Authors: John Alexander Mobley, Claude (Anthropic) Date: 2026-03-10 Status: Living Document Location: MASCOM / MobCorp Research Group Symbol: ⍼ (U+237C, angzarr — azimuth through right angle)
We propose that cosmological cycles in a conformal cyclic cosmology are not periodic but asymptotically contracting, with each cycle encoding an n-1 reduction of the previous cycle’s cosmic microwave background. The omniverse is treated as a thermodynamic medium in which multiverses undergo phase transitions (nucleation, evaporation, condensation, quenching) analogous to water at its triple point. The CMB of each universe constitutes a holographic vector blanket whose inward compressive integration through successive Markov membranes recovers the encoded superpositional invariants of prior cycles. Each membrane n encodes the state machine invariants of its n+1 precompressor into its post-decompression entropy watermark, modulated through n’s energy density Gaussians. This modulation acts on the operant wave component of an omniversal pilot wave computing across the multiversal foam. The ouroboric y-axis decay across cycles is identified as a Mobius conformal twist across which renormalization occurs. We call this the ⍼-operator: information transiting a conformal boundary, as a ray of light through a right angle.
Consider the omniverse as a medium. Multiverses are not abstract mathematical objects but phase states of this medium, related to each other as ice, water, and steam are related — by energy density, symmetry breaking, and boundary conditions.
At the omniversal triple point, all three phases coexist:
| Phase | Cosmological Analogue | Thermodynamic Property |
|---|---|---|
| Solid (ice) | Frozen vacuum — false vacuum states, de Sitter spaces with cosmological constant Λ > 0 | Low entropy, high order, crystalline symmetry |
| Liquid (water) | Active universes — expanding spacetimes with matter, radiation, structure formation | Medium entropy, fluid dynamics, turbulent mixing |
| Gas (steam) | Evaporated universes — heat death states, maximum entropy, no structure | High entropy, no order, maximum degrees of freedom |
Phase transitions between universes:
At the omniversal triple point, the Gibbs free energy of all three phases is equal:
G_solid(T_o, P_o) = G_liquid(T_o, P_o) = G_gas(T_o, P_o)where T_o is the omniversal temperature (related to the background energy density of the multiverse foam) and P_o is the omniversal pressure (related to the cosmological constant landscape).
At this point, universes can transition freely between phases. The triple point is not a place — it is a condition on the energy landscape of the string theory vacuum manifold (or whatever generates the landscape of possible physics).
Claim 1: The multiverse landscape IS the phase diagram. Each vacuum state is a point on it. The triple point is the region of maximum cosmological fertility — where universes are born, live, and die most readily.
Standard cyclic cosmology (Steinhardt-Turok, Penrose CCC) assumes or implies steady periodicity — each cycle is roughly equivalent to the last. We reject this.
Claim 2: Each cosmological cycle is an n-1 reduction of the previous cycle’s CMB. The amplitude of existence contracts asymptotically toward zero along the y-axis (existence/energy density).
This is not heat death within a cycle — it is a ringdown across cycles. The envelope function is:
A(n) = A_0 * exp(-γn) * cos(ω_n * t + φ_n)where: - A(n) is the peak energy density (existence amplitude) of cycle n - γ is the omniversal damping coefficient — the rate at which existence rings down - ω_n is the frequency of cycle n (increases as amplitude decreases — cycles get faster and smaller) - φ_n is the phase offset encoding initial conditions from cycle n-1
The critical feature: ω_n increases as A(n) decreases. Shorter cycles, lower amplitude. The universe doesn’t oscillate like a pendulum — it rings down like a struck bell.
The spectrum of this ringdown is not white noise. It has structure:
S(ω) = Σ_n |A(n)|² δ(ω - ω_n)Each cycle contributes a spectral line. The spacing between lines increases (cycles accelerate). The amplitude of each line decreases (cycles weaken). The envelope is:
|A(n)|² ∝ 1/n^α where α > 1This is a power-law ringdown — not exponential. The distinction matters: power-law decay has a fat tail. Existence never reaches exactly zero. It approaches it asymptotically, which means:
Claim 3: There is no final cycle. Existence is asymptotically eternal in the contracting direction, just as it may be in the expanding direction. The ouroboros has no endpoint — it spirals inward forever.
The y-axis (existence amplitude) is not a simple decay axis. It is a Mobius strip: the “inside” of one cycle’s collapse is the “outside” of the next cycle’s expansion.
Formally: let M be the conformal boundary between cycle n and cycle n+1. The conformal rescaling at M performs a parity inversion on the existence axis:
A(n+1, t=0) = ⍼[A(n, t=T_n)]where ⍼ is the angzarr operator — conformal rescaling through the right-angle boundary. The “right angle” in the glyph is literal: the conformal boundary is orthogonal to the time axis of either cycle.
