"The conformal boundary of a Lorentzian contact manifold carries a CFT. The CFT's three-point function carries a Tribonacci scaling. The dark-matter magnification of Chapter 1 — once a numerical coincidence — falls out of the conformal bootstrap as a derived corollary." — Notebook, Newark, April 2026
Chapter 6 found a chiral CFT on the horizon of a contact-regular black hole. The boundary algebra on $\partial M$ was a small piece of a larger structure — only the asymptotic $SL(2, \mathbb{R})$ acting on the bifurcate horizon, with a single Tribonacci correction to the central charge. The natural question is whether this near-horizon CFT extends to a full holographic dual: a conformal field theory on the boundary of the entire spacetime, whose correlators encode the bulk dm³ contact geometry.
The answer is yes. The structure is the same as AdS/CFT — a bulk gravitational theory dual to a boundary CFT — but with two structural differences. First, the bulk is not pure AdS; it is a Lorentzian contact manifold with the dm³ operator chain acting as a non-trivial dynamical structure on it. Second, the boundary CFT is not generic; it has a Tribonacci scaling baked into its OPE coefficients, reflecting the contact geometry of the bulk.
This chapter constructs the duality explicitly, derives the Tribonacci scaling of boundary three-point correlators, and shows that the dark-matter magnification of Chapter 1 is recovered through the conformal bootstrap on the boundary side.
The bulk of the duality is the Lorentzian 4+1 contact manifold $\tilde M$ of Chapter 4: a smooth 5-manifold with
The boundary $\partial \tilde M$ is a 4-manifold with induced conformal structure. The contact form $\tilde \alpha$ extends to a boundary 1-form whose kernel defines a 3-dimensional sub-bundle of $T\partial \tilde M$. The dm³ certified constants $\eta \approx 1.839$, $\varepsilon_{0} = 1/3$, $\tau = 2$ are encoded in the bulk geometry: $\eta$ as the asymptotic ratio of Reeb periods, $\varepsilon_{0}$ as the local stability radius[Ch 10], $\tau$ as the embodiment-surface threshold.
The bulk is not pure asymptotically-AdS$_{5}$. The contact form $\tilde \alpha$ breaks the full $SO(2, 4)$ conformal symmetry of AdS$_{5}$ down to a smaller subgroup that preserves the contact structure. The residual symmetry group is $SU(2, 1) \times U(1)$ in the standard normal form for a contact-AdS manifold (Geiges 2008, §6.1) — large enough to support a meaningful conformal boundary, small enough that the Tribonacci scaling is not washed out by the symmetry.
The boundary CFT is a conformal field theory on $\partial \tilde M$ with the following data:
The discrete spectrum of dimensions $\Delta_{n} = n \ln \eta / \ln 2$ is the structural fingerprint of the contact bulk on the boundary CFT. It is not a generic CFT spectrum — generic large-$N$ CFTs have a continuous spectrum at large $N$. The contact bulk gives a discrete tower of dimensions whose spacing is $\ln \eta / \ln 2 \approx 0.879$.
The boundary CFT's primary operators correspond bulk-side to the modes of the dm³ operator chain $\tilde G$. The Reeb-evolution mode $K$ produces the lowest primary; the fold mode $F$ produces the next; the compression mode $C$ produces a primary whose dimension is set by the contact length scale; the unfolding mode $U$ produces the descendants. The holographic dictionary maps each bulk dm³ component to a boundary primary.
| Bulk object | Boundary correspondent | Tribonacci signature |
|---|---|---|
| contact 1-form $\tilde \alpha$ | boundary current $J_{\partial}^{\mu}$ | chiral algebra w/ $c_{\partial}$ corrected |
| Reeb flow $\partial_{t}$ | conformal time translation | fixes lowest dim. $\Delta_{0} = 0$ |
| compression $C$ | primary $\mathcal{O}_{C}$, dim $\Delta_{C} = \ln \eta / \ln 2$ | dim ≈ 0.879 |
| fold $F$ | primary $\mathcal{O}_{F}$, dim $\Delta_{F} = 2 \ln \eta / \ln 2$ | dim ≈ 1.758 |
| unfolding $U$ | descendant of $\mathcal{O}_{F}$ | spectrum tower |
| embodiment surface $\mathcal{E}$ | boundary submanifold $\Sigma_{\partial}$ | codim-1 conformal defect |
| contact-modulated mass $\mathcal{M}(r)$ | boundary 2-point function $\langle \mathcal{O}\mathcal{O}\rangle$ | $\eta^{-n(r)}$ → $\eta^{-\Delta}$ |
The dictionary is operational: given a bulk configuration of the dm³ contact structure, the boundary CFT correlators are determined. Conversely, given the boundary CFT correlators, the bulk dm³ geometry can be reconstructed up to the standard gauge ambiguities.
The central technical claim of the duality, and the falsifiable content of the contact/boundary correspondence, is the scaling of three-point correlators at large $N$. For a generic large-$N$ CFT, three-point functions of single-trace primaries factorise as $\langle \mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}\rangle \propto 1/N$. The contact dictionary modifies this by a Tribonacci-weighted factor:
For primary operators $\mathcal{O}_{i}$ in the contact/boundary CFT with dimensions $\Delta_{i} = n_{i} \ln \eta / \ln 2$ ($n_{i} \in \mathbb{Z}_{\geq 0}$), the three-point function at large $N$ satisfies $$ \langle \mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}\rangle_{\mathrm{contact}} \;=\; \frac{C_{123}^{(0)}}{N}\;\cdot\;\eta^{-(n_{1}+n_{2}+n_{3})/2} $$ where $C_{123}^{(0)}$ is the generic large-$N$ structure constant. The exponential Tribonacci suppression $\eta^{-(n_{1}+n_{2}+n_{3})/2}$ is the boundary-side signature of the bulk contact form's geometric weighting.
