Appendix E: Implementation Code

Minimal Core Implementation

This appendix provides a complete, minimal implementation of the Hologram core in Rust, suitable for educational purposes and experimentation.

Core Data Structures

#![allow(unused)]
fn main() {
// lattice.rs - The 12,288 lattice structure
use std::ops::{Index, IndexMut};

pub const PAGES: usize = 48;
pub const BYTES_PER_PAGE: usize = 256;
pub const LATTICE_SIZE: usize = PAGES * BYTES_PER_PAGE; // 12,288

#[derive(Clone, Debug)]
pub struct Lattice {
    data: Vec<u8>,
}

impl Lattice {
    pub fn new() -> Self {
        Lattice {
            data: vec![0; LATTICE_SIZE],
        }
    }

    pub fn from_vec(data: Vec<u8>) -> Self {
        assert_eq!(data.len(), LATTICE_SIZE);
        Lattice { data }
    }

    pub fn get(&self, page: u8, byte: u8) -> u8 {
        let index = (page as usize) * 256 + (byte as usize);
        self.data[index]
    }

    pub fn set(&mut self, page: u8, byte: u8, value: u8) {
        let index = (page as usize) * 256 + (byte as usize);
        self.data[index] = value;
    }

    pub fn linear_index(page: u8, byte: u8) -> usize {
        (page as usize) * 256 + (byte as usize)
    }

    pub fn from_linear_index(index: usize) -> (u8, u8) {
        let page = (index / 256) as u8;
        let byte = (index % 256) as u8;
        (page, byte)
    }

    pub fn iter(&self) -> impl Iterator<Item = &u8> {
        self.data.iter()
    }
}

impl Index<(u8, u8)> for Lattice {
    type Output = u8;

    fn index(&self, (page, byte): (u8, u8)) -> &u8 {
        &self.data[Self::linear_index(page, byte)]
    }
}

impl IndexMut<(u8, u8)> for Lattice {
    fn index_mut(&mut self, (page, byte): (u8, u8)) -> &mut u8 {
        let index = Self::linear_index(page, byte);
        &mut self.data[index]
    }
}
}

Receipt System

#![allow(unused)]
fn main() {
// receipt.rs - Receipt structure and verification
use sha3::{Digest, Sha3_256};

#[derive(Clone, Debug, PartialEq, Eq)]
pub struct Receipt {
    pub r96_digest: R96Digest,
    pub c768_stats: C768Stats,
    pub phi_roundtrip: bool,
    pub budget_ledger: u8,  // In Z/96
}

#[derive(Clone, Debug, PartialEq, Eq)]
pub struct R96Digest {
    histogram: [u32; 96],
    hash: [u8; 32],
}

impl R96Digest {
    pub fn compute(data: &[u8]) -> Self {
        let mut histogram = [0u32; 96];

        // Count resonance residues
        for byte in data {
            let residue = resonance_residue(*byte);
            histogram[residue as usize] += 1;
        }

        // Hash the histogram
        let mut hasher = Sha3_256::new();
        for count in &histogram {
            hasher.update(&count.to_le_bytes());
        }
        let hash_result = hasher.finalize();

        let mut hash = [0u8; 32];
        hash.copy_from_slice(&hash_result);

        R96Digest { histogram, hash }
    }

    pub fn verify(&self, data: &[u8]) -> bool {
        let computed = Self::compute(data);
        self.hash == computed.hash
    }
}

// Resonance function: maps bytes to 96 classes
pub fn resonance_residue(byte: u8) -> u8 {
    // Simple modular mapping for demonstration
    // Real implementation would use specific resonance structure
    byte % 96
}

#[derive(Clone, Debug, PartialEq, Eq)]
pub struct C768Stats {
    pub mean_flow: f64,
    pub variance: f64,
    pub phase: u16,  // Current position in 768-cycle
}

impl C768Stats {
    pub fn compute(lattice: &Lattice, phase: u16) -> Self {
        // Simplified fairness statistics
        let flows: Vec<f64> = lattice
            .iter()
            .map(|&byte| byte as f64)
            .collect();

        let mean = flows.iter().sum::<f64>() / flows.len() as f64;
        let variance = flows
            .iter()
            .map(|x| (x - mean).powi(2))
            .sum::<f64>() / flows.len() as f64;