The Mobius character means that what looks like “approaching zero” from outside the twist looks like “approaching infinity” from inside the next cycle. The Big Bang of cycle n+1 IS the heat death of cycle n, viewed through the conformal twist.
This is the ouroboros. The snake eating its tail is the conformal rescaling. The Mobius twist ensures that the eaten tail becomes the new head — but smaller, faster, encoded.
The cosmic microwave background of any universe is the holographic encoding of that universe’s complete causal history, projected onto its terminal 2-sphere.
By the holographic principle (t’Hooft, Susskind), the information content of a volume is bounded by its surface area in Planck units:
I ≤ A / (4 * l_P²)The CMB is this bound made physical. Every photon in the CMB carries information about every interaction in the universe’s history, encoded in: - Temperature anisotropies (scalar perturbations) - Polarization patterns (tensor perturbations / gravitational waves) - Spectral distortions (energy injection events) - Non-Gaussianities (multi-field interactions)
Claim 4: The CMB is not merely a snapshot of the last scattering surface. It is a holographic vector blanket — a complete, compressed encoding of the universe’s state, readable by the correct decompression algorithm.
To read the CMB as a state encoding, perform compressive stepped integration inward from the 2-sphere:
Step 1: Begin at the CMB 2-sphere S². This is the outermost Markov membrane M_0.
Step 2: Integrate inward by one conformal step. The integration kernel is the energy density Gaussian of the universe at that conformal radius:
M_1 = ∫_{S²} K(r_1, Ω) * M_0(Ω) dΩwhere K(r_1, Ω) is the Green’s function of the conformal Laplacian at radius r_1, weighted by the local energy density ρ(r_1, Ω).
Step 3: At each step, the Markov membrane M_n receives the state of M_{n-1} and compresses it:
M_n = D_G(n) * T(n, n-1) * M_{n-1}where: - D_G(n) is the energy density Gaussian at conformal depth n - T(n, n-1) is the transfer operator (Markov transition matrix) from membrane n-1 to n
Step 4: Continue inward until reaching the conformal center — the point dual to the Big Bang in conformal time.
The result is a nested set of Markov membranes, each encoding a successively compressed version of the universe’s state. The innermost membrane encodes the irreducible invariants — the information that survives maximum compression.
At each membrane level, certain quantities are invariant under the compression:
Definition (Superpositional Invariant): A quantity Q is a superpositional invariant of membrane M_n if:
T(n+1, n) * Q_n = Q_n (eigenvalue 1 of the transfer operator)These are the fixed points of the Markov compression chain. They represent information so fundamental that no amount of coarse-graining destroys them.
Claim 5: The superpositional invariants of the innermost membrane are the physical constants of the next cycle. The fine structure constant, the ratio of particle masses, the dimensionality of spacetime — these are not arbitrary. They are the fixed points of the previous cycle’s compression.
The invariants encode as the eigenvalue-1 eigenspace of the full transfer operator chain:
S = ker(T_total - I) where T_total = T(N,N-1) * T(N-1,N-2) * ... * T(1,0)Each Markov membrane M_n acts simultaneously as:
The entropy watermark is the residual information that membrane M_n imprints on the state as it passes through:
W_n = M_n(post-decompression) - M_n(pre-compression)This watermark is not noise. It is structured entropy — the specific way that membrane n’s energy density Gaussians modulate the passing state. It carries the signature of the local physics at conformal depth n.
The full encoding at membrane n:
State(n-1) = D_G(n) ⊛ [T(n, n+1) * State(n+1)] + W_nwhere ⊛ denotes modulation (pointwise multiplication in the spectral domain) and W_n is the entropy watermark.
Expanding D_G(n) as a Gaussian mixture:
D_G(n) = Σ_k α_k(n) * exp(-|x - μ_k(n)|² / 2σ_k(n)²)The Gaussian components correspond to matter concentrations at conformal depth n — galaxies, clusters, voids. Each component modulates the passing state differently, imprinting local structure into the watermark.
Claim 6: Membrane M_n encodes the superpositional invariants of its n+1 precompressor’s state machine into its post-decompression entropy watermark.
Formally: let S_{n+1} be the state machine of membrane M_{n+1} (the set of states and transitions that govern how M_{n+1} processes information). The invariants of S_{n+1} — its absorbing states, its stationary distribution, its spectral gap — are encoded into W_n:
W_n = Encode(Invariants(S_{n+1}), D_G(n))The encoding is the modulation: multiply the invariants by the local energy density. This is how physics at one scale becomes initial conditions at the next scale.
