Reader’s Guide & Conventions

Mathematical Prerequisites

This book assumes comfort with:

• Discrete Mathematics: Modular arithmetic, equivalence relations, group theory basics
• Automata Theory: Finite state machines, regular languages, decidability
• Type Theory: Typing judgments, inference rules, soundness and completeness
• Denotational Semantics: Mathematical objects as program meanings, compositionality
• Basic Topology: Continuity, compactness (for optimization discussions)

Category theory appears occasionally but is not required—we explain categorical concepts when used.

Notation & Core Objects

The Fundamental Space

The 12,288 Lattice: ℤ/48 × ℤ/256

• Written as T throughout
• Elements: (p,b) where p ∈ [0,47], b ∈ [0,255]
• Linear indexing: i = 256p + b
• Cardinality: |T| = 12,288
• Topology: Toroidal with wraparound

Algebraic Structures

Alphabet: Σ = ℤ₂₅₆ (the byte space)

Resonance Residue: R: Σ → ℤ₉₆

• Partitions bytes into 96 equivalence classes
• Compositional: R(concat(x,y)) determined by R(x) and R(y)

Budget Semiring: C₉₆ = (ℤ₉₆; +, ×)

• Semantic costs compose additively
• Budget 0 represents “fully lawful”

Crush Operator: ⟨β⟩ ∈ {false, true}

• ⟨β⟩ = true ⟺ β = 0 in ℤ₉₆
• Decidable truth from arithmetic

Transformations

Schedule Rotation: σ: T → T

• Fixed automorphism of order 768
• Generates fairness invariants
• Written C768 when discussing the cyclic group

Lift/Projection Pair:

• lift_Φ: boundary → interior
• proj_Φ: interior → boundary
• Round-trip: proj_Φ ∘ lift_Φ = id at budget 0

Gauge Actions:

• Translations on T
• Schedule rotation σ
• Boundary automorphism subgroup G°

Information Structures

Configuration: s ∈ Σᵀ

• Assignment of bytes to lattice sites
• Subject to conservation laws

Receipt: (R₉₆_digest, C₇₆₈_stats, Φ_roundtrip, budget_ledger)

• Verifiable witness of lawfulness
• Compositional under morphism composition

Process Object: Static lawful program denotation

• Geometric path on T
• Characterized by receipts modulo gauge

Reading Conventions

Typography

• Bold for defined terms on first appearance
• Italic for emphasis and meta-level discussion
• Monospace for code and concrete implementations
• SMALL CAPS for system components (e.g., VERIFIER, COMPILER)

Mathematical Style

Definitions are numbered within chapters:

Definition 3.2 (Resonance Class): An equivalence relation on Σ…

Theorems state precise claims:

Theorem 4.7: The address map H is injective on the lawful domain.

Proofs are marked clearly:

Proof: By induction on configuration size…□

Examples and Exercises

Running Examples appear in gray boxes:

Example: 16-site configuration
Sites: (0,0) through (3,3)
Bytes: [0x42, 0x7F, ...]
Residues: [18, 31, ...]
R96 digest: 0xA5F9...

Exercises test understanding:

Exercise 2.3: Prove that receipts are class functions on gauge orbits.

Solutions appear in Appendix D.

Cross-References

• Forward references: “We will see in Chapter 7…”
• Backward references: “Recall from Section 3.2…”
• Margin notes:
  • ⚡ Connection to another chapter
  • 🔬 Open research question
  • ⚠️ Common misconception
  • 💡 Key insight

Pedagogical Approach

Each chapter follows this structure:

1. Motivation: Why does this concept matter?
2. Core Definitions: Precise mathematical foundations
3. CS Analogues: Connections to familiar concepts
4. Theorems & Properties: What can we prove?
5. Running Example: Concrete instantiation
6. Implementation Notes: How to build it
7. Exercises: Test your understanding
8. Takeaways: Key insights to remember

Quick Reference Guides

Symbol Glossary

Concept Map

Information → Intrinsic Structure → Conservation Laws
     ↓              ↓                      ↓
   Bytes     Resonance Classes       Type System
     ↓              ↓                      ↓
  Lattice T    Receipts            Programs as Proofs
     ↓              ↓                      ↓
   CAM/Hash    Verification          Compilation

How Different Readers Should Proceed

For Theoreticians

• Focus on Parts I, II, and IV
• Pay special attention to proofs and exercises
• Explore connections to category theory and type theory

For Systems Builders

• Start with Part III for motivation
• Study Parts I and V carefully
• Focus on implementation notes and Appendix E

For Security Researchers

• Begin with Chapter 9 (Security properties)
• Understand receipt verification (Chapter 3)
• Study collision resistance proofs (Chapter 16)

For Compiler Designers

• Focus on Chapter 8 (Universal cost)
• Study denotational semantics (Chapter 6)
• Examine the mini-compiler (Chapter 12)

Beyond This Book

Active research areas (marked with 🔬) include:

• Expressivity bounds for the 12,288 model
• Quantum extensions preserving conservation laws
• Hardware implementations of receipt verification
• Distributed consensus via receipt agreement

The bibliography provides entry points to the broader literature.

Getting Started

Turn to Chapter 1 to begin with first principles, or jump to Chapter 10 for concrete examples that demonstrate the model in action. Either path will lead you to a new understanding of computation itself.