06. Complete Derivation of Compatibility Conditions

Introduction: Network of Constraints

Previously defined ten components and 11 compatibility conditions (C1-C11), but these conditions are not independent—they form mutually entangled constraint network.

Chapter goals:

1. Derive mathematical details of each compatibility condition one by one
2. Prove self-consistency of constraints (satisfying some implies satisfying all)
3. Calculate dimension of moduli space after constraints

Analogy: Imagine universe as Rubik’s cube:

• Each small cube = one component
• Each rotation = adjusting one parameter
• Solution rules = compatibility conditions

Key insight: Rubik’s cube has only one solution (modulo symmetry)—constraints too strong, degrees of freedom completely fixed.

graph TD
    A["C1: Causality-Geometry"] --> B["C7: IGVP"]
    B --> C["C2: Geometry-Measure"]
    C --> D["C3: Measure Normalization"]
    D --> E["C4: LSZ Reduction"]
    E --> F["C5: Unified Time Scale"]
    F --> G["C6: KMS Condition"]
    G --> B

    H["C8: Entropy-Observer"] --> I["C9: Consensus Condition"]
    I --> J["C10: Realizability"]
    J --> K["C11: CT Thesis"]
    K --> A

    style B fill:#ffe6e6
    style F fill:#e6f3ff
    style I fill:#e6ffe6

Part I: Foundation Triangle (C1-C3)

1.1 Condition C1: Causal-Geometric Alignment

Statement: $$x{⪯}_{\text{evt}}y\iff {\Phi }_{\text{evt}}(x){⪯}_g{\Phi }_{\text{evt}}(y)$$

where:

• Left side: Event causal partial order
• Right side: Causality induced by metric (non-spacelike curves connect)

Theorem 1.1 (Malament-Hawking):

On globally hyperbolic Lorentz manifold $(M,g)$, causality ${⪯}_g$ uniquely determines conformal equivalence class $[g]$.

Proof Outline:

(1) Light Cone Structure Determines Conformal Class:

Define null separation: $$p{\sim }_{\text{null}}q\iff \exists \text{ null geodesic }\gamma :p\to q$$

Lemma: ${\sim }_{\text{null}}$ invariant under diffeomorphisms, and determines $[g]$.

(2) Topology of Causal Future:

Define causal future cone: $$J^{+}(p):=\{q\in M\mid p{⪯}_{g}q\}$$

Key Property: $$J^{+}(p)\cap J^{-}(q)\text{ is compact}\iff (M,g)\text{ globally hyperbolic}$$

(3) Reconstructing Metric:

From topology of $\{J^{+}(p){\}}_{p\in M}$, can reconstruct:

• Manifold topology $M$
• Conformal class $[g]$
• Time orientation

But cannot determine specific conformal factor ${\Omega }^2$ (i.e., $g\sim {\Omega }^{2}g$).

Physical Meaning: Causal structure almost completely determines geometry—only missing one “time scale” degree of freedom.

Non-Triviality of C1:

Counterexample: Consider two metrics: $$g_1=-d{t}^2+d{x}^2,g_2=-e^{2t}d{t}^2+d{x}^2$$

Both on Minkowski spacetime, but:

• Causal structure of $g_1$: Standard light cone
• Causal structure of $g_2$: Conformally equivalent to $g_1$, but light cone “expanded”

Therefore ${\Phi }_{\text{evt}}$ must choose correct conformal factor, determined by Einstein equation (see C7).

1.2 Condition C2: Geometric-Measure Induction

Statement: $$d{\mu }_M=\sqrt{-g}{d}^{4}x$$

where:

• $g=\det (g_{ab})$: Metric determinant
• $\sqrt{-g}$: Scalar density (weight +1)

Derivation:

(1) Coordinate Transformation:

In coordinates $x^{\mu }$, volume element: $$d^{4}x=d{x}^{0}d{x}^{1}d{x}^{2}d{x}^3$$

Coordinate transformation $x\to x^{\prime }$: $$d^4{x}^{\prime }=\left|\frac{\partial x^{\prime }}{\partial x}\right|d^{4}x$$

(2) Scalar Density Transformation: $$\sqrt{-g^{\prime }}={\left|\frac{\partial x^{\prime }}{\partial x}\right|}^{-1}\sqrt{-g}$$

Therefore: $$\sqrt{-g^{\prime }}{d}^4{x}^{\prime }=\sqrt{-g}{d}^{4}x\text{(invariant!)}$$

(3) Physical Interpretation:

In local Lorentz frame: $$g_{ab}={\eta }_{ab}+h_{ab}(h\ll 1)$$

Expansion: $$\sqrt{-g}=1+\frac{1}{2}h+O(h^2),h:={\eta }^{ab}{h}_{ab}$$

Physical Meaning: Spacetime curvature changes “volume element”—gravitational effect.

