Black Hole-Origin Cosmological Model
A Membrane Theory of (Almost) Everything

Part I — The Membrane Universe: Axioms, Story, and Intent

1. Prologue: The Stage Is the Actor

1.1 A universe that moves under us. There is an old habit in physics: paint a stage, place objects on it, push them around with forces, and call the performance reality. It is efficient and often correct. It is also wrong at scale. General relativity does not give us a stage with actors; it gives us a single, self‑interacting sheet of geometry. Mass is not cargo on that sheet. Motion is not choreography on top of it. Everything that exists is a pattern of stress and curvature in the sheet itself.

This paper takes that statement literally and operationalizes it. The “sheet” is spacetime; the “stress” is encoded in quasilocal thermodynamics of horizons; the “patterns” are the deformations we mislabel as matter, force, and expansion. The claim is not mystical. It is technical: when you enforce the horizon version of the first law of thermodynamics, the Friedmann equations drop out as bookkeeping identities, and “dark energy” looks less like a universal fluid and more like a global tension mode of the membrane that we call spacetime. We will prove the first step (thermodynamics ↔ geometry), construct the second (a quasilocal evolution law for the tension, \(\Lambda_{\text{eff}}(a)\)), and show how the only permitted ripples of that tension live at the very largest scales. The rest of physics in the infrared remains exactly what it already is: general relativity plus the Standard Model.

This is the vantage: stop pretending the stage is fixed. Let the sheet move under us, then measure how it moves.

1.2 Why another “theory of everything” isn’t a joke

Scope first, romance later. We are not deriving the couplings of \(\text{SU}(3)\times\text{SU}(2)\times\text{U}(1)\), we are not quantizing gravity, and we are not replacing quantum field theory. What we are doing is unifying, at the level that data can judge today, \(\boxed{\text{gravity}\leftrightarrow\text{thermodynamics}\leftrightarrow\text{information}\leftrightarrow\text{cosmic acceleration}}\) with a single membrane‑centric ontology that obeys general relativity locally, and lets the global vacuum term be a state variable of the membrane.

The ambition is a theory of (almost) everything about spacetime dynamics in the observable universe: a framework in which “dark energy” is membrane tension, black holes are memory‑bearing boundary machines, and ultra‑large‑scale anomalies are the safe, IR‑suppressed residue of horizon noise. If future surveys pin \(w(z)\to -1\) to machine precision and every ultra‑large‑scale signal vanishes, our construction collapses cleanly to \(\Lambda\)CDM. If not, the membrane tells you where to look and what to fit.

2. The Membrane Universe Thought Experiment

2.1 The membrane premise

Premise. The universe is not “stuff in a container.” It is one continuous, tensioned membrane; nothing sits on it or moves through it. Every phenomenon is a deformation of the membrane itself.

We make this precise by mapping each rule to standard GR/BHO objects:

• Mass is curvature. Einstein equation \(G_{\mu\nu} = 8\pi G\,T_{\mu\nu}\): matter‑energy is curvature, not cargo.
• Motion is topology rearranging. Timelike geodesics re‑label as the metric evolves; what looks like “orbiting” is the membrane’s connection coefficients \(\Gamma^\mu_{\nu\rho}\) changing in time.
• Gravity is tension gradient. Small, residual contributions from a tension functional \(\mathcal{T}(x)\) act as an effective, geometry‑encoded vacuum term on top of the usual GR sourcing.
• Black holes are runaway tension collapses. Horizons with large surface gravity \(\kappa\); on the stretched horizon they behave as viscous, resistive fluids (the membrane paradigm).
• Expansion is membrane stretching. A global mode \(\rho_{\Lambda}^\mathrm{eff}(a)\) evolves slowly with the membrane’s state, entering Friedmann as an effective vacuum energy.
• Gravitational waves are ripples. Metric perturbations propagating on the membrane; locally GR‑exact (no change to \(c_\mathrm{GW}\)).
• Hubble tension is sampling a wrinkled sheet. Different probes average different patches/epochs of the membrane’s tension history; mild, bounded drift can bias cross‑calibrations without new fields.

This is not additional structure. It is a refusal to split “stage” and “actors.”

2.2 Formal map to GR and \(\Lambda\)

Define a tension functional of the geometry and matter fields \(\mathcal{T}[g_{\mu\nu}, T_{\mu\nu}, \nabla g, \ldots]\), and write the effective field equation as \(G_{\mu\nu} = 8\pi G\,T_{\mu\nu} - \rho_{\Lambda}^\mathrm{eff}(x)\,g_{\mu\nu}\) with \(\rho_{\Lambda}^\mathrm{eff}(x) \equiv \rho_{\Lambda0} + \delta\rho_{\Lambda}[\mathcal{T}(x)]\).

There is no new bulk force; \(\rho_{\Lambda}^\mathrm{eff}\) is a state variable tied to horizon thermodynamics. In FRW, applying the first law at the apparent horizon reproduces the Friedmann equations with \(S = A/4G\) and \(T = \kappa/2\pi\); allowing slow, non‑equilibrium drift of the horizon state injects a time‑dependent \(\Lambda_\mathrm{eff}(a)\) consistent with the same thermodynamic bookkeeping. We will use the Cai‑Kim/Akbar‑Cai construction (first law ⇄ Friedmann) as the formal bridge.

2.3 Design constraint: background‑free consistency

BHO forbids “forces on a fixed stage” as fundamental. Any effect we add must be re‑expressible as:

• a change in the metric (curvature), and/or
• a change in the global tension mode \(\Lambda_\mathrm{eff}(a)\).

Operationally, that means we keep local GR intact, keep \(c_\mathrm{GW} = c\), and restrict deviations to two small, passivity‑bounded background dials and one ultra‑large‑scale anisotropy knob, all derivable from horizon thermodynamics.

2.4 Historical arc and relation to prior work

1. Einstein as equation of state. Jacobson’s 1995 result that demanding \(\delta Q = T\,dS\) for all local Rindler horizons derives Einstein’s equation, positioning gravity as emergent thermodynamics rather than a fundamental interaction law. That is our license to treat horizon thermodynamics as primary.
2. Horizon first‑law ⇄ Friedmann. In FRW, applying the first law at the apparent horizon reproduces the Friedmann equations, in GR and in higher‑curvature extensions; non‑equilibrium terms appear when the “vacuum” sector evolves. That is our mechanism for a slowly drifting \(\Lambda_\mathrm{eff}(a)\).
3. The membrane paradigm and fluid/gravity. Black‑hole horizons behave like viscous, resistive 2‑D fluids with surface resistivity \(\sim 4\pi\) (≈377 Ω) and shear viscosity \(1/16\pi G\); near‑horizon limits map Einstein dynamics to Navier–Stokes. That is our language for how “tension” behaves and why only ultra‑large scales should retain horizon fingerprints.

BHO keeps all of this orthodoxy and adds one unifying sentence: call the vacuum term what it behaves like—membrane tension—and let it move.

Notation. We use metric signature \((-+++\)), \(c = 1\). Apparent‑horizon quantities carry subscript \(A\); parent‑horizon quantities carry \(S\). \(\Lambda_\mathrm{eff}(a)\) denotes the membrane’s coarse‑grained tension mode entering Friedmann. Details of the evolution law and stochastic corrections appear in Parts II–III. Inline notation keeps hyphenated identifiers legible and spacing explicit, e.g., \(\text{BHO-core}\thinspace: f\negthinspace\big(x_{\text{velocity}}\big)!;\,\Gamma_{\alpha} := \beta_{\text{sheet}}.\) Spacing aliases stay consistent across prose, e.g., \(\text{flow-aware}\medspace \eta_{cross-check}-\zeta_{phase-shift}\enspace;\,\rho.\) Correlators use double‑angle brackets even when typed with ASCII fallbacks, e.g., \(<< T_{\mu\nu},\, S_{\rho\sigma} >>\) or display as \(\left<< T_{\mu\nu} \medspace ,\, S_{\rho\sigma} \right>>_{\enspace\Theta}\).

3. Local Horizons, Thermodynamic Laws, and Einstein as an Equation of State

3.1 Jacobson’s thermodynamic derivation of Einstein gravity

Statement. If every spacetime point admits a local causal horizon whose entropy is proportional to its area, and if the Clausius relation \(\delta Q = T\,dS\) holds for all such horizons, then the field equation governing the metric is the Einstein equation with a cosmological constant as an integration constant.

