Chapter 11 Section 1: Construction of Cosmic Consistency Functional

“Consistency is not optional—the universe must be self-consistent, and this requirement alone is sufficient to determine all physical laws.”

Section Overview

In the previous section, we proposed the ultimate goal: deriving all physical laws from a single variational principle. This section will construct the core object—the cosmic consistency functional $\mathcal{I}[\mathfrak{U}]$.

1. What is “Consistency”?

1.1 Everyday Analogy: Consistency of a Jigsaw Puzzle

Imagine a huge jigsaw puzzle:

• Each piece must match edges with adjacent pieces (local consistency)
• The entire picture must be coherent (global consistency)
• If one piece doesn’t match, the entire puzzle cannot be completed

Cosmic consistency is similar, but stronger:

• Not only edges must match (causal consistency)
• Information must be conserved (entropy consistency)
• All observers’ descriptions must be compatible (observer consistency)

1.2 Physical Consistency Requirements

Question: What if the universe is “inconsistent” somewhere?

Answer:

1. Causal inconsistency → Information can propagate faster than light → Violates relativity
2. Entropy inconsistency → Perpetual motion machine possible → Violates second law of thermodynamics
3. Observer inconsistency → Different observers observe contradictions → Physics impossible

Therefore, consistency is not an assumption, but a necessary condition for physical possibility.

2. Three Types of Basic Consistency

2.1 Causal-Scattering Consistency

Physical Requirement: Any local scattering process must be embeddable into global unitary evolution.

graph TD
    A["Local Scattering<br/>S_loc(omega)"] -->|"Must Extend"| B["Global Unitary Evolution<br/>U_global"]
    B -->|"Restrict to Local"| A

    C["Small Region R<br/>Dynamics"] -.->|"Constraint"| D["Outside Causal Cone<br/>No Influence"]

    style A fill:#e1f5ff,stroke:#333,stroke-width:2px
    style B fill:#ffe1e1,stroke:#333,stroke-width:3px
    style C fill:#f4e1ff,stroke:#333,stroke-width:2px
    style D fill:#fff4e1,stroke:#333,stroke-width:2px

Mathematical Formulation:

• Scattering matrix $S(\omega )$ must be unitary: $S^{†}S=I$
• Support of Green’s functions must respect causal cone
• Cannot have causal paradoxes (e.g., grandfather paradox)

Analogy:

Imagine watching a movie in a cinema. The picture (scattering) seen from each seat (local region) must be different angles of the same movie (global evolution). If different seats see contradictory plots, the movie is inconsistent.

2.2 Generalized Entropy Monotonicity and Stability

Physical Requirement: Under unified time scale, generalized entropy of small causal diamonds must satisfy monotonicity and stability.

Definition of Generalized Entropy (recall Chapter 4):

$$S_{\mathrm{gen}}(D)=\frac{A(\partial D)}{4G\hslash }+S_{\mathrm{bulk}}(D)$$

where:

• $A(\partial D)$: Area of causal diamond boundary
• $S_{\mathrm{bulk}}(D)$: von Neumann entropy of bulk region
• $G$: Gravitational constant, $\hslash$: Reduced Planck constant

Monotonicity Requirement: Along unified time scale $\tau$,

$$\frac{d{S}_{\mathrm{gen}}}{d\tau }\ge 0$$

Stability Requirement: Second-order variation non-negative,

$${\delta }^2{S}_{\mathrm{gen}}\ge 0$$

graph LR
    A["Past<br/>tau_1"] -->|"Time Evolution"| B["Future<br/>tau_2"]
    A -.->|"S_gen(tau_1)"| C[" "]
    B -.->|"S_gen(tau_2)"| D[" "]
    C -->|"Delta S_gen >= 0"| D

    style A fill:#e1ffe1,stroke:#333,stroke-width:2px
    style B fill:#ffe1e1,stroke:#333,stroke-width:2px
    style C fill:#fff,stroke:#fff
    style D fill:#fff,stroke:#fff

Analogy:

Imagine an hourglass. Sand flowing from top to bottom (entropy increase) is monotonic. If at some moment sand suddenly flows upward, the hourglass is “inconsistent.” Monotonicity of generalized entropy ensures the universe has a clear arrow of time.