Necessity of C2:

Theorem 1.2: On Lorentz manifold, unique diffeomorphism-invariant measure is $\sqrt{-g}{d}^{4}x$.

Proof:

• Assume exists another measure $\mu$
• By invariance, $\mu =f(x)\sqrt{-g}{d}^{4}x$, $f$ is scalar
• But $f$ must equal 1 in all coordinate systems (otherwise breaks invariance)
• Therefore $\mu =\sqrt{-g}{d}^{4}x$ ∎

1.3 Condition C3: Measure Normalization

Statement: $${\int }_{\Sigma }\text{tr}({\rho }_{\Sigma })d\sigma =1$$

where:

• $\Sigma$: Spacelike Cauchy hypersurface
• ${\rho }_{\Sigma }$: Density matrix on Cauchy surface
• $d\sigma$: Induced volume element

Derivation:

(1) Induced Metric:

On $\Sigma$, pullback metric: $$h_{ij}=g_{\mu \nu }\frac{\partial x^{\mu }}{\partial y^i}\frac{\partial x^{\nu }}{\partial y^j}$$ where $y^i$ are coordinates on $\Sigma$.

(2) Normal Vector:

Define future-pointing timelike unit normal vector $n^a$: $$g_{ab}{n}^a{n}^b=-1,n^a{\nabla }_{a}t>0$$

(3) Volume Element Relation: $$d\sigma =n_{\mu }d{x}^{\mu }\wedge d^{3}y=\sqrt{h}{d}^{3}y$$

(4) Probability Conservation:

Quantum state normalization: $$⟨\Psi |\Psi ⟩={\int }_{\Sigma }|\Psi {|}^{2}d\sigma =1$$

Generalize to density matrix: $$\text{tr}({\rho }_{\Sigma })=\sum _n⟨n|{\rho }_{\Sigma }|n⟩=1$$

Physical Meaning of C3:

Theorem 1.3 (Probability Conservation):

If ${\Sigma }_1,{\Sigma }_2$ are two Cauchy hypersurfaces, and quantum evolution unitary: $${\rho }_{{\Sigma }_2}=\mathcal{U}{\rho }_{{\Sigma }_1}{\mathcal{U}}^{†}$$

Then: $${\int }_{{\Sigma }_1}\text{tr}({\rho }_{{\Sigma }_1})d{\sigma }_1={\int }_{{\Sigma }_2}\text{tr}({\rho }_{{\Sigma }_2})d{\sigma }_2$$

Proof:

• Unitarity: $\text{tr}({\rho }_{{\Sigma }_2})=\text{tr}(\mathcal{U}{\rho }_{{\Sigma }_1}{\mathcal{U}}^{†})=\text{tr}({\rho }_{{\Sigma }_1})$
• Current conservation: ${\nabla }_{\mu }{j}^{\mu }=0$, $j^{\mu }=n^{\mu }\text{tr}(\rho )$
• By Gauss theorem, integrals on two Cauchy surfaces equal ∎

Part II: Dynamical Closed Loop (C4-C7)

2.1 Condition C4: LSZ Reduction Formula

Statement: $$S_{fi}=\underset{t_{\text{in}}\to -\infty ,t_{\text{out}}\to +\infty }{\lim }⟨f,\text{out}|i,\text{in}⟩$$

where:

• $|i,\text{in}⟩$: Asymptotic free state as $t\to -\infty$
• $|f,\text{out}⟩$: Asymptotic free state as $t\to +\infty$
• $S_{fi}$: Scattering matrix element

LSZ Formula (single particle case): $$⟨p^{\prime }|S|p⟩=i\int d^{4}x{e}^{-i{p}^{\prime }\cdot x}(\square +m^2)\int d^{4}y{e}^{ip\cdot y}(\square +m^2)⟨0|T\{\varphi (x)\varphi (y)\}|0⟩$$

Derivation Outline:

(1) Asymptotic Conditions:

As $t\to \pm \infty$, field operators approach free fields: $$\varphi (x)\stackrel{t\to \pm \infty }{\to }{\varphi }_{\text{in/out}}(x)$$

Satisfy free Klein-Gordon equation: $$(\square +m^2){\varphi }_{\text{in/out}}=0$$

(2) Reduction Formula:

$$a_{\mathbf{p}}^{\text{in}}=i\int d^{3}x{e}^{-i\mathbf{p}\cdot \mathbf{x}}\overleftrightarrow{{\partial }_0}{\varphi }_{\text{in}}(x)$$

Key Step: Express $\varphi$ in terms of ${\varphi }_{\text{in}}$, then extract scattering amplitude.

(3) Multi-Particle Generalization:

$$⟨p_1^{\prime }\cdots p_n^{\prime }|S|p_1\cdots p_m⟩=\prod _{i=1}^n(i)({\square }_i+m^2)\prod _{j=1}^m(i)({\square }_j+m^2)⟨0|T\{\varphi (x_1^{\prime })\cdots \varphi (y_m)\}|0⟩$$

Physical Meaning of C4:

Theorem 2.1 (Scattering Equivalence):

All dynamical information encoded in $S$ matrix: $$\mathcal{U}(+\infty \to -\infty )\leftrightarrow S(\omega )$$

Proof:

• Knowing all $⟨f|S|i⟩$ $\Rightarrow$ know operator $\hat{S}$
• $\hat{S}$ reconstructs $\mathcal{U}(t)$ through Haag-Ruelle theory ∎

2.2 Condition C5: Unified Time Scale

Statement: $$\kappa (\omega )=\frac{{\phi }^{\prime }(\omega )}{\pi }={\rho }_{\text{rel}}(\omega )=\frac{1}{2\pi }\text{tr}Q(\omega )$$

Term-by-Term Derivation:

(1) Scattering Phase Shift and Density of States:

Define cumulative density of states: $$N(\omega ):={\int }_0^{\omega }\rho ({\omega }^{\prime })d{\omega }^{\prime }$$

Krein Spectral Shift Formula: $$N(\omega )-N_0(\omega )=\frac{1}{\pi }\phi (\omega )$$

where $N_0$ is unperturbed case.

Differentiate: $${\rho }_{\text{rel}}(\omega )=\rho (\omega )-{\rho }_0(\omega )=\frac{1}{\pi }\frac{d\phi }{d\omega }$$

(2) Wigner Delay and Phase Shift:

Define: $$Q(\omega )=-i\hslash S^{†}(\omega )\frac{dS(\omega )}{d\omega }$$

For single channel: $S(\omega )=e^{2i\delta (\omega )}$,

$$Q=-i\hslash e^{-2i\delta }\frac{d}{d\omega }(e^{2i\delta })=2\hslash \frac{d\delta }{d\omega }$$

Therefore: $$\frac{1}{2\pi }\text{tr}Q=\frac{\hslash }{\pi }\frac{d\delta }{d\omega }$$

(3) Modular Flow Time Scale:

From KMS condition (see C6), inverse temperature: $$\beta (\omega )=\frac{2\pi \hslash }{\text{tr}Q(\omega )}$$

Define modular flow rate: $$\kappa (\omega ):=\frac{1}{\beta (\omega )}=\frac{\text{tr}Q(\omega )}{2\pi \hslash }$$

Synthesis: All three consistent!

Deep Meaning of C5:

Theorem 2.2 (Uniqueness of Time):

Universe has only one intrinsic time scale (modulo affine transformation): $$[\tau ]=\{T_{\text{cau}},{\tau }_{\text{geo}},t_{\text{scat}},t_{\text{mod}}{\}}_{\text{affine equivalence}}$$

Proof:

• From C5, all time definitions connected through $\kappa (\omega )$
• $\kappa$ is physical invariant (observable)
• Therefore all time definitions differ only by linear transformation ∎

2.3 Condition C6: KMS Condition

Statement:

State $\omega$ in thermal equilibrium at inverse temperature $\beta$ $\iff$ for any operators $A,B$, exists analytic function $F_{AB}(z)$ satisfying: $$F_{AB}(t)=\omega (A{\sigma }_t(B)),F_{AB}(t+i\beta )=\omega ({\sigma }_t(B)A)$$

Equivalent Formulation (Fourier transform): $$\omega (A{B}_{\omega })=\int d\omega e^{-\beta \omega }\rho (\omega )⟨A{⟩}_{\omega }⟨B{⟩}_{\omega }$$

Derivation:

(1) Modular Operator:

From Tomita-Takesaki theory, define: $$\Delta :=S^{\ast }S,S\Omega =A^{\ast }\Omega$$

where $\Omega$ is cyclic separating vector.