Sketch. Consider a spacetime point \(p\) and the local Rindler horizon generated by the past light sheet of a small spacelike 2‑surface through \(p\). Let \(k^\mu\) be the null generator, \(\lambda\) its affine parameter, and \(\theta\) the expansion. The Raychaudhuri equation for a hypersurface‑orthogonal null congruence, \(\tfrac{d\theta}{d\lambda} = -\tfrac12 \theta^2 - \sigma_{\mu\nu}\sigma^{\mu\nu} - R_{\mu\nu}k^\mu k^\nu\), together with \(\theta\vert_{\lambda=0}=0\) and small \(\lambda\), yields \(\delta A \propto -\int R_{\mu\nu}k^\mu k^\nu\,\lambda\, d\lambda\, dA_0\). Assign horizon entropy \(S=\eta A\) and Unruh temperature \(T=\hbar\kappa/(2\pi)\) to the local Rindler wedge. The energy flux across the horizon is \(\delta Q=\int T_{\mu\nu}\chi^\mu d\Sigma^\nu \sim \int T_{\mu\nu}k^\mu k^\nu\,\lambda\, d\lambda\, dA_0\). Imposing the Clausius relation for all such horizons forces \(R_{\mu\nu} + \Phi\, g_{\mu\nu} = 8\pi G\, T_{\mu\nu}\), with Newton’s constant determined by the entropy density \(\eta = 1/(4\hbar G)\) and \(\Phi = -\tfrac12 R + \Lambda\). The result is the Einstein equation with cosmological constant—an operational bridge from local horizon thermodynamics to field equations. (Physical Review Links)

3.2 Cai–Kim / Akbar–Cai horizon thermodynamics in FRW

Apparent horizon and quasilocal bookkeeping. In a spatially homogeneous and isotropic FRW spacetime with scale factor \(a(t)\) and curvature \(k\), the apparent horizon has radius \(R_A = 1/\sqrt{H^2 + k/a^2}\), area \(A_A = 4\pi R_A^2\), and (convention‑dependent) surface gravity \(\kappa_A\). Assign horizon temperature \(T_A = \kappa_A/(2\pi)\) and horizon entropy \(S_A = A_A/(4G)\). Define the Misner–Sharp energy inside \(R_A\) as \(E = \rho V\) with \(V=\tfrac{4\pi}{3}R_A^3\) for a perfect fluid, and work density \(W = (\rho - p)/2\). The unified first law on spherically symmetric backgrounds, \(dE = T_A\,dS_A + W\,dV\), when projected along the apparent horizon, is equivalent to the two Friedmann equations. The same horizon‑first‑law structure also reproduces the correct Friedmann dynamics in Gauss–Bonnet and Lovelock gravity despite generalized entropies. (arXiv)

• Misner–Sharp (quasilocal) energy inside \(R_A\): \(E = \rho V\).
• Work density: \(W = (\rho - p)/2\).
• Apparent‑horizon first law: \(dE = T_A\,dS_A + W\,dV\) with \(S_A = A_A/(4G)\) and \(T_A = \kappa_A/2\pi\).
• Projection: applying the first law along the apparent horizon reproduces the Friedmann equations in GR; generalized entropies map to generalized Friedmann equations. (arXiv)

We keep the quasilocal stress‑energy bookkeeping explicit when translating horizon mechanics into membrane dynamics:

$$\boxed{\begin{aligned}\nabla_\mu T^{\mu\nu}_{\text{mem}} &= 0 \ \kappa\,\delta\!A &= \int_{\partial M} K^{\mu}{}_{\nu} n_\mu u^\nu \end{aligned}}$$

3.3 Non‑equilibrium corrections and apparent‑horizon entropy production

When the “vacuum sector” is not strictly constant, the Clausius relation gains an entropy production term, \(\delta Q = T\,dS + T\,d_i S\), and the horizon first law becomes non‑equilibrium. Eling, Guedens, and Jacobson showed that curvature corrections to horizon entropy (e.g., in \(f(R)\) gravity) require such treatment, with \(d_i S\) interpretable as bulk/shear dissipation of the horizon “fluid.” In a cosmological setting, the same logic applies: a slowly evolving effective vacuum energy \(\Lambda_\mathrm{eff}(a)\) can be encoded as an irreversible entropy production rate \(\dot S_\mathrm{irr}\) at the apparent horizon. Operationally, we keep the GR field equations intact and account for small, slow, boundary‑driven drift of the vacuum term through this \(d_i S\) channel. (Physical Review Links)

3.4 The membrane as a thermodynamic object

• Small scales: Jacobson horizons act as the “atoms” of spacetime thermodynamics.
• Cosmological scales: the FRW apparent horizon is the coarse‑grained membrane patch whose free‑energy accounting reproduces Friedmann evolution.

This two‑tier picture justifies treating the vacuum term as a state variable—a global tension mode—whose slow evolution is permitted by non‑equilibrium horizon thermodynamics, while local dynamics remain those of GR. (Physical Review Links)

4. The Membrane Paradigm and Horizon Hydrodynamics

4.1 The stretched horizon as a fluid

In the membrane paradigm, a black‑hole horizon is replaced by a timelike “stretched horizon,” a physical 2‑surface endowed with fluid‑like transport: surface resistivity \(\rho_s = 4\pi \approx 377~\Omega\), shear viscosity \(\eta = 1/(16\pi G)\), and a negative bulk viscosity fixed by GR. On this surface, the gravitational field equations reduce to Navier–Stokes‑type equations for the membrane, with well‑defined energy‑momentum fluxes and Ohmic response. This is formal equivalence in the near‑horizon limit, not mere analogy. (Physical Review Links)

4.2 Fluid/gravity correspondence near horizons

Work on fluid/gravity duality makes the equivalence explicit: in an appropriate near‑horizon scaling, the vacuum Einstein equations reduce to the incompressible Navier–Stokes equation for a boundary fluid, with transport coefficients determined by the gravitational background. This legitimizes treating horizon dynamics as 2‑D hydrodynamics feeding into 4‑D geometry and underwrites using horizon transport to parameterize how a global tension mode can drift without spoiling local GR. (ResearchGate)

4.3 Horizon Capillary number and flow regimes

We introduce a Horizon Capillary number \(\mathrm{Ca}_H \equiv \mu_H U_H / \sigma_H\), where \(\mu_H\sim \eta\) is the horizon’s effective viscosity, \(U_H\) a characteristic horizon “drift speed” (set by the slow evolution of the parent and/or apparent horizon, e.g., \(U_H\sim H_0 R_H\)), and \(\sigma_H\propto \kappa\) an effective surface tension (surface gravity).

• \(\mathrm{Ca}_H \ll 1\): tension‑dominated, near‑de Sitter response; \(\Lambda_\mathrm{eff}\) is essentially constant.
• \(\mathrm{Ca}_H \sim 1\): incipient flow instability permits a single, small pulse in \(\Lambda_\mathrm{eff}(a)\) (e.g., from a discrete parent event) before the system re‑laminarizes.
• \(\mathrm{Ca}_H > 1\): excluded by passivity/second‑law constraints; the membrane cannot sustain supercritical drift without dissipative blow‑up.

This dimensionless control captures when a boundary event can leave a fossil in the cosmological background while remaining consistent with horizon hydrodynamics.

4.4 Impedance matching and Blandford–Znajek circuits

Horizon electrodynamics naturally organizes into a circuit. For rotating black holes, the Blandford–Znajek mechanism extracts rotational energy when the load impedance matches the horizon’s surface impedance \(\rho_s\sim 377~\Omega\); DC power transfer is then optimal. The membrane’s DC response to slow boundary evolution is the small drift of \(\Lambda_\mathrm{eff}(a)\), with amplitude limited by dissipative matching to the interior. This picture motivates both the smallness and the smoothness of any allowed drift. (OUP Academic)

5. The BHO \(\Lambda_\mathrm{eff}(a)\) Evolution Law

5.1 From parent horizon evolution to \(\Lambda_\mathrm{eff}(a)\)

Let \(R_S(t)\) be the (Schwarzschild‑like) radius of the parent horizon that contains our FRW interior, and \(R_A(a)\) the apparent horizon of our FRW spacetime. We postulate that the coarse‑grained membrane tension responds only to slow changes of these boundaries and to a stochastic IR‑suppressed noise source:

\(\tfrac{d}{dt}\,\Delta\Lambda_\mathrm{BHO}(t) = \mu\,\tfrac{d}{dt}\!\left(\tfrac{1}{R_S(t)}\right) + \nu\,\tfrac{d}{dt}\!\left(\tfrac{1}{R_A(t)}\right) + \xi(t,\mathbf x).\)

Here \(\mu\) encodes sensitivity to the parent horizon (evaporation/accretion/merger), \(\nu\) encodes FRW apparent‑horizon feedback, and \(\xi\) is the stochastic gravity term, with an IR‑hyperuniform spectrum that ensures only ultra‑large scales are affected (Part III). This ansatz is the minimal, quasilocal way to let the membrane’s global tension mode move without inventing new bulk fields.