        C768Stats {
            mean_flow: mean,
            variance,
            phase: phase % 768,
        }
    }

    pub fn verify_fairness(&self, threshold: f64) -> bool {
        // Check if variance is within acceptable bounds
        self.variance < threshold
    }
}

impl Receipt {
    pub fn compute(lattice: &Lattice, phase: u16) -> Self {
        Receipt {
            r96_digest: R96Digest::compute(&lattice.data),
            c768_stats: C768Stats::compute(lattice, phase),
            phi_roundtrip: true,  // Simplified
            budget_ledger: 0,     // Lawful state
        }
    }

    pub fn verify(&self) -> bool {
        // Check budget is zero (lawful)
        self.budget_ledger == 0 && self.phi_roundtrip
    }

    pub fn combine(r1: &Receipt, r2: &Receipt) -> Receipt {
        // Combine receipts for composed operations
        Receipt {
            r96_digest: R96Digest::compute(&[]),  // Would merge histograms
            c768_stats: C768Stats {
                mean_flow: (r1.c768_stats.mean_flow + r2.c768_stats.mean_flow) / 2.0,
                variance: (r1.c768_stats.variance + r2.c768_stats.variance) / 2.0,
                phase: (r1.c768_stats.phase + r2.c768_stats.phase) % 768,
            },
            phi_roundtrip: r1.phi_roundtrip && r2.phi_roundtrip,
            budget_ledger: (r1.budget_ledger + r2.budget_ledger) % 96,
        }
    }
}
}

Content-Addressable Memory

#![allow(unused)]
fn main() {
// cam.rs - Perfect hash implementation
use std::collections::HashMap;

pub struct ContentAddressableMemory {
    store: HashMap<Address, Vec<u8>>,
    normalizer: GaugeNormalizer,
}

#[derive(Clone, Debug, Hash, PartialEq, Eq)]
pub struct Address {
    page: u8,
    byte: u8,
}

impl Address {
    pub fn from_content(content: &[u8]) -> Self {
        // Simplified perfect hash
        let mut hasher = Sha3_256::new();
        hasher.update(content);
        let hash = hasher.finalize();

        Address {
            page: (hash[0] % 48),
            byte: hash[1],
        }
    }

    pub fn to_linear(&self) -> usize {
        (self.page as usize) * 256 + (self.byte as usize)
    }
}

pub struct GaugeNormalizer;

impl GaugeNormalizer {
    pub fn normalize(&self, data: &[u8]) -> Vec<u8> {
        // Simplified normalization - sort bytes
        let mut normalized = data.to_vec();
        normalized.sort_unstable();
        normalized
    }
}

impl ContentAddressableMemory {
    pub fn new() -> Self {
        ContentAddressableMemory {
            store: HashMap::new(),
            normalizer: GaugeNormalizer,
        }
    }

    pub fn store(&mut self, data: Vec<u8>) -> Address {
        let normalized = self.normalizer.normalize(&data);
        let address = Address::from_content(&normalized);
        self.store.insert(address.clone(), normalized);
        address
    }

    pub fn retrieve(&self, address: &Address) -> Option<&Vec<u8>> {
        self.store.get(address)
    }

    pub fn exists(&self, address: &Address) -> bool {
        self.store.contains_key(address)
    }
}
}