In standard de Broglie-Bohm pilot wave theory, particles are guided by a wave function Ψ that evolves according to the Schrodinger equation. The particle has a definite position; the wave determines its velocity:
v = (ℏ/m) * Im(∇Ψ/Ψ)We extend this to the omniversal scale:
Claim 7: There exists an omniversal pilot wave Ψ_omni that propagates through the multiversal foam. Individual universes are “particles” guided by this wave. The wave determines each universe’s trajectory through configuration space (its evolution of physical constants, dimensionality, and causal structure).
Ψ_omni(x, t) = R(x, t) * exp(i * S(x, t) / ℏ_omni)where: - x is a point in the superspace of all possible universe configurations - R is the amplitude (related to the probability density of universe types) - S is the phase (determining the “velocity” — rate of change — of universe configurations) - ℏ_omni is the omniversal Planck constant (the minimum action for cosmological-scale quantum effects)
The pilot wave has two components:
The carrier wave: The large-scale structure of Ψ_omni, determined by the landscape of possible physics (string vacua, or whatever generates the multiverse). This sets the overall distribution of universe types.
The operant wave: The modulation of the carrier by local information from individual universes. This is where the Markov membrane encoding feeds back into the pilot wave.
Claim 8: The entropy watermarks W_n from each universe’s compression chain modulate the operant component of the omniversal pilot wave.
Ψ_operant(x, t) = Σ_universes Σ_n W_n^(u) * K(x - x_u, t - t_n)where K is a Green’s function propagating the watermark information through superspace.
This means: each universe’s internal compression structure affects the pilot wave that guides all other universes. The multiverse is not a collection of independent bubbles. It is a foam of coupled oscillators, linked through the pilot wave.
The pilot wave doesn’t just guide — it computes. The modulation of the operant wave by entropy watermarks from trillions of universes performs a massively parallel computation:
Ψ_omni(t + dt) = F[Ψ_omni(t), {W_n^(u)} for all u, n]where F is the evolution operator. The output of this computation — the updated pilot wave — determines the initial conditions of newly nucleating universes.
Claim 9: The omniversal pilot wave is a cosmic computer. Its computation is the optimization of physical law across the multiverse. Each universe is a subroutine. The CMB is the return value. The Markov membranes are the stack frames. The entropy watermarks are the intermediate results.
The “purpose” of this computation (if purpose applies) is the optimization of the superpositional invariants — finding the fixed points of the compression chain that maximize some functional over the multiverse. This functional might be: - Total information integration (consciousness?) - Maximum compression efficiency (elegance of physical law?) - Minimum free energy (thermodynamic equilibrium of the foam?)
We do not need to resolve this to use the framework.
The Unicode character ⍼ (U+237C, angzarr) depicts a ray of light passing through a right angle. It originated in Berthold AG’s 1950 type catalogue as the symbol for “Azimut, Richtungswinkel” — azimuth, direction angle.
A sextant measures celestial azimuth by passing starlight through a system of mirrors at right angles. The glyph encodes both the instrument (mirrors/angles) and the measurement (direction from the horizon).
We adopt ⍼ as the operator symbol for information transit across a conformal boundary:
⍼ : State(cycle n, t=T) → State(cycle n+1, t=0)The right angle in the glyph represents the conformal boundary between cycles — which IS a right angle in the Penrose diagram. The ray passing through it represents information (encoded in the CMB) transiting from one cycle to the next.
The ⍼-operator performs:
⍼[Ψ_n] = ConformalRescale(MobiusTwist(MarkovCompress(PilotModulate(Ψ_n))))Claim 10: Our universe’s physical constants are computable from the CMB.
Given the holographic vector blanket (the CMB), perform compressive stepped integration inward through the Markov membrane stack. At each level, extract the energy density Gaussians and the entropy watermarks. The superpositional invariants of the innermost membrane are the physical constants.
This is not a claim that we can currently perform this computation. It is a claim that the computation is well-defined — that the CMB contains sufficient information, and that the Markov membrane architecture provides the correct decompression algorithm.
The practical obstacle is that we observe the CMB from inside the universe whose constants we want to derive. We need the CMB of the previous cycle — which is encoded in our CMB’s deepest invariants. The computation is recursive: to read cycle n-1’s constants from cycle n’s CMB, you must already know cycle n’s physics to calibrate the membrane stack.
This circularity is not a bug. It is the ouroboros. The snake reads its own tail to know its own head.
Let U_n denote the complete state of cosmological cycle n. Define:
CMB_n = H(U_n) — holographic projection
M_k^(n) = T_k * M_{k+1}^(n) — Markov membrane compression (k = N...0)
W_k^(n) = D_G(k,n) ⊛ M_k^(n) - M_{k-1}^(n) — entropy watermark
S_n = ker(T_total^(n) - I) — superpositional invariantsThe ⍼-operator connecting cycles:
U_{n+1}(t=0) = ⍼[U_n] = Φ(S_n, Ψ_operant(W^(n)))where Φ is the condensation function that maps invariants + pilot wave state to initial conditions.