A simple interior–exterior smoothing that respects this setup uses a piecewise horizon‑tension profile:

$$\sigma(r)=\begin{cases}\sigma_0 \left(1-\dfrac{r}{r_h}\right)^2 & r

5.2 Effective background parametrization

It is convenient to integrate to an energy‑density form: \(\rho_\Lambda^\mathrm{eff}(a) = \rho_{\Lambda0} + \tfrac{3\nu}{8\pi G}\big(H^2-H_0^2\big) + \tfrac{\sigma}{8\pi G}\,\dot H + \rho_\mathrm{step}(a)\). Here \(\sigma\) is an effective combination of \((\mu,\nu,R_S,R_A)\), and \(\rho_\mathrm{step}(a)\) allows for a single small, smoothed pulse capturing a discrete parent event (e.g., a merger). The first two terms are adiabatic responses to boundary drift; the third term captures mild lag; the step term is optional and tightly bounded by passivity.

5.3 Thermodynamic‑length bound and passivity

We constrain drift using thermodynamic length: in driven systems, the excess dissipation along a protocol is quadratic in the “speed” through state space with a Riemannian metric set by response functions. Translating this to \((\nu,\sigma)\) control yields \(\mathcal L^2 = \int (\alpha_\nu \nu^2 + 2\alpha_{\nu\sigma}\nu\sigma + \alpha_\sigma \sigma^2)\, d\ln a \le \varepsilon_\mathrm{diss}\ll 1\), so the membrane does not produce more irreversible entropy than its horizon transport allows. A separate passivity requirement (no net extractable work over a cycle) enforces a convex, monotone \(w(z)\) that asymptotes to \(-1\) as boundary evolution saturates, keeping BHO close to \(\Lambda\)CDM unless data compel otherwise. (Physical Review Links)

5.4 Mapping to effective \(w(z)\) and CPL‑like parameters

Define \(w(a) \equiv -1 - \tfrac{1}{3}\,\tfrac{d\ln \rho_\Lambda^\mathrm{eff}}{d\ln a}\). Given the parameterization above, \(w(a)\) is a small, smooth function determined by \((\nu,\sigma,\rho_\mathrm{step})\) and \(H(a)\). For comparison with survey pipelines we supply the CPL mapping \(w(a)\approx w_0 + w_a(1-a)\) obtained by a Taylor expansion about \(a=1\) with priors that implement \(\mathcal L^2\) and passivity; this provides drop‑in compatibility with DESI/Euclid/LSST likelihoods while keeping the BHO structure explicit. (Details in Appendix B; summary constraints in Part VII.)

5.5 Simple examples and toy models

• Pure running \((\sigma=0,\nu>0)\): \(\rho_\Lambda^\mathrm{eff}(a)=\rho_{\Lambda0}+\tfrac{3\nu}{8\pi G}(H^2-H_0^2)\) gives \(w(a)>-1\) with a tiny, convex drift that decays at high redshift; CMB‑consistent.
• Pure \(\dot H\) response \((\nu=0,\sigma>0)\): sensitive to the late‑time deceleration‑to‑acceleration transition; produces a very small shoulder in \(w(z)\) near \(z\sim 0.5\) while leaving the early universe intact.
• One pulse \(\rho_\mathrm{step}(a)\): a single smoothed step at \(a_\star\) (e.g., due to a parent merger) generates a localized feature in \(w(z)\) and a correlated ultra‑large‑scale anisotropy axis (from the noise kernel), both bounded by \(\mathcal L^2\) and passivity. This is the only BHO case where \(w(z)\) may briefly change curvature; it remains small and testable.

All three classes preserve local GR and Kerr behavior, alter only the background and ultra‑large‑scale sectors, and are immediately fit‑ready in expansion data, void AP anisotropy, low‑\(\ell\) CMB, standard sirens, and redshift‑drift programs.

References — Part II Back to Part II

6. Stochastic gravity: fluctuations of the membrane

6.1 Primer: from semiclassical to Einstein–Langevin

Semiclassical gravity closes the geometry with the expectation value of the stress tensor, \(G_{\mu\nu}[g]=8\pi G\,\langle \hat T_{\mu\nu}\rangle\), but this neglects stress–tensor fluctuations.

Stochastic gravity upgrades the dynamics to the Einstein–Langevin equation, \(G_{\mu\nu}[g+h]+\Lambda g_{\mu\nu} = 8\pi G\big(\langle \hat T_{\mu\nu}\rangle+\xi_{\mu\nu}\big) + \!\int\!\mathrm d^4y\, H_{\mu\nu}{}^{\alpha\beta}(x,y)\, h_{\alpha\beta}(y)\), where \(h\) is the metric perturbation, \(H\) is a nonlocal dissipation kernel, and \(\xi_{\mu\nu}\) is a Gaussian stochastic source with zero mean and covariance \(N_{\mu\nu\alpha\beta}(x,y) \equiv \tfrac12\langle \delta \hat T_{\mu\nu}(x), \delta \hat T_{\alpha\beta}(y)\rangle\) and \(\delta\hat T \equiv \hat T-\langle \hat T\rangle\).

• The pair \((H,N)\) satisfies a fluctuation–dissipation relation that enforces consistency of dissipation and noise for the open gravitational system.
• Hu, Verdaguer, and collaborators construct \(N\) in curved spacetimes (including Schwarzschild) and derive the Einstein–Langevin equation for black hole backreaction and cosmology. (Annual Reviews)

The correlators we propagate through this system keep full tensor scripts intact:

$$<< T^{\mu}_{\nu}{}^{\rho}_{\sigma}(x) \, T^{\alpha}_{\beta}(y) >>^{\kappa}_{\lambda}$$

Relevance to BHO. In our membrane ontology, “vacuum” is not mute; it contributes colored noise filtered by horizon hydrodynamics. The coarse‑grained \(\Lambda_\mathrm{eff}\) of Part II inherits slow, correlated, horizon‑shaped fluctuations from \(\xi_{\mu\nu}\), with the dissipation kernel furnishing the passivity bounds already imposed via the thermodynamic‑length constraint. (arXiv)

6.2 Parent-horizon noise as an open‑system drive

Near an event horizon, quantum fields excite stress–tensor fluctuations whose backreaction appears as horizon‑width fluctuations and nonlocal dissipation in the effective dynamics. Formally, treat the interior FRW membrane as the system and the parent horizon as the bath; integrating out bath fields yields a colored stochastic drive \(\xi\) entering the interior’s Einstein–Langevin equation.

1. The literature shows noise kernels with finite horizon support.
2. Near‑horizon fluctuation–dissipation relations link Hawking flux fluctuations and metric dissipation.
3. The horizon effectively broadens into a fluctuating layer rather than a sharp null surface. (arXiv)

We model the spatially homogeneous piece of this drive as the \(\xi(t,\mathbf x)\) that perturbs \(\Lambda_\mathrm{eff}(t)\) in Part II, and the inhomogeneous piece as a hyperuniform spectrum on super‑Hubble scales (Sec. 6.3).

6.3 A hyperuniform noise kernel for the membrane

Design principle. On the largest scales we want fluctuations that are present but IR‑suppressed, consistent with observed near‑isotropy and with the suppressed long‑wavelength variance characteristic of hyperuniform media (structure factor \(S(k)\to 0\) as \(k\to 0\)). Torquato’s program formalizes this across crystals, quasicrystals, and special disordered systems, and recent galaxy‑field analyses strongly suggest a hyperuniform lean at very large scales. (courses.physics.ucsd.edu)

We therefore posit a membrane‑noise power in comoving Fourier space: \(\boxed{\langle \xi,\xi\rangle (k) = C k^{\alpha} \exp\!\big[-(k/k_\star)^{\beta}\big],\quad \alpha>0}\), with \(k_\star\) a transition scale at a fraction of the apparent‑horizon wavenumber and \(\alpha\) controlling IR suppression. This functional form is minimal, hyperuniform in the IR, and admits UV damping for numerical stability.

Motivation from horizon hydrodynamics. The stretched horizon behaves like a 2D viscous fluid with fixed transport coefficients \((\eta=1/16\pi G,\ \rho_s=4\pi)\), so long‑wavelength excitations are governed by effective 2D hydrodynamics. In 2D turbulence, energy exhibits an inverse cascade toward larger scales while enstrophy cascades to small scales. Such dynamics generally fill large scales with correlated structure but do not diverge in the IR when bounded by global constraints, naturally yielding softly rising \(k^{\alpha}\) spectra with \(\alpha>0\). (Physical Review Links) The combination “horizon fluid + IR‑tamed inverse cascade + hyperuniform target” fixes the kernel class above.