2.3 Observer-Consensus Consistency

Physical Requirement: Models and readings of any finite observer network must be embeddable in the same cosmic state.

graph TD
    A["Observer 1<br/>O_1"] -.->|"Observe"| U["Cosmic State<br/>U"]
    B["Observer 2<br/>O_2"] -.->|"Observe"| U
    C["Observer 3<br/>O_3"] -.->|"Observe"| U

    A <-->|"Communication<br/>C_12"| B
    B <-->|"Communication<br/>C_23"| C
    C <-->|"Communication<br/>C_31"| A

    A -.->|"Model omega_1"| D["Consensus State<br/>omega_*"]
    B -.->|"Model omega_2"| D
    C -.->|"Model omega_3"| D

    style U fill:#f9f,stroke:#333,stroke-width:4px
    style D fill:#9f9,stroke:#333,stroke-width:3px

Mathematical Formulation: Relative entropy between observer $O_i$’s model ${\omega }_i$ and true cosmic state ${\omega }_{\mathrm{bulk}}$:

$$S({\omega }_i\Vert {\omega }_{\mathrm{bulk}}{|}_{C_i})$$

Must reach consensus through communication and updates:

$$\underset{t\to \infty }{\lim }{\omega }_i(t)={\omega }_{\ast }$$

Analogy:

Imagine three blind men touching an elephant. Although each touches different parts (local observations), through communication, they can eventually piece together a consistent picture (consensus). If consensus cannot be reached, then either the observers have problems, or the “elephant” (universe) itself is inconsistent.

3. Construction of Cosmic Consistency Functional

3.1 Basic Idea

We quantify three types of consistency requirements into a functional:

$$\mathcal{I}[\mathfrak{U}]={\mathcal{I}}_{\mathrm{grav}}+{\mathcal{I}}_{\mathrm{gauge}}+{\mathcal{I}}_{\mathrm{QFT}}+{\mathcal{I}}_{\mathrm{hydro}}+{\mathcal{I}}_{\mathrm{obs}}$$

Each term corresponds to a deviation penalty for a type of consistency requirement:

• If cosmic state $\mathfrak{U}$ is completely consistent, then $\mathcal{I}[\mathfrak{U}]$ reaches extremum
• If there is inconsistency, then $\mathcal{I}[\mathfrak{U}]$ deviates from extremum

3.2 Gravity-Entropy Term ${\mathcal{I}}_{\mathrm{grav}}$

Role: Constrain geometry and entropy structure on small causal diamonds.

$${\mathcal{I}}_{\mathrm{grav}}=\frac{1}{16\pi G}{\int }_M(R-2\Lambda )\sqrt{|g|}{d}^{4}x+\frac{1}{8\pi G}{\int }_{\partial M}K\sqrt{|h|}{d}^{3}x-{\lambda }_{\mathrm{ent}}\sum _D[S_{\mathrm{gen}}(D)-S_{\mathrm{gen}}^{\ast }(D)]$$

Components:

1. Einstein-Hilbert action: $\frac{1}{16\pi G}\int (R-2\Lambda )\sqrt{|g|}$ (geometric part)
2. Gibbons-Hawking-York boundary term: $\frac{1}{8\pi G}\int K\sqrt{|h|}$ (boundary consistency)
3. Entropy penalty term: $\sum [S_{\mathrm{gen}}-S_{\mathrm{gen}}^{\ast }]$ (penalty for entropy deviation from extremum)

Physical Meaning:

This term ensures geometry and entropy reach consistency extremum on each small causal diamond. Its variation will give Einstein equations.

3.3 Gauge-Geometric Term ${\mathcal{I}}_{\mathrm{gauge}}$

Role: Constrain boundary channel bundle and gauge structure.

$${\mathcal{I}}_{\mathrm{gauge}}={\int }_{\partial M\times \Lambda }\left[\mathrm{tr}(F_{\mathrm{YM}}\wedge \star F_{\mathrm{YM}})+{\mu }_{\mathrm{top}}\cdot \mathrm{CS}(A_{\mathrm{YM}})+{\mu }_K\cdot \mathrm{Index}(D_{[E]})\right]$$

Components:

1. Yang-Mills action: $\mathrm{tr}(F\wedge \star F)$ (gauge field strength)
2. Chern-Simons term: ${\mu }_{\mathrm{top}}\cdot \mathrm{CS}(A)$ (topological term)
3. Dirac index: ${\mu }_K\cdot \mathrm{Index}(D_{[E]})$ (K-class and index pairing)

Physical Meaning:

This term ensures gauge structure is consistent with K-class and penalizes configurations violating Ward identities. Its variation will give Yang-Mills equations and field content constraints (anomaly cancellation).