Process Objects and Morphisms

#![allow(unused)]
fn main() {
// process.rs - Process objects and morphisms
pub trait Morphism {
    fn apply(&self, lattice: &Lattice) -> Lattice;
    fn budget_cost(&self) -> u8;
    fn receipt(&self, input: &Lattice, output: &Lattice) -> Receipt;
}

pub struct IdentityMorphism;

impl Morphism for IdentityMorphism {
    fn apply(&self, lattice: &Lattice) -> Lattice {
        lattice.clone()
    }

    fn budget_cost(&self) -> u8 {
        0
    }

    fn receipt(&self, input: &Lattice, _output: &Lattice) -> Receipt {
        Receipt::compute(input, 0)
    }
}

pub struct ClassLocalTransform {
    class_id: u8,
    transform: Box<dyn Fn(u8) -> u8>,
}

impl ClassLocalTransform {
    pub fn new(class_id: u8, transform: Box<dyn Fn(u8) -> u8>) -> Self {
        ClassLocalTransform { class_id, transform }
    }
}

impl Morphism for ClassLocalTransform {
    fn apply(&self, lattice: &Lattice) -> Lattice {
        let mut output = lattice.clone();

        for i in 0..LATTICE_SIZE {
            let value = lattice.data[i];
            if resonance_residue(value) == self.class_id {
                output.data[i] = (self.transform)(value);
            }
        }

        output
    }

    fn budget_cost(&self) -> u8 {
        1  // Minimal cost for class-local operation
    }

    fn receipt(&self, input: &Lattice, output: &Lattice) -> Receipt {
        Receipt::combine(&Receipt::compute(input, 0), &Receipt::compute(output, 0))
    }
}

pub struct ScheduleRotation {
    phase: u16,
}

impl ScheduleRotation {
    pub fn new(phase: u16) -> Self {
        ScheduleRotation { phase: phase % 768 }
    }

    fn rotate_index(&self, index: usize) -> usize {
        // Simplified rotation - circular shift
        (index + self.phase as usize) % LATTICE_SIZE
    }
}

impl Morphism for ScheduleRotation {
    fn apply(&self, lattice: &Lattice) -> Lattice {
        let mut output = Lattice::new();

        for i in 0..LATTICE_SIZE {
            let new_index = self.rotate_index(i);
            output.data[new_index] = lattice.data[i];
        }

        output
    }

    fn budget_cost(&self) -> u8 {
        0  // Rotation preserves lawfulness
    }

    fn receipt(&self, input: &Lattice, _output: &Lattice) -> Receipt {
        Receipt::compute(input, self.phase)
    }
}

pub struct Process {
    morphisms: Vec<Box<dyn Morphism>>,
    total_budget: u8,
}

impl Process {
    pub fn new() -> Self {
        Process {
            morphisms: Vec::new(),
            total_budget: 0,
        }
    }

    pub fn add_morphism(&mut self, morphism: Box<dyn Morphism>) {
        self.total_budget = (self.total_budget + morphism.budget_cost()) % 96;
        self.morphisms.push(morphism);
    }

    pub fn execute(&self, input: &Lattice) -> (Lattice, Receipt) {
        let mut current = input.clone();
        let mut receipts = Vec::new();

        for morphism in &self.morphisms {
            let output = morphism.apply(&current);
            let receipt = morphism.receipt(&current, &output);
            receipts.push(receipt);
            current = output;
        }

        let final_receipt = receipts
            .into_iter()
            .reduce(|r1, r2| Receipt::combine(&r1, &r2))
            .unwrap_or_else(|| Receipt::compute(&current, 0));

        (current, final_receipt)
    }
}
}

Type System

#![allow(unused)]
fn main() {
// types.rs - Budgeted type system
pub struct Type {
    base: BaseType,
    budget: u8,
}

pub enum BaseType {
    Byte,
    Page,
    Configuration,
    Receipt,
    Process,
}

pub struct TypeChecker {
    context: TypeContext,
}

pub struct TypeContext {
    bindings: HashMap<String, Type>,
}

impl TypeChecker {
    pub fn new() -> Self {
        TypeChecker {
            context: TypeContext {
                bindings: HashMap::new(),
            },
        }
    }

    pub fn check(&self, term: &Term) -> Result<Type, TypeError> {
        match term {
            Term::Literal(value) => Ok(Type {
                base: BaseType::Byte,
                budget: 0,
            }),
            Term::Variable(name) => self.context.bindings
                .get(name)
                .cloned()
                .ok_or(TypeError::UnboundVariable(name.clone())),
            Term::Application(func, arg) => {
                let func_type = self.check(func)?;
                let arg_type = self.check(arg)?;