(2) Modular Hamiltonian: $$K:=-\log \Delta$$

Modular flow: $${\sigma }_t(A)={\Delta }^{it}A{\Delta }^{-it}=e^{itK}A{e}^{-itK}$$

(3) KMS of Thermal State:

For Gibbs state: $${\rho }_{\beta }=\frac{e^{-\beta \hat{H}}}{Z(\beta )}$$

Direct verification: $$\text{tr}({\rho }_{\beta }A{e}^{it\hat{H}}B{e}^{-it\hat{H}})=\text{tr}({\rho }_{\beta }{e}^{i(t+i\beta )\hat{H}}B{e}^{-i(t+i\beta )\hat{H}}A)$$

Using cyclicity: $$\text{tr}(e^{-\beta \hat{H}}{e}^{it\hat{H}}B{e}^{-it\hat{H}}A)=\text{tr}(e^{-\beta \hat{H}}A{e}^{it\hat{H}}B{e}^{-it\hat{H}})$$

Holds! ∎

Connection Between C6 and C5:

Theorem 2.3 (KMS $\Rightarrow$ Unified Time Scale):

If $\omega$ satisfies KMS condition, then: $$\beta =\frac{2\pi \hslash }{\text{tr}Q(\omega )}$$

Proof Outline:

• From KMS, phase shift satisfies ${\phi }^{\prime }(\omega )=\beta ⟨n(\omega )⟩$
• From optical theorem, $\text{tr}Q=2\pi \hslash {\phi }^{\prime }$
• Combining gives $\beta =2\pi \hslash /\text{tr}Q$ ∎

2.4 Condition C7: IGVP

Statement: $$\delta S_{\text{gen}}=0\iff G_{ab}+\Lambda g_{ab}=8\pi G⟨T_{ab}⟩$$

Detailed Derivation (already given in Chapter 04, here supplement details):

(1) Generalized Entropy Variation: $$\delta S_{\text{gen}}=\delta \left(\frac{A}{4G\hslash }\right)+\delta S_{\text{out}}$$

(2) Geometric Entropy Term: $$\delta \left(\frac{A}{4G\hslash }\right)=\frac{1}{4G\hslash }{\int }_{\partial \Sigma }\frac{1}{2}\sqrt{h}{h}^{ij}\delta h_{ij}$$

Through Gauss-Codazzi equation: $$=\frac{1}{8G\hslash }{\int }_{\Sigma }\sqrt{-g}(R^{ab}-\frac{1}{2}R{g}^{ab})\delta g_{ab}$$

(3) Matter Entropy Term:

From first law: $$\delta S_{\text{out}}=\beta \delta E=\beta {\int }_{\Sigma }\sqrt{-g}⟨T_{ab}⟩\delta g^{ab}$$

(4) Total Variation Zero: $$\frac{1}{8G\hslash }(R^{ab}-\frac{1}{2}R{g}^{ab})+\beta ⟨T^{ab}⟩=0$$

Choose $\beta =\frac{1}{8\pi G\hslash }$ (unit convention), get: $$R^{ab}-\frac{1}{2}R{g}^{ab}=8\pi G⟨T^{ab}⟩$$

That is Einstein equation! ∎

Necessity of C7:

Theorem 2.4 (Uniqueness of IGVP):

Under reasonable physical assumptions, unique variational principle leading to gravity is IGVP.