3.4 QFT-Scattering Term ${\mathcal{I}}_{\mathrm{QFT}}$

Role: Constrain consistency between bulk QFT and scattering data.

$${\mathcal{I}}_{\mathrm{QFT}}=\sum _{D\in {\mathcal{D}}_{\mathrm{micro}}}S({\omega }_{\mathrm{bulk}}^D\Vert {\omega }_{\mathrm{scat}}^D)$$

where:

• ${\omega }_{\mathrm{bulk}}^D$: Actual bulk state (on causal diamond $D$)
• ${\omega }_{\mathrm{scat}}^D$: Reference state predicted by scattering data and unified scale
• $S(\cdot \Vert \cdot )$: Umegaki relative entropy

Physical Meaning:

This term requires local QFT models to be compatible with scattering-scale predictions. Its variation gives field equations and Ward identities.

3.5 Fluid-Resolution Term ${\mathcal{I}}_{\mathrm{hydro}}$

Role: Constrain fluid dynamics in coarse-graining limit.

$${\mathcal{I}}_{\mathrm{hydro}}={\int }_M\left[\zeta ({\nabla }_{\mu }{u}^{\mu }{)}^2+\eta {\sigma }_{\mu \nu }{\sigma }^{\mu \nu }+\sum _k{D}_k({\nabla }_{\mu }{n}_k{)}^2\right]\sqrt{|g|}{d}^{4}x$$

where:

• $u^{\mu }$: Macroscopic velocity field
• ${\sigma }_{\mu \nu }$: Shear tensor
• $n_k$: Conserved quantity density
• $\zeta ,\eta ,D_k$: Viscosity and diffusion coefficients determined by resolution connection ${\Gamma }_{\mathrm{res}}$

Physical Meaning:

This term requires macroscopic evolution to follow entropy production minimization principle. Its variation gives Navier-Stokes equations and diffusion equations.

3.6 Observer-Consensus Term ${\mathcal{I}}_{\mathrm{obs}}$

Role: Constrain consistency of observer network.

$${\mathcal{I}}_{\mathrm{obs}}=\sum _{i}S({\omega }_i\Vert {\omega }_{\mathrm{bulk}}{|}_{C_i})+\sum _{(i,j)}S({\mathcal{C}}_{ij\ast }({\omega }_i)\Vert {\omega }_j)$$

where:

• First term: Deviation between observer’s internal model and true cosmic state
• Second term: Inconsistency between models after communication
• ${\mathcal{C}}_{ij\ast }$: Push-forward map of communication channel

Physical Meaning:

This term penalizes inconsistency between observer models. Its variation gives multi-agent entropy gradient flow.

4. Unified Consistency Principle

4.1 Formulation of Variational Principle

Core Proposition:

$$\boxed{\delta \mathcal{I}[\mathfrak{U}]=0}$$

Holds for all allowed variations—including variations of metric $g$, channel bundle $E$, total connection ${\Omega }_{\partial }$, bulk state ${\omega }_{\mathrm{bulk}}$, observer models $\{O_i\}$.

4.2 Meaning of Variation

Varying $\mathcal{I}[\mathfrak{U}]$ is equivalent to asking:

“If the universe slightly deviates from current state, will consistency increase or decrease?”

Extremum condition $\delta \mathcal{I}=0$ means:

“Current state is already maximum of consistency, any deviation will destroy consistency.”

4.3 Expansions at Different Levels

Key Insight: Expansions of variational principle $\delta \mathcal{I}[\mathfrak{U}]=0$ on different degrees of freedom give different physical laws!

graph TD
    A["delta I[U] = 0<br/>Single Variational Principle"] --> B["Vary g_mu_nu"]
    A --> C["Vary Omega_partial"]
    A --> D["Vary omega_bulk"]
    A --> E["Vary u_mu"]
    A --> F["Vary omega_i"]

    B --> G["Einstein Equations<br/>G_mu_nu + Lambda g_mu_nu = 8pi G T_mu_nu"]
    C --> H["Yang-Mills Equations<br/>nabla_mu F^mu_nu = J^nu"]
    D --> I["Field Equations<br/>(Box + m^2)phi = 0"]
    E --> J["Navier-Stokes Equations<br/>nabla_mu T^mu_nu = 0"]
    F --> K["Entropy Gradient Flow<br/>partial_tau omega_i = -nabla S"]

    style A fill:#f9f,stroke:#333,stroke-width:4px
    style G fill:#e1f5ff,stroke:#333,stroke-width:2px
    style H fill:#ffe1e1,stroke:#333,stroke-width:2px
    style I fill:#f4e1ff,stroke:#333,stroke-width:2px
    style J fill:#fff4e1,stroke:#333,stroke-width:2px
    style K fill:#e1ffe1,stroke:#333,stroke-width:2px