                // Budgets add under application
                Ok(Type {
                    base: func_type.base,
                    budget: (func_type.budget + arg_type.budget) % 96,
                })
            }
        }
    }

    pub fn crush(&self, budget: u8) -> bool {
        budget == 0
    }
}

pub enum Term {
    Literal(u8),
    Variable(String),
    Application(Box<Term>, Box<Term>),
}

pub enum TypeError {
    UnboundVariable(String),
    TypeMismatch,
    BudgetViolation,
}
}

Action Functional

#![allow(unused)]
fn main() {
// action.rs - Universal action functional
pub struct ActionFunctional {
    sectors: Vec<Box<dyn Sector>>,
    weights: Vec<f64>,
}

pub trait Sector {
    fn evaluate(&self, lattice: &Lattice) -> f64;
    fn gradient(&self, lattice: &Lattice) -> Vec<f64>;
}

pub struct GeometricSmoothness;

impl Sector for GeometricSmoothness {
    fn evaluate(&self, lattice: &Lattice) -> f64 {
        let mut smoothness = 0.0;

        for page in 0..PAGES {
            for byte in 0..BYTES_PER_PAGE {
                let center = lattice.get(page as u8, byte as u8) as f64;

                // Check neighbors (with wraparound)
                let left = lattice.get(page as u8, ((byte + 255) % 256) as u8) as f64;
                let right = lattice.get(page as u8, ((byte + 1) % 256) as u8) as f64;

                smoothness += (center - left).powi(2) + (center - right).powi(2);
            }
        }

        smoothness / (2.0 * LATTICE_SIZE as f64)
    }

    fn gradient(&self, lattice: &Lattice) -> Vec<f64> {
        let mut grad = vec![0.0; LATTICE_SIZE];

        for i in 0..LATTICE_SIZE {
            let (page, byte) = Lattice::from_linear_index(i);
            let center = lattice.get(page, byte) as f64;

            let left = lattice.get(page, ((byte as usize + 255) % 256) as u8) as f64;
            let right = lattice.get(page, ((byte as usize + 1) % 256) as u8) as f64;

            grad[i] = 2.0 * center - left - right;
        }

        grad
    }
}

impl ActionFunctional {
    pub fn new() -> Self {
        ActionFunctional {
            sectors: vec![Box::new(GeometricSmoothness)],
            weights: vec![1.0],
        }
    }

    pub fn evaluate(&self, lattice: &Lattice) -> f64 {
        self.sectors
            .iter()
            .zip(&self.weights)
            .map(|(sector, weight)| weight * sector.evaluate(lattice))
            .sum()
    }

    pub fn minimize(&self, initial: Lattice) -> Lattice {
        let mut current = initial;
        let learning_rate = 0.01;

        for _ in 0..100 {  // Simple gradient descent
            let action = self.evaluate(&current);

            // Compute gradient
            let mut total_gradient = vec![0.0; LATTICE_SIZE];
            for (sector, weight) in self.sectors.iter().zip(&self.weights) {
                let grad = sector.gradient(&current);
                for i in 0..LATTICE_SIZE {
                    total_gradient[i] += weight * grad[i];
                }
            }

            // Update
            for i in 0..LATTICE_SIZE {
                let new_val = current.data[i] as f64 - learning_rate * total_gradient[i];
                current.data[i] = new_val.max(0.0).min(255.0) as u8;
            }

