Planck Quantization

The Temporal Tick and Temporal Stutter

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1. The Planck Tick

1.1 Definition

The Planck Tick is the minimum resolvable unit of time:

Δτ_min = t_P = √(ℏG/c⁵)

1.2 Numerical Value

t_P ≈ 5.391 × 10⁻⁴⁴ seconds

1.3 Planck Units Context

Planck Quantity	Symbol	Value
Planck Length	ℓ_P	1.616 × 10⁻³⁵ m
Planck Time	t_P	5.391 × 10⁻⁴⁴ s
Planck Mass	m_P	2.176 × 10⁻⁸ kg
Planck Energy	E_P	1.956 × 10⁹ J

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2. Physical Meaning

2.1 The Update Cycle

Each Planck tick corresponds to one complete recursive update:

n → n+1  ⟺  Δτ = t_P

The update sequence within one tick:

1. Sample: Read Φ_n(x)
2. Compute: Calculate 𝒞_{i,n} from folding rules
3. Store: Update 𝓜_{ij,n+1}
4. Advance: Write Φ_{n+1}(x)

2.2 Resolution Limit

No physical process can distinguish temporal intervals smaller than t_P:

Δt < t_P  ⟹  Indistinguishable

This is not a measurement limitation—it is a fundamental bound on temporal resolution.

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3. The Delta Threshold Inequality

3.1 Statement

Δt ≥ ℏ / (κ_g · Tr(𝓜))

Where:

• Δt: Minimum distinguishable time interval
• κ_g: Temporal coupling constant
• Tr(𝓜): Trace of Memory Tensor

3.2 Interpretation

This inequality sets a local bound on temporal resolution, depending on memory depth:

• Deep memory (large Tr(𝓜)): Finer temporal resolution possible
• Shallow memory (small Tr(𝓜)): Coarser temporal resolution

3.3 Derivation Sketch

From uncertainty principles:

ΔE · Δt ≥ ℏ

The energy scale is set by memory:

E_𝓜 = κ_g · Tr(𝓜)

Therefore:

Δt ≥ ℏ/E_𝓜 = ℏ/(κ_g · Tr(𝓜))

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4. Temporal Stutter

4.1 Definition

Temporal stutter occurs when the recursion index n approaches maximum depth n_max:

n → n_max  ⟹  Stutter

In this regime:

• Update rate becomes irregular
• Time production fluctuates
• Before/after distinction blurs

4.2 Mathematical Description

Near n_max, the drift scalar fluctuates:

σ_θ(n) = σ₀ + δσ(n)

Where δσ(n) is a stochastic component that grows as n → n_max.

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5. Quantized Time Intervals

5.1 Discrete Levels

Time accumulates in quantized steps:

T_n = n · ΔT_tick

5.2 Non-Uniform Ticks

If drift varies with n:

T_n = Σ_{k=0}^{n-1} ΔT_k

Where:

ΔT_k = [∫_{Ω} σ_θ(x, k) d³x] · t_P

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6. Worked Examples

How Many Planck Ticks in One Second?

N = 1 s / t_P = 1 / (5.391 × 10⁻⁴⁴) ≈ 1.855 × 10⁴³

About 10⁴³ recursive updates per second.

Deep Memory Resolution

If Tr(𝓜) = 10⁶ · E_P:

Δt_min = t_P / 10⁶ ≈ 5.4 × 10⁻⁵⁰ s

Deeper memory allows finer temporal resolution.

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7. Summary

The Planck Tick:

Δτ_min = t_P = √(ℏG/c⁵) ≈ 5.39 × 10⁻⁴⁴ s

The Delta Threshold:

Δt ≥ ℏ / (κ_g · Tr(𝓜))

Key Results:

1. Fundamental bound: Time cannot be resolved below t_P
2. Discrete updates: Each tick is one recursive cycle
3. Stutter: Near n_max, temporal production becomes irregular
4. Memory-dependent: Resolution scales with memory depth
5. Macroscopic limit: Continuous calculus is an excellent approximation

Time is quantized at the Planck scale—not continuous, but discrete in its fundamental nature.

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