6.4 An ultra‑large‑scale (ULS) anisotropy template

A small anisotropic leakage of the kernel into the interior provides an axisymmetric quadrupolar modulation of the large‑scale density field: \(\delta(\mathbf k) \rightarrow \big[1 + A_\mathrm{ULS} P_2(\hat{\mathbf k}\!\cdot\!\hat{\mathbf n})\big]\delta(\mathbf k)\), with amplitude \(A_\mathrm{ULS}\) and axis \(\hat{\mathbf n}\). This is the same harmonic content used in CMB/LSS analyses of quadrupolar statistical anisotropy (often parameterized as \(P(k)!=!P_\mathrm{iso}(k)[1+g_\ast(\hat{\mathbf k}\!\cdot\!\hat{\mathbf n})^2]\)); Planck and large‑scale structure catalogs already constrain \(g_\ast\) at the percent level, providing immediate bounds on \(A_\mathrm{ULS}\) for BHO. (A&A )

• Data hook. The kernel parameters \((\alpha,\beta,k_\star)\) map to a scale‑dependent \(A_\mathrm{ULS}(k)\). We can forecast and fit to low‑\(\ell\) CMB TT/TE/EE couplings, galaxy‑field multipolar anisotropy estimators in BOSS/DESI, and void‑shape/filament‑alignment statistics sensitive to coherent shear on gigaparsec scales (see also the disordered‑heterogeneous descriptors already applied to galaxy surveys). (Astrophysics Data System)
• Quadrupolar template. An axially anisotropic correction to the kernel produces \(P(\mathbf k)=P_\mathrm{iso}(k)\big[1+g_\ast(\hat{\mathbf k}\!\cdot\!\hat{\mathbf n})^2\big]\); our \(A_\mathrm{ULS}\) maps linearly to \(g_\ast\). Current limits \(|g_\ast|\!\lesssim\!\mathcal O(10^{-2})\) at CMB scales bound \(A_\mathrm{ULS}\) at similar levels. (arXiv)

7. Dimensional cascade and genealogical priors (speculative, yet tethered)

7.1 Near‑horizon effective dimensionality

Two complementary facts suggest a “dimensional cascade” at horizons:

• Hydrodynamic reduction. The membrane paradigm projects Einstein’s equations onto a 2D Damour–Navier–Stokes system on the stretched horizon with fixed transport, making long‑wavelength horizon dynamics effectively two‑dimensional. (Physical Review Links)
• Conformal control. Multiple derivations of black‑hole entropy invoke a near‑horizon 2D CFT (Virasoro algebra, Brown–Henneaux–type structures, Cardy counting), indicating that universal horizon microphysics is governed by 2D conformal symmetry across broad classes of black holes. (Physical Review Links)

We borrow both as structural priors: close to the parent horizon, fluctuations and information flow are effectively 2D, and those statistics feed the interior membrane via the stochastic drive of Sec. 6.

7.2 Universes from black holes and selection effects

Smolin’s cosmological natural selection posits that black holes spawn new universes whose parameters are slightly mutated, leading over many generations to parameter values that enhance black‑hole formation. Whether or not one accepts the full biological analogy, BHO can exploit a minimal replicator–mutator prior: cosmological parameters \(\boldsymbol\theta\) drift across generations toward regions of parameter space that raise heavy‑seed formation and BH number density, with small Gaussian “mutations.” (arXiv)

We only need this as a hierarchical prior on the interior membrane’s large‑scale statistical state (Sec. 7.3), not as a new degree of freedom.

7.3 Fractal/hyperuniform projection to 3‑D observables

If (i) horizon microphysics is effectively 2D and hyperuniform‑leaning, and (ii) genealogical priors bias toward parameter regions that amplify early massive BH formation (hence stronger near‑horizon control of IR), then the projected 3D point field of galaxies can inherit a weak hyperuniform signature at the largest scales. Recent analyses of galaxy distributions using disordered‑media descriptors find enhanced large‑scale order consistent with hyperuniform tendencies on very large scales, while small scales are anti‑hyperuniform due to nonlinear clustering. This mixed behavior is precisely what a membrane with IR‑suppressed kernel and UV damping would generate. (Astrophysics Data System)

We do not claim “the Universe is hyperuniform”; we claim BHO naturally supplies a tunable hyperuniform bias at ULS that existing summaries have started to see.

7.4 Constraints, not claims

Everything in Sec. 7 is a prior, not a new freedom. The adjustable pieces are already encoded in the kernel \((\alpha,\beta,k_\star)\) and in the tiny anisotropy amplitude \(A_\mathrm{ULS}\). Planck’s isotropy tests and LSS quadrupolar searches cap these amplitudes sharply; any BHO fit must respect those caps. (A&A )

Technical appendix (brief): from kernel to observables

In linear response, the stochastic source \(\xi\) perturbs the background through the retarded Green function of the FRW background; schematically \(\delta H \sim \mathcal G\!\star\!\xi\). With the kernel above, the induced isotropic power spectrum behaves like \(k^{\alpha}\) at \(k\ll k_\star\), yielding suppressed IR variance consistent with hyperuniform expectations. (courses.physics.ucsd.edu)

Horizon‑hydro rationale. The 2D inverse cascade fills the largest scales with correlated power but, under global tension constraints (membrane passivity/area law), avoids IR divergence, justifying \(\alpha>0\). (Annual Reviews)

What’s firm, what needs tightening

• Firm. The Einstein–Langevin apparatus and the noise kernel in curved spacetime; near‑horizon hydrodynamics and membrane transport; near‑horizon 2D conformal control of horizon microphysics; hyperuniformity as the precise language of IR suppression; observational pipelines for quadrupolar statistical anisotropy. (Annual Reviews)
• Bridging assumption. The parent horizon acts as the dominant bath for the interior membrane, sourcing a colored, IR‑tamed, weakly anisotropic noise with the kernel given in Eq. (†). This is consistent with stochastic gravity and horizon hydrodynamics but needs explicit calculation of \((\alpha,\beta,k_\star)\) from a near‑horizon EFT or numerical relativity with quantum noise surrogates. (arXiv)

Actionable next steps

1. Compute the Einstein–Langevin response on FRW with an evolving apparent horizon, using known noise‑kernel approximations to obtain \(\langle \delta\Lambda_\mathrm{eff}\delta\Lambda_\mathrm{eff}\rangle\) and the induced \(A_\mathrm{ULS}(k)\). (courses.physics.ucsd.edu)
2. Derive \(k_\star\) from the horizon capillary number and surface gravity, then fix \(\alpha\) by matching to 2D inverse‑cascade spectra under passivity. (Annual Reviews)
3. Fit \((\alpha,\beta,k_\star,A_\mathrm{ULS})\) jointly to Planck DR4 isotropy tests and DESI multipolar estimators, plus the hyperuniform descriptors of the 3D galaxy field (nearest‑neighbor tails, percolation thresholds, pair‑connectedness). (InspireHEP)

References — Part III Back to Part III

11. Shadows, waves, and impedance constraints

11.1 EHT shadows of M87* and Sgr A*

What’s measured. The Event Horizon Telescope (EHT) has now imaged horizon‑scale structure in two supermassive black holes: M87* (2017, with reprocessing and 2018 follow‑up) and Sgr A* (2017–2018), including linear polarization maps that probe ordered magnetic fields near the horizon. Multi‑epoch analysis of M87* shows a persistent ring‑plus‑shadow morphology with a diameter consistent across years to within a few percent, and a stable azimuthal asymmetry consistent with GRMHD flows. For Sgr A*, polarization fractions at the 25 percent level favor magnetically arrested or strongly magnetized accretion states and are consistent with a high spin viewed at a moderate inclination, again within Kerr expectations. These “shadow‑scale” observables limit large departures from the Kerr metric and from GR near the horizon.

How we use it. Let \(\delta_\mathrm{sh}\) be a generic parameter capturing fractional deviations of the shadow (or lensing‑ring) diameter from the Kerr prediction at fixed \(M/D\). EHT multi‑epoch results for M87* tightly confine \(\delta_\mathrm{sh}\) around zero (with systematics dominated by mass‑distance calibration and flow morphology). We therefore treat EHT as a prior that keeps all BHO strong‑field behavior strictly Kerr‑like: the membrane tension mode does not alter the local solution, transport coefficients, or photon geodesics. In practice we impose a narrow Gaussian prior on \(\delta_\mathrm{sh}\) centered at 0 with width taken from the multi‑epoch “persistent shadow” analyses, and propagate that into the small‑amplitude BHO background parameters \((\nu,\sigma)\). This locks the near‑horizon sector to GR while allowing cosmological drift in \(\Lambda_\mathrm{eff}(a)\) as in Parts II–IV.

Non‑Kerr one‑offs. A large literature constrains specific deviations (e.g., charges, Yukawa tails, non‑Kerr metrics) using shadow size, photon‑ring structure, or polarization morphology. The bottom line that matters here: current EHT bounds already exclude order‑unity shadow‑scale deviations, and improvements in VLBI baselines and multi‑band imaging will reduce systematics further; none of these results push us away from “Kerr locally, drift cosmologically.”

11.2 GW170817 and the speed of gravity

Constraint. The binary neutron‑star event GW170817 and its prompt gamma‑ray counterpart GRB 170817A arrived within \(\sim\)1.7 s of each other after \(\sim\!10^8\) years of flight, implying \(\big|\tfrac{c_\mathrm{GW}-c}{c}\big|\ \lesssim\ 10^{-15}\), ruling out a broad family of modified‑gravity models with dispersive or subluminal tensor propagation. Later LVK catalog tests have continued to find no evidence for dispersion or extra polarization modes.

