that implies the classical clauses of the target: 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 one of them; the converse fails. The condition fixes the slope only, so the even part, which the remainder sees, stays free.

The target and every candidate tested · The vacuum condition, live

Quantum gravity paper, Table 1 and the box of Section 6; Section 10, item 1; slope paper, Section 6, Theorem 6.4 and Proposition 7.2

Part 3 · Section 5

The measure

How much each geometry counts in the sum. A weight read from the ledger’s cost of curvature, under two identifications, gaps the four doublers and leaves the classical limit alone.

Proved: At Gaussian order a weight whose exponent scales like the action enters only through one positive semidefinite form 𝒟, and the regulated propagator has a limit exactly when 𝒟 and the action’s quadratic form have no common null vector (Theorems 5.1 and 5.2). Under the two identifications below, the ledger’s cost of the curvature at each hinge gives 𝒟 with no free real coefficient once the channel count is fixed (Theorem 5.4). It vanishes on every flat deformation, is of fourth order in the wave number on long waves, and is positive on the doubler at all four axis corners (Theorem 5.5); away from the origin it leaves no massless mode except the graviton (Theorem 5.6). On the uncut cone of geometries with its Haar measure the weight has infinite mass (Proposition 5.7).
Identified: (I1) The cost of a hinge record reads its area as the action does. (I2) The Born weight of a geometry is the Gibbs weight e^−C of its ledger cost C.
Proof complete, verification pending: The ledger’s histories carry a normalized probability law, so a law read from them is never the uncut Born weight.
Open: The channel count at each hinge, which the dual cycle fixes by a definition, not yet as a consequence of the ledger; which alternatives the Born weight of a geometry is normalized over; and the reference measure on squared lengths.
What would settle it: A derivation of (I1), (I2) and the channel count from the ledger.

Constrains, and is constrained by, part 7, unitarity. Parts 1, 2, 5 and 6 wait for it.

The weight on the four doublers, live

Quantum gravity paper, Table 1 and the box of Section 5; Section 10, item 4

Part 4 · Section 4

Einstein gravity at long wavelengths

Whether the lattice gives back Einstein’s gravity for long waves. At linear and second order it does; solutions at finite strength are open.

Proved: From the earlier paper: the linearized Einstein field as the long-wave limit of the linear lattice field; exact solutions of the full Regge equations on every edge with a resting body and a model medium; Newton’s potential for compact bodies with G = 1/(8πs); and solutions at fixed small strength with an explicit second-order limit. New in this paper: the same second order with worldline matter, on the lattice and as a statement about continuum fields; every long-wave vertex of the action is Einstein’s, in that on a sampled metric jet the metric part of the lattice field equations is −½ times the full Einstein density (Theorem 4.2); the lattice’s vertex moves stop being a symmetry at second order in the field (Theorem 4.3); and the scaled second differences of a lattice field with a uniform jet-norm bound are consistent with continuum derivatives (Theorem 4.1).
Premises: κ = 1, identified, for the theorems stated at κ = 1; a model medium that supplies stability, or a posting gas or worldline matter in its place under the identifications of the least-work law.
Open: The nonlinear Einstein equations as the long-wavelength limit: lattice solutions whose long-wave limit solves G_μν = 8πG T_μν at finite amplitude, not order by order. The vertices are Einstein’s; what is missing is the solutions.
What would settle it: A convergence theorem for lattice solutions at fixed finite strength, with matter that has equations of its own, to a solution of the full Einstein equations.

The paper’s list of what remains names no other part for this one.

Since this paper. The slope paper (4 November 2026) asks which hinge charges share these long-wave vertices. If flat space stays a vacuum at every stretch of the time diagonal beyond κ = 1, a charge’s slope is Regge’s, and its long-wave quadratic and cubic vertices around flat space are σ(1)/ℏ times Einstein–Hilbert’s, with Newton’s constant free. The premise is equivalent to Regge’s slope, and the nonlinear theory is not covered.

