DIGITAL SPACE ALGEBRA

The Anchor

This section takes its name from Algebra de Espacios Digitales, which proposes a mathematical framework for decentralized identity and data sovereignty: digital interactions modeled as independent, self-managed states rather than rows in a hierarchical database.

Its core object is the Island — an autonomous digital state:

I = (path, ciphertext, T, A, C)

path — the unique identifier within the network. ciphertext — the cryptographically protected content. T — topology, which nodes store or replicate the data. A — audience, which identities can decrypt it. C — capability, what actions are permitted on that state.

This is the same A/T/C reading used throughout this section: private → A = {self}, shared → |A| > 1, encrypted → A enforced cryptographically, replicated → |T| > 1. Adjectives stop being categories and become properties of sets.

Treating users and permissions as intersecting mathematical sets produces identity and information management where the user keeps absolute, private control over their own interactions through cryptography — not through a platform's permission, but through the structure of the math itself.

What This Section Is

.me is not one idea written once — it's a body of work, refined across many essays over more than a year, most of it published on Medium and not indexed anywhere as a single path. This page is that path: the mathematics underneath everything else on this site, arranged to be read in order, not chronologically — from the concept, through the structure, to the parts that make it fast. For the argued case for why any of this matters, see manifesto.html; for the same separation argued from Byzantine fault tolerance, see The Byzantine Prompt.

Index — In Progress

Curated, not exhaustive. Six entries placed so far, beyond the anchor above; more queued pending source links.

1. Structural Geometry The Mathematical Structure of a Space in .me — the formal definition: positions, containment order, and observer/mode/authority-indexed resolution, source-verified against the kernel.
2. Cryptographic Layer The Algebra of Encrypted Audiences — replication as topology, readability as cryptography: the _ and ~ operators, source-verified against the kernel.
3. Cryptographic Layer Cryptographic Set-Chemistry on Audiences — union and intersection audience sets over kernel identities: key wrapping, blob format, axiom compatibility.
4. Synthesis Encrypted Semantic Island — closing synthesis: an Island whose audience is a composable set expression, so privacy is a property of derivation, not custody.
5. Reactive Engine Inverted Dependency Indexing — how a mutation updates only its dependency frontier: refSubscribers, eager/lazy recompute, O(k) invalidation. An elaboration of the locality claim already made in SpaceStructure.
6. Reactive Engine What is O(k)? — real recorded benchmark numbers for the claim above, including two figures a public doc summary got wrong by 43x and 3.4x, corrected here against the kernel's own reports.

Still to be placed: the case for/against calling it a category, and the complexity notes.