Proof Outline:

• Require variational principle diffeomorphism invariant
• Require derive second-order differential equation (Einstein equation)
• Possible actions: Einstein-Hilbert + higher-order corrections
• IGVP equivalent to Einstein-Hilbert (in classical limit) ∎

Part III: Information-Observer-Category (C8-C11)

3.1 Condition C8: Additivity of Entropy

Statement: $$S_{\text{gen}}=S_{\text{geom}}+\sum _{\alpha \in \mathcal{A}}S({\rho }_{\alpha })$$

Derivation:

(1) Entropy Decomposition:

Total system $\Sigma$ divided into observer regions $\{C_{\alpha }\}$: $$\Sigma =\bigcup _{\alpha \in \mathcal{A}}{C}_{\alpha }$$

(2) Additivity of Entanglement Entropy:

For disjoint regions $C_{\alpha }\cap C_{\beta }=\varnothing$: $$S({\rho }_{C_{\alpha }\cup C_{\beta }})=S({\rho }_{C_{\alpha }})+S({\rho }_{C_{\beta }})$$

(3) Contribution of Geometric Entropy:

Boundary area: $$A(\partial \Sigma )=\sum _{\alpha }A(\partial C_{\alpha })-\text{repeated counting}$$

Define Wald Entropy: $$S_{\text{geom}}=\frac{A_{\text{net}}}{4G\hslash }$$

Physical Meaning of C8:

Theorem 3.1 (Strong Additivity of Entropy):

If observer network covers entire spacetime ($\bigcup C_{\alpha }=\Sigma$), then: $$S_{\text{gen}}(\Sigma )=\underset{\{{\rho }_{\alpha }\}}{\max }\left[S_{\text{geom}}+\sum _{\alpha }S({\rho }_{\alpha })\right]$$

Proof:

• Maximum corresponds to minimum information constraint
• By quantum Darwinism, classical degrees of freedom copied to all observers
• Therefore entropy additive ∎

3.2 Condition C9: Observer Consensus

Statement: $${\rho }_{\text{global}}={\Phi }_{\text{cons}}(\{{\rho }_{\alpha }\})$$ satisfying: $${\text{tr}}_{{\bar{C}}_{\alpha }}({\rho }_{\text{global}})={\rho }_{\alpha },\forall \alpha$$

Derivation:

(1) Maximum Entropy Reconstruction:

$${\rho }_{\text{global}}=\arg \underset{\rho }{\max }S(\rho )\text{s.t.}{\text{tr}}_{{\bar{C}}_{\alpha }}(\rho )={\rho }_{\alpha }$$

Lagrange Multiplier Method: $$\mathcal{L}=-\text{tr}(\rho \log \rho )-\sum _{\alpha }{\lambda }_{\alpha }[{\text{tr}}_{{\bar{C}}_{\alpha }}(\rho )-{\rho }_{\alpha }]$$

Variation: $$\delta \mathcal{L}=0\Rightarrow {\rho }_{\text{global}}=\exp \left(-\sum _{\alpha }{\lambda }_{\alpha }{\mathcal{P}}_{\alpha }\right)$$

where ${\mathcal{P}}_{\alpha }$ is projection operator onto $C_{\alpha }$.

(2) Consistency Conditions:

To ensure $\{{\rho }_{\alpha }\}$ compatible, must: $${\text{tr}}_{{\bar{C}}_{\alpha \beta }}({\rho }_{\alpha })={\text{tr}}_{{\bar{C}}_{\alpha \beta }}({\rho }_{\beta }),C_{\alpha \beta }:=C_{\alpha }\cap C_{\beta }$$

Physical Meaning: Marginalizations of overlapping regions must agree—“puzzle edges match”.

Relation Between C9 and Terminal Object:

Theorem 3.2 (Consensus $\iff$ Terminal Object):

Observer consensus condition holds $\iff$ $\mathfrak{U}$ is terminal object of category $\mathbf{Univ}$.

Proof:

• $(\Rightarrow )$: If consensus holds, all observer states collapse to unique ${\rho }_{\text{global}}$, corresponding to unique universe $\mathfrak{U}$
• $(\Leftarrow )$: If $\mathfrak{U}$ is terminal object, any “candidate universe” $V$ has unique morphism $\varphi :V\to \mathfrak{U}$, inducing unique consensus map ∎

3.3 Condition C10: Realizability Constraint

Statement: $$\text{Mor}(\mathfrak{U})=\{\varphi \mid \text{Real}(\varphi )=1\}$$

Derivation:

(1) Realizability Definition:

Morphism $\varphi :V\to W$ is realizable $\iff$ exists physical process realizing $\varphi$.

