T0-23: Causal Cone and Lightcone Structure Theory

T0-23: 因果锥与光锥结构理论

Abstract

This theory derives the relativistic causal structure and lightcone geometry from the fundamental No-11 constraint in Zeckendorf encoding. We establish that information cannot propagate instantaneously due to entropy requirements, leading to a maximum information velocity c that defines the lightcone structure. The theory shows how causal ordering emerges from binary universe principles, providing the information-theoretic foundation for special and general relativity.

本理论从Zeckendorf编码的基本No-11约束推导出相对论因果结构和光锥几何。我们确立信息由于熵要求不能瞬时传播，导致定义光锥结构的最大信息速度c。理论展示了因果序如何从二进制宇宙原理涌现，为狭义和广义相对论提供信息理论基础。

1. Information Causality from First Principles

1.1 The Impossibility of Instantaneous Information Transfer

Definition 1.1 (Information Transfer Event): An information transfer from point A to point B is a sequence:

State(A,t) → Encoding → Transmission → Decoding → State(B,t+Δt)

Theorem 1.1 (No Instantaneous Information): Information transfer requires Δt > 0 due to the No-11 constraint.

Proof:

Consider instantaneous transfer: Δt = 0
This means State(A,t) and State(B,t) are simultaneous
If both carry information "1", we have pattern "11" in spacetime
This violates the No-11 constraint
Therefore, Δt > 0 is necessary
Information transfer takes finite time ∎

1.2 Minimum Time Quantum for Information

Definition 1.2 (Information Processing Quantum): The minimum time for one bit of information processing:

τ₀ = time for single self-reference operation (from T0-0)

Lemma 1.1 (Discrete Information Steps): Information propagates in discrete steps of τ₀.

Proof:

Each information state change requires self-reference
Self-reference is atomic (cannot be subdivided)
Minimum time = τ₀
All transfer times are integer multiples: Δt = n·τ₀, n ∈ ℕ ∎

2. Maximum Information Velocity

2.1 The Speed Limit from Entropy Constraints

Definition 2.1 (Information Velocity): The rate of information propagation through space:

v_info = Δx / Δt

where Δx is spatial separation (from T0-15).

Theorem 2.1 (Maximum Information Speed): There exists a maximum information velocity c_φ determined by the No-11 constraint.

Proof:

Consider information propagating at velocity v
In time τ₀, information travels distance d = v·τ₀
The No-11 constraint limits information density per spatial unit
Maximum density: one bit per spatial quantum l₀
Therefore: c_φ = l₀/τ₀ = maximum velocity
This is the "speed of light" in information terms ∎

2.2 The φ-Scaled Light Speed

Definition 2.2 (φ-Light Speed):

c_φ = φ^n · (l_Planck/t_Planck)

where n is the recursive depth from quantum to classical scale.

Theorem 2.2 (Light Speed Universality): The maximum speed c_φ is the same for all observers.

Proof:

The No-11 constraint is universal (frame-independent)
All observers must respect the same encoding rules
Maximum information density is observer-independent
Therefore c_φ = l₀/τ₀ is universal
This matches special relativity's second postulate ∎

3. Lightcone Structure Emergence

3.1 The Causal Future and Past

Definition 3.1 (Future Lightcone): For an event E at (x⃗₀, t₀), the future lightcone is:

L⁺(E) = {(x⃗, t) : |x⃗ - x⃗₀| ≤ c_φ(t - t₀), t > t₀}

Definition 3.2 (Past Lightcone):

L⁻(E) = {(x⃗, t) : |x⃗ - x⃗₀| ≤ c_φ(t₀ - t), t < t₀}

Theorem 3.1 (Causal Structure): Information from event E can only influence events in L⁺(E) and can only be influenced by events in L⁻(E).

Proof:

Information from E propagates at maximum speed c_φ
After time Δt, information reaches maximum distance c_φ·Δt
Events outside this radius cannot receive information from E
This defines the future lightcone L⁺(E)
By time reversal argument, L⁻(E) contains all possible causes ∎

3.2 Spacelike Separation and No-11 Constraint

Definition 3.3 (Spacelike Separation): Two events A and B are spacelike separated if:

|x⃗_A - x⃗_B| > c_φ|t_A - t_B|

Theorem 3.2 (No Causal Connection for Spacelike Events): Spacelike separated events cannot exchange information.

