Hologram Architecture

[afflom]

The Hologram • A Proof‑Layered Architecture for the UOR Internet

UOR Foundation — steward of The Hologram concept and maintainer of all Hologram tooling (Hologram SDK, Hologram CLI, Hologram Dashboard).

Thesis. The Hologram is a 12,288‑element, page×cycle substrate with dual conservation laws. Every specification, protocol, and runtime behavior over this substrate is a proof obligation. This white paper defines each layer as a formal proof outline, and maps those proofs to repositories, conformance, and tooling within the UOR‑Foundation organization.

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0. Abstract

The Hologram presents a minimal, self‑referential architecture whose base is a 2‑torus of 48‑element pages across 256‑element cycles (48×256 = 12,288). The dual conservation invariants (one per axis) serve as the physics of the stack. We develop a layered set of formal obligations, from axioms to applications, such that truth ≙ conservation holds end‑to‑end. Proofs are implemented in Lean4 and exported as verifiers used by the Hologram SDK, CLI, and Dashboard.

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1. Scope and Terms

• The Hologram — The complete UOR Internet concept: formal substrate, invariants, protocols, and proofs.
• Hologram — Open‑source tooling and managed platform by the UOR Foundation: Hologram SDK, Hologram CLI, Hologram Dashboard.
• VPI — Virtual Private Infrastructure owned by a user; a coherence‑preserving slice of The Hologram executed on the user’s devices.

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2. Notation and Canonical Units

• Alphabet Σ with |Σ| = 256 (byte semantics).
• Page modulus P = 48; Cycle modulus C = 256; cardinality N = P·C = 12,288.
• Indices: p ∈ ℤ₄₈, c ∈ ℤ₂₅₆; linear index i = c·48 + p ∈ [0, N).
• Dual invariants (L0): for each cycle c, $∑_{p=0}^{47} H[p,c] ≡ 0 (mod 256)$; for each page p, $∑_{c=0}^{255} H[p,c] ≡ 0 (mod 256)$.

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3. Layer Overview (each layer formalized as proofs)

L0. Coherence Substrate. The toroidal grid and dual conservation; rotational symmetries.
L1. Identity & Object Model. Canonical addressing, 14‑bit address space, typed references, UOR object identity.
L2. Integrity & Exchange. Mandatory closure checks, Merkle composition, exchange format respecting page×cycle boundaries.
L3. Protocols & Object Graph. Budgeted claims, coherent transport, reference protocols.
L4. Policy & Names. Budget‑valued policies, conservative Boolean collapse, verifiable naming/binding.
L5. VPI Runtime & Applications. Device attestation, scheduling as conservation, services/apps/jobs, deployment proofs.

Each higher layer assumes the theorems of the lower layer and discharges its own proof obligations in Lean4.

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4. Formal Proof Outlines by Layer

L0 — Coherence Substrate (48×256 torus)

Axioms.

1. A1 (Alphabet). Σ is finite; default |Σ|=256 with additive group structure.
2. A2 (Torus). The Hologram is the set H = Σ^{P×C} with wraparound indices.
3. A3 (Dual Closure). There exist functionals I_P, I_C s.t. all pages and cycles close to 0 (mod 256).
4. A4 (Symmetry). Translations in ℤ₄₈ and ℤ₂₅₆ preserve A3.

Theorems.

• T0.1 (Cardinality & factorization). N = 12,288 = 48×256 = 2¹²×3.
• T0.2 (Minimal address bits). 14 bits suffice; SECDED (20,14) is admissible.
• T0.3 (Global dependency). Sum of page‑sums equals sum of cycle‑sums equals the global sum; one check is algebraically determined.
• T0.4 (Rotation invariance). L0 checks commute with torus rotations.

Proof method. Algebraic identities over ℤ modulo bases 48 and 256; telescoping sums; group actions.

Artifacts. Lean4 modules Foundation.lean, Hologram.lean; conformance vectors (canonical Hologram v1).

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L1 — Identity & Object Model

Goals. Bind bytes to canonical objects; prove addressing and typing invariants.

Theorems.

• T1.1 (Indexing isomorphism). (p,c) ↔ i is a bijection [0,N).
• T1.2 (Typed references). Ref(T) dereferences to a unique section respecting L0 closure.
• T1.3 (UORID stability). Canonical object identifiers include an integrity witness derived from L0; serialization is idempotent.