Constraints:

• Energy conservation: $⟨\hat{H}{⟩}_V=⟨\hat{H}{⟩}_W$
• Entropy non-decreasing: $S(W)\ge S(V)$
• Unitarity (quantum): ${\varphi }^{†}\varphi =1$

(2) Examples of Unrealizable:

Faster-Than-Light Signal: $${\varphi }_{\text{FTL}}:C_{\alpha }(t)\to C_{\beta }(t),C_{\alpha }\cap J^{-}(C_{\beta })=\varnothing$$ Violates causality, $\text{Real}({\varphi }_{\text{FTL}})=0$.

Infinite Energy: $${\varphi }_{\infty }:|0⟩\to \sum _{n=0}^{\infty }|n⟩$$ Violates energy boundedness, unrealizable.

Necessity of C10:

Theorem 3.3 (Closure of Realizability):

Set of realizable morphisms closed under composition: $$\text{Real}({\varphi }_1)=1,\text{Real}({\varphi }_2)=1\Rightarrow \text{Real}({\varphi }_2\circ {\varphi }_1)=1$$

Proof:

• If ${\varphi }_1,{\varphi }_2$ both physically realizable
• Then execute ${\varphi }_1$ then ${\varphi }_2$ also realizable
• Therefore ${\varphi }_2\circ {\varphi }_1$ realizable ∎

3.4 Condition C11: Physical Church-Turing Thesis

Statement: $${\mathcal{C}}_{\text{phys}}\subseteq {\mathcal{C}}_{\text{Turing}}$$

Argument:

(1) Physical Processes Encodable:

Any physical system state can be described by finite precision bit string: $$|\psi ⟩\approx \sum _{i=1}^N{c}_i|i⟩,c_i\in \mathbb{C}\cap \mathbb{Q}[i]$$

(2) Evolution Discretizable:

Time evolution: $$|\psi (t)⟩=e^{-iHt}|\psi (0)⟩\approx {\left(e^{-iH\delta t}\right)}^{t/\delta t}|\psi (0)⟩$$

Can approximate with Trotter decomposition: $$e^{-iH\delta t}\approx \prod _k{e}^{-i{H}_k\delta t}\text{(Suzuki formula)}$$

Each step simulable by Turing machine.

(3) Quantum Gravity Correction:

Conjecture: At Planck scale, possibly: $${\mathcal{C}}_{\text{phys}}⊊{\mathcal{C}}_{\text{Turing}}$$

Reasons:

• Spacetime foam causes continuity breaking
• Topological phase transitions uncomputable
• Singularities correspond to halting problem

Openness of C11:

Question: Do “super-Turing” physical processes exist?

Candidates:

• CTC (closed timelike curves): Can solve halting problem?
• Black hole interior: Information unextractable = uncomputable?
• Quantum gravity: Spacetime emergence = new computation model?

Currently no conclusion.

Part IV: Constraint Algebra and Dirac Analysis

4.1 Commutator Relations of Constraints

Define constraint operators: $${\hat{C}}_1=\text{causal-geometric alignment},{\hat{C}}_2=\text{IGVP},\dots$$

Poisson Bracket: $$\{f,g\}=\frac{\partial f}{\partial q^i}\frac{\partial g}{\partial p_i}-\frac{\partial f}{\partial p_i}\frac{\partial g}{\partial q^i}$$

Commutator of Constraints:

Calculate $\{{\hat{C}}_1,{\hat{C}}_7\}$ (causality vs IGVP):

$$\{\text{causality},\text{IGVP}\}=\int \left(\frac{\delta (x⪯y)}{\delta g^{ab}}\right)\left(\frac{\delta S_{\text{gen}}}{\delta g_{ab}}\right)$$

Result: $$\propto \int \left(\frac{\delta {⪯}_g}{\delta g}\right)G_{ab}\sim 0\text{(on-shell)}$$

Physical Meaning: After satisfying Einstein equation, causal constraint automatically satisfied—constraints first-class closed.

4.2 Dirac Bracket Construction

Dirac Procedure:

(1) Classify Constraints:

• First Class: $\{{\hat{C}}_{\alpha },{\hat{C}}_{\beta }\}\ne 0$ (generate gauge symmetry)
• Second Class: $\{{\hat{C}}_{\alpha },{\hat{C}}_{\beta }\}=0$ (fix degrees of freedom)

(2) Define Dirac Bracket: $$\{f,g{\}}_D:=\{f,g\}-\{f,{\hat{C}}_{\alpha }\}{C}^{\alpha \beta }\{{\hat{C}}_{\beta },g\}$$

where $C^{\alpha \beta }$ is inverse of constraint matrix $C_{\alpha \beta }=\{{\hat{C}}_{\alpha },{\hat{C}}_{\beta }\}$.

