Mathematical Foundations of Opaque State Infrastructure

Mathematical Foundations of Opaque State Infrastructure

Formal Model and Comparative Analysis of the Cryptographic Lockbox Transaction Substrate

Part I: Formal Model (Abstract, Non-Operational)

I.1 Entities and Sets

L — Set of lockboxes

S — Set of stimuli (broadcast inputs)

C — Set of commitments

W — Set of witnesses (bounded proofs)

N — Set of substrate nodes

T — Totally ordered event index (time/order)

I.2 Lockbox Definition

A lockbox ℓ ∈ L is a deterministic state machine:

ℓ := (x, δ, R, K, κ)

Where:

  • x — private internal state
  • δ — deterministic transition function
  • R — set of receptor predicates
  • K — internal key material
  • κ — commitment function mapping state to C

Properties:

  • x is never externally observable
  • δ is total and deterministic
  • R is private and non-discoverable
  • κ(x) binds future transitions

I.3 Stimulus Model

A stimulus s ∈ S is a uniform broadcast object with no addressability.

Each lockbox evaluates:

R(ℓ, s) → {match, no-match}

Only if match, authorization is evaluated.

I.4 Authorization and Transition

If R(ℓ, s) = match, then:

(x′, w) := δ(x, s, K)

Where:

  • x′ — new private state
  • w ∈ W — bounded authorization witness

The lockbox emits:

c′ = κ(x′)

I.5 Substrate Event

A substrate event is a tuple:

e := (T, c′, w)

Substrate nodes verify:

  • witness validity,
  • ordering monotonicity,
  • commitment continuity.

They do not interpret x, s, or δ.

I.6 Bounded Witness Invariant

There exists a global bound B such that:

∀w ∈ W, |w| ≤ B

Verification complexity is bounded and independent of lockbox history length.

I.7 Forgetting (State Collapse)

A lockbox may apply a collapse function:

collapse: (x) → (x̂)

Such that:

  • κ(x̂) remains valid
  • prior internal states are irrecoverable
  • prior witnesses are non-operational

Externally, collapse is indistinguishable from any other transition.

I.8 Distribution and Integrity

Let N be the set of substrate nodes.

Each node stores:

  • commitments C,
  • witnesses W,
  • ordering T.

No node stores:

  • private state x,
  • receptor logic R,
  • keys K,
  • semantic interpretations.

Integrity holds if:

∀e ∈ T, valid(w) ∧ monotonic(c)

I.9 Key Semantics

Keys are evaluated only inside δ.

Externally:

  • all keys are indistinguishable,
  • access, destruction, or no-op outcomes produce valid commitments.

No event reveals intent.

I.10 Formal Summary

The system is a set of deterministic private state machines whose authorized transitions are globally verifiable via bounded witnesses, under a broadcast-only stimulus model, with optional state collapse and no external semantic leakage.

Part II: Comparative Taxonomy (What This Is and Is Not)

This section systematically distinguishes the lockbox substrate from existing cryptographic and distributed systems, highlighting its unique structural properties.

II.1 Versus Public-Key Encryption

Aspect Public-Key Crypto Lockbox Substrate
Primary unit Message State transition
Addressability Explicit sender/receiver None
Metadata Preserved Eliminated
History Logged Optional
Proof Decryption ability Authorization without meaning

PK crypto protects content.
This system protects absence of narrative.

II.2 Versus Secure Messaging (Signal, etc.)

Aspect Secure Messaging Lockbox Substrate
Communication Point-to-point Global broadcast
Identifiers Required None
Logs Exist Non-semantic
Seizure outcome Recoverable graphs Commitments only

II.3 Versus Blockchains / Ledgers

Aspect Blockchain Lockbox Substrate
Meaning Global Local
State Shared Private
History Mandatory Optional
Transparency Core feature Explicitly absent
Value Native None

Blockchains distribute meaning.
This distributes integrity only.

II.4 Versus Zero-Knowledge Systems

Aspect ZK Proofs Lockbox Substrate
Proof scope Statement validity Transition authorization
State Often implicit Explicit but private
History Usually accumulative Collapsible
Semantics Still defined Undefined globally

ZK is a tool this system can use; it is not the system itself.

II.5 Versus MPC / Trusted Execution

Aspect MPC / TEE Lockbox Substrate
Trust model Cryptographic or hardware trust Structural opacity
Participants Defined Undefined
Failure mode Leakage Ambiguity
Semantics Known Unknown

II.6 Versus Databases / Logs

Aspect Database Lockbox Substrate
Queryability Core Impossible
Schema Required None
Audit Semantic Integrity-only
Deletion Exceptional First-class

II.7 Unique Structural Properties

This system uniquely combines:

  • Broadcast without addressability
  • Verification without interpretation
  • Integrity without history
  • Authorization without attribution
  • Destruction without evidence
  • Distribution without shared meaning

No existing category subsumes it.

II.8 Taxonomic Characterization

A distributed integrity substrate for private deterministic state machines, operating under a broadcast-only stimulus model with bounded witnesses and optional state collapse, where global verification never yields semantic reconstruction.

Comments

Post a Comment

Popular posts from this blog

What is Zen Neoplatonism - Attempting to make Sense of John Vervaeke via AI

Image
 I don't understand anything John Vervaeke says, so I asked Claude.ai to make sense of it. Have fun.  Zen Neoplatonism: A Path to Ultimate Reality John Vervaeke, a leading thinker addressing the modern "meaning crisis," introduces Zen Neoplatonism as a framework for integrating the spiritual depth of Zen Buddhism and Neoplatonism . By merging the contemplative traditions of the East and West, Zen Neoplatonism aims to foster a profound relationship with Ultimate Reality and address contemporary existential challenges. The Advent of the Sacred Vervaeke begins by highlighting what he calls the “Advent of the Sacred,” a resurgence in interest around the concept of the sacred and how humans relate to Ultimate Reality. This interest stems from a deep need to find meaning in a time when traditional systems of belief are in decline. He proposes Zen Neoplatonism as a way to engage with this emerging sacredness by creating a dialogue between Zen and Neoplatonism. These two tradit...
Read more
Cryptographic Lockbox Transaction Substrate Core Idea A public, tamper-evident infrastructure that records when authorized state changes occur, without revealing what changed, why it changed, or what value moved. The system is not money, not a ledger, and not a payment rail. It is transactional infrastructure for private state. Structural Properties 1. Opaque State Each participant controls one or more sealed "lockboxes." The contents (data, contracts, balances, logic) are private and unreadable to outsiders. Only key-holders can inspect or modify internal state. 2. Public Integrity, Not Public Meaning The system exposes cryptographic commitments to each lockbox. Observers can verify: that a change occurred, that it was authorized, that history was not rewritten. Observers cannot see: what changed, how much changed, or what the change represents. 3. Mutation Visibility Without Legibility The only globally ...
Read more

Towards a Post-Capitalist Theory of the Firm

  Summary of the Post-Capitalist Theory of the Firm Core Concept: The Post-Capitalist firm reconceptualizes the company as an economic epistemic engine. The founder's role is to initiate a functional economic system that becomes self-perpetuating and self-organizing, similar to the formation of political parties in democratic societies. Foundational Rules: Founder Declaration: Upon company registration, the founder must publicly declare two numbers: Baseline Average Income : The minimum average income level for all employees before the founder can begin extracting rent. Founder's Rent Multiple : The maximum multiple of the average employee income that the founder will be entitled to as rent once the baseline average income is achieved. Compensation Transparency: All employee compensation must be fully transparent. Worker-Determined Pay: The workers colle...
Read more