How we use it. BHO’s construction does not modify the propagation equation for gravitational waves. The background drift in \(\Lambda_\mathrm{eff}(a)\) is thermodynamic and quasilocal; it does not introduce new dynamical degrees of freedom in the tensor sector or any Lorentz‑violating operators. We therefore adopt GW170817’s constraint as a hard prior: all BHO variants must keep \(c_\mathrm{GW}=c\) and the standard two polarizations, with any cosmological signatures entering only through \(H(z)\) and ULS noise (Parts III–V).

11.3 Echoes/ECO phenomenology

Status of searches. “Gravitational‑wave echoes” would indicate partially reflective structure outside the would‑be event horizon, as in some exotic compact object (ECO) models. Broad, template‑agnostic and targeted searches in O1–O3 have reported no statistically significant evidence for echoes; LVK’s GR tests with GWTC catalogs similarly find no robust post‑merger excess. Ongoing O4 analyses and white‑paper roadmaps continue to treat echoes as a null so far.

BHO stance. In BHO the horizon is absorbing (the usual membrane resistivity and viscosity) and there is no reflective shell. That means no echoes by construction. If future high‑significance detections of echoes survive data‑quality and trials‑factor scrutiny, those BHO variants are ruled out. Conversely, continued nulls are an automatic consistency check for the “impedance‑matched,” absorbing‑horizon picture that underpins the thermodynamic priors we use across Parts II–V.

What these tests buy us (and what they don’t)

• Buy: Strong‑field locality. EHT locks the near‑horizon metric and transport to Kerr; GW170817 locks the tensor sector to GR with \(c_\mathrm{GW}=c\); echo nulls lock the boundary to be absorbing. Together these enforce that BHO lives only in the background and ULS‑stochastic sectors it claims.
• Don’t buy: Microstate derivations or photon‑ring substructure predictions beyond current resolution; EHT systematics and flow‑model degeneracies still limit how precisely we can translate ring size to deformation parameters. These are instrument‑limited, not conceptual, and improve with longer baselines and multi‑frequency campaigns.

References — Part VI Back to Part VI

12. Parameter inference framework

12.1 Parameter space

We split parameters into the baseline cosmology and the BHO membrane sector.

• Cosmology (baseline \(\Lambda\)CDM block). \(\Theta_\mathrm{cosmo} = {\Omega_b, \Omega_c, H_0, n_s, A_s, \tau}\).
• BHO (background + ULS + horizon‑hydro block). \(\Theta_\mathrm{BHO} = {\nu, \sigma, \rho_\mathrm{step}, A_\mathrm{ULS}, \hat n, \alpha_\mathrm{HU}, \mathrm{Ca}_H}\).

Where:

• \(\nu\) is the adiabatic \(H^2\) response in \(\rho_\Lambda^\mathrm{eff}\).
• \(\sigma\) is the mild \(\dot H\) “lag” response.
• \(\rho_\mathrm{step}\) is a single, smoothed pulse (optional) capturing a discrete parent‑horizon event.
• \(A_\mathrm{ULS}\) is the ultra‑large‑scale quadrupole amplitude; \(\hat n\) its axis (uniform over \(S^2\)).
• \(\alpha_\mathrm{HU}\) is the IR exponent of the structure factor tail, \(S(k) \propto k^{\alpha_\mathrm{HU}}\) as \(k \to 0\).
• \(\mathrm{Ca}_H\) is the Horizon Capillary number controlling whether a one‑off pulse is hydrodynamically allowed.

Derived functions are computed from these primitives:

• Background \(E(a)\) from the ODE in Part IV, §8.1.
• Effective \(w(a) = -1 - \tfrac{1}{3}\,\tfrac{d\ln \rho_\Lambda^\mathrm{eff}}{d\ln a}\).
• ULS template \(\delta(\mathbf k) \to [1 + A_\mathrm{ULS} P_2(\hat{\mathbf k} \cdot \hat{\mathbf n})] \delta(\mathbf k)\).
• Structure‑factor tail \(S(k \to 0) \sim k^{\alpha_\mathrm{HU}}\).

12.2 Priors from theory and observations

We impose structured priors that encode the membrane physics and strong‑field constraints.

• Thermodynamic length (background smoothness). \(\mathcal L^2 = \int (\alpha_\nu \nu^2 + 2\alpha_{\nu\sigma} \nu \sigma + \alpha_\sigma \sigma^2)\, d\ln a\) \(\Rightarrow \Pi_\mathrm{th.len}:\ \mathcal L^2 \le \varepsilon_\mathrm{diss} \ll 1\). Operationally: a soft prior with support sharply decaying as \(\mathcal L^2\) exceeds \(\varepsilon_\mathrm{diss}\). This enforces small, smooth departures from \(\Lambda\).
• Passivity (no net extractable work). A convex‑monotone prior on \(w(a)\) with \(w(a) \to -1\) at late times. In practice we enforce \(\tfrac{d^2 w}{d(\ln a)^2} \ge 0\) and \(\lim_{a \to \infty} w(a) = -1\), implemented as inequality constraints or a penalty functional.
• Strong‑field GR (EHT, GW170817, echo nulls).
  • Kerr shadow prior: a narrow Gaussian on a generic shadow‑deviation parameter \(\delta_\mathrm{sh}\) centered at 0; pushforward to exclude any near‑horizon transport inconsistent with GR (keeps \(\mathrm{Ca}_H\) in the laminar regime).
  • GW speed prior: delta‑like support at \(c_\mathrm{GW} = c\).
  • Absorbing horizon prior: exclude reflective inner boundary conditions (no echoes); hard prior in the likelihood block.
• ULS and structure‑factor priors.
  • \(A_\mathrm{ULS}\): truncated symmetric prior, \(|A_\mathrm{ULS}| \le A_\mathrm{max}\), with \(A_\mathrm{max}\) set by current CMB low‑\(\ell\) limits.
  • \(\hat n\): isotropic over \(S^2\).
  • \(\alpha_\mathrm{HU} \ge 0\) with broad support and a weak preference for \(\alpha_\mathrm{HU} \lesssim \mathcal O(1)\) (hyperuniform lean, not a crystal).
  • Kernel consistency prior tying \(\alpha_\mathrm{HU}\) to the IR index \(\alpha\) used in the stochastic kernel of Part III (if sampling that layer).
• Capillary bound. \(\mathrm{Ca}_H\) log‑normal prior peaked at \(\ll 1\). If \(\mathrm{Ca}_H \ge 1\), only one short pulse is allowed and its amplitude must still satisfy the thermodynamic‑length budget.

12.3 Likelihood construction

We build a factorized but coupled joint likelihood:

\(\mathcal L(\Theta) = \mathcal L_\mathrm{CMB}\, \mathcal L_\mathrm{BAO/RSD}\, \mathcal L_\mathrm{SNe}\, \mathcal L_\mathrm{void\text{-}AP}\, \mathcal L_{S(k)}\, \mathcal L_\mathrm{sirens}\, \mathcal L_\mathrm{drift}\, \mathcal L_\mathrm{EHT}\, \Pi_\mathrm{th.len}\, \Pi_\mathrm{pass.}\, \Pi_\mathrm{ULS/HU}\, \Pi_{\mathrm{Ca}_H}\).

Blocks and forward models (summary):

• CMB (low‑\(\ell\) focus): pseudo‑\(C_\ell\) TT/TE/EE likelihood with mask coupling; optional inclusion of standard high‑\(\ell\) spectra via compressed distance priors to avoid double counting with BAO. The ULS template multiplies primordial power by \([1 + A_\mathrm{ULS} P_2(\hat k \cdot \hat n)]\) at \(k \lesssim k_\star\) only.
• BAO/RSD: distances \(\{D_M/r_d, H, r_d/c\}\) and growth \(f\sigma_8(z)\) generated from the BHO background ODE; full survey covariances. Background‑only nature of BHO is exploited by testing the BAO–RSD correlation predicted by GR.
• SNe: distance‑modulus residuals with calibration and light‑curve nuisance parameters; anchored so as not to double count with BAO/CMB \(r_d\).
• Void AP: 2D void–galaxy cross‑correlation multipoles \(\xi_\ell(s)\) with a calibrated RSD model; AP stretch fixed by \(\{D_A, H\}\); an additive ULS quadrupole of fixed shape allowed at the largest scales.
• Structure factor \(S(k)\): window‑convolved estimator in the survey volume; fit the \(k \to 0\) tail for \(\alpha_\mathrm{HU}\) after validating on \(\Lambda\)CDM mocks; include integral‑constraint and shot‑noise corrections.
• Standard sirens: Gaussian posteriors in \(\{H_0, H(z)\}\) from BNS/BBH with EM counterparts or statistical host methods; directly probe the background.
• Redshift drift: Gaussian likelihoods for \(\dot z(z)\) in pre‑specified redshift bins (low‑z SKA; high‑z ELT/ANDES). This block cleanly separates geometry‑only drift from models that also modify forces.
• EHT: Gaussian prior on \(\delta_\mathrm{sh}\) (shadow deviation), plus optional polarization‑based constraints folded in as nuisance‑marginalized bounds. This enforces local Kerr and GR transport for the membrane.

