Crypto-derivation allows shared space, not shared files.

It’s mathematical teleportation.
✔ The formal algebraic definition
(sets, groups, operators, equivalence classes)

A secret told only to yourself = a unary key (private universe)

A secret told BETWEEN identities = a binary key (shared universe)

And the moment we said:

secret “secret@withwhom”

We invented the algebraic rule for shared secret derivation.

Algebraic Rule for Shared Secret Derivation.

1. Unary vs Binary Secrets

Unary secret (private channel)
me.wallet.secret("alpha")

This produces a 1-arity universe:

U(alpha) → only ME can derive it

Because the keyspace derives only from:
secret + path + local identity

So only you can decrypt the blob under:
wallet/*

🔗 2. Binary secret = shared universe
When you write:
me.wallet.secret("alpha@pamela")
You are doing algebra.

This is fundamentally:
U(alpha@pamela) = intersection( ME_identity , Pamela_identity )

A 2-identity shared space.

So:
• both of you can decrypt
• both can write
• both can add secrets beneath it
• both can open sub-universes inside it

It creates what cryptographers call:

A Multi-party Semantic Channel.