Check Einstein’s equation live · Which actions are Einstein’s, live · The second-order defect, live · Newton’s law · Waves

Quantum gravity paper, Table 1 and the box of Section 4; Section 10, item 7; slope paper, Theorems 3.1 and 5.1

Part 5 · Sections 3 and 6

A massless graviton

The graviton, gravity’s quantum, has to be massless. The lattice’s native graviton fails that test in the one-loop model; one that passes is the open target.

Proved: The native remainder of the model determinant does not have ordered limit zero: the native graviton is not massless (Theorem 3.3).
Computed: In the one-loop model the remainder is nonzero for every computed candidate (numerical).
Open: A massless graviton. Given an action that meets Target 6.8 and the measure of part 3, the model’s graviton is massless at one loop if the doublers stay gapped and the remainder vanishes for the combined path sum, weight included. A ghost or gauge measure derived from the ledger, in place of the model’s unit-weight gauge term with no ghost line, would have to pass the same test.
What would settle it: The same as for part 2, with which it shares its box: a phase built from the ledger’s hinge records that meets Target 6.8, with the doublers gapped by the measure; or a proof that no phase the ledger can supply meets it.

Waits for part 2, the action, and part 3, the measure; reading it as a massless pole of a propagating graviton also needs part 7, unitarity. Part 9 waits for it.

The target and every candidate tested

Quantum gravity paper, Table 1 and the box of Section 6; Theorem 3.3; Section 10, item 2

Part 6 · Section 7

Finiteness

Whether the one-loop quantum corrections come out finite. In the stated model, with the derived weight, their real parts do.

Proved: For any lattice graviton family and any smooth vertex data, the ordered one-loop limit at zero momentum exists, provided three checkable conditions hold: the degenerate zeros away from the origin and the corners are hyperbolic, the corners are regular, and the origin has a finite local limit (Theorems 7.1 to 7.3). With the derived weight of part 3, the anisotropy that diverges for the native path sum has a finite real ordered limit, and so do the real part of the full weighted anisotropy, each probe’s contribution and each diagram; no cancellation between them is used (Theorem 7.4).
Proof complete, verification pending: The imaginary part of the full weighted anisotropy has a finite ordered limit too (Proposition 7.5).
Premises: κ = 1, identified; the Feynman regulator, a model; and the weight is the ledger’s only under (I1) and (I2). The theorem holds for every positive coupling of the weight.
Open: The value of the limit, which is not computed: a finite limit is not a massless graviton. The full path sum and higher loops, and finiteness for an action that also meets Target 6.8.
What would settle it: The same three conditions checked for the repaired action, together with the local statement at the origin for the native form.

Waits for part 2, the action, and part 3, the measure.

The logarithm the doublers cause, live

Quantum gravity paper, Table 1 and the box of Section 7; Section 10, item 5

Part 7 · Section 8.1

Unitarity

A quantum theory needs a positive norm that each tick of record time keeps. The identified path sum has none: a solution of its linearized equation grows by √2 every tick.

Proved: Ordering the path sum by record time does not remove the doublers (Theorem 3.2). At identified κ = 1 the linearized Regge equation has a solution that is not a vertex move and grows by √2 per record tick, at every spatial period, so no positive norm on its physical solutions is preserved by the tick (Theorem 8.1); the same holds at κ = 3/2, with the factor (1 + √10)/3. A weight is a damping: in a record-time stable weighted evolution it vanishes on every physical mode. With the derived weight, at each fixed strength, the record-time roots are stable in the continuum limit, the long-wave branch growing per tick by at most the cube of its momentum, given (I1), (I2) and an inference about lattice-scale modes off the unit circle; on the lattice itself it is not record-time stable. These are statements about roots, not about a norm-preserving evolution.
Open: A record-time evolution of the path sum with a positive norm on its physical solutions that the tick preserves: for every spatial momentum, every physical record-time frequency real.
What would settle it: An action and measure, derived as for parts 2 and 3, whose physical record-time roots lie on the unit circle or approach it in the continuum limit; or a proof that none exists.