Samplers and constraints. Any of HMC‑NUTS, dynamic nested sampling, or tempered SMC is fine; we recommend nested sampling when exploring the sharp inequality constraints (passivity, \(\mathrm{Ca}_H\) regimes). Inequalities are best implemented as soft barriers with analytic gradients (e.g., log‑barrier on concavity violations of \(w(a)\)) so the sampler never walks into unphysical basins.

Reporting. Show both native BHO posteriors \((\nu, \sigma, \rho_\mathrm{step}, A_\mathrm{ULS}, \alpha_\mathrm{HU}, \mathrm{Ca}_H)\) and the derived \(w_0\text{–}w_a\) map for easy comparison. Always include posterior‑predictive checks: BAO–RSD consistency, low‑\(\ell\) polarization coherence, mock‑to‑data \(S(k)\) tail stability.

12.4 Falsifiers (explicit “kill switches”)

We pre‑register the following decision rules. If any holds at high significance, the corresponding BHO sector is ruled out; if several hold, the framework collapses to \(\Lambda\)CDM or is excluded outright.

1. Background drift absent at decisive precision (degeneracy collapse). If combined BAO+SNe+sirens+redshift‑drift constrain \(w(z) \equiv -1\) with \(|\Delta w| < 10^{-3}\) over \(0 < z \le 2\) and \(\dot z(z)\) matches \(\Lambda\)CDM within forecast errors across the same range, then \(\nu \to 0\), \(\sigma \to 0\), \(\rho_\mathrm{step} \to 0\). Outcome: BHO reduces to \(\Lambda\)CDM (not a failure, but no distinguishable content).
2. Passivity violation (thermodynamic no‑go). If the best‑fit \(w(a)\) inferred from distances requires non‑convex or oscillatory behavior that violates the passivity prior (e.g., multiple sign flips in \(d^2 w/d(\ln a)^2\) inconsistent with the thermodynamic‑length budget), BHO’s membrane‑tension dynamics are falsified.
3. Growth–background inconsistency demanding new forces. If RSD/WL data require scale‑ or time‑dependent growth that cannot be explained by the BHO background \(H(a)\) alone (e.g., significant gravitational slip or \(\gamma(z)\) inconsistent with GR), BHO is falsified (it does not modify perturbation forces).
4. ULS stochasticity absent at forecast sensitivity. If joint CMB low‑\(\ell\) + void‑AP + galaxy \(S(k \to 0)\) constrain \(|A_\mathrm{ULS}| < 10^{-3}\) and \(\alpha_\mathrm{HU} \le 0\) at high significance, then the stochastic kernel’s IR content is observationally zero; BHO’s ULS “wrinkle” is ruled out and the model collapses to its purely background limit (or to \(\Lambda\)CDM if Rule 1 also holds).
5. Echo detection or reflective inner boundary. A robust, population‑level detection of gravitational‑wave echoes (surviving pipeline and trials‑factor scrutiny) is incompatible with BHO’s absorbing horizon. Outcome: horizon sector falsified.
6. Non‑Kerr strong‑field requirement. If EHT (or future horizon‑scale VLBI) demands persistent non‑Kerr lensing or transport at the shadow scale (beyond the current error budget) for M87*/Sgr A*, BHO’s assumption of GR membrane transport is falsified.
7. Tensor‑sector anomaly. Any confirmed deviation in the GW propagation equation (speed, dispersion, extra polarizations) at astrophysical frequencies falsifies BHO’s GR‑faithful tensor sector.
8. Capillary runaway. If fits drive \(\mathrm{Ca}_H \gg 1\) to explain data, the horizon hydrodynamics enter a regime BHO does not allow (would violate passivity/dissipation bounds). Outcome: BHO excluded unless independent strong‑field evidence simultaneously revises the membrane transport itself (which current EHT+GW constraints do not support).

Practical deliverables

• Posterior surfaces for \((\nu, \sigma, \rho_\mathrm{step})\) with and without thermodynamic‑length priors to show how much the physics compresses parameter volume.
• Axis posteriors for \(\hat n\) across TT/TE/EE and void AP; check for persistence and alignment with known systematics.
• Structure‑factor tail plots \(S(k)\) vs \(k\) on log–log with \(\Lambda\)CDM mocks, data, and BHO kernel bands overplotted.
• Posterior‑predictive checks: BAO–RSD consistency diagnostics; low‑\(\ell\) polarization phase‑coherence residuals; siren‑vs‑BAO distance residuals; redshift‑drift bands.

13. What we’ve unified

13.1 One object, many faces.

The central move has been to treat spacetime as a single, tensioned membrane and to insist that apparently different phenomena are just different projections of its state.

• Gravity as geometry. The local dynamics are exactly GR; no new long‑range force is introduced at any scale.
• Dark energy as membrane tension. The effective vacuum term \(\Lambda_\mathrm{eff}(a)\) is a global state variable constrained by quasilocal thermodynamics (first law at the apparent horizon) and by non‑equilibrium entropy production.
• Stochastic fluctuations as horizon noise. The Einstein–Langevin layer endows the background with an IR‑suppressed, hyperuniform noise kernel that only touches ultra‑large scales.
• Large‑scale structure (and low‑\(\ell\) CMB) as the imprint of that IR‑tamed kernel. The signature is a fixed‑shape, small quadrupolar modulation and a weak hyperuniform lean in the structure‑factor tail.

13.2 From thermodynamics to geometry.

Three pillars hold this together:

1. Einstein as an equation of state. Local horizons with \(\delta Q = T\,dS\) recover the field equations.
2. FRW as horizon bookkeeping. The apparent‑horizon first law yields the Friedmann equations, with non‑equilibrium terms when \(\Lambda_\mathrm{eff}\) drifts.
3. Horizon hydrodynamics. The membrane paradigm and fluid/gravity map supply transport, dissipation, and a physically meaningful “Horizon Capillary number” that regulates whether any pulse in \(\Lambda_\mathrm{eff}\) is even allowed.

This is why the framework is compact: one ontology, three mature technologies.

13.3 Empirical anchors, not vibes.

The construction was wired to measurable sectors from the start:

• Background. A first‑order ODE for \(E^2(a)\) that recovers \(\Lambda\)CDM as \((\nu,\sigma,\rho_\mathrm{step})\to 0\), with thermodynamic‑length and passivity bounding any drift in \(w(z)\).
• ULS sector. A single quadrupolar template with fixed spectral shape and two free quantities \((A_\mathrm{ULS},\hat n)\), jointly fit to CMB low‑\(\ell\), void AP, and the galaxy \(S(k\!\to\!0)\) tail.
• Strong‑field. EHT shadows, \(c_\mathrm{GW}=c\), and echo‑nulls impose local‑Kerr and absorbing‑horizon priors, ensuring BHO lives only where it claims to: background and ULS.

13.4 Testability and graceful failure.

Everything has a kill switch: background drift that must be smooth, convex, and small; ULS features that must agree across TT/TE/EE and LSS; strong‑field locality that must remain exactly Kerr. If the switches flip, the model doesn’t waffle; it collapses to \(\Lambda\)CDM with no residue. That’s a feature.

14. What we haven’t done (yet)

14.1 Standard Model from the membrane.

We have not derived \(\text{SU}(3)\times\text{SU}(2)\times\text{U}(1)\) content, couplings, or hierarchies from horizon microstructure or edge modes. We have a suggestive scaffold (near‑horizon 2D control; entanglement‑based microstate counting), but no microscopic derivation of particle spectra or Yukawas. That is open terrain.

14.2 Dynamic‑horizon microstates.

We lack a full quantum description of time‑dependent horizon microstructures that would compute the stochastic kernel \(\langle\xi\xi\rangle\) ab initio, including its IR slope, anisotropy leakage, and non‑Gaussianities. The effective kernel we used is physically motivated and easy to falsify, but not derived.

14.3 Microscopic \(\Lambda_\mathrm{eff}\) dynamics.

We engineered \(\Lambda_\mathrm{eff}(a)\) from thermodynamic and hydrodynamic constraints plus a minimal boundary‑evolution ansatz. A complete theory would compute \((\nu,\sigma)\) and any single pulse \(\rho_\mathrm{step}\) from a microphysical code (e.g., near‑horizon EFT + numerical relativity with stochastic sources), not treat them as effective couplings.

14.4 Everything else.

We did not revise early‑universe inflation, primordial spectra, or reheating; we did not modify neutrino physics; and we did not attempt UV completion. Those can be made consistent with the membrane picture but were outside this scope.