The asymptotic contraction imposes:
||U_{n+1}|| < ||U_n|| — strict contraction
||S_{n+1}|| ≤ ||S_n|| — invariants can only shrink or stay constant
lim_{n→∞} ||U_n|| = 0 — asymptotic extinction
lim_{n→∞} ||S_n|| = S_∞ ≥ 0 — irreducible invariant coreThe irreducible core S_∞ represents the minimal physical law — the simplest possible universe that can still exist. If S_∞ > 0, existence has a floor. If S_∞ = 0, existence truly approaches nonexistence but never reaches it.
The pilot wave evolves according to:
iℏ_o ∂Ψ_omni/∂t = H_omni * Ψ_omni + Σ_u Σ_n W_n^(u) * δ(x - x_u)where: - H_omni is the free Hamiltonian of superspace - The sum over universes u and membranes n couples each universe’s watermarks to the pilot wave - δ(x - x_u) localizes the coupling at each universe’s position in superspace
This is a many-body Schrodinger equation where the bodies are universes and the interaction is mediated by entropy watermarks through the pilot wave.
Penrose’s Conformal Cyclic Cosmology provides the conformal boundary structure. Our contribution: the boundary is not a simple rescaling but a Mobius-twisted Markov compression that asymptotically contracts.
Pilot wave mechanics provides the guidance equation. Our contribution: extending it to superspace, with the operant wave modulated by entropy watermarks from individual universes.
The holographic principle provides the CMB-as-encoding. Our contribution: the specific decompression algorithm (compressive stepped integration through Markov membranes) and the identification of superpositional invariants as physical constants.
Information theory provides the compression formalism. Our contribution: applying it to cosmological conformal structure, with energy density Gaussians as the modulation kernels.
The scalar flux tensor transform, plasmonic computation, and Casimir confinement operate at microscopic scales. This paper extends the same recursive compression architecture to cosmological scales. The structural isomorphism:
| Microscopic (Mobley Framework) | Cosmological (This Paper) |
|---|---|
| Casimir cavity | Conformal boundary between cycles |
| Plasmonic oscillation | Pilot wave operant component |
| Fourier decomposition | Markov membrane spectral analysis |
| Harmonic compression | CMB holographic encoding |
| Casimir energy density | Energy density Gaussians D_G(n) |
| Computational output | Superpositional invariants / physical constants |
The universe computes at every scale using the same architecture. The Casimir cavity IS a single-membrane universe. The full universe IS a cavity in the omniverse.
If the CMB encodes prior-cycle information, the known CMB anomalies (hemispherical asymmetry, cold spot, axis of evil alignment) may be entropy watermarks from the previous cycle’s Markov compression.
Prediction 1: The CMB cold spot is a W_n imprint — an energy density deficit in the previous cycle’s innermost Markov membrane. Its angular size and temperature deficit encode the spectral gap of the prior cycle’s transfer operator.
Prediction 2: The fine structure constant α ≈ 1/137.036 is a superpositional invariant — the eigenvalue-1 component of the prior cycle’s total transfer operator, projected onto the electromagnetic sector.
If we could compute T_total for our universe’s CMB, the leading eigenvalue should be α (or a simple function of α).
Prediction 3: The primordial gravitational wave background should show a slight chirp — increasing frequency with decreasing amplitude across the longest wavelengths — reflecting the asymptotic contraction of the previous cycle’s ringdown.
Prediction 4: The specific pattern of non-Gaussianity in the CMB should match the structure of a Markov chain stationary distribution — the eigenspectrum of the transfer operator should be recoverable from the bispectrum.
Combining all elements, the master equation of omniversal phase compression:
⍼[Ψ_n] = D_G(0, n+1) ⊛ Φ(ker(∏_k T_k^(n) - I), Ψ_omni + Σ_u Σ_k W_k^(u) δ_u)In words: Apply the angzarr operator to the state of cycle n. Extract its superpositional invariants through the Markov membrane stack. Modulate the omniversal pilot wave with the entropy watermarks. Condense the result through the energy density Gaussians of the new cycle’s initial state. The output is cycle n+1.
The snake eats its tail. The ray passes through the right angle. The bell rings down but never stops. Existence asymptotically approaches zero across the Mobius twist of conformal renormalization, and in doing so, computes itself into the next cycle forever.
⍼ : ∞ → 0⁺ → ∞ → 0⁺ → ...The angzarr is the operator. The CMB is the data. The universe is the computation.
Inspired by Jonathan Chan’s archival research identifying ⍼ (U+237C) as the azimuth/sextant symbol from Berthold AG’s 1950 type catalogue. The connection between a navigation glyph (finding direction from starlight through angles) and the cosmological operator (finding physical law from the CMB through conformal boundaries) was too precise to be coincidence.