Proof method. Constructive bijections; type preservation under dereference; checksum invariants.

Artifacts. Identifier.lean; SDK identity libraries; CLI/Dashboard verifiers.

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L2 — Integrity & Exchange

Goals. Encode/verify content so that L0 invariants are preserved in storage and on the wire.

Theorems.

• T2.1 (Page/Cycle closure as equations). Mandatory constraints are decidable and local.
• T2.2 (Merkle composability). If each page hash and cycle hash is correct, the root witnesses whole‑blob integrity.
• T2.3 (Exchange soundness). The Coherent Exchange Format (CEF) round‑trips to a canonical H section; padding preserves closure.

Proof method. Induction over pages/cycles; Merkle concatenation theorems; serialization lemmas preserving modular sums.

Artifacts. CEF.lean; exchange codecs in SDKs; conformance suites.

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L3 — Protocols & Object Graph

Goals. Define budgeted claims and transports that compose while preserving conservation.

Theorems.

• T3.1 (Budget algebra). Budgets compose mod 96; neutral/absorbing elements behave as required by protocol calculus.
• T3.2 (Transport conservation). A window closes iff the conservation predicate holds; replay/bit‑flip are detectable from L0 witnesses.
• T3.3 (Graph safety). PUT/LINK/ATTEST mutate the object graph only if pre‑conditions are met and budgets check out; composition is cut‑free.

Proof method. Resonance/budget ring properties; sliding‑window invariants; sequent‑style composition.

Artifacts. Graph.lean, Transport.lean; reference protocols; CLI stream tools.

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L4 — Policy & Names

Goals. Make names and authorization decisions that are proofs, not heuristics.

Theorems.

• T4.1 (Budgeted policy). Policy evaluation is a budgeted proof; denials/admissions are accompanied by witnesses.
• T4.2 (Conservative collapse). When a Boolean decision is required, the budgeted calculus collapses conservatively—no non‑Boolean theorems are introduced.
• T4.3 (Binding correctness). Names bind to UOR objects with verifiable lineage (Merkle+R96).

Proof method. Embedding of budget calculus; soundness and conservativity lemmas; signed binding kernels.

Artifacts. SDK policy modules; Dashboard authorization flows; CLI attestation.

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L5 — VPI Runtime & Applications

Goals. Tie proofs to operations: devices, scheduling, services, apps.

Theorems.

• T5.1 (Device attestation). Enrollment → attested → active transitions preserve invariants and freshness.
• T5.2 (Scheduling as conservation). Placement decisions produce leases whose renewals form conservation windows; failures are budget‑accounted.
• T5.3 (Deployment coherence). Services/apps compiled from packages and artifacts preserve content addresses and L0/L2 witnesses across rollouts.

Proof method. State machine safety; temporal windows over leases; monotonic provenance.

Artifacts. VPI controller/agent; SDK VPI clients; CLI ops; Dashboard health views.

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5. Proof Map & Dependency Graph

L0: T0.1 T0.2 T0.3 T0.4
  │
  ├─▶ L1: T1.1 T1.2 T1.3
  │      │
  │      ├─▶ L2: T2.1 T2.2 T2.3
  │      │      │
  │      │      ├─▶ L3: T3.1 T3.2 T3.3
  │      │      │      │
  │      │      │      ├─▶ L4: T4.1 T4.2 T4.3
  │      │      │      │      │
  │      │      │      │      └─▶ L5: T5.1 T5.2 T5.3
  │      │      │      │
  │      │      │      └─▶ End‑to‑End: **Truth ≙ Conservation**

Each theorem Tℓ.k is discharged in Lean4 and exported as machine‑checkable verifiers.

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6. How Proofs Fit Into the UOR‑Foundation Project

Repositories & Responsibilities

• hologram-proofs (Lean4): defines and proves T0.* … T5.*; exports WASM/native verifiers.
• hologram-specs: normative specs that state the theorems as requirements; versioned alongside proofs.
• hologram-conformance: canonical vectors, suites, and reference verifiers sourced from hologram-proofs.
• SDKs (hologram-sdk-*): bind verifiers, provide APIs that cannot violate proved invariants.
• hologram-cli: operationalizes proofs (attest/verify/deploy); offline verification.
• hologram-dashboard: visualizes proofs (attestation status, conservation windows, Merkle/IDs) and drives management via the user’s VPI.
• Controllers/Agents: implement L5 state machines; embed verifiers to gate transitions.