15. Relations to other “TOE” attempts

15.1 Entropic/emergent gravity.

Entropic‑gravity programs (and related emergent‑gravity efforts) make deep use of horizon thermodynamics and information. We share the thermodynamic foundation but do not alter local GR or galaxy‑scale dynamics. No MOND‑style phenomenology is introduced or needed; the membrane’s global tension mode is the only addition, and it acts as background‑level bookkeeping, not as a new force.

15.2 Degravitation and vacuum sequestering.

Degravitation/sequestering ideas reinterpret the cosmological term through global constraints or IR filters on vacuum energy. BHO is geometrically adjacent but not identical: we do not filter sources; we re‑identify the vacuum term as a state variable of the membrane and regulate its motion with horizon thermodynamics and hydrodynamic passivity. It’s a different lever.

15.3 Early dark energy (EDE) and dynamical dark energy (DDE).

EDE/DDE models change the background with one or more scalar dofs. BHO reproduces the same data‑facing knobs at leading order in \(w(z)\) but with a strict hierarchy: smooth, small, convex drift only; no extra propagating dofs, no modified‑gravity growth kernels, no \(c_\mathrm{GW}\neq c\). This is how the joint BAO+RSD+drift+ULS test really distinguishes them.

15.4 Modified gravity (Horndeski, \(f(R)\), massive gravity, etc.).

These frameworks alter linear perturbations, lensing, and the tensor sector at some level. BHO never does: Kerr locally, GR waves, standard slip (\(=0\)). If growth or GW propagation is found to deviate, BHO loses by design. If the deviations are purely background and ULS, BHO is the minimal geometric explanation that survives strong‑field constraints.

15.5 What BHO is (and is not).

BHO is a geometric and thermodynamic completion of \(\Lambda\)CDM: it keeps \(\Lambda\)CDM’s empirical core intact and supplies a principled, testable origin for “dark energy” as membrane tension with a tiny, disciplined amount of motion. It is not a replacement for quantum field theory, nor a universal solvent for every anomaly. It is a very sharp knife for a specific class of questions.

Prologue

If precision cosmology finds nothing but a flat line — \(w(z)\to -1\) at the \(10^{-3}\) level, redshift‑drift glued to \(\Lambda\)CDM, no ULS quadrupole, no hyperuniform lean beyond what survey windows fake — then the membrane does its quiet job as an interpretation: Einstein from horizons, Friedmann from the first law, and “dark energy” as a name for a constant tension. The paper then functions as a unification of language, not a revision of physics.

If, however, data resolve even small, coherent shifts — a convex \(w(z)\) drift bounded by passivity, a reproducible ULS axis at sub‑percent amplitude across TT/TE and void stacks, a redshift‑drift curve nudging the \(\Lambda\)CDM line in precisely the BHO direction — then the minimal story that fits everything while leaving GR and Kerr untouched is the one we have built here. The stage is not a stage. It is the actor. And the “dark” in dark energy is only the part of the membrane you haven’t measured yet.

Premise for the program

From here, the path is almost mechanical:

1. Run the joint inference with the priors we encoded.
2. Publish the null‑hypothesis compression if that’s what nature gives.
3. Pin the amplitude and axis of the ULS wrinkle and the magnitude of \((\nu, \sigma)\) if it doesn’t.

Either way, we learn something falsifiable about the thermodynamics of spacetime — and that was always the point.

BHO in One Page: Predictions & Kill Switches

If BHO is even roughly right, future data should nudge the universe this way:

• Background drift (but tiny, smooth, and boring).
  • A small, convex deviation of the dark‑energy equation of state \(w(z) = -1 + \delta w(z)\) with \(|\delta w| \sim 10^{-2}\)–\(10^{-1}\), no oscillations, and \(w(z)\to -1\) at late times.
  • No wild early‑time behavior, no sharp features.
• Redshift drift with a specific “lean.”
  • A redshift‑drift curve \(\dot z(z)\) that is slightly more negative than \(\Lambda\)CDM at \(2 \lesssim z \lesssim 5\).
  • Slightly less positive at \(z \lesssim 1\).
  • The zero‑crossing shifted by \(\Delta z = \mathcal{O}(10^{-2})\), not more.
• Ultra‑large‑scale wrinkle, not a scar.
  • A sub‑percent quadrupolar modulation of large‑scale structure and low‑\(\ell\) CMB, on the same axis across TT/TE and void stacks, with amplitude small enough to hide under cosmic variance but not identically zero.
• Hyperuniform lean in the galaxy field.
  • Structure factor \(S(k) \propto k^{\alpha_\mathrm{HU}}\) on the largest scales with \(\alpha_\mathrm{HU} > 0\) (but small) after you’ve beaten window functions and shot noise into submission.
• Absolutely boring gravity on all the fun scales.
  • Black‑hole shadows consistent with Kerr within current EHT errors.
  • Gravitational waves with \(c_\mathrm{GW} = c\), two polarizations, no dispersion.
  • No gravitational‑wave echoes, no reflective shells, no extra forces in growth or lensing.

If any of this happens, BHO is in trouble:

• Background is too perfect.
  • Joint BAO + SNe + standard sirens + redshift‑drift all nail \(w(z) = -1\) with \(|\Delta w| < 10^{-3}\) over \(0 < z \le 2\), and \(\dot z(z)\) lands exactly on the \(\Lambda\)CDM prediction within next‑gen errors → the membrane tension doesn’t move, and BHO collapses cleanly to vanilla \(\Lambda\)CDM.
• Passivity violation.
  • Best‑fit \(w(z)\) from distances requires sharp features, sign flips in curvature, or oscillations incompatible with a smooth, dissipative tension mode → horizon thermodynamics + BHO evolution law are ruled out as a package.
• Growth demands new forces.
  • RSD/weak lensing/clustering insist on modified gravity — scale‑dependent growth, gravitational slip, or extra tensor dofs that cannot be explained by changing \(H(a)\) alone → BHO loses by design, because it never touches the force law.
• ULS sector is dead quiet.
  • Combined CMB low‑\(\ell\) + void AP + galaxy \(S(k \to 0)\) force \(A_\mathrm{ULS} \approx 0\) and \(\alpha_\mathrm{HU} \le 0\) at high significance → the stochastic “wrinkle” of the membrane is observationally zero; only the background dial can survive (and that can still shrink to \(\Lambda\)CDM by the previous bullet).
• Strong‑field heresy actually wins.
  • Robust gravitational‑wave echoes.
  • Persistent non‑Kerr lensing at the shadow scale for M87*/Sgr A*.
  • Confirmed deviation in \(c_\mathrm{GW}\) or tensor dispersion → the “Kerr‑plus‑thermodynamics” membrane story is wrong in its foundations.

Prologue

Curated citations sit in the same hierarchy used throughout Montopia: short lead-ins, nested lists, and inline claims so \(c_\mathrm{GW} = c\) and similar tests stay readable without preformatted blocks.

Premise

The references are grouped by the pillars of the membrane program—thermodynamics, transport, stochastic gravity, hyperuniform structure, cosmological probes, and strong-field tests—so each observational thread maps cleanly to its theoretical anchor.

Mapping

Follow the ordered list below to trace each citation cluster and the observational levers it informs: background drift, void anisotropy, \(H(a)\) and redshift-drift checks, or direct constraints on \(c_\mathrm{GW}\) and echo amplitudes.

1. Horizon thermodynamics & GR foundations

   • Jacobson 1995 — Einstein as equation of state. Thermodynamics of spacetime: The Einstein equation of state. Physical Review Letters, 75, 1260–1263. DOI: 10.1103/PhysRevLett.75.1260. arXiv: gr-qc/9504004.
   • Hayward 1998 — Unified first law. Unified first law of black-hole dynamics and relativistic thermodynamics. Classical and Quantum Gravity, 15, 3147–3162. DOI: 10.1088/0264-9381/15/10/017. arXiv: gr-qc/9710089.
   • Brown & York 1993 — Quasilocal energy. Quasilocal energy and conserved charges derived from the gravitational action. Physical Review D, 47, 1407–1419. DOI: 10.1103/PhysRevD.47.1407.
   • Eling, Guedens & Jacobson 2006 — Nonequilibrium spacetime thermodynamics. Physical Review Letters, 96, 121301. DOI: 10.1103/PhysRevLett.96.121301. arXiv: gr-qc/0602001.
   • Chirco & Liberati 2010 — Gravitational dissipation. Non-equilibrium thermodynamics of spacetime: the role of gravitational dissipation. Physical Review D, 81, 024016. DOI: 10.1103/PhysRevD.81.024016. arXiv: 0909.4194.
   • Cai & Kim 2005 — First law ⇄ FRW. First law of thermodynamics and Friedmann equations of Friedmann–Robertson–Walker universe. Journal of High Energy Physics, 02, 050. DOI: 10.1088/1126-6708/2005/02/050. arXiv: hep-th/0501055.
   • Akbar & Cai 2007 — Apparent horizon thermodynamics. Thermodynamic behavior of Friedmann equations at the apparent horizon of FRW universe. Physical Review D, 75, 084003. DOI: 10.1103/PhysRevD.75.084003. arXiv: hep-th/0609128.