Constrains, and is constrained by, part 2, the action: with the normalization of a unitary kernel it would make the imaginary part of the remainder vanish (inference). And part 3, the measure: a weight is a damping, so the measure can be compatible with it only at long wavelengths. Part 5 needs it to read a massless graviton as a propagating one.

Watch record time multiply each wave by its root

Quantum gravity paper, Table 1 and the box of Section 8.1; Section 10, item 6

Part 8 · Section 8.2

Matter

How matter enters the lattice. Two kinds of matter carry a body through second order, under identifications; the matter law itself is not yet derived.

Proved: For every κ > ½: a posting gas held by its least-work support, and worldline matter that conserves postings, each carry a body at fixed small strength through second order.
Identified: The least-work law of the recognition cost, read at the flat reference; the ratios of the reference occupation across the timelike classes, which rest on one premise about which ledger entries carry a posting; and the split between body and carrier.
Open: A matter coupling and an equation of state that follow from the cost J without these identifications.
What remains: For the reference occupation, a statement of the ledger about which recognition events carry a posting.

The paper’s list of what remains names no other part for this one.

Quantum gravity paper, Table 1 and the box of Section 8.2; Section 10, item 8

Part 9 · Section 8.3

Predictions

What the theory predicts that a measurement could see. At tree level nothing at any reachable scale; quantum predictions wait for the action.

Proved: At tree level every non-gauge long-wave mode propagates on the light cone up to a relative correction of order (k_phb)², and the static Newton kernel is correct to second order: no observable signature for lattice spacings up to 10⁶ Planck lengths.
Computed, as a conditional diagnostic: If the lattice spacing and the map from the remainder to a mass term are read through two identifications, the computed remainders of the unrepaired path sum correspond to a one-loop mass scale near 10²⁸ eV, about fifty-one orders of magnitude above the measured bound (numerical). It is a diagnostic of the unrepaired sum in the one-loop model, not a prediction of the theory.
Open: Every quantum prediction of a repaired path sum: the one-loop correction to the Newtonian potential for the lattice path sum itself, the graviton’s one-loop mass and width, and lattice-scale corrections, which need the spacing derived.
What would settle it: Parts 2 and 5 first; then the static two-mass amplitude of the repaired path sum, Born-subtracted, with its logarithmic coefficient.

Waits for part 2, the action, and part 5, a massless graviton.

How close to light · The one-loop correction’s size

Quantum gravity paper, Table 1 and the box of Section 8.3; Section 10, item 9

Part 10 · Section 9

One derivation

One chain from the foundation to every claim. Every result already lists its premises.

Done: Every result of the paper is stated with its premises, and its Table 2 sorts them: what holds for every κ > ½, what needs the identification κ = 1, what rests on a model ingredient, and what rests on an open premise. For the earlier paper’s results, and for parts (i) and (ii) of Theorem 7.4, a single theorem already states every result from the foundation’s premises, with κ = 1 and the observational bounds as stated premises.
Open: The same single derivation for every claim of the paper, with each identification named where it enters.

It covers every claim, so it comes last.

The full account’s status table · Run the checks

Quantum gravity paper, Table 1, Section 9 and Table 2; Section 10, item 10

What rests on what

Switch off a premise and see what still stands.

The paper’s Table 2 lists every result with the value of κ it needs and its other premises. Switch a premise off: a result whose row names it is no longer established, and one whose row names it for only part of the result holds in part. The rows are the paper’s own words; the other premises that stay are each result’s own conditions.