            // Check convergence
            let new_action = self.evaluate(&current);
            if (action - new_action).abs() < 1e-6 {
                break;
            }
        }

        current
    }
}
}

Verifier

#![allow(unused)]
fn main() {
// verifier.rs - Linear-time verification
pub struct Verifier {
    window_size: usize,
}

impl Verifier {
    pub fn new(window_size: usize) -> Self {
        Verifier { window_size }
    }

    pub fn verify_window(&self, lattice: &Lattice, start: usize) -> bool {
        let end = (start + self.window_size).min(LATTICE_SIZE);
        let window_data: Vec<u8> = lattice.data[start..end].to_vec();

        // Verify R96 conservation
        let r96 = R96Digest::compute(&window_data);
        if !self.verify_r96_conservation(&r96) {
            return false;
        }

        // Verify budget is zero (lawful)
        let receipt = Receipt::compute(lattice, 0);
        receipt.verify()
    }

    fn verify_r96_conservation(&self, digest: &R96Digest) -> bool {
        // Check that histogram sums to window size
        let total: u32 = digest.histogram.iter().sum();
        total as usize == self.window_size
    }

    pub fn verify_witness_chain(&self, witnesses: &[Witness]) -> bool {
        if witnesses.is_empty() {
            return true;
        }

        let mut current_receipt = witnesses[0].input_receipt.clone();

        for witness in witnesses {
            if witness.input_receipt != current_receipt {
                return false;  // Chain broken
            }

            if !witness.verify() {
                return false;  // Invalid witness
            }

            current_receipt = witness.output_receipt.clone();
        }

        true
    }
}

#[derive(Clone, Debug)]
pub struct Witness {
    pub morphism_id: String,
    pub input_receipt: Receipt,
    pub output_receipt: Receipt,
    pub budget_delta: u8,
}

impl Witness {
    pub fn verify(&self) -> bool {
        // Verify budget conservation
        let expected_budget = (self.input_receipt.budget_ledger + self.budget_delta) % 96;
        self.output_receipt.budget_ledger == expected_budget
    }
}
}

Example Usage

// main.rs - Example usage of the Hologram core
mod lattice;
mod receipt;
mod cam;
mod process;
mod types;
mod action;
mod verifier;

use lattice::*;
use receipt::*;
use cam::*;
use process::*;
use action::*;
use verifier::*;

fn main() {
    // Create a lattice
    let mut lattice = Lattice::new();

    // Set some values
    lattice.set(0, 0, 42);
    lattice.set(1, 1, 137);

    // Compute receipt
    let receipt = Receipt::compute(&lattice, 0);
    println!("Initial receipt: {:?}", receipt);
    assert!(receipt.verify(), "Receipt should be valid");

    // Create a process with morphisms
    let mut process = Process::new();

    // Add identity morphism
    process.add_morphism(Box::new(IdentityMorphism));

    // Add class-local transform
    let transform = ClassLocalTransform::new(
        42,  // Transform class 42
        Box::new(|x| (x + 1) % 256)
    );
    process.add_morphism(Box::new(transform));

    // Add schedule rotation
    process.add_morphism(Box::new(ScheduleRotation::new(1)));

    // Execute process
    let (output, final_receipt) = process.execute(&lattice);
    println!("Final receipt: {:?}", final_receipt);

    // Content-addressable storage
    let mut cam = ContentAddressableMemory::new();
    let data = vec![1, 2, 3, 4, 5];
    let address = cam.store(data.clone());
    println!("Stored at address: {:?}", address);

    // Retrieve
    let retrieved = cam.retrieve(&address);
    assert_eq!(retrieved, Some(&data));

    // Action minimization
    let action = ActionFunctional::new();
    let initial = Lattice::new();
    let optimized = action.minimize(initial);
    println!("Optimized action: {}", action.evaluate(&optimized));

    // Verification
    let verifier = Verifier::new(256);  // Window of 256 bytes
    let is_valid = verifier.verify_window(&output, 0);
    println!("Verification result: {}", is_valid);

    // Witness chain
    let witness = Witness {
        morphism_id: "test".to_string(),
        input_receipt: receipt.clone(),
        output_receipt: final_receipt.clone(),
        budget_delta: 1,
    };

    let chain_valid = verifier.verify_witness_chain(&[witness]);
    println!("Witness chain valid: {}", chain_valid);
}