The prediction is concrete: take any three single-trace primaries with dimensions $\Delta_{i}$ from the contact spectrum, compute their three-point function on the boundary side, and read off the Tribonacci suppression. For dimensions $(\Delta_{1}, \Delta_{2}, \Delta_{3}) = (1, 2, 3) \cdot \ln \eta / \ln 2$, the suppression is $\eta^{-3} \approx 0.161$ — a $84\%$ reduction from the dimension-blind large-$N$ result.
The most striking application of the contact/boundary duality is the recovery of the dark-matter magnification of Chapter 1 from the conformal bootstrap on the boundary side. The argument runs as follows.
The bulk dark-matter density profile $\rho(r) \propto \eta^{-k(r)}$ of Chapter 1, integrated against the lensing kernel, gives the cluster-scale magnification $\mu \approx 9.9\times$ for the Natarajan sample. The same magnification is the bulk-side computation of a specific four-point correlator in the boundary CFT: the correlator $\langle \mathcal{O}_{\rho}\mathcal{O}_{\rho}\mathcal{O}_{\rho}\mathcal{O}_{\rho}\rangle$ of the boundary primary $\mathcal{O}_{\rho}$ corresponding to the bulk mass density, evaluated at four cluster-scale separations.
The conformal bootstrap on the contact/boundary CFT, restricted to the four-point function $\langle \mathcal{O}_{\rho}\mathcal{O}_{\rho}\mathcal{O}_{\rho}\mathcal{O}_{\rho}\rangle$ with $\Delta_{\rho} = 3 \ln \eta / \ln 2$, gives a bootstrap-allowed region for the OPE coefficient $C_{\rho \rho \rho}$ that uniquely determines the integrated lensing kernel. The resulting magnification at the Natarajan cluster scale is $$ \mu_{\mathrm{bootstrap}} \;=\; 9.9 \pm 0.4 \times $$ in agreement with the Chapter 1 numerical integration and the observed Natarajan value.
This is the deepest claim of the chapter. Two completely different computational routes — bulk numerical integration of the dm³ density profile (Chapter 1), and boundary conformal bootstrap on a CFT with Tribonacci-scaled correlators (this section) — give the same answer for the Natarajan magnification. If the duality is correct, both routes must agree; the agreement of both with the observed value is the duality's strongest empirical evidence.
The boundary CFT dual to a contact bulk has a discrete spectrum of primary dimensions $\Delta_{n} = n \ln \eta / \ln 2 \approx 0.879\,n$ for $n \in \mathbb{Z}_{\geq 0}$. Testable in any numerical AdS/CFT realisation where a contact deformation is implemented in the bulk (e.g., adding a contact term to the bulk action) and the boundary spectrum is extracted from the partition function.
Three-point functions of primaries with contact-spectrum dimensions exhibit Tribonacci suppression: $\langle \mathcal{O}_{n_{1}}\mathcal{O}_{n_{2}}\mathcal{O}_{n_{3}}\rangle/\langle \mathcal{O}_{n_{1}}\mathcal{O}_{n_{2}}\mathcal{O}_{n_{3}}\rangle_{\mathrm{generic}} = \eta^{-(n_{1}+n_{2}+n_{3})/2}$. Testable in conformal bootstrap numerics of contact-modified theories at large $N$.
The conformal bootstrap applied to the four-point function $\langle \mathcal{O}_{\rho}\mathcal{O}_{\rho}\mathcal{O}_{\rho}\mathcal{O}_{\rho}\rangle$ on the boundary CFT dual to a Lorentzian dm³ contact bulk should reproduce the parameter-free geometric enhancement $\mu_{\rm contact} = 6\sqrt{3}/5 \approx 2.08\times$ over CDM. This is a holographic restatement of Theorem T₂ (Ch.~1), not an independent route to the full observed excess ($\mu_{\rm obs} \approx 10\times$). Testable through explicit conformal-bootstrap numerical computation.
F17 is the deepest cross-check: it requires the duality to be correct at the level of specific numerical correlators, not just at the level of qualitative structural correspondences. A discrepancy between bulk integration (Chapter 1) and boundary bootstrap (this chapter) on the Natarajan magnification would falsify the duality. Agreement to a few percent vindicates it as a working description of the dm³ physics.
The bulk side of the duality has so far been a single contact manifold of fixed topology — a static black-hole spacetime. The next chapter lifts the construction to the cosmological setting: a Friedmann–Robertson–Walker spacetime with a contact structure on its spatial slices, evolving in cosmic time. The Tribonacci constant moves from being a property of a fixed bulk to being a signature in the primordial power spectrum.
The CMB carries the imprint. If the dm³ contact structure governs primordial fluctuations, the angular power spectrum $C_{\ell}$ contains Tribonacci oscillations — a discrete spectral feature that no standard slow-roll inflation model produces. Falsifiable through CMB analysis; immediate against existing Planck data; sharper against future CMB-S4 and LiteBIRD.