Information-Theoretic Views of Names, Paths, and Gates

Step 1 – From Names and Gates to Information

Step 2 – Entropy and Redundancy for a 22-Letter Alphabet

Step 3 – Worked Example: Entropy of a Biased Letter Set

Step 4 – Names and Gates as Codes and Channels

Step 5 – Mutual Information: How Parts Constrain Each Other

Step 6 – Quick Mutual Information Thought Exercise

Consider a simplified dataset of 3-letter Names built from letters {A, B, G}.

You observe the following 9 Names, each equally frequent:

• AAG, AAB, ABA
• BAA, BAB, BGA
• GAA, GAB, GBA

1. Look at position 1 vs. position 2

• Position 1 letters: {A, B, G} (roughly balanced).
• Position 2 letters: {A, A, B, A, B, G, A, A, B}.

Thought questions (no exact calculation required):

1. If you know the first letter is G, are some second letters more likely?

• Scan the G-starting Names: GAA, GAB, GBA.
• How does this compare to Names starting with A or B?

1. If you ignore position 1, what is your uncertainty about position 2?

• Roughly, which letter appears most often in position 2?

1. Does knowing position 1 reduce your uncertainty about position 2?

• If yes, then I(position1; position2) > 0.

Your task:

• Describe in 2–3 sentences (mentally or in notes) how the pattern of co-occurrence between positions 1 and 2 indicates non-independence.
• Identify one pairing (e.g., starting with B) where the second letter distribution looks different from the overall distribution.

This is exactly what mutual information captures numerically: the degree to which positions “talk” to each other via constraints.

Step 7 – Constraint Networks over Letters, Sefirot, and Paths

Step 8 – Hands-On: Compute Entropy and Redundancy in Python

Use this minimal Python example to compute entropy and redundancy for a small set of Names.

We will:

• Define a list of 3-letter Names.
• Estimate letter frequencies.
• Compute entropy per letter and redundancy relative to 22 letters.

```python

import math

from collections import Counter

Example: a tiny subset of 72 Names (replace with your own list)

names = [

"YHV", "YHY", "HVY", "VHY", "YVV", "HHY", "YHH", "VYH", "HYV", "VVY"

]

Flatten into a list of letters

letters = [ch for name in names for ch in name]

Count frequencies

counts = Counter(letters)

N = len(letters)

print("Letter counts:", counts)

Compute empirical probabilities

probs = {l: c / N for l, c in counts.items()}

print("Probabilities:", probs)

Shannon entropy (base 2)

def entropy(p_dict):

return -sum(p * math.log2(p) for p in p_dict.values())

H = entropy(probs)

print(f"Entropy per letter: {H:.3f} bits")

Max entropy for 22 letters

Hmax22 = math.log2(22)

R22 = 1 - H / Hmax_22

print(f"Redundancy relative to 22 letters: {R_22:.3f}")

Optional: entropy of first-letter distribution

first_letters = [name[0] for name in names]

firstcounts = Counter(firstletters)

firstprobs = {l: c / len(firstletters) for l, c in first_counts.items()}

Hfirst = entropy(firstprobs)

print(f"Entropy of first position: {H_first:.3f} bits")

```

Try this:

1. Replace `names` with 10–20 Names from your own correspondence scheme or a subset of the 72 Names.
2. Compare entropy of all letters vs. entropy of only first letters.
3. Interpret:

• Where is entropy lower? That position is more constrained.
• Higher redundancy means stronger structural rules in that part of the system.

Step 9 – Check Your Understanding: Entropy and Constraints

Answer this question to test your understanding of entropy and redundancy in Kabbalistic letter systems.

Step 10 – Key Term Review

Flip these cards (mentally or with a partner) to reinforce the core concepts from this module.