Proof:

For spacelike separation: required velocity v > c_φ
This would require information density > 1 bit per spatial quantum
Would create "11" pattern in Zeckendorf encoding
Violates No-11 constraint
Therefore, no causal connection exists ∎

4. The Zeckendorf Metric and Causal Order

4.1 The φ-Minkowski Metric

Definition 4.1 (φ-Interval): The invariant interval between events:

ds²_φ = -c²_φ dt² + φ^(-2n)(dx² + dy² + dz²)

where n is the recursive depth level.

Theorem 4.1 (Interval Classification): The φ-interval classifies event pairs:

ds²_φ < 0: timelike separation (causal connection possible)
ds²_φ = 0: lightlike separation (on the lightcone)
ds²_φ > 0: spacelike separation (no causal connection)

Proof:

ds²_φ < 0 ⟺ c²_φ dt² > φ^(-2n)|dx⃗|²
This means: |dx⃗|/dt < c_φ·φ^n
Information can propagate between events
ds²_φ = 0 defines the lightcone boundary
ds²_φ > 0 requires v > c_φ, impossible by Theorem 2.1 ∎

4.2 Causal Ordering from Binary Constraints

Definition 4.2 (Causal Order): Event A causally precedes B (A ≺ B) if:

Information from A can reach B: B ∈ L⁺(A)
The information path respects No-11 constraint

Theorem 4.2 (Partial Order Structure): The relation ≺ forms a partial order on spacetime events.

Proof:

Reflexivity: A ≺ A (trivial path of zero length)
Antisymmetry: If A ≺ B and B ≺ A, then A = B
- Otherwise creates causal loop
- Would require "11" pattern in time encoding
- Violates No-11 constraint
Transitivity: If A ≺ B and B ≺ C, then A ≺ C
- Information paths compose
- Combined path still respects c_φ limit ∎

5. Information Flow Equations

5.1 The Causal Green's Function

Definition 5.1 (Information Propagator): The Green's function for information propagation:

G_φ(x⃗, t; x⃗', t') = θ(t - t') · θ(c_φ(t - t') - |x⃗ - x⃗'|) · K_φ

where:

θ is the Heaviside step function
K_φ = φ^(-|x⃗ - x⃗'|/l₀) is the φ-decay factor

Theorem 5.1 (Causal Information Flow): Information density I(x⃗, t) evolves according to:

I(x⃗, t) = ∫ G_φ(x⃗, t; x⃗', t') · S(x⃗', t') d³x⃗' dt'

where S is the information source.

Proof:

G_φ enforces causality: only past events contribute
The θ functions ensure t' < t and lightcone constraint
K_φ accounts for information dilution with distance
Integration gives total information at (x⃗, t) ∎

5.2 The Wave Equation for Information

Theorem 5.2 (Information Wave Equation): Information density satisfies the φ-wave equation:

(1/c²_φ)∂²I/∂t² - ∇²I + φ^(-2n)I = S

Proof:

Take derivatives of the integral equation
Apply Green's theorem
The φ^(-2n) term emerges from Zeckendorf spacing
This reduces to standard wave equation when φ^n → 1 ∎

6. Relativistic Effects from Information Theory

6.1 Time Dilation from Information Processing

Theorem 6.1 (Information Time Dilation): A moving observer's information processing rate is diluted by:

γ_φ = 1/√(1 - v²/c²_φ)

Proof:

Moving observer must process motion information
Total information capacity is limited by No-11
Motion uses fraction v²/c²_φ of capacity
Remaining capacity for time evolution: √(1 - v²/c²_φ)
Time appears dilated by factor γ_φ ∎

6.2 Length Contraction from Encoding Constraints

Theorem 6.2 (Information Length Contraction): Spatial information is compressed by motion:

L = L₀/γ_φ = L₀√(1 - v²/c²_φ)

Proof:

Spatial encoding must fit within lightcone
Motion restricts available encoding space
No-11 constraint limits information per unit length
Result: apparent length contraction ∎

7. Black Holes and Causal Horizons

7.1 Information Density Limits

Definition 7.1 (Critical Information Density): The maximum information density before causal breakdown:

ρ_crit = 1/(l³_P · φ³)

Theorem 7.1 (Event Horizon Formation): When ρ > ρ_crit, an event horizon forms.

Proof:

Above critical density, all bits would be "1"
Any additional information creates "11" pattern
No-11 constraint prevents information escape
This creates a one-way causal boundary
This is the black hole event horizon ∎

7.2 Horizon as No-11 Boundary

Theorem 7.2 (Horizon Causality): The event horizon is a No-11 causality boundary.