(3) Physical Phase Space: $${\mathcal{P}}_{\text{phys}}=\{(q,p)\mid {\hat{C}}_{\alpha }(q,p)=0\}/\text{gauge}$$

GLS Case:

Theorem 4.1 (Complete Closure of Constraints):

All 11 constraints form first-class constraint algebra: $$\{{\hat{C}}_i,{\hat{C}}_j\}=f_{ij}^k{\hat{C}}_k$$

Corollary: Constraints generate infinite-dimensional gauge symmetry (diffeomorphisms + quantum gauge transformations).

Part V: Dimension Calculation of Moduli Space

5.1 Initial Degree of Freedom Count

(1) Metric Degrees of Freedom: $$\dim {\mathcal{M}}_{\text{Riemann}}=10\times \infty \text{(10 independent components)}$$

(2) Quantum State Degrees of Freedom: $$\dim \mathcal{H}=\infty \text{(Fock space)}$$

(3) Observer Degrees of Freedom: $$\dim {\mathcal{M}}_{\text{obs}}=|\mathcal{A}|\times \dim \mathcal{H}$$

Formal Sum: $\infty \times \infty \times |\mathcal{A}|$

5.2 Dimension Reduction by Constraints

(1) Causal Constraint (C1): $$-\dim (\text{C1})\sim \infty \text{(light cone alignment, pointwise constraint)}$$

(2) Einstein Equation (C7): $$-\dim (\text{C7})=10\times \infty \text{(10 independent equations)}$$

(3) Consensus Condition (C9): $$-\dim (\text{C9})=(|\mathcal{A}|-1)\times \dim \mathcal{H}$$

(4) Gauge Redundancy: $$-\dim (\text{Diff})=4\times \infty \text{(diffeomorphisms)}$$

5.3 Net Dimension Estimate

Formal Calculation: $$\dim {\mathcal{M}}_{\text{univ}}=(\infty -\infty -10\infty -4\infty )+\text{finite corrections}$$

After Regularization:

Theorem 5.1 (Finiteness of Moduli Space):

For fixed topology $M\cong {\mathbb{R}}^4$ and boundary conditions: $$\dim {\mathcal{M}}_{\text{univ}}=d<\infty$$

where $d$ may be:

• $d=0$: Completely fixed (unique universe)
• $d=1$: One parameter (e.g., $\Lambda$)
• $d\ge 2$: Multi-parameter family (unlikely)

Proof Outline:

• Use Atiyah-Singer index theorem
• Ellipticity of Einstein equation
• Closure of constraint algebra ∎

Physical Meaning:

Corollary 5.1 (No Free Lunch):

Cannot freely specify:

1. Spacetime geometry
2. Quantum state
3. Observer network
4. Computational complexity

At most specify one, others determined by compatibility conditions.

Summary and Outlook

Core Points Review

1. C1-C3: Foundation triangle (causality-geometry-measure)
2. C4-C7: Dynamical closed loop (field theory-scattering-modular flow-entropy)
3. C8-C11: Information-observer-category-computation
4. Constraint Closure: Satisfying some implies satisfying all
5. Moduli Space Finite: $\dim {\mathcal{M}}_{\text{univ}}<\infty$

Core Formula: $${\delta }_{\text{total}}=\sum _{i=1}^{11}{\lambda }_i{\hat{C}}_i=0\iff \mathfrak{U}\text{ unique}$$

Connections with Subsequent Chapters

• 07. Complete Proof of Uniqueness Theorem: Supplement application of Atiyah-Singer index theorem
• 08. Observer-Free Limit: Degeneration when $|\mathcal{A}|\to 0$
• 09. Chapter Summary: Panoramic summary of ten-component theory

Philosophical Implication

Universe is not “assembled”, but self-consistent necessity:

• 11 constraints mutually imply
• Change one must change all
• Unique solution (terminal object)

This is answer to “why universe obeys mathematics”—mathematics is logical self-consistency, universe is unique realization of self-consistency.

Next Article Preview:

• 07. Complete Proof of Uniqueness Theorem: From Index Theorem to Terminal Object
  • Application of Atiyah-Singer index theorem
  • Moduli space of elliptic operators
  • Categorical proof of terminal object