Appendix A: Glossary

Core Terms

12,288 Lattice (𝕋) The universal finite state space (ℤ/48)×(ℤ/256) that serves as the carrier for all computation. Has toroidal topology with wraparound at boundaries.

Action (S) Universal cost functional that determines compilation, optimization, and learning. Minimizing action subject to constraints defines lawful computation.

Active Window The subset of the lattice currently being processed or verified. Enables streaming computation with bounded memory.

Address Map (H) Deterministic function from normalized objects to lattice coordinates. Provides perfect hashing on the lawful domain.

Mathematical Components

Budget (β) Semantic cost in ℤ/96. Budget 0 corresponds to fully lawful computation. Non-zero budgets quantify deviation from ideal lawfulness.

Budget Semiring (C₉₆) The algebraic structure (ℤ/96; +, ×) used for budget arithmetic and composition.

C768 Canonical schedule rotation automorphism of order 768. Ensures fairness in distributed computation.

Carrier The underlying set or space on which operations act. For Hologram, this is the 12,288 lattice.

Configuration An assignment of bytes to lattice sites, representing a state of computation. Elements of Σ^𝕋.

Content-Addressable Memory (CAM) Storage system where objects are addressed by their content rather than location. Enables perfect deduplication.

Crush (⟨β⟩) Boolean function mapping budgets to truth values. ⟨β⟩ = true iff β = 0 in ℤ/96.

System Architecture

Gauge Group of symmetry transformations that preserve semantic meaning. Includes translations, rotations, and boundary automorphisms.

Gauge Invariance Property preserved under gauge transformations. Receipts and lawfulness are gauge-invariant.

Lawful Object Configuration whose receipts verify at budget 0 and passes Φ round-trip test.

Lift Operator (lift_Φ) Morphism from boundary to interior that preserves information at budget 0.

Morphism Structure-preserving transformation between configurations. Basic computational operations.

Normal Form (NF) Canonical representative of a gauge equivalence class. Used for unique addressing.

Verification Components

Φ Operator Lift/projection pair ensuring coherence between boundary and interior. Round-trip preserving at budget 0.

Process Object Static lawful representation of a computation. Geometric path on 𝕋 with receipts.

Projection Operator (proj_Φ) Morphism from interior to boundary. Inverse of lift at budget 0.

R96 System of 96 resonance equivalence classes on bytes. Provides compositional semantic labeling.

Receipt Verifiable witness tuple containing R96 digest, C768 stats, Φ round-trip bit, and budget ledger.

Resonance Intrinsic semantic property of bytes determining their equivalence class under R.

Types and Semantics

Budgeted Typing Type system where judgments carry explicit semantic costs. Form: Γ ⊢ x : τ [β].

Denotation Semantic meaning of a program as a geometric object. Written ⟦P⟧.

Observational Equivalence Programs with identical receipts modulo gauge. Semantic equality.

Poly-Ontological Object Entity simultaneously inhabiting multiple mathematical categories with coherence morphisms.

Type Safety Property that ill-typed configurations cannot physically exist in the lattice.

Witness Chain Sequence of receipts proving correct execution. Enables verification and audit.

Algorithmic Concepts

Algorithmic Reification Making abstract computation concrete as verifiable process objects.

Class-Local Transform Morphism operating within a single resonance equivalence class.

Fairness Invariant Statistical property preserved by C768 rotation ensuring balanced resource usage.

Linear-Time Verification O(n) verification complexity for n-sized active window plus witnesses.

Perfect Hash Collision-free hash function on lawful domain via content addressing and normalization.

Schedule Rotation (σ) Fixed automorphism implementing round-robin scheduling without external clock.

Implementation Terms

Action Density Local contribution to global action functional. Used in optimization.

Compilation as Stationarity Program compiles iff it satisfies δS = 0 under constraints.

Conservation Law Invariant preserved by lawful computation. Examples: R96, C768, Φ-coherence, budget.

Gauge Fixing Selection of canonical representative from equivalence class.

Incremental Verification Verifying only changed portions of configuration.

Window-Constrained (WC) Complexity class for operations verifiable in bounded window.

Distributed Systems

Byzantine Fault Tolerance System property of maintaining correctness despite malicious nodes. Achieved through receipts.

Content Routing Network routing based on content addresses rather than locations.

Receipt Consensus Agreement protocol using receipts as votes and proofs.

Shard Partition of lattice for distributed storage or computation.

State Machine Replication Maintaining consistent replicas through receipt chains.

Database Concepts

Index-Free Architecture Database design without auxiliary index structures. Uses CAM for direct access.

Merkle DAG Directed acyclic graph with content-addressed nodes.

MVCC (Multi-Version Concurrency Control) Concurrency through content-addressed snapshots.

Query as Proof Query results include cryptographic proof of correctness.

Schema-Free Storage Storage supporting dynamic types through poly-ontology.

Compiler Terms

Action-Based Code Generation Selecting instructions that minimize action functional.

Gauge Alignment Linking process that aligns gauge across compilation units.

Optimization Pass Transformation that reduces action while preserving semantics.

Universal Optimizer Single optimizer handling all programs through action minimization.

Variational Compilation Compilation as solving variational problem δS = 0.

Machine Learning

Action Flow Learning dynamics as gradient flow of action functional.

Convergence Certificate Receipt-based proof of optimization convergence.

Gradient-Free Optimization Optimization using receipts without computing gradients.

Lyapunov Function Action serving as stability guarantor for learning.

Single Loss Function All ML tasks minimize the same universal action.

Formal Properties

Church-Rosser Property Confluence of reductions modulo gauge equivalence.

Expressivity Class of functions denotable in the 12,288 model.

PAC Learning Bound Sample complexity bound using receipt dimension.

Strong Normalization Termination guarantee under budget discipline.

Type Inhabitation Existence of terms with given type at specified budget.

Network Protocol

Authenticated Message Network message carrying verifiable receipt and witness.

Epidemic Broadcast Probabilistic message propagation with receipt confirmation.

Lattice-Aware Protocol Network protocol exploiting lattice topology.

Receipt-Coordinated Transaction Distributed transaction using receipts for coordination.

Sybil Resistance Protection against fake identity attacks via receipts.

Security Properties

Collision Resistance Cryptographic hardness of finding address collisions.

Information-Theoretic Security Security based on information theory rather than computational hardness.

Memory Safety Absence of pointer errors through content addressing.

Non-Interference Property that secret data doesn’t affect public observations.

Replay Immunity Protection against replay attacks through receipt binding.

Research Frontiers

Categorical Semantics Category-theoretic interpretation of Hologram model.

Convexity Analysis Study of action landscape convexity properties.

Embedding Theory How to embed other computational models in 12,288.

Quantum Extensions Extending model to quantum computation via Φ.

Stability Theory Analysis of fixed points and attractors in configuration space.