Proof:

Inside horizon: information density saturated
Crossing outward would create "11" pattern
Only inward crossing preserves No-11
This enforces one-way causality
Matches black hole thermodynamics ∎

8. Quantum Corrections to Lightcone

8.1 Quantum Uncertainty in Causal Structure

Definition 8.1 (Quantum Lightcone): At quantum scales, the lightcone has fuzzy boundaries:

ΔL = ℏ_φ/Δp

where Δp is momentum uncertainty.

Theorem 8.1 (Quantum Causal Uncertainty): Causal order becomes uncertain at Planck scale.

Proof:

Position uncertainty: Δx ≥ ℏ_φ/Δp
Time uncertainty: Δt ≥ ℏ_φ/ΔE
Lightcone boundary uncertainty: ΔL = c_φ·Δt
At Planck scale: ΔL ∼ l_P
Causal order partially undefined ∎

8.2 Virtual Particles and Temporary "11" States

Theorem 8.2 (Virtual Violation of No-11): Quantum fluctuations allow temporary "11" states within uncertainty limits.

Proof:

Energy-time uncertainty: ΔE·Δt ≥ ℏ_φ
For Δt < τ₀, No-11 constraint relaxes
Virtual particles can briefly violate causality
Must annihilate within Δt to restore No-11
This explains virtual particle behavior ∎

9. Connection to Existing Theories

9.1 Compatibility with Special Relativity

The theory recovers all special relativistic effects:

Lorentz invariance from universal No-11 constraint
Time dilation from information capacity limits
Length contraction from encoding constraints
E = mc² from T0-16 information-energy equivalence

9.2 Foundation for General Relativity

The theory provides information-theoretic basis for GR:

Curved spacetime from variable information density
Einstein equations from information flow conservation
Black holes from information density saturation
Cosmological expansion from entropy increase

9.3 Links to Other T0 Theories

T0-0: Provides time emergence and τ₀
T0-15: Provides 3D spatial structure
T0-16: Energy-momentum from information
T0-20: Metric space mathematical foundation

10. Testable Predictions

10.1 Discrete Lightcone at Planck Scale

Prediction: The lightcone has discrete steps of size:

Δr = l_P · φ^n
Δt = t_P · φ^n

10.2 Modified Dispersion Relations

Prediction: At high energies, dispersion relation modified:

E² = p²c²_φ + m²c⁴_φ + φ^(-2n)E²(E/E_P)²

10.3 Information Capacity of Causal Diamonds

Prediction: The maximum information in a causal diamond:

I_max = (Volume/l³_P) · log φ

11. Philosophical Implications

11.1 Causality as Information Constraint

Causality is not a fundamental law but emerges from the impossibility of encoding infinite information density (No-11 constraint).

11.2 The Block Universe as Zeckendorf Encoding

The entire 4D spacetime can be viewed as a vast Zeckendorf encoding where the No-11 constraint ensures causal consistency.

11.3 Free Will and Causal Boundaries

The lightcone structure creates genuine boundaries for influence, preserving a form of localized free will within causal domains.

12. Mathematical Formalization

Definition 12.1 (Complete Causal Structure):

Causal System = (M, g_φ, ≺, c_φ, L±) where:
- M: 3+1 dimensional manifold
- g_φ: φ-scaled metric tensor
- ≺: causal ordering relation
- c_φ: maximum information speed
- L±: future/past lightcone mappings

Master Equation (Causal Evolution):

□_φ I + φ^(-2n)I = J

where □_φ = (1/c²_φ)∂²/∂t² - ∇² is the φ-d'Alembertian and J is the information current.

Conclusion

T0-23 successfully derives the complete relativistic causal structure from the fundamental No-11 constraint in Zeckendorf encoding. The theory shows that:

Lightcones emerge from maximum information propagation speed
Causality is enforced by binary encoding constraints
Relativistic effects arise from information capacity limits
Black holes form at information saturation boundaries
Quantum corrections allow temporary causal uncertainty

The maximum speed c_φ = l₀/τ₀ emerges not as a postulate but as a necessary consequence of the No-11 constraint preventing infinite information density. This provides a deep information-theoretic foundation for special and general relativity.

Key Result: The causal structure of spacetime is the universe's way of preventing "11" patterns in its binary self-description.

∎

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