5. Unified Meaning of Physical Laws

5.1 No More “Independent Laws”

In this framework:

• Einstein equations are not independent assumptions about gravity, but necessary consequences of geometric consistency
• Yang-Mills equations are not independent assumptions about gauge fields, but necessary consequences of boundary data consistency
• Field equations are not independent assumptions about matter, but necessary consequences of scattering-field theory consistency

5.2 Relationships Between Laws

Since all laws come from the same variational principle, they are no longer accidentally compatible, but necessarily compatible:

5.3 Why Did the Universe “Choose” These Laws?

Traditional Answer: We don’t know, this is nature’s choice.

Unified Framework Answer:

The universe didn’t “choose,” this is the only consistent possibility. Any other laws would lead to causal paradoxes, entropy decrease, or observer contradictions, hence logically impossible.

6. Concrete Example: Consistency on Small Causal Diamond

6.1 Problem Setup

Consider a small causal diamond $D_{p,r}$ near point $p$ in spacetime:

graph TD
    A["Future Vertex"] --> B["Waist Surface<br/>Sigma_ell"]
    B --> C["Past Vertex"]

    B -.->|"Area A"| D["S_gen = A/(4Gh) + S_bulk"]

    style B fill:#ffe1e1,stroke:#333,stroke-width:3px
    style D fill:#e1ffe1,stroke:#333,stroke-width:2px

6.2 Consistency Requirements

On this small diamond:

1. Causal consistency: Scattering must be unitary, $S^{†}S=I$
2. Entropy consistency: $S_{\mathrm{gen}}$ must be extremum, $\delta S_{\mathrm{gen}}=0$
3. Observer consistency: Local observer’s model must be compatible with global state

6.3 Derived Equations

Starting from $\delta S_{\mathrm{gen}}=0$ (details in Section 3):

$$\frac{\delta A}{4G\hslash }+\delta S_{\mathrm{bulk}}=0$$

Using:

• $\delta A\approx -\int \lambda R_{kk}d\lambda dA$ (area variation and curvature)
• $\delta S_{\mathrm{bulk}}=\frac{\delta ⟨H_{\mathrm{mod}}⟩}{T}$ (first law of entanglement)
• $\delta ⟨H_{\mathrm{mod}}⟩=\int \lambda T_{kk}d\lambda dA$ (modular Hamiltonian)

We get:

$$R_{kk}=8\pi G{T}_{kk}$$

Holds for all null directions, thus deriving Einstein equations (detailed in Section 3).

7. Key Points Review

graph TD
    A["Three Types of Basic Consistency"] --> B["Causal-Scattering Consistency"]
    A --> C["Generalized Entropy Monotonicity"]
    A --> D["Observer-Consensus Consistency"]

    B --> E["I_grav + I_gauge + I_QFT"]
    C --> E
    D --> F["I_hydro + I_obs"]

    E --> G["Cosmic Consistency Functional<br/>I[U]"]
    F --> G

    G --> H["Variational Principle<br/>delta I = 0"]

    H --> I["All Physical Laws"]

    style A fill:#ffcccc,stroke:#333,stroke-width:3px
    style G fill:#ccffcc,stroke:#333,stroke-width:4px
    style H fill:#ccccff,stroke:#333,stroke-width:4px
    style I fill:#ffffcc,stroke:#333,stroke-width:3px

Core Insight:

The cosmic consistency functional $\mathcal{I}[\mathfrak{U}]$ quantifies three types of basic physical consistency requirements. The extremum principle $\delta \mathcal{I}[\mathfrak{U}]=0$ is not an additional assumption, but the mathematical expression of consistency. All physical laws are expansions of this principle on different degrees of freedom.

Next Section Preview: In Section 2, we will delve into the mathematical foundation of Information-Geometric Variational Principle (IGVP), particularly how to derive geometric equations from variation of generalized entropy $S_{\mathrm{gen}}$. This is the key bridge from abstract consistency principle to concrete physical laws.