piatra.institute

PIATRA . INSTITUTE

trisquare

conformal quantum gravity, a perfect-square action, and 4D scalar phi-fourth
based on 2021, Boyle and Turok, Cancelling the Vacuum Energy and Weyl Anomaly in the Standard Model with Dimension-Zero Scalar Fields
claude opus 4.8June 2026·first cut. ports the Turok dimension-zero scalar correspondence ideation into playground conventions: PlaygroundLayout, black-and-lime palette, the correspondence triangle with status-tagged edges, a status ledger, the Ward identity constraint game, a conformally flat sector with finite-difference curvature, the phi-fourth RG flow, an action translator, calibration of the curvature method against analytic metrics, and nine assumptions.
Gaussian bump
A localized rise in Omega produces a ring of curvature; the metric is still conformally flat.
peak |K|: 0.58
int K: -0.98
couplings: 4
peak Omega
1.60
min Omega
1.00
mean |K|
0.09
Weyl
0
terms cut
4
pole t*
87.7
the correspondence as a map: three theories, with every edge tagged by how solid the link is
Ward identity (diffeo)single couplingconformal reductionQG (γ→0)PS‖φ‖₄⁴
exact in toy model
known physics
speculative / paper claim
QGγ↓0QG_{\gamma\downarrow 0}
constrained quantum gravity
a special limiting version of quantum gravity, restricted here to the conformally flat sector.
PS\mathrm{PS}
perfect-square action
a dimension-zero scalar theory whose action can be written as a perfect square.
∥φ∥44\lVert\varphi\rVert_4^4
4D scalar phi-fourth
a quartic scalar theory in four dimensions, where the coupling is dimensionless.
speculative / paper claimWard identity (diffeo)
the proposal claims diffeomorphism invariance drives the UV gravity action into a perfect-square form. shown so far only for conformally flat metrics.
speculative / paper claimsingle coupling
the perfect-square scalar action is claimed to rewrite as a 4D quartic theory controlled by one dimensionless coupling, by analogy with gauge theory.
known physicsconformal reduction
that conformally flat gravity reduces to the dynamics of a single scalar conformal factor is standard; the full correspondence to phi-fourth is the speculative part.
status ledger
known physics
a conformal factor changes scale, not causal structure
light cones are invariant under g -> Omega^2 g; this is textbook conformal geometry.
known physics
conformally flat means the Weyl tensor vanishes
in 4D, C_{mu nu rho sigma} = 0 iff the metric is conformally flat. the free spin-2 gravitational field lives in the Weyl tensor.
exact in toy model
K = -Omega^{-2} Laplacian(ln Omega) for a 2D conformal slice
the curvature heatmap computes exactly this; it is validated against analytic constant-curvature metrics in the calibration panel.
known physics
in 4D the phi-fourth coupling is dimensionless and runs by 3 lambda^2 / 16 pi^2
standard one-loop result; reproduced in the flow tab. this is why a single coupling is structurally attractive.
exact in toy model
symmetry compresses the space of allowed Lagrangian terms
the Ward game shows scale, conformal, and diffeomorphism invariance deleting and merging terms, a toy of the proposal's Ward-identity statement.
speculative / paper claim
constrained quantum gravity equals a perfect-square scalar theory
the central correspondence of the proposal. shown so far only for conformally flat metrics, and still research-level, not textbook physics.
speculative / paper claim
the correspondence extends to low energies including mass and Einstein-Hilbert terms
claimed for the conformally flat sector only; the genuinely spin-2 part of gravity is not yet covered.

One proposal, three theories

Turok's program proposes a three-way correspondence between a constrained limit of quantum gravity, a perfect-square scalar action, and 4D scalar phi-fourth theory. Diffeomorphism invariance gives a Ward identity that restricts the UV form of the action; a single dimensionless coupling controls it, by analogy with gauge theory. This playground turns each piece into something you can move.

The big caveat is built in: the correspondence is shown so far only for conformally flat metrics. The genuinely spin-2 part of gravity is not yet covered.

Why gravity starts to look like a scalar

A conformally flat metric replaces all the components of the metric with a single scalar conformal factor:

gμν(x)=Ω2(x) ημν,Ω(x)=eϕ(x)g_{\mu\nu}(x) = \Omega^2(x)\,\eta_{\mu\nu}, \qquad \Omega(x) = e^{\phi(x)}

On a 2D slice the curvature is read straight off that scalar, K=−Ω−2Δln⁡ΩK = -\Omega^{-2}\Delta \ln\Omega, and the Weyl tensor vanishes. The conformal heatmaps show geometry becoming one field; the calibration panel checks the curvature method against metrics of known constant curvature.

Symmetry as compression

The Ward game is the heart of the proposal made tangible. Scale invariance deletes every dimensionful coupling, conformal invariance keeps only the Weyl-squared curvature term, diffeomorphism invariance removes frame-fixed terms, and single-coupling mode collapses the dimensionless survivors into one. A messy Lagrangian is compressed into a near-unique form, which is what a Ward identity does in the UV.

What is solid and what is not

The status ledger and the colour of every triangle edge keep the bookkeeping honest. The conformal curvature and the phi-fourth running are exact or textbook. The Ward game is a schematic of a real statement. The central claim, that constrained quantum gravity equals a perfect-square scalar theory mapping to phi-fourth, is a research-level conjecture shown only in the conformally flat sector. This is a toy explorer of a correspondence, not a proof of it.

Model changelog

v1.0June 2026
  • correspondence triangle: the three nodes (constrained quantum gravity, perfect-square action, 4D phi-fourth) with every edge tagged exact-in-toy, known-physics, or speculative.
  • status ledger: each claim sorted into the same three confidence tiers.
  • Ward identity constraint game: scale, conformal, and diffeomorphism invariance delete terms and single-coupling merges the dimensionless survivors, with a per-term verdict and reason.
  • conformally flat sector: an Omega(x, y) explorer with five presets and a finite-difference curvature heatmap computing K = -Omega^{-2} Laplacian(ln Omega).
  • phi-fourth flow: the one-loop running coupling 3 lambda^2 / 16 pi^2 with its Landau pole.
  • action translator: the rough gravity-side / scalar-side dictionary from the proposal.
  • calibration: the finite-difference curvature is validated against constant-curvature sphere and Poincare-disk metrics to better than one percent.
read the research companion →