corner theorem, four readings, two worlds Theorems 2.1 to 2.3 · Part 1, the lattice and κ
Value of κ: all κ
Other premises: none
divergence of the anisotropy Theorem 3.1 · Part 6, finiteness
Value of κ: κ = 1
Other premises: gauge term with unit weight, flat measure, no ghost line, regulator (models)
record time keeps the doublers Theorem 3.2 · Part 7, unitarity
Value of κ: κ = 1
Other premises: regulator
the native model graviton is not massless Theorem 3.3 · Part 5, a massless graviton
Value of κ: κ = 1
Other premises: regulator
linear field, G = 1/(8πs) earlier paper, Theorem 4.6 · Part 4, Einstein at long wavelengths
Value of κ: all κ
Other premises: conserved stress family on a non-null cone of waves
exact solutions, Newton’s potential earlier paper, Theorems 4.8 to 4.10 · Part 4, Einstein at long wavelengths
Value of κ: κ = 1
Other premises: model medium; odd sides not divisible by 105
second order with worldline matter Section 4 · Part 4, Einstein at long wavelengths
Value of κ: all κ
Other premises: least-work law at the flat reference, ratios of c⁰, body and carrier split (identifications); point postings, non-gravitating reference (models); in the continuum form, the zero wave of the second order balanced by a free constant
every long-wave vertex is Einstein’s Theorem 4.2 · Part 4, Einstein at long wavelengths
Value of κ: all κ
Other premises: a star-good background; sampled smooth jets (a statement about the field equations, not about solutions)
the vertex symmetry fails at second order Theorem 4.3 · Part 4, Einstein at long wavelengths
Value of κ: κ = 1
Other premises: side-8 torus and the infinite lattice, local at one record; also the point mass’s Taylor fields; its far field (numerical)
second differences are consistent Theorem 4.1 · Part 4, Einstein at long wavelengths
Value of κ: any
Other premises: a jet-norm bound independent of the mesh
measure at Gaussian order, precision criterion Theorems 5.1 and 5.2 · Part 3, the measure
Value of κ: κ = 1; none
Other premises: the two classes of Theorem 5.1, separately
the derived weight Theorem 5.4 · Part 3, the measure
Value of κ: κ = 1
Other premises: (I1), (I2); channel count n_h (definition); the scaling of the phase
properties of the weight Theorem 5.5 · Part 3, the measure
Value of κ: κ = 1
Other premises: dual-cycle counts; (iii) as marked
the weight on the uncut domain Proposition 5.7 · Part 3, the measure
Value of κ: κ = 1
Other premises: fixed channel counts; the whole admissible cone with its Haar measure
kernel of the derived form Theorem 5.6 · Part 3, the measure
Value of κ: κ = 1
Other premises: dual-cycle counts
Ward criterion, defect identity, real parts Theorems 6.2 to 6.5 · Part 2, the action
Value of κ: κ ≠ ¼; all
Other premises: differentiability; for the lattice path sum, the regulator
no weight, no cubic correction Theorems 6.6 and 6.7 · Part 2, the action
Value of κ: κ = 1
Other premises: sides divisible by 4 for the failure of the symmetry
higher deficits leave the response unchanged Theorem 6.10 · Part 2, the action
Value of κ: κ = 1
Other premises: same slope, weight and regulator; regular determinants for the defects
what refinement does Theorem 6.9 · Part 2, the action
Value of κ: κ = 1
Other premises: (i) a tree-level coarse-graining exists (proved on the side-4 torus, open on larger tori); (iii) refinement invariance (identification)
one refinement step, the exact tension pending Section 6.5 · Part 2, the action
Value of κ: κ = 3/2; κ = 1
Other premises: side 4 to side 2
exact blocking multiplies the remainder by 16; no blocked fixed point, nor for block averages Sections 6.5 and 6.7 · Part 2, the action
Value of κ: all κ
Other premises: records on edges (identification); damping fixed on the fine lattice; regular fine determinants; with the damping fixed on the coarse lattice (pending): κ = 1, zone-averaged
computed remainders numerical Section 6.3 · Part 2, the action
Value of κ: κ = 1
Other premises: numerical; for the weight, (I1), (I2), dual cycle, phase scale a = ln 2/(2π) (model)
the corner condition pending Section 6.4 · Part 2, the action
Value of κ: κ = 1
Other premises: even torus side; even curvature of class C³ vanishing at the flat areas
the holonomy route pending Section 6.7 · Part 2, the action
Value of κ: κ = 1
Other premises: a loop’s phase read from its postings, one cycle per face (identifications); regulator
an action whose model graviton is massless open Target 6.8 · Part 2, the action
Value of κ: κ = 1
Other premises: open
finiteness theorems Theorems 7.1 to 7.3 · Part 6, finiteness
Value of κ: any family
Other premises: (H), (D), (O), or the blow-up
the weighted responses are finite Theorem 7.4 · Part 6, finiteness
Value of κ: κ = 1
Other premises: coupling a > 0; real parts; the weight is the ledger’s under (I1), (I2)
the complex observable pending Proposition 7.5 · Part 6, finiteness
Value of κ: κ = 1
Other premises: coupling a > 0; the weight is the ledger’s under (I1), (I2)
record-time norm obstruction Theorem 8.1 · Part 7, unitarity
Value of κ: κ = 1
Other premises: spatial period N ≥ 1
continuum-limit stability of the record-time roots with the derived weight with an inference Section 8.1 · Part 7, unitarity
Value of κ: κ = 1
Other premises: (I1), (I2), dual-cycle counts; Gaussian order; fixed weight strength; lattice-scale modes off the unit circle decouple (inference)
the obstruction with postings null Proposition 8.2 · Part 7, unitarity
Value of κ: κ = 1
Other premises: spatial period N ≥ 1; linearized bare equation, open record time
tree-level propagation earlier paper, Theorems 5.1 and 5.4 · Part 9, predictions
Value of κ: κ = 1
Other premises: none beyond the lattice