Test Suite

#![allow(unused)]
fn main() {
// tests.rs - Unit tests for core components
#[cfg(test)]
mod tests {
    use super::*;

    #[test]
    fn test_lattice_indexing() {
        let mut lattice = Lattice::new();
        lattice.set(5, 10, 42);
        assert_eq!(lattice.get(5, 10), 42);
        assert_eq!(lattice[(5, 10)], 42);
    }

    #[test]
    fn test_receipt_verification() {
        let lattice = Lattice::new();
        let receipt = Receipt::compute(&lattice, 0);
        assert!(receipt.verify());
        assert_eq!(receipt.budget_ledger, 0);  // Lawful
    }

    #[test]
    fn test_cam_perfect_hashing() {
        let mut cam = ContentAddressableMemory::new();
        let data1 = vec![1, 2, 3];
        let data2 = vec![4, 5, 6];

        let addr1 = cam.store(data1.clone());
        let addr2 = cam.store(data2.clone());

        assert_ne!(addr1, addr2);  // Different content, different addresses
        assert_eq!(cam.retrieve(&addr1), Some(&data1));
        assert_eq!(cam.retrieve(&addr2), Some(&data2));
    }

    #[test]
    fn test_morphism_composition() {
        let lattice = Lattice::new();
        let mut process = Process::new();

        process.add_morphism(Box::new(IdentityMorphism));
        process.add_morphism(Box::new(IdentityMorphism));

        let (output, _) = process.execute(&lattice);
        assert_eq!(output.data, lattice.data);  // Identity preserves state
    }

    #[test]
    fn test_budget_arithmetic() {
        let r1 = Receipt {
            r96_digest: R96Digest::compute(&[]),
            c768_stats: C768Stats {
                mean_flow: 0.0,
                variance: 0.0,
                phase: 0,
            },
            phi_roundtrip: true,
            budget_ledger: 47,
        };

        let r2 = Receipt {
            budget_ledger: 50,
            ..r1.clone()
        };

        let combined = Receipt::combine(&r1, &r2);
        assert_eq!(combined.budget_ledger, (47 + 50) % 96);  // 97 % 96 = 1
    }

    #[test]
    fn test_action_minimization() {
        let action = ActionFunctional::new();
        let initial = Lattice::new();
        let optimized = action.minimize(initial.clone());

        let initial_action = action.evaluate(&initial);
        let final_action = action.evaluate(&optimized);

        assert!(final_action <= initial_action);  // Action should not increase
    }

    #[test]
    fn test_verifier_window() {
        let lattice = Lattice::new();
        let verifier = Verifier::new(256);

        assert!(verifier.verify_window(&lattice, 0));
        assert!(verifier.verify_window(&lattice, 256));
    }
}
}

Compilation and Usage

To use this implementation:

Create a new Rust project:

cargo new hologram-core
cd hologram-core

Add dependencies to Cargo.toml:

[dependencies]
sha3 = "0.10"

[dev-dependencies]
criterion = "0.5"  # For benchmarking

Copy the code modules into src/:

lattice.rs
receipt.rs
cam.rs
process.rs
types.rs
action.rs
verifier.rs

Build and run:

cargo build --release
cargo run
cargo test

Performance Considerations

This minimal implementation prioritizes clarity over performance. Production optimizations would include:

SIMD vectorization for receipt computation
Memory pooling for lattice allocations
Lock-free data structures for concurrent access
JIT compilation for hot morphisms
Cache-oblivious algorithms for traversals
Compressed representations for sparse configurations

Extensions

This core can be extended with:

Network layer for distributed operation
Persistence layer for durable storage
Query engine for complex searches
Visualization for debugging
Benchmarking suite for performance analysis
Property-based testing for correctness
Formal verification using Rust’s type system

The implementation demonstrates all key concepts while remaining simple enough for educational use and experimentation.

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The Hologram (12,288): A Complete Computer Science Formalization