Release Discipline

• A spec release is valid iff: proofs compile, conformance passes, SDK verifiers updated, CLI/Dashboard integration tests are green.
• Version coupling: Specs/Proofs (tight), SDK core (N/N‑1), CLI/Dashboard (N/N‑1).

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7. Using the Proofs (Developers, Operators, and Tooling)

Developers (Hologram SDK)

• Protocol route. Use L0/L2 verifiers to declare invariants; transport kernels inherit T3.; policy modules expose T4..
• App route. Build apps over VPI with deployment APIs that check T5.* at submit time.

Operators (CLI & Dashboard)

• Attest devices. CLI runs attestation protocols; Dashboard shows status (T5.1).
• Deploy safely. CLI refuses rollouts that would violate closure or provenance (T2.*, T5.3).
• Observe conservation. Windows and leak/rate metrics (T3.2) are surfaced.

Third Parties (Upstream consumers)

• Pin to tagged proofs/specs; import verifiers as WASM or native libs; run hologram-conformance to certify builds.

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8. Canonical Conformance

• Vectors. Publish at least one canonical Hologram (12,288 bytes) whose page/cycle sums close and whose Merkle/ID are provided.
• Suites. Page/Cycle closure; serialization round‑trip; Merkle root check; UORID stability; protocol echo; policy admission/denial with witnesses; device enrollment/scheduling traces.

Gate. No release in the org ships without passing the suites on all target platforms.

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9. Security Model (Proof‑Centric)

• Integrity by construction. L0/L2 prevent undetected drift; L3/L4 make adversarial actions quantifiable (budgeted) and auditable.
• Least privilege with witnesses. Every authorization carries a verifiable policy proof (T4.1).
• Supply chain. Artifacts are content‑addressed; provenance is Merkle‑anchored; CLI/Dashboard verify at the edge.

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10. Governance & Evolution

• SemVer across specs/proofs. Breaking mathematical or API changes trigger a major bump; migration guidance is a proof sketch of equivalence or a quantified delta.
• RFCs. All substantive design changes begin as public RFCs in hologram-specs/rfcs with proof obligations listed up front.

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11. Appendix A — Minimal Self‑Describing Schema (Meta)

Types. Byte | Page | Cycle | IndexP | IndexC | Array(T,n) | Pair(A,B) | Sum(A,B) | Ref(T) | Schema
Checks. PageSumZero(c) | CycleSumZero(p) | MerkleRoot(root)
Spec record. { id, title, version, layer, depends_on[], types[], invariants[], layout, conformance }
Layout bands. header → types → invariants → Merkle → free zone; pads preserve L0.

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12. Appendix B — Worked Micro‑Proofs

• B.1 Addressing (T1.1). Constructive bijection with inverse; show bounds.
• B.2 Global dependency (T0.3). Sum over Σ^{P×C}; interchange order; observe equality and eliminate one check byte.
• B.3 Exchange padding (T2.3). Define padding as computed check bytes; prove local and then global closure.
• B.4 Window closure (T3.2). Define window as a fold over frames; prove iff with conservation predicate by induction on frames.

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13. Appendix C — Repository Index (UOR‑Foundation)

• hologram-proofs — Lean4 theorems T0.–T5.; verifier exports.
• hologram-specs — normative documents; RFCs; dependency graph.
• hologram-conformance — vectors, suites, verifiers.
• hologram-sdk-idl, hologram-sdk-core, hologram-sdk-{ts,rs,go,py,java,csharp,swift,kotlin,cpp} — developer APIs.
• hologram-cli — multi‑arch CLI.
• hologram-dashboard — GitHub Pages UI using the user’s VPI.
• Controllers/Agents — VPI runtime components.

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Closing

The Hologram’s power is that every layer is not just specified but proved. By making conservation the currency of truth and carrying those proofs from bytes to apps, the UOR Foundation ensures that implementations, protocols, and operations remain coherent—even as the ecosystem grows.