“All κ” means every κ > ½. The regulator is the Feynman damping +iε, a model wherever it enters; “numerical” marks computed values; “pending” marks a result whose proof is complete and whose independent verification is not yet finished. Target 6.8 is the open target, not a result. The part each row belongs to is this page’s reading of the paper’s sections.

Quantum gravity paper, Table 2 and Section 9

How to read it

The words, the foundation, and the order.

1 · The words

Proved, identified, open, pending.

Proved means proved as stated in the section named, with the premises listed there. An identification is a physical premise that is not a consequence of the definitions. Open marks a precise statement that is neither proved nor refuted. A step whose proof is complete and whose independent verification is not yet finished is never listed as proved: it has its own category, and no result rests on one without saying so. Computed values are marked numerical.

Every status word, in the glossary

Quantum gravity paper, Table 1 and Section 1
2 · The foundation

The ledger enters in three places.

The lattice is the foundation’s picture of events as records on four counters, which gives the Kuhn triangulation and the family of flat metrics; the value κ = 1 is an identification, not a consequence. The cost J(x) = ½(x + 1/x) − 1 enters twice: as the cost of a hinge cycle in the measure, under (I1) and (I2), and as the least-work law of matter, under its own identifications. Everything else, the Regge action, the analysis of lattice waves, the Ward identity and the finiteness theorem, is mathematics about the lattice and holds whatever the foundation says.

Quantum gravity paper, Section 9
3 · The order

The action comes first.

The paper lists the open statements in the order in which they decide the outcome: the action, a massless graviton, the value of κ, the measure’s identifications, finiteness of the repaired theory, unitarity, the nonlinear Einstein equations, matter from the cost, a prediction, and one derivation. Each is stated so that it cannot be met trivially: the zero action, the quadratic truncation of the Regge action and the Regge action itself are each checked against it. The Regge action meets every classical requirement and, numerically, fails the first; the zero phase has no classical limit. A repair must change the quadratic action at the lattice scale while keeping its long-wave limit, with every coefficient derived. If no phase the ledger supplies can do this, proving that is a result about the programme, and the paper would state it as one.

Quantum gravity paper, Section 10
4 · The map

Only the links the paper names.

The ring shows the paper’s own status of each part, in version 1.2, and draws a line only where its list of what remains, its Table 1 or a part’s box names another part. Einstein gravity at long wavelengths and matter have no line because none is named for them.

Quantum gravity paper, Table 1, the boxes of Sections 2 to 9, and Section 10

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; the earlier paper it recalls is Linearized Einstein gravity from Lorentzian Regge calculus on the Kuhn lattice.

Words on this page, in plain terms: proved · proof complete, verification pending · identification · model · inference · numerical · open.

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