2. Membrane paradigm, fluid/gravity & horizon transport

   • Thorne, Price & Macdonald 1986 — The membrane paradigm. Black Holes: The Membrane Paradigm. Yale University Press.
   • Bredberg et al. 2012 — Navier–Stokes from Einstein. Journal of High Energy Physics, 07, 146. DOI: 10.1007/JHEP07(2012)146. arXiv: 1101.2451.
   • Bhattacharyya et al. 2008 — Nonlinear fluid dynamics from gravity. Journal of High Energy Physics, 02, 045. DOI: 10.1088/1126-6708/2008/02/045. arXiv: 0712.2456.
   • Blandford & Znajek 1977 — BH energy extraction. Monthly Notices of the Royal Astronomical Society, 179, 433–456. DOI: 10.1093/mnras/179.3.433.
   • Penna 2015 — Impedance matching. Black hole jet power from impedance matching. Physical Review D, 92, 084017. DOI: 10.1103/PhysRevD.92.084017. arXiv: 1504.00360.
   • Boffetta & Ecke 2012 — 2D turbulence. Annual Review of Fluid Mechanics, 44, 427–451. DOI: 10.1146/annurev-fluid-120710-101240.

3. Stochastic gravity, noise kernels & horizon microphysics

   • Hu & Verdaguer 2008 — Stochastic gravity review. Living Reviews in Relativity, 11, 3. DOI: 10.12942/lrr-2008-3.
   • Phillips & Hu 2003 — Noise kernel in curved spacetime. Physical Review D, 67, 104002. DOI: 10.1103/PhysRevD.67.104002. arXiv: gr-qc/0209056.
   • Sinha, Raval & Hu 2002 — BH fluctuations & backreaction. Foundations of Physics, 33, 37–64. DOI: 10.1023/A:1022233302007. arXiv: gr-qc/0210013.
   • Carlip 2002 — BH entropy from CFT. Physical Review Letters, 88, 241301. DOI: 10.1103/PhysRevLett.88.241301. arXiv: gr-qc/0203001.

4. Hyperuniformity, disordered media & galaxy fields

   • Torquato 2018 — Hyperuniform states of matter. Physics Reports, 745, 1–95. DOI: 10.1016/j.physrep.2018.03.001.
   • Philcox & Torquato 2023 — Disordered heterogeneous universe. Physical Review X, 13, 011038. DOI: 10.1103/PhysRevX.13.011038. arXiv: 2207.00519.

5. Cosmological natural selection & genealogical priors

   • Smolin 1997 — Life of the Cosmos. Oxford University Press.
   • Smolin 2007 — Status of cosmological natural selection. arXiv: hep-th/0612185.

6. CMB, anomalies & isotropy

   • Planck 2018 — Cosmological parameters (A6). Astronomy & Astrophysics, 641, A6. DOI: 10.1051/0004-6361/201833910. arXiv: 1807.06209.
   • Planck 2018 — Isotropy & statistics (A7). Astronomy & Astrophysics, 641, A7. DOI: 10.1051/0004-6361/201935201. arXiv: 1906.02552.
   • Jung et al. 2024 — Revisiting large-scale CMB anomalies. Astronomy & Astrophysics, 692, A180. DOI: 10.1051/0004-6361/202451238. arXiv: 2406.11543.
   • Ramazanov et al. 2016 — Quadrupolar statistical anisotropy. Journal of Cosmology and Astroparticle Physics, 03, 039. DOI: 10.1088/1475-7516/2016/03/039. arXiv: 1512.05752.

7. Large-scale structure, BAO, evolving dark energy

   • DESI DR2 — BAO & cosmological constraints. Physical Review D, 112, 083502. DOI: 10.1103/PhysRevD.112.083502. arXiv: 2503.14738.
   • Lodha et al. 2025 — Extended dark energy analysis with DESI DR2. Physical Review D, 112, 083511. DOI: 10.1103/PhysRevD.112.083511. arXiv: 2503.14743.
   • Efstathiou 2025 — Evolving dark energy or supernovae systematics? Monthly Notices of the Royal Astronomical Society, 538, 875–889. DOI: 10.1093/mnras/staf301. arXiv: 2408.07175.
   • Efstathiou 2025 — BAO from a different angle. arXiv: 2505.02658.
   • Keeley et al. 2025 — Transitional dark energy & growth. arXiv: 2502.12667.
   • Kunz 2009 — Dark degeneracy. Physical Review D, 80, 123001. DOI: 10.1103/PhysRevD.80.123001. arXiv: astro-ph/0702615.
   • Joyce, Lombriser & Schmidt 2016 — DE vs modified gravity (review). Annual Review of Nuclear and Particle Science, 66, 95–122. DOI: 10.1146/annurev-nucl-102115-044553. arXiv: 1601.06133.
   • Nazari Pooya 2024 — Growth in interacting dark energy. Physical Review D, 110, 043510. DOI: 10.1103/PhysRevD.110.043510. arXiv: 2407.03766.

8. Redshift drift & future expansion tests

   • Liske et al. 2008 — Cosmic dynamics with ELTs. Monthly Notices of the Royal Astronomical Society, 386, 1192–1218. DOI: 10.1111/j.1365-2966.2008.13090.x. arXiv: 0802.1532.
   • Martins et al. 2024 — Cosmology with ELT–ANDES. Experimental Astronomy, 58, 1–40. DOI: 10.1007/s10686-023-09974-6. arXiv: 2311.16274.
   • Trost et al. 2025 — ESPRESSO redshift-drift experiment. Astronomy & Astrophysics, in press. arXiv: 2505.21615.
   • Bessa, Marra & Castro 2024 — Fluctuations in redshift drift. arXiv: 2409.09977.

9. Voids, AP tests & LSS anisotropy

   • Sutter et al. 2014 — First void AP measurement. Monthly Notices of the Royal Astronomical Society, 443, 2983–2990. DOI: 10.1093/mnras/stu1392. arXiv: 1404.5618.
   • Hamaus et al. 2016 — Voids as clean AP probes. Physical Review Letters, 117, 091302. DOI: 10.1103/PhysRevLett.117.091302. arXiv: 1602.01784.
   • Radinović et al. 2023 — Void–galaxy cross-correlation forecasts. Astronomy & Astrophysics, 678, A131. DOI: 10.1051/0004-6361/202346343. arXiv: 2304.06709.
   • Radinović et al. 2024 — AP effect on void-finding. arXiv: 2407.02699.

10. Black-hole imaging, shadows & strong-field tests

    • EHT M87 first results (shadow). Astrophysical Journal Letters, 875, L1. DOI: 10.3847/2041-8213/ab0ec7.
    • EHT — Persistent shadow of M87*. Astronomy & Astrophysics, 683, A265. DOI: 10.1051/0004-6361/202348887.
    • EHT Sgr A polarization. Astrophysical Journal, 964, L26. DOI: 10.3847/2041-8213/ad1b0f.
    • Lockhart et al. 2022 — How narrow is the M87 ring? Monthly Notices of the Royal Astronomical Society, 517, 2462–2476. DOI: 10.1093/mnras/stac2792.
    • Younsi 2023 — BH images & tests of GR (review). General Relativity and Gravitation, 55, 9. DOI: 10.1007/s10714-022-02984-6. arXiv: 2111.01752.

11. Gravitational waves, \(c_\mathrm{GW}\), and echo constraints

    • Baker et al. 2017 — Speed of gravity from GW170817. Physical Review Letters, 119, 251301. DOI: 10.1103/PhysRevLett.119.251301. arXiv: 1710.06394.
    • LIGO–Virgo–KAGRA 2023 — Tests of GR with GWTC-3. Physical Review X, 13, 041039. DOI: 10.1103/PhysRevX.13.041039. arXiv: 2111.03606.
    • Uchikata et al. 2023 — Echo search in O3. Physical Review D, 108, 104040. DOI: 10.1103/PhysRevD.108.104040. arXiv: 2304.14648.
    • Miani et al. 2023 — Constraints on echo amplitude. Physical Review D, 108, 064018. DOI: 10.1103/PhysRevD.108.064018. arXiv: 2302.12158.

12. Miscellaneous but structurally important

    • Bekenstein 1973 — BH entropy. Physical Review D, 7, 2333–2346. DOI: 10.1103/PhysRevD.7.2333.
    • Bekenstein & Hawking — Area law. Particle creation by black holes. Communications in Mathematical Physics, 43, 199–220. DOI: 10.1007/BF02345020.
    • Weinberg 1989 — Cosmological constant problem. Reviews of Modern Physics, 61, 1–23. DOI: 10.1103/RevModPhys.61.1.
    • DESI instrument overview. The DESI Experiment, a whitepaper for Snowmass 2013. arXiv: 1308.0847.